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y > z q> (N) X i/^x y z
interaxial angles, cell parameters, i = 1, 2, 3 matrix representation of the group element gk Kronecker symbol electron density distribution golden mean angles characterizing the unit face (111) Euler numbers character of a representation angles between the normal of a face (mno) and the axes of the crystal coordinate system
1, 2, 3 ... N 2l9...,Nm T 3, 4,... JV *
AT-fold rotation axes AT-fold screw axes center of symmetry, inversion center rotoinversion axes symbols referring to the reciprocal lattice, e.g., the basis vectors, are marked with the index *
GDM HRTEM k MI PBC SHG t
generalized dual-grid method high-resolution transmission electron microscope klassengleich (characterization of subgroups of a spacegroup) morphological importance periodic bond chain second-harmonic generation translationengleich (characterization of subgroups of a spacegroup)
1.1 Introduction
1.1 Introduction The regular polyhedral shape of crystals has long fascinated the observer by their beauty and brightness and by the perfect planarity of their faces (Fig. 1-1), which exceeds the proficiency of the work of artisans. The beliefs of Babylonians and Egyptians in the magical and healing powers of minerals and gemstones has been passed on to other civilizations, a revival taking place for instance in the Middle Ages. Indeed, the important learned man and bishop "doctor universalis" Albertus Magnus (1193-1280) dedicated a part of his book De Mineralibus et Rebus Metallicis Libri V, which appeared in 1276, to the curative properties of crystals. The more rational minds of antiquity dealt with the problems of the formation and composition of minerals. This is reflected in the application of the word %Q\)<j%(xX'koq,Figure 1-1. Rock-crystal (quartz, SiO2). The single crystals show plane faces and trigonal symmetry which means something like "solidification (from Hochleitner, 1981). by freezing" and had originally only been used for ice, to rock-crystal (quartz) during the time of Platon (428-348 B.C.). Following some of the ideas of antiquity with general validity, however, by JeanGeorgius Agricola (1494-1555) was one of Baptiste Rome de Vlsle (1736-1790) as late the learned men to overcome the mystical as 1783. He was able to verify this hypothesis by angular measurements with a conassumptions of medieval times. His books tact goniometer (Fig. 1-2), which was conare not only a collection of the empirical structed in 1780 by his assistant Maurice mining knowledge about minerals of his Carangeot, thus opening the way to quantime but they contain many hypotheses titative crystal morphology. In 1820 about crystal growth and properties. In the William Hyde Wollaston (1766-1829) following centuries the external shape of greatly increased the possible accuracy of crystals, their morphology, attracted more measurement with his optical goniometer and more interest. Thus Niels Stensen (from about 1° to 1'). (1636-1686) found from his crystallization studies that a correlation exists between With the work of the Abbe Rene Just crystal growth and form - and that the Haiiy (1743-1822) a new chapter of crysmorphology of crystals was not accidental. tallography began. A structural way of The law of constant angles between equiva- thinking received strong impetus from his work relating the internal structure of cryslent crystal faces was noticed by Stensen tals to their external shapes. From the ob1669 on the examples of quartz and heservation that the fragments obtained by matite. It was formulated explicitly and
1 Elements of Symmetry in Periodic Lattices, Quasicrystals
Figure 1-2. A historical contact goniometer for measuring the interfacial angles of crystals (from Haiiy, 1801).
repeatedly cleaving a crystal preserved the initial crystal form, Haiiy derived primitive forms ("molecules integrantes") for the basic building units of all crystals. His decrescency theory described the formation of different crystal shapes from basic parallelepipeds (Fig. 1-3). Haiiy's book Traite de Miner alogie (Haiiy, 1801) soon became the standard work on crystallographic mineralogy of the nineteenth century. The morphological school received new stimulus from Christian Samuel Weiss (1780-1856) who polemized against the atomistic basis of Haiiy in an appendix to his German translation of the Traite de Mineralogie. Weiss focused on the dynamic character of matter and the dominating influence of the external crystal form. He discovered the vectorial nature of some physical properties. From the symmetrical arrangement of sets of crystal faces he derived the existence of 2-, 3-, 4- and 6-fold zone axes. He then described the faces by their integer intercepts with these axes to form three-dimensional coordinate systems. As a consequence, Weiss formulated the law of ratio-
nal parameter coefficients which had implicitly already been found by Haiiy in 1784. The more convenient way of indexing used today, based on the reciprocal values of the intercepts, was introduced by William Hallowes Miller (1801-1880). Franz Ernst Neumann (1798 -1895) found a correlation between the morphology of a crystal and the anisotropy of its physical properties. His work on crystal physics was continued by his former student Woldemar Voigt (1850-1919). Interest in crystal symmetry was initiated by the symmetric arrangement of crystal faces. By way of analyzing the morphology of crystals, in 1830 Johann Friedrich Christian Hessel (1796-1876) ordered them into 32 possible crystal classes according to their symmetry. Using the concept of a mathematical point lattice, Auguste Bravais (1811-1863) deduced in 1848 the 14 possible 3-D space lattices in 7 groups which correspond to the 7 crystal systems detected by Weiss. The symmetry of these point lattices (holohedries) was too high in many cases to explain the symmetry properties of the respective elastic tensors determined experimentally. This contradiction could be overcome by occupying the lattice nodes with point complexes, so lowering the symmetry. After the preliminary work of Leonhard Sohncke (18421897), who in 1879 detected 65 space groups (the subset containing symmetry operations of the first kind only) using group theoretical tools, in 1891 all of the 230 possible space group symmetries were derived by Evgraf Stepanovic Fedorov (1853-1919) and Arthur Schonflies (18531929) independently. The atomistic structural theory based on the space lattice concept was confirmed by the first X-ray diffraction experiment (Fig. 1-4) which was suggested by Max von Laue (1879-1960) in 1912.
1.1 Introduction
(b) (d) Figure 1-3. Haiiy's decrescency theory of crystal growth: crystals of the same chemical composition but with a different habit are built from the same basic parallelepipeds ('molecules integrantes'). Schematic drawings of the crystal forms and of their construction using cubic unit cells are shown: (a) and (b) rhomb-dodecahedron, (c) and (d) pentagon-dodecahedron with inscribed cubes (from Hauy, 1801).
1 Elements of Symmetry in Periodic Lattices, Quasicrystals
#, ;
* igure 1-5. Monochromatic zero-layer X-ray precession photograph of the decagonal quasicrystal Al 70 Co 15 Ni 15 showing clearly non-crystallographic tenfold rotational symmetry.
Figure 1-4. One of the very first X-ray photographs taken by Laue, Friedrich and Knipping (1912). The diffraction pattern of zinc blende (ZnS) reflects the fourfold rotation symmetry along one of the main axes of this cubic crystal (from Laue, 1961).
1.2 Symmetry of Crystals 1.2.1 Morphology
The theory of crystal symmetry, i.e., of symmetric transformations in 3-D space under restrictions imposed by the existence of the periodic crystal lattice, appeared to be a rather closed part of crystallography until 1984, when the sensational discovery of quasicrystals by Shechtman, Blech, Gratias and Cahn occurred. The understanding of well-ordered crystals yielding diffraction patterns with non-crystallographic (icosahedral, decagonal, ...) symmetry, i.e., incompatible with a 3-D periodic translation lattice, has been a new challenge for crystallography (Fig. 1-5).
A very extensive and rich collection of crystal drawings was edited by Victor Mordechai Goldschmidt (1852-1933) in the years 1913-1923. He ordered the published information about the morphology of natural crystals, systematically in a nine volume atlas. Figure 1-6 shows one page of Vol. VIII illustrating different natural crystal forms of silver. Single crystals always have a convex polyhedral form, concave parts indicate that two single crystals are grown together. If these two individuals can be transformed into each other by a particular symmetry operation then we call the crystal twinned. A plane of intergrowth, for instance, may correspond to a reflection plane, and we say that this crystal is twinned according to that particular
1.2 Symmetry of Crystals
Figure 1-6. One page of the famous Atlas der Krystallformen in nine volumes edited by Goldschmidt during the years 1913-1923. Silver crystals with different habits grown under different conditions are shown in schematical drawings (from Goldschmidt, 1913 — 1923).
plane. Some of the drawings in Fig. 1-6 show twinned crystals, e.g., the last one of row one, and the second and third ones in row three. The polyhedra characterizing the shape of crystals grown under equilibrium conditions tend to show a particular symmetry, i.e., they are invariant under particular motions around symmetry elements centered in the crystal. These motions, which may be a rotation of the crystal around an axis or a reflection on a mirror plane or on a point (center of sym-
metry), transform a symmetrical object into itself. The transformed object is not distinguishable from the untransformed one, its position in space and its shape coincide with the original ones. Phenomenologically, crystals are defined as chemically homogenous materials with anisotropic physical properties. The most conspicuous manifestation of this anisotropy is the formation of plane faces reflecting the internal symmetry of the crystal. Whereas the crystal habit, defined by
10
1 Elements of Symmetry in Periodic Lattices, Quasicrystals
the relative sizes of the faces, may vary widely between different crystal individuals of the same material due to different growth conditions, the interfacial angles remain constant (law of constant angles). Examples of crystals with the same crystal form of a cuboctahedron but with a different habit are shown in Fig. 1-6 (the second and third drawing in row two). The one polyhedron shows large hexahedron faces h and small octahedron faces o, for the other polyhedron the ratio is inverse. In both cases, however, the interfacial angles are the same as well as the directions of the face normals. The interfacial angles can be measured by means of a contact goniometer or, more accurately, by means of an optical two-circle goniometer, as was developed by Fedorov and Goldschmidt in 1892, where a crystal is mounted on a goniometer head so that the rotation axis of the goniometer is parallel to the edges formed by the intersecting faces to be measured. A collimated light beam falls on the crystal, and in each case, when a face is rotated to a reflecting position a signal is seen in a telescope. The light beam, the normal to the reflecting face and the telescope have to be in a plane perpendicular to the rotation axis of the goniometer. Hence, during one complete revolution of the crystal the angles between the faces of one zone can be measured. The angle between two face normals equals n minus the interfacial angle. The complete set of interfacial angles allows establishment of the eigensymmetry (corresponding to the group of point symmetry operations bringing the crystal into self-coincidence) of the crystal. It is even possible to derive a kind of morphological unit cell which is characteristic for a material with a given chemical composition. It is represented by a parallelepiped with edge lengths a\ br, d in relative units and
angles a, /?, y. Usually the ratio a':b':d is given normalized tofe'= l. The disturbing influence of the individual crystal habit may be eliminated by representing the faces by their normal vectors. The commonly used graphical method for the representation of a crystal form is the stereographic projection of its pole figure. Figure 1-7 (a) shows how the face poles result from the intersection of the face normals with a circumscribed sphere having a common center with the crystal. In the next step, the face poles of the northern hemisphere are connected with the south pole and those of the southern hemisphere with the north pole [Fig. l-7(b)]. The stereographic projection of the face poles is obtained by projecting the poles along the connecting lines upon the equatorial plane [Fig. l-7(c)]. The stereographic projection of the face poles is independent of individual variations of the relative dimensions of crystal faces, it shows the inherent symmetry of the crystal form in an unbiased way. For the practical application of the stereographic projection, it is useful to perform the construction on a Wulffs net, i.e., a stereographic projection of meridians and parallels with 2° divisions (Fig. 1-8). The construction and evaluation is facilitated by the fact that interfacial angles of a crystal form appear as true angles in the projection whilst circles on the sphere appear as circles in the projection. 1.2.2 Crystallographic Axes
The symmetrical arrangement of faces (zone-faces) bounding a crystal grown under equilibrium conditions led Weiss to the idea to refer all crystal faces to a 3-D coordinate system formed by three non-coplanar symmetry axes (zone-axes) (Fig. 1-9). Usually they are given by the basis vectors a, A, c with lengths a, b, c, coordinates x, y9
1.2 Symmetry of Crystals
11
Figure 1-8. Wulff's net: a stereographic projection of meridians and parallels.
z, and interaxial angles a, /?, y. Another notation frequently used is: basis al9 a 2 , a 3 , lengths a1, a2, a 3 , and interaxial angles a x , a2 , a3 , respectively. The metric properties can also be represented by the metric tensor G, a (3 x 3) square matrix of elements gik = (at - ak) with z, k = 1, 2, 3, i.e., the scalar products of all pairs of basis vectors:
(
a- a ab a- c\ b a b b b c\ c-a
cb
(1-1)
c c)
Figure 1-7. (a) Representation of a crystal form by spherical projection. The normals on the crystal faces intersect a circumscribed sphere in so-called face poles (black dots if they are above, white dots if they are below the plane of drawing). All faces with poles on one meridian belong to one zone, (b) The principle of the stereographic projection: the face poles (e.g., PJ on each hemisphere are connected with the opposite pole (e.g., S) of the equatorial projection plane (dashdotted). The projection of the face poles along these connecting lines upon the equatorial plane (e.g., P/) is called a stereographic projection (c).
12
1 Elements of Symmetry in Periodic Lattices, Quasicrystals
Table 1-1. The seven crystallographic coordinate systems : the names, metrical relationships of the lattice parameters and the orientation of the basis vectors with regard to unique symmetry elements are given.
Crystal system
Lattice parameters
Triclinic Monoclinic Monoclinic Orthorhombic Tetragonal Rhombohedral Hexagonal
a^b^c a^b^c a^b^c
Orientation
a ^/?^y^90° a = /? = 9 0 V y c\\2, 1st setting a: = y = 90 V P A || 2, 2nd setting
a^b^c a = P = y = 90° a, b, c||2's a = b^c a; = j3 = y = 9O° c||4 a — b — c; =a j5 = y^9O° a = b^c en: = j5 = 9O°,
(a + 6 + c)||3 c||6
y- = 120° a = b = c a, = P = y = 90° (a + b + c)W5
Cubic
1.2.3 Crystal Faces - Miller Indices Figure 1-9. (a) Schematical drawing of a crystal form and some of its symmetry elements: 4-, 3-, and 2-fold rotation axes. The basis vectors of a coordinate system for this cubic crystal form are oriented parallel to the 4-fold axes, (b) General (triclinic) right-handed coordinate system spanned by the basis vectors a, b, c.
For orthogonal bases the diagonal elements, only, do not equal zero. Table 1-1 lists the seven crystallographic coordinate systems, their metrics and the characteristic orientation of particular unique symmetry elements. In the case of one unique rotation axis N, it is chosen parallel to c, in the monoclinic case another setting with the axis parallel to b is in use too. Instead of rhombohedral axes, with the 3-fold rotation axis parallel to (a + b + c), the hexagonal coordinate system with rotation axis 3 parallel to c is often used. The 3-fold axes of the cubic system always have to be set parallel to the space diagonals (a + b -f c) of the cube.
On the basis of the seven crystallographic coordinate systems all crystal faces can be given in terms of their intercepts with the three axes (Fig. 1-10). The resulting numbers may be expressed as integral multiples m, n, o of the respective unit lengths on these axes yielding the so-called Weiss indices {mno). Face indices are always given in parentheses. The equation of the plane (m n 6) can be written (x, y, z in units of a,fe,c): x
y
z
(1-2)
-+-+-=1 m n o the largest common divisor for Finding 1/m, 1/n and \jo we obtain
nox 4- moy + mnz = mno
(1-3)
and with h = no, k = mo, I = mn and mno =j we end up with
hx + ky + Iz =j
(1-4)
/z, k, I are called the Miller indices and they define the face symbol (h k I) which is more
1.2 Symmetry of Crystals
Figure 1-10. A plane defined by its intercepts on a crystallographic coordinate system. The intercepts at x = 2, y = 3, z = 2 lead to the Weiss indices (232) and the Miller indices (323). The vector normal to the plane H is indicated.
commonly used than the Weiss symbol. j may be a positive or negative integer number. The derivation of the Miller indices may be illustrated by the example given in Fig. 1-10. The face intersects a at x = 2, b at y = 3 and c at z = 2 leading to the Weiss indices (232). Taking the reciprocal of the intercepts 2, 3, 2 we get 1/2, 1/3 and 1/2. Reducing to a common denominator we find 3/6, 2/6, 3/6, and writing the numerator only, the Miller indices (323) result. The indices, on the other hand, define a vector normal to the plane H given on a reciprocal basis (see Sec. 1.3.4). Another plane, parallel to the just mentioned one but two times the distance from the origin, is characterized by the Weiss indices (464) and the Miller indices 1/4, 1/6, l/4=>3/12, 2/12, 3/12 => (323). Both parallel planes have the same Miller indices and generally, the Miller indices (h k I) define an infinite set of parallel planes (lattice or net planes) (Fig. 1-11). Any two crystal faces with Weiss indices (mno) and (m'n'o') have a rational ratio m/mr: njri: ojo' of their intercepts on the crystallographic axes ("law of rational
13
parameters") as a direct consequence of the fact that the numbers m, m', n, n\ o, d are integers. In the case of a trigonal or hexagonal coordinate system it is advisable to use the fourfold Bravais-Miller symbol (hkil) with h + k + i = 0 to facilitate the derivation of symmetrically equivalent indices. A threefold rotation, for instance, acting on a hexagonal coordinate system, transforms the axes a 1? a 2 , a3 into a 2 , — (a1+a2), a3 then into — {a1 + a2\ al9 a 3 and finally back into al9 a2, a3. Thereby, the face indices (h k i I) are changed to (k i h /), (i hkl) and (h k i /), respectively (Fig. 1-12). Generally, the Weiss or Miller indices transform like the axes. The angles i/fx, \jjy, \j/x between the normal of a face (m n o) and the axes of the crystallographic coordinate system obey the relations (see Fig. 1-10) OP ma OP
(1-5)
OP me
(110)
(010)
Figure 1-11. Point lattice on the basis a, b, c (c is perpendicular to the plane of drawing) with different sets of lattice (net) planes (hkO) parallel to c. The density of lattice points on (410), for instance, is smaller than on (010), it is proportional to the inverse of the respective interplanar spacings dhk0.
14
1 Elements of Symmetry in Periodic Lattices, Quasicrystals
absolute scale they are referred to b = l9 generally. We get, consequently, a=
C O S (£>v
- and c =
C O S (7>v
cos
(1-9)
Therefore the determination of the relative cell parameters reduces to a determination of direction cosines. Experimentally, the interfacial angles may be measured with high accuracy using a two-circle reflection goniometer. Figure 1-12. Hexagonal crystallographic axis system under the action of a trigonal rotation. The 3-fold rotation axis is parallel to a 3 , i.e. perpendicular to the paper plane. The motion of a net plane (hkO) parallel to a3 and the variation of the Weiss (upper line) and the Miller indices (lower line) are shown.
with OP being the length of the normal to the face from the origin O of the coordinate system to its intersection point P with the face. For the ratios we obtain cosi/fY:cosi/fvy:cosi/f7 = — : —- : —
ma nb oc
(1-6)
i.e., the ratio of the direction cosines is proportional to that of the reciprocal intercepts. Referring the direction cosines of any face (h k I) to those of the unit face (111) we obtain cosxjjx cosxj/ cos\j/z cos cpx cos cp cos cpz
1
1
A set of faces which have, parallel to their intersecting lines, one direction (the zone axis) in common, constitute a zone. The normals to the planes are all coplanar and perpendicular to the zone axis (Fig. 1-13). Therefrom, it follows that a face belongs to a zone if the scalar product of the normal to the face vector with the zone axis vector is equal to zero. The zone axis [uvw] corresponds to a vector z = ua + vb + wc, with w, v, w as integers, and a face (hkl) has a normal vector H=ha* + kb* + lc* [see Sec. 1.3.4 and Eq. (1-36)] leading to ^_ 1 0 ) H' z = (hkl)-[uvw] = hu + kv + lw = O
la . 1 6 . 1 c ma' nb'oc
(1-7) m n o q>x,
1.2.4 Zones and Forms
1
cos cpx cos cp cos cpz
(1-8)
Since the cell parameters of the morphological unit cell cannot be derived on an
(110)
Figure 1-13. The set of four faces generated by the action of a fourfold rotation around c from the face with Miller indices (110). This set is called crystal form {110}. The four faces form a zone with a zone axis [001] parallel to c.
1.2 Symmetry of Crystals
The indices of the zone axis resulting from two intersecting faces {hlk1ll) and (h2 k2 l2) may easily be calculated from the cross product of their normal vectors, since the zone axis is always perpendicular to the normals of the faces defining the zone z~H^H2~
(1-11)
where the indices in the brackets are in terms of the direct lattice vectors a1,a2,a3. For the example illustrated in Fig. 1-13, we can derive the zone axis belonging to the faces (110) and (HO) using Eq. (1-11) z = [l 0 - 1 0 , 0 - 1 - 0 1, I T - 1 1 ] = = [002] (1-12) Equation (1-12) denotes a direction, it can be reduced, therefore, to z = [001], i.e., the zone axis of the two faces coincides with c. The general indices of any face belonging to that zone can be derived applying Eq. (1-10) (1-13)
Equation (1-13) is fulfilled for any h and /c, and for / = 0. Consequently, the general indices for all faces with the zone axis [001] are(fefcO). Goldschmidt 1897 defined the complication rule: if two faces (hx kx l±), (h2 k212) define a zone [u v w], all other faces belonging to the same zone can be derived by successive additions of the form
with arbitrary integer numbers for Xt. For the indices (h k I) of a face which is defined by the two zone axes zx = [u1 v± w j , z2 = [u2 v2 w2] the equations hux + kVi 4- lw1 = 0, hu2 + kv2 w2 = 0
must be valid simultaneously. Since the normal to the face (h k I) has to be perpendicular to both zone axes, it results from the cross product of the zone axis vectors 2=
=
(1-16)
[v1w2-v2wuw1u2-w2u1,ulv2-u2v1]
And for the two zone axes z1 = [100] and z2 = [010], for instance, we find
~[k1l2-k2l1j1h2-l2huhlk2-h2k1]
h•0+k•0+/•1 =0
15
(1-15)
(hkl) = ( 0 - 0 - 1 - 0 , 0 - 0 - 0 -1,1 - 1 - 0 - 0 ) = = (001) (1-17) A general condition for three crystal faces to belong to one zone is that the determinant formed by the face indices equals zero h2k2l2 =0 h3 k3 l3
(1-18)
The condition that three zone axes are coplanar, i.e., that the zone axes have one crystal face in common, can be written u1 vx wt u2 v2 w2 u3v3w3
(1-19)
A set of symmetrically equivalent faces is said to constitute a crystal form {hkl} (the indices are enclosed by braces). A general crystal form (holohedry: full set of faces) is only generated if the faces show the eigensymmetry (inherent symmetry) 1, the identity operation. In the case of higher symmetry, a special crystal form (hemihedry: one half and tetartohedry: one quarter of the full set, e.g.) is constituted. Thus, depending on the orientation of the face relative to the symmetry element, several characteristic crystal forms may exist for each point group leading to 47 different simple crystal forms, 30 of them closed and 17 open ones. The closed forms are isohedral polyhedra whose respective inspheres touch all the faces. Examples of general,
16
1 Elements of Symmetry in Periodic Lattices, Quasicrystals
special, open, closed, simple and combined forms are given in Fig. 1-14. Weiss found in the year 1819 an algorithm to derive all crystal faces possible in a particular crystal form. Start with at least four faces, each pair of them must belong to a different zone. New faces are defined by each pair of zone axes. By appropriate combination of these new faces, new zone axes can be derived, and so on. In
{hkl}
{100}+{001} Figure 1-14. Some examples of particular crystal forms referring to the point symmetry group 4/m (tetragonal system) in perspective drawings and stereographic projections: (a) general closed simple form {hkl} (tetragonal dipyramid); (b) special open simple form {001} (pinacoid or parallelohedron); (c) limiting open single form {100} (tetragonal prism); (d) combined closed form {100} -f {001} (tetragonal prism).
this way, all possible crystal faces with rational indices can be obtained. 1.2.5 Symmetry Elements
The symmetry operations which leave all natural equilibrium crystal shapes (the crystal forms) invariant, i.e., all transformations mapping the crystals onto itself, are elements of the 32 crystallographic point groups. They are said to be point symmetry operations since they leave at least one point in space unmoved after applying them repeatedly. A symmetry element corresponding to a particular symmetry operation may be the point, straight line or plane which remains unmoved for this operation. Thus the crystallographic symmetry elements are the proper rotation axes N and the improper rotation or rotoinversion (inversion-rotation) axes N. The axes N are: 1, called the identity operation leaving an object unmoved, 2, 3, 4 and 6 corresponding to rotations through angles 2K/N. The rotoinversion axes N are: T, called the center of symmetry or inversion center, 2 = m (instead of 2 the equivalent symmetry element m, the mirror or reflection plane is used), 3, 4 and 6 = 3/m (Fig. 1-15). A proper motion, or motion of the first kind, transforms an object congruently. This means that a left hand, for instance, remains a left hand after performing the symmetry operation. The improper motions, or motions of the second kind, transform an object into its reflection image, consequently, a left hand becomes a right hand. The N rotoinversion axes consist of a JV-fold rotation followed by an inversion on a point lying on the axis. It is useful to represent symmetry operations by transformation matrices R with elements rtj, ij = 1, 2, 3, acting on a vector r = xa + yb + zc, defined on a crystallographic basis a, A, c, and represented by its
1.2 Symmetry of Crystals
2
17
2
360°
(h)
components (x y z) or by the coordinates x, y, z of the point at the end of the vector r r
r
li
12
r
13
21
22
r
23
*32
r
31
(1-20)
33/
The matrix of a general rotation through an angle a on a 3-D orthogonal basis can be written for the case where the rotation axis is parallel to c Figure 1-15. The proper crystallographic symmetry elements and the set of symmetrically equivalent objects belonging to them: (a) 1, identity operation, (c) 2-fold, (e) 3-fold, (g) 4-fold, and (i) 6-fold rotation axes. The improper crystallographic symmetry elements: (b) I, center of symmetry (inversion center), (d) 2 = m reflection (mirror) plane, (f) 3, (h) 4 and (j) 6 = 3/m (reflection plane perpendicular to the 3-fold axis) rotoinversion axes.
/cos a — sin a 0\ R (a) = I sin a cos a 0 | 0 0 1J
(1-21)
Its determinant det [i?(a)] = cos2 a + sin2 a = 1 and its trace, the sum over the diagonal matrix elements, 1 + 2 cos a are invariant to basis transformations. The matrix
18
1 Elements of Symmetry in Periodic Lattices, Quasicrystals
representations for the proper and improper crystallographic rotation axes are given in Table 1-2. It can be seen, for instance, that the traces of the matrices for the 3-fold rotations are equal in both the hexagonal and the cubic coordinate systems. Another characteristic feature is that the matrix elements all correspond to integers only. This is due to the fact that the action of a rotation on a symmetry
adapted coordinate system results in the cyclic permutation of basis vectors, i.e., each basis vector is transformed into a linear combination of other basis vectors. Consequently, all rotations which leave a crystallographic basis invariant, merely have integer matrix elements when defined on such a basis. The trace of the matrix is also an integer. Since the trace of a matrix is independent of the basis we can set the trace of the matrix Eq. (1-21) equal to an integer
Table 1-2. Matrix representations of the proper and improper crystallographic rotation axesa. N
Matrix Multi- Trace N Matrix Multi- Trace plicity plicitiy
-3
= n n = 0, ± 1 , ± 2 , . . .
(1-22)
and we get cosa = 0, ±1/2, ± 1
(1-23)
Thus the only rotations compatible with the invariance of a crystallographic basis are those through the angles a = 0, 7i/3, 7c/2, 2TC/3, n, 4TT/3,
-1
3TC/2, 5TC/3, 2 7i
(1-24)
The matrices of the rotoinversions N (compare Table 1-2) result from the multiplication of the matrix R{N) of the isogonal rotation N by the diagonal matrix D(l) inverting all signs R(N) = R(N)-D(1) -1
which
results
in the elements
(1-25) r'ik =
X rtj djk = — rik because dik = — 1 for j = k j
-2
a
The axes are parallel to c, the 3-fold axes in the rhombohedral and cubic systems are parallel to the space diagonal [111]. The first row of matrices for 3 and 3 refers to a hexagonal basis, the second one to a cubic and a rhombohedral one, respectively. The multiplicity, i.e. the number of symmetrically equivalent points generated from one point by repetitive action of a symmetry operation, is given as well as the trace of each matrix.
and 0 for j ^ k. The determinants of symmetry matrices Rl of motions of the first kind (proper motions) are equal to + 1 , and that of motions of the second kind (improper motion) are equal to —1, in all cases, det(J?!) =
det(/?n) = -
(1-26)
The successive action of two or more proper motions leads to a proper motion again, since for the product of two symmetry ma-
19
1.2 Symmetry of Crystals
trices R3 = Rt- R2 the determinant is (1-27) n improper motions give a proper one for n even, and an improper one for n odd for n even, (— 1)" = — 1 for n odd
(1-28)
Proper motions, consequently, can result from two improper motions, but not vice versa. A reflection on a mirror plane, e.g., is an improper motion. If we have two mirror planes which form an angle of a/2 = n/N with each other, then the reflection of one point on these two mirror planes corresponds to a rotation through an angle a = 2 n/N (Fig. 1-16). The rotation axis N corresponds to the intersection line of the two mirror planes. Say, for instance, there is one mirror plane perpendicular to a and one perpendicular to b of an orthogonal coordinate system with basis a, A, c; the planes form an angle a/2 = n/2 with each other. The action of the two reflection planes can be represented by the product of their symmetry matrices
Figure 1-16. The successive reflection of an object (star) on two mirror planes ml9m2 including an angle a/2 = n/N with each other is equivalent to a rotation through an angle a = 2 n/N around the rotation axis N formed along their intersection line.
a third twofold axis perpendicular to them 2x-2y=
(
1 0 0\
(1-30) /I
0 0\
/TO
0\
0 T O l i o 1 01 = 10 T 0 ] = 2 z 0 0 1/
\0 0 1/
\0 0 1/
The general case of two rotation axes Nx, N2 including an angle q>3 with each other is illustrated in Fig. 1-17. Two successive rotations through the angles OL± around Nx and a2 around N2 can be substituted by one rotation through the angle a3 around N3. For the angles the following equation holds cos(a3/2) = sin(a1/2) • sin(a2/2) • cos(
leading to the matrix for a 2-fold axis parallel to c, the intersection line of the mirror planes. Another example of the generation of a third symmetry element, now by two proper motions, is the case of two rotation axes intersecting at one point. The successive rotations around the two axes are equivalent to one rotation around another axis generated by the initial axes. Thus, if we have, for instance, two twofold axes 2X, 2y parallel to a and A, respectively, of an orthogonal coordinate system, then we get
q?2
Figure 1-17. The combination of a rotation through an angle a t around N1 and a rotation through an angle a2 around N2 is equivalent to a rotation through an angle a 3 around N 3 . A projection of the spherical triangle obtained by the intersection of planes passing through all pairs of axes Nt, Nj and the sphere is shown.
20
1 Elements of Symmetry in Periodic Lattices, Quasicrystals
By cyclically permutating the angles in Eq. (1-31), the angles q>1 and cp2 between the axes N2, N3 and Nl9N3, respectively, can be calculated. Since the possible angles at are restricted to the ones given in Eq. (1-24), the possible angles cpt between the rotation axes are restricted also. The six resulting combinations are given in Table 1-3. Table 1-3. The six possible combinations of crystallographic rotations. JV\ and N2, with interaxial angle cp3, generate N3, which has an angle cp2 with JV1} and an angle q>± with N2.
2 3 4 6 2 2
iV2
N3
2 2 2 2 3 3
2 2 2 2 3 4
*i
90° 60° 45° 30° 70.53° 54.73°
cp2
90° 90° 90° 90° 54.73° 45°
90° 90° 90° 90° 54.73° 35.27°
The odd rotoinversion axes 1 and 3 can be substituted by two separate symmetry operations: 1 and 3, respectively, and the inversion center. Thus, if the 3 operation is acting on a point then three symmetrically equivalent points are generated by the rotation axis 3, and three more by T. Since six equivalent points result from this operation, we say 3 has the multiplicity six. This separation is not possible for N = even rotoinversions. The points remaining unmoved in the case of the rotation operations are straight lines (the rotation axes), for rotoinversion operations one point only on the rotoinversion axis, the location of the center of symmetry, is not transformed. 1.2.6 External Form and Internal Structure
The crystal habit, the size of the crystal and the relative dimensions of its faces, strongly depend on the external conditions
during crystal growth. The interfacial angles and the orientation of the faces with respect to an appropriate crystal coordinate system, however, are material constants and are characteristic for the chemical composition. This fact is expressed by the two laws of geometrical crystallography: the law of constant angles ("interfacial angles between corresponding faces of differently grown crystals of the same chemical composition are constant") and the law of rational parameters ("all faces of a crystal form of a given substance can be characterized by a set of rationally related integers") and has its origin in the lattice character of crystal structures. The relative morphological importance (MI) of a crystal form {/z/c/}, which depends on its size, frequency of occurrence and presence as a cleavage form, can be derived from the laws formulated by Bravais (1850), Friedel (1911), Donnay and Harker (1937): The observed crystal faces are parallel to the net planes with the highest density of lattice points. The density is inversely proportional to the area of the unit meshs on the net planes. Since the volume V of the unit cells defining the space lattice is given by the product of the area of a unit mesh times the interplanar distance dhkl between two neighbouring net planes of a set (hkl), a high density of lattice points directly corresponds to a large interplanar spacing dhkl. The morphological importance of a crystal face (hkl) and the crystal form {hkl}, respectively, increases with increasing dhkl. In the presence of particular centering translations, glide planes or screw axes additional lattice planes are generated which halve the interplanar distance thus reducing the morphological importance of the face. For instance, in the case of a body centered lattice with centering translation (1/2) (a + b + c) an additional plane with intercepts a/2, b/2 and c/2 is generated beside
1.3 Crystal-Lattice Symmetry
(111) with intercepts la, lb and \c. Its Miller indices are (222) and the interplanar spacing d112 = dill/2. The interplanar distances of the set of netplanes parallel to (111) may be smaller now than those of the netplanes parallel to (121), for instance, and hence its MI will be lower. In those cases with additional net planes, the socalled multiple indices which exactly correspond to the lowest order X-ray reflections, which are compatible with the particular space-group symmetry, have to be used along with the resultant systematic extinction rules (see Sec. 1.5.6). The underlying physics of this geometrical picture can be discussed briefly: The interatomic distances in densely packed net planes are often smaller than the interplanar distances corresponding to stronger bonds in the planes than between them. Thus, the energy for cleaving a crystal parallel to such a set of net planes is smaller than for cleaving it not parallel to them. In this way the preferred formation of crystal faces can be correlated to the directions of the strongest bonds as in the periodic-bondchain (PBC) theory of Hartmann and Perdock (1955). The actual shape of a single crystal can be referred to one of the 47 simple crystal forms or to a combination of two or more simple ones. In 19 cases it is possible to determine the crystal class from the crystal form uniquely, and in some rare cases even the space group can be derived (Donnay and Harker, 1937).
21
Figure 1-18. High-resolution transmission electron microscopic (HRTEM) image of silicon viewed in the [110] direction showing the lattice periodicity on an atomic scale (from Spence, 1988).
is said to be the unit cell, and the crystal pattern can be built up by setting unit cell on unit cell like bricks in a wall. All corners of such an infinite framework of unit cells form a point lattice in the point space [Fig. 1-19 (a)]. Decorating these lattice points with groups of atoms gives a lattice array of atom groups. Changing to vector space and interpreting the lattice points as endpoints of vectors emanating from the origin of the coordinate system, we get a vector lattice [Fig. l-19(b)]. Each lattice u
(a)
1.3 Crystal-Lattice Symmetry 1.3.1 Crystal Patterns, Vector and Point Lattices
Characteristic for crystals in an idealized description is their translational periodicity (Fig. 1-18). The smallest repetition unit
(b) Figure 1-19. (a) Point lattice with one unit cell and (b) vector lattice.
22
1 Elements of Symmetry in Periodic Lattices, Quasicrystals
vector r can be referred to a basis (ax ... an) as r = n1a1 + ... + nnan. For regular crystals n = 3, quasicrystals and incommensurately modulated phases show translational symmetry when embedded in n-dimensional spaces with n = 4, 5 or 6. For a crystal structure with n-D translational periodicity of the electron density distribution function the relation holds: Q(r) = Q(r + rf) with r, rr representing n-D lattice vectors. The three basis vectors a1,a2, a3 span a parallelepiped, the unit cell of the lattice. If it does not contain any lattice points in the interior it is called primitive. For a given lattice, it is possible to select an infinite number of different unit cells taking different lattice vectors to span the parallelepipeds. The volume of all possible primitive unit cells of a given lattice is the same. If we take non-primitive unit cells, we find that their volumes are multiples of those of the primitive cells, depending on the number of lattice points located in one unit cell. For the selection of the appropriate unit cell out of the infinite number of possible ones, some conventions exist which are sometimes based on conflicting symmetry and metric considerations: e.g., the basis vectors should have a particular orientation with respect to the orientation of the symmetry elements and form a righthanded coordinate system, a
1.3.2 The 7 Crystal Systems On the basis of the morphologically derived seven crystallographic-axes systems, seven fundamental point lattices can be constructed, which are called the seven crystal systems (syngonies): triclinic (or anorthic) a, monoclinic m, orthorhombic o, tetragonal t, trigonal (or rhombohedral) r, hexagonal /z, and cubic c, respectively. Another classification scheme commonly used distinguishes between six crystal families only, combining trigonal and hexagonal symmetries into the hexagonal crystal family. The unit cells of these point lattices are parallelepipeds with metrics corresponding to those given for the crystallographic coordinate systems listed in Table 1-1. The eigensymmetry of the lattices, i.e., the group of symmetry operations leaving the lattice invariant, can be given by a group of primitive translations T={tl9t29...}, with ti = mia1+nia2
+ ota3
(1-32)
and a group K of point symmetry operations, the holohedral point groups (corresponding morphologically to the full set of crystal faces which are possible for a given crystallographic basis) (see Sec. 1.4.3). 1.3.3 The 14 Bravais Lattices According to the derivation of all possible different 3-D space lattices by Bravais in 1848, 14 point lattices exist, i.e., seven more than those found from morphological considerations (see Sec. 1.3.2). These additional point lattices result from socalled centering translations which lead to lattice points in the interior of the primitive unit cells of the seven point lattices, as discussed in the foregoing section. The combinations of the different translation groups with the holohedral point groups lead to
1.3 Crystal-Lattice Symmetry
the 14 different Bravais groups (Table 1-4). The action of the symmetry operations, which are elements of these groups, upon one single point produces the 14 Bravais lattices (Fig. 1-20). Beside the 7 primitive (without any interior lattice points) space lattices which are spanned by the basis vectors of the 7 crystallographic coordinate systems under the action of the group T9 there are 7 more lattice types as a result of centering translations: side-centered A: (1/2) (b + c\ B: (1/2) (a + c); base-centered C:
Table 1-4. The 14 Bravais groups with centering translations and group symbols. The trigonal centering translations are given on a hexagonal basis. Crystal system
Symbol
Triclinic
P
(0 0 0)
Monoclinic
P A
(0 0 0) (0 0 0) + (0 1/2 1/2)
P2/m A2/m
Orthorhombic
P C
(0 0 0) (0 0 0) + (1/2 1/2 0) (0 0 0) + (1/2 1/2 1/2) (0 0 0) + (1/2 1/2 0) + (1/2 0 1/2) + (0 1/2 1/2)
Pmmm Cmmm
PA/mmm IAImmm
I F
Centering translation
International symbol
Tetragonal
P I
(0 0 0) (0 0 0) + (1/2 1/2 1/2)
Trigonal
R
(0 0 0) + (2/3 1/3 1/3) + (1/3 2/3 2/3)
Hexagonal
P P I
(0 0 0)
Cubic
F
(0 0 0) (0 0 0) + (1/2 1/2 1/2) (0 0 0) + (1/2 1/2 0) + (1/2 0 1/2) + (0 1/2 1/2)
PI
23
(1/2) {a + b)\ body-centered /: (1/2) (a + b + c); all these centered cells contain one additional lattice point, they are called double primitive. The representation of the rhombohedral unit cell on a hexagonal basis R: (l/3)(-fl + A + c) and ( l / 3 ) ( - a - 2 * + c) leads to two interior lattice points (triple primitive). The face-centered lattice F: (1/2) (a + A), (1/2) (a + c), (1/2) (b + c) is fourfold primitive. Each centered lattice can be transformed to a primitive one with a smaller unit cell but at the cost of the rectangular symmetry of the centered lattice (Fig. 1-21). The volume of the unit cell, a parallelepiped with edge lengths a, b, c and angles a, /?, y is given by V = a b c (1 — cos 2 a — cos 2 /? — cos 2 y + + 2cosacosj3cosy) 1/2
(1-33)
or in vector notation V = a - {b x c) = b - {c x a) = c • (a x b) (1-34) or by
Immm Fmmm
F=[det(G)] 1/2 = a-a ab ac 1/2 - b a b b be c a cb c c with the metric tensor G.
(1-35)
1.3.4 The Reciprocal Lattice R3m
P'6Immm Pmlm Jm3m Fm3m
Say we have a vector lattice M on the basis a, 6, c and construct another lattice M* on the basis a*, 6*, c*. a* is taken perpendicular to the (6,c)-plane, and its length is equivalent to the reciprocal of the interplanar spacing d100 of the lattice planes (100) (Fig. 1-22). In an analogous manner we find A* perpendicular to (a,c) and c* perpendicular to (a, b\ and according to the definition of the vectors we ob-
24
\/ iA
1 Elements of Symmetry in Periodic Lattices, Quasicrystals
A a/-^T aP
mP
(a)
(b)
mA
7
(c)
oP
(d)
/I 7
AX,
b
b
o/
7
(e)
oC
\
7
(i)
7\ c\ \
Cl
oF
(g)
7
(h)
7
/
a//
,7
(f)
(j)
i/
\ (m)
hP
7
CF
(n)
CP
(1)
Figure 1-20. Unit cells of the 14 Bravais lattice types. The symbols consist of two letters, the first denotes the crystal system: a triclinic (anorthic), m monoclinic, o orthorhombic, t tetragonal, h hexagonal, c cubic. The second gives the centering type: P primitive, A side-centered, C base-centered, / body-centered, F face-centered, R rhombohedral cell on hexagonal coordinate system, (a) Triclinic primitive, (b) monoclinic primitive, (c) monoclinic side-centered, (d) orthorhombic primitive, (e) orthorhombic body-centered, (f) orthorhombic base-centered, (g) orthorhombic facecentered, (h) tetragonal primitive, (i) tetragonal bodycentered, (j) hexagonal primitive, (k) hexagonal rhombohedral, (1) cubic primitive, (m) cubic body-centered, and (n) cubic-face-centered lattice.
1.3 Crystal-Lattice Symmetry
Figure 1-21. Different choices of unit cells for a given point lattice: centered rectangular with basis vectors a, by and primitive oblique with basis vectors a', b, respectively.
25
sponds to that of the normals of the net planes of the direct lattice. Thus both lattices have the same holohedral point symmetries. The Bravais lattice types of the direct lattice and its corresponding reciprocal lattice are different in two cases: a body-centered direct lattice / corresponds to a face-centered reciprocal lattice /* = F, and vice versa F* = I. Generally, a reciprocal lattice vector H=ha* + kb* + Ic* is always perpendicular to the direct lattice plane (hkl). This will be demonstrated: Figure 1-23 shows a lattice plane (hkl) with intercepts a/h, b/k and c/l. The two vectors b/k — a/h and b/k — c/l lie in the plane and each vector normal to both vectors is, at the same time, normal to the plane. Thus, we have only to prove that the scalar products of both vectors with the reciprocal lattice vector H equal zero:
k h Figure 1-22. Direct lattice spanned by the basis vectors a, b and c (c is perpendicular to the plane of drawing), and the lattice reciprocal to it with basis a*, b*, c*. The distance between two lattice planes (net planes) of the set (h k I) is denoted by dhkl, the interplanar spacings di00 and d010 are marked.
k
k
— -a b* — Ta-c* = h h
=0+1+0-1-0-0 =
h
(1-38)
tain the scalar products a a* = 1 ,
a ** = 0 ,
A-a* = 0 ,
A-6* = 1, b-c*=0
c
a -c* = 0
(1-36)
• a * = 0 , c • A* = 0 , cc* = 1
using a basis af, a%,a%, we can write in a shorter notation araf = 5ij9
ij = 1,2, 3
(1-37)
with the Kronecker symbol btj = 1 for i =j, or zero. A/* is said to be reciprocal to the direct lattice M and vice versa. The point symmetry of the reciprocal lattice corre-
Figure 1-23. Lattice plane (hkl) with its normal vector H = h a* + k b* + 1 c*.
26
1 Elements of Symmetry in Periodic Lattices, Quasicrystals
and
= af-(ajxaf) h k = -b a*+-A 6* - b - c* — - c - a* — k k k I
(1-44)
For the volumes of the direct and reciprocal unit cells 7-7* = l
(1-45)
k I — - c- b* — - c- c* =
The metric relation between a direct lattice and its reciprocal lattice may become (1-39) =0+1+0-0-0-1=0 clearer if we consider that the volume of a general parallelepiped (Fig. 1-24) can be since the scalar products a • a*, b • 6*, c • c* calculated from the area of one of its limitare all equal to 1 and the mixed ones are ing faces times the height, which correequal to zero. sponds to the interplanar spacing dhkl. For The reciprocal basis vectors can be calinstance, taking the area of the paralleloculated from the direct ones by the followgram spanned by at and a2, and the height ing: The volume of a unit cell (parald we find 00l9 lelepiped spanned by al9 a2, a3) is given, for instance, by V = a a sin(a ) d (1-46) x
V = a1-(a2x a3) = a3 • {a1 x a2) = = a2-(a3xai)
(1-40)
Multiplication of both sides of the first term in Eq. (1-40) by af gives * V= = l-(a2x«3) a* = (a2 x a3)/V, af = a2 a3 si a* = (a3 x a^/F, a% = a3 a± sina 2 /K (1-42) a% = (a1 x a2)/V, af = a1 a2 sina 3 /F
3
0{0 1
Rewriting Eq. (1-46) gives = l/d 001
(1-47)
This expression is equal to the third one of Eq. (1-42) and we end up with
= V^ooi
(1-41)
and, generally,
2
(1-48)
showing that the norm of a% is equal to the interplanar spacing d001 between the set of net planes (001). The same holds for spacings of arbitrary planes (h k I) H = 1/dhkl
(1-49)
for the angles we get = (cos a 2 cos a 3 — cos aj^sin a 2 sin a3) cosaf = = (cosa 3 cosa 1 — c o s a ^ s i n a ! sina 3 )
a3
/
\
/
dooil
cosaf = (1-43) = (cos oc1 cos a 2 — cos a3)/(sin oc1 sin a2) The volume V* of the reciprocal unit cell (which is represented by a parallelepiped also) can be written analogously to Eq. (1-40)
/
/
Figure 1-24. Triclinic unit cell with the interplanar spacing d001 perpendicular to the (a1,a2)-plane marked.
1.3 Crystal-Lattice Symmetry
Hence, we can calculate the interplanar spacing dhkl from the scalar product
+ 2/c/6*-c*
(1-50)
Performing the scalar products gives
2/c//>*c*cosa*
(1-51)
the general formula for the calculation of interplanar spacings dhkl in triclinic systems. The formula becomes simpler for the monoclinic case with a* = y* = 90° l/d2kl = /*2(a*)2 + k2(b*)2 + /2(c*)2 +
27
tance of crystal faces in a simpler way: The morphological importance of a crystal form {hkl} increases with decreasing length H. Crystal forms {hkl} only appear if the indices are not affected by systematic extinction rules due to particular translations. 1.3.5 Topological Properties of Lattices
The topological properties of a point lattice can best be represented and analyzed by means of domains of influence (Wirkungsbereiche, Dirichlet or Voronoi domains, Wigner-Seitz cells). Such a domain for a special lattice point can be obtained by connecting that point with all other points of the lattice. A subset of the planes normal to these connecting lines and centered halfway on them, form a convex polyhedron - the domain of influence (Fig. 1-25). It contains all points in space
(1-52) and much more so for the orthogonal orthorhombic system l/d2kl = /i2(a*)2 + fc2(b*)2 + /2(c*)2 (1-53) or the tetragonal lattice with a* = b* l/d2kl = (h2 + fc2)(a*)2 + /2(c*)2
(1-54)
the hexagonal system with a* =fr*and cosy* = cos 60° = 1/2
and for the isometric cubic lattice (1-56) The concept of reciprocal space is very important for the diffraction theory. Diffracted intensities [reflections from netplanes (hkl) with interplanar spacings dhkl] can be ascribed to reciprocal lattice vectors H=ha* + kb* + lc*. Another important application, discussed in Sec. 1.2.6, is the formulation of the morphological impor-
Figure 1-25. Wigner-Seitz cells (Voronoi domains) for an oblique plane lattice. The lattice points are marked by black dots.
which are nearer to the selected lattice point then to any other one. In reciprocal space, or more familiarly in this connection, A:-space or wavevector space, the Wigner-Seitz cells correspond to the Brillouin zones (Fig. 1-26). The symmetry of the Wigner-Seitz cells corresponds to the holohedral point symmetry of the lattice. The construction of Wigner-Seitz cells for a given point lattice is unique, contrary to the selection of a conventional unit cell.
28
1 Elements of Symmetry in Periodic Lattices, Quasicrystals
For the metric tensor G the relation G =Pl G P
(1-61)
l
Figure 1-26. Brillouin zone (Wigner-Seitz cell of the reciprocal space) for the face-centered cubic lattice.
1.3.6 Lattice Transformations: Axes, Indices and Coordinates It may be necessary to select an alternate coordinate system for a given crystal lattice (Fig. 1-27). The new basis can be expressed by a linear combination of the old basis vectors
holds (P is the transposed form of P). The Miller indices of crystal faces transform with the same matrix P, they are covariant quantities (h'k'lf) =
The reciprocal basis (a*' A*' c*'), the coordinates {%' y z') of a point in a unit cell and the indices [u v w] of a direction in the direct space are contravariant quantities and transform with P " 1 , the inverse of P, (x'y'z') (u'v'W)
a = r11a + r12b + f13c b' = P21a + P22 b + P23c c' = P31a + P32 b + P33c
(1-58)
(1-59) P \
The determinant det(P) should be positive to preserve the sense of the coordinate system. The volume V of the transformed unit cell amounts to V'=V-det(F)
(1-63)
1
=(uvw)- P'
1.4 Crystallographic Point-Group Symmetry
with the transformation matrix
P
=(xyz)'P-1
(1-57)
or in matrix notation {a'b'c') = (abc) P
(1-62)
(1-60)
Figure 1-27. Unit cells with acute and obtuse angles between al9 a2 and a\, a2, respectively.
1.4.1 Group-Theoretical Terminology Point groups like the Bravais groups or space groups, mathematical groups G = {gt, #2 > • • •» 9k} with f ° u r fundamental properties (group axioms): (1) Each product of a multiplication of two or more group elements is an element of the same group again gi(g)gj = gkE K
(1-64)
All of the products of the multiplication of two elements of a particular group are listed in a so-called group-multiplication table. Each row (and column) of such a table contains each group element only once. An example for the point group mm2 is given in Table 1-5. (2) The consecutive multiplication of group elements is associative (gt ® gj) ®gk = gt ® (#,- ® gk)
(1-65)
29
1.4 Crystallographic Point-Group Symmetry
Table 1-5. Multiplication table for the point group mm 2. mm!
1
2Z
mx
my
1 2Z mx my
1 2Z mx my
2Z 1 my mx
mx my 1 2Z
my mx 22 1
Some groups, like the translation group T are commutative (Abelian) with Qi®9j = gj®Gi
(1-66)
i.e., the result of a multiplication is independent of the order in which it is performed. (3) Each group has a unit element e with e®9i = gi
(1-67)
For the group of symmetry operations, e corresponds to the identity operation 1. (4) For the group element gt the inverse one Q^1 exists, so that 0i<
=1
A subgroup G' = {g'l9g'2,...9gk,}, k'
(1-68)
The order k of a point group K is given by the number of its group elements, the order k of a point group element gt is given by the relation g\ = 1. The point group K = mm2 = {l,2 z ,m x ,m y }, for instance, is of the order k = 4, the order of its elements is 1, 2, 2,2. Each element of mm! is its own inverse element (the diagonal elements of Table 1-5 are all 1). A group G = {gl9g2,.--,gk} i s s a i d t o b e isomorphic to a group H = {h1,h2,...,hk} if a one-to-one correspondence exists between the elements of both groups as well as between the products of the elements. Naturally, both groups must be of the same order k. If the order of H is lower, several elements of G correspond to each element of H, and only a unidirectional correspondence exists; we say G is homomorphic to H.
(
T 0 0\ / I 0 0 0 T 0 0 1 0 0 0 1/ \0 0 1
A point group K may be represented by matrices of coordinate transformations which correspond to the given point symmetry operation. Thus the point group mra2, for instance, has the matrix representation r(mm2) mm2 = {l,2 2 ,m I ,m y }
(1-71)
T(mm2) = <\ 0 0\ / I 0 0\ / I 0 0 \ / I 0 0 10 1 0 , 0 T 0 , 0 1 0 , 0 T 0
lo o 1/ \o o 1/ \o o 1/ \o o
30
1 Elements of Symmetry in Periodic Lattices, Quasicrystals
Consequently, the successive action of symmetry operations refers to matrix multiplication. Depending on the basis of the coordinate systems, different but equivalent matrix representations are possible. All equivalent representations, however, must have the same character representations. The character x(gd of a matrix representation of a group element gt is defined as the trace of the matrix. Thus the representation for a 3-fold axis on a hexagonal coordinate system (cf. Table 1-2) has the character x(3) = 0 — 1 + 1 = 0, on the rhombohedral basis #(3) = 0 + 0 + 0 = 0, for instance. Another example: the character representation of the point group mm! corresponds to x(mm2) = {3,T, 1,1} whatever the basis might be. A shorter way to represent a point group is to give, instead of the set of transformation matrices, their action on a point with general coordinates x, y9 z. For the point group mm! we find four symmetrically equivalent points with coordinates x, y9 z; x, y, z; x, y, z; x, y, z. This set of points is called a point form and corresponds to a rectangle in our example. If the multiplicity of the point form is equal to the order of the generating point group then it is called a general point form, otherwise it is a special one. Then its site symmetry is higher than 1, and its multiplicity is equivalent to the order of the point group divided by the order of the site symmetry group. The action of the point group mm2, for instance, on a point in the special position 0, y, z transforms it to 0, y, z and back to 0, y, z again. The multiplicity of this special position is 2, therefore, and the site symmetry corresponds to mx, i.e., the point lies on the reflection plane mx. The point form is a line segment. The point form of a point group corresponds to the orbit of a space group (see Sec. 1.5.4). If the point group is acting on a crystal face (hkl) a crystal form
{hkl} is generated. General and special crystal forms are defined analogously to the point form. The faces of a general crystal form have an eigensymmetry of 1, those of a special one show higher eigensymmetry. The action of the group mm! leads, for a face (h k I) in a general position, to the set
of equivalent faces: {hkl), (hkl\ (hkl), (hkl) corresponding to a rhombic pyramid {hkl}. A face (010) with eigensymmetry m is transformed by this group into (0T0) forming a parallelohedron {010}. 1.4.2 Symmetry Operations
Point-group symmetry operations leave at least one singular point invariant. The set of all symmetry operations mapping an object onto itself forms the symmetry group of that object. Thus a benzene molecule, for instance, has point symmetry 6/mmm and an antiprismatic ferrocene molecule 52 m (Fig. 1-28). The number of
Figure 1-28. Antiprismatic ferrocene molecule with non-crystallographic point symmetry 52 m.
general point groups is infinite, those only consisting of symmetry elements which are compatible with 3-D point lattices are called the 32 crystallographic point groups. Crystallographic point-symmetry elements are restricted to the proper and improper rotation axes discussed in Sec. 1.2.5. Graphical evidence that only these rotation axes are compatible with all 3-D space lattices is given in Fig. 1-29: we rotate in a given 2-D lattice using the basic translations tl9 tl9 the lattice point P through the angle +
ap (R^) X X (q) ij/ (r) Q Qo *)2 + - R] (3-26) An^l map, considerable resolution is obtained among the binary systems in which where / is a concentration (and molar volumes) dependent function, A<£* =
homogeneous-electron gas exchange-correlation energy per electron dielectric response function bond angle inverse screening length entering pair potential shear modulus nth moment of local electronic density of states normalised nth moment electron density pair potential pair potential between a and /? atoms on sites i and j , a distance Rtj apart chemical scale Lindhard response function wave function atomic volume equilibrium atomic volume
au b.c.c. CPA d.h.c.p. f.c.c. h.c.p. KKR-CPA LDF NFE RBA SRO TB
atomic unit body centred cubic coherent potential approximation double hexagonal close packed face centred cubic hexagonal close packed Korringa, Kohn and Rostoker coherent potential approximation local density functional nearly free electron rigid band approximation short-range order tight binding
2.1 Introduction
Note on the Choice of Units The energy and length scales which are appropriate to electron theory are those set by the ionisation potential and first Bohr radius of the hydrogen atom. In SI units the energy and radius of the nth Bohr stationary orbit are given by 1 me n2\32n2s2h2 and an =
me2
where m is the electronic mass, e is the magnitude of the electronic charge, e0 is the permittivity of free space and h is Planck's constant divided by 2 n. Substituting in the values m = 9.1096 x 10~ 31 kg, e = 1.6022 x 0
and h = 1.0546 x 10 " 3 4 Js, we have 2.1799 x l O ~ 1 8 E J »= n2 and an = 5.2918 x l O ^ ^ m Therefore the ground state of the hydrogen atom, which corresponds to n = 1, has an energy of - 2 . 1 8 x l O ~ 1 8 J and an orbital Bohr radius of 0.529 x 10" 1 0 m or 0.529 A. The first value defines the Rydberg unit (Ry), the latter one the atomic unit (au). Thus, in atomic units we have En=
-n2Ry
and an = n2 au
where 1 Ry = 2.18 x 10" 1 8 J = 13.6 eV and 1 au - 5.29 x 10" X1 m = 0.529 A. It follows from the first, second, fifth and sixth equations that h2/(2 m) = 1 and e2/(4 nso) = 2 in atomic units.
65
The total energy of the bulk metal will usually be given in either Ry/atom or eV/atom. Conversion to other units may be achieved by using 1 mRy/atom = 1.32 kJ/mol - 0.314 kcal/mol.
2.1 Introduction Crystal structure and materials' properties are intimately linked. Currently, alloy developers are searching for new cubic alloys with (hopefully) good mechanical properties (see, e.g., Liu, 1984), new tetragonal alloys with (hopefully) good permanent magnetic properties (see, e.g., Mooij and Buschow, 1987), and new perovskite ceramics with (hopefully) good superconducting properties (see, e.g., Pickett, 1989, and references therein). Thus, the ability to understand the origins of structural stability and to predict which alloying additions might stabilise a required structure type is central to materials science and technology. This chapter reviews the dramatic developments in electron theory which now allow the quantitative prediction of the simpler crystal structures of elements and binary or ternary compounds. As recently as 1962 Hume-Rothery was stressing "the extreme difficulty of producing any really quantitative electron theory". The breakthrough came, in fact, two years later with the advent of local density functional theory, which transformed the many-electron problem into an effective one-electron problem which could be solved (Hohenberg and Kohn, 1964; Kohn and Sham, 1965). This recent ability to make accurate predictions has been accompanied by the development of simple, yet reliable nearly free electron or tight binding models, which have provided physical insight into the origin of bonding and structure at the
66
2 Electron Theory of Crystal Structure
atomistic level (see, e.g., Hafner, 1989; Majewski and Vogl, 1989; and references therein). This allows direct contact to be made with well-known factors such as electronegativity difference, atomic size, angular character of the bonding orbitals, and electron-per-atom ratio which are argued to control structural stability (see, e.g., Pearson, 1972). We begin in Sec. 2.2 with an overview of the experimental data base on the ground state structures of the elements and binary compounds with the AB, AB2 and AB 3 stoichiometries. In Sec. 2.3 some examples are given of the ability of local density functional theory to predict reliable ground state structures and charge density for metals, semiconductors and insulators. In Sec. 2.4 the nearly free electron approximation is introduced and applied to predicting the structures of simple metal elements and their compounds. In Sec. 2.5 the tight binding approximation is presented and applied to understanding the structural trends within sp-bonded elements and sdbonded rare-earth and transition metals and their compounds. In Sec. 2.6 the structural stability of solid solutions is discussed with particular reference to the famous Hume-Rothery electron phases within noble-metal-sp-valent alloys.
2.2 Experimental Data Base The experimental data base on the ground state structures of binary compounds with a given stoichiometry may be presented and ordered within a single two-dimensional structure map. This is achieved by running a one-dimensional string through the two-dimensional periodic table as shown in Fig. 2-1 (Pettifor, 1988 a). Pulling the ends of the string apart orders all the elements along a one-dimen-
sional axis, their sequential order being termed the Mendeleev number Jt. This simple procedure, which defines a purely phenomenological co-ordinate Jl, is found to provide excellent structural separation of all binary compounds with a given stoichiometry AmBn within a single map Figure 2-2 shows the AB ground state structure map using the experimental data base of Villars and Calvert (1985). The bare patches correspond to regions where compounds do not form due to either positive heats of formation or the competing stability of neighbouring phases with different stoichiometry. The boundaries do not have any significance other than they were drawn to separate compounds of different structure type. In regions where there is a paucity of data the boundary is usually chosen as the line separating adjoining groups in the periodic table. We see that excellent structural separation has been achieved between the 52 different AB structure types that have more than one representative compound each. The two most common structure types, namely Bl (NaCl) and B2 (CsCl), are well separated, the NaCl lattice being found only outside the region defined by JfA, J?B < 81, which encloses the main CsCl domain. There is only one exception, namely the very small region of Cs-containing salts. The AB structure map successfully demarcates even closely related structure types such as B27 (FeB) and B33 (CrB); B&± (NiAs) and B31 (MnP); or B3 (cubic ZnS, zincblende) and B4 (hexagonal ZnS, wurtzite). Moreover, coherent phases with respect to the b.c.c. lattice, namely B2 (CsCl), Bll (CuTi), and B32 (NaTl) are also well separated, as too are the close-packed polytypes cubic L l 0 (CuAu) and hexagonal B19 (AuCd). As might be expected, neighbouring domains often have similar or related local
2.2 Experimental Data Base
HA
IMA
I V A V A VIA V11A
Villa Vlllb
67
Vlllc
P H PP P 60 Mn
'T
17 1 Yb
/ 18 I Eu
33
32
31
I
I
30
29
28
27
26
24
La—Ce— Pr—
Nd — Pm — Sm-l(Eu )
Gd — Tb - - - Oy
48
47
45
41
Ac—
Th — Pa —
46
44
43
U — M p — Pu —
42
40
Am — Cm — Bk —
39
23 —
Ho 38
22 —
21
Er — Tm 37
36
20 -J (Y b ) L- Lu 35
34
Cf — Es — f m - M d — No — Lr -
Figure 2-1. The string running through this modified periodic table puts all the elements in sequential order according to the Mendeleev number. (Pettifor, 1988 a). Note that group IIA elements Be and Mg have been grouped with IIB, divalent rare earths have been separated from trivalent, and Y has been slotted between Tb and Dy.
co-ordination polyhedra. Very recently Villars et al. (1989) have assigned local coordination polyhedra to all binary structure types with more than five representative compounds each. Figure 2-3 illustrates some of the more commonly occurring coordination polyhedra, labelled according to a generalised Jensen notation [see Table 5 of Jensen (1989), Figs. 3 and 4 of Villars et al. (1989), and also Sec. 3.3.5.1 of this Volume]. Structure types can then be characterised not simply by their stoichiometric formulae, such as NaCl and NiAs, but by their crystal co-ordination formulae, such as 3 [NaCl6/6] and 3 [NiAs6 6,]
(Jensen, 1989). This immediately informs us that NaCl and NiAs are infinite threedimensional framework structures. Moreover, the symmetry of the co-ordination polyhedron about the Na, Cl or Ni sites is that of an octahedron (denoted by 6), whereas that about the As site is that of a trigonal prism (denoted by 6'), as can be seen from Fig. 2-3. It is therefore not surprising that the NaCl and NiAs domains adjoin each other in Fig. 2-2 and that there is a small domain of 3 [NbAs 676 /| stability in their midst. Figure 2-4 shows that excellent structural separation has been achieved be-
68
2 Electron Theory of Crystal Structure
EuLu Ho Tb Pm Ce Nc Es Cm Np Th EZA ^AlZEAMAMaTHIbWc
IA— IIA
D
100-
EPBB8aonaaaDDB
IB EB
I B IVBB VB SIB C 2HB NOFH
/
Baon
90
80
70
60
50-
30
20-
|/f... i . . . . I . . . .
I . . . . I.... I . . . . I . . . . I . . . . . . . . I
WC
NaCl
D
BaCu
a MoC
m PbO B
CsO
•
NbAs
A CuTi
• MnP
O ZnS c u b
V CoSn
•
• ZnShex O CrB
NiAs
< AsS
• NaC
A AlCu
> Til
•
HgCl TiAs
EB
NaO
V NaP
M
NiS
^
B
HgO
o
LiRh
v PtS
•
CdSb
B
NiO
o
BiSe
t, PdS
•
GaTe
a CoO
• Other
AB Figure 2-2. The AB structure map (Pettifor, 1988 a).
O FeB <•> AsGe <> M o B • GaMg
r n i_
FeSi GeS GaS
HgS + HgMn
j
*
IrV NaPb
•
AuCd
X
o CsCl G CuAu NaTl e SeTl LiAs
©
KGe c AlCe e AIDy
®
69
2.2 Experimental Data Base
41
61
101
12
121
121
14
141
141
16
Figure 2-3. Local co-ordination polyhedra with generalised Jensen notation (after Villars et al., 1989).
70
2 Electron T h e o r y of Crystal Structure
10 1
• i•
IA
20 • *• •
HA
I
EuLu
T T T I
Ho
30 '
Tb
'i
Pm Ce
nunn uuuu G no no
100
40 • '
No
50
i
Es
r^•
Cm Np Th
HA
60 •
70
r
'
i
Y A YIA WAWaYflTbYIIIc
n
80 •
IB
•
HIB
com CD m m » » i n H )iil»u l m mm mm a m CHT^WT
•
'
—
i
j
JLi
90
I
IB
ii
EBB
LI** • nrmtn* 1
i
YB
100 '
i
•
"SZEB C Y U B N O F H
• • • • • • • • 4 *
• /
i '
/
y
/
/
•
/
-
C 3ZTB
90 B
EZB
- ILB IB
70
YflLc
YULQ
60
Jk,
2LA YA _EZ"A
50
Th Np
40-
Cm Es
1
No Ce
Tb Ho
20 —
Lu Eu
i
10
IA
I
D CaF2 Q CaC2 m TiO2 H Cdl2 BB Ag2O H Li2Sb 0ThH 2 s BQS 2
n
a H a B
H B
I .....I 1 I , , ,,!,.,• I Cu2Sb nCS2 AMgZn 2 A CoSb2 u Pd2As CQC12 V Cu2Mg RuB2 V0 2 <MgNi2 =i DySe 2 t> Caln 2 CuCl2 • FeS2(m) c GdSi2 ^ Cd2Ce ZrO 2 H MoS2 r MoCI2 P7 SiS2 TQ 2 P T HoSb2 • OsGe2 K GaCl2 NbTe2 -L Ll2GQ • Au2V A PdCU SrnSb2 I- ZrGa2 <>CuP2 H FeSu
O ZnP2 • NbSb2
AB9
Figure 2-4. The AB2 structure map (Pettifor, 1988 a).
I...... :
PtHg2 ^ V2N < NdAs2 >CaSb 2 -z. CuMg2 sEuS2 N Au 2 Nd M NbS2 m Li2Sb iu ReB7
nCaSi 2 rCdCI 2 L As2Ge JFe 2 N tFeSb2 j PdSe2
PbO2 PtoGa
I A CeCu2 V ZrSb 2 o HfGa 2 o Ir Se2 O ZrSi 2 <3> Ti Si2 O ThSi2 0 AIB 2 • Co2Si • Ofher
o MoSi 2 0 A!2Cu © Ni 2 ln e Fe 2 P e CrSi 2 ® La2Sb * MoPf2 o SrBr2 © Srl2 © Ag2Te
2.2 Experimental Data Base
tween the 84 different AB2 structure types that have more than one representative compound each. The 8:4 co-ordinated fluorite structure Cl (CaF2) with crystal coordination formula 3 [CaF8/4] is observed in the ionic regions at the top left and bottom right of Fig. 2-4, although it also occurs elsewhere, for example in the small domains centred on Mg2Si, Rh 2 P and Al2Pt. The latter domains are metallic. In these cases it is more sensible to define the local co-ordination polyhedron about the fluorine site by including the six next nearest neighbour fluorine atoms in addition to the four tetrahedrally configured calcium first nearest neighbours. This leads to the ten-atom configuration polyhedron 10IV, which is shown in Fig. 2-3 (Villars et al., 1989). A similar type of assignment must be made for the B2 (CsCl) lattice: in the ionic region only first nearest neighbour unlike atoms are retained so that the crystal co-ordination formula is 3 [CsCl8/8], whereas in the very large metallic domain of Fig. 2-2 the six second nearest neighbour like atoms (which are only 14% more distant) are also included to define the fourteen-atom configuration polyhedron 14, which is drawn in Fig. 2-3 [see Sec. 2.8 of Jensen (1989)]. The 6:3 co-ordinated rutile structure C4 (TiO2) with its crystal co-ordination formula 3 [TiO 6/3 ] is stable only with the very electronegative constituents hydrogen, oxygen and fluorine. The corresponding compounds with the less electronegative halogens form the two-dimensional layer structures C19 (CdCl2) and C6 (Cdl2), in which the metal atom is sixfold octahedrally co-ordinated. They thus have the crystal co-ordination formulae 2 [CdCl6/3] and 2 [CdI6/3], respectively. These two structures are very similar in energy because they differ only in the stacking of the
71
composite layers, which are held together by weak van der Waals interactions. CdCl2 has the halogen atoms arranged on a f.c.c. lattice, whereas Cdl 2 has them on a h.c.p. lattice. The metal atoms can also be sandwiched so that they have sixfold trigonal symmetry; a- and P-MoS2 with crystal coordination formula 2 [MoS673] correspond to the different stacking sequences 3R and 2H t respectively [see Fig. 4.11 of Wells (1975)]. They are found amongst the early transition metal sulphides, selenides and halides. The two forms of FeS 2 , namely C2 (pyrites) and C18 (marcasite), are well separated in Fig. 2-4. They have the same sixfold octahedral co-ordination about the Fe site and are characterised by the S atoms occurring in pairs. The pyrites structure is derived from the NaCl structure by replacing Na with Fe and Cl with S 2 dimers pointing along <111> directions, so that it may be assigned the crystal co-ordination formula 03 Fe [S 2 ] 6/6 [see Table 13 of Jensen (1989)]. C l l b (MoSi2), C40 (CrSi2), and C54 (TiSi2) are poly types which result from stacking close-packed planes of AB2 stoichiometry in a b.c.c. (110) stacking sequence, so that it is not surprising that MoSi2 is characterized by the distorted b.c.c. local co-ordination polyhedron 14 and CrSi2 and TiSi2 by the variant 14' (see Fig. 2-3). The C16 (CuAl2) structure type neighbours both these polytypes and CaF 2 , which is not unexpected since it can be derived from the b.c.c. lattice (Burdett, 1982). Finally, the well-known space-filling AB2 Laves phases are found almost in their entirety above the diagonal in Fig. 2-4 and are seen to be well separated from each other amongst the C14 (MgZn2), C15 (MgCu2), and C36 (MgNi2) structure types. The larger A atom is surrounded by
72
2 Electron Theory of Crystal Structure
16 atoms within the Frank-Kasper co-ordination polyhedron 16, whereas the smaller B atom is surrounded icosahedrally by 12 atoms as illustrated by 12" in Fig. 2-3. Figure 2-5 shows that excellent structural separation has also been achieved between the 52 different AB 3 structure types that have more than one representative compound each. The simplest three-dimensional framework structure with AB 3 stoichiometry that can be built from the octahedral AB6 complex is cubic ReO 3 , in which every octahedron is joined to six others through their vertices. This structure with crystal co-ordination formula 3 J
oo
[ReO6/2'] is adopted by the trifluorides ZrF 3 , TaF 3 and NbF 3 . Like their AB2 counterparts, the less-electronegative halogens form two-dimensional layer structures, but with the A atoms occupying only two-thirds of the octahedral holes within the close-packed sandwich of B atoms. 2 [AlCl6/2] has the halogen atoms arranged on a f.c.c. lattice, whereas 2 [FeCl6/2] has them on a h.c.p. lattice. The halogens with group IVA form the one-dimensional chain structures TiCl3 and Til 3 , in which the AB6 octahedra share opposite faces. Distorted tricapped trigonal prismatic co-ordination is shown by the halogens with the actinides and some rare-earth and group IIIA elements; YF 3 , LaF 3 and UC13 structure types have ninefold co-ordination about the metal sites. It is clear from Fig. 2-5 that many 1:3 stoichiometric compounds take closepacked structure types with either the (distorted) cubic 12 or hexagonal 12' local co-ordination polyhedron (see Fig. 2-3). Consider frist the close-packed layer of MN 3 stoichiometry with a triangular arrangement of the M atoms [see Fig. 7.15 of Pearson (1972)]. These close-packed layers
may be stacked one above the other in the usual close-packed positions A, B or C, so that the M atoms have only N atoms as nearest neighbours. The following polytypes have been marked explicitly on the structure map: Ll 2 (Cu3Au) with cubic ABC stacking sequence, D0 19 (Ni3Sn) with hexagonal close-packed AB stacking sequence, Ni 3 Ti with double hexagonal close-packed sequence ABAC, and BaPb 3 with the hexagonal cell containing nine close-packed layers in the sequence ACACBCBAB. The CuTi 3 structure type is a tetragonal distortion of Cu3Au. Finally, the structure types D0 22 (Al3Ti) and Cu3Ti are polytypes which are based on the stacking of close-packed MN 3 layers with rectangular arrangement of the M atoms [see Fig. 7.21 of Pearson (1972)]. They are well separated from each other and from another close-packed superstructure D0 2 4 (Al3Zr). Finally, most AB3 compounds with B from groups IVA, VA or VIA take the A15 (Cr3Si) structure type. The A atoms form a b.c.c. lattice through which lines of B atoms run parallel to the edges of the cubic cell. The A15 structure is compact with the A atoms surrounded icosahedrally by twelve B atoms with co-ordination polyhedron 12". The B atoms sit at the centre of a fourteen-atom polyhedron 14'" which is formed from four A and ten B nearest neighbours. The D0 3 (BiF3) structure type is an ordered structure based on the b.c.c. lattice with the local co-ordination polyhedron 14. The pure elements would lie along the diagonal line JiK = JiB in these binary structure maps. It is therefore not surprising that the diagonal in Fig. 2-2 cuts through the CsCl, CuAu, and cubic ZnS domains where the elemental b.c.c, f.c.c. and diamond lattices are stable, whereas in Fig. 2-5 it passes through the Cu3Au,
73
2.2 Experimental Data Base
10
20
' ' " " l " " ' " " l
30 I
40
50
" " I
60
• • • • ) ' • , , , , , , . ,
70
80
|,.,.|....,
90
100
,.,,.,••••,
<-IA- HA EuLu HoTb PmCe No Es Cm NpTh ETA^AWWHA^-VIII—IB EB BIB IVBB VB VIBCVEBNOFH/ — ^»« 100-
90
SOI
Figure 2-5.
The AB3 structure map (Pettifor, 1988
a).
/
74
2 Electron Theory of Crystal Structure
Ni3Sn, and Cr3Si domains where the elemental f.c.c, h.c.p. and (3-W lattices are found. Thus the phenomenological Mendeleev number should also order the elements according to their structure type. This is broadly the case as can be seen from Table 2-1, where the string runs from the close-packed noble gas and metallic elements through the more open metalloid elements to the halogens and hydrogen, which solidify as dimers held together on the lattice by very weak van der Waals interactions. The importance of running the string from right to left through the lanthanides and actinides is now apparent if the continuity of structure is to be maintained. We will see in Sec. 2.5.3 that this is in accordance with quantum mechanical theory, which predicts the structural trend from h.c.p. to d.h.c.p. to f.c.c. as the core size and corresponding number of valence d electrons increases through these trivalent systems (Duthie and Pettifor, 1977). Similarly, the string runs from the top (or near the top) of group IIB (IIIB) to the bottom of group IIIB (IVB) for structural continuity. It should be noted that Li and Na are b.c.c. at room temperature and Yb f.c.c. We will see in Sec. 2.5.3 that the anomalous structure of a-Mn and b.c.c. Fe (compared with the 4 d and 5 d elements in the same group) results from the presence of magnetism (e.g., Hasegawa and Pettifor, 1983). The more complex structures of some of the early actinides are a result of the importance of the f electrons in the elemental bonding (Skriver, 1985). This ordering of the experimental binary compound structural data base within twodimensional maps has led to their use in the search for new pseudobinary alloys with a required structure type. For example, in the aircraft industry new lightweight titanium-aluminium-based alloys are be-
Table2-1. Structures of elements (Pettifor, 1988 a). Mendeleev no. Jt 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 a
Element
Structure3
He Ne Ar Kr Xe Rn Fr Cs Rb K Na Li Ra Ba Sr Ca Yb Eu Sc Lu Tm Er Ho Dy Y Tb Gd Sm Pm Nd Pr Ce La Lr No Md Fm Es Cf Bk
h.c.p. f.c.c. f.c.c. f.c.c. f.c.c. — — b.c.c. b.c.c. b.c.c. h.c.p. c.p. — b.c.c. f.c.c. f.c.c. h.c.p. b.c.c. h.c.p. h.c.p. h.c.p. h.c.p. h.c.p. h.c.p. h.c.p. h.c.p. h.c.p. c.p. d.h.c.p. d.h.c.p. d.h.c.p. f.c.c. d.h.c.p. — — — — _ d.h.c.p. d.h.c.p.
s.c, simple cubic; b.c.c, body centred cubic; f.c.c, face centred cubic; h.c.p., hexagonal close packed; d.h.c.p., double hexagonal close packed; c.p., close packed stacking variant; tetr., tetragonal; orth., orthorhombic; rhom., rhombohedral; compl, complex; dia., diamond; gra., graphite; lay., chain, ring and dim. are structure types built up from puckered layers, helical chains, rings and dimers, respectively.
2.2 Experimental Data Base
Table 2-1. Continued. Mendeleev no. M 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87
Table 2-1. Continued.
Element
Structurea
Mendeleev
Element
Structure3
As P Po Te Se S C At I Br Cl N O F H
lay. compl. s.c. chain chain ring gra dim. dim. dim. dim. dim. dim. dim. dim.
no. Ji
Cm Am Pu Np U Pa Th Ac Zr Hf Ti Ta Nb V W Mo Cr Re Tc Mn Fe Ru Os Co Rh Ir Ni Pt Pd Au Ag Cu Mg Hg Cd Zn Be Tl In Al Ga Pb Sn Ge Si B Bi Sb
d.h.c.p. d.h.c.p. compl. orth. orth. tetr. f.c.c. f.c.c. h.c.p. h.c.p. h.c.p. b.c.c. b.c.c. b.c.c. b.c.c. b.c.c. b.c.c. h.c.p. h.c.p. compl. b.c.c. h.c.p. h.c.p. h.c.p. f.c.c. f.c.c. f.c.c. f.c.c. f.c.c. f.c.c. f.c.c. f.c.c. h.c.p. rhom. h.c.p. h.c.p. h.c.p. h.c.p. tetr. f.c.c. compl. f.c.c. dia. dia. dia. compl. lay. lay.
90 91 92 93 94 95 96 97 98 99 100 101 102 103
ing sought with the cubic Ll 2 (Cu3Au) structure type, which may have better mechanical properties than the brittle noncubic Ti3Al or TiAl3 binary systems (Liu et al, 1989; Subramanian et al, 1989). In the magnet industry new rare-earth-ironbased alloys are being sought with the tetragonal BaCd l l 9 ThMn 12 or NaZn 13 structure types for use as good, cheap permanent magnets (Mooij and Buschow, 1987; Pettifor, 1988 b). The structure maps suggest which alloying elements might move the parent binary phase from its given structural domain to a neighbouring domain with the desired structure type. However, the phenomenological structure maps perform only a limited function. They cannot guarantee a priori that the desired pseudobinary will be stabilised (rather than an unwanted neighbouring phase with different stoichiometry), nor can they guarantee that the pseudobinary, if stabilised, has the required physical properties (Pettifor, 1991). A more fundamental quantum mechanical understanding of phase stability is required.
76
2 Electron Theory of Crystal Structure
2.3 Ab Initio Prediction of Crystal Structure 2.3.1 Local Density Functional Theory
The crystal structure that a given phase adopts depends directly on the bonding between the constituent atoms. Thus, in order to provide a fundamental theory of structure, it is necessary to understand the behaviour of the valence electrons which bind the atoms together. The theory which describes the electrons in a solid is couched, however, in a conceptual framework that is very different from our everyday experience, since the microscopic world of electrons is governed by quantum mechanics rather than the more familiar classical mechanics of Newton. Rather than solving Newton's laws of motion the quantum theorist solves the Schrodinger equation (2-1) m where V2 = 82/8x2 + d2/dy2 + 82/8z2, m is the electronic mass and h is Planck's constant divided by 2 n. The term - (h2/2m) V2 represents the kinetic energy and v(r) the potential felt by the electron which has energy E. \j/(r) is the wave function of the electron, where \il/(r)\2 gives the probability density of finding the electron at some point r = (x, y, z). The power of the Schrodinger equation is illustrated by solving Eq. (2-1) for the case of the hydrogen atom. As is well known (see, e.g., Schiff, 1968), solutions exist only if the wavefunction \j/ is characterized by three distinct quantum numbers rc, /, and m. A fourth quantum number ms, representing spin, arises from the relativistic extension of Eq. (2-1). Thus, the existence of different orbital shells and hence the chemistry of the periodic table is under-
pinned directly by the Schrodinger equation. Nevertheless, the problem remained of how to solve the Schrodinger equation for the many-body problem, which is encountered when atoms are brought together to form the solid (see, e.g., Cottrell, 1988). In 1928 Hartree made the simplest assumption that the individual electrons move independently of each other, so that each electron feels the average electric field of all the other electrons in addition to the potential from the ionic lattice. This approximation, however, fails to describe the bonding between atoms. For example, the bond in aluminium is predicted to be more than two orders of magnitude smaller than that observed experimentally. In 1930 Fock extended the theory by taking account of Pauli's exclusion principle, which states that no two electrons can be in the same quantum state. This automatically introduces correlations between parallelspin electrons which have the same spin quantum state. Unfortunately, the HartreeFock approximation still makes a sizeable error because correlations between antiparallel spin electrons are neglected. This error leads to the electronic specific heat of metals, for example, varying with temperature as T/(log T), whereas experimentally metals display a simple linear temperature dependence. But to go beyond the HartreeFock approximation seemed very hard, as Hume-Rothery noted in the preface of his textbook Atomic Theory for Students of Metallurgy (Hume-Rothery, 1962). The breakthrough came two years later when Hohenberg, Kohn and Sham proved that the total ground state energy of a many-body system is a functional of the electron density (Hohenberg and Kohn, 1964; Kohn and Sham, 1965). This seemingly simple result, by focusing on the electron density rather than the many-body
2.3 Ab Initio Prediction of Crystal Structure
wave function, allowed them to derive an effective Schrodinger equation that was directly analogous to Hartree's except that each electron now feels an additional attractive potential. In principle, density functional theory takes account of all the correlations between the electrons so that any given electron is surrounded by a mutual-exclusion-zone or hole from which other electrons are kept out. In practice, the exact shape of this hole is not known. Hohenberg, Kohn and Sham therefore proposed replacing the exact hole by the hole which an electron would have in a homogeneous free electron gas with the same density as that seen locally by the given electron at any particular instant the so-called local density functional (LDF) approximation. Relatively recent reviews and books on the foundations, applications and limitations of the LDF approximation are by Lunquist and March (1983), Callaway and March (1984), Jones and Gunnarsson (1989) and Parr and Yang (1989). The effective one-electron Schrodinger equation within LDF theory may therefore be written in the form (2-2)
h2
where vH(r) is the usual Hartree potential arising from the ions and the average electrostatic field of all the electrons and vxc (r) is the attractive potential arising from the mutual-exclusion-zone or exchange-correlation hole around the ith electron. Slater (1951) gave a simple physical argument to show that the magnitude of the latter potential should vary as [@(r)]1/3 in the first approximation, where Q (r) is the local density, namely r)\2
(2-3)
77
with i running over all the occupied electronic states. The total energy cannot be written simply as the sum over all the occupied oneelectron energies Et because the eigenvalue Et of the fth electron contains the potential energy of interaction with the jth electron and vice versa. Thus Et + Ej double-counts the Coulomb interaction energy between electrons i and j . The total LDF energy is therefore given by
- J Q (r) R e [Q W] - £xc [Q Ml} d r + Uion_ion where e is the electronic charge, £0 is the permittivity of free space, £xc (g) is the exchange and correlation energy per electron of a homogenous electron gas of density Q, and vxc (Q) is the exchange-correlation potential entering Eq. (2-2). The second and third contributions to Eq. (2-4) correct for the double-counting of the Coulomb and exchange-correlation energies respectively, whereas the last contribution represents the ion-ion Coulomb interaction. The accurate solution of the LDF Schrodinger equation in order to find the band structure Ek and hence the total binding energy U of crystalline solids requires non-trivial computer programs which are usually based on pseudopotentials, augmented plane waves or muffin tin orbitals. The interested reader is referred to Chap. 1 of Vols. 3 and 4 in this Series for a detailed discussion of these first principles techniques. In this chapter we present some selected results of these accurate first principles calculations and then discuss the underlying origins of structural trends within the simpler and more intuitive nearly free electron (NFE) and tight binding (TB) approximations.
78
2 Electron Theory of Crystal Structure
Table 2-2. Comparison of the predicted and experimental values of the equilibrium atomic volume Qo, cohesive energy Ucoh and bulk modulus K of the 3d transition elements (after Paxton et al., 1990). Element
Q0(k 3)
Structure
Sc Ti V Cr Mn Fe Co Ni Cu
h.c.p. h.c.p. b.c.c. b.c.c. h.c.p. h.c.p. f.c.c. f.c.c. f.c.c.
(eV/atom)
K (Mbar)
Theory
Expt.
Theory
Expt.
Theory
Expt.
23.0 16.8 13.0 11.2 10.6 10.3 10.4 10.9 11.9
25.0 17.7 13.8 12.0 12.2 11.8 11.1 10.9 11.8
4.87 5.98 5.83 4.58 4.61 5.90 5.96 5.29 3.89
3.93 4.86 5.30 4.10 2.98 4.29 4.39 4.44 3.50
0.6 1.2 2.0 2.8 2.9 3.0 2.6 2.0 1.6
0.44 1.05 1.62 1.90 0.60 1.68 1.91 1.86 1.37
2.3.2 Elements
An excellent example of the power of LDF theory for making reliable structural predictions is provided by the binding energy curves for the 3 d transition metals in Fig. 2-6 (Paxton et al., 1990). These curves were calculated from first principles, the only input into the computer being the atomic number and crystal structure. The latter was chosen to range from the closepacked structure types f.c.c, b.c.c. and h.c.p. through the more open simple hexagonal and simple cubic lattices to the fourfold co-ordinated diamond cubic structure type. The calculations were performed neglecting magnetic contributions in the bulk. We see that Fig. 2-6 predicts the well-known structure trend across the non-magnetic 3 d, 4d and 5d transition metal series, namely h.c.p. -»b.c.c. -• h.c.p. -• f.c.c. (see Table 2-1 with Jt running from 49 to 72). The accuracy of the LDF binding energy curves can be gauged from Table 2-2, where the predicted cohesive energy, equilibrium atomic volume and bulk modulus are compared with experiment (Paxton et al., 1990). We see immediately that the cohesive energy is substantially overestimated, being 25% too large for Sc and rising to nearly
60% for Mn. The source of this error is probably due to the poor treatment of the free atom within the local density approximation (see, e.g., Jones and Gunnarsson, 1989). The binding energy is, of course, the difference in energy between that of the solid, Eq. (2-4), and that of the isolated free atoms. Fortunately, in materials science we are usually interested in predicting the relative stability of one bulk phase with respect to another bulk phase, so that the free atom state is not of immediate import and systematic errors in the total LDF energy of bulk phases tend to cancel. This is evidenced by the correct structural predictions in Fig. 2-6 for the non-magnetic 3 d elements. We will see other examples of this in the next section, where the predicted LDF heats of formation are found to agree well with experiment. We see from Table 2-2 that the equilibrium volume is predicted to be about 10% too small away from the noble metal end of
Figure 2-6. The predicted LDF binding energy curves of the 3 d transition elements in the non-magnetic state [from Paxton et al. (1990); reproduced with permission ]. Q/Qo is the ratio of the atomic volume to the equilibrium value.
2.3 Ab Initio Prediction of Crystal Structure
79
Sc -2.5 -
-2.5
-3.0
-4.5
-5.0
-5.5 -6.1 -6.5
0.8 i
i
i
08
i
'
1.0
0.9 i
i
12
10 i
1.4
1.6
-3.5
Mn -2.5 \
n
\
-3.0
1. cubic -45
a s.cubic -3.5
- X o _ [ _a \j_^T
s.hex
-50
-4.3
-4.0 -4.4
-4.5
~ ^-ftR^
hcp
"^ ^ -4.6
-55
bcc
V
,fcc 0.8
0.9
i
i
i
-5.0
0.8 0.8
1.0
1.2
0.9 1.4
1.0 1.6
-5.5 0.8
1.0
1.2
1.4
1.6
0.8
10
1.2
14
1.6
80
2 Electron Theory of Crystal Structure 1
2.7 _-
1
_
-X-..
""••x.
3 -
- 0 II-""" .O—
X
S2.6 co
-x... .•X""
o—
*—-D
2.5 2.4
(b)
(a)
Cr
1
1
1
Mn
Fe
Co
Ni
1-
_
'••x-*'
1
1
1
Cr
Mn
Fe
Co
Ni
Figure 2-7. The equilibrium Wigner-Seitz radius S (a) and bulk modulus K (b) across the magnetic 3d transition metals. The crosses, circles and squares are the experimental, spin-polarized LDF and non-magnetic LDF results, respectively (after Janak and Williams, 1976).
graphite structure is 1 kJ/mol more stable than the diamond cubic structure for carbon whereas it is 70 kJ/mol less stable for silicon (Yin and Cohen, 1983). The accuracy of the LDF predictions for the cohesive energy, equilibrium lattice constant and bulk modulus is much better for the sp-bonded elements than the sd-bonded transition elements. This can be seen by comparing the results in Table 2-3 for C, Si and Ge with the diamond cubic structure with those in Table 2-2 for the 3 d transition elements. Recently ab initio LDF calculations have been used to study the transformation path from b.c.c. to h.c.p. in barium under pressure at the absolute zero of temperature (Chen et al., 1988; Ho and Harman, 1990). As illustrated in Fig. 2-8, the b.c.c. to h.c.p. transformation involves atomic displacements corresponding to the zone boundary [110] T1 phonon mode and an additional lattice shear (Burgers, 1934).
the series. Figure 2-7 a shows that the larger than expected atomic volumes taken by the ferromagnetic elements Fe, Co and Ni can be partially accounted for by the inclusion of spin polarization within the LDF calculations (Janak and Williams, 1976). Such calculations predict magnetic moments in good agreement with experiment. Finally we see the errors in the bulk modulus of the non-magnetic transition elements can approach 35%. The very large discrepancies, which are observed for Mn and Fe, are mainly magnetic in origin, as Fig. 2-7 b indicates. Similar curves to Fig. 2-6 have also been computed for sp-bonded elements. For example, in their classic paper Yin and Cohen (1982) predicted correctly that the diamond cubic lattice is more stable than the diamond hexagonal lattice for silicon and germanium. In addition the transition to P-Sn under pressure is faithfully reproduced. As expected, they found that the
Table 2-3. Comparison of the predicted and experimental values of the equilibrium lattice constant a, cohesive energy Ucoh and bulk modulus K of the sp-bonded elements C, Si and Ge with the diamond structure (after Yin and Cohen, 1982, 1983).
a(k)
Element
C Si Ge
£/coh(eV/atom)
K (Mbar)
Theory
Expt.
Theory
Expt.
Theory
Expt.
3.60 5.45 5.66
3.57 5.43 5.65
7.57 4.67 4.02
7.37 4.63 3.85
4.41 0.98 0.73
4.43 0.99 0.77
2.3 Ab Initio Prediction of Crystal Structure
The dashed lines in Fig. 2-8 b show that a displacement 5 = ^/2a/12 in this b.c.c. phonon mode creates a nearly hexagonal geometry, the perfect geometry being achieved in Fig. 2-8 c through a subsequent shear which changes the angle 9 from 109.47° to 120°. Figure 2-9 displays the calculated total energy contours as a function of both co-ordinates 5 and 8 for barium at its equilibrium atomic volume £20,0.793 Qo and 0.705 Qo, respectively. The latter volume corresponds to a pressure of 38.4 kbar. We see that at Q = Qo the upper contour plot shows that b.c.c. barium is more stable than h.c.p., in agreement with experiment. However, as pressure is applied, the h.c.p. phase has its energy lowered with respect to b.c.c. The middle contour plot shows that at Q = 0.793 Qo their energies are approximately equal, with an energy barrier between them of about 4 meV/atom. The lower contour plot shows that at Q = 0.705 Qo the energy barrier has gone and the b.c.c. phase is no longer metastable.
81
The predicted T = 0 transformation pressure is 11 kbar, corresponding to the b.c.c. and h.c.p. lattices having equal enthalpies. However, at low temperatures the system would not be able to overcome the energy barrier so that the b.c.c. phase would probably remain metastable until the Tx N-point phonon mode became soft at 31 kbar. Experimentally the phase transformation occurs at a pressure of 55 kbar at room temperature so that the LDF predicted pressure appears too low, reflecting the intrinsic errors in the local approximation to density functional theory (see Tables 2-2 and 2-3). 2.3.3 Compounds
A good example of the reliability of LDF theory for predicting intermetalhc heats of formation and structural stability at absolute zero is given by the aluminium-lithium system. This system has recently come to the fore with the successful development of lightweight aluminium-lithium alloys for aerospace applications. Figure 2-10 gives the predicted LDF heats of formation of different ordered structures with respect to either the f.c.c. or b.c.c. lattices (Sluiter et al., 1990). The AB structure map, Fig. 2-2, shows that LiAl with Mendeleev co-ordinate {Jtu = 12, JtK1 = 80) falls in the B32 (NaTl) domain. On the other hand, LiAl3 and Li3Al are metastable phases which fall
(a)
&—& —€>
(b)
(IIO)bcc
c)
Figure 2-8. Illustration of the b.c.c. to h.c.p. phase transformation. The arrows in (a) and (b) indicate the atomic displacements in the b.c.c. lattice corresponding to the polarisation vector of the Ti N-point phonon mode. A final long-wavelength shear changes the angle from 109.47° to 120° to obtain the h.c.p. lattice in (c) [from Ho and Harmon (1990); reproduced with permission].
82
2 Electron Theory of Crystal Structure
107.5°
127.5°
0.02 0.04 0.06 0.08 0.1 Atomic displacement /V2a
0.12
0 Al
0.2 0.4 0.6 0.8 concentration of lithium
Figure 2-10. The predicted heat of formation of f.c.cand b.c.c.-based lithium-aluminium ordered compounds (after Sluiter et al., 1990).
0.02
0.04
0.06
0.08
0.1
0.12
Atomic displacement /V2a 107.5°
127.5< 0.02
0.04
0.06
0.08
0.1
0.12
Atomic displacement /V2a
Figure 2-9. Contour plots of the LDF energy for barium as a function of the atomic displacement S corresponding to the T t N-point phonon mode and the angle 9 of the shear motion. The upper, middle and lower panels correspond to the volumes Qo, 0.793 Qo and 0.705 Qo, where Qo is the observed equilibrium volume at ambient pressure. The energy contours are in steps of 0.5 mRy/cell [from Ho and Harmon (1990); reproduced with permission].
near the boundaries of the L l 2 (Cu3Au) and DO 3 (BiF3) domains, respectively, within the AB3 structure map (Fig. 2-5). We see from Fig. 2-10 that the LDF heats of formation indeed predict the B32 LiAl phase to be much more stable than the B2 or L l 0 equiatomic phases. In addition, this strong stability of the B32 phase is responsible for the metastability of the neighbouring LI 2 LiAl3 and D0 3 Li3Al phases. Another important class of intermetallics are the titanium-aluminides. Figure 211 shows the predicted LDF heats of formation of different ordered structures with respect to either the f.c.c. or h.c.p. lattices (van Schilfgaarde et al., 1990). We see that the theory predicts the correct most stable ground state structure for Ti3Al and TiAl3, namely hexagonal D0 1 9 and tetragonal D0 2 2 , respectively. Moreover, Fig. 2-11 shows that whereas the metastable cubic LI 2 phase is very close to the ground state energy for TiAl3 it is much further removed for Ti3Al. This accounts for the fact that whereas TiAl3 has been stabilised as a
2.3 Ab Initio Prediction of Crystal Structure
cubic pseudobinary by suitable alloying addition, it has not been possible to stabilise the cubic phase of Ti3Al even though the phenomenological AB3 structure map of Fig. 2-5 suggests that this might be the case (Liu et al., 1989). This illustrates the importance of the first principles LDF calculations: they provide information not only about the ground state (which is usually already known experimentally) but also about the metastable phases (which have often not been directly accessed by experiment). Contrary to initial expectations the cubic pseudobinary phase of TiAl3 is found to be brittle even though it has the same close packed L l 2 crystal structure as the ductile single crystals of Cu3Au and Ni3Al. This is not due to the detrimental influence of the alloying additions. It appears to be an intrinsic feature of the transition metal tri-aluminides in this region of the AB 3 structure map since ScAl3, which takes the LI 2 structure (see Fig. 2-5), also cleaves
I expt. a L1 0 (CuAu) O D019 (Ni3Sn)
D L12(Cu3Au) y D022 (AI3Ti)
Figure 2-11. The predicted heat of formation of f.c.cand h.c.p.-based titanium-aluminium ordered compounds (after van Schilfgaarde et al., 1990).
83
transgranularly. A clue as to why cubic scandium and titanium tri-aluminides are brittle even though they have the same crystal structure as the ductile single crystals of Ni3Al may be provided by their elastic constants. Fu and Yoo (1990) and Fu (1989) have recently calculated these within LDF theory. Their predicted values for the cubic intermetallics are given in Table 2-4 together with the experimental values for nickel, aluminium and silicon for comparison. The ratio of the appropriate shear modulus for slip on the close-packed planes to the bulk modulus, namely /i/K, is given in the last column, as this has proved an effective criterion for deciding whether a sharp crack cleaves or blunts. Cottrell (1989) has shown that ductile f.c.c. and b.c.c. metals generally have fi/K < 0.4, whereas brittle cubic metals have fx/K > 0.5. We see from Table 2-4 that the intermetallic Ni3Al satisfies the ductile criterion fi/K < 0.4, whereas the tri-aluminides TiAl3 and ScAl3 do not (Cottrell, 1991). The large value of ja/K for the early transition metal tri-aluminides reflects the importance of the angular character of the bonding, which can be seen in their values of the Cauchy pressure C 12 — C 44 given in Table 2-4. Whereas the elemental metals nickel and aluminium and the intermetallic Ni3Al have positive values of the Cauchy pressure, the tri-aluminides have negative values that are comparable to that of silicon. This reflects the nature of the bonding at the atomistic level. If the bonding is describable by simple pairwise potentials such as the Lennard-Jones potential then the Cauchy pressure will be zero. If the bonding is more metallic, in that spherical atoms are embedded in the electron gas of the surrounding neighbours, then the Cauchy pressure will be positive (Johnson, 1988). A negative Cauchy pressure usually indicates angular character in the bonding.
84
2 Electron Theory of Crystal Structure
Table 2-4. Predicted values of the elastic constants C^C = ( C n - C12)/2; for n/K see the text] for three Ll 2 intermetallics compared with the experimental values (in brackets) of Ni, Al, Si and Ni3Al (after Fu and Yoo, 1989; Fu 1990). Anisotropy
Cauchy pressure c c (10"N/m2)
H/K
C 12 (lO^N/m 2 )
^44
(lO^N/m 2 )
C44/C
(10 n N/m 2 ) Ni Al Si
[2.61] [1.14] [1.66]
[1.51] [0.62] [0.64]
[1.32] [0.32] [0.80]
[2.41] [1.23] [1.57]
[0.19] [0.30] [-0.16]
[0.36] [0.35] [0.59]
Ni3Al
2.35 [2.30] 1.77 1.89
1.45 [1.49] 0.77 0.43
1.32 [1.32] 0.85 0.66
2.93 [3.25] 1.70 0.90
0.13 [0.17] -0.08 -0.23
0.33 [0.30] 0.53 0.77
TiAl3 ScAl3
Direct evidence for the angular character of the bonding in the titanium-aluminides is given in Fig. 2-12, which shows the LDF bond charge density for the LI 0 TiAl phase (Fu, 1989). We see that strong pd bonding
takes place in the [001] direction between the individual aluminium and titanium (001) layers. This is very different to the nickel aluminides such as Ni3Al, where the valence charge density is found to be much
Figure 2-12. The predicted bond charge density for the Ll 0 phase of TiAl. Note that on bringing the constituent atoms together to form the solid electronic charge has flown from the Al and Ti sites (corresponding to the negative contours, i.e., dashed lines) into the bonding region between the sites (corresponding to the positive contours, i.e., solid lines). The charge density contours are in units of 10 ~3 electrons/(au)3 [from Fu (1990); reproduced with permission].
85
2.3 Ab Initio Prediction of Crystal Structure
Zincblende
-0.5 1.0
I 0.5 •SI >> O)
g 0 LU
_J
A' -
I
L.
II-VI
(a) ^S
Zincblende
-0.5 1.0
1
1
I-VII
|
^
^
""p-Sn 0.5 -
\ 0
/
^ y ^
Zincblende
NaCI -0.5
I
100
is central to understanding the behaviour of sp-valent octet compounds A ^ B 8 ^ (Phillips, 1973). Figure 2-13 a gives the LDF binding energy curves for the archetypal III-V compound GaAs with the three different structures types (Chelikowsky and Burdett, 1986). The calculations correctly predict that the tetrahedrally co-ordinated zincblende structure is the most stable. The lower two panels, however, show that as the ionicity is increased to model the behaviour of II-VI and I-VII compounds the octahedrally co-ordinated NaCI structure is eventually realised. In addition, we see from Fig. 2-13 that under pressure the III-V compound transforms to (3-Sn whereas the II-VI compound transforms to NaCI. This behaviour of the AB octet compounds can be displayed explicitly in the schematic phase diagram of Fig. 2-14, which is found to be satisfied experimentally except for 2p-valent constituents such as BN (Chelikowsky, 1987). The increasing ionicity in going from IV-IV to III-V to II-VI semiconducting compounds is evident in Fig. 2-15, where the LDF valence electron density with respect to the zincblende structure is shown (Christensen et al., 1987). This change in ionicity is particularly noticeable for the
(c) i
1
I
1
140 120 Atomic volume (au)
300
Figure 2-13. The binding energy curves of archetypal (a) III-V, (b) II-VI and (c) I-VII octet compounds with respect to the three structure types zincblende, NaCI and (3-Sn (after Chelikowsky and Burdett, 1986).
more spherically symmetric about the nickel sites. The competition between the fourfold co-ordinated zincblende and the sixfold co-ordinated rocksalt or (3-Sn structures
IV-IV
III-V
II-V
I-VII
Charge configuration Figure 2-14. Schematic phase diagram for the spvalent octets (after Chelikowsky and Burdett, 1986).
86
2 Electron Theory of Crystal Structure C BN
BeO (b)
0.5-| 2.5-
Be-
0.5-
Mg-
GaAs
ZnSe
0.5
Figure 2-15. The predicted valence electron charge density in the (110) plane for IV-IV, III-V and II-VI octet compounds from different periods. The charge density contours are in units of 10" 2 electrons/(au)3 [from Christensen et al. (1987); reproduced with permission].
2.3 Ab Initio Prediction of Crystal Structure InSb
(k)
cdTe
87 (i)
05 -
Cd-
Figure2-15. Continued
2p-bonded compounds across the top row of Fig. 2-15, where the angularly dependent covalent sp 3 hybrids of carbon are almost totally replaced by spherically symmetric ionic charge clouds in BeO. Comparing the III-V compounds BN and GaAs we see that the former is much more ionic than the latter, so that it is not surprising that BN does not fit into the schematized phase diagram of Fig. 2-14 as mentioned previously. The structural properties of the new high temperature ceramic superconducting perovskites have also been investigated recently within LDF theory. Cohen et al.
0.05
0.05
(a)
0.04
(1989) have found that the stoichiometric ternary compound La 2 CuO 4 is predicted to have the correct equilibrium atomic volume, internal structural parameters and bulk modulus in the tetragonal phase. In addition, their frozen-phonon calculations, which are shown in Fig. 2-16, display the known anharmonicity of the oxygen Eu mode about the zone centre and the instability of the tilt mode about the X point which drives the tetragonal to orthorhombic distortion. The vibrational frequencies of the stable modes are in excellent agreement with the Raman and neutron scattering data. However, we must stress that al-
Breathing
0.04
LaA 1
|
; 0.03
0.03
o) 0.02
0.02
iS 0.01
0-01
o
Quadrupolar
fe Axial
0
-0.01 0.5
-0.5 Displacement (A)
-0.01 -0.5
0,5 Displacement (A)
Figure 2-16. The change in predicted energy versus the amplitude of the frozen phonon mode of La 2 CuO 4 for (a) Brillouin zone-centre displacements and (b) Brillouin zone-corner displacements [from Cohen et al. (1989); reproduced with permission].
88
2 Electron Theory of Crystal Structure
though LDF theory has predicted good structural behaviour, the local approximation fails to describe the anti-ferromagnetic state of La 2 CuO 4 so that a better treatment of the correlations between the electrons is required (see Sec. XI of Pickett, 1989).
2.4 Nearly Free Electron Systems 2.4.1 Simple Metal Elements The trends in crystal structure amongst the elemental sp-bonded simple metals may be understood within the nearly free electron (NFE) approximation (see, e.g., Cottrell, 1988). Figure 2-17 shows the densities of states n (E) of group I A, II A, IIB and III B metals from the first four periods which were calculated within LDF theory by Moruzzi et al. (1978). We see that Na, Mg and Al across the second period and Al, Ga and In down group III B are good NFE metals because their densities of states are given by only very small pertur-
bations from the free electron density of states (2-5)
[using atomic units, where h2/(2m) = 1], where V is the volume of the crystal. Their structural properties are therefore well described by second-order perturbation theory (see, e.g., Harrison, 1966; Heine and Weaire, 1970; Hafner, 1989). The heavier alkali metals K and Rb and the alkaline earths Ca and Sr are seen in Fig. 2-17 to have their occupied energy levels affected by the presence of the unfilled 3d or 4d band which lies just above the Fermi energy. The group IIB elements Zn and Cd, on the other hand, have their valence states strongly distorted by the presence of the filled d band. Nevertheless, Moriarty (1982, 1983, 1988) has shown that they may still be treated within second-order perturbation theory provided the unfilled or filled d band is explicitly taken into account.
Na
Figure 2-17. The density of states of sp-bonded simple metals (after Moruzzi etal., 1978). Energy (2 eV interval scale)
2.4 Nearly Free Electron Systems
This NFE-like behaviour is due to the scattering of the free electron gas by the ion cores being very much weaker than expected (see, e.g., Cottrell, 1988). This weak scattering is a direct consequence of the orthogonality constraints imposed on the valence electrons by the core states. The valence electrons are thereby excluded from the core so that they do not see the strongly attractive Coulomb potential in this region. The resulting scattering may be modelled by a pseudopotential such as the very simple Ashcroft (1966) empty-core potential drawn in Fig. 2-18. It is defined by
\2Z/r
for for
r
where Z is the valence, Rc is the Ashcroft ion core radius, inside of which the potential vanishes, and the prefactor 2 is e2/4 n s0 in atomic units. This potential has Fourier components ^°sn (q) =
^L- cos q Rc
/
\ \ \ \ \
(2-6)
89
/ / /
1 1
Figure 2-18. The Ashcroft empty-core pseudopotential.
order perturbation theory as 2
„
(2-7) (2-8)
where Q is the volume per atom. In the absence of the core (Rc = 0) the Fourier components are negative as expected for an attractive ionic potential — 2 Z/r. However, in the presence of the core the Fourier components oscillate in sign and may, therefore, take positive values. The oscillatory behaviour of the Fourier components of the screened aluminium potential vps (q) is illustrated in Fig. 2-19 for the more sophisticated Heine-Aberenkov pseudopotential (Heine and Aberenkov, 1964). We see that the f.c.c. zone boundary Fourier components i?ps(lll) and i;ps(200) are indeed small (and positive), thereby providing the justification for the NFE approximation. The sum of the occupied one-electron energies Ek (normalised by the number of atoms Jr) may be written within second-
+
|S((7)|2|<S"(G)|2
[k2-(k + G)2]
-0.2
-0.4
-0.6 L Figure 2-19. The Heine-Aberenkov pseudopotential for aluminium. q0 gives the position of the first node. The two large dots mark the zone boundary Fourier components i; ps (lll) and i;ps(200).
90
2 Electron Theory of Crystal Structure
where kF is the Fermi wave vector, k2 is the free electron kinetic energy (in atomic units since h2/2m = 1), the prefactor 2 accounts for spin degeneracy and S (G) is the structure factor corresponding to the reciprocal lattice vector G Therefore, the structuredependent part of this energy (coming from the second-order contribution) takes the form ^(i2)=- £
\S(G)\2\v%n(G)\2x(G)
(2-9)
G+0
where x(q) is the Lindhard density response function namely 1-1
3Z/1 2£ F \2
1-x 2 1 +x In Ax 1 -x
(2-10)
with EF the Fermi energy and x = q/2 kF. This response function displays the wellknown weak logarithmic singularity in its slope at q = 2 kF as seen in Fig. 2-20. The total energy must also include the double-counting contribution [cf. Eq. (2-4)]. This modifies Eq. (2-9) to give the so-called band structure energy (Heine and Weaire, 1970)
uhs=- Z
2
^
2
(2-11)
where s(q) is the dielectric response function. We see, therefore, that the magnitude of the attractive band structure energy will be very sensitive to where the first few reciprocal lattice vectors for a given lattice fall with respect to q0 in Fig. 2-19, where q0 locates the first node in the Fourier transform of the ionic pseudopotential. It follows from Eq. (2-7) that for the Ashcroft empty core pseudopotential q0 Rc = n/2, i.e., (2-12)
Figure 2-20. A comparison of the Lindhard response function Eq. (2-10) (solid curve) with the approximation used to obtain the three-term analytic pair potential Eq. (2-16) (from Pettifor and Ward, 1984).
Heine and Weaire (1970) have argued convincingly that gallium, indium and mercury take distorted structures in order to guarantee that the magnitude of their nearest neighbour reciprocal lattice vectors does not lie close to q0, so that a sizeable, non-vanishing band structure energy contribution is gained through Eq. (2-11). Most structural trends amongst the simple metals can most easily be understood, however, within a real space rather than reciprocal space formalism (Hafner and Heine, 1983, 1986; Hafner, 1989; McMahan and Moriarty, 1983; Moriarty, 1988; Pettifor and Ward, 1984). The structrual energy is determined by summing over a central pair potential ${R\ where for a local pseudopotential
2.4 Nearly Free Electron Systems
R
sinqR
(2-13)
^ion(^) is the normalized pseudopotential matrix element [(Q q2/(4 n e2 Z)] iA°sn (q). The pair potential thus represents the interaction between a given ion and another ion and its screening cloud of electrons a distance R away. Equation (2-13) contracts down to the single contribution
The weak logarithmic singularity of the Lindhard function at q = 2 kF gives rise to the very long range Friedel oscillations so that the pair potential behaves asymptotically (Friedel, 1952) as
cos (2 kF R)
(2-15)
Figure 2-21 shows the pair potentials for Na, Mg, Al and Si that result from using non-local pseudopotentials (Moriarty and McMahan, 1982; McMahan and Moriarty, 1983). The oscillatory nature of the pair potential with respect to the first few nearest neighbour shells of atoms is clearly seen. The origin of these nearest neighbour oscillations, however, is not due to the weak logarithmic singularity at 2 kF since this determines the very long range asymptotic behaviour of Eq. (2-15). Instead they are determined by the overall shape of the response function in Fig. 2-20. By writing the Lindhard function as a rational polynomial the pair potential may be expressed analytically at metallic densities as the sum of damped oscillatory terms (Pettifor, 1982), namely (2-16) 2Z 2 -
91
The wave vector kn and the screening length x~l depend only on the density of the free electron gas, whereas the amplitude $tn and the phase shift (xn depend also on the nature of the ion core. Figure 2-22 shows the pair-potentials for Na, Mg and Al which result from keeping the first three terms in Eq. (2-16) and using the simple Ashcroft empty core pseudopotential (Pettifor and Ward, 1984). We see that the total pair potentials reflect the characteristic behaviour of the more accurate potentials calculated by McMahan and Moriarty in Fig. 2-21. Figure 2-22 shows that all three metals are characterised by a repulsive hard-core contribution # ! (R) (short-dashed curve), an attractive nearest-neighbour contribution
R
(2-17)
where fc3 - 0.96 kF and x3 = 0.29 fcF. The Fermi wave vector kF may be written ex-
92
2 Electron Theory of Crystal Structure
cc E
I •^
3
1.5
2.5
3.5
2.5
1.5
3.5
6
16 (c) Al (d)Si
5 5
c ?
LLin 2.8
3.6
2.0
3.6
Relative separation r/S
Figure 2-21. Interatomic pair potentials for (a) Na, (b) Mg, (c) Al and (d) Si as a function of the interatomic separation in units of the Wigner-Seitz radius S. The positions of the f.c.c. first and b.c.c. first and second nearest neighbours are marked. Since f.c.c. and ideal h.c.p. structures have identical first and second nearest neighbours, their relative structural stability is determined by the more distant neighbours marked in the figures [from McMahan and Moriarty (1983); reproduced with permission].
plicitly in terms of either the Wigner-Seitz radius S or the radius rs of the average
spherical volume occupied by one electron, namely
3T^Z\1/3
/9K\1/3Z1/3
(2-18)
2.4 Nearly Free Electron Systems
93
Thus, both k3 R and x3 R are proportional to Z 1 / 3 R/S and hence for a given crystal structure take fixed values proportional to Z 1/3 . For a given valence Z the structural stability between f.c.c, b.c.c. and h.c.p. is, therefore, controlled by the phase shift a 3 , which for the Ashcroft empty core pseudopotential is given by ,~ A ^ a3 = <53 - 2 tan" 1 [tan(2 k3 Rc) tanh (x3 Rc)]
Figure 2-22. The analytic pair potential (solid curve) for Na, Mg and Al, the short-range, medium-range and long-range contributions being given by the short-dashed, long-dashed and dotted curves respectively. The arrows mark the position of the twelve nearest neighbours in the close-packed f.c.c. and ideal h.c.p. lattices. The values of Rc and rs (the radius of the average spherical volume occupied by one electron) in Au are written (Rc,rs) for each metal (Pettifor and Ward, 1984).
Since <53 is approximately constant at metallic densities, it follows from Eqs. (2-18) and (2-19) that a 3 is approximately a function of RJrs alone. Figure 2-24 a shows the predicted structure map (Z, a3) resulting from the longrange potential <J>3CR) (Wyatt et al., 1991). This map allows the structural trends given by Fig. 2-23 to be interpreted. Under pressure jRc/rs increases, thereby causing the phase shift a 3 to decrease through Eq. (2-19). Hence, as indicated by the arrows in Fig. 2-24, Na, Mg and Al all move down into the neighbouring b.c.c. domain under pressure as first predicted by Moriarty and McMahan (1982). They predicted metallic silicon, on the other hand, to transform from h.c.p. to f.c.c. under pressure, as has recently been verified experimentally by Duclos et al. (1987). The asymptotic Friedel oscillations, Eq. (2-15), arise from the weak logarithmic sin-
Mg /
fee
hep
-to
/
/bee
\ -20
A/
10 0
• . 1 1
0-5
1-0 0
Relative atomic volume
Figure 2-23. The energy of the b.c.c. and h.c.p. lattices with respect to the f.c.c. lattice for Na, Mg and Al as a function of their atomic volume relative to the observed equilibrium volumes predicted by the analytic pair potentials (Pettifor and Ward, 1984).
94
2 Electron Theory of Crystal Structure
steep slope in the immediate vicinity of 2 fcF is somewhat flattened by the approximate response function. The Friedel oscillatory contributions from very distant neighbours will, therefore, only sum up constructively to give a non-negligible contribution for those cases where the reciprocal lattice vector G is close to 2fcF, thereby sampling explicitly the weak logarithmic singularity. Following Mott and Jones (1936), 2kF = G for electron-per-atom ratios e/a = 1.36 and 2.09 for f.c.c. lattices, 1.48 for b.c.c. lattices, and 1.14, 1.36 and 1.65 for h.c.p. lattices. Because of the steeper slope of the Lindhard function about 2 kF in Fig. 2-20, we expect from Eq. (2-11) that the stability of the appropriate lattice will be enhanced (or reduced) over that predicted by the analytic threeterm potential for an electron-per-atom ratio slightly greater (or slightly less) than these critical e/a values at which 2 k F = G. Figure 2-24 b shows the predicted structure map (Z,a F ) resulting from summing over the potential cos (2 kF R + oeF) R~3 Figure 2-24. (a) The predicted structure map (Z, a3) resulting from the exponentially damped potential # 3 (R). The three dots indicate the values of the phase shift oc3 for Na, Mg and Al, corresponding to Z = 1, 2 and 3, respectively, the arrows indicating the direction the phase shift changes under pressure. The small triangle below the h.c.p. field at the lower left corner is b.c.c. (after Wyatt et al., 1991). (b) The predicted structure map (Z,aF) resulting from an oscillatory potential decaying with distance as the inverse cube, see Eq. (2-20) (after Krause and Morris, 1974).
gularity in the slope of the Lindhard function at 2 kF. These very long range oscillations are neglected if only the first three exponentially damped terms in the analytic potential, Eq. (2-16), are retained, since, as can be seen from Fig. 2-20, the
(2-20)
in which the Friedel oscillatory contribution has been generalised by the inclusion of a phase shift aF (Krause and Morris, 1974). We see that the topology of the domains of structural stability is similar to that predicted by the exponentially damped potential # 3 (i?) in the upper panel. We notice, however, that the boundaries on the left-hand side of the structure map have become much more vertical around aF = 0. This increasing dominance of the electron-per-atom ratio or Z in controlling structural stability reflects the crucial contribution from the many distant neighbours in summing together constructively for 2 kF ~ G.
95
2.4 Nearly Free Electron Systems 20
11
1
1 1 -
1
cc E 10
5
.<£
0
(a)
cc(6)
15
'
Ca
-bee (8) fee (12)
As expected from Fig, 2-17, the structural stability of the divalent alkaline earths Ca and Sr and the group IIB elements Zn and Cd can only be understood by including the unfilled or filled d band explicitly within second-order perturbation theory. Figure 2-25 shows the resultant pair potentials for Ca and Zn (Moriarty, 1983). We see that the unfilled d band causes the calcium potential to deepen, thereby stabilising the f.c.c. structure, whereas the filled d band in zinc suppresses the nearest neighbour minimum in &(R\ thereby causing the ideal h.c.p. lattice to relax to the much larger axial ratio of 1.86. This suppression of the local minimum becomes even more marked as one proceeds down group IIB so that Hg is unstable with respect to all three lattices, b.c.c, f.c.c. and ideal h.c.p., as illustrated in Fig. 2-26 for Q/Qo = 0.824 (Moriarty, 1988). The body-centred-tetragonal phase (3-Hg is predicted to be most stable, in agreement with experiment. The relative stability of the a, p, y, and 5 (simple cubic, b.c.c, f.c.c. and h.c.p.) phases of mercury as a function of volume is shown in Fig. 4 of Moriarty (1988). This change in shape of the pair potential as one proceeds down a group has been modelled by Hafner and Heine (1983) using the Ashcroft empty core pseudopotential. Assuming that RJrs decreases down a group, the phase shift a3 will increase through Eq. (2-19), thereby driving the outer maximum of <£3 (cf. Fig. 2-22) in through the nearest neighbour closepacked distance. This results in the instability of the close-packed lattice which we have discussed above for group IIB and which is also observed when moving down group IIIB from Al to Ga. In the latter case, on moving further down the group to Tl, the stability of the close-packed lattice is regained as the maximum in
-
•
\
\
S "5-10
Vy
-
1.5
i 2.0
i 2.5
i 3.0
3.5
Relative separation (r/R w s )
2.0
2.5
3.0
3.5
Relative separation (r/R w s )
Figure 2-25. The interatomic pair potentials for (a) Ca and (b) Zn. The full and dashed curves correspond to including and excluding the explicit d band contribution, respectively. The positions of the f.c.c. first and b.c.c. first and second nearest neighbours are marked in the upper panel. The positions of ideal h.c.p. first four nearest neighbours are given in the lower panel, the horizontal arrows indicating the directions these neighbours move as the c/a axial ratio is increased above ideal. Rws is the Wigner-Seitz radius [from Moriarty (1983); copyright © Wiley 1983; reprinted by permission of John Wiley & Sons, Inc.].
96
2 Electron Theory of Crystal Structure I 15
10
i
I
I
I
I
f
bcc
-
A
-
5 --
_ -5 -
I;/
(a) bet
6-Hg
2.4.2 Simple Metal Compounds
v
-10 - -
-
P-Hg
I
i 0.8
0.6
I 1.0
I
I
1.2
1.4
I 1.6
c/a ratio
50
55
60
65
70
1
I
75
Angle (deg)
1
6
1
/
7-
\ 4 -
^
(c) hep
_
2 0
/
—
fee
\
-4 -
V
\
-6 - 8 -r 1.2
I 1.4
1
1
1
1.6
1.8 c/a ratio
2.0
The alloys of simple metals, which lie in the same or neighbouring groups of the periodic table, are also well described by the NFE approximation. Figure 2-27 shows the density of states of the ordered CsCl binary compounds NaMg, NaAl and MgAl which were calculated within the LDF scheme by Gelatt et al. (1980). We see that NaMg and MgAl with AZ = 1 both have densities of states which are only weakly perturbed from the free-electron values. NaAl with AZ = 2 is more strongly perturbed, as expected. Second-order perturbation theory may therefore be used for discussing the cohesion and structural stability of these binary systems [see, e.g., Pettifor (1987), Hafner (1989) and references therein]. The total binding energy per atom for the alloy ACA BCB may be written within the real space representation (Brovman et al., 1970; Finnis, 1974; Hafner, 1989) U = (Ueg-±QKeg)+Upa+Ustruci
/
-2 -
through the nearest neighbour closepacked distance and is absorbed in the short-range repulsive part of the potential. Thus, the structural trends amongst the elemental simple metals can be rationalised within second-order perturbation theory.
/ / 1
-7-
2.2
2.4
Figure 2-26. Relative binding energies of Hg at Qj Qo = 0.824 for three major structural families: (a) body centred tetragonal, (b) simple rhombohedral and (c) hexagonal close packed [from Moriarty (1988); reproduced with permission].
(2-21)
where C/eg and Keg are the energy and bulk modulus of a free electron gas with the average density Q = (cA Z A + cB ZB)/Q, where Q is the average volume per atom of the alloy. Upa represents the binding energy of the screened pseudoatoms, namely f/pa = \ [cA
2.4 Nearly Free Electron Systems
97
equilibrium atomic volume Q of the compound is assumed equal to \ (QA + QB) by Zen's law, then the electron-gas term (in eV/atom) may be expanded to second order as AHeg = Z / V / 3 ) ( A e 1 / 3 ) 2
(2-25)
where Z is the average number of valence electrons per atom, AQ113 = QBJ3 — gA13 and
-10
-5 E-E F (eV)
0
Figure 2-27. The LDF density of states of the ordered CsCl binary compounds NaMg, NaAl and MgAl (after Gelatt etal, 1980).
pair potentials, namely (2-23)
Struct = iocjp
where a, /? takes the appropriate label A or B depending on the occupancy of site ij. Jf gives the total number of atoms. The heat of formation may therefore be written as the sum of three terms AH = AHeg + Atfpa + Atfs
(2-24)
where the electron-gas, pseudoatom and structure-dependent contributions correspond to the different terms resulting from Eq. (2-21). Pettifor and Gelatt (1983) have pointed out that the trend within the heats of formation of simple metal compounds is dominated by the electron-gas contribution. For equiatomic AB compounds, if the
/ (x) = - 43.39 + — + °^ (2-26) x x The three terms in Eq. (2-26) are the kinetic, exchange and correlation contributions respectively. Mixing together two electron gases of density QA and QB respectively to form a new electron gas of average density Q lowers the kinetic energy but raises the exchange and correlation energies. Figure 2-28 shows the predicted behaviour of the electron-gas contribution AHJ[Z(AQ1/3)2] as a function of Q113. We see that as expected from Eq. (2-26) it is positive at lower densities, where the repulsive exchange-correlation contribution dominates, but negative at higher densities, where the attractive kinetic energy contribution dominates. The first principles LDF values of the heat of formation for the Na, Mg, Al, Si, P series with respect to the CsCl (b.c.c.) lattice show the same trend as Eq. (2-26) but are displaced somewhat upwards from the free electron gas result due primarily to the neglect of the repulsive pseudoatom contribution A/Jpa in Eq. (2-21) [see Fig. 13 of Pettifor (1987)]. Hafner (1989) has shown that the trend in the experimental heats of formation of equiatomic liquid simple metal alloys is accurately reflected by the theoretical electron gas contribution, Eq. (2-25) [see Fig. 18 of Hafner (1989)]. The structural contribution AHstruct in Eq. (2-24) can play a crucial role in deter-
98
2 Electron Theory of Crystal Structure
25
K\ Na ONaK NaRbO
20
Mg
Al
Si P
qNaMg \ \ \ \ \ \ 6V NaAl
15
I10
i \ \NaSi0 OMgA| CD
\NaP%MgSi \ \ oAlSi IN
--10
X
\
OAIP
i \aPo\
-15 -20_ 0.1
P1/3 (au1) 0.2 0.3
X . 0.4^
-25 -
Figure 2-28. The normalised heat of formation AH/[Z(AQ113)2] as a function of the cube root of the electron density @1/3. The solid curve is the electron gas contribution Eq. (2-25). The open circles are the LDF results for ordered CsCl compounds (from Pettifor and Gelatt, 1983).
mining the sign of the heat of formation. For example, we see from Fig. 2-28 that the equiatomic alkali metal binaries NaK and NaRb are predicted both by LDF theory and the simple electron-gas expression to have a positive heat of formation. This is in agreement with the absence of any compounds at the bottom left hand corner of the AB structure map, Fig. 2-2. However, we see that the hexagonal MgZn 2 Laves structure type is taken by Na 2 Cs, Na 2 K and K 2 Cs at the bottom left hand corner of the AB2 structure map, Fig. 2-4. Hafner (1977) has shown that the negative heat of formation for these alkali metal Laves phases can be understood within secondorder perturbation theory. Figure 2-29 shows his predicted heats of formation of A2B alkali metal compounds for both the disordered b.c.c. phase and the ordered MgZn 2 Laves phase as a function of the ratio of their Ashcroft empty core pseudopotential radii. We see that whereas the disordered b.c.c. phases have positive heats of formation, the ordered Laves phases Na 2 K, K 2 Cs and Rb2Cs are predicted to have negative heats of formation. This is
0.03 0.3-
-0.0
Figure 2-29. The predicted heats of formation of A2B alkali metal alloys for (a) the disordered b.c.c. phase and (b) the ordered MgZn2 Laves phase as a function of the ratio of their core radii R^/Rc (after Hafner, 1977).
2.5 Tight Binding Systems
due to the positioning of the nearest-neighbour atoms with respect to the minima in the appropriate pair potentials ^ap(Rij) [see Fig. 35 of Hafner (1989)]. We note that the minimum in the predicted AH curve of Fig. 2-29 b is close to the radius ratio K A /# B = (3/2)1/2 = 1.225 that is expected from simple space-filling arguments. In concluding this section on nearly free electron systems we should note that the predictions of second-order perturbation theory become increasingly unreliable as the valence difference between the constituent elements of the binary compound increases. We have already seen in Fig. 2-27 that the density of states of NaAl with AZ = 2 is starting to deviate from a simple weak perturbation of the free electron gas result. This deviation becomes even more marked for LiAl because the lithium pseudopotential is no longer weak due to the absence of p electrons in the ion core (compare, for example, the densities of states for Li, Na and Al in Fig. 2-17). Thus, secondorder perturbation theory fails (Inglesfield, 1971) to predict the B32 (NaTl) Zintl structure for LiAl which we see displayed in the AB structure map of Fig. 2-2 and predicted by LDF theory in Fig. 2-10. The B32 (NaTl) structure is coherent with the B2 (CsCl) structure in that they both order with respect to an underlying b.c.c. lattice. However, whereas the eight nearest neighbour sites in CsCl are all occupied by unlike atoms, in the NaTl structure four are occupied by like and four by unlike atoms. The stability of this structure is due to the strong sp 3 hybrids which form between aluminium and its four tetrahedrally co-ordinated aluminium neighbours [see Sec. 7.3 of Hafner (1989) and references therein]. Just like tetrahedrally co-ordinated silicon this requires a theory that goes beyond the second-order NFE approximation. The simplest approach is the tight binding ap-
99
proximation, which is discussed in the next section.
2.5 Tight Binding Systems 2.5.1 Structural Trends Within the sp-Bonded Elements
The sp-bonded elements exhibit a broad range of crystal structure (Donohue, 1974), as is seen in Table 2-5. The alkaline earths Ca, Sr, Ba and Ra have been excluded from the table because their structural stability is strongly influenced by the proximity of the transition metal d band (see Sec. 2.4.1) and Be and Mg have been grouped with Zn, Cd and Hg as suggested by the Mendeleev number (see Fig. 2-1). The first three groups (I, II and III) usually take the close-packed metallic structure types f.c.c, h.c.p. or b.c.c. The elements of group IV show the trend from threefold co-ordinated graphite through fourfold co-ordinated diamond to twelvefold co-ordinated f.c.c. on moving down the column from carbon through silicon, germanium and tin to lead. Apart from the dimeric form of nitrogen, the group V pnictides take structures based on the stacking of three-fold co-ordinated buckled layers of atoms, whereas the group VI chalcogenides take structures based on two-fold co-ordinated helical chains. The group VII halogens crystallise as dimers which are held together on the lattice by very weak van der Waals interactions. These structural trends within the spbonded elements can be understood within the tight binding (TB) approximation (Allan and Lannoo, 1983; Cressoni and Pettifor, 1991; Lee, 1991 a, 1991 b). An introduction to the TB Hiickel description of energy bands can be found in Pettifor (1983, 1987, 1990), Hoffmann (1988), Ma-
100
2 Electron Theory of Crystal Structure
Table 2-5. The ground-state structures of the sp-bonded elements (Donohue, 1974; Hafner, 1989). (dimer), (octomer), (chain) or (layer) indicates that the structure comprises dimers, octomers, helical chains or buckled layers weakly bound together in the solid. 2
3
4
5
6
7
Li h.c.p.
Be h.c.p.
B complex
C graphite
N (dimer)
O (dimer)
F (dimer)
Na h.c.p.
Mg h.c.p.
Al f.c.c.
Si dia.
P (layer)
S (octomer)
Cl (dimer)
K b.c.c.
Zn h.c.p.
Ga complex
Ge dia.
As (layer)
Se (chain)
Br (dimer)
Rb b.c.c.
Cd h.c.p.
In f.c.t.
Sn dia.
Sb (layer)
Te (chain)
I (dimer)
Cs b.c.c.
Hg b.c.t.
Tl f.c.c.
Pb f.c.c.
Bi (layer)
Bi s.c.
At (dimer)
1 H (dimer)
dia., diamond; f.c.c, face centred cubic; f.c.t., face centred tetragonal; b.c.t., body centred tetragonal; s.c, simple cubic.
jewski and Vogl (1989) and Ducastelle (1991). Within the TB approximation the total binding energy per atom may be written as the sum of three terms, namely
u = ur
u
bond
u prom
(2-27)
The repulsive energy C/rep is assumed to be pairwise in character (Ducastelle, 1970) so that (2-28) where Jf is the number of atoms in the system. The attractive covalent bond energy t/bond is evaluated within the two-centre orthogonal TB approximation (Slater and Koster, 1954). For the case in which all sites are equivalent and the crystal field shifts are orbital independent (2-29) where na (E) are the local a — s, p electronic density of states, Ea are the effective s,p
atomic energy levels, and EF is the Fermi energy. The promotion energy Uprom is driven by the change in the relative s:p occupancy and is given by Uprom = (Ep -£ s )AiV p
(2-30)
where AiVp is the change in the number of p electrons on bringing the reference atoms together to form the bond. We should note that in practice Eq. (2-27) gives the binding energy with respect to some reference free atom state, which usually differs from the true atomic ground state due to, for example, the neglect of spin-polarisation or the shift in the atomic energy levels arising from the renormalisation of the wave functions in the bonding situation. The form of Eq. (2-27) may be justified from first principles by working within the Harris-Foulkes approximation (Harris, 1985; Foulkes and Haydock, 1989) to density functional theory [see, e.g., Sutton et al. (1988) and references therein]. It is important to realise that the usual crystal field
2.5 Tight Binding Systems
shifts in the atomic band energy levels have been removed from the band energy Uhand=
j£na(£)d£
(2-31)
[cf. the first term on the right-hand side of Eq. (2-4), where the electron energies E are defined with respect to the crystalline potential v through the Schrodinger Eq. (2-2)] and grouped together with the first term of Eq. (2-27), [7rep (Allan and Lannoo, 1983; Pettifor, 1990). The remaining bond and promotion energies depend only on the electronic energies relative to £ s and Ep. The energy difference £ sp = Es — Ep is itself assumed to be environment independent, the crystal field effects giving a uniform shift in the on-site energy levels. It follows from Eqs. (2-29) to (2-31) that the change in the bond and promotion energies on going from one structure type to another is equivalent to the change in the band energy under the constraint that the atomic energy levels Es and £ p keep fixed values (Pettifor, 1976, 1978; Pettifor and Varma, 1979; Andersen, 1980), i.e. A[/bond + AUprom = (AC/band)A£s p = 0
(2-32)
The band energy depends on the density of states through Eq. (2-31). This is evaluated within the orthogonal TB approximation by assuming that the a and n bond integrals display the same distance dependence h(R) (Cressoni and Pettifor, 1991) so that they may be written in the following simple form:
ppa(R)(
ppn(R) I
cept for pp7i, which has been increased by 30% in order to stabilise the close-packed structures with respect to the dimer for the alkali metals. The explicit form of the distance dependence h(R) is not required within a first nearest neighbour bond model if the structural energy difference theorem (Pettifor, 1987) is used to predict the relative stability of the different structure types. This theorem states that the total energy difference A U between two different structure types in equilibrium under a binding-energy law of the type given by Eq. (2-27) is, to first order in A17/17, At/ = (A[/bond + AL/pprom r o m ))A C / r e p =
(2-33)
spa(R)J The ratios ssa:ppa:pp7i:spa implicit in Eq. (2-33) have been chosen equal to Harrison's (1980) solid-state table values ex-
(2-34)
That is, the difference in the total energy between two lattices is simply the difference in the bond plus promotion energies provided the bond lengths have been adjusted so that the two lattices have identical repulsive energies. It follows from Eqs. (2-32) and (2-34) that AU = (AUh ) A
=
(2-35)
The repulsive energy Urep corresponding to a lattice with co-ordination z is given from Eq. (2-28) by Urep = ±z$(Rz)
(2-36)
where Rz is the nearest neighbour distance for that lattice. For the study of the structural trends within Table 2-6 the repulsive potential may be assumed to fall off with distance as the square of the bond integrals, i.e., \ = Ah2(R)
x h(R)
101
(2-37)
where A is a constant. This approximation appears to be a reasonable approximation for sp-bonded systems. For example, Goodwin et al. (1989) have recently fitted the LDF binding energy curves of diamond, P-Sn, simple cubic and f.c.c. silicon (Yin
102
2 Electron Theory of Crystal Structure
Table 2-6. Contributions to the normalised fourth moment /}4(z)//i4(l) where (L = \IJ\I\ and z is the local co-ordination. nt and w^ give the number and normalised weight of the ith type of contribution to the fourth moment respectively. The numbers in brackets in the last three columns given the total s, p or sp normalised fourth moments for each structure type. Two fourfold co-ordinated lattices are given, namely the three-dimensional diamond lattice and the two-dimensional square lattice; only the latter contains four-membered rings (see Fig. 2-3). type j
1
=
ni
1
T « = =
3
<
4 / 4 = *
0
' II
p
sp
s
P
sp
1.000
1.000
1.000
1.000 (1.000)
1.000 (1.000)
1.000 (1.000)
1.000
1.000
1.000
0.500
0.500
0.500
1.000
0.224
0.167
1.000 (1.500)
0.224 0.724
0.167 (0.667)
3
1.000
1.000
1.000
0.333
0.333
0.333
12
1.000
0.418
0.079
1.333 (1.667)
0.557 (0.890)
0.105 (0.438)
4
1.000
1.000
1.000
0.250
0.250
0.250
24
1.000
0.311
0.084
1.500 (1.750)
0.467 (0.717)
0.126 (0.376)
4
1.000
1.000
1.000
0.250
0.250
0.250
1.000
1.000
0.211
0.500
0.500
0.106
16
1.000
0.224
0.167
1.000
0.224
0.167
8
1.000
0.224
0.134
0.500 (2.250)
0.112 (1.086)
0.067 (0.590)
U
1 AAA l.UUU
\ AAA l.UUU
1 AAA l.UUU
A i &1
A 1 £.1
A 1 CH
12
1.000
1.000
0.211
1.000
0.224
0.167
0.333 1.333
0.333 0.299
0.070 0.223
1.000
0.224
0.134
0.667 (2.500)
0.149 (0.948-
0.089 (0.549)
8
Ii
S
48
and Cohen, 1982) with a short-ranged twocentre orthogonal TB model in which
U.lo /
U.lo /
U.lo/
where \i2 is the second moment of the local density of states, namely (2-39) This follows since the second moment of a TB density of states may be expressed in terms of all hopping paths of length two
2.5 Tight Binding Systems
which start from and end on a given atom (Cyrot-Lackmann, 1968). Formally this may be stated using the Dirac notation as follows:
a
fi
j
where |ia> represents the atomic orbital of angular character a on site i and (ioi\H\jf}y is the hopping integral between the a-orbital on site i and the /?-orbital on site j . Thus, the second moment of the density of states on site i is obtained by summing over all paths which involve two hops, namely those from atom i to a nearest neighbour atom j (i.e.,
h2(R)]ccAUn
(2-40)
where the last proportionality follows from Eqs. (2-36) and (2-37). Thus we have the very important result in Eq. (2-38) that the structural energy difference A U may be obtained by simply comparing the band energies once the bond lengths have been adjusted so that the respective densities of states have the same moment or variance (Pettifor and Podloucky, 1984; Burdett and Lee, 1985; Lee, 1991 a, b). This constraint fixes the relative values of the bond integrals between the different structure types. Taking the simple cubic lattice with z = 6 as a reference with an equilibrium nearest neighbour value of h(R) denoted by h0, we have from Eq. (2-40) that the appropriate nearest neighbour bond integrals for any
103
other lattice with co-ordination z will be given by Eq. (2-33) with (2-41) Figure 2-30 shows the resultant pure s, pure p and hybridised sp densities of states with Esp = Es — Ep = 0 for structures with nearest neighbour co-ordinations ranging from z = 2 (zig-zag chain with 90° bond angles) through z = 3 (single graphitic sheet or honeycomb lattice), z = 4 (both diamond-cubic and diamond-hexagonal lattices), z = 6 (simple cubic) and z = 8 (simple hexagonal) to z = 12 (both f.c.c. and ideal h.c.p. lattices). In addition the densities of states are shown for the b.c.c. lattice with z = 14 corresponding to eight first and six second nearest neighbours, the bond integrals being evaluated under the assumption that h(R2)/h(R1) = 0.33. (The subscripts 1 and 2 stand for first and second nearest neighbours.) The densities of states were calculated using the recursion method of Haydock et al. (1972) to nine exact levels. The energy is in units of h0 so that the simple cubic s band, for example, runs from — 6 to + 6 as expected. Similar densities of states have been calculated for non-vanishing values of the atomic energy level separation Esp [see Fig. 4 of Cressoni and Pettifor (1991)]. Figure 2-31 shows the structural energy as a function of band filling which results from occupying the densities of states corresponding to £ sp = 0 in Fig. 2-30. The structural energy is defined as the difference between the band energy for a given structure [see Eq. (2-31)] and that corresponding to a reference rectangular density of states with the same second moment [see Eq. (2-38)]. This procedure allows the very small energy differences between the different structure types to be displayed more
104
2 Electron Theory of Crystal Structure
Figure 2-30. (a), (b) and (c) give the s, p and sp densities of states respectively for the different lattices in energy units of h0. The broken curves give the integrated density of states provided the numbers on the vertical scale are multiplied by five for (a), by six for (b) and by eight for (c) (Cressoni and Pettifor, 1991).
clearly. We see that as a function of band filling pure s-bonded systems are predicted to show the following structure trend: close-packed -* zig-zag chain -> dimer -» zig-zag chain -> simple cubic, whereas pure p-bonded systems are predicted to
show the structure trend close-packed -> diamond-hexagonal -> diamond-cubic -» zig-zag chain -» simple cubic -• dimer -> simple cubic. On the other hand, spbonded systems with £ s p = 0 are predicted to show the trend close-packed -» simple
2.5 Tight Binding Systems
105
Figure 2-30. Continued
hexagonal -> honeycomb -• diamond-cubic -> honeycomb -> simple cubic -> zigzag chain -» dimer. We note that there is no difference in energy between the cubic and hexagonal close-packed or diamond lattices within the nearest neighbour s-band model. It is the angular character of the valence orbitals which distinguishes between cubic and hexagonal structure types in the p and sp panels of Fig. 2-31. These predicted structural trends are compared directly with experiment in Fig. 2-32, where structure maps corresponding to the atomic energy level mismatch E sp = Esp/(12/i0) versus band filling N are plotted. We see that the predicted structural trend corresponding to E sp = 0 follows that of the sp panel in Fig. 2-31 whereas that corresponding to £ sp = — oo follows the pure s band from N equals 0 to 2 and the pure p band from N equals 2 to 8. A comparison of the two panels in Fig. 2-32 shows that the theory predicts most of the broad features displayed by the
experimental structure map. In particular, beginning on the right-hand side of the figure where the assumptions of the nearest neighbour TB model are most appropriate, we see that the theory predicts correctly that the most stable structures of the halogens are built from dimers, whereas those of the chalcogens are based on zig-zag linear chains [see, e.g., Fig. 3-12 of Harrison (1980)]. The exceptions are oxygen with its dimeric behaviour and polonium with its simple cubic structure (sulphur exhibits structures based on helical chains at high temperatures). Nevertheless we see that both dimeric and simple cubic domains adjoin the theoretical zig-zag domain centred on N — 6. Relaxation of the constraint that the repulsive pair potential varies as the square of the bond integrals in Eq. (2-37) would change the theoretical predictions, a softer repulsive core favouring lower co-ordinations, a harder repulsive cure favouring higher co-ordinations (Abell, 1985).
106
2 Electron Theory of Crystal Structure '
1
s
0.4-
/
o -
76
L
I
"••••'
V V
-0.4-
C< D C "I "l — 1 i
-
i
P
1' V
c
jtrucTure
CD
-
\/
/
/
\
\
• • ' • • • •
• .
S—X ' - • — ' ' < r < \
/
v A
••/'
0-
-1-
•ooc 0
1
1
2
3
4
5
6
-2-
••OO • *•<• • - ( D H 0 1
2
3 4 5 Band filling
6
—
dimer
•
sc
~1
zig - zag
^
hex
—<
h.comb
+ 0
dia(c) dia(h)
•
•
fee
O
hep
7
8
The theoretical predictions for the pnictides with N = 5 are poor since a buckled threefold co-ordinated layer should also have been considered theoretically in Figs. 2-30 and 2-31 in addition to the planar honeycomb lattice (see Allan and Lannoo, 1983; Lee, 1991a). The dimeric form of nitrogen, like oxygen, is probably due to the softer repulsive core which results from the absence of core p electrons. The group IV B elements are predicted to change from the open four-fold co-ordinated diamond-cubic structure to the close-packed twelve-fold co-ordinated f.c.c. structure as £ sp becomes increasingly negative. This is consistent with the observation that silicon, germanium and tin are diamond cubic whereas lead is f.c.c. The latter has a larger negative value of E sp due to a 3 eV relativistic contribution (Herman and Skillman, 1963) which weakens the strength of the sp 3 hybrids due to a sizeable positive promotion energy [see Eq. (2-30)]. Again the 2p element is exceptional, carbon taking the lower co-ordinated graphite structure as might be expected for a less steep repulsive potential. Even though the sp-bonded simple metals with N = 1, 2 and 3 are not expected to be described accurately by a nearest neighbour TB model (compare, for example, the densities of states in Fig. 2-30 with those in Fig. 2-17), we see that the simple theory predicts correctly the occurrence of close-packed structures in this region. Moreover, the trend from h.c.p. to f.c.c. as N increases across a period from
Figure 2-31. The structural energy in units of hc as a function of band filling for the pure s, pure p and sp (Esp = 0) cases (Cressoni and Pettifor, 1991). Note that in the sp case the cubic and hexagonal diamond curves are almost indistinguishable, leading to the apparently single solid cusp-like curve.
2.5 Tight Binding Systems
107
1 Be -0.2-
-0.4-
r oeH ii
£ -0.8
B
Nail ZnQ Al m ° l N m Si I O E3 H m Cd* Ga* p 9^Hg m D Gej K Tl n * S •*• As-*-
?b !
Rb* Cs * dimer
Sb-^
zig - zag
S
h.cx>mb
E3 fee
dia(hex)
•
hep
BB dia(cub)
•
fee/hep
*
other
s.cubic
hex
-0.32
F
m
SeH a m Te g^ Po m Br | 3
--0.73
-1.38
At m --3.08 Expt
-1
--0.32
--0.73
--1.38
-3.08
3
4 5 Band filling N
the alkali metals and the trend from f.c.c. to h.c.p. as £ s p becomes more negative down group III B are well reproduced. The dimer becomes most stable for N = 1 as £sp becomes large and negative since then the s bonding dominates. 2.5.2 Interpretation of Structural Trends in Terms of Moments
A powerful link between structural stability, which is displayed in curves such as Fig. 2-31, and the underlying topology of
Figure 2-32. A comparison of the experimental (a) and theoretical (b) structure maps for the sp-bonded elements (Cressoni and Pettifor, 1991).
the lattice is provided by the moments of the local density of states (Cyrot-Lackmann, 1968; Ducastelle and Cyrot-Lackmann, 1971; Burdett and Lee, 1985; Ducastelle, 1991; and references therein). Following the discussion after Eq. (2-39) we have already inferred that the n-th moment jun of the density of states associated with a given atom is related directly to the sum of all paths of length n that start from and end on that atom. Thus, for example, the third moment of the density of states on site i will be given by summing over all
108
2 Electron Theory of Crystal Structure
paths of length three, which will involve nearest neighbour hops from atom i to atom j9 from atom j to atom /c, and from atom k back to atom i. This means that there will be a non-vanishing third moment only if the lattice contains three-atom nearest neighbour triangles i j k about site i or, equivalently, three-membered rings. If the lattice contains no three-membered rings, as, for example, in the diamond or in the simple cubic lattice, then the third moment from intersite hopping will be zero. This is an important result. It immediately implies that all lattices which contain only even-membered rings will have all their odd moments vanishing, so that the resultant densities of states must be symmetric (assuming no on-site hopping contributions, which is true for Es = Ep = 0). This explains the symmetric behaviour of the densities of states in Fig. 2-30 for the zig-zag, honeycomb, diamond and simple cubic lattices, which contain only evenmembered rings. We see, on the other hand, that the simple hexagonal, f.c.c, h.c.p. and b.c.c. lattices have densities of states skewed to lower energies due to the presence of the odd three-membered rings and non-vanishing third moments ^ 3 . This accounts for the asymmetric behaviour of the close-packed and hexagonal structural energy curves in Fig. 2-31. Thus, we have the important result that the close-packed structures are more stable than the more open structures for less than half-full bands due to the presence of three-membered rings, which are absent in the latter structure types (Burdett and Lee, 1985). The relative stability of different structure types with even-membered rings can sometimes be deduced by looking at the relative strengths of their normalised fourth moments ju4 = \ij\i\. In Table 2-6 we show the different types of paths which contribute to the fourth moment of the dif-
ferent lattices, including the two-dimensional square lattice for later comparison with the three-dimensional diamond lattice. The fourth moment with respect to a given site can be written as jLt4 = Y,ni I**, i
(2-42)
i
where i runs over all the different types of four-path contributions about that site, and nt and \xAi give the number of such contributions and the corresponding fourth moment, respectively. For example, on the threefold co-ordinated honeycomb lattice there are two types of contribution, the one (i = 1) corresponding to four hops back and forth between neighbouring pairs of atoms A and B (symbolized by A = B), the other (i = 2) corresponding to four hops between three atomic neighbours A, B and C (A = B = C). The two-atom diagram enters z times, where z is the local coordination, so that nx = 3 for the honeycomb lattice with z = 3. The three-atom diagram enters 2z(z —1) times since there are z(z — 1) contributions of the type A —• B -• C -> B -> A and z(z — 1) contributions of the type A - > B ^ A - > C - > A s o that n2 = 12, as given in Table 2-6. For the case of angularly independent s orbitals all types of paths have the same weight ssa 4 so that from Eq. (2-42) ^ = ^2>.w. Z
(2-43)
i
where the normalized weight of each path is unity and is thus independent of distance or structure type. Summing the non-ring paths we have z two-atom contributions and 2z(z —1) three-atom contributions, making a total of 2 z2 — z contributions, so that we have .^s _ /j _ r*4 — \^
-J I \ , s J. / ^ / i r^4ring
V
O-AA\ • •/
where the final term on the right-hand side represents the ring contributions to the
109
2.5 Tight Binding Systems 1
1
i
i
i i i normalized fourth moment. In Fig. 2-33 fi4 versus z is plotted for the different lattices. N We see that for the s orbital case (2 — 1/z) 3.0 is the sole contribution for the dimer, zig. square ^^ zag, honeycomb and diamond lattices, the - V«) ; large deviation from the dotted curve for 2.0 the square, simple cubic, hexagonal and diamond close-packed structures being due to the p : presence of four-membered ring terms (see 1.0 Table 2-6). \ m sp For the case of angularly dependent p t i i 0.0 orbitals or sp hybrids the weight fiAi is a 1 2 3 4 5 7 8 9 10 11 12 function not only of the hopping integrals but also of the type of path i. If 0t is the Figure2-33. The normalized fourth moment (tA(z)/ /}4(1) versus local coordination z for the pure s, pure relevant bond angle in the three-atom conp and sp (£ sp = 0) cases. The dashed curve gives the s tribution in Table 2-6, then it follows from orbital result in the absence of ring terms. The inset the angular dependence of the energy inteshows the oscillatory behaviour of the energy differgrals in Slater and Koster (1954) that ence between two structures with different fourth mo•
jn4i = [ssa4 + spa 4 + pp7i4 + 2ssa 2 spa 2 + 2pp7i2 (spa 2 + ppa 2 )] 2
2
1
'
«o
1
<
1 1
ments such that 5//4 > 0 (Cressoni and Pettifor, 1991).
(2-45) 2
+ 2spa (ssa — 2ssappa + ppa ) cos 9t + [spa 4 + ppa 4 + pp7i4 + 2spa 2 ppa 2 -2pp7i 2 (spa 2 + ppa 2 )] cos2 9t Figure 2-34 shows the resultant angular dependence for the p and sp cases with E = 0, which look very similar to the numerical results of Carlsson (1989) for the angular dependence of an effective threebody potential derived from p or sp 3 hybrids. We see that in the pure p case fi4i has a minimum corresponding to 90°, whereas in the sp case the minimum is close to 117° due to hybridization. We will see that this is important for stabilising the diamond structure with 9 = 109° or the graphite structure with 6 = 120° when the bands are half-full. Finally, Eq. (2-43) can be generalised to the sp case by writing (2-46)
where w^ = JU4J/JU41. It follows that the normalized weight wx for the two-atom contributions is unity, as shown in Table 2-6. For the s case /}4(z = l ) = 1 . 0 so that Eqs. (2-43) and (2-46) are equivalent. Table 2-6 gives the corresponding normalised weight
1.25
:
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1.00 -" 0.75
s
pure
sp
0.50 '\
/
0.25 "-
/
V
0.00
/ '.
pure p
:
/
/
/
/ /
/
:
/ 1
^ — :
:
0°
30°
60° 90° 120° 150° 180° Bond angle
Figure 2-34. The angular dependence of the threeatom fourth moment contribution, Eq. (2-45), for the pure s, pure p and sp (£ sp = 0) cases (Cressoni and Pettifor, 1991).
110
2 Electron Theory of Crystal Structure
wt for the different orbitals and types of path and their contributions to the sum in Eq. (2-46). Figure 2-33 shows that the angular dependence of the p orbitals severely decreases the value of /i4(z)/ji4(l) compared with the s orbital case. In particular, we see that the fourfold co-ordinated diamond lattice has the lowest value of /24 for both the p and sp cases, whereas the dimer had the lowest value for the pure s case. The value of /z4 reflects the shape of the density of states in that a large value suggests a central peak or unimodal behaviour whereas a small value suggests two wellseparated peaks or bimodal behaviour [see, e.g., Fig. 1 of Gaspard and Lambin (1985)]. This is illustrated by Fig. 2-30a for the s bands. As the local co-ordination z increases, //4 increases rapidly and the densities of states clearly change from having a bimodal to a unimodal distribution. On the other hand, for the sp case, as z increases /x4 decreases until a minimum is reached for the diamond lattice (see Fig. 2-33). We see in Fig. 2-33 c that this corresponds to the opening up of a hybridisation gap, thereby stabilising the diamond lattice for a half-full band. Thus, if two lattices have different normalised fourth moments, the lattice with the smaller moment will be the more stable for approximately half-full bands, whereas the lattice with the larger moment will be the more stable for nearly full or empty bands, as is illustrated schematically by the inset in Fig. 2-33. The structural trends shown in Fig. 2-31 are consistent with the behaviour of the third and fourth moments. For less than half-full bands the close-packed structures are stabilised by the presence of threemembered rings. For more than half-full bands the trends dimer -> zig-zag -> simple cubic for s orbitals, diamond -• zig-zag chain -> simple cubic -» dimer for p orbitals, and diamond -> honeycomb -> sim-
ple cubic -> zig-zag for the sp orbitals are in the direction of increasing fourth moment. The higher moments are necessary, however, for predicting the precise shape of the curves in Fig. 2-31. In particular, the relative stability of the cubic versus hexagonal close-packed or diamond lattices requires a knowledge of fi5 or fi6, which can be seen from the fact that the cubic and hexagonal curves cross each other at least three or four times for the p and sp cases in Fig. 2-31 (Ducastelle and Cyrot-Lackman, 1971). Finally, the qualitative features of the theoretical structure map in Fig. 2-32 can be deduced by combining the predictions for Esp = 0 with those for the pure s and p bands corresponding to E sp = — oo. 2.5.3 Structural Trends Within the sd-Bonded Elements
The sd-bonded transition elements in Table 2-1 show the well-defined structural trend h.c.p. -• b.c.c. -• h.c.p. -> f.c.c. across the 3d, 4d and 5d series. This trend is driven by the change in number of valence d electrons, as is clearly evidenced by Fig. 2-35, where the TB d-bond energies of the f.c.c, b.c.c. and h.c.p. lattices are compared as a function of band filling Nd (Pettifor, 1972). Apart from the incorrect predicted stability of the b.c.c. phase at the noble metal end of the series the observed structure trend amongst the non-magnetic transition elements is correctly reproduced. The strong stability of the b.c.c. phase in V and Cr, Nb and Mo, Ta and W, when the d band is just less than half-full, is due to the marked bonding-antibonding separation in the b.c.c. density of states compared to f.c.c. and h.c.p. [see, e.g., Fig. 26 of Pettifor (1983)]. Turchi and Ducastelle (1985) have shown that the oscillatory behaviour seen in Fig. 2-35 is essentially due to differences in the fifth moment between the structure types.
111
2.5 Tight Binding Systems
The 3 d transition element iron is interesting in that all three phases, b.c.c, f.c.c. and h.c.p., are found within its pressuretemperature phase diagram (see the inset of Fig. 2-36). The occurrence of the a (b.c.c), y (f.c.c), 5 (b.c.c.) and 8 (h.c.p.) phases can be understood qualitatively within a finite temperature theory of band magnetism (Hasegawa and Pettifor, 1983). The isovalent non-magnetic 4d and 5d elements Ru and Os take the h.c.p. structure as expected from Fig. 2-35 for a d band filling around 7 electrons per atom. The 3d element Fe, however, has a much narrower bandwidth than its 4 d or 5 d counterparts so that its density of states at the Fermi level n (£F) is sufficiently large to satisfy the Stoner criterion for ferromagnetism, namely / n (EF) > 1, where / is the exchange integral. The resultant magnetic energy stabilises the b.c.c. phase over h.c.p. (Madsen et al., 1976). Under pressure the d band widens and the density of states at the Fermi energy falls, thereby decreasing the magnetic energy of the b.c.c oc phase until, at a sufficiently high pressure, the non-magnetic h.c.p. 8 phase is stabilised as seen in Fig. 2-36. The occurrence of the f.c.c y phase at higher temperatures is driven by large magnetic fluctuations, as first suggested by the 2y-state model of Weiss (1963). The return to the b.c.c 8 phase at still higher temperatures is controlled by entropy contributions, which arise from either the softer phonons in the b.c.c. lattice (Zener, 1947) or the loss of magnetic short-range order above the Curie temperature (Hasegawa and Pettifor, 1983). Magnetism also helps to stabilise the ground state structure of ot-Mn, since it leads to two types of atoms, those with large atomic volume, corresponding large magnetic moments, and those with small atomic volume, corresponding to small magnetic moments [see Fig. 5 of Hoistad and Lee (1991)].
\
NH
Figure 2-35. The d bond energy of the b.c.c. (solid line) and the h.c.p. (dotted line) lattices with respect to the f.c.c. lattice as a function of band filling Nd (Pettifor, 1972).
/
2000
1400
exp.
6
/
1500
1200
y
^1000
-~
500
1000
a
\
50
100 150 P (kbar)
e
Y *
°c)
800 600
\
v— \
400 -
200
200
a
^—-—'
8
i_
50
100
150
200
250
P (kbar)
Figure 2-36. The predicted phase diagram of iron compared with the experimental one (inset) (Hasegawa and Pettifor, 1983).
112
2 Electron Theory of Crystal Structure
discussion around Eq. (2-6)]. The bottom of the d band, B d , on the other hand, rises only slowly in this region as the increasing d-band width under compression compensates for the upward shift in the centre of the d band, C d . Thus, as shown in Fig. 2.37 a, the number of d electrons increases under compression. Moreover, since La has a larger ion core than Lu, the number of d electrons also increases on moving from right to left across the rare earth series from Lu to La. This increase in the number of d electrons drives the structural trend h.c.p. -• Sm type -• d.h.c.p. -» f.c.c. as is demonstrated by Fig. 2-37 b which compares the TB dband energy of the four different closepacked lattices. We see, therefore, that the
The trivalent rare earth crystal structure sequence h.c.p. -• Sm type -> d.h.c.p. -• f.c.c, which is observed for decreasing atomic number and increasing pressure, is also determined by the d-band occupancy JVd (Duthie and Pettifor, 1977; McMahan and Young, 1984). Figure 2-37 a shows the self-consistent LDF energy bands of La in the f.c.c. structure as a function of the atomic volume which were calculated neglecting the hybridisation between the NFE sp band and the TB d band. We see that the bottom of the sp band, Bs, moves up rapidly in energy in the vicinity of the equilibrium atomic volume as the free electrons are compressed into the ion core region from where they are repelled by orthogonality constraints [cf. the earlier
(b)
rJ
\
<
i
2
/
//A
3
/
structure
cla = 1.63
c la = 1.58
E CO
f.c.c.
Stable
V \
f.c.c.
-0.01
d.h.c.p.
0
d.h.c.p.
-
\
1
0.01
h.c.p.
-
h.c.p.
2
VIV0 Figure 2-37. (a) The energy bands of La about the equilibrium atomic volume Vo and the corresponding d-band occupancy Nd of La and Lu. T d , Cd and Bd are the top, centre and bottom of the d band, Bs is the bottom of the sp band, and £ F is the Fermi energy (Duthie and Pettifor, 1977). (b) The relative d-band energies in units of the d-band width Wd of h.c.p. (full curve), d.h.c.p. (dashed curve) and Sm-type (dot-dashed curve) with respect to f.c.c. as a function of d-band occupancy Nd. The resulting stable structures for the ideal and a non-ideal axial ratio are also shown (Duthie and Pettifor, 1977).
113
2.5 Tight Binding Systems
running of the string in Fig. 2-1 backwards through the periodic table from Lu to La is consistent with its direction through the transition elements, since both are in the direction of increasing Nd. The string had to run from La to Lu through the rare earths in order to obtain the perfect separation of the rare earth silicides into the respective FeB and CrB domains in Fig. 2-2.
07 25
0-9
1-1
D
T7TO D
20
1-5
HIA ETA YA "HA "SEA
OTa"2HIbWc
IB
2.5.4 Structural Trends Within the Binary Compounds
The origin of some of the structural trends within the AB, AB2 and AB 3 structure maps has been investigated recently within the TB approximation. In particular, Majewski and Vogl (1986, 1987, 1989) have studied the relative stability of the NaCl, CsCl and cubic zincblende structure types amongst the sp-bonded AB octet semiconductors and insulators within a TB model that includes explicitly the ionic Madelung contribution. Pettifor and Podloucky (1984, 1985) have predicted the relative stability of the seven most common structure types amongst the pd-bonded AB intermetallics within a locally charge neutral TB model. Ohta and Pettifor (1989) have examined the different roles played by atomic size and electronic factors in stabilising the dd-bonded AB2 Laves phases against the two competing phases MoSi2 and CuAl2. Lee (1991a, 1991b) has predicted the relative stability of ten different structure types amongst the sp-bonded AB2 intermetallic compounds. Turchi et al. (1983) have examined the stability of the A15 (Cr3Si) structural domain within ddbonded systems and Bieber and Gautier (1981) have determined the relative stability of L l 2 (Cu3Au) and D0 2 2 (Al3Ti) structure types. The role played by the difference in size of the atomic constituents was analysed by
Figure 2-38. The upper panel shows the structure map (xp, Xd) f° r 1^9 pd-bonded AB compounds, where Xp and Xd a r e values for the A and B constituents of a certain chemical scale x which orders the elements in a similar way to the Mendeleev number Jl in Fig. 2-1. Each transition metal group comprises columns corresponding to 3d, 4d and 5d elements; each III B to VIIB group comprises rows corresponding to 3p, 4p, 5p and 6p elements. The 2p elements B, C, N, O and F are not included in the figure (but see Fig. 2-2). The lower panel shows the theoretical structure map (iVp, Nd) where Np and Nd are the number of p and d valence electrons associated with atoms A and B, respectively (Pettifor and Podloucky, 1984).
Pettifor and Podloucky (1984, 1986) in their study of the structural trends within the pd-bonded AB compounds, which are displayed in the upper panel of Fig. 2-38. They generalised the TB model of the elements presented in Sec. 2.5.1 to the case of binary compounds.They found that the fractional change in volume (AV)/V between one structure type and another with the same repulsive energy [cf. Eq. (2-25)] was simply a function of the relative size
114
2 Electron Theory of Crystal Structure
factor ^ which they had defined through the strength of the pp repulsive pair potential compared to the dd repulsive pair potential. Figure 2-39 shows the resultant fractional change in volume with respect to the CsCl lattice as reference. As expectd, the NaCl lattice has the smallest volume at either end of the 01 scale, because as the size of either the p-valent atom or the d-valent atom shrinks to zero the repulsion will be dominated by one or other of the closepacked f.c.c. sublattices. On the other hand, in the middle of the scale, where the nearest neighbour pd repulsion dominates, the volume of the NaCl lattice with six nearest neighbours is about 13% larger than the CsCl with eight nearest neighbours. The packing of hard spheres rather than the softer atoms would have led to the much larger volume difference of 30%. Using the structural energy difference theorem, Eq. (2-35), the structural stability fan"1/? _ U/K
-0-1 0
37^8
TJ/J
NaCl
0-2
0-5
R 1
5
Figure 2-39. The fractional change in volume (AV)/V with respect to the CsCl lattice versus the relative size factor & (see text). The upper and lower NiAs curves correspond to c/a = 1.39 and (8/3)1/2 respectively (Pettifor and Podloucky, 1984).
t2 ! "o
frfl • LSLl NQCI FGSI
-s o
T
'
'
'
—
NiAs — ! / ,
'%/
ID
/
MnP FeB CrB
0
N
12
.
16
Figure 2-40. The structural energy as a function of band filling N for the seven different crystal lattices with Epd = 0 (Pettifor and Podloucky, 1984).
of the pd-bonded AB compounds may be predicted by comparing the TB bond energy of the different lattices at the volumes determined by a relative size factor M that guarantees identical second moments ^ 2 , namely 0i — 0.8. Figure 2-40 shows the resultant structural energies as a function of the band filling N for the case where the atomic p level on the A site and the atomic d level on the B site are equal, i.e., Epd = Ep — Ed = 0. As N increases, we find the structural sequence CsCl -• FeSi -• CrB -> NaCl -• NiAs -• (MnP) -> NiAs -+ NaCl, where MnP, a distorted NiAs structure, has been put in parentheses because it does not quite have the lowest energy for AT-9. From the discussion in Sec. 2.5.2, we see that the low band filling stability of the CsCl, FeSi and CrB lattices is due to a sizeable third moment /i 3 , which arises from the presence of many three-membered rings in these close-packed structure types. On the other hand, the absence of nearest neighbour three-membered rings in the more open NaCl and NiAs structure types accounts for their stability for N > 5.5. The NaCl and NiAs structural energy curves are fairly similar, as might be
2.5 Tight Binding Systems
expected from the similarity in the NaCl 6/6 and NiAs6/6, structure types (cf. Fig. 2-3). The minimum in the above two curves at N ~ 6 corresponds to a minimum in their density of states at the Fermi energy when all the pd-bonded orbitals are occupied [see Fig. 3 of Pettifor and Podloucky (1986)]. The structural energy depends not only on the electron-per-atom ratio or N but also on Epd = Ep — Ed, which is a measure
of the electronegativity difference. Curves similar to Fig. 2-40 have therefore been calculated for values of the atomic energy level difference in the range from — 10 to + 5 eV (in steps of 2.5 eV). However, rather than plotting the most stable predicted structure on a structure map of Epd versus N, Pettifor and Podloucky (1984, 1986) used the rotated frame of Np versus Nd9 in order to make direct comparison with the experimental results in the upper panel of Fig. 2-38. Np and Nd are the number of p and d valence electrons associated with atoms A and B, respectively. The resulting theoretical structure map is shown in the lower panel of Fig. 2-38. We see that the TB model predicts the broad topological features of the experimental pd-bonded AB structure map. In particular, NaCl in the top left-hand corner adjoins NiAs running across to the right and boride stability running down to the bottom. MnP stability is found in the middle of the NiAs domain and towards the bottom right-hand corner, where it adjoins CsCl towards the bottom. The main failure of this simple pd TB model is its inability to predict the FeSi stability of the transition metal silicides, which is probably due to the total neglect of the valence s electrons within the bonding. Lee (1991 a, 1991 b) has recently investigated the structural stability of AB2 compounds, where A is an electropositive ele-
115
ment from the first four columns of the periodic table and B is a more electronegative sp-bonded group B element. In particular he examined the ten major structural families MgCu 2 , MoSi 2 , CeCd 2 , CeCu 2 , MgAgAs, Caln 2 , A1B2, ThSi 2 , ZrSi2 and Cu 2 Sb (see Fig. 2-4). The electropositive element A is considered to donate all its electrons to the B sites, so that only the bonding between the B atoms need be considered. The resultant covalent network within the ten different structure types is shown in Fig. 2-41. We should note that (e) MgAgAs corresponds to diamond cubic, whereas (f) Caln 2 corresponds to diamond hexagonal. The band energy resulting from the bonding between the B sites only was then computed within the TB Hiickel approximation under the constraint of equal second moments for the different lattices [cf. Eq. (2-40)]. The experimental and predicted ranges of structural stability as a function of the number of electrons per AB2 unit are shown in Fig. 2-42. We see that good overall agreement is obtained. Not unexpectedly, the diamond sublattices of MgAgAs and Caln 2 are most stable for a half-full band corresponding to 8 electrons per AB2 unit. Again, structural stability may be linked directly to the local topology through the moments. Lee (1991 a, 1991 b) has pointed out that on moving from (a) to (j) in Fig. 2-41 the following trend is observed: (a) MgCu2 contains many triangles of bonded atoms; (b) MoSi 2 contains both triangles and squares; (c) CeCd2 and (d) CeCu2 contain both squares and hexagons; (e) MgAgAs, (f) Caln 2 and (g) A1B2 contain only hexagons; (h) ThSi2 contains no small rings; and (i) ZrSi2 and (j) Cu 2 Sb contain only square sublattices. This is consistent with the simplest s-valent moment arguments that would predict a trend from structures with three-membered rings to
116
2 Electron Theory of Crystal Structure
Experiment MoSi 2 — CeCd 2 •CeCu 2 — Caln 2 , MgAgAs ThSi 2 — ZrSi 2
—SmSbg Cu2Sb Theory -MgCu 2 MoSi 2 — CeCd 2 — CeCu2 Caln 2l MgAgAs -ThSi2 — ZrSi 2 — SmSb2 Cu2Sb
12
1 6
N Figure 2-42. A comparison of the theoretical and experimental ranges of stability with respect to band filling N of the ten AB2 structure types shown in Fig. 2-41 [reprinted with permission from Lee (1991 b); copyright 1991 American Chemical Society].
six-membered rings to four-membered rings as a function of band filling [see Fig. 3 of Lee (1991 b) and Fig. 3 of Pettifor and Aoki (1991)]. However, just as with the sp-bonded elemental case, the angular character will also be important in determining the size of the non-ring contributions to the moments (cf. Figs. 2-33 and 2-34). Thus, the simple TB model allows an understanding of structural trends that are found within the experimental structure maps for binary compounds. Figure 2-41. The covalent networks found in the (a) MgCu2, (b) MoSi2, (c) CeCd2, (d) CeC 2 , (e) MgAgAs, (f) Caln 2 , (g) A1B2, (h) ThSi2, (i) ZrSi2 and (j) Cu2Sb structure types [reprinted with permission from Lee (1991 a); copyright 1991 American Chemical Society],
2.6 Structural Stability of Solid Solutions As illustrations of the role that electron theory can play in understanding the behaviour of solid solutions, we consider the
2.6 Structural Stability of Solid Solutions
two examples illustrated by the two phase diagrams in Fig. 2-43. Panel (a) shows the Cu-Pd phase diagram, where a continuous solid solution is found for temperatures above about 600 °C. On cooling down from the melt short-range order (SRO) develops, which in the vicinity of Cu 3 Pd acts as a precursor for the long-range ordered superstructures a' and a". Figure 2-43 b shows the famous Cu-Zn phase diagram, where the oc phase primary solid solution gives way to the (3, y, 5 and 8 phases as a function of alloying concentration or electron-per-atom ratio. We will see that electron theory can both predict the nature of the SRO within the Cu-Pd f.c.c.-based solid solution (Gyorffy and Stocks, 1983) and, in principle, account for the structural transitions between the different disordered Hume-Rothery phases (Massalski and Mizutani, 1979). The average electronic structure of a disordered solid solution is well described within the coherent potential approximation (CPA) [see, e.g., Chap. 6 of Ducastelle (1991) and references therein]. This meanfield approximation has been implemented for cubic f.c.c. and b.c.c. disordered alloys using the scattering theory first developed by Korringa, Kohn and Rostoker for ordered lattices, resulting in the so-called KKR-CPA method [see Gyorffy et al. (1991) and references therein]. Provided the constituent atoms of the disordered alloy are not too chemically dissimilar, the Fermi surface will remain well-defined and can be mapped out experimentally using positron annihilation (Berko, 1979). The predicted behaviour of the Fermi surface in disordered Cu-Pd alloys is given in Fig. 2-44 (Gyorffy and Stocks, 1983; Gyorffy et al, 1991). Panel (a) shows that in the FXWKWXr plane of the Brillouin zone the Bloch spectral function A (k, E{) at the Fermi energy is indeed nar-
Weight Percent Palladium 0 10 20 30 40 50 60 70 80 90
117
100
1500
0 10 20 30 40 50 60 70 80 90 100 Cu Atomic Percent Palladium Pd Weight Percent Zinc 10 20 30 40 50 60 70 80 90 100 1200
' 0 10 20 30 40 50 60 70 80 90 100 Cu Atomic Percent Zinc Zn Figure 2-43. The Cu-Pd (a) and Cu-Zn (b) phase diagrams (after Massalski, 1986).
row, thereby indicating that the Fermi surface remains sharp even in the disordered state. But most importantly Fig. 2-44 b shows the perpendicular to the FK direction there is a very flat sheet of Fermi surface whose spanning wave vector changes with alloy concentration c or average number of electrons per atom e/a. The nesting across these flat sheets of Fermi surface will lead to a tendency to form concentration waves that are incommensurate with the underlying f.c.c. lattice. Figure 2-44 c shows that the predicted incommensurability agrees very well with that found by diffuse electron scattering experiments from Cu-Pd solid solutions (Oshima and
118
2 Electron Theory of Crystal Structure (c)
(a)
0.4
0.2
-S.
X
W
0 02 O4 OX^ Concentration of palladium
K
Figure 2-44. (a) The Bloch spectral function A (k, EF) at the Fermi energy EF in the FXWKWXF plane of the Brillouin zone for Cu0 75 Pd 0 25 alloy (see also Fig. 5-1). (b) Evolution of the predicted Fermi surface with concentration, (c) Variation of the incommensurability ra = 2 [y/l — 2 /cF(011)] with concentration; the theoretical and experimental points fall on the same curve [from Gyorffy et al. (1991); reproduced with permission].
Watanabe, 1976). Very recently these KKR-CPA calculations of incommensurate SRO have been used as the basis for predicting the low-temperatue transitions to the commensurate phases a' and a" in Fig. 2-43 (Cedar et al., 1992). We should note that LDF calculations correctly predict that at the absolute zero of temperature the 50-50 alloy will order as the b.c.cbased CsCl structure type rather than f.c.c.-based CuAu structure type (Lu et al., 1991). The occurrence of the different HumeRothery electron phases, which are displayed in Fig. 2-43 b for the Cu-Zn system, was sought by Jones (1937) within the rigid band approximation (RBA). This assumes that the effect of alloying is to change the occupancy of the bands but not their shape. Provided the band width is scaled to take account of lattice expansion the RBA has been found to be an excellent approximation for Cu-Zn alloys, both experimentally from specific heat and positron annihilation experiments and theoretically from a comparison of the CPA and RBA predictions for the shape of the Fermi surface and density of states (see, e.g., Massalski, 1989). The structural trends predicted by the rigid band approximation for Cu-Zn are
shown in Fig. 2-45 a, where we find the expected trend from f.c.c. (a phase) to b.c.c. (p phase) to h.c.p. (e phase) as a function of the electron-per-atom ratio (Paxton et al., 1992). This trend as a function of the band filling TV is a direct consequence of rigidly occupying the copper densities of states n(E) in the middle panel and comparing the resultant band energies, i.e., En{E)dE]
(2-47)
where N = J n(E)dE
(2-48)
It follows from Eq. (2-47) that d
= A£ F (2-49)
since on differentiating Eq. (2-48) with respect to N we have immediately d£ F n(EF) = (2-50) dJV Further, it follows from Eqs. (2-49) and (2-50) that diV
r(Al/) = .
(2-51)
Thus, as first pointed out by Jones (1962), the shape of the band energy differ-
2.6 Structural Stability of Solid Solutions
Figure 2-45. Analysis of f.c.c, b.c.c. and h.c.p. relative structural stability within the rigid band approximation for Cu-Zn alloys, (a) The difference in band energy as a function of band filling N with respect to elemental rigid copper bands, (b) The density of states at the Fermi level EF for f.c.c, b.c.c. and h.c.p. lattices as a function of band filling N. (c) The difference in the Fermi energies AEF as a function of band filling N [from Paxton et al. (1992); reproduced with permission].
119
ence curves in Fig. 2-45 a can be understood in terms of the relative behaviour of the densities of states in the middle panel. In particular, from Eq. (2-49) the stationary points in the upper curve correspond to band occupancies for which A£ F vanishes in panel (c). Moreover, whether the stationary point is a maximum or a minimum depends on the relative values of the density of states at the Fermi level through Eq. (2-51). In particular, the b.c.c. — f.c.c. energy difference curve has a minimum around N = 11.6, where the b.c.c. density of states is lowest, whereas the h.c.p. — f.c.c. curve has a minimum around N = 11.9, where the h.c.p. density of states is lowest. The f.c.c. structure is most stable around N = 11, where A£ F ~ 0 and the f.c.c. density of states is lowest. The structural trends in these HumeRothery electron phases are driven by the van Hove singularities in the densities of states which arise from band gaps at specific Brillouin or Jones zone boundaries [see Fig. 8-17 of Massalski and Mizutani (1979) for impressive correlation between theory and experiment for the y phase ]. It is therefore not totally surprising that the NFE second-order perturbation theory results of Stroud and Ashcroft (1971) and Evans et al. (1979) found energy difference curves that are very similar to those in the top panel of Fig. 2-45 (that is, away from the copper-rich end where local pseudopotentials predict the incorrect structure). The strong curvature of the b.c.c. — f.c.c. and h.c.p. — f.c.c. curves as a function of band filling can be reproduced only by including explicitly the weak logarithmic singularity in the slope of the Lindhard response function at q = 2 kF (see Fig. 2-20). It is for this reason that these Hume-Rothery alloys are correctly termed electron phases since this singularity is driven solely by the electronper-atom ratio (through 2 kF) and does not
120
2 Electron Theory of Crystal Structure
depend on the particular chemical constituents (through the pseudopotential).
2.7 Outlook This chapter has reviewed the recent developments in understanding the origins of structural stability within elements, binary compounds and disordered alloys at the absolute zero of temperature. We can expect rapid progress during the next decade in the first principles prediction of phase diagrams where electron theory (discussed in this chapter) and statistical mechanics (discussed by Binder in Chap. 3 of Vol 5 of this Series) will be integrated together. Initial steps in this direction have been reviewed in an excellent book by Ducastelle (1991).
2.8 References Abell, G. C. (1985), Phys. Rev. B31, 6184. Allan, G., Lannoo, M. (1983), J. Phys. (Paris) 44, 1355. Andersen, O. K. (1980), in: Electrons at the Fermi Surface: Springford, M. (Ed.). Cambridge: Cambridge University Press, Sec. 5.3. Ashcroft, N. W. (1966), Phys. Lett. 23, 48. Berko, S. (1979), in: Electrons in Disordered Metals and at Metallic Surfaces: Phariseau, P., Gyorffy, B. L., Scheire, L. (Eds.). New York: Plenum, p. 239. Bieber, A., Gautier, F. (1981), Solid State Commun. 38, 1219. Binder, K. (1991), in: Materials Science and Technology, Vol.5: Haasen, P. (Ed.). Weinheim: VCH, Chap. 3. Brovman, E. G., Kagan, Y., Kholas, A. (1970), Sov. Phys.-JETP 30, 883. Burdett, X K. (1982), J. Solid State Chem. 45, 399. Burdett, J. K. (1988), Ace. Chem. Res. 21, 189. Burdett, J. K., Lee, S. (1985), J. Am. Chem. Soc. 107, 3063. Burgers, W. G. (1934), Physica 1, 561. Callaway, X, March, N. H. (1984), Solid State Phys. 38, 135. Carlsson, A. E. (1989), in: Atomistic Simulations of Materials: Beyond Pair Potentials: Vitek, V., Srolovitz, D. X (Eds.). New York: Plenum, p. 103.
Cedar, G., de Fontaine, D., Dreysse, H., Nicholson, D. M., Stocks, G. M., Gyorffy, B. L. (1992), to be published. Chelikowsky, X R. (1987), Phys. Rev. B35, 1174. Chelikowsky, X R., Burdett, X K. (1986), Phys. Rev. Lett. 56, 961. Chen, Y, Ho, K. M., Harmon, B. N. (1988), Phys. Rev. B37, 283. Christensen, N. E., Satpathy, S., Pawlowska, Z. (1987), Phys. Rev. B36, 1032. Cohen, R. E., Pickett, W. E., Krakauer, H. (1989), Phys. Rev. Lett. 62, 831. Cottrell, A. H. (1988), Introduction to the Modern Theory of Metals. London: Institute of Metals. Cottrell, A. H. (1989), Mater. Sci. Technol. 5, 1165. Cottrell, A. H. (1991), Mater. Sci. Technol. 7, 981. Cressoni, X C , Pettifor, D. G. (1991), /. Phys.: Condens. Matter 3, 495. Cyrot-Lackmann, F. (1968), J. Phys. Chem. Solids 29, 1235. Donohue, X (1974), The Structure of the Elements. New York: Wiley. Ducastelle, F. (1970), /. Phys. (Paris) 31, 1055. Ducastelle, F. (1991), Order and Phase Stability in Alloys. Amsterdam: North-Holland. Ducastelle, E, Cyrot-Lackmann, F. (1971), /. Phys. Chem. Solids 32, 285. Duclos, S. X, Vohra, Y K., Ruoff, A. L. (1987), Phys. Rev. Lett. 58, 775. Duthie, X C , Pettifor, D. G. (1977), Phys. Rev. Lett. 38, 564. Evans, R., Lloyd, P., Rahman, S. M. M. (1979), J. Phys. F9, 1939. Finnis, M. W. (1974), J. Phys. F4, 1645. Foulkes, W. M. C , Haydock, R. (1989), Phys. Rev. B 39, 12520. Friedel, X (1952), Phil. Mag. 43, 153. Fu, C. L. (1989), Philos. Mag. Lett. 58, 199. Fu, C. L. (1990), Phil. Mag. Lett. 62, 159. Fu, C. L., Yoo, M. H. (1989), Mater. Res. Soc. Symp. Proc. 133, 81. Fu, C.L., Yoo, M.H. (1990), Philos. Mag. Lett. 62, 159. Gaspard, X P., Lambin, P. (1985), in: The Recursion Method and Its Applications: Pettifor, D. G., Weaire, P. L. (Eds.), Springer Ser. Solid-State Sci., Vol. 58. Berlin: Springer, p. 75. Gelatt, C. D., Moruzzi, V. L., Williams, A. R. (1980), unpublished. Goodwin, L., Skinner, A. X, Pettifor, D. G. (1989), Europhys. Lett. 9, 701. Gyorffy, B. L., Stocks, G. M. (1983), Phys. Rev. Lett. 50, 374. Gyorffy, B. L., Stocks, G. M., Ginatempo, B., Johnson, D. D., Nicholson, D. M., Pinski, F. X, Staunton, X B., Winter, H. (1991), Philos. Trans. R. Soc. London 334, 515. Hafner, X (1977), Phys. Rev. B 15, 617. Hafner, X (1989), in: The Structures of Binary Compounds: de Boer, F. R., Pettifor, D. G. (Eds.). Amsterdam: North-Holland, p. 147.
2.8 References
Hafner, I, Heine, V. (1983), J. Phys. F13, 2479. Hafner, I, Heine, V. (1986), J. Phys. F16, 1429. Harris, J. (1985), Phys. Rev. B31, 1770. Harrison, W. A. (1966), Pseudopotentials in the Theory of Metals. New York: Benjamin. Harrison, W. A. (1980), Electronic Structure and the Properties of Solids. San Francisco, Freeman. Hasegawa, H., Pettifor, D. G. (1983), Phys. Rev. Lett. 50, 130. Haydock, R., Heine, V., Kelly, M. J. (1972), /. Phys. C 5, 2845. Heine, V., Abarenkov, I. (1964), Philos. Mag. 9, 451. Heine, V., Weaire, D. (1970), Solid State Phys. 24, 1. Herman, R, Skillman, S. (1963), Atomic Structure Calculations. Englewood Cliffs, NJ: Prentice Hall. Ho, K. M., Harmon, B. N. (1990), Mater. Sci. Eng. A 127, 155. Hoffmann, R. (1988), Solids and Surfaces: A Chemist's View of Bonding in Extended Structures. New York: VCH. Hohenberg, P., Kohn, W. (1964), Phys. Rev. 136 B, 864. Hoistad, L. ML, Lee, S. (1991), /. Am. Chem. Soc. 113, 8216. Hume-Rothery, W. (1962), Atomic Theory for Students of Metallurgy. London: Institute of Metals. Inglesfield, J. E. (1971), J. Phys. C4, 1003. Janak, J. R, Williams, A. R. (1976), Phys. Rev. B 14, 4199. Jensen, W. B. (1989), in: The Structures of Binary Compounds: de Boer, F. R., Pettifor, D. G. (Eds.). Amsterdam: North-Holland, p. 105. Johnson, R. A. (1988), Phys. Rev. B37, 3924. Jones, H. (1937), Proc. Phys. Soc. A 49, 250. Jones, H. (1962), J. Phys. Radium 23, 67. Jones, R. O., Gunnarsson, O. (1989), Rev. Mod. Phys. 61, 689. Kohn, W, Sham, L. J. (1965), Phys. Rev. 140, A1133. Krause, C. W, Morris, J. W. (1974), Acta Metall. 22, 767. Lee, S. (1991a), J. Am. Chem. Soc. 113, 101. Lee, S. (1991b), Ace. Chem. Res. 24, 249. Liu, C. T. (1984), in: High-Temperature Alloys: Theory and Design: Stiegler, J. O. (Ed.). New York: AIME, p. 289. Liu, C. T, Horton, J. A., Pettifor, D. G. (1989), Mater. Res. Soc. Symp. Proc. 133, 37. Lu, Z. W., Wei, S.-H., Zunger, A. (1991), Phys. Rev. B44, 3387. Lundquist, S., March, N. H. (Eds.) (1983), Theory of the Inhomogeneous Electron Gas. New York: Plenum. Madsen, I, Andersen, O. K., Poulsen, U. K., Jepsen, O. (1976), in: Magnetism and Magnetic Materials, Becker, J. I, Lander, G. H. (Eds.). New York: American Institute of Physics, p. 327. Majewski, J. A., Vogl, P. (1986), Phys. Rev. Lett. 57, 1366. Majewski, J. A., Vogl, P. (1987), Phys. Rev. B35, 9679.
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Majewski, J. A., Vogl, P. (1989), in: The Structures of Binary Compounds: de Boer, F. R., Pettifor, D. G. (Eds.). Amsterdam: North-Holland, p. 287. Massalski, T. B., (Ed. in Chief) (1986), Binary Alloy Phase Diagrams, Vol. 1: Murray, J. L., Bennett, L. H., Baker, H. (Eds.). Metals Park, OH: American Society for Metals. Massalski, T. B. (1989), Metall. Trans. 20 B, 445. Massalski, T. B., Mizutani, U. (1979), Prog. Mater. Sci. 22,151. McMahan, A. K., Moriarty, J. A. (1983), Phys. Rev. B27, 3235. McMahan, A. K., Young, D. A. (1984), Phys. Lett. 105 A, 129. Mooij, B., Buschow, K. H. J. (1987), Philips J. Res. 42, 246. Moriarty, J. A. (1982), Phys. Rev. B 26, 1754. Moriarty, J. A. (1983), Int. J. Quantum Chem., Quantum Chem. 17, 541. Moriarty, J. A. (1988), Phys. Lett. 131, 41. Moriarty, J. A., McMahan, A. K. (1982), Phys. Rev. Lett. 48, 809. Moruzzi, V. L., Janak, J. R, Williams, A. R. (1978), Calculated Electronic Properties of Metals. New York: Pergamon. Mott, N. E, Jones, H. (1936), Properties of Metals and Alloys. New York: Dover, Chap. 7. Ohta, Y, Pettifor, D. G. (1989), /. Phys.: Condens. Matter 2, 8189. Olijnyk, H., Holzapfel, W. B. (1985), Phys. Rev. B 31, 4682. Oshima, K., Watanabe, D. (1976), Acta Crystallogr. A 32, 883. Parr, R. G., Yang, W. (1989), Density Functional Theory of Atoms and Molecules. New York: Oxford University Press. Paxton, A. T., Methfessel, M., Polatoglou, H. M. (1990), Phys. Rev. B41, 8127. Paxton, A. T., Methfessel, M., Pettifor, D. G. (1992), in preparation. Pearson, W. B. (1972), The Crystal Chemistry and Physics of Metals and Alloys. New York: Wiley. Pettifor, D. G. (1972), in: Metallurgical Chemistry: Kubachewski, O. (Ed.), London: HMSO, p. 191. Pettifor, D. G. (1976), Commun. Phys. 1, 141. Pettifor, D. G. (1978), /. Chem. Phys. 69, 2930. Pettifor, D. G. (1982), Phys. Scr. Tl, 26. Pettifor, D. G. (1983), in: Physical Metallurgy: Cahn, R. W, Haasen, P. (Eds.). Amsterdam: North-Holland, Chap. 3. Pettifor, D. G. (1986), /. Phys. C 19, 285. Pettifor, D. G. (1987), Solid State Phys. 40, 43. Pettifor, D. G. (1988 a), Mater. Sci. Technol. 4, 2480. Pettifor, D. G. (1988 b), Physica B 149, 3. Pettifor, D. G. (1990), in: Many-Atom Interactions in Solids: Nieminen, R. M., Puska, M. I , Manninen, M. (Eds.), Springer Proc. Phys., Vol. 48. Berlin: Springer, p. 64.
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2 Electron Theory of Crystal Structure
Pettifor, D. G. (1991), in: Intermetallic Compounds Structure and Mechanial Properties: Izumi, 0. (Ed.)- Sendai: Japan Institute of Metals, p. 149. Pettifor, D. G., Aoki, M. (1991), Philos. Trans. R. Soc. London A 334, 439. Pettifor, D. G., Gelatt, C. D. (1983), in: Atomistics of Fracture: Latanision, R. M., Pickens, J. R. (Eds.). New York: Plenum, p. 296. Pettifor, D. G., Podloucky, R. (1984), Phys. Rev. Lett. 53, 1080. Pettifor, D. G., Podloucky, R. (1986), /. Phys. C 19, 315. Pettifor, D. G., Varma, C. M. (1979), /. Phys. C12, L253. Pettifor, D. G., Ward, M. A. (1984), Solid State Commun. 49, 291. Phillips, J. C. (1973), Bonds and Bands in Semiconductors, New York: Academic. Pickett, W. E. (1989), Rev. Md. Phys. 61, 433. Schiff, L. I. (1968), Quantum Mechanics, New York: McGraw-Hill, van Schilfgaarde, M., Paxton, A. T., Pasturel, A., Methfessel, M. (1990), Materials Res. Soc. Symp. Proc. 186, 107. Skriver, H. L. (1985), Phys. Rev. B31, 1909. Slater, J. C. (1951), Phys. Rev. 81, 385. Slater, J. C , Koster, G. F. (1954), Phys. Rev. 94,1498. Sluiter, M., de Fontaine, D., Guo, X. Q., Podloucky, R., Freeman, A. J. (1990), Phys. Rev. B42, 10460. Stroud, D., Ashcroft, N. W. (1971), /. Phys. Fl, 113. Subramanian, P. R., Simmons, J. P., Mendiratta, M. G., Dimiduk, D. M. (1989), Mater. Res. Soc. Symp. Proc. 133, 51. Sutton, A. P., Finnis, M. W, Pettifor, D. G., Ohta, Y. (1988), J: Phys. C21, 35. Turchi, P., Ducastelle, F. (1985), in: The Recursion Method and Its Applications, Pettifor, D. G., Weaire, D. L. (Eds.), Springer Ser. Solid-State Sci., Vol. 58, Berlin: Springer, p. 104. Turchi, P., Treglia, G., Ducastelle, F. (1983), /. Phys. F13, 2543. Villars, P., Calvert, L. D. (1985), in: Pearsons Handbook of Crystallographic Data for Intermetallic Phases, Vols. 1, 2, 3. Metal Park, OH: American Society for Metals.
Villars, P., Mathis, K., Hullinger, F. (1989), in: The Structures of Binary Compounds: de Boer, F. R., Pettifor, D. G. (Eds.). Amsterdam: North-Holland, p. 1. Weiss, R. J. (1963), Proc. Phys. Soc. 82, 281. Wells, A. F. (1975), Structural Inorganic Chemistry. Oxford: Clarendon. Wyatt, T. K., Pettifor, D. G., Jacobs, R. L. (1991), The Structural Stability of sp-bonded Metals. Third Year Project, Imperial College, London. Yin, M. T., Cohen, M. L. (1982), Phys. Rev. B26, 3259. Yin, M. T., Cohen, M. L. (1983), Phys. Rev. Lett. 50, 2006. Zener, C. (1947), Phys. Rev. 71, 846.
General Reading Cottrell, A. H. (1988), Introduction to the Modern Theory of Metals. London: Institute of Metals. Ducastelle, F. (1991), in: Cohesion and Structure, Vol. 3: Order and Phase Stabilities in Alloys: de Boer, F. R., Pettifor, D. G. (Eds.). Amsterdam: North-Holland. Hafner, J. (1987), From Hamiltonians to Phase Diagrams. Berlin: Springer. Harrison, W A. (1980), Electronic Structure and the Properties of Solids. San Francisco: Freeman. Majewski, J. A., Vogl, P. (1989), "Quantum Theory of Structure: Tight-Binding Systems" in: Cohesion and Structure, Vol. 2: The Structures of Binary Compounds, de Boer, F. R., Pettifor, D. G. (Eds.). Amsterdam: North-Holland, Chap. 4. Pettifor, D. G. (1983), in: Physical Metallurgy: Cahn, R. W, Haasen, P. (Eds.). Amsterdam: North-Holland, Chap. 3. Pettifor, D. G., Cottrell, A. H. (Eds.) (1992), Electron Theory in Alloy Design. London: Institute of Materials.
3 Structure of Intermetallic Compounds and Phases Riccardo Ferro and Adriana Saccone
Istituto di Chimica Generale, Universita di Genova, Genova, Italy
List of 3.1 3.1.1 3.1.2 3.2 3.3 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.3.5.1 3.3.5.2 3.3.5.3 3.3.5.4 3.3.5.5 3.4 3.4.1 3.4.2 3.4.3 3.4.4 3.4.5 3.5 3.5.1 3.5.2 3.5.2.1 3.5.2.2 3.5.2.3
Symbols and Abbreviations Introduction Preliminary Remarks Definition of an Intermetallic Phase Chemical Composition of the Intermetallic Phase and Its Compositional Formula Crystal Structure of the Intermetallic Phase and Its Representation General Remarks Structure Types Unit Cell Pearson Symbol Trivial Structure Names Rational Crystal Structure Formulae Coordination and Dimensionality Symbols in the Crystal Coordination Formula Layer Stacking Sequence Representation Polyhedra Assembling Bauverband (Connectivity Pattern) Approach An Exercise on the Use of Alternative Structure Notations Relationships Between Structure Types (Structure Families) Degenerate and Derivative Structures (Defect, Filled-Up, Derivative Structures) Antiphase Domain Structures Homeotect Structure Types (Polytypic Structures) "Chimney-Ladder" Structures (Structure Commensurability, Structure Modulation) Recombination Structures, Intergrowth Structure Series Elements of Structure-Type Systematic Description Introductory Remarks and General References Description of a Few Selected Structure Types cI2-W, cP2-CsCl, cF16-MnCu2Al, cF16-Li3Bi, cP52-Cu9Al4 and cI52-Cu5Zn8 Type Structures and Martensite cF4-Cu, cP4-AuCu3, tP2-AuCu(I), oI40-AuCu(II), tP4-Ti3Cu Types; hP2-Mg, hP4-La and hR9-Sm Types; hP6-CaCu 5 Type cF8-C (Diamond), tI4-pSn, cF8-ZnS Sphalerite, tI16-FeCuS2, hP4-C Lonsdaleite, hP4-ZnO (or ZnS Wurtzite), oP16-BeSiN2 Types and SiC Polytypes
Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. Allrightsreserved.
125 127 127 129 130 133 133 135 136 137 139 140 143 147 149 150 154 154 155 156 157 159 160 160 162 162 165
170
124
3 Structure of Intermetallic Compounds and Phases
3.5.2.4 cF8-NaCl, cF12-CaF 2 , and cF12-AgMgAs Types 3.5.2.5 hP4-NiAs, hP3-CdI 2 , hP6-Ni 2 In, oP12-Co2Si, oP12-TiNiSi Types; hP2-WC, hP3-AlB2, hP3-coCr-Ti; hP6-CaIn 2 , hP9-Fe 2 P Types and tI8-NbAs, tI8-AgTlTe2 and tI10-BaAl4 (ThCr2Si2) Types 3.5.2.6 Frank-Kasper Structures (a-Phases, Laves Phases) and Samson Phases . . . 3.6 Regularities in Intermetallic Compound Structures and Semi-Empirical Approaches to the Prediction of Compound Formation 3.6.1 Preliminary Remarks 3.6.2 Factors Controlling the Structure of Intermetallic Phases 3.6.2.1 Chemical Bond Factor and Electrochemical Factor 3.6.2.2 Energy Band Factor and Electron Concentration 3.6.2.3 Geometrical Principles and Factors 3.6.2.4 Atomic Dimensions and Structural Characteristics of the Phases 3.6.2.5 Alternative Definitions of Coordination Numbers 3.6.2.6 Atomic-Environment Classification of the Structure Types 3.6.3 Semi-Empirical Approaches to the Prediction of Compound and Structure Formation in Alloy Systems 3.6.3.1 Stability Diagrams and Structure Maps 3.6.3.2 The Savitskii-Gribulya-Kiselova Method 3.6.3.3 The Villars and Villars and Girgis Approaches 3.6.3.4 Miedema Theory and Structural Information 3.6.3.5 Gschneidner's Relations 3.6.3.6 Pettifor's Chemical Scale and Structure Maps 3.7 References
172
174 179 185 185 188 189 191 191 192 201 203 204 205 206 206 209 209 211 212
List of Symbols and Abbreviations
a, b, c a), b), etc. CN d dXY dmin dr ^ Dx eA,ec
Lt M Me, X, Y n n nt nws PN Rt RCN 12, t Rz t Rn S Tc 7^ v_ A^form Fat F cell F sph VE VEC X XMB x, xt x,y,z
125
lattice parameters Wyckoff symbols of the atomic position in the unit cell coordination number interatomic distance (generally reported in picometers) interatomic distance between X and Y atoms minimum interatomic distance reduced interatomic distance (dr = d/dmin) dimensionality index atomic diameter of the X atom valence electron number (of anions or cations) effective coordination number of the j species according to Hoppe's scheme (see Sec. 3.6.2.5) formation enthalpy sublimation heat of species i Mendeleev number indication of unspecified elements number of valence electron per ligand average quantum number number of type i atoms electron density at the boundary of the Wigner-Seitz cell Schlafli symbol radius of atom i radius of atom i for the reported coordination 12 Zunger's pseudopotential radius of element i radius for bond order (bond strength, number of valence electron per ligand) n reduced strain parameter critical temperature Gschneidner's reduced temperature vacant site formation volume (difference between the average atomic volume in the compound and in the components) atomic volume volume of the unit cell atomic volume of a "spherical atom" valence electrons valence electron concentration electronegativity electronegativity (according to the Martynov-Batsanov scale) molar fraction (molar fraction of species i) symbol for the crystal axes and, also, symbols used for the fractional coordinates of the atoms in the unit cell
126
Atj e
3 Structure of Intermetallic Compounds and Phases
difference between computed and experimental interatomic distance of atoms i and j in the Pearson unit cell dimensional analysis radius (or diameter) ratio s = Rx/R\ = DX/DY work function of the element A{ (in the Miedema formula) space filling parameter chemical scale (value for atom A)
RE rare earth element cF, hR, t P , . . . and similar two-letter codes: symbols of lattices (see Table 3-4) cF8, hP6, t P I O , . . . and similar letter-numerical codes: Pearson symbols of the unit cell (see Sec. 3.3.3) P, I, F, W, G, R, etc. lattice complex symbols (see Sec. 3.3.5.4) 3 6 ,3636,3 2 4 2 and similar numerical codes: symbols (Schlafli symbols) of nodes in a net (see Sec. 3.3.5.2) 4 2 1 1 4 0 ... 8 0 3 and similar codes: symbols of atomic-environment (coordination polyhedra, see Sec. 3.6.2.6) 4,4', 4", 4t, 41,6,6', 6o, 6 p , . . . and similar numeric and numeric-letter codes: coordination symbols (see Table 3-5) AuoAuoCuf /2 , CuoCuf / 3 Cu2/3,... and similar codes: symbols for layer stacking sequence representation (see Sec. 3.3.5.2) ABABA ..., c , . . . h , . . . he, etc.: alternative symbolic representation of layer stacking types (see Sec. 3.3.5.2)
3.1 Introduction
3.1 Introduction 3.1.1 Preliminary Remarks
In the field of solid state chemistry an important group of substances is represented by the intermetallic compounds and phases. A few general and introductory remarks about these substances may be presented by means of Figs. 3-1 and 3-2. In binary and multi-component metal systems, in fact, several crystalline phases (terminal and intermediate, stable and metastable) may occur. Simple schematic phase diagrams of binary alloy systems are shown in Fig. 3-1. In all of them the formation of solid phases may be noticed. In Fig. 3-1 a we observe the formation of the AB^ phase (which gen-
erally crystallizes with a structure other than those of the constituent elements) and which has a negligible homogeneity range. Thermodynamically, the composition of any such phase is variable. In a number of cases, however, the possible variation in composition is very small (invariable composition phases or stoichiometric phases, or "compounds" proper). In Figs. 3-1 b and 3-1 c, on the contrary, we observe that solid phases with a variable composition are formed (non-stoichiometric phases). In the reported diagrams we see examples both of terminal (Figs. 3-1 b and c) and intermediate (Fig. 3-1 c) phases. These phases are characterized by homogeneity ranges (solid solubility ranges) which, in the case of the terminal phases, include the pure components
a)
liquid
\
/
|
V
at.%B-
at.% B
127
B
A
at.%B
Figure 3-1. Examples of simple binary diagrams, (a) A stoichiometric, congruently melting, compound is formed at the composition corresponding to the ABX formula, (b) No intermediate phase is formed. The components show a certain limited mutual solid solubility, (c) The two components show limited mutual solid solubility (formation of the oc and P phases). Moreover, an intermediate phase is formed: it is homogeneous in a certain composition range.
128
Ba
3 Structure of Intermetallic Compounds and Phases
BaAAI5 Ba7AI13 BaAIA
Al
a)
-AlTi Au^AI
b)
Figure 3-2. Isobarothermal sections of actual ternary systems (from "Ternary Alloys", Petzow and Effenberg, 1990, Vol. 3). (a) Ba-Al-Ge system. A number of binary compounds are formed in the side binary systems. Moreover, a few ternary phases have been observed. xx: Ba(AlxGe1_JC)2, line phase, stable for 0.41 < x < 0.77, x2: Ba 3 Al 2 Ge 2 , x3: Ba 10 Al 3 Ge 7 , x4: BaAl2Ge2, point phases, (b) Ti-Au-Al system. The binary systems show the formation of several intermediate phases, generally characterized by certain composition ranges (ideal simple formulae are here reported). Two ternary field phases are also formed. Their homogeneity ranges are close to the TiAu2Al (5t) and TiAuAl (52), respectively.
and which, generally, have a variable temperature-dependent extension. [In the older literature, stoichiometric and nonstoichiometric phases were often called "daltonides" and "berthollides", respectively. These names, however, are no longer recommended by the Commission on the Nomenclature of Inorganic Chemistry (IUPAC), Leigh (1990).] More complex situations are shown in Fig. 3-2, where some typical examples of isobarothermal sections of ternary alloy phase diagrams are presented. In the case of a ternary system, such as that reported in Fig. 3-2 a, we notice the formation of several, binary and ternary, stoichiometric phases. In the case shown in Fig. 3-2 b, different types of variable composition phases can be observed. We may differentiate between these phases by using terms such as: "point compounds" (or point phases), that is, phases represented in the composition triangle, or, more generally, in the composition simplex by points, "line phases", "field phases", etc. For all the mentioned phases the identification (and classification) requires information about their chemical composition and structure. To be consistent with the other field of descriptive chemistry, this information should be included in specific chemical (and structural) formulae built up according to well-defined rules. This task, however, in the specific area of the intermetallic phases (or more generally in the area of solid state chemistry) is much more complicated than for other chemical compounds. This complexity is related both to the chemical characteristics (formation of variable composition phases) and to the structural properties (the intermetallic compounds are generally non-molecular in nature, while the conventional chemical symbolism has been mainly developed for the representation of molecular units). As a
3.1 Introduction
consequence there is not a complete, or generally accepted method of representing the formulae of intermetallic compounds. Some details on these points will be given in the next paragraphs. These will then be used for a description of selected common phases and a presentation of a few characteristic general features of intermetallic crystallochemistry. For an exhaustive description of all the intermetallic phases and a comprehensive presentation and discussion of their crystallochemistry, general reference books and catalogues, such as those reported in the list of references, should be consulted. More references to specific topics will be reported in the following paragraphs. Those who are interested in the historical development of the intermetallic compound concept and science may refer to the review written by Westbrook (1977) on the past and future potential of the intermetallic compounds. In this review Westbrook selected the following topics for the examination of their historic roots: (a) the development of the modern concept of the intermetallic compound; (b) the development of the phase diagram; (c) the role of electron concentration in determining intermetallic phase stability; (d) the role of geometrical factors in determining intermetallic phase stability; (e) the point defect concept and its relation to non-stoichiometric compounds; (f) the unusual role of grain boundaries in intermetallic compounds. Westbrook (1977) reported information on the chronological growth in the number of binary metallic phase diagrams studied (starting from ^ 1830 with the systems Pb-Sn, Sn-Bi, etc.) and of the intermetallic compounds. The first problems encountered while studying these substances are pointed out:
129
typically that simple valence concepts were not applicable for rationalizing compound formulation and that several compounds seemed to exist over a range of composition and not at some specific ratio as with ordinary salts. The development of the systematics of the intermetallic phases and of their applications is then discussed and compared with the history of the rise of thermodynamics and crystallochemistry. 3.1.2 Definition of an Intermetallic Phase The identification and crystallochemical characterization of an intermetallic phase requires the definition and analysis of the following points: (a) Chemical composition (and the homogeneity composition range and its temperature and pressure dependence). (b) Structure type (or crystal system, space group, number of atoms per unit cell and list of occupied atomic positions). (c) Values of a number of parameters characteristic of the specific phase within the group of isostructural phases (unit cell edges, occupation characteristics and, if not fixed, coordinate triplets of every occupied point set). (d) Volumetric characteristics (molar volume of the phase, formation volume contraction, or expansion, space filling characteristics, etc.). (e) Interatomic connection characteristics (local atomic coordination, long distance order, interatomic distances, their ratios to atomic diameters, etc.). Clearly, not all the data relevant to the mentioned points are independent of each other. The strictly interrelated characteristics listed under d) and e), for instance, may be calculated from the data indicated in b) and c), from which the actual chemical composition of the phase may also be obtained.
130
3 Structure of Intermetallic Compounds and Phases
For each of the mentioned points (and for their symbolic representation) a few remarks may be noteworthy: these will be presented in the following. (For the involved crystallographic nomenclature see Chap. 1.)
3.2 Chemical Composition of the Intermetallic Phase and Its Compositional Formula Simple compositional formulae are often used for intermetallic phases; these (for instance, Mg 2 Ge, ThCr 2 Si 2 ,...) are useful as quick references, especially for simple, stoichiometric, compounds. The following remarks may be noteworthy: Order of citation of element symbols in the formula The symbol sequence in a formula (LaPb 3 or Pb 3 La) is, of course, arbitrary and, in some particular cases, may be a matter of convenience. Alphabetical order has often been suggested [for example by IUPAC, Leigh (1990)]. A symbol sequence based on some chemical properties, however, may be more useful when, for instance, compounds with analogous structures have to be compared (Mg2Ge and Mg 2 Pb). Recently the Materials Science International Team (MSIT) performing the critical assessment of a new series on ternary alloys edited by Petzow and Effenberg (1990) decided in 1990 to adopt a symbol quotation order based on a parameter introduced by Pettifor (1984, 1986) (see also Chap. 2 of this Volume). This parameter is the so-called Mendeleev number (M) and the correlated "chemical scale / " both of which are shown in Table 3-1. The chemical meaning of these parameters may be deduced not only by their relation to the Periodic Table. By using them, in fact, ex-
cellent separation of similar structures is achieved for numerous Am Bn phases with a given stoichiometry within single two-dimensional MA/MB maps (see Sec. 3.6.3.6). According to this suggestion (which will be generally adopted here) the element E with a lower value M E (or XE) wiU be quoted first. Indication of constituent proportions No special comments are needed for stoichiometric compounds (LaPb 3 , ThCr 2 Si 2 ,...). More complex notation is needed for non-stoichiometric phases. Selected simple examples will be given below and more detailed information will subsequently be reported, when discussing crystal coordination formulae. (a) Ideal formulae While considering a variable composition phase, it is often possible to define an "ideal composition" (and formula) relative to which the composition variations occur (or are considered to occur). This composition may be that for which the ratio of the numbers of different atoms corresponds to the ratio of the numbers of the different crystal sites in the ideal (ordered) crystal structure [as suggested by IUPAC, Leigh (1990)]. These formulae may be used even when the "ideal composition" is not included in the homogeneity range of the phase (Nb3Al for instance, shows a homogeneity range from 18.6at.%A1 which hardly reaches 25 at.% Al; at the formation peritectic temperature of 2060 °C the composition of the phase is about 22.5 at.% Al). (b) Approximate formulae A general notation which has been suggested by IUPAC when only little informa-
131
3.2 Chemical Composition of the Intermetallic Phase and Its Compositional Formula
Table 3-1. Chemical order of the elements (according to G. D. Pettifor, 1986). a) Mendeleev numbers M for the elements in alphabetical order Ac Ag Al Am Ar As At Au B Ba
48 71 80 42 3 89 96 70 86 14
Be Bi Bk Br C Ca Cd Ce Cf Cl
77 87 40 95 16 75 32 39 99
Cm 41 Co 64 Cr 57 Cs 8 Cu 72 Dy 24 Er 22 Es 38 Eu 18 F 102
Fe Fm Fr Ga Gd Ge H He Hf Hg
61 37 7 81 27 84 103 1 50 74
Ho I In Ir K Kr La Li Lr Lu
23 97 79 66 10 4 33 12 34 20
Md Mg Mn Mo N Na Nb Nd Ne Ni
36 73 60 56 100 11 53 30 2 67
No Np O Os P Pa Pb Pd Pm Po
35 44 101 63 90 46 82 69 29 91
Pr Pt Pu Ra Rb Re Rh Rn Ru S
31 68 43 13 9 58 65 6 62 94
Sb Sc Se Si Sm Sn Sr Ta Tb Tc
19 93 85 28 83 15 52 26 59
Te 92 Th 47 Ti 51 Tl 78 Tm 21 U 45 V 54 W 55 Xe 5 Y 25
Yb 17 Zn 76 Zr 49
b) Mendeleev numbers M and chemical scale % for the elements in ascending order of M
M
Element
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
He Ne Ar Kr Xe Rn Fr Cs Rb K Na Li Ra Ba Sr Ca Yb Eu Sc Lu Tm Er Ho Dy Y Tb Gd Sm Pm Nd Pr Ce La Lw No
0.00 0.04 0.08 0.12 0.16 0.20 0.23 0.25 0.30 0.35 0.40 0.45 0.48 0.50 0.55 0.60 0.645 0.655 0.66 0.67 0.675 0.6775 0.68 0.6825 0.685 0.6875 0.69 0.6925 0.695 0.6975 0.70 0.7025 0.705 0.7075 0.71
M
Element
36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69
Md Fm Es Cf Bk Cm Am Pu Np U Pa Th Ac Zr Hf Ti Ta Nb V W Mo Cr Re Tc Mn Fe Ru Os Co Rh Ir Ni Pt Pd
0.7125 0.715 0.7175 0.72 0.7225 0.725 0.7275 0.73 0.7325 0.735 0.7375 0.74 0.7425 0.76 0.775 0.79 0.82 0.83 0.84 0.88 0.885 0.89 0.935 0.94 0.945 0.99 0.995 1.00 1.04 1.05 1.06 1.09 1.105 1.12
M
Element
70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87
Au Ag Cu Mg Hg Cd Zn Be Tl In Al Ga Pb Sn Ge Si B Bi
sb 90 91 92 93 94 95 96 97 98 99 100 101 102 103
As p Po Te Se
s c
At i Br
ci
N O F H
11.16 11.18 11.20 11.28 L32 1.36 1.44 1.50 1.56 1.60 L.66
.68 1.80 1.84 1.90 1.94 :2.00 :2.04 :2.08 \2.16 :2.18 :2.28 :2.32 2.40 2.44 2.50 :2.52 :2.56 :2.64 :2.70 3.00 3.50
: :
100 5.00
132
3 Structure of Intermetallic Compounds and Phases
tion has to be conveyed and which can be used even when the mechanism of the variation in composition is unknown, is to put the sign « (read as circa or approximately) before the formula; for instance « CuZn.
pied by A in the ideal structure, whereas AB represents an atom A in a site normally (ideally) occupied by B.) A formula such as:
(c) Variable composition formulae (Ni, Cu) or N i ^ C u ^ (0 < x < 1) are the equivalent representations of the continuous solid solution between Ni and Cu, homogeneous in the complete range of compositions; other examples are: Ce1_;cLaJCNi5 (0 < x < 1); (Ti 1 _ JC CrJ 5 Si 3 (0 < x < 0.69); etc. Similar formulae may also be used in more complicated cases to convey more information: Am+xBn_xCp ( . . . < * < . . . ) (phase involving substitution of atoms A for B). A1_JCB may indicate that there are Atype vacant sites in the structure. LaNi5H;c (0 < x < 6.7) indicates the solid solution of H in LaNi 5 . (d) Site occupation formulae According to the Recommendations by the Commission on the Nomenclature of Inorganic Chemistry (Leigh, 1990), additional information may be conveyed by using a more complicated symbolism; suggestions have also been given about the indication of site occupation and of their characterization. These points will be discussed in more detail in the following paragraphs; in the meantime we may mention that, for the indication of site occupation, the following criteria have been suggested by the Commission: The site and its occupancy are represented by two right lower indexes separated by a comma. The first index indicates the type of site, the second one indicates the number of atoms in this site. (AA, for instance, means an atom A on a site occu-
represents a disordered alloy (whereas the ideal composition is MN with an ideal MMNN structure). In this notation vacant sites may be represented by • or by v_. The following examples of alloy formulae have been reported: MgMg, 2 - *SnMg shows a partially disordered alloy with some of the Mg atoms on Sn sites, and viceversa. (Bi2_xTe:c)Bi(BixTe3_x)Te shows the composition changes from the ideal Bi2Te3 formula; AIAI, I PdA1 :cPdpd x
_x •
Pd 2x
which shows that in the phase (corresponding to the ideal composition PdAl), every Al is on an Al site, but x Pd atoms are on Al sites (1 — x Pd atoms in Pd sites) and 2x Pd sites are vacant. This type of formulae may be especially useful when discussing thermodynamic properties of the phase and dealing with quasi-chemical equilibria between point defects. (e) Polymorphism descriptors Several substances may change their crystal structure because of external conditions such as temperature and pressure. These different structures (polymorphic forms) may be distinguished by using special designators of the stability conditions. (If the various crystal structures are known, explicit structural descriptors may obviously be added). A very simple, but system-
3.3 Crystal Structure of the Intermetallic Phase and Its Representation
133
Table 3-2. An example of crystallochemical description of an alloy system. Binary solid phases in the Ag-Al system (from Petzow and Effenberg, 1990, Vol. 3). Phase Trivial name, ideal formula temperature range (°C)
Pearson symbol
Lattice parameters (pm)
Maximum composition range (at.% Al)
(Ag) < 961.93
cF4 Cu
a = 408.53 (23 °C)
0 to 20.4 (*450°C)
P-Ag3Al (h) 780-605 u-Ag3Al (r) <448
cI2 W
a = 330.2
20.5 to 29.8 (at 727 °C)
cP20 (3-Mn
a = 693
- 21 to 24
5-Ag2Al <727
hP2 Mg
a = 287.1 (27 at.% Al) c = 466.2
22.9 to 41.9
(Al) < 660.45
cF4 Cu
a = 404.88 (24 °C)
76.5 to 100 (at 567 °C)
atic notation has been introduced by the MSIT (see before) which in the meanwhile has been adopted worldwide (see Introduction of all volumes on "Ternary Alloys" edited by Petzow and Effenberg, 1990). The different temperature modifications are indicated by lower case letters in parenthesis behind the phase designation, with (h) = high temperature modification, (r) = room temperature modification and (1) = low temperature modification; (h1? h 2 , etc. represent different high temperature modifications). In the description of a number of modifications which are stable at different temperatures, the letters are used in the sequence h 2 , h 1? r, 11? 1 2 ,..., in correspondence to the decreasing stability temperature. Table 3-2, taken from Vol. 3 of the series edited by Petzow and Effenberg (1990), shows a few examples of this notation. (In this case, of course, the temperature and composition ranges of stability explicitly indicated for all the phases give additional, more detailed information.) In connection with this group of descriptors we may perhaps remember indicators
such as (am), (vt), etc. for amorphous, vitreous substances. For instance: SiO2(am) amorphous silica; Si(am)HJC amorphous silicon doped with hydrogen.
3.3 Crystal Structure of the Intermetallic Phase and Its Representation 3.3.1 General Remarks
The characterization of a phase requires a complete and detailed description of its structure. As examples of such a description, we may consider the data (as obtained, for instance, from X-ray diffraction experiments) reported in Table 3-3 for stoichiometric and variable composition phases. [For an explanation of the various symbols used in the table see Chap. 1 of this book and the International Tables of Crystallography (Hahn, 1989).] For the CsCl compound the description reported in Table 3-3 corresponds to the atom arrangement presented (with alternative representations) in Fig. 3-3.
134
3 Structure of Intermetallic Compounds and Phases
Table 3-3. Examples of crystallographic description of phase structures (from Villars and Calvert, 1991). CsCl (stoichiometric compound):
Primitive cubic; a = 411.3 pm; space group Pm3m, No. 221. 1 Cs in a): 0,0,0; l C l i n b ) : 1 /2, 1 / 2 , 1 / 2 . (The two special a) and b) Wyckoff positions have no free coordinate parameter.) The two occupancy parameters are 100%. Mg 2 Ge (stoichiometric compound):
Face-centered cubic; a = 638.7 pm; space group Fm3m, No. 225. Equivalent positions (0,0,0; 0, Y2, V2; 72,0, 72; 72, 72,0) + 4 Ge in a): 0,0,0; 8 Mg in c): V4, 74, %; V4, %,3/4. (No free parameters in the atomic positions of Mg and Ge. In this case the two occupancy parameters have been found to be 100%.) MoSi 2 (nearly stoichiometric compound):
Body-centered tetragonal; a = 319.6 to 320.8 pm and c = 787.1 to 790.0 pm, according to the composition; space group I4/mmm, No. 139. Equivalent positions (0,0,0; V2, Vi, 72) + 2 Mo in a): 0,0,0; 4 Si in e): 0,0, z; 0,0, - z; z = 0.333. (The Si position has the free parameter z, for which, in this particular case, the value 0.333 has been determined; the two occupancy parameters are 100%.) « Ce 2 NiSi 3 (disordered structure):
Hexagonal; a = 406.1-407.1 pm; c = 414.9-420.2 pm; space group P6/mmm, No. 191. ICein a): 0,0,0; 2 (Ni + Si) (in a ratio 1:3) in d): V3,2/3, V2; 2/a, V3, Vi. (In this case the atomic sites corresponding to the d) Wyckoff position are randomly occupied by Ni and Si atoms.) Cr 1 2 P 7 (simple structure showing partially occupied sites): Hexagonal; a = 898.1 pm; c = 331.3 pm; space group P6 3 /m, No. 176. 2 P in a): 0,0, V4; 0,0, 3 / 4 ; (occupancy 50%); 6 P in h): x, y, V4; — y, x — y, V4; — x + y, — x, V4; — x, — y, %; y, — x + y, 3/4; x — y, x, 3/4 (x = 0.2851, y = 0.4462); (occupancy 100%);
6 Cr in h): x, y, %; — y, x — y9 %; — x + y, — x, 7 4 ; — x, — y, 3/4; y, — x + y, 3/4; x — y, x, 3/4 (x = 0.5109, y = 0.3740); (occupancy 100%); 6 Cr in h): x, y, 74; — y, x — y, 74; — x + y, — x, V4; — x, — y, 3/4; y, — x + y, 3/4; x — y, x, 3/4 (x = 0.2108, y = 0.0144); (occupancy 50%); 6 Cr in h): x, y, 74; — y, x — y, %; — x + y, — x, 74; — x, — y, 3/4; y, — x + y, 3/4; x — y, x, 3/4 (x = 0.2638, y = 0.0137); (occupancy 50%); In this case several groups of atoms have the same type of Wyckoff position, the h) position, which has free parameters. The values of the free parameter experimentally determined for each atom group are reported. The partial occupancies found for the different positions are also reported. The corresponding number of atoms in the unit cell are: P: 0.5 • 2 + 6 = 7; Cr: 6 + 0.5 • 6 + 0.5 • 6 = 12. Notice that, for instance, in the case of the MoSi2 structure the different atomic positions in the unit cell are the following: Mo in 0,0,0, and in V2, V2, V2; Si in 0,0,0.333; in 0,0,0.667; in V2, Vi,0.833 and in V2, V2,0.167. These positions have been indicated, according to the International Tables of Crystallography conventions, explicitly indicating the centering translations (0,0,0; 72, 72, V2) + before the coordinate triplets. The symbol + means that, in order to obtain the complete Wyckoff position the components of these centering translations have to be added to each of the listed triplets. A similar presentation has been used for the Mg2Ge structure description. Notice that the coordinates are formulated: Modulo 1: x or y, for instance, have the same meaning as 1 + x or 1 + y, respectively. For the meaning of the different crystallographic symbols see Chap. 1.
3.3 Crystal Structure of the Intermetallic Phase and Its Representation
a)
O
-cr
o-
-o
b)
Figure 3-3. Alternative representations of the unit cell of the CsCl compound. (The actual structure of the CsCl, of course, corresponds to a three-dimensional infinite repetition of unit cells.) (a) and (b) sticks and balls models; (c) packed spheres model.
From such a description the interatomic distances may be computed and, consequently, the coordinations and grouping of the various atoms may be derived. (A systematic listing of the crystal data relevant to all the known phases has been reported in a number of fundamental reference books such as Pearson, 1967; LandoltBornstein, 1971; Villars and Calvert, 1985, 1991.) For the criteria to be followed, especially when complex structures are involved, in
135
the preparation and presentation of coordinate lists see Parthe and Gelato (1984). Their paper describes a proposal for a standardized presentation of inorganic crystal structure data with the aim of recognizing identical (or nearly identical) structures from the similarity of the numerical values of the atom coordinates. Different, equivalent (but not easily recognizable) descriptions could, in fact, be obtained by shift of origin of the coordinate system, rotation of the coordinate system, inversion of the basis vector triplet. 3.3.2 Structure Types
Several intermetallic phases are known which have the same (or a similar) stoichiometry and crystallize in the same crystal system and space group with the same occupied point positions. Such compounds are considered as belonging to the same structure type. The reference to the structure type may be a simpler and more convenient way of describing the structure of the specific phase. The structure type is generally named after the formula of the first representative identified: the "prototype". Trivial names are also used in some cases. The various representatives of a specific structure type generally have different unit cell edges, different values of the occupancy parameters and of the free coordinates of the atomic positions and, in the same atomic positions, different atoms (see, for instance, Sec. 3.3.5.5). If these differences are small, we may consider the general pattern of the structure unaltered. On the other hand, of course, if these differences become larger, it might be more convenient to describe the situation in terms of a "family", instead of a single structural type, of different (more or less strictly interrelated) structural (sub) types.
136
3 Structure of Intermetallic Compounds and Phases
According to Parthe and Gelato (1984), some structures may not really be isotypic but only isopointal, which means that they have the same space group and the same occupation of Wyckoff positions with the same adjustable parameters but different unit-cell ratios and different atom coordinations. Very interesting general comments and definitions on this question have been proposed, for instance, by Pearson (1972), and more recently by Lima-de-Faria et al. (1990). According to these authors, two structures are isoconfigurational (configurationally isotypic) if they are isopointal and are similar the corresponding Wyckoff positions and their geometrical interrelationships. Two structures, moreover, are defined crystal-chemically isotypic if they are isoconfigurational and the corresponding atoms (and bonds) have similar chemical/physical characteristics. We have finally to mention that, when considering phases having certain polar characteristics (salt-like "bonding"), the concept type and antitype may be useful. Antitypic phases have the same site occupations as the typic ones, but with the cation-anion positions exchanged (or more generally some important physical/chemical characteristics of the corresponding atoms interchanged). (As examples the structure types CaF 2 and Cdl 2 and their antitypes reported in Sees. 3.5.2.4 and 3.5.2.5 may be considered.) 3.3.3 Unit Cell Pearson Symbol The use of the so-called Pearson notation is highly recommended (IUPAC, Leigh, 1990; "Ternary Alloys", edited by Petzow and Effenberg, 1990) for the construction of a compact symbolic representation of the structure of the phase. As far as possible, it should be completed by a
Table 3-4. Pearson symbols. System symbol
Lattice symbol
a m o t h
triclinic (anorthic) P primitive monoclinic I body centered orthorhombic F all-face centered tetragonal C side face centered hexagonal (and trigonal R rhombohedral and rhombohedral) c cubic
more detailed structural description by using the prototype formula which defines (as previously mentioned) a certain structure type. The Pearson symbol is composed of a sequence of two letters and a number. The first (small) letter corresponds to the crystal system of the structure type involved; the second (capital) letter represents the lattice type (see Table 3-4). The symbol is completed by the number of the atoms in the unit cell. A symbol as tPIO, for example, represents a structure type (or a group of structure types) corresponding to 10 atoms in a primitive tetragonal cell. In this chapter, the Pearson symbol will be used throughout; the convention has been adopted indicating in every case the number of atoms contained in the chosen unit cell. In the case, therefore, of rhombohedral substances for which the (triple primitive) hexagonal cell is reported, the number of atoms is given which is in the hexagonal cell and not the number of atoms in the equivalent rhombohedral cell (Ferro and Girgis, 1990). So, for instance, at variance with Villars and Calvert (1985, 1991), hR9 and not hR3 for Sm-type structure. If the structure is not known exactly, the prototype indication cannot be added to the Pearson symbol. In some cases, moreover, only incomplete Pearson symbols (such as o?60, cF?, etc.) can be used.
3.3 Crystal Structure of the Intermetallic Phase and Its Representation
A criterion similar to Pearson's for the unit cell designation was used by Schubert (1964) in his detailed and systematic description of the structural types of the intermetallic phases and of their classification. A slightly more detailed notation, moreover, for the unit cell of a given structure has been suggested by Frevel (1985). Four items of information are coded in Frevel's notation: - the number of different elements contained in the compound, - the total number of atoms given by the chemical formula, - the appropriate space group expressed in the Hermann-Mauguin notation and - the number of formulae for unit cell. The notation for the CaF 2 structure, for instance, is: 2, 3 Fm3m (4). Possible augmentation of the notation has been discussed by Frevel and its use for classification and cataloguing the different crystal structures suggested. 3.3.4 Trivial Structure Names
A number of trivial names have been used (and are still in use) both as indicators, of a single phase in specific systems or as descriptors of certain structural types (or of families of different interrelated structural types). Among the trivial names, we may mention the use of Greek (and Roman) letters to denote phases. These have often been used to indicate actual phases in specific systems, for instance in a given binary system, phase oc, (3,.., etc., in alphabetical order according to the increasing composition from one component to the other, while in a unary system the oc, (3, etc., symbols have often been used to denote different allotropic forms. For instance, ocFe (cI2), yFe (cF4), 5Fe (cI2), ocPu (mP16), (3Pu
137
(mC34), yPu (oF8), 5Pu (cF4), 5Tu (tI2) and ePu (cI2). Obviously this notation (or other similar ones such as x1? t 2 , x3, denoting 1st, 2nd, etc., phase) may be useful as a quick reference criterion while discussing and comparing phase properties of alloys in a single specific system, but in general cannot be used as a rational criterion for denoting structural types. In a few cases, however, certain Greek (and Roman) letters have assumed a more general meaning (as symbols of groups of similar phases): for instance, the name y phases which is an abbreviation of a sentence such as phases having the y-brass (the yCu-Zn) type structure. A short list, taken from Landolt-Bornstein (1971), of (Greek and Roman) letters which have also been used as descriptors of structural types, may be the following: y : s : £ : T[ : H : a : X : co : E : G : P : R : Tx: T2:
y-brass type or similar structures Mg type Mg type W 3 Fe 3 C or Ti 2 Ni type W 6 Fe 7 type a phase or a-CrFe type oc-Mn or Ti 5 Re 24 type co2-(Cr,Ti) type (similar to the A1B2 type) PbCl 2 or CoSi2 type G phase, Th 6 Mn 2 3 or Cu 16 Mg 6 Si 7 P phase or P-(Cr, Mo, Ni) R phase or R-(Co, Cr, Mo) W 5 Si 3 type Cr 5 B 3 type
In a number of cases, names of scientists are used as descriptors. We may mention the following groups of structures (some of which will be described in more detail later). Frank-Kasper phases. (For all of which the structure can be described as composed of a collection of distorted tetrahedra which fill the space.) This family of phases
138
3 Structure of Intermetallic Compounds and Phases
includes those of the structural types: Laves phases (a family of polytypic structures corresponding to the hP12-MgZn2, cF24-Cu2Mg and hP24-Ni2Mg types (see Sec. 3.5.2.6), tP30 a-phases, oP56-Pphases and hR39-W 6 Fe 7 type phases. Hume-Rothery phases. These designations can be connected to the research carried out as far back as 1926 by HumeRothery, Westgren (Westgren and Phragmen, 1926) etc. They observed that several compounds (electron compounds) crystallize in the same structural type if they have the average number of valence electrons per atom (the so-called VEC: valence electron concentration) included within certain well-defined ranges. Some groups of these phases (brasses, etc.) will be presented in Sees. 3.5.2.1 and 3.6.2.2. Heusler phases. Magnetic compounds of the cF16-MnCu2Al-type. (See Sec. 3.5.2.1 on this structure which can be considered "derivative" of the CsCl type.) Zintl phases. This term was first applied to the binary compounds formed between the alkali or alkaline-earth elements and the main group elements from group 4 on, that is to the right of the "Zintl boundary" of the periodic table. These combinations not only yield some Zintl anions (homopolyatomic anions) in solution but also produce many rather polar or salt-like phases. The most reduced member is usually a classical valence compound in which the more noble member achieves a filled "octet" and an 8-N oxidation state in saltlike structure (for example Na 3 As, Mg2Sn) (Corbett, 1985). An important intermetallic structure discovered by Zintl was that of the cF16-NaTl-type (superstructure of the b.c.c. lattice). The Na and Tl atoms are arranged according to two (interpenetrating) diamond type sublattices; each atom is tetrahedrally coordinated by four like neighbors on the same sublattice and has
four unlike neighbors on the other sublattice. This could be interpreted as a Tl~ array, isoelectronic with carbon in the limit of complete charge transfer. For a critical discussion on the NaTl-type structure, its stability, the role of the size factor, the comparative trend of the stabilities of CsCl and NaTl type structures, the application of modern band-structure techniques, see Hafner (1989). Subsequent applications of the term "Zintl phases" have been based on the structural characteristic of such polar phases. A review on this subject has been published by Corbett (1985). In this paper several phases are mentioned: starting from compounds such as hR18CaSi2 (containing rumple double layers of Si atoms resembling those of the As structure), mP32-NaGe and mC32-NaSi (respectively containing Ge atoms, or Si atom, tetrahedra with the Na atoms arranged in the intervening spaces), up to complex alkali metal-gallium compounds exhibiting complex structures containing large interconnected usually empty gallium polyhedra, reminiscent of boron chemistry. Hdgg phases. Interstitial phases based on the occupancy of interstices in closepacked structures of transition metals by small non-metal atoms: H, B, C, N (see, for instance, Sec. 3.5.2.4 for NaCl type related structures and Sec. 3.5.2.5 for the WC-type structure). Nowotny phases. Chimney-ladder phases (see Sec. 3.4.4). Chevrel phases. A group of compounds having a general formula such as M x Mo 3 S 4 (M = Ag, As, Ca, Cd, Zn, Cu, Mn, Cr, etc.). Many representatives of these structure types are superconducting with critical Tc as high as 10-15 K. Samson phases. Complex intermetallic structures with giant unit cells, based on
3.3 Crystal Structure of the Intermetallic Phase and Its Representation
framework of fused truncated tetrahedra (see Sec. 3.5.2.6). Within this group of names we may also include a few "personal" names such as austenite, ferrite, martensite, etc., and a few mineralogical names such as pyrite, blende, cinnabar, etc. According to the IUPAC recommendations, Leigh (1990), mineralogical names should be used to designate actual minerals and not to define chemical composition. They may, however, be used to indicate a structure type. They should be accompanied by a representative chemical formula: cF8-ZnS sphalerite, hP4-ZnS wurtzite, cF8-NaCl rock salt, cP12-FeS2 pyrite, etc. In closing this paragraph we have to mention the old Strukturbericht designation, no longer recommended by IUPAC, but still used. According to this designation, each structure type was represented by a symbol generally composed of a letter (A, B, C, etc.) and a number (possibly in some cases followed by a third character). The letter was related to the stoichiometry: A: unary systems (or believed to be unary), B: binary phases having a 1:1 stoichiometry, C: binary phases having a 1:2 stoichiometry, etc. In every class of stoichiometries, the different types of structures were distinguished by a number. (For instance the cF4-Cu-type is represented by Al, the cF8NaCl type is represented by Bl). Equivalence tables between the Strukturbericht designation and the Pearson symbol-prototype may be found in Pearson (1972), Massalski (1990). Following is a short list of old Strukturbericht symbols of common structural types: Al: cF4-Cu; A2: cI2-W; A3: hP2-Mg; A4: cF8-C (diamond);...; A12: cI58-oc-Mn;...; A15: cP8-Cr 3 Si;.... The A15 structure was
139
previously considered to be the structure of a W modification (and therefore a unary structure): later on the substance concerned was recognized to be a W oxide: W 3 O (isostructural with Cr3Si). Bl: cF8-NaCl; B2: cP2-CsCl; B3: cF8Zns (sphalerite or zinc blende);...; B8j: hP4-NiAs; B8 2 : hP6-Ni 2 In;...; B20: cP8FeSi;...;B32:cF16-NaTl;.... Cl: cF12-CaF 2 ; Cl b : cF12-AgMgAs; C2: cP12-FeS 2 ;...; Cll a : tI6-CaC2; Cll b : tI6-MoSi 2 ;...; C14: hP12-MgZn 2 ; C15: cF24-Cu2Mg; C15 b : cF24-AuBe5; C16: tI12-CuAl 2 ;...; C32: hP3-AlB 2 ;...; C36: hP24-Ni 2 Mg;.... D0 2 : cI32-CoAs 3 ;...; D0 1 8 : hP8-Na3As; ...; D8j: cI52-Fe3Zn10; D8 2 : cI52-Cu5Zn8; D8 3 : cP52-Cu9Al4; D8 4 : cF116-Cr 23 C 6 ; L l 0 : tP2-AuCu; Ll 2 : cP4-AuCu3; L ^ : cF16-MnCu 2 Al;.... 3.3.5 Rational Crystal Structure Formulae
We know that all of the requisite structural information for a solid phase is contained (either explicitly or implicity) in its unit cell and this can be obtained from the Pearson-symbol-prototye notation (complemented, if necessary, by data on the values of lattice parameters, atomic positions, etc.). A number of features, however, which are especially relevant for chemical-physical considerations, such as local coordination geometries, the existence of clusters, chains or layers, etc., are not self-evident in the mentioned structural descriptions and can be deduced only by means of a more or less complicated series of calculations. It should, however, be pointed out that the same structure can be differently viewed and described (Franzen, 1986; Parthe and Gelato, 1984). The simple rock-salt structure, for instance (see Sec. 3.5.2.4), can be viewed as cubic close packed anions with
140
3 Structure of Intermetallic Compounds and Phases
cations in octahedral holes, as XY6 octahedra sharing edges with resultant short X-X distances, as a stacking sequence of superimposed alternate triangular nets respectively of X and Y atoms or as a cubicclose packed structure of a metal with nonmetals in octahedral interstices. As a further example we may consider the Cu structure which, for instance, could be conveniently compared with those of Mg, La and Sm, or from another point of view, with the AuCu, AuCu 3 structures. In the two cases one would choose a different description and representation of the mentioned Cu structure. In the different cases, some criteria may therefore be useful in order to give (in a systematic and simple way) explicit information on the characteristic structural features. In the following Sections some information will be given on a few complementary, alternative notations. 3.3.5.1 Coordination and Dimensionality Symbols in the Crystal Coordination Formula
Several attempts have been carried out in order to design special formulae (crystal coordination formulae) which (in a convenient linear format) may convey explicit information on the local coordination geometry. A detailed discussion of these attempts and of their development [through the works, inter alios, by Niggli (1945, 1948), Machatschki (1938, 1953), Lima-deFaria and Figueiredo (1976, 1978), Parthe (1980), Jensen (1984)] may be found in a review by Jensen (1989) who presented and systematically discussed a flexible notation for the interpretation of solid-state structures. A short description of Jensen's notation will be given here below. The different symbols used will be briefly presented. For
the notation concerning the common coordination geometries a summary is reported in Table 3-5. A report of the International Union of Crystallography Commission on Crystallographic Nomenclature (Lima-deFaria et al., 1990) presents a concise description of similar alternative notations, a summary of which is also reported in Table 3-5. The symbols suggested by Jensen, based on Niggli's proposals, indicate the local coordination environments by means of coordination number ratios. For instance, a formula AEm/n will indicate a binary compound where m is the coordination number (the nearest neighbor number) of atoms E around A and n will be considered the coordination number of E by A. The ratio m/n will be equal to the stoichiometric compositional ratio. For instance, we will write NaCl 6/6 to represent the hexacoordination (in this case octahedral coordination) of Cl around Na (and viceversa) in sodium chloride. Similarly we will have: ZnS4/4.; -P-"-3/i5 ^ s Clg/8? CaF 8 / 4 ; UCl 9 / 3 ;
etc. According to one of Jensen's suggestions it is possible to add modifiers to the coordination numbers in order to specify not only topological but also geometrical characteristics of the primary coordination sphere. (For example, 6: octahedral; 6': trigonal prismatic; 6": hexagonal planar; etc., see Table 3-5 a and Fig. 2-3 in Chap. 2.) Similar symbols were proposed by Donnay et al. (1964) who suggested adding to the coordination number, one or two letters to indicate the geometry: y, pyramidal; 1, planar; c, cubic; etc. Detailed descriptions of the coordination polyhedra are obtained by means of the Lima-de-Faria (1990) symbols reported in Table 3-5 b. An advantage of the Lima-de-Faria symbolism may be the existence of two alternative sets of symbols: complete and simplified.
3.3 Crystal Structure of the Intermetallic Phase and Its Representation
141
Table 3-5. Suggested notations for common coordination geometries. a) from Jensen (1989) 1 2 2' 3 y 3" 4 4' 4" 5 5' 6 6' 6" 1 7 1"
Terminal Bent CN2 Linear CN 2 Pyramidal or in general non-planar CN 3 Trigonal planar T-planar Tetrahedral Square planar Base of a square pyramid with the central atom as the apex Trigonal bipyramid Square based pyramid with the central atom inside Octahedral or trigonal antiprism Trigonal prism Hexagonal planar Pentagonal bipyramid Monocapped octahedron Monocapped trigonal prism
8 8' 8" 8"' 8 9 10 11 12 12'
Cube Square antiprism Dodecahedron Bicapped trigonal prism Hexagonal bipyramid Tricapped trigonal prism Bicapped square antiprism Monocapped pentagonal antiprism Cubic closest-packed or cuboctahedron Hexagonal closest-packed or twinned cuboctahedron 12" Icosahedron 12 Hexagonal prism n Complex, distorted rc-hedron n Disordered structure in which it is possible to define only an average coordination number n
b) from Lima-de-Faria et al. (1990). Coordination polyhedron around atom A
Complete symbol
Alternative simplified symbols
Single neighbor Two atoms collinear with atom A Two atoms non-collinear with atom A Triangle coplanar with atom A Triangle non-coplanar with atom A Triangular pyramid with atom A in the center of the base Tetrahedron Square coplanar with atom A Square non-coplanar with atom A Pentagon coplanar with atom A Tetragonal pyramid with atom A in the center of the base Trigonal bipyramid Octahedron Trigonal prism Trigonal antiprism Pentagonal bipyramid Monocapped trigonal prism Bicapped trigonal prism Tetragonal prism Tetragonal antiprism Cube Anticube Dodecahedron with triangular faces Hexagonal bipyramid Tricapped trigonal prism Cuboctahedron Anticuboctahedron Icosahedron Truncated tetrahedron Hexagonal prism Frank-Kasper polyhedra with 14 vertices 15 vertices 16 vertices
[11] [21] [2n] [31] [3n] [4y] [4t] [41] or [4s] [4n] [51] [5y] [5by] [6o] [6p] [6ap] [7by] [6plc] [6p2c] [8p] [8ap] [8cb] [8acb] [8do] [8by] [6p3c] [12co] [12aco] [12i] [12tt] [12p]
[1] [2] [2] [3] [3] [4] [t] [4] t [s] [4] s [4] [5] [5] [5] [o] [6] o [p] [6] P [ap] [6] ap [7] [7] [8] [8] [8] [cb] [8] cb [acb] [8] acb [do] [8] do [8] [9] [co] [12] co [aco] [12] aco [i] [12] i [12] [12]
[14FK] [15FK] [16FK]
[14] [15] [16]
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3 Structure of Intermetallic Compounds and Phases
The simplified symbols give only a numerical indication, without any distinction between different geometries, the complete symbols (clearly distinguishable from the previous ones) contain beside the numeric indication a description of the coordination polyhedron. A selection of the Limade-Faria symbols, together with Jensen's suggestions, will be used here. According to Jensen, the dimensionality of a structure (or of a substructure of the same) is indicated by enclosing its compositional formula in square brackets and prefixing an appropriate symbol £. The dimensionality index, d, may be d = 0 for a discrete molecular (cluster, ring) structure, d = 1 for a one-dimensional, infinite chain structure, d = 2 for a two-dimensional, infinite layer structure and d = 3 for an infinite three dimensional, framework structure. More complex symbols such as d~J' or d d '^ will represent intermediate dimensionality (between d and dr) or, respectively the dimensionality indexes of different substructures (d' and d") followed by that of the overall structure (d). A few examples are reported here below: Molecular structures °[HI], £[CO2], linear structures ^[BeCl2], layer structures ^[C] graphite, ^[As], framework structures ^[C] diamond, substructures °Ca[CO 3 ] (finite ions); 4K[PO 3 ] (infinite anionic PO^ chain), etc. If, in an A-B structure, one wishes to show not only the A/B coordination but also the B/B or A/A self-coordinations this is done, according to the suggestion by Jensen via the use of a composite dimensionality index and the relative positions of the various ratios and brackets in the formula, with the last unbracketed ra-
tio always referring to the B/A coordination. So, for instance, °^3[(H2O)4/4] is a compact form for °003[(H2O)(H2O)4/4] to indicate the molecule packing in the ice structure. The formulae ^Al[B 3 / 3 ] 1 2 / 6 or T[Al 8by/ 8by][ B 3i/3i]i2p/6 P
correspond
to
more or less detailed description of the A1B2 type structure where the coordination of B around Al is 12 (12p: hexagonal prismatic) and that of Al around B is 6 (6p: trigonal prismatic). The self coordinations are bipyramidal for Al/Al (8by: hexagonal bipyramidal) and trigonal-planar (31) for B/B (the B atoms form a two-dimensional net). Considering as a further example the compounds AB having the CsCl type structure, we may mention that according to Jensen, the two descriptions 3 3 ' [A 6/6 ][B 6 / 6 ] 8 / 8 and ^[AB8/8] (with and without the indication of the self coordination) may also be used to suggest the bonding type (metallic if the A-A and B-B interactions contribute to the overall bonding, ionic, or covalent, if only A-B interactions have to be considered). A detailed example (AuCu3) of the application of the mentioned notation to the description of a simple intermetallic structure will be presented in Sec. 3.3.5.5 (with the pertinent Figs. 3-11 to 3-14). A few more examples will be reported in the following descriptions of a number of typical structures. In conclusion to this description of "crystal coordination formulae" we have, however, to notice that the term "coordination number" (CN) may be used in two ways in crystallography (Frank and Kasper, 1958). According to the first, the coordination number, as previously mentioned, is the number of nearest neighbors to an atom. According to the other way, the definition of the coordination should be based on an "interpretation" of the
3.3 Crystal Structure of the Intermetallic Phase and Its Representation
structure which depends not only on an evaluation of the interatomic distances to assign bonding versus non-bonding contacts but on considerations on the bonding mechanism (Jensen, 1989). These considerations are particularly important when considering metallic phases where it may be difficult to make distinctions between X-X, X-Y or Y-Y contacts. So, for instance, when considering the b.c. cubic structure of the W type, some authors define the coordination number as 8 (in agreement with the nearest-neighbors definition) but others prefer to regard it as 14 (including a group of 6 atoms at a slightly higher distance). Further considerations on this subject will be reported in Sec. 3.6.2.5 in a discussion on alternative definitions of coordination numbers (weighted coordination number, effective coordination number). In Sec. 3.6.2.6, on the other hand, the atomic-environment types will be introduced, their codes presented and the results of their use in the classification of the cubic intermetallic structure types summarized. 3.3.5.2 Layer Stacking Sequence Representation
A large group of structures of intermetallic phases can be considered to be formed by the successive stacking of certain polygonal nets of atoms (or, in more complex cases, by the successively stacking of characteristic "slabs"). These structural characteristics can easily be described by using specific codes and symbols, which can be very useful for a compact presentation and comparison of the structural features of several structures. Many different notations have been devised to describe the stacking pattern (for a summary see Parthe, 1964; Pearson, 1972). A few of them will be presented here. As an introduction to this
143
Figure 3-4. Close-packed bidimensional arrangement of equal spheres. The A, B, C coding used to indicate different relative positions is shown.
point we may consider Figs. 3-4 to 3-6 where typical simple close-packed structures are shown and presented as built from the superimposition of close-packed atomic layers. If spheres of equal sizes are packed together as closely as possible on a plane surface they arrange themselves as shown in Fig. 3-4. (Their centers are in the points of a triangular net.) Each sphere is in contact with six others. Such layers may be stacked to give three-dimensional close packed arrays. If we label the positions of the (centers of the) spheres in one layer as A, then an identical layer may be superimposed on the first so that the centers of the spheres of the second layer are vertically above the positions B (for two layers, it is insignificant whether we choose the positions B or the equivalent position C). When we superimpose a third layer above the second (B) we have two alternatives: the centres of the spheres may be above either the A or the C positions. The two simplest sequences of layers correspond to the superimpositions ABABAB... and ABCABCABC... (more complex sequences may of course be considered). The sequence ABAB..., corresponding to the socalled hexagonal close-packed structure (Mg-type structure) is shown in Fig. 3-5. The sequence ABCABC... having a cubic
144
3 Structure of Intermetallic Compounds and Phases
— \j±^uy^Ls b) I II in Figure 3-5. Hexagonal close-packing, (a) A few spheres of three superimposed layers are shown. The spheres of layers III are just above those of the first one. (b) Lateral view of the same arrangement. The stacking symbols corresponding to the Mg unit cell description (reported in Sec. 3.5.2.2) are shown. (The .. .BCBCBC... sequence is identical to a ... ABABAB... or ... CACACA... symbol.) The heights of the layers are reported as fractions of the repeat unit along the z axis of the hexagonal cell (distance between 0 and 1).
b)
c)
Figure 3-6. Face-centered cubic close-packed structure of equal spheres, (a) Sphere-packing: a group of eight cubic unit cells is shown, (b) A section of the same structure shown in (a) is presented. The typical arrangement of layers similar to that shown in Fig. 3-4 is evidenced, (c) A lateral view of the stacking of the layers in the f.c.c. structure is presented. The layer positions along the superimposition direction (which corresponds to the cubic cell diagonal) are shown as fractions of the repeat unit.
symmetry, is shown in Fig. 3-6. It is the cubic (face-centered cubic) close packed structure (also described as cF4-Cu type structure). A more complete representation of different layer sequences (which can be used not only for the description of close packed structures) may be obtained by using stacking symbols such as those shown in Fig. 3-7, together with layer stacking indications. Figure 3-7 a shows a network of atoms which can be considered as a triangular net, T net, that is 3 6 net. This symbol, the Schlafli Symbol PN, describes the characteristics of each node in the network, that is the number N of P-gon polygons
surrounding the node. In the reported net all the nodes are equivalent: their polygonal surrounding corresponds to 6 triangles. In the case of the simple 3 6 triangular net the mentioned stacking symbols A, B, C, as can be seen in Fig. 3-7 c relate the positions of the nodes to the origin of the cell defined in Fig. 3-7 b. In the layer stacking sequence full symbol, the component atoms occupying the layers are written on the base line, with the stacking symbols as exponents and the layer spacings given sub as the fractional height of the repetition constant along the direction perpendicular to the layers. In the case of Mg, for instance, with reference to the standard
145
3.3 Crystal Structure of the Intermetallic Phase and Its Representation -o—o—o
/\ O A/\ O
AA/V AAAA
o—o—o—o—o
a)
o
o o
o o
/o\ o
Figure 3-7. Triangular net of points, (a) and (b): the 3 6 net and the corresponding (bidimensional) cell are shown, (c) Different point positions (relative to the cell origin) and corresponding coding: A: the representative point is in 0,0; B: the representative point is V3,2A; C: the representative point is in 2/3, Vs.
choice of the unit cell origin (two equivalent atomic positions for the two Mg atoms in V3, %, V* and 2/3, V3,3/i), the symbol will be Mg? /4 Mg3 /4 (which, with a zero point shift, is equivalent to MgoMg?/2). The symbol CuoCui /3 Cu2 /3 , on the other hand, represents the Cu structure as a stacking sequence viewed along the direction of the unit cell diagonal. A few other nets, based on the hexagonal cell, are of frequent structural occurrence. Following Pearson's suggestions, the corresponding sequence of stacking symbols which has a wide application are here presented. Figure 3-8 shows the hexagonal (honeycomb) net (H net) and the symbols used for relating the different positions of the nodes to the cell origin. (Notice that two nodes are contained in the unit cell.) Figure 3-9 shows the three-ways bamboo weave net, known as kagome, a net of triangles and hexagons (K net, the 3636 net of points). The different positions of the nodes (three nodes in the unit cell) are represented by the symbols shown in Fig. 3-9 b. Several (especially hexagonal, rhombohe-
Figure 3-8. Hexagonal (63) net of points. The net is shown in (a). In (b) the different positions of the points in the unit cell are indicated with the stacking symbols a, b, c. (Notice that the unit cell contains two points.)
-o"—b—o'—bx/
\ /
\ ,
o
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a)
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'\ O
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o o o o o o\/o O
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o/ o \O/
o o o o o o o o o o o o b)
a
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Figure 3-9. The 3636 (kagome) net of points. The net is shown in (a). In (b) the different positions (relative to the hexagonal cell origin) are indicated by the stacking symbols a, P, y (three points of the net are contained in the unit cell).
146
3 Structure of Intermetallic Compounds and Phases
dral and cubic) structures may be conveniently described in terms of stacking triangular, hexagonal and/or kagome layers of atoms. Examples will be given in the following paragraphs. The specification of the spacing between the layers is useful in order to compare different structures, to recognize the close-packed ones (A, B, C symbols with appropriate layer distances) and to deduce atomic coordinations. We have to notice, however, that the A, B, C notation previously described is not the only one devised. Several different symbols have been suggested to describe stacking patterns. (For a description of the more frequently used notations see Parthe, 1964, Pearson, 1972). A very common notation is that by Jagodzinski (1954). This notation involving h and c symbols is applicable only to those structure type groups which allow not more than three possible positions of the unit layer (or more generally of the "unit slabs". See Sec. 3.4.3 on polytypic structures). The h, c notation cannot therefore be applied, for instance, to disilicide types. The letters h and c have the following meaning: The letter h is assigned to a unit slab, whose neighboring (above and below) unit slabs, are displaced sideways, in the same direction for the same amount: for instance
ABABA hhh
or
CBCBCB hhhh
(h comes from hexagonal: this is the stacking sequence of the simplest hexagonal structures such as hP2-Mg, hP4-ZnS wurtzite and hP12-MgZn 2 types). The letter c, on the other hand, is assigned to unit slabs, whose neighboring slabs, have different sideways displacements: for instance
ABCABC cccc
or
CBACBA cccc
(c comes form cubic: this is the stacking sequence found in cubic structures such as cF4-Cu, cF8-ZnS sphalerite and cF24Cu 2 Mg types). To denote the stacking sequence of the different structures it is sufficient to give only one identity period of the h, c symbol series. For instance: cF4-Cu, c (instead of ABC), cF8-ZnS sphalerite, c; hP4-ZnS wurtzite, h; hP4-La, he; hR9-Sm, hhc. As can be seen from the previously reported examples, the identity period of the h, c symbols is generally shorter than the A, B, C... letter sequence. The h, c... symbols may be condensed like hcchcchchc to (hcc)2(hc)2. (If the number of c letters in a Jagodzinski symbol are divided by the total number of letters one obtains the percentage of "cubic stacking" in the total structure.) Another common notation for describing stacking of close-packed 3 6 nets (T nets) is that devised by Zhdanov (1945) (which is a number notation equivalent to the Jagodzinski's notation). A short description of the Zhdanov symbol may be the following. A ' + ' is assigned if the order between a layer and its previous partner follows the sequence corresponding any two subsequent layers in the face-centered cubic type structure, that is A -• B, B -»C, C ->A. Otherwise a ' —' is assigned. For instance, the sequence ' + + -\ ' (shortened 33) corresponds to ABCACBA. Finally as another simple example of description (and symbolic representation) of structures in terms of layer stacking sequence we may now examine structures which can be considered as generated by layer networks containing squares. A typical case will be that of structures containing 4 4 nets of atoms (square net: S net). The description of the structures will be made in term of the separation of the different nets (along the direction perpendicular to their
3.3 Crystal Structure of the Intermetallic Phase and Its Representation I
I
-O
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147
I
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I III I
a)
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2-o—o—o—o—o—oI I I I I I o o o o o O O [—O—i o o p d o o o LOJ
o
o
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o
o
o
o
o
o
o
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b) o
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o
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o °\f>
°
°
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o <(o cy o
o o o o o o o o o 5
6
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o o
cN'o o 7
plane) and of the origin and orientation of the unit cell. Figure 3-10 shows the different symbols (in this case numbers) suggested by Pearson (1972) which will be used to indicate origin and orientation of the nets. These numbers will be reported as exponents of the symbols of the atoms forming the different nets. In this case, too, the relative height of the layers will be indicated by a fractional index. A few symbols of square net stacking sequences are the following: POQ*. the simple cubic cell of Po (containing 1 atom in the origin) corresponds to a stacking sequence of type 1 square nets. WJW\ l2 : the body-centered cubic structure of W (1 atom in 0,0,0 and 1 atom in VijVijVi) corresponds to a sequence of type 1 and type 4 square nets at the heights 0 and Vi, respectively. For more complex polygonal nets, their symbolic representation and use in the de-
Figure 3-10. A square (44) net is shown in (a), (b) Different positions of the representative point in the unit cell are presented and coded (net of points aligned parallel to the cell edge), (c) Codes used for different positions of a square net of points referred to a cell with axes at 45° to the net alignment (and edges equal to Jl times the repeat unit of the net).
scription, for instance, of the Frank-Kasper phases, see Frank and Kasper (1958) and Pearson (1972). (Brief comments on this point will be reported in Sec. 3.5.2.6.) 3.3.5.3 Polyhedra Assembling
A complementary approach to the presentation and analysis of the intermetallic phase structures consists of their description with coordination polyhedra as building blocks. A classification of types of intermetallic structures based on the coordination number, configurations of coordination polyhedra and their method of combination has been presented by Kripyakevich (1963). According to Kripyakevich a coordination polyhedron of an atom is the polyhedron, the vertices of which are defined by the atoms surrounding this atom: a coordination polyhedron should have a form as
148
3 Structure of Intermetallic Compounds and Phases
close as possible to a sphere, that is, it should be convex and have the maximum number of triangular faces. At the vertices of a coordination polyhedron of a given atom (in addition to atoms of different elements) there can also be atoms of the same kind. A considerable variety of coordination polyhedra exists. In some cases, plane coordination polygons have to be considered. The number of vertices may vary from, say, 3 to 24. Generally, the structure consists of atoms with different coordination numbers; according to Kripyakevich, structures are most conveniently classified according to the type of coordination polyhedron of the atoms with the lowest coordination number. (For a general approach to the classification of atomic environment types in terms of coordination polyhedra see also Chap. 3.6.2.6.) An important contribution to the structure analysis of intermetallic phases in terms of the coordination polyhedra has been carried out by Frank and Kasper (1958). They presented several structure types as the result of the interpenetration of a group of polyhedra, which give rise to a distorted tetrahedral close-packing of the atoms. (The Frank-Kasper structures will be presented in Sec. 3.5.2.6.) In particular, Samson (1967,1969) developed the analysis of the structural principles of intermetallic phases having giant unit cells. These structures have been described as arrangements of fused polyhedra rather than the full interpenetrating polyhedra (see a short description in Sec. 3.5.2.6). The principles of describing structures in terms of polyhedra-packing has been considered by Girgis and Villars (1985). To this end they consider, in a given structure, the coordination polyhedra of all the atomic positions; structures are then described by packing the least number of polyhedra
types. All the atoms in the unit cell are incuded in the structure-building polyhedra. The polyhedra considered should not penetrate each other. Structures are then classified mainly on the basis of the following points: - Number of polyhedra types employed in the description of the structure, - characteristics of the polyhedra (number of vertices, symmetry), - types of polyhedra packing (either tridimensional distribution of discrete polyhedra sharing corners, edges or faces, or layerlike distribution of polyhedra. As examples of structures described by packing of one polyhedron type we may mention: cP4-AuCu3 type, tridimensional arrangement of cubooctahedra (CN12); tP30-aCr,Fe-type, layer-like arrangement of icosahedra (CN 12). For a general approach to the problem of structure descriptions in term of polyhedra assembling a paper by Hawthorne (1983) should also be consulted. The following hypothesis is proposed: crystal structures may be ordered or classified according to the polymerization of those coordination polyhedra (not necessarily of the same type) with the higher bond valences. The linkage of polyhedra to form "clusters" is then considered from a graphtheoretic point of view. Different kinds of isomers are described and their enumeration considered. According to Hawthorne, moreover, it has to be pointed out that many classifications of complex structures recognize families of structures based on different arrangements of a fundamental building block or module (see the subsequent paragraph on recombination structures). If this building module is a tightly bound unit within the structure it could be considered, for instance, as the analogue of a molecule in an organic structure. Such
3.3 Crystal Structure of the Intermetallic Phase and Its Representation
modules can be considered the basis of structural hierarchies that include, for instance, simple and complex oxides and complex alloy structures. These modules may be considered as formed by polymerization of those coordination polyhedra that are most strongly bonded and may be useful for a classification and systematic description of crystal structures.
3.3.5.4 Bauverband (Connectivity Pattern) Approach
As summarized in the comprehensive report on the nomenclature of inorganic structure types by Lima-de-Faria et al. (1990) the so-called Bauverband terminology gives a short informative description of crystal structures. The term Bauverband (introduced by Laves, 1930) may be translated as connection pattern, construction pattern, or framework. It may be defined as an arrangement of points in the 3-dimensional periodic space occupied by atoms; it represents a connectivity pattern typical for a given structure and represents (or approximates) a sphere packing with typical selfcoordination. The arrangement of points in a space group with specified size of the unit cell may be desribed by the parameters of one point position (homogeneous Bauverband) or by the parameters of two or more independent point positions (heterogeneous Bauverband). A Bauverband may be described by a combination of applicable, original or transformed invariant lattice complexes. {A lattice complex is the set of all point configurations that may be generated within one type of Wyckoff sets. A set configuration, or crystallographic orbit, is the infinite set of all points that are symmetrically equivalent to a given point with respect to a space group [see International
149
Tables for Crystallography. Hahn (1989) and Chap. 1 of this Volume]. The same lattice complex may occur in different spacegroup types. The number of degrees of freedom of a lattice complex, normally, is the same as that of any of its Wyckoff positions which is the number of coordinate (free) parameters that can vary independently. According to its number of degrees of freedom a lattice complex is called invariant, uni-, bi-, or trivariant.} The invariant lattice complexes in their characteristic Wyckoff positions are represented mainly by capital letters. Those with equipoints at the nodes of the Bravais lattice are designated by their appropriate lattice symbols. (Lattice complexes, from different crystal families that have the same coordinate description for their characteristic Wyckoff positions, receive the same symbol: for instance, lattice complex P corresponding to coordinate 0,0,0. In such a case, unless it is obvious from the context which lattice is meant, the crystal family may be stated by a small letter, preceding the lattice-complex symbol as follows: c = cubic, t = tetragonal, h = hexagonal, o = orthorhombic, m = monoclinic, a = anorthic = triclinic.) Other invariant complexes are designated by letters that recall some structural features of a given complex, for instance D from the diamond structure, E from the hexagonal closepacking. Examples of two-dimensional invariant complexes are G (from graphite layer) and N (from kagome net). A short list of invariant lattice complex symbols is reported in the following. [For a complete list see Chap. 14 in Volume A of the International Tables of Crystallography; Hahn (1989).] For the Bauverbande in the cubic structure types (their symbols and their use in a systematic classification and description of cubic structures) see Hellner (1979).
150
3 Structure of Intermetallic Compounds and Phases
- Lattice complex P: (multiplicity, that is the number of equivalent points in the unit cell, 1); coordinates 0,0,0; (crystal families: c, t, h, o, m, a). - Lattice complex I: (multiplicity 2); coordinates 0,0,0; V2, V2, V2; (crystal families: c, t, o). - Lattice complex J: (multiplicity 3); coordinates 0, V2, V2; V2,0, V2; V2, V2,0; (crystal families: c). - Lattice complex F: (multiplicity 4); coordinates 0,0,0; 0, V2, V2; V2,0, V2; V2, V2,0; (crystal families: c, o). - Lattice complex D: (multiplicity 8); coordinates 0,0,0; 1 / 2 , 1 / 2 ,0; V2,0,V2; O,1/.,1/.; 1/4,1/4,1/4; 3/4,3/4,V4; V^V^A\ V 4 , 3 / 4 ,%; (crystal families: c, o). - Lattice complex W: (multiplicity 6); coordinates 7 4 ,0, V2; V2, 'A, 0; 0, V2, V4; 3 A, 0, V2; V2,3/4,0; 0, V2,3/4; (crystal families: c). - Lattice complex T: (multiplicity 16); coordinates Vs, Vs, Vs; Vs, Vs, %; Vs, Vs, Vs; Vs, Vs, Vs; Vs, Vs, Vs; Vs, Vs, Vs; Vs, Vs, Vs; 3 7 /8, /8,V8; 3/8,5/8,7/s; 3/8,ys,3/8; 7/8,5/8,3/8; 7 /8,V8,7/8; 5/8,7/8,3/s; V8,3/8,7/8; Vs,1/*,1/*; Vs, Vs, Vs; (crystal families: c, o). - Lattice complex E: (multiplicity 2); coordinates V3,2/3, ¥4; 2/3,V3,3/4; (crystal families: h). - Lattice complex G: (multiplicity 2); coordinates V3, %, 0; 2/3, V3,0; (crystal families: h). - Lattice complex R: (multiplicity 3); coordinates 0,0,0; V3, %, 2/3; 2/3, V3, V3; (crystal families: h). The coordinates indicated in the reported (partial) list of invariant lattice complexes correspond to the so called "standard setting". Some of the non-standard settings of an invariant lattice com-
plex may be described by a shifting vector (defined in terms of fractional coordinates) in front of the symbol. The most common shifting vectors have also abbreviated symbols: P' represents V2 V2 V2 P (coordinates V2, V2, V2), J' represents V2 V2 V2 J (coordinates V2,0,0; 0, V2,0; 0,0, V2; F" represents X A Vi Vi F (coordinates %, %, XA; V4,%, 3A; %, %, 3A; 3 /4,%, %) and F'" represents 3 3 A A % F. [The following notation is also used J* = J + J' (complex of multiplicity 6); W* = W + W (multiplicity 12).] It can be seen, moreover, that the complex D corresponds to the coordinates F + F". Simple examples of structure descriptions in terms of Bauverbande may be: CsCl type P + P (Cs in 0,0,0; Cl in 1 /2,1/2,1/2); NaCl type structure: F + F ; ZnS type structure: F + F".
33.5.5 An Exercise on the Use of Alternative Structure Notations In the following, data concerning a few selected structures, will be presented. In this paragraph, by using a simple structural type (cP4-AuCu3, 3 i 3 [Au 6/6 ][Cu 8/8 ] 12/4 , or, in more detail, 333[Au6o/6o][Cu8p/8p]12co/41) a presentation will be given on the different ways of describing the structure. AuCu3 is primitive, cubic. The space group is Pm3m [N. 221 in the International Tables for Crystallography, Hahn (1989)]. In the unit cell there are 4 atoms in the following positions: 1 Au in a) 0,0,0; 3 Cu in c) 0, V2, V2; V2,0, V2; V2, V2,0. Several phases are known which have this structure; in the Villars and Calvert compilation (1985) there are around 390 listed: 2.2% of all the reported phases. This structural type is the 6th in the frequency rank order. A short selection is presented in the following list:
151
3.3 Crystal Structure of the Intermetallic Phase and Its Representation
HfPt 3 Lain 3 La 3 ln Mn3Pt MnZn3 Pt 3 Al Ti 3 Hg TiZn 3 UPb3 YA13 Y3A1
a a a a a a a a a a a
= 398.1 pm = 473.21 pm = 508.5 pm = 383.3 pm = 386 pm = 387.6 pm = 416.54 pm = 393.22 pm = 479.3 pm = 432.3 pm = 481.8 pm
(Note that, in this structure type, in some cases, according to the phase stoichiometry, the same element may occupy, either the a) or the c) position.) In the reported list the unit cell edge has been included. In the following, while discussing the characteristics of this structural type, we will consider the data referring to the prototype itself (a = 374.84 pm). The structure is shown in Fig. 3-11, where the tridimensional sequence of the atoms is suggested by presenting a small group (eight) of contiguous cells. The unit cell itself is shown in Figs. 3-12 a and 312 b, by using two different drawing styles. The subsequent Figs. 3-13a to 3-13d correspond to an analysis of the structure carried out in order to show the different local atomic arrangements (coordinations around the atoms in the two crystal sites). In the analysis of a structure, however, it is also necessary to take into consideration the values of the interatomic distances. It may be useful to consider both absolute and so called "reduced" values of the interatomic distances. In the case of the AuCu 3 phase, the minimum interatomic distance corresponds to the A u - C u distance (Au in 0,0,0 and Cu in 0, V2, V2) which is the same as the C u - C u distance between Cu in 0, V2, V2 and Cu in V2,0, V2. This distance is given by a
Cu Au
o
Figure 3-11. cP4-AuCu3 type structure. A group of eight cells is shown. The light spheres represent Au atoms. In order to get a better view of the atoms inside, the atomic diameters are not to scale.
b)
L.J
o
c) Figure 3-12. The cP4-AuCu3 unit cell is presented in two different drawing styles. In (a) an (approximate) indication of the packing and space filling is given. In (b) the positions of the different atoms are reported in a perspective view of the unit cell and (c) in some sections of the same. (Notice the square net arrangement.)
152
3 Structure of Intermetallic Compounds and Phases
r /
a)
x
J
y
b)
c)
d)
Figure 3-13. cP4-AuCu3 type structure. Different fragments of the structure (generally of a few unit cells) are presented in order to show the various typical coordinations. (Cu atoms are represented by small spheres.) (a) Au - 6Au (octahedral); (b) Au - 12Cu (cuboctahedral); (c) Cu - 8Cu; (d) Cu - 4Au (square). The 8Cu + 4Au at the same distance from Cu form a heterogeneous cuboctahedron. (Compare also with Fig. 3-20.)
For the AuCu 3 phase a = 374.84 pm, and, therefore, dmin = 265.1 pm. This value could be compared, for instance to the value 272 pm, sum of the radii of Cu and Au (as defined for a coordination number of 12) or to the value 256 pm (Cu atom diameter). Reduced interatomic distances (dr = d/dmin) may be defined as the ratios of the actual distance values to the minimum value. A first set of interatomic distances (and coordination) which can be considered in the AuCu 3 phase is that corresponding to the Au coordination around Au atoms (see Fig. 3-13 a): Considering as the reference atom, the atom Au in 0,0,0, the next neighbors Au atoms are the six Au shown in Fig. 3-13 a, having the coordinates 0,0,1; 0,0, T; 0,1,0; 0,1,0; 1,0,0; 1,0,0; all at a distance d = a = 374.8 pm, corresponding to a reduced distance d/dmin = 1.414. In the same group of Au-Au interatomic distances a subsequent set is represented by distances such as those between Au 0?050 , and A u o j ? 1 (or A u M f i , A u o ? l j , Au1>0>1, etc.). This set corresponds to a group of 12 atoms (all at an absolute distance of a^Jl = 530.1 pm, that is, at a reduced distance dT = d/dmin = 2.000).
A compact representation of these data is given by means of the bar-graph in Fig. 3-14 a. A second set of interatomic distances (and coordination) corresponds to the Cu coordination around Au atoms: Considering as the reference atom, the atom Au in 0,0,0, the next neighbors Cu atoms are the 12Cu reported in Fig. 3-13b, i n t h e coordinates: O.Vi^/i; 0, V2,%;
Vi.ViA Vi^/iA all at a distance d = ay/l/l = 265A pm, corresponding to a reduced distance d/dmin = 1.000. Considering also the subsequent sets of Au-Cu distances (24 atoms at d = 459A pm,
dr = d/dmin = yfe = 1.132,
24 Cu at d = 592.7 pm, dT = 2.236, etc.) we obtain the histogram reported in Fig. 3-14b. A third group of interatomic distances (and coordination) which has to be considered is that corresponding to the Au coordination around Cu atoms (see Fig. 3-13 d). Considering as the reference atom one of the three equivalent Cu atoms in c), for instance, the Cu atom in 0, V2, V2, the next neighbor Au atoms are 4 Au in 0,0,0;
3.3 Crystal Structure of the Intermetallic Phase and Its Representation
153
(reduced distance d/dmin = 1.732), to a group of 8 Au (in coordinates such as 0,0,1; 0,1,1; 0,0,2; etc.), at a distance d = 592.7 pm, d/dmin = 2.236, etc. The corresponding coordination bar-graph is presented in Fig. 3-14 c. The fourth (and last) type of interatomic distances (and coordination) characteristic of the AuCu3 structure is given by the Cu coordination around Cu atoms (see Fig. 3-13 c): Considering as the reference atom, the atom Cu in V^V^O, the next neighbors the Cu atoms are the 8 Cu atoms in 1 / 2 ,0,%; 0,y2_//2; l, 1 /^ 1 / 2 ; VU,1^ 1 1 1 1 1 1 V^O, /,; O, /., /.; I, /!, /!; ViX /* all at a distance d = a y/l/2 = 265.1 pm, corresponding to a reduced distance d/dmin = 1.000.
Figure 3-14. cP4-AuCu3 type structure. Coordinations and distances. For each type of coordination the numbers (N) of near-neighbor atoms are plotted as a function of their distances from the central atom. (Relative values of the distances, d/dmin, have been used, where
0,0,1; 0,1,0; 0,1,1, respectively; all at a distance d = a yJl/2 = 265.1 pm, corresponding to a reduced distance d/dmin = 1.000. (The coordination, according to the symbols given in Table 3-5 a, is shown as 4' in Fig. 2-3 of Chap. 2.) Subsequent sets of Cu-Au distances correspond to a group of 8 Au atoms (in coordinates such as 1,0,0; 1,0,1; 1,1,0; etc.) at a distance d = 459.1 pm
The subsequent set of Cu-Cu distances corresponds to a group of 6 Cu atoms (in coordinates such as 1, V2, V2; T, V2, V2; 0, V2,3/2, etc.) at a distance d = 374.8 pm (reduced distance dr = d/dmin = 1.414). Subsequent Cu-Cu distance sets correspond to 16Cu atoms at d = 459.1 pm (dT = 1.732), 12 Cu atoms at d = 530.1 pm (dT = 2.000), 16Cu atoms at 592.7 pm (dr = 2.236), etc. The corresponding histogram is presented in Fig. 3-14d. Lists of coordinating atoms (with distances from the reference atom), coordination polyhedra, and next-neighbor histograms are presented systematically by Daams et al. (1991). Like them, we use here the shortest interatomic distance observed for each coordination set. As a conclusion to the description of the different coordinations we may observe that those corresponding to the first distance sets are summarized in the symbol T[A 6 / 6 ][B 8 / 8 ] 1 2 / 4 (T[Au 6 / 6 ][Cu 8 / 8 ] 1 2 / 4 for the prototype). In terms of polyhedra packing, therefore, this structure may be described as a tridimensional arrangement of cubooctahedra (see Sec. 3.3.5.3).
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3 Structure of Intermetallic Compounds and Phases
3.4 Relationships Between Structure Types (Structure Families)
Figure 3-15. cP4-AuCu3 type structure. The unit cell is viewed along its diagonal. (Au atoms white, Cu black.)
Figure 3-15, on the other hand, shows how for the same structure, alternative descriptions (layer stacking sequence descriptions) may be obtained and, according to Pearson, symbolized. In this figure the structure (viewed along the cube diagonal) is presented as a stacking sequence of triangular and kagome nets. It corresponds to the symbol Au^CuSAu^Cu^Auf^Cuj^. [In the symbol we have the same number of triangular (A, B, C) Au atom nets and of kagome (a, (3, y) Cu atom nets. These two net types are characterized by the presence of 1 and 3 points in the unit cell (see Figs. 3-7 and 3-9). This, of course, corresponds to the total 1:3 stoichiometric ratio.] The same structure, viewed along the unit cell edge direction, corresponds to a square net stacking sequence (see Fig. 3-12). The stacking symbol is AuoCuoCuf/2. With reference to the Bauverband terminology, we may finally note that the AuCu 3 type structure corresponds to P + J complexes. [According to Hellner (1979) this structure may be considered as pertaining to a F-family as a consequence of a particular splitting of the points of the F complex.] A few other comments on the AuCu 3 type structure and some remarks on the relationship with other structural types will be reported in Sec. 3.5.2.2.
The structures corresponding to different types may often be interrelated on the basis of some transformation schemes. These schemes can be used as criteria for classifying structure types and showing structural relationships. A few selected groups of interrelated structural types will be presented in the following paragraphs. 3.4.1 Degenerate and Derivative Structures (Defect, Filled-Up, Derivative Structures) An important and general scheme of structure transformation and interrelation is that described, for instance, by Pearson (1972), by means of the concept of derivative structures and degenerate structures. A derivative structure can be considered to be obtained from a reference structure by ordered atomic substitution, subtraction or addition processes (superstructure formation) or by unit cell distortions (or both). The opposite kinds of transformation correspond to the so-called degeneration processes. A derivative structure has fewer symmetry operations than the reference structure (a degenerate has more). A derivative structure has either a larger cell or a lower symmetry (or both) than the reference structure. It is possible, for instance, that a set of equipoints of a certain structure (considered as the reference structure) has to be subdivided into two (or more) subgroups in order to obtain the description of another (derivative) structure. The structure of the Cu type (cF4-type), for instance, corresponds to 4Cu atoms in the unit cell, placed in 0,0,0; V 2 ,7 2 ,0; V2,0, V2; 0, V2, V2, whereas in the cP4-AuCu3 type structure the same atomic sites are subdivided into
3.4 Relationships Between Structure Types (Structure Families)
two groups with an ordered distribution of the two atomic species (1 Au atom in 0,0,0, and 3Cu atoms in 1/2,1/2,0; ViAVi; 0,1/2,1/2). The AuCu3 type structure can, therefore, be considered as a derivative structure of the Cu type. On the other hand, if we consider the AuCu3 type as the reference structure, we may describe the Cu type as a degenerate structure. For several phases having the AuCu3type structure (and for the prototype itself) the transformation from one structure to the other corresponds to a real process. At low temperature an alloy with the AuCu 3 composition has the ordered cP4-AuCu3 type structure; at higher temperature the two atomic species are, statistically, equally distributed in the four atomic sites which become equivalent: the structure degenerate in the cF4-Cu type. The mentioned subdivision of a set of equipoints into two or more groups may, in some cases, lead to an increase of the unit cell size (formation of a multiple cell). An example may be the structure of the MnCu2Al type (see Fig. 3-19 and Sec. 3.5.2.1) which can be considered a derivative structure (superstructure) of the cI2-W type (or of the cP2-CsCl type) structure. Derived structures may also be formed with the ordered introduction of vacant sites. As an example we may consider the hP3-CdI 2 type structure (see Sec. 3.5.2.5) which can be related to the hP4-NiAs type structure in which the set of equivalent points 0,0,0 and 0,0, Vz is considered as being subdivided into two groups (each of 1 site) 0,0,0, (occupied by 1 atomic species) and 0,0, V2 (vacant). We can, therefore, regard the hP3-CdI 2 type structure as a defect derivative form of the hP4-NiAs type. Similar considerations may be extended to include (besides substitution and subtraction) ordered addition of atoms. In this case stuffed or filled-up derivative struc-
155
tures are considered in which extra-atoms have been added in an ordered way, on sites unoccupied in the reference structure. An example may be the hP6-Ni2In structure, which is a stuffed derivative structure of the previously mentioned NiAs structure. As a comment to these observations we have also to mention that frequently structural distorsions (axial ratio and/or interaxial angle variations) accompany the formation of derivative structures (especially because of the ordered distribution of atoms of different sizes or of vacant sites). 3.4.2 Antiphase Domain Structures
A special case of superstructures may now be considered. A typical example can be observed in the oI40-AuCu(II) type structure (Fig. 3-16 and Sec. 3.5.2.2). We have first to mention that ordering of the Au-Cu face-centered cubic (cF4-Cu type) solid solution, having a 50-50 atomic composition, distributes Cu and Au atoms alternatively on two layers, resulting in a tetragonal structure, tP2-AuCu(I) with the c
Figure 3-16. AuCu type structures, (a) AuCu(I) type structure. Both the tP2 cell (a and c edges) and a tP4 pseudocell (a' and c edges) are shown, the latter one is for easier comparison with the cF4-Cu structure, (b) oI40-AuCu(II) type structure.
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3 Structure of Intermetallic Compounds and Phases
axis perpendicular to the layers (see Fig. 3-16a). The more complex structure, oI40AuCu(II) type, is obtained by a longperiod ordering which results in an orthorhombic cell containing 10 (slightly distorted) AuCu(I) pseudocells (Fig. 3-16b). This ordering corresponds to a periodic shift (every 5 cells along the orthorhombic b axis) of the structure by Vi {a' + c) in the a\ c plane. This out-of step shift corresponds to a "so-called" antiphase boundary. An antiphase domain may correspondingly be defined; in this case it contains 5 AuCu(I) type pseudocells. Several examples of onedimensional long period structures found in 1:1 and 1:3 alloys and of two-dimensional long period structures (characterized by two different domain periods and two steps-shifts) found in 1:3 alloys have been presented by Pearson (1972); the role of the valence-electron concentration in defining the superstructure period has also been discussed. A general presentation of several antiphase-boundaries (not only planar, but also cylindrical) and related structure groups may be found in the book of Hyde and Andersson (1989). 3.4.3 Homeotect Structure Types (Polytypic Structures) According to Parthe (1964), two different structure types of the same formula Xm Yn are called homeotect structure types, if every X atom has the same number of nearest X neighbors and the same number of nearest Y neighbors, and, conversely, if every Y atom has the same number of nearest X and Y neighbor atoms. It is possible that several structure types show this feature. All the different structure types of equal composition, which have (for corresponding atoms) the same kind of surroundings, form a set of homeotect structure types (the
term polytypic structures is also used to denote the relationships observed with homotect structures). According to Parthe (1964) all structure types which belong to a homeotect set can be described as different stacking variants of identical structural unit slabs (minimal sandwiches). All structure types of a set are constructed by stacking identical unit slabs one on top of another. The various types differ only in the relative horizontal displacement of these units. (The vertical unit cell edges of the different types are integer multiples of a common unit which is the height of the unit slab characteristic for the homeotect structure type set.) All structure types which belong to a homeotect set have the same space-filling curve (see Sec. 3.6.2.2). A few important examples of groups of homeotect structure types will be described in the following paragraphs. A short index of the same is the following list (in which the Jagodzinski-Wyckoff notation of the stacking pattern has been inserted): - Close packed element structure types (see Sec. 3.5.2.2): Mg-type (h), Cu-type (c), La-type (he), Sm-type (hhc). - Equiatomic tetrahedral structure types (carborundum structure types) (see Sec. 3.5.2.3): wurtzite-type (h), sphalerite-type (c), SiC polytypes [he, hec, hece, heche, ... (hcc)5(hccc)(hcc)5hc ... (hchcc)17(hcc)2, ... (hcc)43hc ...]. - Laves phases (see Sec. 3.5.2.6): hP12 MgZn2-type (h), cF24 Cu2Mg-type (c), hP24 Ni2Mg-type (he), Laves polytypes (hhc, hhece, etc.). Other important sets of homeotect structure types are those related to disilicide structure types (MoSi2, CrSi 2 , etc.), cadmium halide structure types, etc. (see Parthe, 1964; Hyde and Andersson, 1989). From a general point of view polytypism may be considered a special case of polymorphism: the two-dimensional transla-
3.4 Relationships Between Structure Types (Structure Families)
157
tions within the layers are (essentially) preserved whereas the lattice spacings normal to the layers vary between polytypes and are indicative of the stacking period (Guinier et al., 1984). As evidenced by Zvyagin (1987) we may distinguish various forms of polytypic structures, including (besides close packing of like and unlike atoms) polytypes of tetrahedral, octahedral and prismatic layers packed according to the laws of closest packings. Complex silicate structures, for instance, may be considered which are characterized by much variety in the orientations and displacements of the layers and also structures in which two-dimensional layers are conjoined with one-dimensional band and island groups. The mentioned papers (Guinier et al., 1984; Zvyagin, 1987) contain also suggestions and recommendations on the nomenclature and on the symbolism for use in the general case in either simple or complex polytypic structures.
groups). In these phases, along the c axis, the unit cell (superstructure cell, supercell) contains n pseudocells of T atoms and m interpenetrating pseudocells of X atoms. These phases (Nowotny phases or "chimney-ladder" structures) contain rows of atoms X (the ladder), with variable interatomic spacing from one compound to another, which are inserted into channels (chimneys) in the T array. The T metals in all of the superstructures form a (3-Sn-like array with the number of T metal atoms in the formula of the compound corresponding to the number of (3-Sn-like pseudocells stacked in the c direction of the supercell (see Sec. 3.5.2.3). The arrangement of the atoms in these phases can be compared to that found in the structure of TiSi2. Following is a list of some chimney-ladder phases (phases containing as many as 600 atoms in the unit cell have been described):
3.4.4 "Chimney-Ladder" Structures (Structure Commensurability, Structure Modulation)
The Ru atoms form a (3-Sn-like array with two pseudocells along the c direction of the supercell.
In the cases of ordered alloys, described in the foregoing paragraphs, long period structures were considered in which the near-neighbor coordination of the atoms remains essentially unchanged between one structural modification and another. More complex cases can, however, be considered. As an introduction to this point, we may remember that it is often convenient to describe structures as consisting, for instance, of two interpenetrating substructures (two different atom sets). An interesting group of phases T n X m may be considered which are tetragonal and are formed between transition metals T and p-block elements X (of the Ga and Si
tP20 Ru 2 Sn 3 [a = 617.2 pm, c = 991.5 pm, c/{2 (1-^/2) = 0.568]
tP32 Ir 3 Ga 5 [a = 582.3 pm, c = 1420 pm, c/(3 a-y/2) = 0.573] tP36 Ir 4 Ge 5 [a = 561.5 pm, c = 1831 pm,
tP192-V 17 Ge 31 [a = 591 pm, c = 8365 pm,
c/(17a'y/2) = 0.589] (In V 17 Ge 31 , for instance, there are 17 P-Sn like pseudocells of V atoms and 31 Ge pseudocells stacked along the c axis.) The atomic arrangements in a few chimney-ladder phases are shown in Fig. 3-17 and compared with that found in TiSi2. (This structure corresponds to the orthorhombic cell oF24-TiSi2-type with
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3 Structure of Intermetallic Compounds and Phases
a0 = 825.2 pm, b0 = 478.3 pm, c0 = 854.0 pm. It can be approximately described in terms of a smaller body-centered tetragonal pseudocell, shown in Fig. 3-17 a, having
a' « ajy/l « c0 yjl\ d = b0 and c'/a' «
a)
b)
c)
d)
Figure 3-17. Nowotny phases, chimney-ladder structures (Jeitschko and Parthe, 1967). (a) The reference oF24-TiSi2 type structure presented in terms of the tetragonal pseudo-cell (12 atoms in the pseudocell). (b) tP120-Mn1]LSi19; (c) tP20-Ru2Sn3 and (d) tI156Rh 17 Ge 22 phases. (Notice that the metal atoms, black points, form sequenes of (3-Sn like cells; compare with Fig. 3-26.)
0.577 for the "ideal" structure.) The electron concentration appears to play some role in control of this family of structures as noted by Schwomma etal. (1964a, b), Flieher et al. (1968a, b), Jeitschko and Parthe (1967) and Parthe (1969) and reported by Pearson (1972). In the book of Hyde and Andersson (1989), the Nowotny phases are presented as a special case of a group of one-dimensional, columnar misfit structures which also include compounds such as Bam(Fe2S4)n and other complex sulfides. Layer misfit structures, such as those of some oxidefluorides, arseno-sulfides, etc. are also presented and classified with reference to a concept of structure commensurability based on the analysis, along one or more axes, of the ratios between the different repeat units of various interpenetrating substructures. The coexistence of different kinds of periodicity has also to be considered in the description of a quite different type of structure which is becoming increasingly common. In this, some atomic parameters (and/or the partial occupancy of some sites) vary in a periodic way through the structure. The periodicity may or may not be commensurate with the unit cell of the basic structure. Structures having these characteristics are often termed modulated structures (Hyde and Andersson, 1989). Several non-stoichiometric compounds present such modulations (FeS x , Yb 3 S 4 , etc.). Various modulated structures have also been considered, for instance, for the NiAs-type structure (see Sec. 3.5.2.5). An interesting case of magnetic modulated structure is that reported for EuCo 2 P 2
3.4 Relationships Between Structure Types (Structure Families)
(Reehuis et al., 1992). The positional structure of the atoms (of the atomic nuclei, nuclear structure) corresponds to the tI10-ThCr2Si2 type (see Sec. 3.5.2.5). A magnetic structure has been also determined, which is related to the ordering of the magnetic moments of the Eu atoms. These moments are oriented perpendicular to the c-axis and form an incommensurate spiral with the turning axis parallel to the c-axis. The magnetic moments lie in the basal planes and they order parallel within these planes. Along the c-axis, from one basal plane to the next one, there is a periodic turning of the moments. The ratio, along this axis, of the characteristic lengths of the magnetic and nuclear structures, is slightly dependent on temperature. At 64 K it is close to 5/6 (that is: there are 5 translation lenghts of the magnetic cell for 6 translation lengths of the nuclear structures). At 15 K the ratio was found to be close to %. If this magnetic structure were to be maintained at still lower temperatures, it may correspond to the exact 6 /7 value. The ground state may then be called a commensurate structure with this ratio. 3.4.5 Recombination Structures, Intergrowth Structure Series
Some of the previously reported relationships between structures may be included in the general term "recombination" structures. Such structures (see Limade-Faria et al., 1990) are formed when topologically simple parent structures are periodically divided into blocks, rods or slabs (that is structure portions which are finite or infinite in one or two dimensions, respectively) which are recombined into derivative structures by means of one or more structure building operations. The most important operations are: unit cell
twinning,
crystallographic
shear
159
planes,
intergrowth of blocks, rods or slabs of different structural types (for instance, intergrowth of cF24-Cu2Mg type and hP6-CaCu5 type slabs to obtain the hP36-Ce2Ni7 type structure), periodic outof-plane, antiphase boundaries (AuCuII, as an example), rotation of rods or blocks. The frequency of structure building operators (and, therefore, the size of undisturbed structure portions) can vary by well defined increments, so that many phases may occur as members of homologous series. For a scheme of relationships between inorganic crystal structures based on a systematic "construction" of complex structural types by means of a few operations (symmetry operations, topological transformations) applied to some building units (point systems, clusters, rods, sheets), see Hyde and Anderson (1989). We may add here that, within the "recombination" scheme, a very interesting method of describing, interpreting and interrelating complex structures is that based on the mentioned intergrowth concept (Kripyakevich etal., 1972, 1976, 1979; Grin' et al., 1982, 1990; Parthe et al., 1985; Lima-de-Faria, 1990; Pani and Fornasini, 1990). According to this concept, selected structure types may be considered as belonging to certain intergrowth structure series. The different structure types of an intergrowth series are described as being constructed from structure segments of more simple structures (the so-called parent structures). See Appendix A. Considering, for instance, the particular case of the "linear intergrowth structure series", we may mention that many, binary and ternary, intermetallic phases can be considered members of those series (both homogeneous and inhomogeneous). For example, the structure of the
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3 Structure of Intermetallic Compounds and Phases
oC16-NdNiGa2 type belongs to the series BaAl4-AlB2. Its unit cell contains indeed two BaAl4-type segments and two A1B2type segments. A simple code of this structure may be (BaAl 4 /AlB 2 ) 2 . (In a more complex and detailed notation, however, superscripted indexes may be added to the formulae of the segments in order to specify, for instance, their symmetry (Grin' etal., 1982; Parthe etal., 1985).) See Appendix B.
3.5 Elements of Structure-Type Systematic Description 3.5.1 Introductory Remarks and General References
By means of the considerations previously presented some typical structures will be described here below. On the basis of somewhat arbitrary criteria (such as high frequency of the structural type, existence of phases of considerable practical importance, possibility of presenting some features of general interest, etc.) the types to be described have been selected and presented in a few paragraphs. This description, therefore, should be considered as only an initial introduction to a vast subject. As already mentioned, complete and updated descriptions may be found in some reference books such as: LandoltBornstein (1971), Massalski (1990) and especially Villars and Calvert (1985, 1991). Moreover, for systematic classifications of the structure types, the following monographs may be consulted: In his book "Kristallstrukturen zweikomponentiger Phasen" (Crystal Structures of Binary Phases) Schubert (1964) described a few hundred structural types. He paid great attention to chemical criteria for the description, classification and discussion of the properties of the different
phases. The position of the elements involved in the Periodic Table was considered to be particularly relevant. For this purpose, the elements were subdivided into the following families: A-metals (elements of the s-block of the periodic table), Tmetals (transition metals), B-elements (elements of the p-block of the Periodic Table). The different structural types were then described according to the following chapter subdivision: - Brass-type alloys and close-packed sphere stacking and superstructure variants: AuCu 3 , AuCu, SrPb 3 , ZrAl 3 , ZrGa 2 , Nb 5 Ga 1 3 , etc.; Mg-type structure and superstructures Ni3Sn, etc.; body-centered sphere packing W structure and derivatives Fe3Si, CsCl, NaTl, Cu 5 Zn 8 , Ni 2 ln, etc. - T-T phases (among them the T element structures of the so-called Cr3Si family such as the pU, cI58-oc-Mn, hR39W 6 Fe 7 , cF116-Th 6 Mn 23 , etc., and the Laves phase structures). - B-B phases (structures considered as deformation variants of close packed structures, such as Zn, In, etc., structures of B, graphite, structures of the diamondfamily, of the P and As families, etc.). - A-B phases (several types partly classified according to the stoichiometry: Li3Bi, Mg2Sn, Mg 3 Sb 2 , NaCl, etc.). - T-B phases (T-rich borides, carbides, nitrides, oxides and hydrides, CuAl2, MoSi 2 , NiAs, FeS 2 structures and their variants). Pearson (1972), in his book "The Crystal Chemistry and Physics of Metals and Alloys", discussed the characteristics and specific features (coordination, stability, relationships with other structures, etc.) of about a thousand structure types. He was able to classify all these structures in 12 different families. The most important 10 are summarized here below.
3.5 Elements of Structure-Type Systematic Description
(1) Valence compounds of non-metals (semiconducting compounds with anions forming close packed arrays, polyanionic compounds, polycationic compounds, group IV, V and VI elements and IV-VI and V-VI compounds, etc.). (2) Metastable phases, interstitial phases, martensite (in this group of phases the Hagg interstitial phases formed by transition metal and small non-metal atoms such as H, B, C, N have been especially considered: in these phases the non-metals occupy the interstices, generally the octahedral ones of the close-packed structures of the transition metals). (3) Structures based on the close packing of the 36 close packed nets (Cu and Mg structures and their derivative structures AuCu 3 , AuCu, Ti3Cu, TiAl 3 , ZrGa 2 , MoNi 4 , etc.). (4) Structures derived by filling tetrahedral, octahedral (and other) holes in closepacked arrays of atoms (sphalerite structure and derivative structures oP12-CuAsS, tI16-FeCuS2, tI16-Cu 3 AsS 4 , etc., wurtzite structure and derivative structures, oP16CuSbS 2 , oP16-Cu 3 AsS 4 , hP30-In 2 Se 3 , etc., CaF 2 structure and distorted, defective, superstructures of CaF 2 , NaCl structure and derivative structures of the NaCl type, NiAs structure, etc.). (5) Structure types dominated by triangular prismatic arrangements (hP2-WC, hR9-MoS 2 , tI8-NbAs, tP6-Cu2Sb, oP36Ta2P, hP3-AlB2, hP6-CaIn 2 , hP6-Ni2In and their variants, examples of structure types included in this group). (6) Structures based on simple cubic and body centered cubic packing (in this group the structure types cI2-W, tI2-Pa, martensite, cP6-Cu2O, cP2-CsCl, tP4-TiCu, cF16-Li3Bi, cF16-NaTl, cF16-MnCu2Al, tP3-FeSi2, cI52-Cu5Zn8 and several variants are considered. In this structure fam-
161
ily the Nowotny chimney-ladder phases are also included). (7) Structures generated by square-triangle nets of atoms: cubes and cubic antiprisms (for instance tI12-CuAl2, oP24-AuSn2, mC12-PdP 2 , oC20-PtSn4, tP10-U 3 Si 2 , tP40-FeCu 2 Al 7 , oP16-ThNi, oI20-UAl4, etc.). (8) Structures generated by alternate stacking of triangular and kagome nets. (The structures of hP6-CaCu 5 , tI26ThMn 12 , hP38-Th 2 Ni 17 and their variants are included in this family. The Laves phases cF24-Cu2Mg, hP12-MgZn2 and hP24-Ni2Mg types and several variants are considered in this family. However, they are also described, as Frank-Kasper structures, in the subsequent group.) (9) Structures in which icosahedra and CN 14, 15 and 16 polyhedra play a dominant role. (Laves phases; [i phases: hR39W 6 Fe 7 ; P phases: oP56- Mo-Cr-Ni phase (which, at a composition corresponding to 42 at.% Mo and 18 at.% Cr, has a unit cell containing 56 atoms in partial substitutional disorder); R phases: hR159Mo-Co-Cr, etc. are included in this family, as well as a number of intermetallic phases with giant cells such as the cF1124Cu 4 Cd 3 , cF1192-NaCd2, cF1832-Mg2Al3 types studied by Samson (1969).] (10) Structures with large coordination polyhedra. [Structures are presented in which large coordination polyhedra are contained: for instance cP36-BaHg11 in which Ba is surrounded by 20 Hg, tI92Ce 5 Mg 41 , tI48-BaCd ll9 cF112-NaZn3 in which coordination polyhedra corresponding to coordination numbers (CN) 20, 22 and 24 are present respectively.] A substantially geometrical approach has been adopted in their book by Hyde and Andersson (1989) who presented and discussed the structure of more than a
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3 Structure of Intermetallic Compounds and Phases
thousand inorganic compounds, explicitly ignoring "the artificial barrier between inorganic and mineral structures on the one hand and metallurgical structures (intermetallic compounds, borides, carbides, etc.) on the other". In their treatment and classification of the structural types, they generate complex structures starting with relatively few basic structures and applying to segments of such structures, one or more of a few geometrical operations that are essentially symmetry operators. The "segments" or building units considered may be blocks (or clusters, bounded in 3 dimensions), rods (or columns, bounded in 2 dimensions, infinite in the third), slabs (or lamellae, sheets, layers, the latter bounded in 1 dimension and infinite in the other two). 3.5.2 Description of a Few Selected Structure Types 3.5.2.1 cI2-W, cP2-CsCl, cF16-MnCu 2 Al, cF16-Li 3 Bi, cP52-Cu 9 Al 4 and cI52-Cu 5 Zn 8 Type Structures and Martensite
In this section a few structural types are presented which can be described as related to the simple body-centered cubic structure, cI2-W type. For some of them Fig. 3-18 shows the normalized interatomic distances and the corresponding numbers of equidistant atoms. Structural type: cI2-W Body-centered cubic, space group Im3m, No. 229. Atomic positions: 2W in a) 0,0,0; 1 / 2 , 1 /2, 1 / 2 . Coordination symbol: ^[W 8/8 ]. Layer stacking symbols: Triangular (T) nets: A B c A B c W 1/2 W2/3 W W W WW W0 W 1/6 W 1/3 W 5/6 •
1
1.2
U
1.6
1.8
Figure 3-18. Trends of interatomic distances and coordinations in a group of closely interrelated structures, (a) cI2-W type structure: coordination around W. (b) XY compounds of cP2-CsCl type structure: (+) X-Y (or Y-X) coordination; (*) X-X (or Y-Y) coordination, (c) cF16-MnCu2Al type structure: coordination around Al (or Mn): ( + ) Al-Cu (or Mn-Cu); (•) Al-Mn (or Mn-Al); (o) Al-Al (or Mn-Mn). (d) cF16-MnCu2Al type structure: coordination around Cu: ( + ) Cu-Mn; (*) Cu-Al; (o) Cu-Cu.
Square (S) nets: WjW? /2 . For the prototype itself, W, a = 316.5 pm. This structure can be compared with the CsCl type structure (which can be obtained from the W type by an ordered substitution of the atoms) and the MnCu2Al type
3.5 Elements of Structure-Type Systematic Description
163
a 75% occupation probability, and by Al, with a 25% occupation probability). A number of these phases can be included within the group of the "Hume-Rothery" phases (see Sec. 3.3.4). In the VillarsCalvert compilation 240 phases (about 1.5% of the total number of phases considered) are listed under this structural type (which is the 15th in the frequency order). Structural type: cP2-CsCl
b)
Figure 3-19. cF16-AlCu2Mn type structure. The unit cell is shown in (a). The small cube presented in (b) corresponds to V8 of the unit cell and degenerates into a CsCl type cell if the atoms at the vertices are equal.
Cubic, space group Pm3m, No. 221. Atomic positions: 1 Cs in a) 0,0,0. lClinb) V i , 1 ^ . Coordination formulae: ^[CsCl]8/8 or ^[CsCl] 8cb/8cb (ionic description). T[X 6 / 6 ][Y 6 / 6 ] 8 / 8 or 3 3 ' [X 6o/6o ][Y 6o/6o ] 8cb/8cb (metallic description). Layer stacking symbols: Triangular (T) nets: 0 l / 6 l / 3 l / 2 2 / 3 5 / 6
structure ("ordered" superstructure of the CsCl type): see Fig. 3-19 a and 3-19 b and notice the typical 8 (cubic) coordination. The W-type structure is shown by a number of unary systems: Li, Na, K, Rb, Ba, Eu, Fe, Cr, Mo, V, Ta, W, etc. (as the only form or the room temperature stable form), Be, Ca, Sr, several rare earth elements, Th, etc. (as a high temperature form). The same structure is formed in a number of binary (or ternary) phases, for which a random distribution of the two (or three) atomic species in the two equivalent sites is possible. Typical examples are the pCu-Zn phase (in which the equivalent 0,0,0; 1/2,1//2,1/2 positions are occupied by Cu and Zn with a 50% probability) and the (3Cu-Al phase having a composition around Cu3Al (in which the two crystal sites are occupied, on average by Cu, with
Square (S) nets: CsJCl?/2. For the prototype itself, CsCl, a = 411.3 pm. See also Fig. 3-3. The 8 coordination (cubic) of the two atomic species is apparent. The normalized interatomic distances and numbers of equidistant neighbors are reported in Fig. 3-18. In the same figure data are also reported for the W-type structure, which can be considered a degenerate structure of the CsCl-type structure (in the W-type structure the two atomic sites are equivalent) and of the derivative (superstructure) MnCu2Al type. In the Villars-Calvert compilation, about 400 compounds ( « 2.7% of the total number of phases considered) are listed under this structural type (4th in the frequency rank order); about 300 phases are binary, the others are (more or less disordered) ternary phases. Among the binary
164
3 Structure of Intermetallic Compounds and Phases
phases we may mention 1:1 compounds such as those of alkaline earth and rare earth elements with Mg, Zn, Cd, Hg (and often with In, Tl, Ag, Au), those of Al and Ga with Fe and Pt group metals. The (3' Cu-Zn phase (stable at room temperature) belongs to this structural type; at higher temperature it undergoes the order-disorder transformation into the cI2-W-type structure. Structural type: cF16-MnCu2Al Face centered cubic, space group Fm3m, No. 225. Atomic positions: 4 Al in a) 0,0,0; V2, V2,0; V2,0, V2; 0, V2, V2. 4Mn in b) Vi.Vi.Vi; 0,0, Vi; 0,V2,0; V2,0,0. 8Cu in c) 1/4,1/4,1/4; V ^ A / A ; %9lA9%; X 3 3 3 3 3 3 A9 A9 A; A9 A9%; V*SA A\ A9V*SA9 Coordination formulae: *[AlCu8/4Mn6/6]. Layer stacking symbols: Triangular (T) nets: 1
/ 2/3^ U 3/
Square (S) nets:
Li3Bi types are listed together (see also Sec. 3.3.5.1). They are about 200 ( ^ 1.2% of the total number of phases considered and 17th in the frequency rank order). Among the ternary alloys, we may mention several Me'Me"Me'2" phases (with Me' = Al, Ga, Ge, Sn; Me" = Ti, Zr, Hf, V, Nb, Mn, etc. and Me'" = Co, Ni, Cu, Au, etc.). The compounds which crystallize with the MnCu2Al type structure (and particularly the magnetic compounds having this structure) are called Heusler Phases. In the specific case of the Al-Cu-Mn system this phase is ferromagnetic and stable above 400 °C, but it can be frozen by quenching to room temperature. It is assumed that its whole moment is due to the spin moment of Mn which has an unfilled d shell (5 electrons). Magnetic properties of Heusler phases are strongly dependent on the ordering of the atoms. Structural type: cF16-Li3Bi Face-centered cubic, space group Fm3m, No. 225. Atomic positions: 4 Bi in a) 0,0,0; V2, V2,0; V2,0, V2; 0, V2, V2. 4 Li in b) V2, V2, V2; 0,0, V2; 0, y2,0; y 2 ,0,0.
8 Li in c) VtSA^A; 3A3ASA\ 3
3
3
3
3
3
3
y4, /4, /4; /4, /4, /4; %^A9 A\ For the prototype itself, MnCu2Al, a = 596.8 pm. The structure is shown in Fig. 3-19. In this figure a comparison is also made with the CsCl type structure. It is apparent that if the two a) and b) sites are occupied by the same atomic species, the cell degenerates into a block of 8 equal cells (of the CsCltype). We may also observe that, on the contrary, if a single atomic species were assigned to the b) and c) sites, another ordered structure would be obtained: Li3Bi-type (or BiF3-type). In the Villars-Calvert compilation the phases belonging to the MnCu2Al and
ASA3A\
V^A^A;
ViSASA. Coordination formula: ^[BiLi8/4Li6/6] (ionic description). Layer stacking symbols: Triangular (T) nets: D ; A T -B
Ric r)1
T : C T -A r>iB
T :C
T "A T B
T iA T iB T i c
2/3 J ^ 1 3/4 J ^ 1 5/6 i ^ 1 ll/12-
Square (S) nets: Bi 0 Bi 0 Li 0 Li 1/4 LiJ /4 Li 1/2 Lii /2 Bii /2 Lif /4 L13V For the prototype itself, Li3Bi, a = 672.2 pm. Compare the data with those concerning the previously reported MnCu2Al-type structure.
3.5 Elements of Structure-Type Systematic Description
This structure could also be described as derived from a cubic close-packed array of atoms (Bi atoms) by filling all the tetrahedral and octahedral holes with Li atoms. Comments on the b.c.c. derivative structures In the family of b.c.c. derivative structures we may include several other structural types. As important defect superstructures based on the b.c.c. structure we may mention the cP52-Cu9Al4 type structure (Ag 9 ln 4 , Au 9 ln 4 , Pd 8 Cd 43 , Co 5 Zn 21 , Cu 9 Ga 4 , Li 10 Pb 3 can be considered as reference formulae of selected solid solutions having this structure). The large cell (a = 870.4 pm in the case of Cu9Al4) can be considered to be obtained by assembling 27 CsCl type pseudocells with two vacant sites. One vacant site occurs on each sublattice Al 1 6 Cu 1 0 D and C u 2 6 Q . The y-brass, cI52-Cu5Zn8-type structure can be similarly described as a distorted defect superstructure of the W-type structure, in which 27 pseudocells are assembled together with two vacant sites (corner and body center of the supercell). In this case, however, the atoms are considerably displaced from their ideal sites. The structure could also be described as built up of interpenetrating, distorted, icosahedra (each atom being surrounded by 12 neighbors). This description applies also to the cP52Cu 9 Al 4 type structure. (Ag5Cd8, Li 7 Ag 3 , Ag 5 Zn 8 , V5A18, Au 5 Cd 8 , Au 5 Hg 8 , Fe 3 Zn 10 , NiGa 4 , V 6 Ga 5 , Ni 2 Zn 11? etc. crystallize in the cI52-Cu5Zn8 structural type.) Martensite. The martensite structure can be considered a tetragonal distortion of the body-centered cubic cell of Fe (a = 285 pm, c = 298 pm at w 1 mass% C, in comparison to a = 286.65 pm for aFe, cI2-W type). Carbon is randomly distributed in the octahedral holes having coordinates 0,0, V2
165
and Vi, V2,0. Typically an occupancy of these sites of only a few % has to be considered. [For a 100% occupancy the structure of the tI4-CoO type is obtained with 2 Co in a) 0,0,0; V2, V2, V2; and 2 O in b) 0,0, V2; 1 /2,1/2,0 in the space group I4/mmm, No. 139.] In the martensitic cell the position parameters of the Fe atoms have a range along the fourfold axis, so there is a displacement from the cell corners and body center and an enlargement of the octahedral holes containing carbon.
3.5.2.2 cF4-Cu, cP4-AuCu 3 , tP2-AuCu(I), oI40-AuCu(II), tP4-Ti 3 Cu Types; hP2-Mg, hP4-La and hR9-Sm Types; hP6-CaCu 5 Type
In this Section, a few important elemental structures are described. Particularly the simple cubic (cF4-Cu type) and hexagonal close-packed (hP2-Mg) structures are presented. A few other stacking variants of identical monoatomic triangular nets are also reported. A group of structures which can be considered as derivative structures of Cu are also described. Normalized interatomic distances and numbers of equidistant neighbors are shown in Figs. 3-20 and 3-21. Structural type: cF4-Cu Face-centered cubic, space group Fm3m, No. 225. Atomic positions: 4 Cu in a) 0,0,0; 0, V2, V2; V2,0, V2; V2, V2,0. Coordination formula: <^[Cu12/12]. Layer stacking symbols: Triangular (T) nets: Cu£Cu?/3Cuf/3. Square (S) nets: CUJCUQCUI / 2 . For the prototype itself, Cu, a = 361.46 pm. The atoms are arranged in close packed layers stacked in the ABC sequence (see Sec. 3.3.5.2).
166 24-, N
3 Structure of Intermetallic Compounds and Phases
a)
a) 16
16-
8-I
8-
0
0-
1
d/d,min
b)
N
/
d/dm
24-, b) 16-
0
8-
d/dm
0-
d/dm 1.2 U 1.6 1.8 Figure 3-21. Distances and coordinations in the hexagonal close-packed (Mg-type) structures, (a) Ideal structure, c/a = 1.633 (first coordination shell corresponding to 12 atoms at the same distance), (b) Mgtype structures with c/a = 1.579. The group of the first 12 neighbors is subdivided into 6 + 6 atoms.
c) 1680-I
1 1.2 U 1.6 1.8 Figure 3-20. Distances and coordinations in the cF4-Cu and cP4-AuCu types structures. (Compare also with Figs. 3-13 and 3-14.) (a) cF4-Cu type structure, (b) cP4-AuCu3 type structure: coordination around Au: ( + ) Au-Cu; (*) Au-Au. (c) AuCu3 type structure: coordination around Cu: ( + ) Cu-Cu; (*) Cu-Au (added to Cu-Cu).
Several metals, such as Al, Ag, Au, aCa, ocCe, yCe, aCo, Cu, yFe, Ir, pLa, Pb, Pd, Pt, Rh, ocSr, ocTh and the noble gases Ne, Ar, Kr, Xe crystallize in this structural type. Several binary (and complex) phases having this structure have also been reported (solid solutions with random distribution of several atomic species in the four equivalent positions). Derivative structures may be obtained from the Cu-type structure by
ordered substitution or by ordered addition of atoms. As examples of derivative structures obtained by ordered substitution (and/or distorsion) in the Cu-type we may mention the AuCu 3 , AuCu, Ti3Cu types, which are described here below. (In the specific case of the AuCu3-type structure and the Cu-AuCu 3 types interrelation, see also Sec. 3.3.5.5.) For a systematic description of the derivative structures which may be obtained from the Cu-type by ordered filling-up it may be useful to consider that in a closest packing of equal spheres there are, among the spheres themselves, essentially two kinds of interstices (holes). These are shown in Fig. 3-22. The smallest holes surrounded by a polyhedral group of spheres are those marked by T. An atom inserted in this hole will have four neighbors whose centres lie at the vertices of a regular tetrahedron (tetrahedral holes). The larger holes (octahedral holes) are surrounded by octahedral groups of six spheres. In an infinite assembly of close-
3.5 Elements of Structure-Type Systematic Description
0 T Figure 3-22. Voids in the closest packing of equal spheres; tetrahedral (T) and octahedral (O) holes are evidenced within two superimposed triangular nets.
packed spheres the ratios of the numbers of the tretrahedral and octahedral holes to the number of spheres are, respectively, 2 and 1. Considering the Cu-type structure (in which the 4 close-packed spheres are in 0,0,0; 0,V2,V2; Vi&Vi; 1/2,1/2,0) the centers of the tetrahedral and octahedral holes have the coordinates: 4 octahedral holes in W / i / A ; V2,0,0; 2 sets of 4 tetrahedral holes in %,%,%; %, 3 / 4 , 3 / 4 ; 3A, y 4 , 3 A; 3A, %, %; and in 3/4,3/4,3/4; %,%,%; 1A,3/4,1/4; 1
/4,1/4,3/4.
Several cubic structures, therefore, in which (besides 0,0,0; O.Vi.Vi; Vi.O.Vi, V2, V2,0) one (or more) of the reported coordinate groups are occupied could be considered as filled-up derivatives of the cubic close packed structures. The NaCl, CaF 2 , ZnS (sphalerite), AgMgAs and Li3Bi type structures could, therefore, be included in this family of derivative structures. For this purpose, however, it may be useful to note that the radii of small spheres which fit exactly into tetrahedral and octahedral holes are respectively
167
0.225 ... and 0.414... if the radius of the close-packed spheres is 1.0. For a given phase pertaining to one of the mentioned types (NaCl, ZnS, etc.) if the stated dimensional conditions are not fulfilled, alternative descriptions of the structure may be more convenient than the reported derivation schemes. Similar considerations may be made with reference to the other simple closepacked structure, that is to the hexagonal Mg-type structure. In this case two basic derived structures can be considered: the NiAs type with occupied octahedral holes and the wurtzite (ZnS) type with one set of occupied tetrahedral holes. Structural type: cP4-AuCu3 Cubic, space group Pm3m, No. 221. Atomic positions: 1 Au in a) 0,0,0. 3Cuinc)0, 1 /2, 1 / 2 ; 1/2,0,1/2; 1 / 2 , 1 / 2 ,0. Coordination formula: Layer stacking symbols: Triangular, kagome (T, K) nets: Square (S) nets: AUQCUQCU^. For the prototype itself, AuCu3, a = 374.84 pm. (See also Sec. 3.3.5.5 for a detailed description of this structure.) This structure can be considered a derivative structure (ordered substitution) of the cF4-Cu type. A discussion of the characteristics of a number or ordered layer (super) structures involving a XY3 stoichiometry has been reported by Massalski (1989). Sequences of layer structures (among which those corresponding to the cP4-AuCu3, hP16-TiNi3, hP24-VCo3, hR36-BaPb3 types) as observed in V (or Ti) alloys with Fe, Co, Ni, Cu are described. The relative stabilities of the different stacking sequences have been
168
3 Structure of Intermetallic Compounds and Phases
analyzed in terms of a few parameters which characterize the interactions between various layers. Structural Types: tP2-AuCu(I), oI40-AuCu(II) tP2-AuCu(I) is tetragonal, space group P4/ mmm, No. 123. Atomic positions: 1 Auin a) 0,0,0. l C u i n d ) 1/2?1/2,1/2. Layer stacking symbols: Square (S) nets: Au^C\ j2 . For the prototype itself, AuCu(I), a = 280.4 pm, c = 367.3 pm, c/a = 1.310. The unit cell could be considered either as a distorted CsCl type cell greatly elongated in the c direction or, better, as a deformed (and orderly substituted) Cu-type cell. This is apparent from Fig. 3-16 where the tP2 unit cell and two tP4 supercells having d = a yjl = 396.6 pm, d = c = 367.3 pm are also shown. The larger cell is similar to a Cu-type cell, slightly compressed (cf/af = 0.926) and in which the atoms placed in the center of the sidefaces have been orderly substituted. The coordinates in the tP4 super (pseudo) cell are: Auin 0,0,0, and V^V^O, Cuin Vi^ViandO, 1 /^ 1 /!, and the corresponding square nets stacking sequence is A U Q A U O C U ^ . The long period superstructure of AuCu(I), discussed in Sec. 3.4.2, resulting in the antiphase-domain structure of AuCu(II) is shown in Fig. 3-16 b. Structural type: tP4-Ti 3 Cu Tetragonal, space group P4/mmm, No. 123. Atomic positions: 1 Cuin a) 0,0,0. lTiinc)1^,1/!^. 2Tiine)0, 1 /2, 1 / 2 ; Vi&Vi.
Coordination formula: [Cu] Layer stacking symbols: Square (S) nets: Cu£Ti£Ti*/2. For the prototype itself, Ti3Cu, a = 415.8 pm, c = 359.4 pm, c/a = 0.864. This structure can be described as a tetragonal distortion of the AuCu3-type structure. Structural type: hP2-Mg Hexagonal, space group P63/mmc, No. 194. Atomic positions: 2 Mg in c) V3,2/3, V4; 2/a, V3,3A. Coordination formula: ^[Mg (6 + 6)/(6 + 6)] and ideally: £[Mg 12/12 ]. For the prototype itself, Mg, a = 320.89 pm, c = 521.01 pm, c/a = 1.624. Normalized interatomic distances and numbers of equidistant neighbors are presented in Fig. 3-21 a for an "ideal" hexagonal close-packed structure (c/a = 1.633), which corresponds to 12 nearest neighbors at the same distance, and, in Fig. 3-21 b, for a slightly distorted cells. The atoms are arranged in close packed layers stacked in the sequence ABAB... (see Sec. 3.3.5.2). The corresponding layer symbol (triangular nets) is MgQ 25MgQ 75 . Several metals have been reported with this type of structure, such as: ocBe, Cd, Co, aDy, Er, Ho, Lu, Mg, Os, Re, Ru, Tc, ocY, Zn, etc. Several binary (and complex) phases have also been described with this type of structure. These are generally solid solution phases with a random distribution of the different atomic species in the two equivalent positions. Other stacking variants of close-packed structures are the La-type and Sm-type structures. Characteristic features of these types are presented here below. Structural type: hP4-La Hexagonal, space group P63/mmc, No. 194. Atomic positions:
3.5 Elements of Structure-Type Systematic Description
169
2 La in a) 0,0,0; 0,0, V2. 2 La in c) y 3 , 2 / 3 , V4; 2h,Vz^U. For the prototype itself, oc-La, a = 377.0 pm, c = 1215.9 pm, c/a = 3.225. Layer stacking symbols: Triangular (T) nets: La^La^ 25Lao.5Lao 75 . Structural type: hR9-Sm Rhombohedral, space group R3m, No. 166. Atomic positions: 3 Sm in a) 0,0,0; 2/3, V3, V3; y 3 , 2 / 3 , 2 / 3 . 6 Sm in c) 0,0, z; 0,0, - z; 2/3, V3, V3 + z\ %, V3, V3 - z; V3,2/3, % + z; V3, %, 2/3 - z. For the prototype itself, a-Sm, a = 362.90 pm, c = 2620.7 pm, and z = 0.222. Layer stacking symbols: Triangular (T) nets: ^ m 0.67^ m 0.78^*^0.89 •
The La and Sm type structures belong to the same homeotect type set as Mg and Cu (see Sec. 3.4.3). All these close-packed element structures are stacking variants of identical slab types (monoatomic triangular nets). Structural type: hP6-CaCu 5 Previously a few structures have been considered which can be conveniently described in terms of stacking of 3 6 nets of atoms. As an example of structures in which more complex stacking sequences can be observed we may mention here the hP6-CaCu5 type structure, which is the reference type for a family of structures in which 3 6 nets (and 63) are alternatively stacked with 3636 (kagome) nets of atoms. The hP6-CaCu 5 structure is hexagonal, space group P6/mmm, No. 191, with: 1 Cain a) 0,0,0. 2Cuinc)y 3 , 2 / 3 ,0; 2 / 3 ,y 3 ,0; 3 Cu in g) y>,0, y2; 0, V2, V2; V2, V2, y2; For the prototype, a = 509.2 pm, c = 408.6 pm, c/a = 0.802.
Figure 3-23. Projection of the hP6-CaCu5 type unit cell on the x, y plane.
The layer stacking symbol, triangular (T: A, B, C), hexagonal (H: a, b, c) and kagome (K: oc, p, y) nets its: Ca^Cu^CuS 5 (see Fig. 3-23). A large coordination is obtained in this structure: Ca is surronded by 6 Cu + 12 Cu + 2 Ca at progressively higher distances and the Cu atoms have 12 neighbors (in a non-icosahedral coordination). Several phases belonging to this structure are known (alkali metal compounds such as KAu 5 , alkaline earth compounds, such as LaCo 5 , LaCu 5 , LaNi 5 , LaPt 5 , LaZn 5 , SmCo 5 , which is a good permanent magnet, etc. Ternary phases have been also described, both corresponding to the ordered derivative hP6-CeCo 3 B 2 type [1 Ce in a), 2 B in c) and 3 Co in g)] and to disordered solid solutions of a third component in a binary CaCu5-type phase. According to Pearson (1972) several structures may be described as derived from the CaCu 5 type (for instance, the tI26-ThMn 12 type; hR57-Th 2 Zn 17 type; hP38-Th 2 Ni 17 type; etc.). As for the building principles of the CaCu 5 type some analogies with the Laves phases (see Sec. 3.5.2.6) may be noticed.
170
3 Structure of Intermetallic Compounds and Phases
3.5.2.3 cF8-C (Diamond), tI4-pSn, cF8-ZnS Sphalerite, tI16-FeCuS2, hP4-C Lonsdaleite, hP4-ZnO (or ZnS Wurtzite), oP16-BeSiN2 Types and SiC Polytypes In this paragraph a few typical tetrahedral structures are presented. For the simplest ones, normalized interatomic distances and numbers of equidistant neighbors are shown in Fig. 3-24.
a)
O Zn
O-
b)
Structural type: cF8-C (diamond) Face-centered cubic, space group Fd3m, No. 227. Atomic positions: 8 C in a) 0,0,0; 0, V2, V2; V2,0, V2; V2, V2,0; 1
1
1
3
3
3
3
A, /4, /4; V4, /4, /4; /4,y 4 , /4;
V^A^A.
Coordination formula: ^[C 4/4 ]. Layer stacking symbols: Triangular (T) nets:
© Cu •
Fe
O S
Figure 3-25. (a) cF8-ZnS (sphalerite) and (b) til 6FeCuS2 (chalcopyrite) type structures.
^0^1/4^1/3^7/12^2/3^11/12-
Square (S) nets: CjCSCf /4 Cf /2 C? /4 . For the prototype itself, C diamond, a = 356.69 pm.
a) 16
d/dm
b)
2.2 d/dm 1 1.2 U 1.6 1.8 Figure 3-24. Distances and coordinations in the cF8-C diamond and cF8-ZnS sphalerite types structures, (a) cF8-C (diamond) type structure, (b) XY compounds of cF8-ZnS (sphalerite) type structure: ( + ) X-Y (or Y-X) coordination; (*) X-X (or Y-Y) coordination.
The diamond structure is a 3-dimensional adamantine network in which every atom is surrounded tetrahedrally by four neighbors. The 8 atoms in the unit cell may be considered as forming two interpenetrating face centered cubic networks. If the two networks are occupied by different atoms we obtain the derivative cF8-ZnS (sphalerite) type structure. As a further derivative structure, we may mention the tI16-FeCuS2 type structure (see Fig. 3-25). These are all examples of a family of "tetrahedral" structures which have been described by Parthe (1963, 1991) and briefly presented in Sec. 3.6.2.1. Si, Ge and a-Sn have the diamondtype structure. The tI4~pSn structure (a = 583.1pm, c = 318.1pm) (4Sn in a) 0,0,0; O,1/!,1/*; 1//2,1/2,1/2; 1/2,0,3A; space group I41/amd, No. 141 can be considered a very much distorted diamond type structure. Each Sn has 4 close neighbors, 2 more at a slightly larger (and 4 other at a consid-
3.5 Elements of Structure-Type Systematic Description
171
Compounds isostructural with the cubic cF8-ZnS sphalerite include AgSe, A1P, AlAs, AlSb, AsB, AsGa, Asln, BeS, BeSe, BeTe, BePo, CdS, CdSe, CdTe, CdPo, HgS, HgSe, HgTe, etc. The sphalerite structure can be described as a derivative structure of the diamond type structure. Alternatively we may Figure 3-26. tP4-(3Sn type structure. describe the same structure as a derivative of the cubic close-packed structure (cF4Cu-type) in which a set of tetrahedral holes erably larger) distance. The (3-Sn unit cell is has been filled-in. (This alternative descripreported in Fig. 3-26. tion may be especially convenient, when Structural types: cF8-ZnS sphalerite the atomic diameter ratio of the two speand hP4-ZnO (ZnS wurtzite) cies is close to 0.225: see the comments reported in Sec. 3.5.2.2.) - cF8-ZnS sphalerite In a similar way the closely related hP4Face-centered cubic, space group F43m, ZnO structure can be considered as a deNo. 216. rivative of the hexagonal close packed Atomic positions: structure (hP2-Mg-type) in which, too, a 4 Zn in a) 0,0,0; 0, V2, V2; V2,0, V2; V2, V2,0. 1 1 3 3 3 3 4S in c) %, /4, /4; V*, /*, /*; A^A A\ set of tetrahedral holes has been filled-in. Compounds, isostructural with ZnO include Agl, BeO, CdS, CdSe, CuX (X = H, Coordination formulae: Cl, Br, I), MnX (X = S, Se, Te), MeN ^[ZnS4/4] (ionic or covalent description); 3 3 (Me = Al, Ga, In, Nb), ZnX (X = O, S, Se, ^ [Zn 12/12 ][S 12/12 ] 4/4 (metallic descripTe). tion). In order to have around each atom, four For the prototype itself, ZnS sphalerite, exactly equidistant neighboring atoms, the a -541.09 pm. axial ratio should have the ideal value - hP4-ZnO or ZnS wurtzite 1.633. The experimental values range from 1.59 to 1.66. The ideal values of one of the Hexagonal, P63mc, No. 186. parameters (being fixed at zero the other Atomic positions: 2 Zn in b(l)) V3,2/3,z; 2/3, V3, V2 + z; (z = zi). one by conventionally shifting the origin of the cell) is z = 3/8 = 0.3750. 2 O or 2S in b(2)) V3,2/3,z; %, Vs.Vi + z; The C diamond, sphalerite and wurtzite type structures are well known examples of Coordination formula: ^[ZnO 4/4 ]. the "normal tetrahedral structures" (see For the prototypes themselves, ZnO, Sec. 3.6.2.1). a = 325.0 pm, c = 520.7 pm, c/a = 1.602; Several superstructures and defect suZnS (wurtzite): a = 382.25 pm, c = 626.1 pm, perstructures based on sphalerite and on c/a =1.638. The atomic positions correwurtzite have been described. The til 6spond, for both types of atoms, to similar FeCuS 2 (chalcopyrite) type structure coordinate groups (to the same Wyckoff (tetragonal, a = 525 pm, c = 1032 pm, position) with different values of the z parameter. For ZnO, zZn = 0, z o = 0.3825 c/a = 1.966), for instance, is a superstrucand for ZnS, zZn = 0, zs = 0.371. ture of sphalerite in which the two metals
172
3 Structure of Intermetallic Compounds and Phases
adopt ordered positions. The superstructure cell corresponds to two sphalerite cells stacked in the c direction. The c/(2 a) ratio is nearly 1. As another example we may mention the oP16-BeSiN2 type structure which similarly corresponds to the wurtzite type structure. The degenerate structures of sphalerite and wurtzite (when, for instance, both Zn and S are replaced by C) corresponds to the previously described cF8-diamond type structure and, respectively, to the hP4-hexagonal diamond (or lonsdaleite which is very rare compared with the cubic, more common, gem diamond). While discussing the sphalerite and wurtzite type structures we have also to remember that they belong to a homeotect structure type set (see Sec. 3.4.3). The layer stacking sequence symbols (triangular nets) of the two structures are: Sphalerite: Z<Sf /4 Zn? /3 S« /12 Zn^ /3 S? 1/12 . Wurtzite: ZngSg.38Zn?/2Sg.88. In the first case we have (along the direction of the diagonal of the cubic cell) a sequence ABC of identical "unit slabs" ("minimal sandwiches") each composed of two superimposed triangular nets of Zn and S atoms. The "thickness" of the slabs, between the Zn and S atom nets is 0.25 of the lattice period along the superimposition direction (cubic cell diagonal: a yJ3). It is (0.25 ^/3 • 541) pm = 234 pm. In the wurtzite structure we have a sequence BC of slabs formed by sandwiches of the same triangular nets of Zn and S atoms [their thickness is « 0.37 • c = (0.37 • 626.1) pm = 232 pm]. With reference to the mentioned structural unit slab the Jagodzinski-Wyckoff symbol of the two structures will be: ZnS sphalerite: c; ZnS wurtzite: h. In the same (equiatomic tetrahedral structure type) homeotect set many more structures occur often with very long stack-
ing periods. Several other polytypes of ZnS itself have been identified and characterized. The largest number of polytypic forms and the largest number of layers in regular sequence have, however, been found for silicon monocarbide. A cubic form of SiC is known and many tenths of rhombohedral and hexagonal polytypes. (In commercial SiC a six-layer structure, h.c.c, is the most abundant.) All have the same ahex « 308 pm, the chex of their hexagonal (or equivalent hexagonal) cells are all multiples of «252 pm and range from 505 pm to more than 150 000 pm (up to more than 600 Si-C slabs in a regular sequence). 3.5.2.4 cF8-NaCl, cF12-CaF2, and cF12-AgMgAs Types In this paragraph the NaCl type, CaF 2 type (and the related AgMgAs type) structures are described. In Fig. 3-27 the normalized interatomic distances and the equidistant neighbors are shown for the NaCl and CaF 2 structures. Structural type: cF8-NaCl Face-centered cubic, space group Fm3m, No. 225. Atomic positions: 4 Nain a)0,0,0; 0, V2, V2; V2,0, V2; V2, V2,0', 4 Cl in b) V2, V2, V2; V2,0,0; 0, V2,0; 0,0, V2. Cordination formulae: oo [Na12//12][Cl12//12]6/6 Layer stacking symbols: Triangular (T) nets:
Na£Cl? /6 Na? /3 Clt /2 Na5 /3 Cl? /6 . Square (S) nets: /
/
/
For the prototype itself, NaCl, a = 564.0 pm. We may also describe this structure as a derivative of the cubic close packed struc-
3.5 Elements of Structure-Type Systematic Description
173
Atomic positions: 4 Ca in a) 0,0,0; 0, V2, V2; V2,0, V2; V2, V2,0; 8 F in c) V4,74, V4; V*,1/*,3/*; lA,3A,3A; 1/
3/
1/ . 3/
/4j /4j / 4 ,
1/ 3/ . 3/ 1 /
/ 4 , /4j / 4 ,
1 / . 3/ 3/
/4? /4> /4>
1/ .
/4> / 4 , /4}
Coordination formula: 333r^o
ire1
1
Layer stacking symbols: Triangular (T) nets: A F Fc r r 3/4
11/12-
Square (S) nets:
Ca1Ca4'F6
F7 fa5
F6 F7
^ d o ^ d o r 1j4_r i/4^d-i/2 r 3/4r
16 + *
1
1.2
U
1.6
1.8
* d/dm
Figure 3-27. Distances and coordinations in the cF8NaCl and cF12-CaF2 types structures, (a) XY compounds of cF8-NaCl type structure: ( + ) X-Y (or Y-X) coordination; (*) X-X (or Y-Y) coordination, (b) cF12-CaF2 type structure: coordination around Ca: (*) Ca-Ca; ( + ) Ca-F; (c) CaF 2 type structure: coordination around F: (*) F-F; ( + ) F-Ca.
ture (cF4-Cu-type), in which the octahedral holes have been filled in. This description may be specially convenient when the atomic diameter ratio between the two elements is close to the theoretical value 0.414 (for octahedral interstices). This could be the case of a number of "interstitial compounds". Compounds of the transition metals with relatively large atomic radii with non metals with small radii (H, B, C, N, O) are simple examples of this type. These compounds have been especially studied by Hagg (1931). Structural type: cF12-CaF 2 Face-centered cubic, space group Fm3m, No. 225.
3/4-
For the prototype itself, CaF 2 , a = 546.2 pm. As pointed out in the description of the cubic close-packed structure (cF4-Cu-type) this structure may be described (especially for certain values of the atomic diameter ratio) as a derivative of the Cu-type structure in which two sets of tetrahedral holes have been filled in. Several (more or less ionic) compounds such as CeO 2 , UO 2 , ThO 2 , etc. belong to this structural type. Several Me2X compounds, with Me = Li, Na, K; X = O, S, Se, Te, Po, also belong to this type. In this case, however, the cation and anion positions are exchanged (Me in c) and X in a)) and these compounds are sometimes referred to a CaF2-antitype. Typical (more metallic) phases having this structure are also, for instance, AuAl2, PtAl 2 , Mg 2 Pb, Mg2Sn, Mg 2 Ge, Mg2Si. A ternary ordered derivative variant of this structure is the cF12-AgMgAs type. Structural type: cF12-AgMgAs Face-centered cubic, space group F43m, No. 216. Atomic positions: 4 As in a) 0,0,0; 0, V2, V2; V2,0, V2; V2, 7 2 ,0; 4 Ag in c) 1//4,1/4,1//4; V4.93A93A] 3A^A93A\
174
3 Structure of Intermetallic Compounds and Phases
4 Mg in d) %, 3A, 3A; %, XU, %; V4, %, 'A; 1
1
3
/4, /4, A.
Layer stacking symbols: Triangular (T) nets: /
/
Mg£ / 4 Ag? 1 / 1 2 . Square (S) nets: For the prototype itself, AgMgAs, a = 625.3 pm. In systematic investigations of MeTX ternary alloys (Me = Th, U, rare earth metals, etc., T = transition metal, X = element from the V, IV main groups) several tens of phases pertaining to this structure type have been identified. For the same group of alloys, however, other structural types are also frequently found. The hP6-CaIn 2 type and its derivative types often represent a stable alternative. The relative stabilities of the two structures (especially as a function of the atomic dimensions of the metals involved) have been discussed, for instance, by Dwight (1974), Marazza et al. (1980, 1988), Wenski and Mewis (1986).
3.5.2.5 hP4-NiAs, hP3-CdI2, hP6-Ni2In, oP12-Co2Si, oP12-TiNiSi Types; hP2-WC, hP3-AlB2, hP3-wCr-Ti, hP6-CaIn2, hP9-Fe2P Types and tI8-NbAs, tI8-AgTlTe2 and tI10-BaAl4 (ThCr2Si2) Types In this section a number of important interrelated structures are presented. A first group is represented by the hP3-CdI 2 , hP4-NiAs and hP6-Ni 2 In types. Some comments on the interrelations between these structures have been reported in Sec. 3.4.1. A further comparison may also be made by considering their reported characteristic triangular net stacking sequences:
hP3-CdI 2 C hP4-NiAs N hP6-Ni2In N ^ ?
/ 4
?
/ 4
t
/ 2 / 4
^
/ 4
We see, on passing from Cdl 2 to the NiAs type the insertion of a new layer at 0.5 and, from NiAs to Niln 2 , the ordered addition of atoms at levels 1/4 and 3/4. Structural type: hP4-NiAs Hexagonal, space group P63/mmc, No. 194. Atomic positions: 2Niina)0,0,0;0,0, 1 / 2 . 2Asinc) 1 / 3 , 2 /3, 1 /4; 2 /3, 1 /3, 3 /4.
Coordination formula: ^[Ni 2/2 ]As 6/6 . For the prototype itself, a = 361.9 pm, c = 503.4 pm, c/a = 1.391. According to Hyde and Andersson (1989), the data reported have to be considered as corresponding to an average slightly idealized structure, corresponding for several compounds to the form which is stable at high temperature. At room temperature, in the real structure, there are very small displacements of both Ni and As from their ideal average positions. The structure should, therefore, be better described by: 2 Ni in a) 0,0, z; 0,0, V2 + z, (z = 0). 2 As in b) V3,2/3,z; 2/3, V3, V2 + z, (z « V4) in the space group P63mc, No. 186. The small (probably correlated) displacements of the atoms produce several sort of modulated structures (see Sec. 3.4.4). Structural type: hP3-CdI 2 Hexagonal, space group P3ml, No. 164. Atomic positions: 1 Cd in a) 0,0,0. 2 I i n d ) V3,2/3,z; 2 / 3 , 1 /3,-z. Coordination formula: ^[CdI 6/3 ]. For the prototype itself, Cdl 2 , a = 424.4 pm, c = 685.9 pm, c/a = 1.616 and z = 0.249.
175
3.5 Elements of Structure-Type Systematic Description
Typical phases pertaining to this structural type are CoTe2, HfS2, PtS 2 , etc. and also Ti2O (which, owing to the exchange in the unit cell of the metal/non-metal positions may be considered to be a representative of the Cdl2-antitype).
Hexagonal, space group P63/mmc, No. 194. Atomic positions: 2Niina)0,0,0;0,0, 1 / 2 ; 2Ininc) 1 / 3 , 2 /3, 1 /4; 2 /3, 1 /3, 3 /4. 2Niind) 1 / 3 , 2 /3, 3 /4; 2 /3, 1 /3, 1 /4.
Coordination formula: ^[InNi 6/6 Ni 5/5 ]. For the prototype itself, a = 417.9 pm, c = 513.1 pm,c/a = 1.228. Typical phases assigned to this structural type are, for instance: Zr2Al, Co2Ge, La2ln, Mn2Sn, Ti2Sn and several ternary phases such as: BaAgAs, CaCuAs, CoFeSn, LaCuSi, VFeSb, KZnSb, etc. A distorted variant of the Ni 2 ln type structure is the oP12-orthorhombic structure of the Co2Si, (or PbCl2) type: ^[SiCo6/5Co4/5], that is total coordination 10 of Co around Si with 6/5 + 4/5 - 10/5 = 2 Co atoms for each Si atom. A ternary derivative of this type is the oP12-TiNiSi type (prototype of the so-called E phases). Structural types: oP12-Co2Si(PbCl2) and oP12-TiNiSi Orthorhombic, space group Pnma, No. 62. In this structural type the atoms are distributed in three groups of positions corresponding (obviously with different values of the x and z free parameters) to the same type of Wyckoff positions (Wyckoff position c). in Co2Si 4 Co
4 Co
4Ni
c(3))x,%,z;
4 Si
4 Si
Vz-x^Vz -x,3/4, - z ;
Structural type: hP6-Ni2In
Atomic positions: c(l))x,V 4 ,z; V z - x ^ V z + z; -x,3/4, - z ;
c(2))x,V4,z; V z - x ^ V z + z; - x, 3/4, - z;
in TiNiSi 4Ti
For the prototypes: Co2Si: a = 491.8 pm, b = 373.8 pm, c = 710.9 pm, a/c = 0.692; xc(1) = 0.038, zc(1) = 0.218; xc(2) = 0.174, zc(2) = 0.562; C c(3) = 0.702, zc,3) = 0.611. TiNiSi: a = 614.84 pm, b = 366.98 pm, c = 701.73 pm, a/c = 0.876; xc(1) = 0.021, zc(1) = 0.1803; xc(2) = 0.1420, zc(2) = 0.5609; xc(3) = 0.7651, zc(3) = 0.6229. Co2Si is the prototype of a group of phases (also called PbCl2-type) which can be subdivided into two sets according to the value of the axial ratio a/c which is in the range from 0.67 to 0.73 for one set (for instance, Co2Si, PdAl2, ZrAl2, Rh2Ge, Pd2Sn, Rh2Sn, etc.) and in the range from 0.83 to 0.88 for the other set [for instance PbCl 2 , BaH2(h), Ca2Si, Ca 2 P, GdSe 2 , ThS 2 , TiNiSi, etc.] (Pearson, 1972). The ternary variant TiNiSi-type is also called E-phase structure. Many ternary compounds belonging to a MeTX formula (Me = rare earth metal, Ti, Hf, V, etc., T = transition metal of the Mn, Fe, Pt groups, X = Si, Ge, Sn, P, etc.) have this structure. Other groups of more or less strictly interrelated structures which will be considered in this paragraph are those corresponding to the hP2-WC, hP3-AlB2, hP6-CaIn2 and hP9-Fe 2 P types, and, respectively, to the tI8-NbAs, tI8-AgTlTe2, tI10-BaAl4 (and tI10-ThCr2Si2) types. Structural type: hP2-WC Hexagonal, space group P6m2, No. 187. Atomic positions:
176
3 Structure of Intermetallic Compounds and Phases
1 Win a) 0,0,0. lCindJVa,2/^. For the prototype itself, a = 290.63 pm, c = 283.67 pm, c/a = 0.976. This structure type with the axial ratio c/a close to 1 is an example of the Hagg interstitial phases formed when the ratio between non-metal and metal radii is less than about 0.59. The structure can be described as a tridimensional array of trigonal prism of W atoms (continguous on all the faces). Alternately trigonal prisms are centered by C atoms. Structural type: hP3-AlB2 Hexagonal, space group P6/mmm, No. 191. Atomic positions: 1 Al in a) 0,0,0. 1
2
1
2
1
1
2Bind) / 3 , /3, /2; /3, /3 ? /2.
Coordination formula: 3 ^ 3 [Al 8/8 ][B 3/3 ] 12/6 . For the prototype itself, a = 300.6 pm, c = 325.2 pm, c/a = 1.082. The structure can be considered a filledup WC structure type. The B atoms form a hexagonal net and center all the Al trigonal prisms. The layer stacking sequence symbols of the two above reported structures are: WC-type, triangular (T) nets: A B
w r
W 0^1/2-
AlB2-type, triangular, hexagonal (T, H) nets: A1 A R a
While considering the structural characteristics of the A1B2 type phases, we may mention that boron centered triangular metal prisms are the dominating structural building elements in the crystal structures of simple and complex metal borides. Building blocks of centered triangular prisms as base units for classification of these substances have been considered by Rogl (1985, 1991) in a systematic presentation of the crystal chemistry of borides.
Structural type: hP3-coCr-Ti The co phase, a ubiquitous metastable phase in Ti (or Zr or Hf)-transition metal systems, is approximately isotypic with A1B2. (The axial ratio of the unit cell, however, instead of being close to unity, is very much smaller and has a value of about 0.62.) The components are randomly arranged. One third of the atoms are distributed in a triangular net at z = 0 forming trigonal prisms. Two thirds of the atoms are placed near the centers of the prisms (slightly displaced alternately up and down) forming a rumpled 63 net at z ^ Vi. (The space group is P3ml.) Structural type: hP6-CaIn2 Hexagonal, space group P63/mmc, No. 194. Atomic positions: 2Caina)0,0,V4;0,0, 3 / 4 . 4 Ininf) V3,2/3,z; 2/3, 73,z + V2; 2/3, V3, - z; 1
/3, 2 /3,-z + 1/2.
Layer stacking symbol: Triangular (T) nets: For the prototype itself, a = 489.5 pm, c = 775.0 pm, c/a = 1.583 and z = 0.455. This structure can be described as a distortion (a derivative form) of the AlB2-type structure. Ca atoms form trigonal prisms alternatively slightly off-center up and down by In atoms. In Fig. 3-28 the normalized interatomic distances and the equidistant neighbors are shown for the NiAs and Caln 2 structures. Structural type: hP9-Fe2P Hexagonal, space group P62m, No. 189. Atomic positions: lPintyOAVi. 2 P i n e ) 1 /3, 2 /3,0; 2 / 3 , 1 / 3 ,0. 3 Fe in f) x, 0,0; 0, x, 0; — x, — x, 0. 3 Fe in g) x,0, V2; 0,x, V2; - x, - x, V2.
177
3.5 Elements of Structure-Type Systematic Description
a) 16 8 0 2 2U N
d/d.min
b)
known. To the same structure, however, ternary (or even more complex) phases may be related if different atomic species are distributed in the different sites. This structure can be considered as an example of more complex structures built up by linked triangular prism of Fe atoms. Several ordered ternary phases have structures related to the Fe2P-type.
16 *
Structural types: tI8-NbAs, tI8-AgTlTe2 and tI10-BaAl4(ThCr2Si2)
*
1 c)
N 16 8
h Hh
+
* 1
0 2U d)
Nr 16 8 0
** * + + I *l I 1 1.2
i lil U
1.6
1.8
2
d/c/m
Figure 3-28. Distances and coordinations in the hP4NiAs and hP3-CaIn2 types structures, (a) hP4-NiAs type structure: coordination around Ni: ( + ) Ni-As; (*) Ni-Ni. (b) hP4-NiAs type structure: coordination around As: ( + ) As-Ni; (*) As-As. (c) hP6-CaIn2 type structure: coordination around Ca: ( + ) Ca-In; (*) Ca-Ca. (d) hP6-CaIn2 type structure: coordination around In: ( + ) In-Ca; (*) In-In.
For the prototype itself, a = 586.5 pm, c = 345.6 pm, c/a = 0.589 and x(f) = 0.256 and x(g) = 0.594. In the Fe2P-type structure there are 4 different groups of equipoints. The distribution of P and Fe atoms in different groups of positions is reported. A number of isostructural binary compounds are
The three structural types tI8-NbAs, tI8-AgTlTe2 and tI10-BaAl4 (with its ordered ternary variants such as the tllOThCr 2 Si 2 ) belong to a group of interrelated structures. All these structures contain among their building parts layers of (metal atoms) triangular prisms with specific distributions of the (non-metal) atoms centering the prisms (Pearson, 1972). The prisms are parallel to the basal planes of the tetragonal unit cells. Features of the hP2-WC type structure (characterized by an array of trigonal prisms alternatively centered by C atoms) are, therefore, present in the mentioned structures. (In the hP2-WC structure, of course, the prism axes are laying in the c direction of the hexagonal cell.) Another convenient description of these groups of structures may be in term of 4 4 net layer stacking. The corresponding square net symbols for the 8-layers stacks are the following ones: tI8-NbAs:
tI8-AgTlTe2: TloTeo 13 Ag 0 25 Te 0 37T1O 5Te0 63 Ag 0 lc
0.87-
tI10-ThCr 2 Si 2 : 0 38
75
178
3 Structure of Intermetallic Compounds and Phases
Structural type: tI8-NbAs Body-centered tetragonal, space group I^md, No. 109. 4 Nb in a(l)): 0,0, z; 0, V2, V4 + z; 1 /2,V2,1/2 + z; V2,0,3/4 + z. 4Asina(2)):0,0,z;0, 1 / 2 , 1 /4 + z; 1
/2, 1 /2, 1 /2+r, V2,0,3/4 + Z.
For the prototype itself (NbAs), a = 345.2 pm, c = 1168 pm, c/a = 3.384, z(Nb) = 0, z(As) = 0.416.
Si Cr
Structural type: tI8-AgTlTe2
Th
Body-centered tetragonal, space group I4m2, No. 119. 2 Tl in a): 0,0,0; Vi^/i^/i. 2Aginc):0, 1 / 2 , 1 /4; V2,0,3/4; 4 Te in e): 0,0, z; 0,0, - z; V2, 7 2 , V2 + z; 1
/2, 1 /2, 1 / 2 -z;
For the prototype itself, a = 392 pm, c = 1522 pm, c/a = 3.883 and z(Te) = 0.369. Structural type: tI10-BaAl4 and tI10-ThCr2Si2 The ThCr 2 Si 2 is an ordered ternary variant of the BaAl4-type. The two structures may be described by the following occupation of the same atomic positions in the space group I4/mmm (No. 139).
a) 0,0,0;
in BaAl4 in ThCr2Si2 2Ba 2Th
l
/2,V2,V2
d)0,V 2 , 1 / 4 ;
4A1
4Cr
e) 0,0,z;0,0, - z; 4A1
4Si
o,V 2 , 3 / 4
For the prototypes themselves: BaAl4, a = 453.9 pm, c = 1116.0 pm, c/a = 2.459, z = 0.38.
©
Figure 3-29. Unit cell of the tI10-ThCr2Si2 type structure, a derivative structure of the tI10-BaAl4 type.
ThCr 2 Si 2 : a = 404.3 pm, c = 1057.7 pm, c/ a = 2.616, z = 0.375. The unit cell is presented in Fig. 3-29. Normalized interatomic distances and numbers of equidistant neighbors are reported in Fig. 3-30 for the ternary ThCr2Si2-type. Many ternary alloys MeT 2 X 2 (Me = Th, U, alkaline-earth, rare earth metal, etc., T = Mn, Cr, Pt family metal, X = element of the fifth, fourth and occasionally third main group) have been systematically prepared and investigated (Parthe and Chabot, 1984; Rossi et al., 1975). A few hundreds of them resulted in the ThCr 2 Si 2 (or other Al4Ba derivatives) structure. The peculiar superconductivity and magnetic properties of these materials have been reported. The ThCr2Si2-type structure, can be described as formed by T 2 X 2 layers interspersed with Me layers. The bonding between Me and T 2 X 2 layers has been considered as largely ionic. In the T 2 X 2 layers covalent T-X and some T - T bonding have to be considered. A detailed discussion of this structure and of the bonding involved has been reported by Hoffmann (1987).
3.5 Elements of Structure-Type Systematic Description
tron diffraction investigation of this phase carried out by Reehuis et al. (1992) the positional and the magnetic structures were determined. The ordering of the magnetic moments of the Eu atoms and the relation (commensurability) between this ordering and that of the atomic positions were studied (see Sec. 3.4.4).
a) N
b)
+ +
*
o
*
o
3.5.2.6 Frank-Kasper Structures (cy-Phases, Laves Phases) and Samson Phases
d/dmin
- Tetrahedrally Close-Packed Frank-Kasper Structures
+ 2
c) N * *
Oil*
lo 1
179
1.2
U
1.6
1.8
2
d/dmln
Figure 3-30. Distances and coordinations in the tllOThCr2Si2 type structure, (a) Coordination around Th: ( + ) Th-Si; (*) Th Th; (o) Th-Cr. (b) Coordination around Cr: ( + ) Cr-Si; (*) Cr-Cr; (o) Cr-Th. (c) Coordination around Si: ( + ) Si-Cr; (*) Si-Si; (o) Si-Th.
In the specific case of the RET 2 X 2 phases (RE = rare earth metal) the data concerning ten series (T = Mn, Fe, Co, Ni, Cu; X = Si, Ge) have been analyzed by Pearson (1985). It has been observed that the cell dimensions are generally controlled by RE — X contacts. In the case of Mn, however, the RE — Mn contact has to be assumed to control cell dimensions (see Sec. 3.6.2.3). Magnetic phase transition in RET 2 X 2 phases have been described by Szytula (1992). Structural distortions in some groups of RET 2 X 2 phases (REPt2Sn2), leading to less symmetric cells, have been reported by Latroche et al. (1992). An interesting compound belonging to the RET 2 X 2 family is EuCo 2 P 2 . In a neu-
A number of structures of several important intermetallic phases can be classified as tetrahedrally close-packed structures. As an introduction to this subject we may remember, according to Shoemaker and Shoemaker (1969) that in packing spheres of equal sizes the best space filling is obtained in the cubic or hexagonal close packed structures (or in their variants). In this arrangement there are tetrahedral and octahedral holes (see the comments on this point reported in the description of the cF4-Cu-type structure). The local density (the average space filling) is somewhat higher at the tetrahedral holes than in the octahedral ones. A more compact arrangement might, therefore, be obtained if it were be possible to have only tetrahedral interstices. It is, however, impossible to fill space with regular tetrahedra throughout. By introducing some variability in the sphere dimensions it is possible to obtain packing containing only tetrahedral holes. The tetrahedra are not regular: the ratio of the longest tetrahedron edge to the shortest, however, need not exceed about 4/3 in a given structure. The corresponding crystal structure can be considered to be obtained from the space filling of these tetrahedra (which share faces, edges and
180
3 Structure of Intermetallic Compounds and Phases
vertices). In structures containing atoms of "approximately" the same size, and within the mentioned limits of edge-length ratio, the sharing of a given tetrahedron edge either among 5 or 6 tetrahedra has to be considered the most favored situation (according to the systematic analysis of these structures carried out by Frank and Kasper, 1958, 1959). On the assumption that only 5 or 6 tetrahedra may share a given edge the number of tetrahedra that share a given vertex is limited to the values 12, 14, 15 and 16. The 12 (or 14, 15, 16) tetrahedra sharing a given vertex form, around this point, a coordination polyhedron with triangular faces. The radii of this polyhedron are the edges shared among 5 or 6 component tetrahedra and connect the central atom with the polyhedron (fivefold or six-fold) vertices (in which 5 or 6 faces meet). The four possible coordination polyhedra are shown in Fig. 3-31 and correspond to the following properties: Coordination 12 (regular, or approximately regular, icosahedron): 12 vertices (12 five-fold vertices) and 20 faces. Coordination 14:14 vertices (12 five-fold and 2 six-fold) and 24 faces. Coordination 15:15 vertices (12 five-fold and 3 six-fold) and 26 faces. Coordination 16:16 vertices (12 five-fold and 4 six-fold) and 28 faces. Several structures (Frank-Kasper structures) can be considered in which all atoms have either 12 (icosahedral), 14, 15 or 16 coordinations. These can be described as resulting from the polyhedra presented in Fig. 3-31. These polyhedra interpenetrate each other so that every vertex atom is again the center of another polyhedron. All structures in this family contain icosahedra and at least one other coordination type.
CA/12
CA/U
CAM 5
CAM 6
a)
b)
Figure 3-31. The coordination polyhedra of the Frank-Kasper structures are shown in two different styles: (a) the relative positions of the coordinating atoms (the central atoms are not reported). For the coordination numbers CN = 12 and 14, one atom of the coordination shell is not visible, (b) The corresponding triangulated polyhedra. Vertices in which 5 or 6 triangles meet are easily recognizable.
Frank and Kasper demonstrated that structures formed by the interpenetration of the four polyhedra contain planar or approximately planar layers of atoms. (Primary layers made up by tessellation of triangles with hexagons and/or pentagons were considered, and intervening secondary layers of triangles and/or squares.) For a classification and coding of the nets and of their stacking see Pearson (1972)
3.5 Elements of Structure-Type Systematic Description
181
Table 3-6. Examples of tetrahedral close-packed structures. Structural types
Unit cell dimensions for the reported prototype [pm]
% of atoms in the center of a polyhedron with CN 12
14
a= 455.5
25
75
a= 880.0 c= 454.4 a= 476.4 c = 2585.0
33
53
13
55
15
15
a= 543.3 c= 539.0
43
28
28
a= 930.3 b= 493.3 c = 1626.6
55
15
15
oP56 « Mo 21 Cr 9 Ni 20 a P phases
a = 1698.3 b= 475.2 c= 907.0
43
36
14
hR159»Mo 3 1 Cr 1 8 Co 5 1 a R phases
a = 1090.3 c = 1934.2
51
23
11
15
cI162-M gll Zn 11 Al 6 a
a = 1416.0
61
7
7
25
cF24-Cu2Mg hP12-MgZn2
a= 704.8 a= 518.0 c= 852.0
67 67
33 33
hP24-Ni2Mg
a = 482.4 c = 1582.6
67
33
cP8-Cr3Si (also called W 3 O or |3-W type or A15 type phase) tP30-aCr 46 Fe 54 a a phases hR39-W6Fe7 [i phases hP7-Zr4Al3
M phases
15
16
15
15
Laves phases:
a For these phases the reported formulae generally correspond to an average composition within a solid solution field. This is also in relation with a (partially) disordered occupation of the different sites.
and also Shoemaker and Shoemaker (1969) or Frank and Kasper (1958, 1959). A short summary of structural types pertaining to this family is reported in Table 3-6. For a few of them some details or comments are reported in the following. Structural type: cP8-Cr3Si also called W 3 O or p-W type (it was previously believed to be a W modification instead of an oxide) or A15 type (see Sec. 3.3.4).
Cubic, space group Pm3n, No. 223. Atomic positions: 2 Si in a) 0,0,0; 1/2,1/2,1/2. 6Cr in c) 1/4,0,1/2; 3/4,0,V2; Vi.V^O; 1
/2, 3 /4,0;0, 1 / 2 , 1 /4;0, 1 / 2 , 3 /4.
This structure type is observed for many phases formed in the composition ratio 3:1 by several transition metals with elements from the III, IV, V main group (or with Pt metals or Au). It is the structure of some superconductors with relatively high critical temperatures (for instance, Nb 3 Ge: Tc = 23.2 K).
182
3 Structure of Intermetallic Compounds and Phases
0 phase type structure (tP30-a, CrFe type) In the space group P42/mnm, No. 136, the two atomic species, Cr and Fe, are distributed in several sites with a nearly random occupation. Different atom distributions have been proposed in the literature (also owing to different preparation methods and heat treatments). The following distribution is one of those reported in Villars and Calvert (1991): two atoms in sites a) (with a 10% probability for Cr and 90% for Fe), 4 atoms (70% Cr, 30% Fe) in sites f), 8 atoms (59% Cr, 41% Fe) in a set i) of sites, 8 atoms (13% Cr, 87% Fe) in another set i) and 8 atoms (72% Cr, 28% Fe) in j). The structure can be considered as made up of primary hexagon-triangle layers containing 3636 + 3 2 6 2 and 63 nodes (in a 3:2:1 ratio) at height « 0 and V2 separated (at height « 1/4 and 3/4) by secondary 32434 layers (that is layers, in which every node is surrounded, in order, by 2 triangle, 1 square, 1 triangle and 1 square). As pointed out by Pearson (by studying the near-neighbors diagram) the a phase structure is a good example of a structure which is controlled by the coordination factor: all the known phases are closely grouped around the intersection of lines corresponding to high coordination numbers. (The most favorable radius ratio for the component atoms is included between 1.0 and 1.1.) It is also possible that the electron concentration plays some role in controlling the phase stability. The different phases are grouped in the range 6.2 to 7.5 electrons (s, p and d) per atom. Laves phases: cF24-Cu2Mg (and cF24Cu4MgSn and cF24-AuBe5), hP12-MgZn2 (and hP12-U2OsAl3) and hP24-Ni2Mg types. The Laves phases form a homeotect structure type set (a family of polytypic
structures). In all of them (described in terms of a hexagonal cell) three closely spaced 3 6 nets of atoms are followed (in the z direction of the cell) by a 3636 net (see Figs. 3-7 and 3-8). The 3 6 nets are stacked on the same site as the surrounding 3636 nets (for instance: (3BACyCAB ...). The Laves phases, as Frank-Kasper structures (see Table 3-6), can also be described by alternative stacking of pentagon-triangle main layers of atoms and secondary triangular layers [parallel to (110) planes of the hexagonal cell]. The importance of the geometrical factor in determining the stability of these phases has been pointed out (Pearson, 1972). The role of the electron concentration in controlling the differential stability of the different Laves phase types has been also observed. By studying, for instance, solid solutions of Cu 2 Mg and MgZn2 with Ag, Al, Si (Laves and Witte, 1936; Klee and Witte, 1954) it was observed that for an average VEC (valence electron concentration) included between 1.3 and 1.8 e/a (electrons per atom) the Cu 2 Mg structure is generally formed, for VEC values in the range from « 1.8 to 2.2 e/a generally the MgZn 2 type structure is obtained. The Ni 2 Mg type can be observed for intermediate values of VEC between 1.8 and 2.0. Many (binary and complex solid solutions) Laves phases are known. Typically Laves compounds XY 2 are formed in several systems of X metals such as alkalineearths, rare earths, actinides, Ti, Zr, Hf, etc., with Y = Al, Mg, group VIII metals, etc. Structural type: cF24-Cu2Mg and derivative structures Face-centered cubic cF24-Cu2Mg, space group Fd3m, No. 227. Atomic position:
183
3.5 Elements of Structure-Type Systematic Description
Cu
Be
Au
Figure 3-32. cF24-Cu2Mg type structure (1 unit cell is shown).
a) *
+ *
2
d/dm
2
d/tfmin
b)
+
*
1
1.2
U
1.6
1.8
Figure 3-33. Distances and coordinations in the cF24-Cu2Mg type structure, (a) Coordination around Mg: ( + ) Mg-Cu; (*) Mg-Mg. (b) Coordination around Cu: ( + ) Cu-Cu; (•) Cu-Mg.
8 Mgin a)0,0,0;0, V2, V2; V2909 V2; V2, V2,0;
o
Figure 3-34. Unit cell of the cF24-AuBe5 type structure. (Compare with the cF24-Cu2Mg type structure, Fig. 3-32).
Fig. 3-32 shows the Cu 2 Mg packing spheres structure. Normalized interaction distances and numbers of equidistant neighbors are shown in Fig. 3-33. Ordered variants of this type of structure are the Cu4MgSn type structure and the AuBe5 type structure. The packing spheres structure of AuBe5 is shown in Fig. 3-34. The atomic positions of the two structures correspond to the following occupation of the same equipoints in the space group F43m (No. 216). in Cu4MgSn in Au5Be a)0 ? 0,0;0,y 2 ,y 2 ; 4 Sn 4 Au y 2 ,o,y 2; y 2 , 290 4Mg 4 Be cVASASA\ %, %9%;
16 Cu in d) 5/8,5/8,5/8; %9%9%\ %95/*9%\
e)x,x,x;
Vs9Vs95/8'9 3 /8, 7 /8,y 8 ;
-x,-x,x;
7/
3/
1/ •
/8 5 /8 > /8 j
3
7/
1/
3/ •
/8 5 /8 5 / 8 ?
3
/8, 3 /8, 5 / 8 ; 7 / 8 , 7 /8, 5 / 8 ,
7/
5/
7/ • 3/
/8j /8j /8j
1/
7/ .
/8» /8 5 /8j
/8,5/8,3/8; V 8 , 3 / 8 , 7 / 8 ; y 8 , 7 / 8 , 3 /s; 5 /s, %, 3/8; y8,7/8, y8. Coordination formula: 3 "[Mg4/4][Cu6/6]12/6. For the prototype itself, Cu 2 Mg, a = 704.8 pm.
16 Cu
16 Be
-x,x,-x;x,-x,-x;
v* 1/. I •v" i/» I \-* *^5 /2 J^ -^5 / 2 T^ ^Vj
v
A> -A^j /2
Y" A> I v* -^5 / 2 1^ -^5
- x, y 2 + x, V2 - x; y 2 + x, x, V2 + x; v 2 - x, - x, y 2 + x; y 2 - x, x, y2 - x; V2 + x, Vi + x, x; Va - x, V2 — x, x; V2 - x, y2 + x, - x; x, V2 - x, V2 - x; V2 + x, — x, V2 — x; V2 + x, V2 — x, — x; (x « 0.625 = 5/8).
184
3 Structure of Intermetallic Compounds and Phases
We can see that the 8 atom equipoint of the Cu 2 Mg type structure has been subdivided into two different ordered 4 point subsets in the derivative structures. Layers stacking symbols, triangular, kagome (T, K) nets: Zn
Cu2Mg:
Mg 0.96-
Cu4MgSn:
o
Figure 3-35. Unit cell of the hP12-MgZn2 type structure.
Structural type: hP24-Ni2Mg AuBe5
Structural type: hP12-MgZn2 Hexagonal, space group P63/mmc, No. 194. Atomic positions: 2Znina)0,0,0;0,0, 1 / 2 . 4 Mg in f) V3,2/3,z; %, V3, y2 + z; 2
/3,y3,-z;y3,2/3,y2-z.
6 Zn in h) x, 2x, %; — 2x, — x, %; x, — x, %; — x, — 2x, %; 2x, x, 3A; -X,X,3/4.
Coordination formula: 333 [Mg 4/4 ][Zn 6/6 ] 12/6 . Layer stacking symbols, triangular, kagome (T, K) nets:
Zn£Mg5. 06 Zng. 25 Mgg. 44 Zn£ 50 Mg<;. 56 For the prototyp itself, MgZn 2 , a = 518 pm, c = 852 pm, c/a = 1.645, z = 0.062 and x = 0.830. Figure 3-35 shows the packing spheres structure for the MgZn 2 compound. A ternary ordered variant of this structure corresponds to three different atomic species in the three equipoint set. An example may be U 2 OsAl 3 (2Os in a), 4 U in f) and 6 Al in h).
Hexagonal, space group P63/mmc, No. 194. Atomic positions: 4Mg in e) 0,0, z; 0,0,V2 + z; 0,0, - z; 0,0,V 2 -z. 4 Mginf) V3,3/2,z; 2/3, V3, V2 +z; 2/3, V3, - z ; VsS/^Vi-z. 4Ni in f) V3,2/3,z; 2/3, Vs,1^ + z; 2 / 3 , 1 / 3 ,-z;V 3 , 2 / 3 , 1 / 2 -z. 6 Ni in g) V2,0,0; 0, V2,0; V2, V2,0; V2,0, V2; O,1/,,1/,; V., 1 /., 1 /.. 6Niinh)x,2x, Vi; —2x, — x, %; x, —x, %; — x, —2x,3/4; 2x,x, 3 A; —x,x,3/4. The structure can be described by the following layer stacking sequence triangular, kagome (T, K) nets:
Coordination formula: 3 33 o, [Mg4/4][Ni6/6]12/6. For the prototype itself, Ni 2 Mg, a = 482.4 pm, c = 1582.6 pm and z(e) = 0.094, z(fMg) = 0.8442, z(fNi) - 0.1251, and x(h) = 0.1643. - Structures based on frameworks offused polyhedra, Samson phases In addition to the Frank-Kasper phases, other structures may be considered in which the same four coordination polyhedra prevail although some regularity is
3.6 Regularities in Intermetallic Compound Structures and Semi-Empirical Approaches
185
by Cu atoms and 208 by Cu and Cd atoms in substitutional disorder.)
Figure 3-36. Truncated tetrahedron (Friauf polyhedron) related to the coordination number 16.
lost. Many of these structures and, in particular the giant cell structures studied by Samson (1969) can be described as based on frameworks of fused polyhedra rather than the full interpenetrating polyhedra. Among the most important polyhedra we may mention the truncated tetrahedron: it is shown in Fig. 3-36. It can be related to the CN 16 polyhedron (Friauf polyhedron) of Fig. 3-31. The two polyhedra can be transformed into each other by removing (adding) the 4 six-fold vertices of the CN 16 polyhedron (corresponding to positions out from the center of each of the 4 hexagons of the truncated tetrahedron). Several other coordination polyhedra occur in giant cell structures in addition to the Frank-Kasper polyhedra and to the truncated tetrahedron. (The most important are polyhedra corresponding to CN between 11 and 16.) The following phases represent a few examples of structures to which the mentioned considerations specially apply: cI58-a-Mn (a = 891.4 pm) type structure (and its binary variants, cI58-Ti5Re24 or X-phase and cI58-y-Mg17Al12), cF1124Cu4Cd3-type (a = 2587.1 pm); cFl192NaCd2-type (a = 3056 pm); cFl 832Mg2Al3 (a = 2823.9 pm), etc. (In the giant cell structures partial disorder and/or partial occupancy in some atomic positions have been generally reported, for cF1124Cu 4 Cd 3 , for instance, the structure was described as corresponding to the occupation of 388 atomic positions by Cd atoms, 528
3.6 Regularities in Intermetallic Compound Structures and Semi-Empirical Approaches to the Prediction of Compound Formation 3.6.1 Preliminary Remarks
As already mentioned in the previous paragraphs, several thousands of binary, ternary and quaternary intermetallic phases have been identified and their structures determined. In a comprehensive compilation such as that by Villars and Calvert abut 2200 (in the first edition, 1985) or about 2700 (second edition, 1991) different structural types have been described. The specific data concerning about 17 500 different intermetallic phases (pertaining to the mentioned structural types) have been reported in the 1st edition. As an introductory remark, a little statistical information about the phase and structure type distributions may be interesting. For this purpose, we may consider the group of phases described in the compilation by Villars and Calvert (1985). This, in fact seems to be considered a fairly representative sample even if the number of new intermetallic phases (and structural types) is constantly increasing. As a first observation we may notice that the number of phases pertaining to each structural type is not at all constant. Table 3-7 shows that a very high number of phases crystallize in a few more common structure types. About 25% of the known intermetallic phases belong to the first 10 more common structure types and about 50% of the phases belong to 39 types (that is « 2% of the known structural types).
186
3 Structure of Intermetallic Compounds and Phases
Table 3-7. Intermetallic phases: the most common structural types (from the data reported in Villars and Calvert, 1985). Number (Rank order)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
cF8-NaCl cF24-Cu2Mg tI10-BaAl4 cP2-CsCl hP12-MgZn2 cP4-AuCu3 cF4-Cu oP12-Co2Si hP9-Fe 2 P cI28-Th3P4 hP6-CaCu5 hP2-Mg hP16-Mn5Si3 hP3-AlB2 hP24-Ce6Al3S14 cI2-W cF56-MgAl2O4 cF16-BiF3 tP6-Cu2Sb oC8-CrB cF8-ZnS cF116-Th6Mn23 hP4-NiAs cP8-Cr3Si cP5-CaTiO3 oP16-Fe3C tI26-ThMn12 hP6-CaIn2 hP6-Ni2In hR12-NaCrS2 cF12-CaF2 oI12-CeCu2 hP5-La 2 O 3 oP8-FeB tI12-CuAl2 cF12-AgMgAs tI16-FeCuS2 tP2-AuCu hP38-Th 2 Ni 17
Integral sum
Number of phases belonging to eacrltype
Structural type Total
Binary
674 491 461 401 394 386 369 363 293 280 277 271 264 253 252 241 218 204 193 165 160 158 154 152 139 137 132 128 120 115 112 110 110 106 101 99 95 93 87
307 219 20 288 133 258 318 93 16 99 94 233 170 115 0 209 12 36 57 116 42 37 72 50 1 91 31 12 42 4 84 61 16 70 53 0 0 75 60
Ternary 367 272 441 113 261 128 51 270 277 181 183 38 94 138 252 32 206 168 136 49 118 121 81 102 138 46 101 116 78 111 28 49 94 36 48 99 95 18 27
674 1165 1626 2027 2421 2807 3176 3539 3832 4112(25%) 4389 4660 4924 5177 5429 5670 5888 6092 6285 6450 6610 6768 6922 7074 7213 7350 7482 7610 7730 7845 7957 8067 8177 8283 8384 8483 8578 8671 8758 (50%)
187
3.6 Regularities in Intermetaliic Compound Structures and Semi-Empirical Approaches
This kind of distribution seems to be significant even if Table 3-7 contains only an approximate list. (Some changes may actually be obtained by a more accurate attribution of different phases to a certain structural type or to its degnerate or derivative variants.) The distribution of the phases among the different types is summarized in Fig. 3-37, where (in a double logarithmic scale) the number of phases belonging to each structural type is plotted against the rank order of the type itself. In the same figure a curve is presented which has been computed by fitting the reported data by means of the Eq. (3-1): r0) -B
(3-1)
Log (frequency)
0
0.2
0.4
0.6
0.8 1.0 1.2 Log (rank)
1.4
1.6
1.8
2.0
Log (frequency) J.U
b)
2.5 2.0 1.5
where N{ is the number of phases corresponding to the structure type having rank r (A, B and r 0 are empirical constants whose values have been determined by the fitting). It may be interesting to point out that the mentioned equation is that suggested by Mandelbrot (1951) as a generalization of the Zipf's law (1949), which corresponds to the special case of r0 « 0 and B = 1. This law, in linguistic, relates for a given text the recurrence frequency (Nf) of a word to its rank (recurrence order). The formula had been deduced defining a cost function for the transmission of the linguistic information and minimizing the average cost. (The word cost was considered to be related to the complexity of the word itself.) We note, moreover, the larger numbers of phases having highly symmetric structures (cubic, hexagonal or tetragonal structures). The most frequent orthorhombic and monoclinic structures are the 8th and the 58th respectively in a general list such as reported in Table 3-7. This may be partially related to a certain greater ease in solving highly symmetric structures but
1.0 0.5 0 (D
0.2 0.4 0.6 0.8 1.0 1.2 Log (rank)
1.4 1.6 1.8 2.0
Log (frequency) 30 c)
2.5 2.0 1.5 1.0
~
\
0.5 0 C)
0.2 0.4 0.6 0.8 1.0 1.2 Log (rank)
1.4 1.6 1.8 2.0
Figure 3-37. Distribution of the intermetaliic phases among the structural types. In a double logarithmic diagram the phase numbers are plotted versus the rank order of the structural type. The continuous line corresponds to the Mandelbrot's equation, (a) Number of phases belonging to the overall different structural types. (Compare with Table 3-7). (b) Number of phases belonging to the cubic structural types, (c) Number of phases belonging to the hexagonal structural types.
188
3 Structure of Intermetallic Compounds and Phases
tain extent) of structural types, which, at least ideally, may be related to simple (1:2, 1:1, 1:3, 2:3, etc.) stoichiometric ratios. We have, however, to remark that, considering only selected groups of alloys, quite different stoichiometric ratios may be predominant. As an example we may mention the binary alloys formed by an element such as Ca, Sr, Ba, rare earth metals, actinides, etc., with Be, Zn, Cd, Hg and, to a certain extent, Mg. Many compounds are generally formed in these alloys. Among them phases having very high stoichiometric ratios are frequently observed, such as, for instance: CaBe 13 , LaBe 13 , BaZn 13 , BaCd 11? BaHg 6 , B a H g l l , BaHg 13 , La 2 Zn 17 , LaZn 13 , La 2 Cd 17 , LaCd 11? Th 2 Zn 17 , Pu 3 Zn 22 , Ce 5 Mg 41 , La 2 Mg 17 , LaMg 12 , etc.
Number of structural types 200
51
4000
61
71 81 Atomic %
Number of phases
3000-
2000-
1000
61
71 81 Atomic %
91
Figure 3-38. Distribution of intermetallic phases and structural types, according to the stoichiometry. (a) Distribution of the structural types; (b) distribution of the intermetallic phases.
probably also contains an indication of a stability criterion. Considering then the phase composition as a significant parameter, we obtain the histogram reported in Fig. 3-38 for the distribution of the structural types and of the intermetallic phases (as obtained from the 1st edition of Villars-Calvert) according to the stoichiometry of binary prototypes (that is, for instance, the binary and ternary Laves phases, the A1B2, Caln 2 , etc., type phases are all included in the number reported for the 66 to 67.99 stoichiometry range, even if the real stoichiometry of the specific phase is different). We may note the overall prevalence of phases and (to a cer-
3.6.2 Factors Controlling the Structure of Intermetallic Phases
The origin of structure of crystalline solids and the structural stability of compounds and solid solutions have been presented and discussed by Pettifor (see Chap. 2 of this Volume). In this section a brief sampling of some semiempirical useful correlations and methods of predicting phase (and structure) formation will be summarized. The search for regularities and criteria for the synthesis of new representatives of particular structure types has been carried out by many authors. Several factors were recognized to be important in controlling the structural stability and some of them were used as coordinates for the preparation of "classification and prediction maps", in which various compounds can be plotted and separated into different structure domains. Intermetallic phases, therefore, could be classified following the most important fac-
3.6 Regularities in Intermetallic Compound Structures and Semi-Empirical Approaches
tor which controls their crystal structure (Pearson, 1972; Westbrook, 1977; Girgis, 1983; Hafner, 1989). According to Pearson (1972), the following factors may be evidenced: - Chemical bond factor, - electrochemical factor (electronegativity difference), - energy band factor, electron concentration, - geometrical factors, - size factor. In the following paragraphs a few comments will be reported on this matter. Emphasis, however, will be given only to those aspects which are more directly related to a description of the "geometrical" characteristics of the phases. For the other questions reference should be made to other parts of this Volume (Chap. 2) and to Vols. 3 A, 4 or 11 of this Series. For an introduction to the electronic structure of extended systems, see Hoffmann (1987, 1988). 3.6.2.1 Chemical Bond Factor and Electrochemical Factor A chemical bond factor can be said to control the structure when interatomic distances (and as a consequence unit cell dimensions) can be said to be determined by a particular set of chemical bonds. Two different situations can be considered: bonds having high ionic characteristics (largely non-directional, the larger anions tend to form symmetrical coordination polyhedra subjected to the limitation related to the anion/cation atomic size ratio) or bonds having covalent character (the directional characteristic of which tend to determine the structural arrangement in the phase). To an increasing weight of the chemical bond factor (ionic and/or covalent bond-
189
ing) will, of course, correspond, in the limit, the formation of valence compounds. According to Parthe (1980) a compound Cm An can be called a normal valence compound if the number of valence electrons of cations (ec) and anions (eA) correspond to the relation (normal valence compound rule): m x ec = n x (8 — eA)
(3-2)
If we consider only the s and p block elements, the number of valence electrons of the elements correspond to their traditional group number. In this case (considering that no anions are formed from the elements of groups I, II and III) following formulae can be deduced for the normal valence compounds (formed in binary systems with large electronegativity difference between elements): - 144 - 224 - 3 4 4 3 - 135 - 2 3 5 2 - 35 - 4 3 5 4 - 126 - 26 - 3 2 6 3 - 462 - 5 2 6 5 - 17 - 272 - 373 - 47 4 - 575 - 676 (in these formulae each element is indicated by a number corresponding to its number of valence electrons; for instance: 17 represent NaCl, KC1, etc., 3 2 6 3 A12O3, etc.). In the more general case where some electrons are also considered to be used for bonds between cations and anions we have (general valence compound rule):
mx(ec-
ecc) = nx(8-eA-
eAA)
(3-3)
In this formula, which can only be applied if all bonds are two-electron bonds and additional electrons remain inactive in non-bonding orbitals (or, in other words, if the compound is semiconductor and has not metallic properties) ecc is the average number of valence electrons per cation which remain with the cation either in non bonding orbitals or (in polycationic valence compounds) in cation-cation bonds; similarly eAA can be assumed to be the av-
190
3 Structure of Intermetallic Compounds and Phases
The mentioned rule may be extended to include the defect tetrahedral structures where some atoms have less than four neighbors (general tetrahedral structure): (m x ec + n x eA) = 4 x (m + n)
erage number of anion-anion electron pair bonds per anion (in polyanionic valence compounds). In a more limited field than that of the previously considered general octet rule, it may be useful to mention the tetrahedral structures which form a subset of the general valence compounds. According to Parthe (1963, 1991), if each atom in a structure is surrounded by 4 nearest neighbors at the corner of a tetrahedron, the structure is called normal tetrahedral structure. The general formula of this structure, for the compound CmAM, is {normal tetrahedral structure) (m x ec + n x eA) =
+ Nmox(m
In this formula 7VNBO is the average number
of non-bonding orbitals per atom. By adding the symbol 0 (zero) to the described notation, vacant tetrahedral sites can be represented. Examples of formulae of defect tetrahedral structures are: 40 3 7 4 (Sil 4 , Snl 4 ); 4062 (GeS2); 3 6 05 4 6 3 (Ga 6 As 4 Se 3 ), 1252O64 (CuSbS2); etc. Notice that the mentioned compositional scheme is a necessary condition for building the tetrahedral structures, but not every compound that fulfills this condition is a tetrahedral compound. The influence of other parameters, such as the electronegativity difference, has been pointed out. By means of a diagram as shown in Fig. 3-39, the separation of tetrahedral structures from other structures may be evidenced (Mooser and Pearson, 1959). As a final comment to this point, we may mention that when one component in a binary alloy is very electropositive relative to the other, there is a strong tendency to form compounds of high stability in
(3-4)
For the same elements previously mentioned the possible combinations are: 4X4Y (all compositions, for instance, C, Ge, SiC), 35 (BP, AlSb, etc.), 26 (BeO, MgTe, ZnS), 17 (CuBr, Agl), 326, 337, 25 2 (ZnP 2 , ZnAs 2 ), 2 3 7 2 , 153 and 1263. [Ternary or more complex combinations may be obtained by a convenient addition of different binary formulae; for instance: 14253 = (153 + 44): for instance CuGe 2 P 3 , 1362 = (1 2 6 3 + 326)/2: CuAlS 2 , CuInTe 2 , etc., 122464 = (1 2 6 3 + 26 + 4): for instance Cu 2 FeSnS 4 , etc.]
0.1
0.5
0.9 Ax
1.3
+ n) (3-5)
1.7
Figure 3-39. Mooser-Pearson diagram separating AB compounds into covalent (o) and ionic (•) types after HumeRothery (1967). (The average n, quantum number, is plotted versus the electronegativity difference multiplied by the radius ratio.)
3.6 Regularities in Intermetallic Compound Structures and Semi-Empirical Approaches
which valence rules are satisfied (Pearson, 1972). Such alloys are considered to show a strong electrochemical factor. 3.6.2.2 Energy Band Factor and Electron Concentration The properties of a solid in principle could be calculated on the basis of the states of the electrons in the crystal. The status of the understanding of the structures of the solids and indications on the technical and computational problems have been presented in Chap. 2. We may mention here that if the stable crystal structure is mainly controlled by the number of electrons per atoms, the phase is called an electron compound. An important class of electron compounds (generally showing rather wide homogeneity ranges) are the Hume-Rothery phases. These include several groups of isostructural phases, each group corresponding to a given value of the so-called valence electron concentration (VEC). Three categories of Hume-Rothery phases are generally considered: those corresponding to VEC values of % (that is three valence electrons every two atoms), 21/i3 and 7A, respectively. Representatives of the Hume-Rothery phases are the following: VEC « 3/2, body centered cubic (cI2-Wtype): CuZn, « Cu3Al, « Cu5Sn, etc. VEC « 3/2, complex cubic (cP20-P Mntype): Cu5Si, Ag3Al, Au5Si, etc. VEC « 21/i3, complex cubic, 52 atoms in the unit cell (or superstructures) (cP52: ^Cu 9 Al 4 , ^ C u 9 G a 4 , Ag 9 ln 4 , « Co 5 Zn 21 , etc.; cI52: « Cu 5 Zn 8 , y-brass, « Ag 5 Cd 8 , Ag 5 Zn 8 , Ru 3 Be 10 , etc.; cF408: Fe 11 Zn 39 , etc.). VEC « 7 /4, hexagonal close-packed (hP2Mg-type or superstructures): ^AgZn 3 , « Au3Ge, « Ag5Al3, etc.
191
The VEC in all the mentioned cases, for which approximate "ideal" formulae have been indicated, were calculated assuming the following "valence": transition elements with non-filled d-shells: 0; Cu, Ag, Au: 1; Mg and Zn, Cd, Hg: 2; Al, Ga, In: 3; Si, Ge, Sn: 4; Sb: 5. The given ratios indicate ranges (which can even overlap). (It has to be noted, moreover, that the number of electrons to be considered may be uncertain.) The VEC values, therefore, indicate only a composition range where one of the mentioned structure types may occur. According to Girgis (1983) the existence field of the electron phases may be especially related to the combinations of d elements with the elements of the Periodic Table columns from 11 to 14 (from Cu to Si groups).
3.6.2.3 Geometrical Principles and Factors Laves (1956) when considering the factors which control the structures of the metallic elements presented three principles that are interrelated and mainly geometric in character: (a) The principle of efficient (economical) use of space (space-filling principle). (b) The principle of highest symmetry. (c) The principle of the greatest number of connections (connection principle). These principles may be considered to be valid to a certain extent for the intermetallic phase structures and not only for the metallic elements. a) Space-filling principle The tendency to use the space economically (to form structures with the best space-filling) which is especially exemplified by the closest-packing of spheres is considered to be the result of a specific
192
3 Structure of Intermetallic Compounds and Phases
principle which operates in the metal structures (and also in ionic and, to a lesser degree, in van der Waals structures). This principle is less applicable to covalent crystals because the characteristic interbond angles are not necessarily compatible with an efficient use of the space. Among the metallic elements, 58 metals possess a close packed arrangement (either cubic or hexagonal) which, in the assumption that the metal atoms are indeformable spheres having fixed diameters, corresponds to the best space-filling; 23 of the remaining metals crystallize in another highly symmetric structure, the body-centered cubic, which corresponds to a slightly less efficient space-filling. [The space-filling concept has been analysed and discussed by several authors: we may mention Laves (1956), Parthe (1961), Pearson (1972). A short summary of this discussion will be reported in the following, together with some considerations on the atomic dimension concept itself.] b) The principle of highest symmetry (symmetry principle) According to Laves a tendency to build configurations with high symmetry is evident and is called the symmetry principle. This tendency is particularly clear in metallic structures, especially in the simple ones. However, according to Hide and Andersson (1989), for instance, the validity extension of this principle is difficult to evaluate. As time passes, crystallographers are able to solve more and more complex crystal structures and these tend to have low symmetry. The symmetry principle could perhaps be restated by observing that a crystal structure has the highest symmetry compatible with efficient use of
space and the specific requirements of chemical bonding between nearest neighbors. c) The principle of the greatest number of connections (connection principle) To understand the meaning of this principle it may be at first necessary to define the concept of connection. To this end we may consider a certain crystal structure and imagine connecting each atom with the other atoms present in the structure by straight lines. There will be a shortest segment between any two atoms. We will then delete all links except the shortest ones. After this procedure the atoms that are still connected constitute a "connection" (the connection is homogeneous if it consists of structurally equivalent atoms, otherwise it is a heterogeneous connection). Such connections may be finite or 1, 2, 3 dimensionally infinite and are respectively called islands, chainsy nets or lattices. Symbols corresponding to the letters I, C, N, L (homogeneous connections) or i, c, n, 1 (heterogeneous connections) have been proposed (see also the dimensionality indexes mentioned in Sec. 3.3.5.1). As pointed out by Laves (1967) metallic elements and intermetallic phases show a tendency to form multidimensional (possibly homogeneous) connections (connection principle). 3.6.2.4 Atomic Dimensions and Structural Characteristics of the Phases a) Atomic radii and volumes A few comments about the atomic dimension concept may be useful also in order to present a few characteristic parameters and diagrams (such as space-filling parameters, reduced strain parameters, near-neighbor diagrams, etc.).
3.6 Regularities in Intermetallic Compound Structures and Semi-Empirical Approaches
Quoting from a comprehensive review on this subject (Simon, 1983) we may remember that ever since it has been possible to determine atomic distances in molecules and crystals experimentally, efforts have been made to draw conclusions from such distances about the nature of the chemical bonding and to compare interatomic distances (dimensions) in the compounds with those in the chemical elements. Distances between atoms in an element can be measured with high precision. As such, however, they cannot be simply used in predicting interatomic distances in the compounds. In rational procedure, reference values (atomic radii) have to be "extracted" from the individual (interatomic distances) measured values. Various functions have been suggested for this purpose. In the specific case of the metals it has been pointed out that interatomic distances depend primarily on the number of ligands and on the number of valence electrons of the atoms (Pearson, 1972). Pauling's rule (Pauling, 1947): n
= R1-30\ogn
(pm)
(3-6)
relating radii for bond order (bond strength) n (number of valence electrons per ligand) to that of strength 1, gives a means of correcting radii for coordination and/or for effective valencies. It has been shown (Pearson, 1972; Simon, 1983) that, no matter what the limitations may be of any particular set of metallic radii (or valencies) that is adopted, the Pauling's relation appears to be reliable, giving a basis for comparing interatomic distances in metals. According to Simon (1983) slightly better results could be obtained changing the Pauling's formula to:
R^R^l-Alogn)
(3-7)
where A is not constant but can be represented as a function of the element valency.
193
The subsequent point is to select some system of (a set of) atomic radii which can be used when discussing interatomic distances. The radii given by Teatum et al. (1968) (and reported in Table 3-8, together with the assumed "valencies") are probably the most useful for discussing metallic alloys. These radii have been reported for a coordination number of 12; they were taken from the observed interatomic distances in the f.c. cubic (cF4-Cu type) structure and in the hexagonal close-packed hP2-Mg type structure (averaging the distances of the first two groups of 6 neighbors, if the axial ratio has not the ideal 1.633 ... value) or from the b.c. cI2-W type. Since the coordination is 8 in the cI2-W type structure, for the elements having this structure the observed radii were converted to coordination 12 by using a correction given by the formula: = 1.0316
8
2 (pm) (3-8)
which was empirically obtained from the properties of elements having at least two allotropic modifications, cI2-W type and either cF4-Cu type of hP2-Mg type. The radii in the two structures (calculated at the same temperature by means of the known expansion coefficients) were compared and used to construct the reported equation. For the other metals (that is for the more general problem of the radius conversion from any coordination to coordination number 12) a percentage correction was applied (by using a curve which ranges from about +2.8% for the conversion from CNSto CN12 to about + 20% for the conversion from CN3 to C/V12) as suggested by Laves (1956) in a detailed paper dealing with several aspects of crystal structure and atomic sizes. While dealing with atomic dimension concepts, atomic volumes may also be con-
194
3 Structure of Intermetallic Compounds and Phases
Table 3-8. Radii (CN 12) of the elements (from Teatum et al., 1968)a Element
H Li Be B C N O Na Mg Al Si P S K Ca Sc Ti V Cr Mn Mn Fe Co Ni Cu Zn Ga Ge As Se Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sn
Valence
Radius (pm)
Element
Valence
Radius (pm)
-1 1 2 3 4 -3 -2 1 2 3 4 -3 -2 1 2 3 4 5 6 5 7 8 9 10 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 10 1 2 3 2 4
77.9 156.2 112.8 92.0 87.6 82.5 89.7 191.1 160.2 143.2 132.2 124.1 125.0 237.6 197.4 164.1 146.2 134.6 128.2 130.7 125.4 127.4 125.2 124.6 127.8 139.4 135.3 137.8 136.6 141.2 254.6 215.1 177.3 160.2 146.8 140.0 136.5 133.9 134.5 137.6 144.5 156.8 166.6 163.1 158.0
Sb Te Cs Ba La Ce Ce Pr Nd Pm Sm Eu Eu Gd Tb Dy Ho Er Tm Yb Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po Fr Ra Ac Th Pa U Np Pu Am
5 6 1 2 3 3 4 3 3 3 3 2 3 3 3 3 3 3 3 2 3 3 4 5 6 7 8 9 10 1 2 3 4 5 6 1 2 3 4 5 6 6 5 4
157.1 164.2 273.1 223.6 187.7 184.6 167.2 182.8 182.2 180.9 180.2 204.1 179.8 180.1 178.3 177.5 176.7 175.8 174.7 193.9 174.1 173.5 158.0 146.7 140.8 137.5 135.3 135.7 138.7 144.2 159.4 171.6 175.0 168.9 177.4 280.0 229.4 187.8 179.8 162.6 154.3 152.8 159.2 173.0
The elements are arranged according to atomic number. Noble gases and halogens are not included.
3.6 Regularities in Intermetaliic Compound Structures and Semi-Empirical Approaches
sidered. A value of the volume per atom, Fat in a structure may be obtained from the room temperature lattice parameter data by calculating the volume of the unit cell and dividing by the number of atoms within the unit cell. See also the table reported by King (1983). An equivalent atomic radius could be obtained by computing, on the basis of the space-filling factor of the structure involved, the corresponding volume of a "spherical atom" using the relationship Fsph = 47rK3/3. In the cI2-W type (CN8) structure we have F s p h ^0.68F a t and in the cF4-Cu type, and in the "ideal" hP2-Mg type (CN12) structures we have Fsph « 0.74 Fat. Considering now the previously reported relationship between RCN12 and RCN8 we may compute for a given element, very little volume (Fat) changes in the allotropic transformation from a form with CN12 to the form with CAT 8. (The radius variation is nearly counterbalanced by the change in the space filling.) This generally is in agreement with the experimental observations (Pearson, 1972). We will see that on the basis of the atomic dimensions of the metals involved (expressed, for instance, as Rx — RY or RX/RY) many characteristic structural properties of a XnYm phase may be conveniently discussed and/or predicted (size factor effect). As a further comment to this point we may mention here two "rules", the Vegard's and the Biltz-Zen's rules, which have been formulated for solid solutions and to a certain extent for ordered compounds. These rules, mutually incompatible, are very seldom obeyed; they may, however, be useful either as approximations or for defining reference behaviors. The first one, Vegard's rule (1921), corresponds to an additivity rule for interatomic distances (or lattice parameters or "aver-
195
age" atomic diameters). For a solid solution A x B ^ x between two components of similar structure it takes the form: dAB = x x dA + (1 - x) x dB
(3-9)
The Biltz (1934) (or Zen, 1956) rule has been formulated as a volume additivity rule: V
= x x F + (1 — x) x F
(3-10)
These rules are only roughly verified in the general case (for the evaluation of interatomic distances weighted according to the composition and for a discussion on the calculation and prediction of the deviations from Vegard's rule see Pearson, 1972 and Simon, 1983). As contributions to the general question of an accurate prediction of the variation of the average atomic volume in alloying we may mention a few different approaches. Miedema and Niessen (1982) calculated atomic volumes and volume contractions on the basis of the same model and parameters used for the evaluation of the formation enthalpy of the alloy (see Sec. 3.6.3.4). In a simple model proposed by Hafner (1985) no difference of electronegativity and no charge transfer were considered. Volume (and energy) changes in the alloy formation were essentially related to elastic effects. Good results have been obtained for alloys formed between s and p block-elements. An empirical approach has been suggested by Merlo (1988). Deviations from Biltz-Zen trend have been discussed and represented as a function of a "charge transfer atomic parameter" which correlates with Pauling's electronegativity. This approach has been successfully employed for groups of binary alloys formed by the alkaline earths and the bivalent rare earth elements. Negative experimental deviations from Vegard's rule (and values of the volume
196
3 Structure of Intermetallic Compounds and Phases
contractions) have been sometimes considered as an approximate indication of the formation of strong bonds and related to more or less negative enthalpies of formation (Kubaschewski, 1967). This indication is only very poor in the general case. For selected groups of alloys, however, the existence of a correlation between the formation volume and enthalpy (AFform and Ai/form) has been pointed out (even only as an evaluation of relative trends). This is the case of the rare earth (RE) alloys. As noticed by Gschneidner (1969) considering the trivalent members of the lanthanide series, we may compare the atomic volume decreasing observed in the metals (RE) (lanthanide contraction) with the decreasing of the average atomic volume measured in a series of REMe x compounds. If this diminution is more (less) severe in the compounds than in the RE metal series, this is considered an indication that the bonding strength in the REMe x compounds increases (decreases) relative to that of the metal as we proceed along the series from La to Lu; the heats of formation are expected to increase (decrease) in the same order. To make this comparison the unit cell volumes of the compounds are divided by the atomic volumes of the pure metals. The volume ratio for the series of compounds are then divided for that corresponding to a selected rare earth, this giving a relative scale. If the resultant values increase with the atomic number of the rare earth metal, then the lanthanide contraction is less severe in the compounds (in comparison to the rare earth element) and a decrease of the heat of formation is expected [conversely if the relative volume ratio decreases, an increase of the heat of formation (more negative enthalpy of formation) is expected]. (Examples of this correspondence will be examined in Sec. 3.6.3, see also Fig. 3-47.)
b) Space-filling parameter
(and curves)
The space-filling parameter introduced by Laves (1956) and by Parthe (1961) gives a means of studying the relationships between atomic dimensions and structure. For a compound, it is defined by the ratio between the volume of atoms in a unit cell and the volume of unit cell. (3-11)
q> =
'cell
(ft,, Rt number and radius of type i atoms). To calculate the space filling value for a specific compound, one has to know the radii of the atoms and the lattice constant. Neither of these is needed for the construction of a space filling curve of a crystal structure type, it is sufficient to know the point positions of the atoms and the axial ratios. The curve is based on a hard sphere model of the atoms: the cell edges are expressed as functions of the atomic radii (Rx and Ry for a binary system) for the special cases of X-X, X-Y and Y-Y contacts. The parameter cp can then be given (and plotted) as a function of the RJRy ratio. Considering, for instance, the cF8-ZnSsphalerite type structure (Parthe, 1964) the space filling can be given by: "4TC
3~
(3-12)
where a is the cubic cell edge and Rx and Ry are the radii of the atoms in the a) and c) positions (4 Zn and 4 S, respectively) in the unit cell. (See the description of the structure in Sec. 3.5.2.3.) In the case that the two atoms (or, more accurately, the hard spheres) occupying the Zn and S sites are touching each other, then the sum of the two radii must be equal to one-quarter of the cubic cell diagonal. = a
(3-13)
3.6 Regularities in Intermetallic Compound Structures and Semi-Empirical Approaches
Figure 3-40. Space filling diagram for the CsCl, NaCl, NaTl and ZnS structures (from Parthe, 1961).
By expressing the unit cell volume as a function of the sum of the radii we obtain: (3-14) Introducing the radius ratio e = RJRy one obtains: (« + I) 3
(3-15)
This equation describes the middle section (0.225 < £ < 4.44) of the space-filling curve for the sphalerite type structure plotted (with log scales) in Fig. 3-40. (The other sections, 0<£<0.225 and 4.44 < £ < oo correspond to the cases in which Y-Y atoms or X-X atoms are touching.) In the cp versus s diagram every structure type is generally characterized by its own individually shaped space-filling curve. The space-filling curves, however, of all binary structures belonging to one homeotect structure set coincide with one curve (see Sec. 3.4.3). By assuming appropriate values for the radii Rx and Ry it is possible to compare, with the specific curve of a given structure, the points representing actual compounds.
197
Generally a good agreement is found for ionic structures (and/or compounds) while it is often observed that the q> versus s points for particular metallic phases lie above the space-filling curves, indicating a denser packing and emphasizing the lack of unique radii associated with X-X, X-Y, etc. contacts (compressible atom model) (Pearson, 1972). In the specific case of unary structures (element structures) providing that there are no variable atomic positional parameters or axial ratios, there is a unique space-filling parameter (independent of atomic size for every structure type). For the cF4-Cu type structure, for instance,
(3-16)
Assuming the atoms to be hard spheres a = 2^/2R, then
198
3 Structure of Intermetallic Compounds and Phases
c) Reduced strain parameter and near-neighbors diagrams By comparison between the space-filling theoretical curves and the actual values of intermetallic phases it has been observed that an incompressible sphere model of the atom is unsuitable when discussing metallic structures. Pearson (1972) suggested the use of a model which allows the atoms of a binary X - Y alloy to be compressed until subsequently (and according to the structure geometry) X-X, X-Y, Y - Y contacts are established. The contacts are considered to occur when the X-X, X - Y and Y - Y interatomic distances in the compound structure, dx, dXY and dY are equal to 2RX (=DX), RX + RY and 2 RY (= DY) (Rx, RY, Z)x, DY atomic radii and diameters, respectively). According to Pearson, the metallic radii choosen are those appropriate for the coordination of the atoms (compare with Sec. 3.6.2.4). The distances between all the close atoms in the structure may be expressed in terms of the cell (and atomic site) parameters. (As an example see, for instance, the phases XY 3 , AuCu 3 type, described in Sec. 3.3.5.5 and in Figs. 3-11 to 3-13. In these phases around each X atom there are 6 X atoms at a distance equal to the unit cell edge dx = a. Around the X atoms there are 12 Y atoms at a distance dXY = ay/2/2.) All these distances may thence be expressed as a function of one of them, selected as a reference. In the case of the AuCu 3 type phase, for instance: dXY =
(3-17)
A reduced strain parameter is then defined with reference to the arbitrarily selected set of contacts. With reference to the dx distances the strain parameter is iS = (Dx — dx)/DY. This parameter gives an
indication of the atomic dimension compression. It is computed, as a function of the ratio s = DX/DY = RX/RY, for the different kinds of interatomic contacts. In the mentioned AuCu3-type phases, we have 3 cases corresponding to X-X, X-Y and Y-Y contacts. If X-X atoms are touching dx = D x , then the strain parameter S x _ x will be (D x - DX)/DY = 0 for all the s values. If X-Y atoms are considered to be in contact dXY = dx y/l/2 will be equal to Vi (Dx + DY) so we will have:
=
s
x
x
-
x
D
D
(3-18)
5>
If, on the other hand, the Y-Y atoms are those which are considered to be in contact we will have: (3-19a) and SY Y
~ ~D
= 7 r - V 2 (3-19b)
The values of the strain parameters are then plotted, according to Pearson (1972), as a function of e = RX/RY- Several straight lines are obtained (see Figs. 3.41 to 3-43), the lines corresponding to the reference contact are horizontal and set at zero. What matters is only the relative position of the different straight lines (which does not change by taking another contact as the reference one: a rotation will only be obtained of the whole diagram). The diagram is called near-neighbor diagram. In the diagram, points may also be plotted which represent actual phases. (To this end the experimental dx, dXY, etc., values will be used.) According to Pearson (1972), when a point representing a specific phase has a
3.6 Regularities in Intermetallic Compound Structures and Semi-Empirical Approaches
199
X-X
o -0.2 «_
Figure 3-41. Near neighbor diagram for binary phases with XY3 formula belonging to the cP4-AuCu3 structural type (according to Pearson, 1972). The lines corresponding to the different contacts are shown. The points represent actual phases with the AuCu 3 type structure.
O Q_
*O -0.4
-0.6 0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.2
"oJ
X-X
0
a a c
-0.2
-0.4
Figure 3-42. Near-neighbor diagram for binary phases with MN 2 formula belonging to the cF8-ZnS structural type (according to Pearson, 1972).
-0.6
-0.8 0.6
0.8
1.0 1.2 e = /?x / tf Y
U
1.6
Figure 3-43. Near neighbor diagram for binary phases with XY2 formula belonging to the cF24-Cu2Mg structural type (according to Pearson, 1972).
larger value of the strain parameter than that of a particular contact line, then the contacts corresponding to that line are to be considered (on the basis of the Dx and DY assumed for the components) compressed. If, on the other hand, the experimental points lie below a line then those contacts have not been established. Figures 3-41 to 3-43 represent the data and the trend for a few structure types. For compounds having the cF8-ZnS sphalerite structure (see Sec. 3.5.2.3) it can be seen that the X-Y (Zn-S) bonds (correspond-
200
3 Structure of Intermetallic Compounds and Phases
ing to a tetrahedral coordination) are the most important in controlling the structural characteristics. The different points, representing actual compounds, are very close indeed, for a wide range of diameter ratio and of electronegativity differences to the line corresponding to the X-Y contacts. (The X-X and Y-Y contacts are not formed.) The structure can, therefore, be considered as formed by a skeleton of presumably covalent (and directional in character) X-Y bonds. An X-Y chemical bond can similarly be recognized as important in several compounds having cF12-CaF 2 type (or antitype), cF16-Li3Bi, hP3-CdI 2 , hP8-Na3As, etc., type structures. The different behaviors of more "metallic" phases can be seen in Fig. 3-41 and Fig. 3-43. The AuCu 3 type near-neighbor diagram (Fig. 3-41), on the other hand, shows the importance of contacts corresponding to high coordinations. A similar trend can be observed for the XY2 Laves phases (see Fig. 3-43 for the Cu2Mg-type) for which, moreover, a certain compression of the X-X contacts generally results (the X-X curve is, for s > 1.25, far below the data points). Many near-neighbor diagrams have been presented by Pearson (1972) and systematically discussed for several structure types in order to show the importance of factors such as geometrical bond factors in controlling occurrence and structural characteristics of different phases. d) Unit-cell dimension analysis While discussing the interest in an analysis of the dimensional characteristics of phases with given structures and reconsidering advantages and limitations of the near-neighbor diagrams, Pearson himself has proposed (1985) a new analytical method based on plots as functions of the
i
1
i
a)
0.3 ^Ni-Ge —
0.2 -
A
l«
,
^Ge-Ge
0.1 ^RE-Ge
0 -
-0.1 -
-0.2 --
^RE-Ni
- • —
-
2.45 c/a 2.40 -
2.35 -
Figure 3-44. RENi2Ge2 phases (RE = rare earth) with the tI10-ThCr2Si2 structure (from Pearson, 1985). (a) Plot of Atj [ = V2 (Dt + Dj) - du] versus DRE. (b) Plot of the c/a axial ratio of the cell versus DRE.
CN12 atomic diameters determined from elemental structures and in which attention is paid to the group and period of the component elements in the selection of subsets of the data of phases to be considered together. As an example of such an analysis we may consider the data reported in Fig. 3-44. Phases are considered which pertain to the tI10-ThCr2Si2 type; the structure contains three different position sets, as described in Sec. 3.5.2.5. It is one of the most populous
3.6 Regularities in Intermetallic Compound Structures and Semi-Empirical Approaches
of the different structure types. In particular, there are ten almost complete groups of data for RET 2 X 2 phases given by rare earth metals (RE) with T = Mn, Fe, Co, Ni, Cu and X = Si or Ge. The data reported in Fig. 3-44 are those concerning the RENi 2 Ge 2 compounds. According to Pearson (1985) and Pearson and Villars (1984) the contacts of interest between the three components are defined by the relation: Aij=V2{Di + Dj)-dij
(3-20)
where Dt, Dj are the atomic diameters and dtj is the interatomic distance between / and j atoms (obtained from the experimental structure data). Generally it has been observed that Atj varies linearly with DRE (for series with different RE but the same T and X components). A parameter ftj may thus be defined by:
If a specific ftj is of the order of zero (see, for instance, AKE_Ge in Fig. 3-44) this can be considered an indication that the particular ij contact is independent of change in DRE and therefore it can be assumed to control the cell dimensions (as the size of RE changes in the series of phases having the same T and X components). For the different RET 2 X 2 phases it was observed that / RE _ X ~ 0 for T = Fe, Co, Ni, Cu and X = Si, Ge, whereas / RE _ T ~ 0 for T = Mn. Structural aspects of chemical bonding in another family of phases formed by similar groups, RE-T-X, of elements (1:1:1 compounds) have been analyzed by using the same technique by Bazela (1987). For a general discussion on the dimensional analysis of the structures of the metallic phases with special reference to the hR57-Th 2 Zn 17 , tI26-ThMn 12 and hP6-CaCu 5 type structures see also Pearson (1980).
201
3.6.2.5 Alternative Definitions of Coordination Numbers
We have seen in the previous paragraphs that the determination of the coordination number of an atom in a structure is clearly recognized as an important point in the definition of that atom's contribution to the bulk material properties and in the characterization of the structure itself. Several properties (for instance, atomic size, atomic valence and magnetic properties and species stability and reactivity) are know to be coordination number dependent. In many cases the coordination number (or ligancy) of a central atom is readily obtained by enumerating the number of neighbors; we have seen, however, that there are numerous cases where the criteria for the enumeration procedure may be ambiguous. As an introductory summary of this point see, for instance, Carter (1978), O'Keeffe (1979). As already pointed out by Frank and Kasper (1958) the term "coordination number" has been used in two ways in crystallography. According to the first (more precisely defined, in principle) the coordination number (CN), is the number of the nearest neighbors to an atom. According to this definition in the hexagonal close-packed hP2-Mg type structure CN is 6 unless the axial ratio c/a has exactly the "ideal' value ^ 8 / 3 = 1.63299..., in which case it is 12 (see Fig. 3-21). In this structure the mentioned definition is seldom applied with rigour, the CN in the hP2-Mg type structure is generally regarded as 12, that is, even with c/a slightly different from the "ideal" value, not only the first group but also the very close second group distances are considered together. More difficulties arise in less symmetrical structures and when there is a high coordination number.
202
3 Structure of Intermetallic Compounds and Phases
Near neighbors with slightly different interatomic distances are often found and it may be difficult to determine (and to state in an unambiguous way) how many should be considered as coordinating the central atom. Several schemes for the calculation of an "effective" coordination have been proposed. According to Frank and Kasper (1958) a calculation of the coordination number may be based on the definition of the "domain" of an atom in a structure as the space in which all points are nearer to the center of that atom than to any other. It is a polyhedron (Voronoi polyhedron, Voronoi cell, Wigner-Seitz cell), each face of which is the plane equidistant between that atom and a neighbor. (Every atom whose domain has a face in common with the domain of the central atom is, by the Fank-Kasper definition, one of its neighbors.) The counting of the faces of the domain polyhedron gives the number of neighbors: the set of neighbors is the "coordination shell". [The coordination polyhedron, of course, is the polyhedron whose edges are the lines joining all the atoms in the coordination shell. The domain (Voronoi) polyhedron and the coordination polyhedron, therefore, stand in dual relationship, each having a vertex corresponding to each face of the other.] According to the Fank-Kasper definition the coordination number is unambiguously 12 in the hexagonal closepacked metals and assumes the value 14 in a body-centered cubic metal. Generally in several complex metallic structures this definition yields reasonable values such as 14, even when the nearest neighbor definition would give 1 or 2. According, for instance, to O'Keeffe, however, this definition may lead to some difficulties (the value 14 for the b.c.c. structure, higher than that of closest packing does not seem entirely reasonable; the
difficulty becomes more acute in a structure as that of diamond for which a very high value, 16, is computed according to the mentioned definition). For a better quantification of the coordination number, several alternative schemes have been proposed. For example, a simple procedure is based on the identification of a gap in the list of interatomic distances (and to add atoms up to this gap). A similar procedure (O'Keeffe, 1979) may be to add atoms to the coordination polyhedron in order of increasing interatomic distances and to stop when the next addition would result in a non-convex polyhedron. Brunner and Schwarzenbach (1971) suggested cutting off the coordinating atoms at the largest gap in the list of the interatomic distances (see also Sec. 3.6.2.6). According to Brunner (1977) the largest gap in the list of reciprocal interatomic distances is used to limit the coordination polyhedra. It has also been suggested to weight the contribution of the atoms according a weight that decreases with interatomic distances (Bhandary and Girgis, 1977) or according to a bond strenghts of the Pauling type (Brown and Shannon, 1973). Non-integral coordination numbers may of course be obtained. In relation with the Frank-Kasper proposal, previously reported, O'Keeffe (1979) suggested that coordinating atoms contribute faces to the Voronoi polyhedron around the central atom and their contributions are weighted in proportion to the solid angle subtended by that face at the center. By using this definition increasing values of the (weighted) CN coordination number are obtained for the structures: diamond (4.54), simple cubic (6), body-centered cubic (10.16), face-centered cubic (12) (in agreement with the increasing packing density).
3.6 Regularities in Intermetallic Compound Structures and Semi-Empirical Approaches
A more complex weighting scheme has been suggested by Carter (1978) on the basis of the following assumptions: The interactions of a central atom with its fth neighbor is considered as being measured by a certain parameter At The CN as a function of all the At should satisfy the following conditions: - CN(At) is dimensionless and > 1 if any neighbors with non-zero At exists. - CN (At) is a continuous function of the At (its slope may not be). - If N interactions exist such that A1 = A2 = ... = AN9 for all neighbors with non-zero Ai9 then CN(At) = N. - If some of the A{ are unequal, then CN(At)
203
together. Each contribution ECoNj quickly becomes vanishingly small with increasing atomic distances dj according to an expression such as ECoNj = exp (1 — (dj/dm)% where dm is a reference distance (the "mean fictive" atomic size) which has to be determined beforehand from the structure. (For a discussion on the "effective coordination number" its relation with atomic size, bond-strength, Madelung constant, etc., see also Simon, 1983.) 3.6.2.6 Atomic-Environment Classification of the Structure Types
Daams et al. (1992) in a review have given an important contribution to the problem of the classification of the structural types, reporting a complete description of the geometrical atomic environments found in the structure types of the cubic intermetallic compounds. To define an atomic environment they used the maximum-gap rule (see Sec. 3.6.2.5). The Brunner-Schwarzenbach (1971) method was considered, where all interatomic distances between an atom and its neighbors are plotted in a histogram such as those shown in Figs. 3-14, 3-18, 3-20, etc. (The height of the bars is proportional to the number of neighbors and all distances are expressed by means of the values relative to the shortest one.) In most cases a clear maximum gap is revealed. The atomic-environment is constructed with the atoms to the left of this gap. To avoid, in some particular cases, bad or ambiguous descriptions, however, a few additional rules have been considered. In those cases where two or more, nearly equal, maximum gaps were observed, a selection was made in order to keep the number of different atomic-environment types in a given structure type as small as possible. A convexity criterium for the environment poly-
204
3 Structure of Intermetallic Compounds and Phases
hedron was also considered. The atomicenvironment types are coded by means of symbols representing the number of triangles, squares (and, if that is the case, pentagons and hexagons) that, in this order, join each other in the different vertices (coordinating atoms). For example, a quadratic pyramid has four corners adjoining two triangles and one square (no pentagons or hexagons) and one corner adjoining four triangles; its code, therefore, is 42.1.0.0^.0.0.0
(or?
briefly?
42.44.0)
The
03
cube has the code 8 , the octahedron 6 4 0 and the Fank-Kasper polyhedra have the codes 12 5 0 , 12 5 0 2 6 0 , 12 50 3 6 - 0 and 12 5.o 4 6.o
(see
Sec
35
2 6 and Fig
3_31)
Daams et al. (1992) have analyzed all the cubic structure types reported in Villars and Calvert (1985), after excluding all oxides and a few types with improbable interatomic distances, thus leaving 128 structure types representing 5521 compounds. Their analysis showed that these cubic structure types have 13 917 atomic-environments (point sets). Of those environments 92% belong to one of the 21 most frequently occurring atomic-environment types. 3.6.3 Semi-Empirical Approaches to the Prediction of Compound and Structure Formation in Alloy Systems
In the previous sections a brief sampling of some correlations has been given which relate crystallochemical characteristics of the phase to the properties of the component elements. This group of correlations may be considered as a first reference point for a number of methods of predicting the formation, in a given system, of a compound and/or of a certain structure. It is well known that, in scientific literature, more and more space is dedicated to the question of the forecast of chemical equi-
libria in simple and complex systems. A clear indication of this interest, both from a general and a technological point of view, may be seen in the development and success of a number of monographs and periodic publications and proceedings on this subject. Several approaches to this problem have been considered: we may mention, with special attention to metal systems, the explicit over-all summary already presented by Kaufman et al. (see Kaufman and Bernstein, 1970, and the recent discussion by Massalski, 1989). The role of a thermodynamic approach is well known: a thermodynamic control, optimization and prediction of the phase diagram may be carried out by using methods such as those described by Kaufman and Nesor (1973), Ansara et al. (1978) and very successfully implemented by Lukas etal. (1977, 1982). The knowledge (or the prediction) of the intermediate phases which are formed in a certain alloy system may be considered as a preliminary step in the more general, and complex, problem of assessment and prediction of all the features of the phase equilibria and phase diagrams. Evidence has to be given to the phase stability problem (Massalski, 1989). The significant progress and the limits of the first principles calculations may be mentioned (Hafner, 1989; Pettifor, Chap. 2 of this Volume), the usefulness, however, of a number of semiempirical approaches has to be pointed out. Several schemes and criteria have been suggested to forecast and/or optimize the data concerning certain properties. In the following a short outline will be reported on some prediction methods based on selected correlations between elemental properties and structure formation.
3.6 Regularities in Intermetallic Compound Structures and Semi-Empirical Approaches
3.6.3.1 Stability Diagrams and Structure Maps Several authors have tried to classify and order the numerous data concerning the different intermetallic substances by using two (or three)-dimensional structure maps (stability, existence diagrams). These maps were prepared by selecting coordinates based on those parameters (generally properties of the component elements) which were considered to be determinant factors of the structural stability formation control. As an introductory example to this subject we may remember the well known diagrams developed by Darken and Gurry (1953) for solid solution prediction. In such a diagram (as shown in Fig. 3-45) all elements may be included. The two coordinates represent the atomic size (generally the radius corresponding to CN12) and the electronegativity of the elements. It is
205
well-known that the first table of electronegativity values was introduced by Pauling (1932). Several alternative definitions have since been proposed. A reliable compilation extensively used in discussing the metallurgical behavior is that by Teatum et al. (1968). References to other scales will be reported later. To determine the solid solubility of the different elements in a given metal, in the Darken and Gurry map, the region with the selected metal (Mo, for instance, in Fig. 3-45) in the center can be considered. Generally we observe that elements which have high solubility lie inside a small region around the selected metal. As a rule of thumb an ellipse may be drawn in the diagram (with the selected metal in the center), for instance, with +0.3 electronegativity unity difference in one axis and + 15% atomic radius difference on the other axis. For a review of the application of the Darken and Gurry method to predict solid
RuPdPt Oslr
2.2 B
*>
\
"5
/Cu Ag 1
§1.8 o
Tin
Q) Q)
Io 1-6
Be
Q_
MnAlri Ta
•Zr
Hf
° L Mg
1.0
.Sr
L? Ma*0 100 Rx
180 (CA/12)
Figure 3-45. Darken and Gurry diagram for the Mo element.
*Ba 220
pm
206
3 Structure of Intermetallic Compounds and Phases
solubilities see Gschneidner (1980). An improvement of the method by means of simultaneous use of rules based on the electronic and crystal structures of the metals involved, is also presented. The diagrams reported in Figs. 3-39 and 3-46 are examples of other structure stability maps which have been suggested and successfully used in order to obtain a good separation (classification) of typical alloying behaviors (compound formation, crystallization in a certain structure type, etc.). As an outline of more general approaches along these lines we may mention a selection of a few methods proposed by several researchers. 3.6.3.2 The Savitskii-Gribulya-Kiselova Method
Cybernetic computer-learning methods have been proposed by Savitskii et al. (1980) for predicting the existence of intermetallic phases with a given structure and/ or with certain properties. The computer learning, in this case, is a process of collecting experimental evidence on the presence (or absence) of a property of interest in
various physicochemical systems (defined by means of a convenient selection of the properties of the components). As a result of machine learning a model is produced of characteristic exhibition of property (for instance, the formation of a particular type of chemical compound) which corresponds to a distribution "pattern" of this property in the multidimensional representative space of the properties of the elements. The subsequent pattern recognition corresponds to a criterium for the classification of the known compounds and for the prediction of those still unknown. Examples of this approach reported by Savitskii are the prediction of the formation of Laves phases, of CaCu 5 type phases of compounds XY 2 Z 4 (X, Y any of the elments, Z = O, S, Se, Te), etc. (Data on the electronic structures of the components were selected as input.) 3.6.3.3 Villars and Villars and Girgis Approaches
In an examination of the binary structure types (containing more than five rep-
0.4
0.3
compound forming systems
Liquid miscibility gaps Simple eutectic systems
0.2
0.1
-0.8
Laves phases and other compounds
Figure 3-46. Kubaschewski's plot of the regions of preference for formation of certain type of binary equilibrium diagrams (Rit Lt and Xt are atomic radius, heat of sublimation and electronegativity of element i).
3.6 Regularities in Intermetallic Compound Structures and Semi-Empirical Approaches
resentatives), Villars and Girgis (1982) observed that 85% exhibited the following regularities: (a) linear dependence of interatomic-distances on concentration weighted radii; (b) narrow ranges of the space-filling parameter and of the unit cell edge ratio c/a (and b/a) for the representatives of a given structure type; (c) dependence between the position of the elements in the periodic table (in the s, p, d, f blocks) and their equipoint occupation in the structure; (d) narrow grouping of the phases pertaining to a given structure type, in isostoichiometric diagrams based on the positions of the components in the Periodic Table. These relationships have been used to predict the existence and/or the structure type (and the unit cell characteristics) of binary intermetallic compounds. By using a systematic procedure to find the relevant element properties representing the alloying behavior of binary systems Villars (1983, 1985) defined three expressions for atomic properties which enable systems that form compounds to be separated from those that do not. A systematic elimination procedure was also used by Villars (1983) to find atomic property expressions which could be used to distinguish the crystal structures of intermetallic compounds. 182 sets of tabulated physical properties and calculated atomic properties were considered. These were combined, for binary phases, according to the modulus sums, differences and ratios. The best separations were obtained by using three-dimensional maps, which, for a binary A x B y , x < y compound, were based on the following variables (Villars and Hulliger, 1987; Villars et al., 1989):
207
X VE, sum of the valence electrons of the elements A and B, defined by E VE = [x/(x + y)] VEA + [y/(x + y)] VEB (3-23) AX, electronegativity difference, according to the Martynov-Batsanov (1980) scale defined by AX = [2 x/(x + y)] (XA - XB)
(3-24)
A(rs + r p ) z , difference of Zunger's pseudopotential radii sum (Zunger, 1981), defined by A(rs + r p ) z =
(3-25)
= [2x/(x + y)] [(r8 + r p ) z ? A - (rs + rp)ZfB] The relevant data concerning the different elements have been reported in Table 3-9 (Villars, 1983). Several structural types, corresponding to about 5500 binary compounds and alloys, were considered. 147 structure types were classified as 97 coordination types. The applications of these maps (which, in the most favorable cases, make it possible to predict not only the coordination number and polyhedron but also the structure type of a limited number of possibilities) were discussed. The possible extension to ternary and quaternary phases was also considered. As an example of an investigation of a selected group of ternary alloys we may mention a paper by Hovestreydt (1988). In analogy with the work of Villars a three-dimensional structure stability diagram was constructed. For the equiatomic RETX compounds formed by the rare earth metal (RE) with transition metal (T) and Ga, Si or Ge (X) the variables considered were: the difference in atomic radii rx — rRE, the Martynov-Batsanov electronegativity of the T metal and the expression G T + G x 4- Px> related to the position in the Periodic
Table 3-9. Valence electron number (VE), Martynov-Batsanov electronegativity {XMB) and Zunger's pseudo potential radii sum Rz (from Villars, 1983). ex
HI 2.10 1.25 Lil 0.90 1.61
Be 2 1.45 1.08
B3 1.90 0.795
C4 2.37 0.64
N5 2.85 0.54
O6 3.32 0.465
F7 3.78 0.405
Nal 0.89 2.65
Mg2 1.31 2.03
A13 1.64 1.675
Si 4 1.98 1.42
P5 2.32 1.24
S6 2.65 1.10
C17 2.98 1.01
Kl 0.80 3.69
Ca2 1.17 3.00
Sc3 1.50 2.75
Ti4 1.86 2.58
V5 2.22 2.43
Cr6 2.00 2.44
Mn7 2.04 2.22
Fe8 1.67 2.11
Co 9 1.72 2.02
NilO 1.76 2.18
Cull 1.08 2.04
Znl2 1.44 1.88
Ga3 1.70 1.695
Ge4 1.99 1.56
As 5 2.27 1.415
Se6 2.54 1.285
Br7 2.83 1.20
Rbl 0.80 4.10
Sr2 1.13 3.21
Y3 1.41 2.94
Zr4 1.70 2.825
Nb5 2.03 2.76
Mo 6 1.94 2.72
Tc7 2.18 2.65
Ru8 1.97 2.605
Rh9 1.99 2.52
PdlO 2.08 2.45
Agll 1.07 2.375
Cdl2 1.40 2.215
In 3 1.63 2.05
Sn4 1.88 1.88
Sb5 2.14 1.765
Te6 2.38 1.67
17 2.76 1.585
Csl 0.77 4.31
Ba2 1.08 3.402
La 3 1.35 3.08
Hf4 1.73 2.91
Ta5 1.94 2.79
W6 1.79 2.735
Re 7 2.06 2.68
Os8 1.85 2.65
Ir9 1.87 2.628
PtlO 1.91 2.70
Aull 1.19 2.66
Hgl2 1.49 2.41
T13 1.69 2.235
Pb4 1.92 2.09
Bi5 2.14 1.997
Po6 2.40 1.90
At 7 2.64 1.83
Frl 0.70 4.37
Ra2 0.90 3.53
Ac 3 1.10 3.12
El VE
OD CO -^
c o c
CD
5 0
3
1 o'
O
o3 o cQ . (/> &>
u. Q "0 or 0) C/) CD W
Ce3 1.1 4.50
Pr3 1.1 4.48
Nd3 1.2 3.99
Pm3 1.15 3.99
Sm3 1.2 4.14
Eu3 1.15 3.94
Th3 1.3 4.98
Pa 3 1.5 4.96
U3 1.7 4.72
Np3 1.3 4.93
Pu3 1.3 4.91
Am 3 1.3 4.89
Gd3 1.1 3.91
Tb3 1.2 3.89
Dy3 1.15 3.67
Ho 3 1.2 3.65
Er3 1.2 3.63
Tm3 1.2 3.60
Yb3 1.1 3.59
Lu3 1.2 3.37
3.6 Regularities in Intermetallic Compound Structures and Semi-Empirical Approaches
Table of the T and X elements, where G is the group and P the period number. A good separation was obtained for the 8 structural types considered (corresponding to 202 compounds).
209
empirical one it is important to observe that the model incorporates basic physics. A quantum-mechanical interpretation of Miedema's parameters has already been proposed by Chelikowsky and Phillips (1977, 1978). 3.6.3.4 Miedema's Theory Extensions of the model to complex aland Structural Information loy systems have been considered. As an interesting application we may mention The model for energy effects in alloys the discussion on the stabilities of ternary suggested by Miedema and coworkers is compounds presented by de Boer et al. well known. By assigning two coordinates (
210
3 Structure of Intermetallic Compounds and Phases
(For instance, alloys of the elements of the same group of the Periodic Table.) As an example we may mention the alloys of the rare earth metals (especially the "trivalent ones"). It is well known that, within this family of elements, several properties change according to well-recognized and systematic patterns. The atomic number itself can be used in this case as a simple and convenient chemical parameter. In several instances it has been pointed out that a systematic consideration of the crystal structures (and of the phase diagrams) of alloys formed by analogous elements (as those of the trivalent rare earth family) enables a number of empirical regularities to be deduced and theoretical statements to be made. [See a general discussion on this subject by Gschneidner (1969, 1971), the comments by Yatsenko (1979, 1983), Colinet (1984a, b) on alloys thermodynamics, the papers by Massalski (1989) on the applications of this behavior to phase diagram assessment, by
a)
-30.
1
Parthe and Chabot (1984), Rogl (1984) and by Iandelli and Palenzona (1979) for a systematic crystallochemical description. See also Sereni (1984) for a discussion of the properties of the rare earth metals themselves.] Criteria based on the mentioned characteristics have been used in assessment procedures and in the prediction of phase diagrams and of phase (and structure type) formation. Figure 3-47 may be considered as an example of such typical trends and of their correlations. Special applications (prediction of Pm-alloys) have been described by Saccone et al. (1990). Considering other families of similar compounds we may mention the contributions given by Guillermet et al. (1991,1992) (cohesive and thermodynamic properties, atomic average volumes, etc. of nitrides, borides, etc. of transition metals). They are another example of systematic descriptions of selected groups of phases and of the use of special interpolation and extrapolation procedures to predict specific properties.
-40.
-50
5" -60-
•S'S
1/
H
a o aB
10
~6~
0.9-
1.3-
"a 0.9 -I -a
0.5-
La Ce Pr Nd .
Lu
La Ce Pr ...
Lu
Figure 3-47. Gschneidner's plots for some rare earth (RE) alloys, (a) REIn3 compounds, (b) RET13 compounds. Following data are reported as a function of RE atomic number: Formation enthalpy, volume ratio relative to cerium (see Sec. 3.6.2.4) and reduced melting temperature XR. This is the ratio (Kelvin/Kelvin) of the melting point of the phase and of the melting point of the involved earth metal. (•) experimental values; (o) hypothetical values of TR computed for compounds with a constant melting point. All the diagrams show a decreasing phase stability with an increasing atomic number.
3.6 Regularities in Intermetallic Compound Structures and Semi-Empirical Approaches
3.6.3.6 Pettifor's Chemical Scale and Structure Maps
We have seen that in a phenomenological approach to the systematics of the crystal structures (and of other phase properties) several types of coordinates, derived from physical atomic properties, have been used for the preparation of (two, three-dimensional) stability maps. Differences, sums, ratios of properties such as eleo tronegativities, atomic radii, valence electron numbers have been used. These variables, however, as stressed, for instance, by Villars et al. (1989) do not always clearly differentiate between chemically different atoms. The difference in the bonding behavior of s, p and d electrons is not generally fully taken into account [in the cybernetic computer-learning methods proposed by Savitskii et al. (1980) a description, however, of the different atoms in terms of their electronic composition has also been sug-
211
gested for the preparation of the information matrix describing a physicochemical system]. In order to stress the chemical character of the elements and to simplify their description Pettifor (1984, 1985, 1986) (see also Chap. 2 of this Volume) created a new chemical scale (x) which orders the elements along a simple axis (see Table 3-1). The progressive order number of the elements in this scale (the so-called Mendeleev numbers M) may also be considered. These numbers M (which, or course, are different from the atomic numbers) start, according to Pettifor, with the least electronegative elements Hel, Ne2,... and end with the most electronegative ones ... N100, O 101, F102 up to H 103. For binary compounds (and alloys) XnYw (with a given n:m ratio) two-dimensional Xx> XY ( o r ^x> MY) maps m a Y be prepared (see Fig. 3-48 and also Figs. 2-2, 2-4 and 2-5 of Chap. 2). It has been proved that by using this ordering of the elements
Figure 3-48. Pettifor's map for AB compounds. The elements are arranged according to the Mendeleev number. The existence regions of the NaCl, CsCl and cubic ZnS are evidenced (see also Figs. 2-2, 2-4 and 2-5 of Chap. 2).
212
3 Structure of Intermetallic Compounds and Phases
an excellent structural separation may be obtained of the binary compounds of various stoichiometries (n:m= 1:1, 1:2, 1:3, 1:4,...,1:13, 2:3, 2:5,..., 2:17, 3:4,..., etc.) (Pettifor, 1986). See also Villars et al. (1989) who have updated the Pettifor maps for several stoichiometries. An extension of the application of these maps to the systematic description of certain groups of ternary alloys has been presented also by Pettifor (1988 a, b). Composition averaged Mendeleev numbers can be used, for instance, in the description of pseudobinary, ternary or quaternary alloys. All these maps show well defined domains of structural stability for a given stoichiometry, thus making the search easier for new ternary or quaternary alloys with a particular structure-type [and which, as a consequence, have the potential of interesting properties and applications (Pettifor, 1988 a, b)].
3,7 References Ansara, I., Bernard, C , Kaufman, L., Spencer, P. (1978), CALPHAD2, 1-15. Bazela, W. (1987), /. Less-Common Metals 133, 193200. Beaudry, B. X, Gschneidner Jr., K. A. (1978), in: Handbook on the Physics and Chemistry of Rare Earths, Vol. 1: Gschneidner Jr., K. A., Eyring, L. (Eds.). Amsterdam: North-Holland, pp. 173-232. Bhandary, K. K., Girgis, K. (1977), Ada Cryst. A 33, 903. Biltz, W. (1934), Raumchemie der Fes ten Stoffe, VossVerlag, Leipzig. Brown, I. D., Shannon, R. D. (1973), Acta Cryst. A 29, 266-280 Brunner, G. O. (1977), Acta Cryst. A 33, 226. Brunner, G. O., Schwarzenbach, D. (1971), Z. Krist. 133, 281 -292. Carter, F. L. (1978), Acta Cryst. B34, 2962-2966. Chelikowsky, I R., Phillips, J. C. (1977), Phys. Rev. Lett. 39, 1687. Chelikowsky, J. R., Phillips, J. C. (1978), Phys. Rev. B17, 2453. Colinet, C , Pasturel, A., Percheron-Guegan, A., Achard, J. C. (1984a), J. Less-Common Metals 102, 167.
Colinet, C , Pasturel, A., Percheron-Guegan, A., Achard, J. C. (1984b), /. Less-Common Metals 102, 239. Corbett, J. D. (1985), Chem. Rev. 85, 383-397. Daams, J. L. C , Villars, P., Van Vucht, J. H. N. (1991), Atlas of Crystal Structure Types for Intermetallic Phase. Materials Park, OH: ASM International. Daams, J. L. C , van Vucht, J. H. N., Villars, P. (1992), J. Alloys and Compounds 182, 1-33. Darken, L. S., Gurry, R. W. (1953), Physical Chemistry of Metals. New York: McGraw-Hill. deBoer, F. R., Boom, R., Mattens, W. C. M., Miedema, A. R., Niessen, A. K. (1988), Cohesion in Metals, Transition Metal Alloys. Amsterdam: North Holland. Donnay, J. D. H., Hellner, E., Niggli, A. (1964), Z. Krist. 120, 364-374. Dwight, A. E. (1974), Proc. 11th Rare Earth Res. Conf 1974, Oct. 7-10, Traverse City, MI, U.S.A. p. 642. Elliott, R. P. (1965), Constitution of Binary Alloys, 1st Supplement. New York: McGraw-Hill. Ferro, R., Girgis, K. (1990), J. Less-Common Metals 158, L41-L44. Flieher, G., Vollenkle, H., Nowotny, H. (1968 a), Monatsh. Chemie 99, 877. Flieher, G., Vollenkle, H., Nowotny, H. (1968 b), Monatsh. Chemie 99, 2408. Frank, F. C , Kasper, J. S. (1958), Acta Cryst. 11, 184-190. Frank, F. C , Kasper, J. S. (1959), Acta Cryst. 12, 483-499. Franzen, H. F. (1986), Physical Chemistry of Inorganic Crystalline Solids. Berlin: Springer. Frevel, L. K. (1985), Acta Cryst. B41, 304-310. Girgis, K. (1983), in: Physical Metallurgy: Cahn, R. W., Haasen, P. (Eds.). Amsterdam: North-Holland, pp. 220-269. Girgis, K., Villars, P. (1985), Monatsh. Chemie 116, 417-429. Grin', Yu. N., Akselrud, L. G. (1990), Acta Cryst. A46, Suppl., C-338. Grin', Yu. N., Yarmolyuk, Ya. P., Gladyshevskii, E. I. (1982), Sov. Phys. Cryst. 27, 413-417. Gschneidner Jr., K. A. (1969), J. Less-Common
Metals 17,1-12. Gschneidner Jr., K. A. (1980), in: Theory of Alloy Phase Formation: Bennett, L. H. (Ed.). Proceedings 108th AIME Annual Meeting, New Orleans, USA, February 19-20, 1979, pp. 1-39. Gschneidner Jr., K. A., McMasters, O. D. (1971), Monatsh. Chemie 102, 1499. Guillermet, A. F, Grimvall, G. (1991), /. Less-Common Metals 169, 257-281. Guillermet, A. F , Frisk, K. (1992), Proceedings of CALPHAD XXI, June 14-19, Jerusalem, Israel, p. 38. Guinier, A., Bokij, G. B., Boll-Dornberger, K., Cowley, J. M., Durovic, S., Jagodzinski, H., Krishna,
3.7 References
P., De Wolff, P. M., Zvyagin, B. B., Cox, D. E., Goodman, P., Hahn, Th., Kuchitsu, K., Abrahams, S. C. (1984), Acta Cryst. A40, 399-404. Hagg, G. (1931), Z. Phys. Chem. B12, 33. Hafner, J. (1985), / Phys. F. 15, L43-L48. Hafner, J. (1989), in: The Structures of Binary Compounds, Vol. 2 of Cohesion and Structure: de Boer, F. R, Pettifor, D. G. (Eds.). Amsterdam: NorthHolland, pp. 147-286. Hahn, T. (Ed.) (1989), International Tables for Crystallography, Vol. A. Dordrecht: Kluwer Academic Publishers. Hansen, M., Anderko, K. (1958), Constitution of Binary Alloys. New York: McGraw-Hill. Hawthorne, F. C. (1983), Acta Cryst. A 39, 724-736. Hellner, E. E. (1979), The Frameworks (Bauverbdnde) of the Cubic Structure Types, Vol. 37 of Structure and Bonding. Berlin: Springer. Hoffmann, R. (1987), Angew. Chem. Int. Ed. EngL 26, 846-878. Hoffmann, R. (1988), Solids and Surfaces. New York: VCH Publishers. Hoppe, R. (1979), Z. Krist. 150, 23. Hoppe, R., Meyer, G. (1980), Z. Metallkd. 71, 347. Hyde, B. G., Andersson, S. (1989), Inorganic Crystal Structures. New York: Wiley. Hovestreydt, E. (1988), /. Less-Common Metals 143, 25-30. Hume-Rothery, W. (1926), J. Inst. Metals 35, 295-307. Hume-Rothery, W, Mabbott, G. W., Channel Evans, K. M. (1934), Phil. Trans. Roy. Soc. A233, 1. Hume-Rothery, W. (1967), in: Phase Stability in Metals and Alloys: Rudman, P. S., Stringer, J., Jaffee, R. I. (Eds.). New York: McGraw-Hill, pp. 3-23. Iandelli, A., Palenzona, A. (1979), in: Handbook on the Physics and Chemistry of Rare earths, Vol. 2: Gschneidner Jr., K. A., Eyring, L. (Eds.). Amsterdam: North-Holland, pp. 1-54. Jagodzinski, H. (1954), Ada Cryst. 7, 17-25. Jeitschko, W, Parthe, E. (1967), Acta Cryst. 22, 417. Jensen, W. B. (1984), in: Communicating Solid-State Structures to the Uninitiated. Rochester: Institute of Technology. Jensen, W. B. (1989), in: The Structures of Binary Compounds, Vol. 2: Cohesion and Structure: de Boer, F. R., Pettifor, D. G. (Eds.). Amsterdam: North-Holland, pp. 105-146. Kaufman, L., Bernstein, H. (1970), Computer Calculation of Phase Diagrams, Vol. 4: Refractory Materials. New-York: Academic Press. Kaufman, L., Nesor, H. (1973), Ann. Rev. Mater. Set, Vol. 3: R. Huggins (Ed.), pp. 1. King, H. W. (1983), in: Physical Metallurgy: Cahn, R. W, Haasen, P. (Eds.). Amsterdam: North-Holland, pp. 37-72. Klee, H., Witte, H. (1954), Z. Phys. Chem. 202, 352. Kripyakevich, P. I. (1963), A Systematic Classification of Types of Intermetallic Structures, /. Struct. Chem. 4, 1-35.
213
Kripyakevich, P. I. (1976), Sov. Phys. Cryst. 21, 273276. Kripyakevich, P. I., Gladyshevskii, E. I. (1972), Acta Cryst. A 28, Suppl., S97. Kripyakevich, P. I., Grin', Yu. N. (1979), Sov. Phys. Cryst. 24,41-44. Kubaschewski, O., Evans, E. L. (1958), Metallurgical Thermochemistry. London: Pergamon Press. Kubaschewski, O. (1967), in: Phase Stability in Metals and Alloys: Rudman, P. S., Stringer, J., Jaffee, R. I. (Eds.). New York: McGraw-Hill, pp. 63-83. Landolt-Bornstein Tables (1971), Structure Data of Elements and Intermetallic Phases, Vol. 6: Hellwege, K.-H., Hellwege, A. M. (Eds.). Berlin: Springer. Latroche, M., Selsane, M., Godart, C , Schiffmacher, G., Thompson, J. D., Beyerman, W. P. (1992), /. Alloys and Compounds 178, 223-228. Laves, F. (1930), Z. Krist. 73, 303. Laves, F (1956), in: Theory of Alloy Phases. Cleveland, OH: American Society for Metals, pp. 124-198. Laves, F. (1967), in: Phase Stability in Metals and Alloys: Rudman, P. S., Stringer, I , Jaffee, R. I. (Eds.). New York: McGraw-Hill, pp. 85-99. Laves, F , Witte, H. (1936), Metallwirtschft 15, 840842. Leigh, G. J. (Ed.) (1990), Nomenclature of Inorganic Chemistry. Recommendations 1990. Oxford: Blackwell Scientific Publications. Lima-de-Faria, X, Figueiredo, M. O. (1976), /. Solid State Chem. 16, 1. Lima-de-Faria, I , Figueiredo, M. O. (1978), Garcia de Orto, Ser. Geol. 2, 69. Lima-de-Faria, I, Hellner, E., Liebau, F, Makovicky, E., Parthe, E. (1990), Acta Cryst. A 46, 1-11. Lukas, H. L., Henig, E.-Th., Zimmermann, B. (1977), CALPHAD 1, 225. Lukas, H. L., Weiss, I, Henig, E.-Th. (1982), CALPHAD 6, 229. Machatschki, F. (1938), Naturwissenschaften 26, 6787. Machatschki, F. (1946), Grundlagen der Allgemeinen Mineralogie und Kristallchemie. Berlin: Springer, pp. 146-190. Machatschki, F. (1947), Monatsh. Chem. 77, 333342. Machatschki, F. (1953), Spezielle Mineralogie aufGeochemischer Grundlage. Wien: Springer, pp. 1-11; pp. 298-353. Mandelbrot, B. (1951), C. R. Acad. Sc. Paris 232, 1638-1640. Marazza, R., Rossi, D., Ferro, R. (1980), J. LessCommon Metals 75, P25-P28. Marazza, R., Rossi, D., Ferro, R. (1988), J. LessCommon Metals 138, 189-193. Martynov, A. I., Batsanov, S. S. (1980), Russian J. Inorg. Chem 25, 1737-1739. Massalski, T. B. (1989), Metall. Trans. 20 A, 1295. Massalski, T. B. (Ed.) (1990), Binary Alloy Phase Diagrams, Vol. 1-3, 2nd ed.: Okamoto, H., Subra-
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3 Structure of Intermetallic Compounds and Phases
manian, P. R., Kacprzak, L. (Eds.)- Metals Park, OH: American Society for Metals. Merlo, F. (1988), J. Phys. F: Met. Phys. 18, 19051911. Miedema, A. R. (1973), /. Less-Common, Metals 32, 117-136. Miedema, A. R., Niessen, A. K. (1982), Physica 114B, 367-374. Moffatt, W. G. (1986), The Handbook of Binary Phase Diagrams, Vol. 1-5. Schenectady, N.Y.: Genium. Mooser, E., Pearson, W. B. (1959), Ada Cryst. 12, 1015. Niessen, A. K., de Boer, F. R., Boom, R., de Chatel, P. F, Mattens, W. C. M., Miedema, A. R. (1983), CALPHAD7, 51-71. Niggli, P. (1945), Grundlagen der Stereochemie. Basel: Birkhauser, p. 125. Niggli, P. (1948), Gesteine und Minerallagerstdtten, Vol. 1. Basel: Birkhauser, pp. 42-99. O'Keeffe, M. (1979), Acta Cryst. A 35, 772-775. Pani, M., Fornasini, M. L. (1990), Z. Krist. 190, 127133. Parthe, E. (1961), Z. Krist. 115, 52-79. Parthe, E. (1963), Z. Kristallogr. 119, 204-225. Parthe, E. (1964), Crystal Chemistry of Tetrahedral Structures. New York: Gordon and Breach. Parthe, E. (1969), in: Developments in the Structural Chemistry of Alloy Phases: Giessen, B. C. (Ed.). New York: Plenum Press, p. 49. Parthe, E. (1980), Acta Cryst. B36, 1-7. Parthe, E. (1980), "Valence and Tetrahedral Structure Compounds", in: Summer School on Inorganic Crystal Chemistry, Geneva: Parthe, E. (Ed.). Parthe, E. (1991), J. Phase Equilibria 12, 404-408. Parthe, E., Chabot, B. (1984), in: Handbook on the Physics and Chemistry of Rare Earths, Vol. 6: Gschneidner Jr., K. A., Eyring, L. (Eds.). Amsterdam: North-Holland, pp. 113-334. Parthe, E., Gelato, L. M. (1984), Acta Cryst. A 40, 169-183. Parthe, E., Chabot, B. A., Cenzual, K., (1985),
Chimia39, 164-174. Pauling, L. (1932), /. Am. Chem. Soc. 54, 35-70. Pauling, L. (1947), /. Amer. Chem. Soc. 69, 542. Pearson, W. B. (1967), A Handbook of Lattice Spacings and Structures of Metals and Alloys, Vol. 2. Oxford: Pergamon Press. Pearson, W. B. (1972), The Crystal Chemistry and Physics of Metals and Alloys. New York: Wiley-Interscience. Pearson, W. B. (1980), Z. Kristallogr. 151, 301-315. Pearson, W. B. (1985), /. Less-Common Metals 114, 17-25. Pearson, W. B., Villars, P. (1984), /. Less-Common Metals 97, 119. Pettifor, D. G. (1984), Solid State Communications 51, 31. Pettifor, D. G. (1985), /. Less-Common Metals 114,1. Pettifor, D. G. (1986), New Scientist, May 29, 48.
Pettifor, D. G. (1986), /. Phys. C19, 285. Pettifor, D. G. (1988a), Physica B 149, 3-10. Pettifor, D. G. (1988 b), Materials Science and Technology 4, 675-692. Petzow, G., Effenberg, G. (Eds.) (1990), Ternary Alloys. Weinheim, FRG: VCH. Rajasekharan, T., Girgis, K. (1983), Physical Review B 27, 910-920. Reehuis, M., Jeitschko, W., Moller, M. H., Brown, P. J. (1992), /. Phys. Chem. Solids 53, 687-690. Rogl, P. (1984), in: Handbook on the Physics and Chemistry of Rare Earths, Vol. 6: Gschneidner Jr., K. A., Eyring, L. (Eds.). Amsterdam: North-Holland, pp. 335-523. Rogl, P. (1985), /. Less-Common Metals 110, 283294. Rogl, P. (1991), in: Inorganic Reactions and Methods, Vol. 13: Hagen, A. (Ed.). New York: VCH-Publishers, pp. 85-167. Rossi, D., Marazza, R., Ferro, R. (1979), /. LessCommon Metals 66, P17-P25. Saccone, A., Delfino, S., Ferro, R. (1990), CALPHAD 14, 151-161. Samson, S. (1967), Acta Cryst. 23, 586. Samson, S. (1969), in: Developments in the Structural Chemistry of Alloy Phases: Giessen, B. C. (Ed.). New York: Plenum Press, pp. 65-106. Savitskii, E. M., Gribulya, V B., Kiselyova, N. N. (1980), J. Less-Common Metals 72, 307-315. Schwomma, O., Nowotny, H., Wittmann, A. (1964 a), Monatsh. Chemie 95, 1538. Schwomma, O., Preisinger, A., Nowotny, H., Wittmann, A. (1964b), Monatsh. Chemie 95, 1527. Schubert, K. (1964), Kristallstrukturen zweikomponentiger Phasen. Berlin: Springer. Sereni, J. G. (1984), J. Phys. Chem. Solids 45, 1219 — 1223. Shoemaker, C. B., Shoemaker, D. P. (1969), in: Developments in the Structural Chemistry of Alloy Phases: Giessen, B. C. (Ed.). New York: Plenum Press, pp. 107-139. Shunk, F A. (1969), Constitution of Binary Alloys, 2nd Supplement. New York: McGraw-Hill. Simon, A., (1983), Angew. Chem. Int. Ed. Engl. 22, 95-113. Smith, J. F. (Ed.) (1989), Phase Diagrams of Binary Vanadium Alloys. Metals Park, OH: American Society for Metals. Szytula, A. (1992), J. Alloys and Compounds 178,1-13. Teatum, E. T., Gschneidner Jr., K. A., Waber, J. T. (1968), Compilation of Calculated Data Useful in Predicting Metallurgical Behavior of the Elements in Binary Alloy Systems, Report LA-4003, UC-25, Metals, Ceramics and Materials, TID-4500, Los Alamos Scientific Laboratory. Vegard, L. (1921), Z. Phys. 5, 17. Villars, P. (1983), J. Less-Common Metals 92, 215238. Villars, P. (1985), J. Less-Common Metals 109, 9 3 115.
3.7 References
Villars, P., Calvert, L. D. (1985), Pearson's Handbook of Crystallographic Data for Intermetallic Phases, Vol. 1-3, 1st ed. Metals Park, OH: American Society for Metals. Villars, P., Calvert, L. D. (1991), Pearson's Handbook of Crystallographic Data for Intermetallic Phases, Vol. 1 - 4 , 2nd ed. Metals Park, OH: American Society for Metals. Villars, P., Girgis, K. (1982), Z. Metallhde. 73, 455462. Villars, P., Hulliger, F. (1987), /. Less-Common Metals 132, 289. Villars, P., Mathis, K., Hulliger, F. (1989), in: The Structures of Binary Compounds, Vol. 2: Cohesion and Structure: de Boer, F. R., Pettifor, D. G. (Eds.). Amsterdam: North-Holland, 1-103. Wells, A. F. (1970), Models in Structural Inorganic Chemistry. London: Oxford University Press. Wenski, G., Mewis, A. (1986), Z. Anorg. Allg. Chem. 543, 49-62. Westbrook, J. H. (1977), Metall. Trans. A 8 A, 13271360. Westgren, A. R, Phragmen, G. (1926), Z. Metallkd. 18, 279. Yatsenko, S. P., Semyannikov, A. A., Semenov, B. G., Chuntonov, K. A. (1979), /. Less-Common Metals 64, 185. Yatsenko, S. P., Semyannikov, A. A., Shakarov, H. O., Fedorova, E. G. (1983), J. Less-Common Metals 90, 95. Zen, E-an (1956), Amer. Min. 41, 523. Zhdanov, G. S. (1945), C. R. Acad. Sc. USSR 48, 39-42. Zintl, E., Woltersdorf, G. (1935), Z. Elektrochem. 41, 867. Zipf, G. K. (1949), Human Behavior and the Principle of Least Effort. New York: Addison-Wesley. Zvyagin, B. B. (1987), Sov. Phys. Crystallogr. 32(3),
394-399. Zunger, A. (1981), in: Structure and Bonding in Crystals, Vol. 1: O'Keeffe, M., Navrotski, A. (Eds.). New York: Academic Press, pp. 73-135.
General Reading Hahn, T. (Ed.) (1989), International Tables for Crystallography. Dordrecht: Kluwer. Hyde, B. G., Andersson, S. (1989), Inorganic Crystal Structures. New York: Wiley. Jensen, B. (1989), in: The Structures of Binary Compounds, Vol. 2: Cohesion and Structure: de Boer, F. R., Pettifor, D. G. (Eds.). Amsterdam: NorthHolland, pp. 105-146. Massalski, T. B. (1989), Metall. Trans. 20 A, 1295. Parthe, E. (1990), in: Elements of Inorganic Structural Chemistry: Sutter-Parthe, K. (Ed.). Geneva.
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Pearson, W. B. (1972), The Crystal Chemistry and Physics of Metals and Alloys. New York: WileyInterscience. Schubert, K. (1964), Kristallstrukturen zweikomponentiger Phasen. Berlin: Springer. Villars, P., Calvert, L. D. (1991), Pearson's Handbook of Crystallographic Data for Intermetallic Phases, Vols. 1-4, 2nd ed., and Atlas of Crystal Structure Types, Vols. 1-4. Metals Park, OH: American Society for Metals. Westbrook, J. H. (1977), Metall. Trans. 8 A, 13271360.
Appendix A The structure series are classified according to (a) the kind offragments and (b) the method of construction, (a) Homogeneous intergrowth structures consist of identical fragments, while inhomogeneous intergrowth structures consist of segments (differing in composition and/or coordination) belonging to different parent structures, (b) Intergrowth structure series can be one- (linear), two- or three-dimensional series. In a linear series we have one-dimensional stacking (along one direction) of two-dimensional, infinite segments (slabs) of the parent structures. Two-dimensional intergrowth series are built up from aggregations of several one-dimensional fragments (infinite rods, columns), and three-dimensional intergrowth series are constructed from (zero-dimensional, finite) parent structure blocks stacked in three dimensions.
Appendix B General compositional formulae are often used to represent a series. Mem + nX5m + 3nY2n, for instance, may be the overall formula of a series consisting of intergrown CaCu5-type and CeCo3B2-type slabs. (For the hP6-CaCu5, and its ordered variant hP6CeCo 3 B 2 , see Sec. 3.5.2.2.) Members of this series are: hP12-CeCo4B (corresponding to m = \, n = \), hP18-Ce3Co11B4 (m=l, n = 2), hP24-Ce2Co7B3 (m = \, n = 3), hP18-Nd 3 Ni 13 B 2 (m = 2, /i = l) and hP30-Lu5Ni19B6 (m = 2, « = 3). This series is an example of the close chemical similarity observed among compounds that correspond to several members of a given series. Often, members of a certain intergrowth structure series have representatives in the same (binary or ternary) alloy system. Representatives of the parent structures may also be found in the same system (or in chemically analogous systems). The attractiveness of a crystallochemical description based on the intergrowth concept is thus evident.
4 Structure of Amorphous and Molten Alloys Peter Lamparter and Siegfried Steeb Max-Planck-Institut fur Metallforschung, Institut fur Werkstoffwissenschaft, Stuttgart, Federal Republic of Germany
List of Symbols and Abbreviations 4.1 Introduction 4.2 Basic Equations for the Description of the Structure of Non-Crystalline Systems 4.2.1 Monatomic Systems 4.2.2 Binary Systems 4.2.2.1 Description with Atomic Pairs (Faber-Ziman Formalism) 4.2.2.2 Description with Number-Density and Concentration (Bhatia-Thornton Formalism) 4.2.2.3 Extended Fluctuations 4.3 Experimental Techniques 4.4 Structure of Amorphous Metallic Alloys 4.4.1 Metal-Metalloid Alloys 4.4.2 Metal-Metal Alloys 4.4.3 Structural Inhomogeneities 4.4.3.1 Phase Separation 4.4.3.2 Very Extended Fluctuations 4.4.3.3 Defects at the Outer Surfaces 4.4.4 Relaxation 4.4.5 Structural Models 4.5 Structure of Molten Metallic Alloys 4.5.1 Elements 4.5.2 Alloys 4.5.2.1 Compound-Forming Alloys 4.5.2.2 Segregation Alloys 4.5.2.3 Metal-Nonmetal Transition 4.6 Conclusions 4.7 References 4.7.1 Conference Proceedings 4.7.2 Other Literature
Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.
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4 Structure of Amorphous and Molten Alloys
List of Symbols and Abbreviations c D D* Ds
f(Q) G
g(R) G(R) H k
ko,k M(t,T) N n Q,Q R R R Re RG
Rs
R< RDF(i?) -S S(Q)
sx
T Tc
t
V
vm
V V
Y Z a 5
s = (T-Tc)/Tc
n
scattering amplitude concentration in atomic fractions diameter of an atom coefficient determinant fractal dimension atomic scattering length Gibbs free energy normalized distribution function pair correlation function enthalpy scattering intensity Boltzmann constant wavevector of incoming and of scattered radiation relaxation parameter apparent coordination number number of atoms scattering vector and its modulus atomic position distance molar gas constant correlation length of chemical order Guinier radius radius of a region correlation length of topological order radial distribution function Porod slope structure factor excess stability temperature critical temperature annealing time volume molar volume volume fraction viscosity packing fraction coordination number Warren-Cowley short-range order parameter scattering length density reduced temperature Cargill-Spaepen short-range order parameter isothermal compressibility
List of Symbols and Abbreviations
X v, y Q0 g (R) Qel
wavelength critical exponents number of atoms per unit volume density distribution function electrical resistivity electrical conductivity hard-sphere diameter dilatation factor scattering angle
A-L B-T CSRO DRPHS EXAFS FIM F-Z HNC LMS MC MD MSA NMR O- Z SANS SAXS SCD TEM TPP TSRO XANES
Ashcroft-Langreth formalism Bhatia-Thornton formalism chemical short-range order dense random packing of hard spheres extended X-ray absorption fine structure field ion microscopy Faber-Ziman formalism hypernetted chain approximation Laue monotonic scattering Monte Carlo calculation molecular dynamics calculation mean spherical approximation nuclear magnetic resonance Ornstein-Zernike small-angle neutron scattering small-angle X-ray scattering stereochemically defined model transmission electron microscopy trigonal prismatic packing topological short-range order X-ray absorption near edge structure
219
220
4 Structure of Amorphous and Molten Alloys
4.1 Introduction In a catalog of the different states of matter the liquid state and the amorphous state can be placed between gaseous materials and crystalline materials. Their specific properties have frequently been illustrated by comparison with those of gases and crystals. The absence of longrange order, as in the case of completely disordered gases, distinguishes them from periodically ordered crystals. On the other hand, within a range of several atomic distances liquids and amorphous solids are characterized by distinct ordering effects. Concerning the mobility of the atoms, liquids are closer to gases and amorphous solids are closer to the crystalline state. The local organization of the structure makes the frequently used term "disordered systems" somewhat misleading and, hence, we prefer the term non-crystalline systems. In fact, investigators use the expression "order" more often than the term "disorder" when they discuss structural properties of non-crystalline materials. We distinguish between topological order, which is the only order present in monatomic systems, and chemical order, which is important in alloys. The complex nature of the structure of non-crystalline systems was the reason for the relatively late development of theories on this state of matter. Liquids are usually in thermodynamic equilibrium, in contrast to the metastable amorphous systems, and theories for them are based on the principles of statistical mechanics, as reviewed, e.g., in the textbooks by March (1968), Faber (1972), Croxton (1975), and Shimoji (1977). Experimental exploration of the structure of molten metals by X-ray diffraction started in the 1930s (Debye and Menke, 1930).
A relatively new group of amorphous materials are the amorphous metals, also called metallic glasses. The first metallic glass, nickel-phosphorus, was obtained by electroless deposition by Brenner and Riddell in 1947 (Brenner and Riddell, 1947). However, it was the discovery of Duwez and co-workers that metallic glasses can be produced from the melt by rapid quenching which catalyzed the ever increasing interest in these materials (Klement et al., 1960; Duwez and Willens, 1963). Amorphous metals as a specific state of matter were a challenge for scientists to study both experimentally and theoretically. From a practical point of view metallic glasses are also very interesting. Due to their special mechanical, electronic, and magnetic properties, which are often superior to those of crystalline materials, they have increasingly found industrial applications. Series of international conferences concerned with liquid and amorphous metals have taken place in the last few decades: The Conferences on Liquid and Amorphous Metals, LAM (at Brookhaven, 1966; at Tokyo, 1972; at Bristol, 1976; LAM IV at Grenoble, 1980; LAM V at Los Angeles, 1983; LAM VI at GarmischPartenkirchen, 1986; LAM VII at Kyoto, 1989; LAM VIII at Vienna, 1992). The Conferences on Rapidly Quenched Metals, RQM (Metastable Metallic Alloys at Brela, 1970; RQM II at Cambridge, Mass., 1975; RQM III at Brighton, 1978; RQM IV at Sendai, 1981; RQMV at Wurzburg, 1984; RQM VI at Montreal, 1987; RQM VII at Stockholm, 1990). The Conferences on Non-Crystalline Materials, NCM (NCM 1 and NCM 2 at Cambridge, 1976 and 1982; NCM 3 at Grenoble, 1985; NCM 4 at Oxnard, 1988; NCM 5 at Sendai, 1991). The atomic structure is the basis for understanding the nature of the liquid and
4.2 Basic Equations for the Description of the Structure of Non-Crystalline Systems
amorphous state of matter. In this chapter we present an overview of the state of the art in this field as far as metallic systems are concerned. Emphasis is given to the point of view of an experimentalist, whereas theoretical work is referenced as necessary for comprehension of the essentials. The aim is not to give a comprehensive review of the available data, but rather to illustrate the principles by means of selected examples. Liquid metals and amorphous metals have many structural properties in common. However, there are characteristic differences, and hence they are discussed in separate sections.
4.2 Basic Equations for the Description of the Structure of Non-Crystalline Systems In this section a summary of the relationships needed for the characterization of the structure of non-crystalline materials is given. Only the static aspects are considered; i.e., time dependent phenomena, such as diffusion and collective excitations, which require inelastic scattering experiments, are beyond the scope of this presentation. The aim is not to present a detailed derivation of the equations, but certain steps are outlined which seem to be important for understanding the peculiarities in the description of non-crystalline alloys. A comprehensive development of the framework of equations may be found in the papers by Huijben (1978) and Wagner (1978). The two essential functions are the pair distribution function g(R), describing the atomic arrangement, and the structure factor S (Q), which is the Fourier transform of g(R) and which is accessible by a diffrac-
221
tion experiment. These two functions contain essentially the same structural information, but they emphasize different aspects of the structure. In many cases it is the structure factor which elucidates special structural properties rather than the pair distribution function. 4.2.1 Monatomic Systems Consider an atom at the position R relative to the origin (Fig. 4-1). An incoming radiation wave with wavevector k0 will excite a scattered wave from the atom with wavevector k and amplitude: A(Q) = /(Q)exp[— iQR]
(4-1)
Q = k — k0 is the scattering vector. For the case of elastic scattering, i.e., no energy transfer,
= \ko\=2n/X and 4TT
= Q = ~r sin 0 A
(4-2)
where X = wavelength of the radiation and 2 0 = scattering angle. f(Q) is the scattering amplitude of an isolated atom, also called the scattering length or scattering factor. It incorporates the interaction of an atom with radiation and depends on the kind of radiation. For X-rays and electrons it decreases monotonically with increasing scattering vector Q, whereas for neutrons it does not depend on Q. The Q-dependence of f(Q) will be omitted in the following equations. The phase factor exp[ — iQ-R] with phase angle QR is given by the path length difference (X/2n) (k R-k0 R) with respect to the scattering from an atom at the origin (see Fig. 4-1). The amplitude of the scattered wave of an ensemble of n atoms is: (4-3)
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4 Structure of Amorphous and Molten Alloys
Figure 4-1. Illustration of the path length difference [2/(2 n)] (k R-k0 R) between the waves scattered from an atom at the origin and from an atom at position R, respectively {k = ko = 2n/X).
and the intensity: (4-4)
This step clearly shows that the structure, if probed by diffraction experiments where in principle the intensity is recorded, is represented by atomic pairs v — /x, i.e., in terms of (two body) pair correlations. Higher-order correlations, like triplet correlations, are not accessible, although they of course contain important information about the three-dimensional structure, such as bond angles. Separating the terms where v = \i, we obtain for a monatomic system:
(4-5) In this equation the sum
was replaced
by taking rc-times the space (and time) average (denoted by <...» over configurations where in each case the origin is fixed at a central atom v with RVjll = R0^. This step shows that diffraction yields a spatial average for the structure, which in non-crystalline systems varies from site to
site for a selected central atom v. ROfi will be replaced by R in the following, but we have to bear in mind that an atom has to be assumed to be at R = 0. The average of the sum X o v e r the atoms neighboring a central atom can be replaced by an integral extending over the volume V of the system: I(Q) =
nf2(l+$Q(R)exp[-iQ-R]dR) (4-6)
The density distribution function Q(R) gives the number of atoms per unit volume at a distance R from a central atom. Its meaning is a statistical one, i.e., it represents a probability function. If the structure of a non-crystalline system is isotropic, Q(R) does not depend on the direction of R, Q (R) = Q (R), and the integration in Eq. (4-6) with dR = R2 dR dcj) dS can be performed with respect to the two polar coordinates cj) and 3 yielding the scalar expression: I(Q)=
(4-7)
The contribution associated with the average atomic density Q0 = n/V is subtracted, because this contribution is the volume
4.2 Basic Equations for the Description of the Structure of Non-Crystalline Systems
scattering of the entire system and is concentrated in the primary beam (zero-angle scattering). The structure factor is defined as: (4-8)
Equation (4-8) represents a Fourier transformation of Q(R) into S(Q) and this can be inverted to give: Q(R) =
QR Equations (4-8) and (4-9) are the basic equations for diffraction from non-crystalline systems; they relate the scattered intensity to the atomic distribution by Fourier transformation. S(Q) is a diffuse scattering function with a first peak followed by damped oscillations (see e.g. the examples in Fig. 4-37, Sec. 4.5.1). With increasing scattering vector Q it approaches unity. In practice S(Q) is only measured up to a maximum attainable Q value, which leads to so-called truncation errors when Q(R) is calculated from S(Q) by Fourier transformation: Experimental curves g(R) show additional nonphysical oscillations (spurious ripples). In the region of small R, preceding the nearest neighbor peak, g(R) is theoretically zero and thus the ripples are easily discernible. However, at larger R the artificial oscillations are superimposed on the physical ones. Furthermore, the peaks of Q (R) are affected by broadening, in addition to their intrinsic widths. In a reliable experiment nowadays S(Q) has to be recorded up to a Q value where the function Q2 [S{Q) — 1] in Eq. (4-9) attains negligible values.
223
The density distribution function Q(R) shows fluctuations around the average atomic density Q0 illustrating a certain degree of topological order in a non-crystalline system. The subsequent maxima show that the neighboring atoms can be assigned to a first, a second, and so on, coordination shell. However, in contrast to crystalline materials, this assignment is not unique because between the maxima Q(R) is never zero, and the atoms located near the minima can hardly be attributed to a certain shell. At larger distances the atomic positions are not spatially correlated and Q(R) approaches the average value Q0. The normalized pair distribution function g(R) = Q(R)/Q0 becomes 1. Below a minimum distance Rmin, Q{R) is zero because two atoms can only approach to a certain distance due to repulsive forces. Besides Q(R) and g(R) there are other definitions of atomic distribution functions in use, which, however, provide no additional information: G(R)
= 4KR[Q{R)-QO]
=
pair correla-
tion function. This definition has no obvious physical meaning, but is often used because G(R) is obtained directly as the Fourier transform of S(Q\ as can be seen from a simple rearrangement of Eq. (4-9), without needing to know the number density Q0 of the system. Contrarily, it yields an estimate of the possibly unknown Q09 because G(R) is given by — 4TT i^ ^ 0 at
= radial distribution function. RDF(#) dR represents the number of atoms within a spherical shell with thickness djR at a distance R. RDF(R) oscillates around The structural parameters are obtained from the distribution functions: The nearest neighbor distance is usually taken from the position Rl of the main
224
4 Structure of Amorphous and Molten Alloys
maximum of g(R) or G(R). The term "nearest distance" is somewhat misleading because Rl is the most probable distance, and is usually about 5 to 10% larger than the nearest possible distance Rmin. The coordination number Z 1 is obtained from the RDF(R) by integration over the main maximum:
Zl= J 4nR2Q(R)dR
(4-10)
Ri
where Rx and Ru define the first coordination shell. In practice problems arise because the coordination shells are not defined uniquely, as mentioned above, and hence the choice of the upper integration limit Ru, separating the first from the second shell, is somewhat arbitrary. Often a fitting procedure using Gaussian curves is performed. Note that Gaussian fitting should be performed using g(R) functions because the main peak of g(R) is expected to be more symmetric, whereas G(R) and RDF(i?) functions are wider at the right-hand side of their main peak. Due to these ambiguities coordination numbers can be derived with restricted accuracy to the order of 10%. An important structural parameter is the width AR1 of the nearest neighbor distribution, measured as the full maximum half width of the main peak of Q (R). In the case of amorphous alloys the width AR1 contains the structural disorder width AJR' and the thermal disorder width ARlh: (AR1)2 = (ARl)2 + (ARlh)2
(4-11)
A further parameter is the correlation length Rt of the topological ordering. Sometimes it is taken as the distance at which the oscillations of the correlation function G(R) fall below a certain level compared with the amplitude of the first maximum. Alternatively, jRt is calculated according to the Scherrer formula from the
full width at half maximum AQ1 of the first maximum of the structure factor S(Q): (4-12) 4.2.2 Binary Systems The formalisms to describe the structure of a system with K components are more complicated, requiring K(K+ l)/2 partial structure factors and partial pair correlation functions, respectively. In the following we restrict the discussion to binary systems, because here the essentials can be discussed using three partial functions. As the principles for the derivation of the diffraction equations are the same as for a monatomic system they will not be repeated in detail, but rather an intuitive approach is given. For several years two different approaches to characterize the structure of non-crystalline alloys have been in use; one based on atomic pairs, and the other based on the concept of local density and concentration. The two approaches are complementary and, as it turns out, adopting both methods for an investigated alloy system yields an extended insight into its structure. 4.2.2.1 Description with Atomic Pairs (Faber-Ziman Formalism) For a binary system A-B the total structure factor SFZ(Q), as measured in a diffraction experiment, is a weighted sum of three F - Z partial structure factors Stj as introduced by Faber and Ziman (1965):
clfl AA
(0
,
r\2
;S BB (0 (4-13)
where = c A / A + c B / B . Correspondingly, the total pair distribution function gFZ(R) is a sum of three
225
4.2 Basic Equations for the Description of the Structure of Non-Crystalline Systems
weighted F - Z partial pair distribution functions gtj: rL 2 f2
FZ
r22 frl C BJB
g (R) = AJ A
(4-14)
where gAB(R) = These relations can be understood intuitively, as an i-type atom in an i —j atomic pair contributes to the total structure function with its concentration c{ and with its scattering factor f{, normalized to the mean scattering factor . Within the Faber-Ziman formalism the total structure factor is defined as:
where = cA/A2 + cB/B2. In this definition the Laue monotonic scattering
=
cAcB(fA-fB)2
(4-16)
is subtracted. For the case of statistical distribution of two types of atoms with equal sizes the LM S scattering in fact appears as a structureless contribution. However, ordering effects and size effects cause oscillations of this Laue term, as will be described in Section 4.2.2.2. With the Faber-Ziman formalism the general behavior of the structure factors S{Q) and correlation functions g(R)9 total as well as partial ones, is the same as for a monatomic system, approaching a constant level of 1 at large Q and R, respectively. The relation between the structure factors and the pair distribution functions, for the total as well as the partial ones, is given by the same Fourier transformation as for a monatomic system [Eqs. (4-8), (4-9)],
where in the case of partial functions Q{R) has to be replaced by Qij{R)/cj = Qogij(R). Qij(R) is the number ofj-type atoms per unit volume at a distance R from a central i-type atom. At large distances it approaches CjQ0. The alternative definitions of the atomic distributions are: 4nQoR[gij(R)-l]
(4-17)
) = 4nR2Qij(R) = = cj[RGij(R)
Qo]
(4-18)
The partial coordination numbers Ztj are obtained from Qij(R) according to Eq. (4-10). It is important to note that for a multicomponent system the total functions S(Q) and g(R) depend on the scattering factors of the constituents, and thus on the radiation used in the diffraction experiment. Therefore structural parameters evaluated from total functions, i.e., the nearest neighbor distance and the coordination number, also depend on the scattering factors and thus generally have no obvious physical meaning. If we introduce gtj(R) = Qij(R)/(Q0Cj) and g(R) = Q(R)/Q0 in Eq. (4-14), multiply by 4nR2 and integrate from R = R{ up to R = Ru, we obtain the relation for the apparent coordination number Nl in terms of the partial coordination numbers Ztj: Nl
C A / A rr _ ^AJA z AA
, .
C
BJB
"+"
C
AJAJB
(4-19) AB
where the denotation "apparent" means that Nl depends on the scattering factors, i.e., on the radiation. This relation only makes sense if the size effect is small, i.e., if the first maxima of the three partial gtj{R) fall reasonably well into the same integration range RX
226
4 Structure of Amorphous and Molten Alloys
is even more complicated, and in many cases not possible, to set up a relation between the apparent nearest neighbor distance, taken as the maximum position Rl of g (R\ and the three individual distances #AA> #BB> and
RAB.
The aim of a detailed structural investigation of a binary system is to establish the complete set of the three partial structure factors and correlation functions. Characterization of the atomic ordering in a non-crystalline binary alloy is frequently performed by comparison of its structural parameters with the corresponding values expected in the hypothetical case of a statistical distribution of both atomic species. The definition of a statistical distribution model is not unique. In a strict sense it requires that the atoms have the same diameter. However, several models have been proposed which allow for a size effect. One could for example simulate the structure of a statistical reference alloy by a model calculation (see Section 4.4.5) using proper atomic diameters and no preferred chemical interactions between the different atomic pairs A-A, B-B, and A-B. The short-range order parameter is the most widely used parameter to characterize the degree of order in an alloy. The Warren-Cowley short-range order parameter a1 for the first coordination shell is defined on the basis of the partial coordination numbers (Cowley, 1950; Warren, 1969), and was first applied to non-crystalline materials in the following form (Steeb and Hezel, 1966): a1 = 1 -
(4-20)
where Z = total coordination number. For the case of statistical distribution ZAB is simply given by cBZ and a1 is zero. For the case of preferred hetero-coordination ZAB >cBZ and a1 < 0, and for preferred ho-
mologous coordination a > 1. The extreme case is a1 = + 1 where the alloy is segregated completely into its two constituents (ZAB = 0). For alloys with negative shortrange order parameters, if cA is small the minimum possible value is attained if ZAA = 0, i.e., ZAB = Z, and a1 = - cA/cB. At higher concentration cA, the minimum possible value is a1 > — cA/cB because even for strong hetero-coordination Z AA = 0 can no longer be realized (Spaepen and Cargill, 1985). In alloys with distinctly different atomic diameters the total coordination numbers around an A atom and around a B atom are not the same, ZA = ZAA + ZAB ^ Z B = ZBB + ZBA, and consequently the shortrange order parameter is different with respect to a central A atom and to a central B atom, respectively. For the case where ZA^ZB, an alternative definition of the short-range order parameter has been introduced by Cargill and Spaepen (1981): st
-1
(4-21)
where the coordination number for the case of statistical distribution of both atoms ZAB follows from entropy considerations as: C
A
ZA + cB ZB
(4-22)
Obviously, rjl has been defined with an opposite sign with respect to the corresponding a1, Eq. (4-20). For compound-forming alloys rjl is positive and the maximum possible value is attained if Z AB = ZA: wmax
(4-23)
where cA ZA
227
4.2 Basic Equations for the Description of the Structure of Non-Crystalline Systems
For the description of the structure with atomic pairs an alternative definition of the total structure factor is used in the socalled Ashcroft-Langreth formalism (Ashcroft and Langreth, 1967):
s AL «2) =
4.2.2.2 Description with Number-Density and Concentration (Bhatia-Thornton Formalism) Bhatia and Thornton (1970) introduced a complementary formalism to describe the structure of a binary non-crystalline alloy where the local atomic number density n(R) and the local concentration c(R) are the essential parameters. They showed how the diffracted intensity can be expressed in terms of the distance correlations between the fluctuations of the density and the concentration around their average values Q0 and c, respectively. The total structure factor, as defined originally by Ashcroft and Langreth [S BT (Q) = S A L ( 0 ] , is a weighted sum of the three Bhatia-Thornton partial structure factors:
, cAcB(A/)2
SNN(Q)=
Topological short-range order (TSRO) structure factor. It describes the correlations between density fluctuations.
(4-24)
We note that the A - L formalism uses a set of partial atomic-pair structure factors which are defined in a different way compared with the F - Z formalism. However, as these partial structure factors contain no further information, in contrast to those of the Bhatia-Thornton formalism described in the following section, they shall not be presented here [refer to e.g. Huijben (1978) and the textbook by Waseda (1980)].
SBT(Q) =
With this formalism the atomic ordering effects are separated into a topological and a chemical effect:
(4-25)
SCc (Q) = Chemical short-range order (CSRO) structure factor. It describes the correlations between concentration fluctuations. SNC(<2) = Size-effect structure factor. It describes the cross correlations between density fluctuations and concentration fluctuations. In real space the three Bhatia-Thornton partial correlation functions are: GNN(R) = 4nR[QNN(R)-Qo]
=
= 4nQoR[gNN(R)-l]
(4-26)
GCC(R) =4KRQcc(R)
=
4KQoRgcc(R)
G NC (R) = 4nRgNC(R)
= 4KQ0
RgNC(R)
They are related to the structure factors by Fourier inversion: GtJ(R) = ~R J Q2 [Stj(Q)-
StJ]
(4-27) where 3tj is the Kronecker symbol (5tj = 1 for i=j; dij = 0 for i^j). Of course the two sets of the F - Z and the B - T partial correlation functions are not independent, but can be converted into each other: GNN = c\ GAA + c\ GBB
2 cA cB GAB
= cA cB [GAA + GBB - 2 GAB] where A / = / A - / B .
dQ
=
C
A
( G AA -
G
AB)
~ CB ( G BB ~
(4-28)
228
4 Structure of Amorphous and Molten Alloys
These relations illustrate the physical significance of the B-T correlation functions. For an example see Fig. 4-13. The density-density correlation function GNN (R) describes the distribution of neighboring atoms around a central atom without considering their chemical nature. This function has in principle the same meaning as the pair correlation function of monatomic systems and yields a quite similar scattering contribution SNN(<2) with the usual main peak at Q1 followed by damped oscillations. The concentration-concentration correlation function GCC(R) involves the contributions of the like and the unlike atomic pairs, respectively, with opposite signs. For a statistical alloy these contributions cancel each other out and G c c = 0. The corresponding partial structure factor is S c c = 1. In this case the contribution of S c c to the intensity yields the Laue monotonic scattering LMS = c A c B (A/) 2 . In alloys with preferred hetero-coordination, so-called compound-forming alloys, GAB dominates in the first coordination shell and thus G cc < 0. In the second shell like next nearest neighbors are preferred and G cc > 0, whereas in the third shell G cc < 0 again, etc. This generates a larger period in GCC(R) compared with GNN(#), and hence a peak in the scattering contribution S c c (Q) at a smaller scattering vector Qp than that of the main peak at Ql. This "prepeak" preceding the main peak in a measured total structure factor was taken as evidence for a compound-forming tendency in an alloy before the B-T formalism was introduced (Steeb and Entress, 1966). The modulation of the Laue term in Eq. (4-25) due to the oscillations of SCC(Q) involves for Q->0 a decrease below the level c A c B (A/) 2 /. For the opposite case of a segregation tendency, i.e., dominating like nearest
neighbors A-A and B-B, GCC(R) is a positive function gradually approaching zero with increasing R. The corresponding scattering contribution SCC(Q) shows a rise towards small Q values, the so-called smallangle scattering effect. The decay of GCC{R) with increasing R is characterized by the correlation length Rc of the concentration fluctuations. The significant features of Scc(<2), depending on the type of preferred coordination in an alloy, are the reasons why in many cases the total structure factor already allows classification of the type and strength of the CSRO effect rather than the Fourier transform, without the knowledge of the individual partial structure factors. However, it is important to note that the CSRO contributes with the weight (A/) 2 to the total structure factor, and the investigator has to provide a sufficient difference in the scattering factors of the two components by choice of a suitable contrast variation technique, as described in Sec. 4.3. The density-concentration correlation function GNC(R) is zero if the atomic volumes of both types of atoms in an alloy are the same. If not, interchange of an A and a B atom at the same time would affect the local atomic density. This means that fluctuations in the density and the concentration are not independent, but correlated to a certain degree. An alloy with no preferred chemical interaction, but with a size effect, already has an oscillating GCC(R) function besides the oscillating GNC(R) function. The general behavior in this case can be visualized by imagining the Gfj functions in Eq. (4-28) with maxima of the same heights but at different positions Rtj. The short-range order parameter a1 according to the Warren-Cowley definition can also be obtained from the B-T partial distribution functions (Ruppersberg and Egger, 1975):
229
4.2 Basic Equations for the Description of the Structure of Non-Crystalline Systems
i
a =
The free energy-term can be expressed as:
4nR2gcc(R)dR
f
(4-29)
52G
kT
5c2
J
where the integration limits define the range of the first coordination shell given by the main maximum of QNN(R). For compound-forming systems QCc(R) < 0 a n d a* becomes negative.
4.2.2.3 Extended Fluctuations
In this section some equations are presented which are frequently adopted to describe structural features occurring on length scales larger than atomic distances. The long-wavelength limits of the B-T partial structure factors are related to the mean square fluctuations of the number density <(An)2> and of the concentration <(Ac)2> in subsystems much larger than the range of atomic distances. For noncrystalline alloys in thermal equilibrium, these limits are also related to thermodynamic properties (Bhatia and Thornton, 1970): SNN(0) = (4-30)
= /cT[52G/8cT
(4-31)
s NC (0) =
(4-32)
where k = Boltzmann constant T = temperature xT = isothermal compressibility 9 = (6VJ5c)/Vm = dilatation factor Vm = molar volume G = Gibbs free energy per atom
kT c(l-c)
(4-33)
where Sx is the excess stability function (Darken, 1967). For a random alloy Sx = 0 and Scc(0) = c(l—c). For a compoundforming alloy Sx > 0 and Sic(0) < c(l -c\ whereas for a segregation alloy Sx < 0 and ^cc(O) > c ( l ~c)- The extreme case of a compound-forming A-B alloy is one where the atoms form stable molecules. In this case, in a subsystem which is large compared with atomic distances, no concentration fluctuations exist because the ratio cA/cB is fixed. Therefore Ac = 0 in Eq. (4-31) and consequently ScC(0) = 0. The long-wavelength limit of the total structure factor is: S BT (0) =
-gokTxT +
(4-34) Scc(O)
In small angle studies of segregating alloys this relation is sometimes also assumed to hold for Q > 0 in order to estimate Scc(0 from an experimentally obtained SBT(Q) at small Q values. Amorphous alloys are not in thermodynamic equilibrium. However for those alloys which were obtained from the liquid state by rapid quenching the thermodynamic relations were nevertheless frequently applied, taking the temperature as the glass transition temperature at which the liquid structure is frozen in. Long-range fluctuations in segregating molten alloys can often be described by an Ornstein-Zernike (1918) behavior (see e.g. Stanley, 1971): S%(Q) = Scc(0)/(l + R2 Q2)
(4-35) (4-36)
230
4 Structure of Amorphous and Molten Alloys
where Rc = correlation length. Near a critical point, above a temperature Tc on the liquidus line in the phase diagram of a binary system, SQC(0) and Rc diverge according to: f (4-37) Rc = RcOs~v (4-38) where e = (T - Tc)/Tc is the reduced temperature, y and v are the so-called critical exponents, which can be calculated based on different theories. In solid non-crystalline alloys a variety of different non-equilibrium extended fluctuations may exist, giving rise to a smallangle scattering effect, which may depend substantially on the production conditions of the specimen and its thermal history. Of course it is not possible to give a description in terms of general equations covering all possible types of fluctuations for all kinds of specimens. The most simple case of inhomogeneities is well-defined particles embedded in a matrix, like crystallites in an amorphous alloy. Then the well-known small-angle scattering equations apply for the analysis of the intensity 7 ( 0 (see e.g. the textbook by Guinier and Fournet, 1955). The innermost part (QRG<1) of 7 ( 0 follows the Guinier approximation: 7 ( 0 = 7(0) e x p ( - Q 2 7^/3) (4-39) where RG = Guinier radius of the particles (radius of gyration), which for homogeneous spherical particles with radius Rs is RG = ^/3/5Rs. From the slope of the Guinier plot In 7 ( 0 versus Q2, RG can be determined. For a two-phase system, the integrated intensity, the so-called invariant, is given by: lQ2I(Q)dQ = 2n2v1v2(61-52)2 (4-40) where vt = volume fractions and 6t = scattering length densities of the two phases.
Equation (4-40) is used for the treatment of phase-separated systems. If the scattering length densities of the two phases are known, i.e., if their compositions and their atomic densities are known, the volume fractions vx and v2 = l—v1 can be determined from the integrated intensity.
4.3 Experimental Techniques The most direct method to explore the structure of matter on an atomic scale is the diffraction technique using different radiations. However, there are also other possibilities, most of them more indirect, which may provide additional information and thus are complementary. Nuclear magnetic resonance (NMR) and Mossbauer spectroscopy are nuclear methods used to probe the local order in a system. For applications of both techniques to metallic glasses see the proceedings of the "Int. Conf. on Amorphous Systems Investigated by Nuclear Methods" (1981), and the paper by Janot (1983), and for Mossbauer spectroscopy, the paper by Gonser and Preston (1983). In the present chapter we confine ourselves to diffraction methods. The wide-angle scattering regime contains information about the structure over a length of several atomic distances, whereas small-angle scattering, that is, in the range Q < 5 n m " 1 , is sensitive to extended fluctuations, often termed as medium range structure. Of course there is no strict separation, but the small-angle regime and the wide-angle regime overlap continuously in the range where structural inhomogeneities occur on a scale of atomic distances. The intensity curves measured with amorphous or liquid materials have to be subject to several correction procedures to obtain the corrected coherent scattering
4.3 Experimental Techniques
intensity I (Q) and finally the structure factor S(Q\ according to Eq. (4-15) or Eq. (4-24). For details see Wagner (1972,1978), Waseda (1980), Chieux (1978), Sadoc and Wagner (1983), and in particular, Huijben (1978). In these articles the different methods for the determination of partial structure factors by diffraction experiments are described, and are summarized briefly in the present section. In order to elucidate the atomic-scale structure of a non-crystalline alloy the atomic distances and the partial coordination numbers must be determined from the partial pair distribution functions gij(R). The gij(R) are the Fourier transforms of the partial structure factors S l 7 (0. In a diffraction experiment we obtain one total structure factor S(Q). In the case of a binary alloy S(Q) is a weighted sum of three partial structure factors Stj(Q)9 where the weighting factors contain the concentrations and the scattering lengths of the constituents [Eq. (4-13) and Eq. (4-25)]. Variation of the scattering length of at least one component in a given alloy-system is called contrast-variation. This variation may be achieved by manipulation of the specimen or by manipulation of the radiation. The diffraction experiments yield total S(Q) functions with different weighting factors of the partial S/7((2), and thus yield independent equations for the determination of the partial functions. Applying the contrast-variation method to the case of a binary alloy, three diffraction experiments must be performed with three specimens in order to obtain a system of three equations for the Faber-Ziman or the Bhatia-Thornton partial Sl7(<2). The crucial point is the degree of possible variation, which may be very different depending on the applied method. A quantitative measure for the degree of variation, and thus for the reliabil-
231
ity of the resulting partial functions, is the normalized determinant of the coefficients, i.e., the weighting factors of the partial functions in the equations. The larger the value of the determinant, the better conditioned the equations. In the ideal case, never achieved in practice, this value is one. With the isotopic substitution technique the fact that the isotopes of a specific element may have different neutron-scattering lengths is used. In some favorable cases the variation is quite strong, especially if one uses isotopes with negative scattering lengths, such as 62 Ni and 162 Dy. The magnetic neutron scattering of atoms with a magnetic moment, in addition to the nuclear scattering, can be varied by the orientation of an external magnetic field and by the polarization of the neutrons. The anomalous dispersion technique makes use of the fact that near its absorption edge the X-ray scattering length of a constituent in an alloy becomes wavelength dependent. Variation of the contrast is achieved by performing diffraction experiments at different, properly-tuned Xray wavelengths. Often the combination of neutron data and X-ray data provides a contrast-variation which gives information about partial structure factors and correlation functions. This is the case when the ratio of the scattering lengths of the constituents of an alloy is distinctly different for the two radiations. Isomorphous substitution can be applied if the replacement of an element in an alloy by a chemically-related element does not change the structure of the alloy. Provided the scattering lengths of the mutually substituted elements are sufficiently different, a contrast-variation can be achieved in this way. With the high resolution technique the structure factor is measured up to very
232
4 Structure of Amorphous and Molten Alloys
large scattering vectors, Q > 200 nm x, thus suppressing the artificial broadening of the peaks in the Fourier transform G(R). In those cases where the atomic diameters of the constituents of an alloy are sufficiently different, the individual atomic pairs may appear in the total correlation function G(R) as more or less resolved peaks. This method is not a real contrastvariation method, but like those it is aimed to obtain information about partial coordinations. A rather new technique is EXAFS (extended X-ray absorption fine structure). The X-ray absorption coefficient at the highenergy side of the absorption edge of an element in an alloy shows oscillations which are determined by the arrangement of its neighboring atoms. Thus, by EXAFS the coordination of specific elements can be probed separately by selecting their characteristic edges. For the application of EXAFS to amorphous solids see the papers by Wong (1981) and Gurman (1981, 1982).
4.4 Structure of Amorphous Metallic Alloys The amorphous state of matter is generally in a metastable equilibrium with respect to crystalline phases. Therefore upon heating up to the crystallization temperature, an amorphous solid will transform into a crystalline one. From this it is understandable that alloys which are relatively stable can be obtained more easily in the amorphous state. These are alloys which exhibit, to a certain degree, a compoundforming tendency, and thus a negative enthalpy of mixing. The preferred chemical interaction between the different atomic species making up an alloy leads to a local equilibrium state in the amorphous system.
This explains why pure amorphous metals can only be obtained at extremely low temperatures (except for semimetals such as Ge). Amorphous films of Co, Cr, Fe, and Mn were produced by vapor deposition on a substrate at liquid helium temperatures [see Wright (1977) and references there]. Many techniques have been developed to produce amorphous metallic alloys. These techniques are very different, and can be divided into three main groups. By rapid solidification, such as the most widely used melt spinning technique, certain liquid alloys can be quenched into an amorphous state. With deposition techniques amorphous films are deposited on substrates by sputtering, vapor deposition, or electro-chemical plating. With solid state reactions crystalline starting materials are transformed into the glassy state by mechanical alloying, ion implantation, irradiation with electrons or neutrons, and by hydrogen-loading. The fact that a certain metallic glass may be produced by very different methods is evidence that the amorphous state is a well-defined and quite stable one, rather than just an accidental one. The glasses discussed in the following Sections were produced by rapid solidification unless specified otherwise. Amorphous metallic alloys can be classified into two main groups, namely metalmetalloid and metal-metal alloys. Besides common structural features these two groups also exhibit different properties, and will therefore be discussed in separate Sections. 4.4.1 Metal-Metalloid Alloys Most of the metal-metalloid (T-M) alloys contain as the metallic component a transition metal, and their composition is close to a eutectic concentration. Examples are Fe 8 0 B 2 0 , Pd 8 0 Si 2 0 , and
4.4 Structure of Amorphous Metallic Alloys
Ni 6 4 B 3 6 . Their structural properties have been investigated extensively in the past, nevertheless the number of systems where reliable partial atomic distribution functions have been established is still limited, especially with respect to correlations between the metalloid atoms. In Table 4-1 structural parameters are listed for some T-M glasses. Most of them are derived from partial distribution functions. As an example, the results of an isotopic substitution neutron diffraction experiment with amorphous melt spun Ni 80 P 20 are presented in the following (Lamparter and Steeb, 1985). Three Ni8oP2o samples were produced using different Ni isotopes. One with natural nickel natNi, which has a large positive neutron scattering length /( nat Ni) = 1.03 x 10 ~ 12 cm, one with the isotope 62 Ni, which has a negative scattering length /( 6 2 Ni) = — 0.88 x 10~ 12 cm, and one with a zeroscattering Ni-isotopic mixture for which /(°Ni) = 0. The zero isotopic mixture was produced by alloying 62 Ni and 60 Ni. The latter sample was designed in such a way that the weighting factors of the Ni-Ni and the N i - B correlations in Eq. (4-13) were zero, in order to observe directly the metalloid-metalloid correlations in a metal-metalloid glass. (However, the real scattering length of °Ni turned out to be slightly larger than zero, / = 0.025 x 10" 1 2 cm. Therefore, even with this sample all three correlations contributed to the scattering.) The three corresponding, experimental, total F-Z structure factors S(Q) (Fig. 4-2) are represented in terms of the F-Z partial structure factors S NiNi (0, S P P (0 and S N i P (0 according to Eq. (4-13) as: S(Q) = O.79S NiNi (0 + 0.01 S P P (0 + O.2OSNiP(0
233
0
Ni 8 0 P 2 0 : S(Q) = O.O3SNiNi(0 + 0.70 SPP ( + O.27S NiP (0 62
Ni 8 0 P 2 0 : S(Q) = 139SNiNi(0 + 0.03 Spp(Q) -O.42S N i P (0 The isotopic substitution causes a strong contrast-variation, as can be seen from the very different weighting factors in the equations and correspondingly from the very different shapes of the S(Q) functions in Fig. 4-2. The value of the normalized coefficient determinant Dc = 0.48 is quite large, indicating a good separation of the three terms Stj. This set of equations was solved for the F-Z partial structure factors from which also follow the B-T partials. The results are shown in Figs. 4-3 and 4-4. Figures 4-5 and 4-6 represent the corresponding corre-
50
100
150
Q (nm-1)
Figure 4-2. Amorphous Ni 8 0 P 2 0 : Total F - Z structure factors obtained by isotopic substitution - neutron diffraction (Lamparter and Steeb, 1985). The two upper curves are shifted by 2.5 and 5, respectively.
234
4 Structure of Amorphous and Molten Alloys
Table 4-1. Amorphous transition metal (T)-metalloid (M) alloys: Structural parameters. RtJ = atomic distance. Ztj = partial coordination number, ARl = width of the pair distribution. rjl (rjlrel) = (relative) Cargill-Spaepen short-range order parameter. Applied methods: N = Neutron diffraction. X = X-ray diffraction. I = isotopic substitution. M = magnetic scattering. H = high resolution in R-space. E = EXAFS. * Authors mention correction for Fourier transformation broadening. Alloy T-M Ref. Method
RTT [nm] Z TT ARlTT [nm]
#MT
Nold et al. (1981), NI,X
0.257 12.4 0.04*
0.214 8.6 0.02*
0.357 6.5
Ni 81 B 19 Lamparter et al. (1982), NI
0.252 10.8 0.034*
0.211 8.5 + 0.8 0.032*
0.329 3.6
Ni 80 B 20 Suzuki et al. (1985), NH
0.251 10.8
0.208 5.8
Ni 67 B 33 Ishmaev et al. (1987), NI Ni 67 B 33 Wong and Liebermann (1984), E Ni 64 B 36 Cowlam et al. (1984), NI Lamparter and Steeb (1985), NI Sadoc and Dixmier (1976), NM,X
0.253 9.4 0.036* 0.224-0.263 11.2 0.255 9.2
[nm] ARlMM [nm]
0.208 6.4 + 2.6 0.026*
0.15 1.0 0.402 3.7
0.17 1.0
0.187 0.9 0.034
0.329 7.4
0.402 8.0
0.23 0.65
0.211 7.6 0.212 8.7
0.172 1.1
0.24 0.60
0.256 9.4 0.039* 0.255 10.1
0.228 9.3 0.032* 0.232 8.9
0.373 5.3 0.053
0.430 3.3 0.055
0.20 1.0 0.18 1.0
Ti 84 Si 16 Lamparter et al. (1986), N,X
0.290 11.5 0.045 *
0.264 9.4 0.033*
Pd 80 Si 20 Fukunaga and Suzuki (1981), NH
0.280 10.6 0.042
0.242 6.6 0.024
Pd 80 Ge 20 Hayashi et al. (1982), NH
0.281 10.3 0.046
0.253 5.6 0.023
Pd 78 Ge 22 Hayes et al. (1978), E
nl x
0.249 8.6
235
4.4 Structure of Amorphous Metallic Alloys
50
Q (nm-1)
100
150
Figure 4-3. Amorphous Ni 8 0 P 2 0 : Partial F - Z structure factors.
50
Q (nrrr1)
100
show a splitting of the second coordination shell which is found for the majority of metallic glasses. The second nearest neighbors are in quite well-defined positions at Rlhl around 1.7R1 and at Rn>2 around 2.0 R1. The nearest neighbor distribution of the T-M correlation, compared with the T-T correlation, is characterized by a smaller distance and by a sharper bond length distribution (see ARjj values Table 4-1). After the main peak GTM(R) again drops down to the — 4UQ0R line, which means that there is a range where e no QTM(R) — 05 i- -> T-M pairs at all occur in this range. These pronounced features give evidence that the short-range order in T-M glasses is governed by the chemical interaction between the metal and the metalloid atoms. Whilst the T-T distances are very close to the Goldschmidt diameters Dj of the T atoms, the apparent diameters of the M atoms in metal-
150
Figure 4-4. Amorphous Ni 8 0 P 2 0 : Partial B-T structure factors.
lation functions Gtj. The F-Z partial correlation functions of amorphous melt-spun Ni 8 1 B 1 9 (Lamparter etal, 1982), derived by the same method, are also shown in Fig. 4-5. The atomic distances and the partial coordination numbers determined by Gaussian fitting to the main peak of the Gtj(R) are listed in Table 4-1. The common features in both systems are: The T-T and the T-M distributions
0.5
1.0 R (nm)
Figure4-5. Amorphous Ni 8 0 P 2 0 and Ni 8 1 B 1 9 : Partial F - Z pair correlation functions. The — 4TZRQ0 line is shown for GNiP.
236
4 Structure of Amorphous and Molten Alloys
Table 4-2. Amorphous transition metal (T)-metalloid (M) alloys: Atomic distances compared with atomic diameters. RTT(= DT) = T - T distance. Dj= atomic (Goldschmidt) diameter. RTM = T-M distance. DM = apparent atomic diameter. D^ = covalent diameter. D£J = atomic diameter. 5#TM = contraction of T-M distance. Alloy rP R 80 20 Ni 81 B 19 Ni 67 B 33 Ni 64 B 36 Ni 8 0 P 2 0 Co 80 P 20 Ti 84 Si 16 Pd 80 Si 20 Pd 80 Ge 20 rc
D
RTT = [nm]
[nm]
0.257 0.252 0.253 0.255 0.256 0.255 0.290 0.280 0.281
0.252 0.248 0.248 0.248 0.248 0.250 0.294 0.274 0.274
r\co
Dat
[nm]
DM [nm]
[nm]
[nm]
0.214 0.211 0.208 0.212 0.228 0.232 0.264 0.242 0.253
0.171 0.170 0.163 0.169 0.200 0.209 0.238 0.204 0.225
0.164 0.164 0.164 0.164 0.212 0.212 0.222 0.222 0.244
0.196 0.196 0.196 0.196 0.256 0.256 0.264 0.264 0.274
metalloid pairs, calculated from DM = 2RTM-R TT , are distinctly smaller than their atomic diameters D^S and are closer to their tetrahedral covalent diameters D^ (Table 4-2). A measure of the strength of the chemical interaction in non-crystalline alloys is the contraction 5R of the distance in unlike atomic pairs with respect to the value expected in the case of ideal mixing,
lO"
161412E10-
£ 8-
\l
l\ /\ _ NN n /\ A /\,
AAAAA KM
+11
" \
r\ / N - N ^ _ £ ^ _
+7
£ 642-
I
0-9-
0
0.5
1.0 R (nm)
Figure 4-6. Amorphous Ni 8 0 P 2 0 : Partial B-T correlation functions.
DJDT
5K TM [%]
0.67 0.67 0.64 0.66 0.78 0.82 0.82 0.73 0.80
5.5 5.8 7.4 6.0 11.0 9.2 4.7 11.0 8.8
i.e., for additive atomic diameters. The values of bRTM in Table 4-2 were calculated by taking RTT as the diameter of the T atoms and D$ as the diameter of the M atoms according to 5RTM= (RTT + D$-2RTM)/ (RTT + D^). Contractions of 5 to 10% were observed. It is interesting to note that the size effect function GNC(R) of Ni 80 P 20 (Fig. 4-6) exhibits distinct oscillations comparable to those of the Ni-B glasses (Fig. 4-8). This is because in amorphous Ni 80 P 20 the diameter of the P atoms (DP = 0.200 nm) is smaller than their atomic diameter (Dp1 = 0.256 nm) and thus smaller than the diameter of the Ni atoms (DNi = 0.256 nm). Important structural features are presented by the M - M correlations in Fig. (4-5). The small metalloid atoms do not occur as direct neighbors but exhibit a very pronounced distance correlation at two different distances around 0.4 nm. The following oscillations are extended as far as those of the T-T and the T-M correlations up to at least 1.4 nm. From these features it must be concluded that the metalloid atoms are not distributed randomly in a framework of metal atoms, and are thus not just filling up the available free spaces. This space-filling picture corresponds to
237
4.4 Structure of Amorphous Metallic Alloys 35-
0.6 R (nm)
0.8
0.2
0.4
0.6 R (nm)
0.8
1.0
1.2
Figure 4-7. Amorphous Ni 81 B 19 and Ni 67 B 33 : Partial F - Z pair correlation functions. ( ) 19 at.% B (Lamparter et al., 1982), ( ) 33 at.% B (Ishmaev etal., 1987).
Figure 4-8. Amorphous Ni 81 B 19 and Ni 67 B 33 : Partial B-T correlation functions. ( ) 19 at.% B, ( ) 33 at.% B.
interstitial packing, as proposed by Polk (1972) in the early stages of studies on T-M glasses. In actual fact the metalloid atoms play a dominant role in the construction of the amorphous structure by establishing their own neighborhood by chemical interaction with the metal atoms. Comparing the metalloid-metal coordination numbers in Table 4-1, in most cases we find about nine metal atoms around a metalloid atom independent of the size ratio DM/DT of the constituents. It is suggested that this structural feature is independent of the metalloid concentration, as indicated by the Ni-B alloys (Table 4-1). This again illustrates that the metal metalloid correlation is governed by distinct chemical interaction rather than by simple interstitial packing. The partial correlation functions in Fig. 4-5 indicate that besides common structural properties there may also be larger
differences among T 80 M 20 glasses than are simply explained by varying atomic sizes. The amplitudes of the double peak in the P - P correlation are different from those in the B-B correlation, corresponding to different M - M coordination numbers. In the range around 0.6 nm the P - P correlation shows a complicated structure, reflecting that the medium range ordering in the two glasses is different despite the strong similarities in the nearest neighbor ordering. The concentration dependence of the short-range order in T-M glasses has not been investigated thoroughly hitherto as far as partial correlation functions are concerned. In Fig. 4-7 the partial pair correlation functions of the glass Ni 8 1 B 1 9 (Lamparter et al., 1982) are compared with those of Ni 6 7 B 3 3 , which were established by Ishmaev et al. (1987) by isotopic substitution neutron diffraction. The most important difference is the occurrence of a peak at a
238
4 Structure of Amorphous and Molten Alloys
small distance RBB = 0.19 nm in the B-B curve of the alloy with higher B concentration, which corresponds to about one nearest B neighbor around each B atom. This value is still small compared with that for random occupation, in which case, at cB = 0.33, out of the ten neighboring atoms in the first coordination shell of a central B atom, 3.3 B atoms would be expected. Thus the avoidance of direct metalloidmetalloid contact, though not strictly, is a composition-independent property of T-M glasses. For the T 80 M 20 alloys, where ZMM = 0, the Cargill-Spaepen short-range order parameter rjl [Eq. (4-21)] takes its maximum possible value, i.e., rj\el = 1, whatever the values of Z TT and Z MT are. However, for the glasses with a larger boron content Ni 6 7 B 3 3 and Ni 6 4 B 3 6 , where ZBB = 1, the short-range order parameter takes only 65% and 60%, respectively, of the maximum possible value (Table 4-1). In Fig. 4-8 the partial Bhatia-Thornton correlation functions of two glasses with different B-concentration are compared. The chemical ordering effect is reflected by a pronounced negative peak in the GCC(R) function at the Ni-B distance. The G NC (JR) function shows a distinct size effect, as expected in T-M glasses, which appears to be independent of the metalloid-concentration. In both alloys the oscillations of GCC(R) and GNC(R) are damped more rapidly with increasing R than those of the topological ordering GNN(R). The range of chemical ordering reaches up to about 1 nm. The same observation was made for the Ni 80 P 20 glass (Fig. 4-6). The application of EXAFS to amorphous systems is still at the development stage as far as the data evaluation is concerned. Therefore it is interesting to compare the results obtained using EXAFS with those obtained using conventional diffraction techniques. Wong and Lieber-
40
60
80
100
120
140
wavevector (nnr1)
Figure 4-9. Amorphous Ni 67 B 33 : EXAFS spectrum (Wong and Liebermann, 1984). (ooo) experimental, ( ) fitted function.
mann (1984) used three Ni subshells and one B shell around a central Ni atom to fit the measured Ni-edge EXAFS of amorphous Ni 6 7 B 3 3 (Fig. 4-9). The resulting structural parameters are listed in Table 4-1. Results from isotopic substitution neutron diffraction for amorphous Ni 6 7 B 3 3 (Ishmaev et al., 1987) and for Ni 6 4 B 3 6 (Cowlam etal., 1984) are also listed. EXAFS yielded 11.2Ni and 3.8 B atoms around a central Ni atom. (The value Z BNi = 7.6 listed in the table follows from c C B Z B NI = m ^NiB •) Neutron diffraction using the Ni-B glasses with a high boron content (33 and 36 at.%, respectively) yielded less Ni neighbors around Ni (9.4 and 9.2, respectively) and more Ni neighbors around B (9 and 8.7, respectively). The Ni-B distance, around 0.21 nm, is in good agreement, whereas the weighted average Ni-Ni distance - 0.243 nm from EXAFS is smaller than 0.253-0.255 nm, as derived from neutron diffraction. Wong and Liebermann (1984) compared the EXAFS results for the Ni 6 7 B 3 3 glass with those obtained for crystalline Ni 2 B and stated substantial differences in the short-range order. This is in agreement with NMR measurements by Panissod et al. (1983) where the site symmetry of the B atoms in
4.4 Structure of Amorphous Metallic Alloys
crystalline Ni 2 B was not observed in the corresponding glass. 4.4.2 Metal-Metal Alloys Binary amorphous metal-metal (ml-m2) alloys can be subdivided into several subgroups. The most extensively investigated alloys are those where ml and m2 are a late and an early transition metal, respectively (Co-Ti, Ni-Zr, Ni-Nb). Other types are simple metal glasses (Mg-Ca, Mg-Cu, Mg-Zn, Ca-Al), simple metal-transition metal alloys (Mg-Ni, Be-Ti), and transition metal-rare earth alloys (Fe-Tb, Ni-Dy, Co-Gd). Of course systems with more than two constituents, like Mg-Ni La, are numerous and cannot be classified uniquely. Usually the glass formation concentration ranges of metal-metal systems are larger than those of T-M systems, and therefore it is easier to investigate the composition dependence of their properties. The partial atomic distribution functions have been established for a variety of ml-m2 alloys up to now, however only in a few cases over a range of concentrations. In Table 4-3 the resulting structural parameters are compiled for a number of selected alloys. In those cases where the authors reported a comparison with structural data of related crystalline phases, the data are also listed in Table 4-3. In the following, structural results for Ni-Nb glasses will be discussed in more detail as a representative example for an amorphous ml-m2 alloy system. Figure 4-10 shows the F-Z partial structure factors of Ni^NbiOQ.^ glasses at four different compositions x = 40, 50, 56, 63, determined from isotopic substitution neutron diffraction (Lamparter et al, 1990), and Fig. 4-11 the corresponding Fourier transforms. The general features of the correlation functions Gtj(R) of ml-m2 glasses are
239
quite similar to those of T=M glasses: The second maximum shows characteristic splitting into two well-defined subpeaks at RU1 around 1.1 Rl and at R112 around 2.0R l . According to Fig. 4-13 the range of the topological ordering extends up to about 2 nm. Also a chemical short-range ordering effect is generally observed in these glasses, as shown by pronounced oscillations in the concentration-concentration correlation functions SCC(Q) (Fig. 4-12) and GCC(R) (Fig. 4-13). The chemical interaction between unlike atoms causes preferred hetero-coordination and also a contraction of the distance in unlike atomic pairs of the order of 5% compared with the mean value of the distances in like atomic pairs. Comparison of the values of the short-range order parameters in Table 4-3 with those in Table 4-1 shows that the CSRO in ml-m2 glasses is distinctly weaker than in T-M glasses. Also the contraction of the unlike-pair distances tends to be smaller. The apparent atomic diameters Dm = Rmm of the two constituents are in general different and cause a considerable size effect in the functions SNC(<2) (Fig. 4-14) and GNC{R). The range of the chemical ordering is smaller than that of the topological ordering; for Ni-Nb alloys the oscillations of GCC(R) and of GNC(R) vanish at about 1 nm (Fig. 4-13). But it should be noted that this correlation range already involves some hundred atoms, so the idea of small ordered clusters of some atoms is certainly too simple for the description of the chemical short-range order. In the past it has been suggested by several investigators that amorphous ml-m2 alloys are substitutional. More recent structural results, however, revealed that the short-range order is different with respect to a central ml atom and a central m2 atom, respectively. This is already illus-
240
4 Structure of Amorphous and Molten Alloys
Table 4-3. Amorphous (a) and crystalline (c) metal-metal alloys: KtJ = atomic distance. 5RAB = contraction of the distance RAB compared with (RAA + RBB)/2. Ztj = partial coordination number, r]1 = Cargill-Spaepen shortrange order parameter. A = anomalous dispersion. S = isomorphous substitution. The other symbols denoting the methods are the same as in Table 4-1. Alloy A-B Reference
Method
R AA nml
RBB nml
RAB nml
NI, X
0.250
0.265
0.260
XA
0.250
0.30
0.2490.288
0.288 0.323
8.33 2.9
10.7
-1.0
4.5
6.4
6.0
0.06
0.07
8.4
0.02
0.06
Muller etal. (1987) dQ
^PUn8 3 , 3 Y * 16.7
Laridjani etal. (1987) c Cu5Y
10.4
2.8
3.66
NI
0.3 ••• 0.4
0.356
0.286
a Ni 31 Dy 69 Wildermuth et al. (1985)
NI
0.254
0.350
0.285
6.3
3.0
12.4
10.8
0.06
0.18
a Ni 62 Nb 38 Svab et al. (1988)
NI
0.248
0.305
0.263
5.0
6.1
5.5
5.7
0.11
0.15
0.25 0.248 0.250
0.306 0.302 0.302
0.262 0.264 0.264
5.8 4.0 4.3 2.9
6.6 5.5 5.0 3.8
5.6 6.5 7.5 9.0
5.9 6.6 7.4 8.2
0.11 0.11 0.09 0.06
0.16 0.11 0.11 0.11
4
0.33
1.0
n 33 Y dQ P v_.u i 67
Maretetal. (1987)
a Ni 63 Nb 37 a Ni 56 Nb 44 a Ni5ONb5O a Ni 40 Nb 60 Lamparter etal. (1990) c Ni 3 Nb
NI 0.2550.26 NI
Fukunaga etal. (1984) c NiTi9
0.263
0.301
0.287
0.291- 0.2490.299 0.289
0.260
4.1
no nearest neighbor
2.3
8.1
7.9
0.17
0.33
3
9
9
0.08
0.19
0.356
0.286
5.5
1.9
9.5
9.3
0.16
0.40
0.252
0.328
0.267
7.9
6.0
5.8
5.0
0.06
0.08
0.252
0.326
0.266
8.0
6.0
5.0
5.0
0.10
0.13
XA
0.263
0.328
0.270
8.6
6.4
6.6
5.3
0.04
0.05
NI
0.263
0.332
0.273
8.2
3.3
7.8
6.7
0.13
0.19
4
8
7
0.10
0.14
NI
^0.25
8
0.2550.26
Maret et al. (1987) a Ni 63 7 Zr 36 . 3 Lefebvre et al. (1985 and 1988)
NI
Sadoc and Calvayrac (1986) a
Ni
67Zr33
de Lima etal. (1988) Fukunaga et al. (1985) c NiZr
0.249- 0.326- 0.2660.326 0.346 0.287
4.4 Structure of Amorphous Metallic Alloys
241
Table 4-3. Continued. Alloy A B Reference
Method
a Ni 36 Zr 64 Mizoguchi et al. (1985) a Ni35(Zr,Hf)65 Lee etal. (1984) c NiZr2
zAA
#AA
^BB
^AB
[nm]
[nm]
[nm]
NI
0.245
0.330
0.285
0.9
3.3
11
N,X,S
0.266
0.315
0.269
7.4
2.3
9
0.262
0.298- 0.276 0.343
0.264
0.324
XA
0.277
5.8
-^BB
zAB 8.6
0.03
0.07
5.4 -0.06
2
11
8
0.07
0.20
2.3
10.8
7.5
0.02
0.07
de Lima (1989)
trated by the size effect. However, the chemical ordering is different as well. The Warren-Cowley short-range order parameter of Ni 5O Nb 5O , calculated from the partial coordination numbers, is a1 = — 0.2 for a central Ni atom, whereas for a central Nb atom it is a1 = 0, which means that more unlike first neighbors, compared with the statistically expected number, are found only around an Ni atom. This in turn means less Ni-Ni nearest neighbors than expected for the statistical distribution. This non-symmetrical behavior of the chemical ordering with respect to the two atomic species is reflected also in the F-Z partial structure factors of Ni 5O Nb 5O , as shown in Fig. 4-10, where the prepeak is much more pronounced in the Ni-Ni function than in the N b - N b function. The tendency to more or less avoid direct neighborhood between the smaller atoms is generally the case for ml-m2 glasses. Figure 4-17 shows a compilation of partial G22{R) functions, including also T-M glasses, where 2 denotes the smaller constituent in each alloy. For all alloys the first peak, if present at all, is rather small compared with the split up, second maximum. This behavior, whether more or less pronounced, and strongest for T 80 M 20 glasses, seems to be common for
all metallic glasses. Splitting of the maximum at larger distances with a higher subpeak at R11'1 also seems to be a general feature. The occurrence of these common features cannot be understood from the point of view that the short-range order in metallic amorphous systems is close to that in related individual crystalline phases. The concentration dependence of the structural properties of ml-m2 glasses, derived from the partial pair distribution functions, is only known for a few alloy systems (see Table 4-3). In amorphous Ni x Nb 1 0 0 _ x alloys (40 < x < 63) the effect of the chemical short-range order increases with increasing Ni concentration, as shown by the increasing oscillations and the sharpening of the first peak of the partial structure factor SCC{Q) in Fig. 4-12. On the other hand, interestingly, the size effect structure factor SNC(Q) is almost independent of the composition, i.e., of the degree of chemical ordering. The dashed SNC(Q) curve in Fig. 4-14 was calculated by a simple hard-sphere model, and it already represents quite well the experimental function. The topological ordering function GNN(R) in Fig. 4-13 shows a sharper main peak at higher Ni concentration, and also the strong negative peak of GCC{R),
242
4 Structure of Amorphous and Molten Alloys
12-
Ni40Nb60
10-
Ni - Nb
+8
Ni - Ni
+4
86 O
Nb-Nb
(a)
-2 12
Ni63Nb37
10-
Ni - Nb
+8
8
O of
6-
Ni - Ni
+4
4-
2-
Nb-Nb
(d)
-2 50
100 150 Q (nnr1)
200
50
100 150 Q (nnr1)
200
Figure 4-10. Amorphous Ni^Nb alloys: Partial F - Z structure factors (Lamparter et al., 1990). (a) 40 at.% Ni, (b) 50 at.% Ni, (c) 56 at.% Ni, (d) 63 at.% Ni.
representing preferred Ni-Nb correlation, becomes more pronounced. In Fig. 4-15 the composition dependence of several properties, calculated from the diffraction results, is presented. The increase in
the short-range order parameters a1 [Eq. (4-29)] and rf [Eq. (4-21)] is associated with a parallel increase in the Darken excess stability Sx/kT, calculated from Scc(0) using Eq. (4-33). The increasing contraction
4.4 Structure of Amorphous Metallic Alloys
243
30 |
20-
Ni 40 Nb 60
A/\ f\ /\ ^ . \j v —
Ni5oNb5o
Ni -Ni
\\ iv i\ ^ ^.
Ni -Nb
Ni - Nb +20
i
+10
Ni- Ni
Nb -Nb
+20
+10
Nb- Nb
(b)
(a)
Ni 63 Nb 37
Ni- Nb
+20
Ni- Ni
+10
Nb- Nb
(d) 0.5
1.0
1.5
R (nm)
2.0 0
0.5
1.0
1.5
2.0
R (nm)
Figure 4-11. Amorphous Ni-Nb alloys: Partial F - Z pair correlation functions, (a) 40 at.% Ni, (b) 50 at.% Ni, (c) 56 at.% Ni, (d) 63 at.% Ni.
of the Ni-Nb distance 5JRNiNb is further evidence of increasing chemical interaction at higher Ni content. Ni-Zr glasses have been investigated by several authors using different techniques.
In almost all cases a slightly positive Cargill-Spaepen short-range order parameter was observed. The short-range order parameter is largest at the composition of Ni 50 Zr 50 where a quite stable crystalline
244
4 Structure of Amorphous and Molten Alloys
5
0 50
100
150
200
Q (nm-1)
Figure 4-12. Amorphous Ni Nb alloys: Partial Scc (Q) structure factors.
phase exists. However, SCC(Q) functions, calculated by Pasturel et al. (1988) from available neutron diffraction data as well as calculated on the basis of a theoretical model, pointed to a stronger ordering in the Ni-rich range of the Ni-Zr system. Sakata et al. (1980, 1981, 1982) derived he Bhatia-Thornton partial correlation unctions GNN(R) and GCC(R) of amor3hous Cu^TijQQ.^ alloys ( 3 5 < x < 7 0 ) from a combination of X-ray and neutron diffraction. Due to the negative neutron scattering length of Ti, together with the positive one of Cu, the weighting factor of SCC(Q) in Eq. (4-25) is large for neutrons, whereas SNN(<2) is mainly measured by X-rays. Neglecting the contribution of SNC(<2), they solved the two equations for S r r and SNN and obtained from the corresponding Fourier transforms values for the Warren-Cowley short-range order parameter [Eq. (4-29)] in the range a1 = - 0.06 to — 0.2. An interesting correlation was found between a1 and the crystallization tempera-
Figure 4-13. Amorphous Ni 40 Nb 60 and Ni 63 Nb 37 : Partial B-T correlation functions, (a) 40 at.% Ni, (b) 63 at.% Ni.
ture, which is a measure of the stability of the glass. They both show a parallel composition dependence with a maximum near 65 at.% Cu. The most stable crystalline phases within (or close to) the covered composition range are Ti2Cu and TiCu,
245
4.4 Structure of Amorphous Metallic Alloys 10T 8
-0.5 100
150
200
Q (nnr1)
Figure 4-14. Amorphous Ni-Nb alloys: Partial SNC (Q) structure factors. The dashed curve was calculated using the Percus-Yevick hard-sphere model.
50
60
70
80
Ni concentration (at.%)
Figure 4-16. Amorphous Ni-Nb alloys and crystalline Ni 3 Nb (75 at.% Ni). Coordination numbers.
14-
/
1
12-lr
•6
12-
-5
co
R/R
1010
/
o -4
8/
//
.ft.
y
6 - --3
6-
4-
40
50 60 Ni concentration (at.%)
70
Figure 4-15. Amorphous Ni-Nb alloys: Concentration dependence of structural properties, (o) Sx/k T = excess stability, (•) 5KNiNb = contraction of the Ni-Nb distance referred to 0.5 (# NiNi + # NbNb ), (x) a1 = Warren-Cowley short-range order parameter, calculated from the B-T correlation functions, (A) rf = CargillSpaepen short-range order parameter, calculated from the partial coordination numbers.
i.e., the strongest CSRO in the Cu-Ti system is not correlated with the stability of related crystalline phases. The question as to what extent the atomic structure of metal-metal glasses is related to crystalline counterparts is still
under discussion. Since some agreement of atomic distances and coordination numbers can be found between amorphous and crystalline phases (see Table 4-3), several authors suggest a close structural relationship. Of course the atomic distances, usually close to the Goldschmidt diameters, provide a much less crucial criterion than the coordination numbers. As an example, in Fig. 4-16 the coordination numbers are given for Ni-Nb glasses and for crystalline Ni 3 Nb. Extrapolation of the coordination numbers of the glasses to the composition of Ni 3 Nb would yield values close to the crystalline ones for Z NiNi and Z NiNb , but not for Z NbNb , which is zero in the crystalline phase because there are no N b - N b nearest neighbors. Consequently, any similarity in the structure of a hypothetical Ni 3 Nb glass and the crystalline phase, if existent at all, would be confined to the neighborhood of the Ni atoms. 4.4.3 Structural Inhomogeneities
Most of the metallic glasses investigated hitherto are not homogeneous, but contain fluctuations of local structural properties
246
4 Structure of Amorphous and Molten Alloys
30-
Figure 4-17. Amorphous alloys: Partial F - Z pair correlation functions G22(R\ where 2 denotes the smaller atom in a 1-2 alloy. The distances are normalized to the position of the subpeak JR IU in the split second maximum, a: Zr 64 Ni 36 ; b: Ti 60 Ni 40 ; c: Dy 69 Ni 31 ; d: Y 67 Ni 33 ; e: Ni 64 B 36 ; f: (dashed) Ni 67 B 33 ; g: Ni 8 1 B 1 9 ; h: Ni 80 P 20 - For references see Table 4-3.
on length scales larger than the scale of atomic distances. This has been established by small-angle scattering experiments with neutrons (SANS) and with X-rays (SAXS) and in some cases it was also concluded from more indirect probes, like electron microscopy, Mossbauer spectroscopy, field ion microscopy, etc. (see references in Lamparter and Steeb, 1988). These structural features are usually termed as medium range though they may extend up to several 100 nm. In contrast to atomic shortrange order, the inhomogeneous properties in glasses seem to depend considerably on their production conditions as well as on the history of thermal treatments. This is of importance in conjunction with technical applications of these materials be-
cause their mechanical and magnetic properties depend on structural inhomogeneities. Figure 4-18 shows as an example for a monatomic glass the scattering curves at low Q of amorphous Ge prepared by different deposition techniques (Shevchik and Paul, 1972). Electrodeposited Ge shows no small-angle scattering effect and therefore is homogeneous. On the other hand, the sputtered and even more the vapor-deposited Ge films show a small-angle scattering effect, reflecting fluctuations in the density of these materials. In amorphous alloys the inhomogeneous distribution of quite different local properties may give rise to fluctuations of the scattering length density for X-rays and/or neutrons, such as the atomic density, the concentration, and the density and orientation of magnetic moments in amorphous ferromagnets. Also, crystallites may be embedded in an amorphous matrix. These imperfections may be located in the bulk or at the outer surface of the material and may occur over a large length scale, from one up to several hundred nanometers. The variety of possible inhomogeneities renders the interpretation of smallangle scattering data complicated, and consequently we find quite different con-
•s CO
O
5 \
0
5
10 Q (nrrr1)
15
20
Figure 4-18. Amorphous Ge: Small angle X-ray scattering curves from samples as prepared by different methods (Shevchik and Paul, 1972). ( ) electrodeposited, ( ) sputtered, ( ) vapor-deposited.
247
4.4 Structure of Amorphous Metallic Alloys
elusions in the literature. In this section some examples were selected out of the available material in order to illustrate the state of the art. SANS experiments with binary amorphous T-M alloys revealed that we have to distinguish between two kinds of fluctuations occurring on different length scales: phase separation and very extended fluctuations.
Phase separation in amorphous alloys, preceding the crystallization process, has been found or suggested frequently. Figure 4-19 shows the Guinier plot of the SANS of melt spun Fe 8 0 B 2 0 in the as-quenched state as well as after annealing well below the crystallization temperature for 1 h at 300 °C (Lamparter and Steeb, 1988). In the region Q > 1 nm" 1 the curves are linear with slopes corresponding to the Guinier radii JRG [Eq. (4-39)] of 0.48 nm for the asquenched and 0.92 nm for the annealed sample. Thus we state that amorphous Fe 8 0 B 2 0 contains regions with diameter 2RS = 2 v /5/3i? G of about 1.3 nm, which upon annealing grow by a factor of 2. During the crystallization of many T-M glasses with 20 at.% or more of metalloid content T3M compounds are formed as the first product. This led to the idea that the phase separation into two cotnpositionally different amorphous phases corresponds to the formula: 8oM2o
where T ^ ^ . M , is a metalloid-depleted phase. In some glasses phase separation may have taken place already during the quenching process. An FIM study of an Fe 4 0 Ni 4 0 B 2 0 glass by Piller and Haasen (1982) yielded direct evidence for regions, about 3 nm in diameter, containing
300°C/1h
Fe4oNi4oB2o
as - q u e n c h e d
1 Q2
Figure 4-19. Amorphous Fe 80 B 20 and Fe 40 Ni 40 B 20 : Guinier plot of the SANS curves of as-quenched samples and after annealing (Lamparter and Steeb, 1988). In the range Q2 > 1 nm" 2 straight lines are fitted to
25 at.% boron. SANS with Fe l o o _ x B x glasses (Faigel and Svab, 1985, Fig. 4-20) yielded Guinier radii of 0.6 (±0.1) nm for x < 25. At x = 25 the SANS vanished, supporting the idea of regions with the composition Fe 7 5 B 2 5 in alloys with x < 25. Direct evidence that the SANS is due to compositional fluctuations rather than to density fluctuations has been found for Ni8OP2o (Schild et al., 1985). Using isotopic substitution neutron diffraction, the Bhatia-Thornton partial structure factors were established in the small-angle scattering regime (Fig. 4-21). In the range
at.% B
o 6-
CO
s
0 14 x 19
•
.ri T
as - quenched
Q Fe80B20
-3
un.)
4.4.3.1 Phase Separation
& Fe80B20
© 25
o r-
—- 5 '
x
\ X
D
X
O
X
X
x
• » ^
B X
X
o
X
%
X
o
4" 2
4
6
8
10
O (nm1 )
Figure 4-20. Amorphous F e i o o - XBX alloys: SANS curves (Faigel and Svab, 1985).
248
4 Structure of Amorphous and Molten Alloys
0.5-rt
O CO
1
2 Q
3
(nnr 1 )
Figure 4-21. Amorphous Ni 8 0 P 2 0 : Partial B-T structure factors in the small angle scattering regime 0.5 nm- J < Q < 2.7 nm" x (Schild et al., 1985).
Q > 1 nm 1 only the concentration-concentration structure factor SCC(Q) contributes to the small-angle scattering effect, proving that concentration fluctuations are associated with small inhomogeneities of 2 nm in diameter. On the basis of Ni 75 P 25 as the average composition of the regions it follows from Eq. (4-40) that about 40 vol.% of the material is contained in these regions. Note that, using the term phase separation, a statistical view of these inhomogeneities in terms of concentration fluctuations has to be adopted rather than the picture of clusters with a defined surface, otherwise interparticle interference effects would be expected, giving rise to oscillations in the SANS curves. However, in a SANS study using amorphous Fe 40 Ni 40 P 20 by Gerling et al. (1988) a volume fraction of only 5 vol.% (Fe,Ni) 75 P 25 yielded an interference peak, implying that in this case the regions were spatially correlated. In electrodeposited Ni l o o _ x P x glasses, where x < 20, metal-rich precipitates were detected by SAXS, TEM, and magnetic measurements (Sonnenberger etal., 1986; Dietz et al., 1988; Dietz and Schneider, 1990). Their volume fraction was estimated to be a few percent, and on annealing they became more Ni-rich and
their sizes increased from around 1.8 nm to 5 nm. The authors note that the precipitates might be crystalline. In some cases of ternary T ^ T ^ B alloys, like Fe 4 0 Ni 4 0 B 2 0 (see Fig. 4-19) and Ni 3 2 Pd 5 2 P 1 6 , which form a glass more easily than the corresponding binary alloys, no evidence for the phase separation as described above has been found by neutron diffraction. This supports the idea that during rapid quenching of the molten alloy, depending on the cooling rate, phase separation may be suppressed in easy glass formers, thus yielding a more homogeneous structure compared with the binary T—M alloys. Comparing the SANS results with the FIM results (see above) for Fe 4 0 Ni 4 0 B 2 0 , it appears that the inhomogeneous structure of an amorphous alloy system may not be unique, but dependent on its history. In conclusion, it must be noted that for phase separated alloys the structural properties on an atomic scale, as derived from wide-angle scattering, merely represent an average over the two phases. 4.4.3.2 Very Extended Fluctuations
The second range of the scattering vector Q to be considered in small-angle scattering experiments with metallic glasses is Q < 1 nm" 1 . Here the scattering does not usually show a Guinier regime. For many transition metal-metalloid as well as metal-metal glasses the scattering at small Q follows the power law I(Q) = const Q -s
(4-41)
where the power — S is between —3 and -4. SANS curves of T-M glasses are shown in Fig. 4-22 (Ni 32 Pd 52 P 16 : Schaal etal., 1989; other curves: Lamparter and Steeb, 1988). The curves are linear on a
4.4 Structure of Amorphous Metallic Alloys
249
Figure 4-22. Amorphous metal-metalloid alloys: log-log plot of the SANS
quenched, d: 260 °C, 334 °C; e: 297 °C (for 20 h); f: Fe 80 B 20 as-quenched, g: 300 °C/ 7h; h: Fe 80 B 16 Si 2 C 2 as-quenched; i: 300°C/147 h. For references see text log [Q (nm 1 )]
log I(Q) — log Q scale at small Q values. The continuous increase towards very small scattering vectors Q < 1 0 ~ 2 n m ~ 1 , without any crossover to a Guinier regime at the lowest Q values, reflects very extended fluctuations in the amorphous state on length scales up to at least 100 nm. Note that the curves for melt-spun and electrodeposited Ni 80 P 20 exhibit essentially the same SANS signal. In the case of amorphous Ni 80 P 20 it could be proved that these fluctuations are compositional: S c c (<2) is larger than SNN (Q) by more than one order of magnitude in Fig. 4-23. Several physical interpretations for the power-law small angle scattering have been given in the past. Boucher et al. (1983) proposed a model for amorphous sputtered Tb 22 Cu 78 consisting of a random network of large domains with diameters of the order of some 100 nm. Labeling of the domain walls with appropriate concen-
tration profiles was used to explain the power-law SANS with - S = - 3. In that model the concentration profiles were attributed to a non-uniform distribution of
-1.2
-1.0 -0.8 log [Q (nm1)]
-0.6
Figure 4-23. Amorphous Ni 8 0 P 2 0 : log-log plot of the partial B-T structure factors SNN (Q) and S cc (Q) in the small angle scattering regime Q <0.25 nm" 1 (Schild et al., 1985). Straight line with slope - S = - 3.
250
4 Structure of Amorphous and Molten Alloys
impurities such as hydrogen accumulating near the inner surfaces. However, as power-law scattering is observed with many amorphous alloys, independent of their specific production method, it seems to be an individual feature of the medium range structure of metallic glasses. For a quantitative discussion of relations between power-law scattering and spatial variations of the scattering length density see Boucher et al. (1990). An alternative interpretation is based on the view of fractal geometry. Observation of power-law scattering on a wide Q-scale strongly suggests that the associated concentration fluctuations are self-similar on a correspondingly wide size scale (scale invariance). Self-similarity or dilatation symmetry means that the fluctuations look the same on changing the "magnification" of the SANS instrument, i.e., by changing the observed g-window. Fractal geometry is a recent concept to describe structures displaying self-similarity. Schaal et al. (1989) considered surface fractals which yield a power-law scattering (Martin and Hurd, 1987): I(Q) = const Q~(6~Ds) = i
distribution of the different atomic species in their vicinity. Annealing treatments at temperatures below the crystallization points of metallic glasses affect the extended fluctuations, but the effects are not unique. The SANS signals of the Fe-based alloys in Fig. 4-22 increased on annealing, and a bump developed near log Q = — 0.8, indicating that the population of correlation lengths around 10 nm is favored by the annealing procedure. On the other hand, annealing (20 h) of Ni 3 2 Pd 5 2 P 1 6 at 260 °C and also at 297 °C decreased the SANS and also the slope, which from the fractal point of view means that the inner surfaces became rougher. At 334 °C annealing again caused an increase in such a way that the SANS was the same as that for the 260 °C anneal. This illustrates that the atomic redistribution phenomena upon annealing may be quite complicated, and general rules cannot be given. Loading of metallic glasses with hydrogen leads to a ductility loss and at the same time enhances the small-angle neutron scattering signal (Fig. 4-24, Lamparter and Boucher, 1992). The increase in the SANS
(4-42)
Ds is the fractal or Hausdorff dimension, which in the case of smooth surfaces is D s = 2, yielding a slope — S = — 4, i.e., the Porod law. For 2 < Ds < 3 the surfaces have fractal properties: the rougher they are the closer Ds is to 3, and they yield scattering laws with 3 < S < 4, as observed for many glasses. In those cases where the extended fluctuations involve the concentration rather than the density, such as for Ni 8 0 P 2 0 , the roughness of the inner surfaces has to be considered as compositional. Within this picture, metallic glasses contain inner self-similar rough surfaces which are characterized by non-uniform
-1.4
-1.2
-1.0
-0.8
-0.6
log [Q (nm-1)]
Figure 4-24. Amorphous Cu5OTi5O: log-log plot of the SANS curves (Lamparter and Boucher, 1992). •: unloaded, x: loaded with 50at.% hydrogen, •: loaded with 50at.% deuterium. Straight line with slope —3.5.
251
4.4 Structure of Amorphous Metallic Alloys
is evidence for a non-uniform distribution of the hydrogen atoms, accumulating at the inner surfaces. 4.4.3.3 Defects at the Outer Surfaces An important question always in connection with small-angle scattering experiments is to what extent scattering from the sample surfaces has to be considered. Due to their preparation methods, metallic glasses are usually foils with thicknesses of only some ten micrometers, and thus they have a rather larger surface-to-bulk ratio. Therefore, any type of surface imperfection involving fluctuations of the scattering length density may cause a substantial scattering contribution. In a few investigations the contrast matching method has been applied by immersing the sample in a liquid for which the scattering length density has been adjusted to that of the sample. In this case the surface should yield no contrast and thus no scattering contribution. The results were quite different. Rodmacq et al. (1984) first used this method for amorphous Pd 80 Si 20 and found the signal to be due to the surface, whereas Maret et al. (1988) detected a minor surface contribution with amorphous Ni 33 Y 67 . Figure 4-25 shows the SANS of the alloy 5 4 Fe 4 4 6 2 Ni 3 6 1 1 B 2 0 . By isotopic substitution the resulting mean scattering length density of this alloy was adjusted to 3 = 0. The small-angle scattering was measured while the sample was immersed in different C 2 H 5 OH-C 2 D 5 OD mixtures (Trauble et al., 1992). In the case where the immersion liquid has almost the same scattering length density as the sample, i.e., 3 close to zero, the SANS signal was almost completely suppressed. Because of the zero scattering length density of the sample this would have also been obtained by measuring this sample in air.
-1.5
-1.0 log [Q (nnr 1 )]
Figure 4-25. Amorphous 54 Fe 44 62 Ni 36 curves measured with the sample in different immersion liquids. AS = contrast of the scattering length density between sample and liquid (in units of 1010 cm" 2 ) (Trauble et al., 1992).
From this surface matching experiment it was concluded that the SANS in this case is due to imperfections at the sample surfaces. On the other hand, Lamparter and Boucher (1992) measured amorphous Ni 16 Ti 68 Si 16 in air. This alloy was also designed to have a resultant zero scattering length density. Therefore the surface contrast was expected to be very low. However, quite a strong SANS signal was observed (Fig. 4-26) and hence, in this case, the SANS was attributed to the bulk of the material.
-1.5
-1.0
-0.5
0
log [Q (nnr1)]
Figure 4-26. Amorphous Ni 16 Ti 68 Si 16 with a mean scattering length density of zero: log-log plot of the SANS curve (Lamparter and Boucher, 1992). Straight line with slope —3.5.
252
4 Structure of Amorphous and Molten Alloys
Concluding this section, it must be stated that more small-angle scattering work with metallic glasses has to be done in order to get a more complete overview of all the different, possible kinds of structural inhomogeneities. The investigation of surface effects is important. 4.4.4 Relaxation Metallic glasses in their as-prepared state are generally not in configurational equilibrium. This is due to the necessity that during their preparation, whatever the specific technique, the mobility of the atoms has to be reduced very quickly, or kept low, in order to prevent crystallization. During rapid quenching of molten alloys at the glass transition temperature, which depends on the cooling rate, the mobility of the atoms is no longer sufficient to follow the changing equilibrium state of the supercooled liquid, and the configuration is frozen in as a glass. On annealing below their crystallization temperature metallic glasses lower their free energy and relax towards an internal metastable equilibrium state, which depends on the temperature. The structural relaxation processes involve reduction of frozen-in free volume and defects. Hereby, the increase in the density is of the order of 0.5%. Relaxation affects many macroscopic properties, some of them significantly, such as electrical resistivity, Curie temperature, ductility, Young's modulus, etc. On a microscopic level, a range of processes associated with a spectrum of activation energies has to be considered. Based on the concept of a distribution of activation energies Gibbs et al. (1983) developed a model for the time and temperature dependence of property changes during relaxation. Two types of behavior were distinguished: Irreversible relaxation takes
place in glasses annealed for the first time after their production. Reversible relaxation is observed in cyclical annealing experiments between two temperatures below the glass transition temperature, where the glass at each temperature develops towards the corresponding equilibrium state. Most of the relaxation measurements have been carried out on macroscopic properties, where scanning calorimetry plays the most important role. On the other hand, only a few direct studies of the atomic structure during relaxation have been reported so far. Examples are: Fe 4 0 Ni 4 0 P 1 4 B 6 (Egami, 1978), Pd 80 Si 20 (Waseda and Egami, 1979), Pd 82 Si 18 (Chason et al, 1985), (Ni,Pd) 80 P 20 (Biihler et al, 1988), by X-ray diffraction, Ni 40 Ti 60 (Ruppersberg et al, 1980), Mg 70 Zn 30 (Mizoguchi et al, 1984), Ni 3 2 Pd 5 2 P 1 6 (Schaal 1988; Schaal et al, 1988 a, b), by neutron diffraction, Fe 90 Zr 10 (Maeda et al, 1982), Fe 79 Si 11 B 10 (Yu et al, 1988), by EXAFS. In most cases the observed changes in the structure factor and in the correlation function were very small, of the order of only a few percent, but they generally showed that during irreversible relaxation the degree of ordering in metallic glasses is enhanced. In N i - P d - P glasses exceptionally large relaxation effects were found. Figure 4-27 shows S(Q) of melt spun Ni 3 2 Pd 5 2 P 1 6 in the as-quenched state, and the change AS (Q) after annealing at 570 K (Schaal, 1988; Schaal et al, 1988 a). The change in S (Q) after the 0.2 h annealing is different from that at later stages: Initially AS (Q) in the range of the main peak shows an asymmetric oscillation which corresponds to a shift of the first moment of the peak of S (Q) towards a larger Q value by a factor of (1 + /?). This corresponds to a change of the i^-scale from R to R (1 — /?) and means densification of the glass by a factor (1 + /?)3
4.4 Structure of Amorphous Metallic Alloys
253
as the increase of an oscillation of the structure function Q(S(Q) — 1), can be de-
scribed by M(t, T) = A(T)logt + B(T) T.
Figure4-27. Amorphous Ni 3 2 Pd 5 2 P 1 6 : Total F - Z structure factor of an as-quenched sample and its change after relaxation, annealed at 297 °C (AS = Sann.-SM_q.) (Schaal et al., 1988 a).
without a change in the topological arrangement of the atoms. In the case of Ni 32 pd 52 p i6> P = 0.0023, which means an increase in the density of 0.7%. At later stages AS (Q) is larger and more symmetric and its shape shows that the main peak of S(Q) has increased by about 30% and has become sharper. This behavior implies a rearrangement of the atoms leading to a more defined and more extended topological ordering in the glass. The correlation function G (R) after annealing for 20 h shows enhanced oscillations at larger distances of about 100% (Fig. 4-28). A quantitative description of both effects on S ( 0 , densification and increasing order, due to relaxation, has been developed by Bruning (1990) and Bruning and Strom-Olsen (1990). For the Fe 4 0 Ni 4 0 P 1 4 B 6 glass Egami (1978) has shown that the evolution of a quantitative parameter M (t, T), measured
In the relaxation experiments with Ni 3 2 Pd 5 2 P 1 6 by Schaal et al. (1988a) a corresponding investigation of the time and temperature behavior of the structural relaxation showed that a certain relaxed state of this glass can be obtained by different heat treatments. It turned out that the development of the structure is the same if followed either on a linear temperature scale at constant time or followed on a logarithmic time scale at constant temperature. From this it was concluded that the time and temperature behavior follows the theory developed by Gibbs et al. (1983). Relaxation phenomena may affect the topological short-range order and/or the chemical short-range order. Those irreversible processes which involve reduction of the free volume, i.e., the densification of the glass, are associated with the TSRO. The degree of CSRO decreases with increasing temperature, which follows from entropy considerations, and the relaxation of the CSRO during temperature changes is reversible. So far we have very little in-
o
2-
CD
0
Figure 4-28. Amorphous Ni 3 2 Pd 5 2 P 1 6 : Total F - Z correlation function. ( ) as quenched, ( ) annealed for 200 h at 297 °C.
254
4 Structure of Amorphous and Molten Alloys
formation about the microscopic processes taking place during structural relaxation in metallic glasses. This requires detailed investigations in terms of the partial structure factors and correlation functions, but little work has been done up to now in this field. Calvayrac et al. (1986) and Lefebvre et al. (1988) determined the partial F Z and B-T structure factors of amorphous Ni 63 7 Zr 3 6 3 by isotopic substitution neutron diffraction in the as-quenched state as well as after relaxation annealing. They report the main effect to be correlated with the TSRO function S N N (0, but also an effect in the CSRO function SCC(Q\ where both functions displayed an increase in the oscillations. The absolute increase in S c c (Q) was much smaller than in SNN (Q), however in both functions the height of their main peak increased by about 2%. Jergel and Mrafko (1984 and 1986) investigated structural relaxation in Ti-Cu-Ni-Si glasses by X-ray diffraction. They report the occurrence of a prepeak at Qp = 0.6 Ql for the as-quenched glasses, which is characteristic of a CSRO effect. On annealing the height of the prepeak increased significantly and its position shifted gradually to smaller Q values, down to Qp = 0.52Ql. The authors infer from this shift that during relaxation, in addition to the CSRO effect, the TSRO controls more and more the shape and position of the prepeak by forming small clusters of icosahedral type, which are embedded in a less-ordered matrix. Schaal et al. (1988 b) used isotopic substitution neutron diffraction to evaluate the atomic distances and the partial coordination numbers in glassy Ni 3 2 Pd 5 2 P 1 6 in the as-quenched state as well as after relaxation annealing at different temperatures. The observed changes were very small and of the same order as the experimental accuracies. However, there is a ten-
dency towards increasing coordination between Ni and Pd (5%) and between P and Pd (10%) and decreasing Ni-Ni coordination (5%) during relaxation, accompanied by a decrease in the Ni-Ni distance (2%). The most substantial changes were found in the range of the second Ni-Ni coordination sphere between 0.37 nm and 0.57 nm (Fig. 4-29). The features marked by a, b, c, d, e show that relaxation involves considerable reorganization of the Ni-Ni correlations at larger distances, whereas the nearest neighbor correlations are almost unaffected. This is evidence that the relaxation process is mainly associated with the redistribution of bond angles, leading to sharper defined coordination shells at larger distances. The fact that the correlations involving Pd and P, respectively, do not show such strong effects, can be under-
0.2
0.3 0.4 R (nm)
0.5
0.6
Figure 4-29. Amorphous Ni 32 Pd52Pi 6 : Development of the partial radial distribution function RDF NiNi (R) during relaxation annealing for 2 h. Dashed lines show Gaussian fits (Schaal et al, 1988 b).
4.4 Structure of Amorphous Metallic Alloys
stood by the restricted mobility of these atoms. In the case of Pd because of its larger diameter and in the case of P because it is trapped in the structure by chemical bonding. 4.4.5 Structural Models
Since the experimental evaluation of structural properties of amorphous metals many attempts have been made to develop structural models which are able to explain the experimental data. Mainly glasses belonging to the T - M group have been subjected to model calculations up to now. These were based on quite different concepts. In the present section some selected models for T^M glasses are described, each representing one of these different concepts. For a review of models of T - M glasses see Gaskell (1983). In earlier times models were set up to simulate total G(R) functions only. These were mainly measured by X-ray diffraction, which for T - M glasses essentially presents the distribution of the T atoms only. It turned out that simple packing of hard spheres with a diameter crHS, representing the metal atoms, can describe the general features of experimental distribution functions, such as the splitting of the second maximum into two subpeaks near 1.7 aHS and 2.0 aHS. This can be understood from the point of view of close packing of tetrahedra, which have to be distorted in order to fill space. If we start with a regular tetrahedron of 4 atoms in close contact and locate 2 more atoms onto 2 triangular faces, these additional atoms will be apart a distance 1.67 <7HS. The second subpeak near 2.0 aHS corresponds to 3 collinear atoms. Ichikawa (1975) has shown by constructing clusters of single-size hard spheres that the degree of splitting of the second maximum in the structure factor as well as
255
in the pair correlation function of glasses is related to the degree of the distortions of the tetrahedra making up a cluster. With increasing distortion the splitting becomes less pronounced, until finally a liquid-like situation is reached. In the meantime, models for binary metallic glasses have to be judged on the basis of their ability to represent the complete set of the partial structure factors of an alloy as well as the partial correlation functions. The experimental establishment of these partial functions for several glasses in the last decade again stimulated the model constructions of these materials. In the very early stages the question was whether or not an amorphous solid is simply an assembly of randomly oriented microcrystals of one crystalline phase or a mixture of two crystalline phases. The absence of long-range order, and, accordingly, diffuse scattering, would then be due to the finite sizes of the crystallites, confined to a few coordination shells. Pair distribution functions have been simulated by Gaussian broadening of the atomic distances in crystalline phases thereby introducing a certain degree of disorder. Alternatively, the structure factor was calculated from the distances using the Debye expression, which corresponds to Eq. (4-4), written for the case of random orientation. However, with these simple approaches it was not possible to describe the experimental G(R) satisfactorily. One of the problems here is presented by the surfaces of the crystallites. A considerable number of the atoms would necessarily be located in the vicinity of these surfaces and the contribution of the associated pair correlations, which are not defined, is not considered in this type of model. Within the microcrystalline picture the glass is not a particular state of matter, but the extreme state within a series of states starting from
256
4 Structure of Amorphous and Molten Alloys
a single crystal and going on to polycrystals with decreasing grain sizes down to say 2 nm. Correspondingly, all possible intermediate stages should be realized in the diffraction patterns of glasses, for example, of melt spun samples at varying cooling rates. In reality, however, at low cooling rates no intermediate stages are observed, but diffraction patterns show two defined phases, namely sharp crystalline peaks superimposed on the diffuse scattering of the amorphous phase. On the other hand, a group of materials with extremely small grain sizes does in fact exist, i.e., nanocrystals, which have gained increasing interest recently. Structural models for non-crystalline systems can be classified into two main classes: (i) Models which are based on the theory of the liquid state. (ii) Models which are based on three-dimensional clusters. (i) Concerning the models which are based on the theory of the liquid state, structural functions are calculated from suitable pair potential functions using one of the existing theoretical equations which interrelate both functions. The most wellknown one is the Percus-Yevick hardsphere model (Percus and Yevick, 1958; Ashcroft and Langreth, 1967), for which the potential is given by the hard-core repulsion between the atoms. Refinements were achieved by also taking into account chemical interactions (Hafner et al., 1984; Pasturel et al., 1988). As a model for the amorphous state, it is assumed that it can be represented by the corresponding supercooled liquid just above the glass-transition. (ii) In contrast to theoretical analytical models of liquid alloys, which are in ther-
modynamical equilibrium, most of the models for metallic glasses belong to the second main class. Here three-dimensional clusters of atoms are constructed. In the past these were laboratory-built clusters using balls. Nowadays the clusters are defined by a set of coordinates of several thousand atoms which are constructed using a computer. These clusters contain more structural information than the experimental one-dimensional G (R) functions, e.g., about bond angles, local symmetries, and certain coordination polyhedra. For comparison the models have to be projected into one dimension, and furthermore an average over the individual atomic sites has to be taken. Thus comparison with the experiment necessarily involves a considerable loss of information. An important advantage of a cluster model, provided it fits the experimental G(R), is that it may serve as a basis for the calculation of other physical properties, such as electronic and mechanical properties of a glass, which again can be compared with experimental data. Of course the density is the first parameter to be checked against a model. The development of a suitable cluster model comprises two steps: 1. Construction of an initial set of atomic positions as a starting structure. 2. Refinement of the structure. It is not possible to classify existing cluster models unambiguously into different groups. However, as a rough classification we can distinguish between four main groups: Dense random packing of hard spheres (DRPHS), stereochemically defined models (SCD), molecular dynamics calculations (MD), and Monte Carlo simulations (MC). For DRPHS and SCD models the construction of the starting cluster is the crucial step whereas for MC and MD models the refinement is the essential procedure.
257
4.4 Structure of Amorphous Metallic Alloys
a) DRPHS models: Strictly speaking, suitable models are neither random, nor are the spheres hard after refinement. Often the chemical ordering effects observed in real glasses are already incorporated to a certain extent in the initial packing of hard spheres, which involves some intuition based on available knowledge about the structure to be simulated. A computer packing algorithm by Bletry (1978) was used for amorphous Ni 8 1 B 1 9 by Lamparter et al. (1982) taking CSRO into account. 5000 hard spheres of two different sizes were successively packed together using the ordering rule that the small spheres (the metalloid atoms) are surrounded only by large spheres (the metal atoms). Thereafter, softness of the spheres was simulated by introducing a temperature factor, i.e., the peaks of the resulting Gtj(R) curves were broadened, but no relaxation refinement was included. The density of the model, and thus also the coordination numbers, turned out to be distinctly smaller than those of the real glass, and a multiplication factor was used for the ordinate to match the amplitude of the first maximum of the experimental partial Gtj(R). A comparison between experiment and model is shown in Fig. 4-30 a. The agreement is poor, but the model already shows the main features such as atomic distances and the characteristic peak splitting in the B-B correlation. Obviously, these features can be reproduced by this very simple model with forbidden B-B nearest neighbors. (Figs. 30b and c will be discussed later in this section.) Better agreement is achieved by a subsequent relaxation procedure. Using appropriate pair potential functions, the atoms are moved sequentially in the direction of the resulting force until after a certain number of cyclic iterations minimum final potential energy is achieved. Fujiwara
0.2
0.4
0.6
0.8
1.0
1.2
R (nm)
Figure 4-30. Amorphous Ni 81 B 19 : Model calculations of the partial F - Z pair correlation functions (solid lines) and comparison with the experimental curves (dashed lines) from Fig. 4-5. (a) DRPHS model (Lamparter et al., 1982). (b) SCD model (Dubois et al., 1985), (c) MD model (Beyer and Hoheisel, 1983).
258
4 Structure of Amorphous and Molten Alloys
et al. (1982) used a model to obtain the partials Gtj(R) of Fe 8 0 B 2 0 and compared them with the experimental ones of Nold et al. (1981) (Fig. 4-31). The starting cluster was a random packing of 1500 hard spheres of two sizes. The initial structure was relaxed using Morse-type potentials. The ordering effects were forced by choosing the Fe-B interaction stronger than the Fe-Fe interaction and a weak B-B interaction at a distance distinctly larger than the size of the B atoms. The relaxation refinement yields much better agreement with the experimental data compared with the unrelaxed Ni-B model. However, the coordination numbers of the model as reported by the authors are still smaller by about 10%, and the details in the G0(K) functions differ, the real glass appearing to be more ordered, especially in the range of the second coordination shell. A similar computer simulation for amorphous
30
0.2
0.4 0.6 R (nm)
0.8
Figure 4-31. Amorphous Fe 80 B 20 : Model calculation (DRPHS) of the partial F - Z pair correlation functions (solid lines) by Fujiwara et al. (1981) and comparison with experimental curves (dashed lines) from Nold et al. (1981).
Fe 8 0 B 2 0 by Lewis and Harris (1984) also yielded good agreement with the experimental data, but again the coordination numbers obtained were smaller by 15 to 20%. b) SCD models: Here no single atoms are packed together but structural units consisting of several atoms which already contain a stereochemically defined short-range order. Trigonal prismatic packing (TPP) models by Gaskell (1983) are based on the fact that those transition metal-metalloid systems, which form glasses, build crystalline phases, like Fe 3 P and Ni 3 B, where the metalloid atoms are coordinated by 6 metal atoms forming a trigonal prism and by 3 further metal atoms capping the three rectangular faces of the prism (Fig. 4-32). This apparently quite stable unit was suggested to define the metalloid coordination also in corresponding T - M glasses, in that case more distorted, of course. In support of this view is the evidence of strong T - M interaction in the glasses, suggesting a well defined T - M correlation, and the frequently observed coordination number ZMT = 9, which is most probably independent of the metalloid concentration. It is possible to pack trigonal prismatic units together in different ways. One is shown in Fig. 4-32 with two edge-sharing prisms, where one face-capping atom is at the same time at the vertex of the neighboring prism. To model the structure of amorphous Ni 8 0 B 2 0 , Dubois et al. (1985) constructed a cluster of some thousand atoms where trigonal prisms were packed together using the concept of chemical twinning, followed by a relaxation under the influence of Lennard-Jones potentials. The authors report that distorted capped prisms also persist as the dominant coordination polyhedron of the M atoms in the final relaxed cluster. Comparison with the experimental Gtj(R)
4.4 Structure of Amorphous Metallic Alloys
o M
Figure 4-32. Structural metal-metalloid unit: Two trigonal prisms are connected by two edge sharing atoms and by one atom of the second prism, this being one of the face capping atoms of the first prism. (Only one of the face capping atoms is shown.) The metalloid atoms are located at the center of the prisms.
of Ni 8 1 B 1 9 in Fig. 4-30b shows good agreement, but also differences, particularly in the B-B correlation at larger distances. c) MD models: In these models not only positions but also certain velocities are attributed to the atoms in a starting cluster. Further development of the model occurs in sequential steps by solving the Newtonian equations of motion using selected pair potentials. To simulate quenching from a liquid to an amorphous state the velocities of the atoms were reduced in some cases during the development. As an example, the MD model for Ni 8 1 B 1 9 by Beyer and Hoheisel (1983) is presented in Fig. 4-30 c. The model cluster consisted of 2048 atoms. Applying Lennard-Jones potentials, the Ni-B interaction used was not especially strong compared with the Ni-Ni interaction, but
259
the B-B equilibrium pair distance was taken as large. Good agreement with the experimental Gij(R) is observed, where again the main differences occur in the B B correlation at larger distances. d) MC models: With this method no potentials are required. The refinement procedure is performed on the basis of experimental distribution functions and/or structure factors. The calculation starts from a certain initial configuration of atoms generated by a computer. A new configuration is then created by random displacement of a randomly chosen atom. If the partial structure functions calculated for the new configuration show improved agreement with the experimental data it is accepted as a new initial cluster. Otherwise it is accepted only with a certain probability. After a certain number of steps no further improvement is obtained and the final cluster is taken as a model of the amorphous alloy. Obviously, the quality of the experimental data is a crucial condition for the success of this approach. A MC computer algorithm by McGreevy and Pusztai (1988) was used to simulate the partial pair correlation functions of amorphous Ni 6 7 B 3 3 (Pusztai, 1991). The results are shown in Fig. 4-33, and very good agreement with the experimental data is observed. Since all the different models used for the metallic glass Ni 8 1 B 1 9 gave almost the same agreement with the experimental partial correlation functions Gij(R) (Fig. 4-30), it is difficult to decide which approach is most suitable. This agreement can be explained either by the assumption that the relaxation process of the various models leads to comparable structures or by the suggestion that different three-dimensional models may lead to similar Gtj functions. Thus, comparison of models should not only be made via the experi-
260
4 Structure of Amorphous and Molten Alloys
0.4
0.5
0.6
0.7
R (nm)
Figure 4-33. Amorphous Ni 67 B 33 : Model calculation (MC) of the partial F - Z pair correlation functions (solid lines) by Pusztai (1991) and comparison with experimental curves (dashed lines) from Ishmaev et al. (1987).
mental reference functions, but also directly using the same three-dimensional analysis criteria for each model and comparing the distributions of local structural features. However, such comparative studies have rarely been performed up to now. Boudreaux and Frost (1981) analyzed DRPHS models for binary T - M alloys as a function of composition. The models (Boudreaux, 1978) were computer generated in two steps. The starting clusters were random packings of hard spheres of two different sizes, for which care was taken to avoid nearest M - M contact. Modified Lennard-Jones potentials were used for relaxation, the T - M interaction being twice as strong as the T - T interaction, and the M - M interaction very weak and at a larger equilibrium distance. It is reported that the metalloid coordinations are quite regular with octahedra and trigonal prisms as the dominant coordination polyhedra, and that the number of metal nearest neighbors around the M atoms does not depend on the composition.
Fujiwara and Ishii (1980) constructed DRPHS clusters for F e ^ ^ R , alloys, where close P - P contact was forbidden, and applied a relaxation process. They analyzed the geometrical configurations around the P atoms and found the coordination number Z PFe = 9 to be independent of the concentration x, and about 50% of the P atoms to be coordinated in trigonal face-capped prisms. Of course the result of an analysis with respect to special coordination polyhedra will depend on the maximum degree of distortion which is tolerated by the investigator for a specific type before it is rejected. Kizler (1988) and Kizler et al. (1988) analyzed existing models for Fe 8 0 B 2 0 and Ni 8 1 B 1 9 with respect to coordination numbers and bond angles. Fig. 4-34 shows the histograms of the distribution of the first sphere coordination number Z BT for crystalline Fe3B and five different models. With the exception of Z week's model they show a maximum at Z BT = 8, and do not exhibit any significant differences from model to model. The smaller value of Z BT compared with Z BT = 9, as found in most of the experimental studies, can be explained by the lower density of the model clusters compared with the real glass. Symmetry properties should show up in triplet correlation functions, which contain information about angular correlations. Fig. 435 shows the histograms of the T - B - T bond angle distributions. They show distinct features with a two-peak structure, but again no significant differences except for some details. These observations strengthen the idea that the different models become rather similar after refinement. Bearing in mind that diffraction experiments and conventional EXAFS only yield a one-dimensional image of a non-crystalline structure, it is important for testing the models to develop experimental ap-
261
4.4 Structure of Amorphous Metallic Alloys 0.5
0.08-1
- 1.0
0.05Fe3B :
(e)
r
i
(0
s
0.05-
a. (c) _ p
(a)
^j-
I
(d)
I
10
(b) o
J
_r
I
I 10
5 CO
0.05-
15
ZBT 60
120
0 60 Angle a
120
180
Figure 4-34. Amorphous Fe 80 B 20 , Ni 81 B 19 , and crystalline Fe3B: Frequency distribution of the coordination number ZBT (T = Fe, Ni) according to different models, a: Beyer-Hoheisel (MD), b: Dubois-Gaskellle Caer (SCD), c: Zweck (DRPHS), d: Fujiwara (DRPHS), e: Brandt (MD), f: Fe3B. (Kizler et al., 1988, and references given there.)
Figure 4-35. Amorphous Fe 80 B 20 , Ni 8 1 B 1 9 , and crystalline Fe3B: Frequency distribution of the bond angle aTBT (T = Fe, Ni) according to different models (Kizler et al., 1988, and references given there). For labels of the curves see Fig. 4-34.
proaches which provide information about higher order correlations, i.e., about the three-dimensional atomic arrangements. A rather new method is XANES (X-ray absorption near edge structure). The near edge part of the absorption spectrum of a constituent embedded in a structure extends up to about 50 eV above the edge. At higher energies a continuous transition occurs into the EXAFS regime, which extends up to about 700 eV. In the XANES region, corresponding to low-energy photo-electrons, multiple scattering within the neighboring shells becomes important. As multiple scattering involves more than one neighboring atom at the same time, the features of a XANES spectrum are sensitive to higher order correlations, i.e., bond
angles and the symmetry of the environment. A XANES investigation of amorphous Fe 8 0 B 2 0 (melt spun) and Ni 8 0 B 2 0 (sputtered) has been performed by Kizler et al. (1989). The results are shown in Fig. 4-36. The experimental spectra were compared with spectra calculated theoretically for a number of available model clusters, where the XANES computer algorithms of Durham et al. (1981 and 1982), in their updated version by Vvedensky et al. (1986), were used. The calculated spectra in each case represent the average for 40 to 50 central T atoms selected out of the cluster. The experimental spectra of the Ni-B and the Fe-B glass are much the same, consisting of two peaks with a small bump in the
262
4 Structure of Amorphous and Molten Alloys
Figure 4-36. Amorphous Fe 8 0 B 2 0 , Ni 8 0 B 2 0 : Comparison of experimental and calculated XANES spectra (Kizler et al., 1989). The dashed curve was calculated without multiple scattering. Curve (f) is for crystalline Fe3B. For labels of the curves see Fig. 4-34.
middle. The slightly smaller peak separation for the Fe-B alloy is explained by the larger atomic diameter of Fe compared with that of Ni, causing larger atomic distances in the Fe-B alloy. The calculated XANES curves agree quite well with the experimental ones as far as the general features are concerned. From this it was concluded that the XANES theory is able to yield realistic results for disordered alloys and that the dif-
ferent structural models present the main features of the short-range order more or less with the same agreement. However, there are differences in the details concerning the peak separation and the region between the peaks in Fig. 4-36: the Dubois-model (b) and, less distinctly, the Zweck-model (c) exhibit an indication of the characteristic bump (marked by an arrow) as observed in the experimental curves. The distance between the peaks (29.8 eV for Fe 8 0 B 2 0 and 30.5 eV for Ni 80 B 20 ) is best represented by the Dubois-model. An important result of this study was that the XANES spectrum is not simply related to the bond angles between a central atom and the nearest neighbors, but that it was essential to include also at least the second coordination shell. The dashed curve (bs) in Fig. 4-36 was calculated without multiple scattering, i.e., only on the basis of pair correlations. Comparison with curve (b), which includes multiple scattering effects, illustrates that the details, but not the overall shape of the XANES spectra, are determined by the specific three-dimensional atomic arrangement. From further investigations of experimental and calculated XANES spectra for amorphous and molten alloys useful structural information, complementary to diffraction methods, is to be expected. In conclusion to this section concerning model calculations, two general remarks are given: (i) Refinement of model clusters is usually performed using spherical potentials. In real metal-metalloid glasses the distinct chemical interaction probably involves directional bonding and thus may cause the M - T coordination polyhedra to be more defined. Thus, a narrower distribution of the M - T coordination numbers compared with the models may exist in real glasses.
263
4.5 Structure of Molten Metallic Alloys
(ii) Phase separation phenomena, explained by the tendency of the glass to establish locally a defined stoichiometry combined probably with a higher degree of ordering in regions with a size of some nm, are not taken into account by the existing models. Note that the diameter of a model cluster with 5000 atoms is of the order of some nm only.
4.5 Structure of Molten Metallic Alloys Comparison of molten alloys with molten elements is useful for the understanding of the specific structural features of molten alloys. Therefore, in the first part of this section the structure of molten elements is described briefly. 4.5.1 Elements
Since the first diffraction experiment with a liquid element by Debye and Menke (1930), who investigated liquid mercury, almost all of the elements in the liquid state have been studied by diffraction methods. Figure 4-37 shows as examples the structure factors S (Q) of some liquid elements and Fig. 4-38 their Fourier transforms G (R). S (Q) for pure elements represents the density-density Bhatia-Thornton structure factor SNN (Q). At Q = 0 it is given by the compressibility term in Eq. (4-30). The general shape of S(Q) as well as of G {R) of the elements Zn and Cs (100 °C) is typical for all simple metals in the liquid state near their melting point. In the case of a few elements such as Zn, Cd, and Hg the main maximum is asymmetric being flatter on the left-hand side. In Table 4-4 structural parameters for some selected liquid metals are compiled, each representing a group of the periodic table. A similarity in
20
40 Q
60 (nm 1 )
80
100
120
Figure 4-37. Liquid elements: Structure factors. Zn: Biihler et al. (1987), Cs: Martin et al. (1980a), Bi,Sb: Lamparter et al. (1976), Se: Waseda (1980), X = X-ray diffraction, N = neutron diffraction. The dashed curve was calculated for Cs using the Percus-Yevick hard-sphere model.
1210-
Zn
Bi
+4.5
Bi
+3
Sb
+1.5
Se
-2 0.5
1.0
1.5
R (nm)
Figure 4-38. Liquid elements: Pair correlation functions calculated from the structure factors in Fig. 4-37.
264
4 Structure of Amorphous and Molten Alloys
Table 4-4. Liquid metals: Structural parameters (selected from the literature). R1 = first neighbor distance. Dat = atomic diameter. Z1 = coordination number. Z\ = coordination number in the solid state. Rt = correlation length. <7HS = hard-sphere diameter in the Percus-Yevick model. Y = packing fraction. Element T[°C]
^[nm] Dat [nm]
Z1
Na 105
0.381 0.38
10.4 8
1.9
0.87 0.46
Mg 700
0.32 0.32
10.9 12
1.4
0.89 0.46
Ti 1700
0.317 0.294
10.9 12
Fe 1550
0.258 0.252
10.6 8
1.4
0.86 0.44
Cu 1150
0.257 0.256
11.3 12
1.3
0.88 0.46
Zn 450
0.27 0.276
10.8 12
1.4
0.89 0.46
Ga 50
0.28 0.282
10.4 7
1.4
0.95 0.43
Ge 960
0.263 0.274
6.5 4
Sb 650
0.299 0.318
5.6 6
1.4
Se 250
0.236 0.28
2.1 6
0.6
Rt [nm] Y
A
0.80 0.44
0.9 0.38 0.96 0.40
the general features can be seen. The width AQ1 of the first peak in S (Q) is a measure for the spatial extension of the topological order, the so-called correlation length Rt, according to Eq. (4-12). The figures in Table 4-4 show that the ordering extends to about Rt = 1.5 to 2 nm. The nearest neighbor distances Rl agree well with the Goldschmidt diameter Dat tabulated for the elements in the solid state. This means that atomic distances are not a sensitive enough parameter for the
investigation of structural differences or similarities between the liquid and the solid state of an element. However, the coordination numbers Z1 change distinctly on melting. For typical liquid metals they lie between 9 and 11 and thus are smaller than the value 12 for a close-packed solid structure. The coordination number is reduced on melting in those cases where the solid was closepacked. This is explained by defects introduced at the melting point. In those cases where the solid is not close-packed, such as the body-centered cubic crystals with Z1 = 8 or the diamond structure with Z1 = 4, the coordination number tends to increase on melting. In the field of theories on liquid metals based on the principles of statistical mechanics several equations have been developed which relate the pair distribution function to the pair potential. These relations, however, are generally only approximate because the interaction between two atoms also depends on the remaining atoms, and this has to be taken into consideration somehow. On the other hand, computer experiments, such as the molecular dynamics method and the Monte Carlo method, have been applied to simulate the atomic structure, and hence the distribution function, on the basis of a specific pair potential. Thus, computer experiments provide a tool for critical tests of theoretical equations. For detailed presentations of the theories on the liquid state as well as of computer simulations see the textbooks, e.g., by Rice and Gray (1965), Faber (1972) and Shimoji (1977). A frequently used, though not very realistic, theoretical model is the PercusYevick hard-sphere model (Percus and Yevick, 1958; Ashcroft and Langreth, 1967). It yields an analytical expression for the structure factor as a function of the
265
4.5 Structure of Molten Metallic Alloys
hard-sphere diameter aHS and the packing fraction (4-43)
A computer code for the application of the Percus-Yevick model to binary alloys may be found in the book by Waseda (1980). Fitting this model to experimental structure factors just above the melting point by adjusting aHS yields aHS = (0.85 ... 0.9) x Rl. (Note that Rl is the most frequent distance, whereas crHS is the minimum possible distance.) The values of the packing fraction Y calculated from Q0 and
(4-44)
For a compilation of numerical values for the parameters Yo and Yx see the textbook by Waseda (1980). The structure of semi-metals in the molten state is different from that of simple liquid metals, depending on the specific element, and cannot be described by a hardsphere model. The structure factor has a more complicated shape and in many cases (Ga, Sn, Ge, Si, Bi, Sb) exhibits a shoulder or even an additional peak sitting on the high-<2 side of the main peak, as can be seen for Bi and Sb in Fig. 4-37 and for Sn, Ge and Si in Fig. 4-39. The structure factors of the liquid group IVB elements in Fig. 4-39 (Gabathuler and Steeb, 1979) show that this behavior becomes more prominent on going from the more metallic Pb to the less metallic Si, while at the same time the covalent nature of the bonding in the solid state increases. In several investigations this doublepeak structure factor has been attributed to two different first neighbor distances,
40 Q
60
80
100
(nm-1)
Figure 4-39. Liquid group IV B elements: Structure factors from neutron diffraction (Gabathuler and Steeb, 1979).
266
4 Structure of Amorphous and Molten Alloys
which belong to two types of topological ordering coexisting in the liquid state. Orton (1975, 1980) described liquid polyvalent metals showing a shoulder in the structure factor in terms of the PercusYevick model using a system of hard spheres of two different diameters. A shorter atomic distance was attributed to directional p-type bonding, and a larger distance to metallic bonding. In a similar approach, based on the presence of covalent bonding in the crystalline as well as in the amorphous state of these elements, it was also suggested that in the molten state the atoms fluctuate locally between a metallic and a covalent structural state. In model calculations for Sb and Bi (Lamparter et al, 1976) and for Pb, Sn, and Ge (Gabathuler and Steeb, 1979), the metallic part was described by a hardsphere model, whereas the covalent part was described by tetrahedral units. From this viewpoint the low coordination numbers in these melts (Table 4-4), in contrast to simple metals, are explained as being the mean value of the coordination of 3 or 4 within the tetrahedra and a value of about 10 within the metallic part. In the case of liquid Sb the two Sb-Sb distances were found to be 0.278 nm and 0.317 nm, which are in fact the covalent and the atomic diameter, respectively, of Sb. The temperature dependence of the structure factors of these semi-metals is characterized by a lowering of the shoulder with increasing temperature (cf. Bi in Fig. 4-37) indicating that the melt becomes more metallic. Theoretical considerations have been applied to explain this shoulder on the basis of specific pair potentials (Silbert and Young, 1976; Levesque and Weis, 1977; Oberle and Beck, 1979; Beck and Oberle, 1980; Mon et al., 1979; Regnaut et al., 1980; Hafner, 1984). Oberle and Beck (1979) gave
a qualitative view of the connection between the potential and S (Q): the main features, such as the principal peak, are determined by the repulsive part of the potential and can be well simulated by a hard core repulsion. The long-range tail of the potential shows oscillations with a dominant wavelength, which is determined by the Fermi wavevector, i.e., by the number of valence electrons. In the case of metals with a valence greater than 3 the Fourier component of this wavelength is located on the high-Q side of the principal peak of S (Q). The structure factors of liquid semi-conductors like Se (Fig. 4-37) and Te differ even more from those of simple liquid metals. The very small coordination numbers (2 for Se and 3 for Te) have been explained by a chain- or a ring-like structure. For details see the textbook by Waseda (1980). 4.5.2 Alloys Molten metallic alloys can be classified into three main groups according to their structural, electronic and thermodynamic properties (Sauerwald, 1943 and 1950): Statistical, compound-forming, and segregation alloys. In statistical alloys the atoms of different types are distributed at random. This is the case when their chemical properties and their sizes are quite similar. The partial structure factors and also the partial correlation functions are much the same and do not change significantly with composition. The structural parameters of an alloy can then be predicted from those of its pure elements by interpolation. The heat of mixing in such alloys is small. In the following some selected binary alloys which deviate from the statistical distribution will be discussed. Although structural studies of molten metallic alloys were started about two decades before amor-
267
4.5 Structure of Molten Metallic Alloys
phous metallic alloys were investigated, the number of liquid systems where the partial structural functions have been established is still quite small. This is due to the experimental difficulties in handling liquids at high temperatures. In most cases structural properties were deduced only from total structure factors rather than from partial ones (see e.g. Steeb, 1968). 4.5.2.1 Compound-Forming Alloys
In compound-forming alloys the stronger chemical interaction between unlike atoms compared with that between like atoms causes a short-range order with preferred unlike atomic pairs, i.e., hetero-coordination. These alloys have a negative heat of mixing. In most cases a distinct difference between the electronegativities of the constituents is the source of the chemical interaction and involves a certain degree of charge transfer and affects the atomic distances and the coordination numbers. Many of these alloys are composed of an alkaline or an alkaline earth element and an element from the higher groups of the periodic system, such as Li-Pb, Mg-Sn, and Cs Au alloys. In Table 4-5 some structural parameters of some compound-forming alloys are compiled. Frequently the strongest tendency for compound formation is found at those compositions where a stable intermetallic phase exists up to quite high temperatures in the solid state. The Mg-Sn system was one of the first in which strong hetero-coordination was observed by X-ray diffraction (Steeb and Entress, 1966). Figure 4-40 shows the total F - Z structure factors of liquid Mg-Sn alloys. The most significant feature is the occurrence of a prepeak near Q = 16 nm" 1 , in front of the main peak, which is not observed with pure elements and proves the existence of chemical ordering in these
80
Figure 4-40. Liquid Mg-Sn alloys: Total F - Z structure factors from X-ray diffraction (Steeb and Entress, 1966).
alloys. As we know nowadays, the prepeak is the principal peak of the Bhatia-Thornton partial structure factor S c c (Q\ whereas the main peak belongs to the partial structure factor SNN(<2). At the composition Mg2Sn the Scc peak has the largest amplitude, i.e., the ordering effect is strongest. In Fig. 4-41 structural parameters are compared with several other properties. Both the nearest neighbor distance and the coordination number show a negative deviation from the straight line which would be expected for statistical distribution. The chemical interaction between the Mg and the Sn atoms causes a contraction of the Mg-Sn distance compared with the mean value of the atomic diameters of Mg and Sn in their elemental liquid state. The phase diagram contains the stable intermetallic compound Mg2Sn with its pronounced melting point maximum. Obviously, the chemical interaction creating this compound in the solid state persists
268
4 Structure of Amorphous and Molten Alloys
Table 4-5. Liquid compound-forming alloys: Structural parameters. T = temperature. Qp = position of the first Scc peak. Q1 = position of the first SNN peak. Qp/Q1 = ratio of the positions. Scc (Qp) = height of the first S c c peak (* estimated from total S(Q)). Rt, Rc = correlation length of the topological and chemical ordering, respectively. aJ(a[ei) = (relative) Warren-Cowley short-range order parameter. X = X-ray diffraction, N = neutron diffraction. T[°C]
Mg 7O Zn 3O Biihler et al. (1987), X
near Tm
0.59 2.0*
1.4 0.90
Mg 6 7 Sn 3 3 Steeb and Entress (1966), X
near Tm
0.65 2.1*
0.93 0.98
Mg7OBi3O Boos and Steeb (1977), X, N
near Tm
0.64 2.6
0.9 1.2
-0.44 1.0
Li 7O Mg 3O Chieux and Ruppersberg (1980)
near Tm
Qp=153nm-1 1.24
_ 0.70
-0.04 0.09
near Tm
fip=17.7nm-1
— 0.80
-0.15 0.39
Qp/Ql
scc(QP)
Rt [nm] Rc [nm]
a1
Alloy Ref., Method
«lel
N, =0 Li
72 A g28
Chieux and Ruppersberg (1980) N, < / > = 0
1.58
Li 77 Ga 23 Reijers et al. (1989c), N, = 0
475
QF=15nm~1 1.33
_ 0.66
-0.09 0.30
Li 7 2 5 Sn 2 7
750
e^l^nm-1 1.44
— 1.8
-0.16 0.42
0.63 2.67 — 2.38 — 2.06
2.0 — 1.8 — 1.6
0.61 1.29
0.82 0.79
5
Ablasetal. (1984), N, = 0 Li80Pb20 Ruppersberg and Reiter (1982) N,=0
722 802 952
Cu 6 6 Ti 3 4 Fenglaietal. (1986), N a
950
-0.25 1.0
-0.08 0.16
With SNC = 0 and SNN from Percus-Yevick hard-sphere model.
into the molten state. This is associated with a maximum in the electrical resistivity, a maximum in the viscosity, i.e., a minimum in the atomic mobility, and maxima for the thermodynamic functions, the enthalpy of mixing and the excess stability. The width of the prepeak AQP is as small as that of the main peak AQl in Fig. 4-40 implying the same extension of the chemical
and the topological ordering in real space for Mg-Sn melts. For those compound-forming alloys where the deviations of the distances and the coordination numbers from the statistical values are moderate, it was suggested that the topological order is not affected much by the chemical effect. These alloys belong to the type of substitutional com-
4.5 Structure of Molten Metallic Alloys .33
269
150-r
.32
100-
.31 *
.30
50-
(a)
3: o
200 20
40 60 Sn concentration (at.%)
100
Figure 4-41. Liquid Mg-Sn alloys: Comparison of structural and other physical properties, (a) Apparent atomic distance R\ (b) apparent coordination number N\ (c) phase diagram, (d) electrical resistivity gel, (e) viscosity v, (f) enthalpy of mixing AH, (g) thermodynamic activities aMg and aSn as well as Darken excess stability Sx (Steeb and Entress, 1966, and references given there).
20 Sn
pound-forming systems comparable to solid solutions with short-range order. In a rough approximation Mg-Sn melts may be classified as this type. The main peak of S(<2), belonging to S N N (0, is not drastically changed where a strong prepeak occurs in Fig. 4-40. There are other compound-forming systems where complicated topological and chemical ordering effects occur and which cannot be classified into one special type. Figure 4-42 shows the total A-L structure factors for Mg-Bi melts, as obtained from
40 60 80 concentration (at.%)
100
neutron and X-ray diffraction (Weber et al., 1979). Neutrons mainly yield SNN(Q) whereas X-rays in this case are also sensitive to SCC(Q) (see below), and for 30 at.% Bi the prepeak at Q = 16 n m ' 1 in the Xray curve is even higher than the main peak. The neutron curve shows the prepeak only as a shoulder on the left hand side of the main peak. At 40 at.% Bi, i.e., at the stoichiometric composition Mg 3 Bi 2 , the main peak has a complicated doublepeak structure besides the prepeak, illustrating that a considerable reorganization
270
4 Structure of Amorphous and Molten Alloys
2-
40 Q
20
60 (nirr1)
80
100
Figure 4-42. Liquid Mg-Bi alloys: Total A-L structure factors. ( ) Neutron diffraction, ( ) X-ray diffraction (Weber et al., 1979).
1970) implies that this alloy is non-metallic. In order to obtain a more detailed picture on the short-range order, e.g., the value of the short-range order parameter, the partial structure factors have to be determined, which are however only known in a few cases for liquid metallic alloy systems. Enderby et al. (1966) evaluated for the first time the partial structure factors of a molten alloy. They investigated liquid Cu 6 Sn 5 using isotopic substitution neutron diffraction. In the case of Mg 70 Bi 30 the partials Scc (Q) and SNN (Q) have been obtained by a combination of neutron and X-ray diffraction (Boos and Steeb, 1977). According to Eq. (4-25), the total A-L structure factors are written in terms of the three BhatiaThornton partial structure factors as: SN = 0.95 SNN + 0.05 S cc - 0.93 SNC
of the topological short-range order has taken place on alloying Mg and Bi. At this composition, where a very stable solid intermetallic compound exists, the coordination number in Fig. 4-43 obtained from the neutron data shows a distinct minimum. Obviously Mg-Bi melts do not belong to the substitutional type. Moreover, the very low electrical conductivity (45Q~ 1 cm~ 1 ) of liquid Mg 3 Bi 2 (Enderby and Collings,
20
40 Bi
60
concentration
80
100
(at. %)
Figure 4-43. Liquid Mg-Bi alloys: Apparent coordination number Nl derived by neutron diffraction. The dashed line would be expected in the case of ideal mixing.
S x = 0.51 SNN + 0.49 S cc - 2.18 SNC In these equations the scattering factors / of Mg and Bi are 12 and 83 for X-rays (given at Q = 0 by the atomic numbers) and 0.54 and 0.85 for neutrons (in units of 10~ 12 cm). The large difference between / Mg and / Bi for X-rays, compared with that for neutrons, makes the contribution of the partial structure factor S cc stronger in the X-ray experiment by one order of magnitude. The two radiations yield two equations for the three partial structure factors. In those cases where the atomic diameters of the constituents are not very different (i.e., the size effect is small) the contribution of SNC is small and is neglected in order to solve the two equations for SNN and Scc. This procedure yielded the partial structure factors S N N (0 a n d ScciQ) i n Fig. 4-44. The strong oscillations of S cc (Q) reflect the concentration fluctuations due to a strong chemical ordering effect. The
271
4.5 Structure of Molten Metallic Alloys
20
40 Q
60 (nm 1 )
Figure 4-44. Liquid Mg 70 Bi 30 : Partial B-T structure factors SNN(Q) and SCC(Q) (Boos and Steeb, 1977).
Warren-Cowley short-range order parameter, calculated from the corresponding Fourier transforms GNN(R) and GCC(R), according to Eq. (4-29), is a1 = — 0.44. This is the largest possible negative value (aInin = ~~ cBi/cMg = "" 0.43) and thus shows that in liquid Mg 70 Bi 30 the highest possible degree of short-range order is present. For some Li-containing alloys so-called zero-alloys were produced by alloying certain amounts of 7Li, which has a negative neutron scattering length, together with the second constituent in such a way that the resulting mean scattering length is zero. According to Eq. (4-25), neutron diffraction with such an alloy yields directly the partial structure factor SCC(Q) (see the review by Chieux and Ruppersberg, 1980, and references there; and Ruppersberg and Reiter, 1982). Figure 4-45 a shows the ^cc (2) functions for liquid Li 8 0 Pb 2 0 at two temperatures, and for Li 72 Ag 28 and Li 70 Mg 30 . The chemical ordering in the Pb containing melt is much stronger than in the other two melts: The main peak of ScciQ) i s higher and sharper, implying stronger and more extended chemical fluctuations, and Scc(0) is lower, corresponding to a larger excess stability [Eqs. (4-31, 4-33)]. This is reflected in the larger shortrange order parameter, which for Li-Pb at
722 °C is a1 = — 0.25, i.e., the maximum value given by — cPb/cLi, while for Li Ag it is al= -0.15, that is about 40% of the maximum possible value. [Figures 4-45 b, c show calculated SCC(Q) functions, which will be discussed later on.] Figure 4-46 displays the Fourier transforms of three S cc functions from Fig. 4-45 a, plotted as 4 7i R2 QCC (R). The strong negative peaks at # = 0.295 nm for Li 80 Pb 20 and at R = 0.22 nm for Li 72 Ag 28 reflect the pref-
Li80Pb20
722 °C
Li 80 Pb 20
952 °C
Li80Pb20
722 °C
MSA model HNC model
30 Q (nm-1)
Figure 4-45. Liquid Li-m alloys (m = Pb, Ag, Mg): Partial B-T structure factors SCC(Q). (a) Experimental (Chieux and Ruppersberg, 1980; Ruppersberg and Reiter, 1982); (b) (Hoshino, 1984) and (c) (Copstake et al, 1984) comparison of the experimental curve of Li 80 Pb 20 at 722 °C with theoretical models.
272
40-
4 Structure of Amorphous and Molten Alloys
Li 8 0 Pb 2 0
722 °C
Li 80 Pb 20
952 °C
E
B 20-
-20-
-40
Figure 4-46. Liquid Li 80 Pb 20 and Li 72 Ag 28 : Partial B T correlation functions 4 n R2 QCC (R), calculated from the SCC(Q) in Fig. 4-45 a.
erence for unlike nearest neighbors. The bond-length # = 0.295 nm for Li-Pb is distinctly smaller than the mean value R = 0.32 nm of the atomic distances in pure Pb (R — 0.34 nm) and pure Li (R = 0.30 nm). This contraction was explained by a charge transfer from the Li atoms to the Pb atoms which, based on the classical valences, suggests an electronic distribution according to the stoichiometric composition (Li 1+ ) 4 Pb 4 ~. Thereby the bonding becomes partially salt-like. With increasing temperature the amplitudes of the oscillations of S cc (Q) and Gcc (R) decrease and the peaks in SCC(Q) become broader. This decrease in the CSRO means that the liquid becomes more metallic. A decrease in the chemical ordering effects with increasing temperature is generally observed in compound-forming alloys. Recently those liquid alloys which can be transformed into metallic glasses by rapid quenching gained special interest and the question was whether the shortrange order responsible for the stability of the glassy state already exists in the molten state. Fig. 4-47 shows the total X-ray structure factors of molten Mg-Zn alloys just above their melting points (Biihler et al., 1987). The occurrence of a prepeak around
Q = 15 nm 1 shows the existence of a short-range order effect with hetero-coordination. From the height of the prepeak in the total function, taking into account the weighting factor of SCC(Q) in Eq. (4-25), a relative measure of the amplitude of the peak height of SCC(Q) was estimated for each concentration and as a result the short-range order was found to be strongest in the Mg 70 Zn 30 alloy, which is a eutectic composition of the Mg-Zn system. The intermetallic compound with the highest melting point in this system is MgZn 2 . This shows that Mg-Zn melts do not exhibit strongest ordering at the composition of the most stable solid phase. Around the eutectic composition the melts can be quenched into a glass. Figure 4-48 shows in solid lines the X-ray structure factors of liquid and amorphous Mg 70 Zn 30 (Biihler et al., 1987). The height of the prepeak as a
o
20
40
60 8C Q (nm 1 )
100
120
Figure 4-47. Liquid Mg-Zn alloys: Total F - Z structure factors at temperatures about 40 K above the liquidus line, measured by X-ray diffraction (Biihler et al. 1987).
273
4.5 Structure of Molten Metallic Alloys
10
20
30 40 Q (nm 1 )
50
60
70
Figure 4-48. Liquid and amorphous Mg 70 Zn 30 and Cu 66 Ti 34 : Total F - Z structure factors. (- ) Mg 70 Zn 30 , X-ray diffraction (Biihler et al., 1987); ( ) Cu 66 Ti 34 , neutron diffraction (liquid: Fenglai et al., 1986; amorphous: Sakata et al., 1981).
relative measure of the strength of the CSRO gives evidence for an enhancement of the ordering during the quench. From the widths of both the prepeak and the main peak it follows for the Mg-Zn system that the correlation lengths of topological (/?t = 1.4nm) and chemical order (,RC = 0.9 nm) are the same in the liquid and the glassy state. Sakata et al. (1981) and Fenglai et al. (1986) investigated liquid Cu-Ti alloys and compared them with results obtained from the amorphous state (Sakata et al., 1982) which is formed on rapid quenching. Due to the negative scattering length of Ti, the contribution of the chemical ordering function SCC(Q) appears strongly in neutron diffraction patterns. The prepeak occurring in S(Q) of liquid Cu 66 Ti 34 (Fig. 4-48) indicates a CSRO effect in the liquid state, which is enhanced in the amorphous state. Furthermore, the authors found the same trend in the variation of the CSRO with composition for the amorphous state and the molten state. As in the Mg-Zn system, the strongest CSRO does not coincide with the composition of a particularly stable crystalline phase.
A specific structural feature of glassforming melts (around 70at.%Mg in Fig. 4-47) is the indication of splitting of the second maximum of S (Q) into two subpeaks (cf. Fig. 4-48). This feature, more distinctly developed in the glass, suggests that glass-forming melts, besides their chemical order, are characterized also by a higher degree of topological order compared with e.g., simple hard-sphere liquids. This behavior of a more detailed topological structure is also observed with glass-forming transition metal-metalloid melts, as shown in Fig. 4-49 for the correlation function G(R) of Fe-B alloys, by the double peak structure of the second maximum (NoldetaL, 1983). Concerning the nature of the concentration fluctuations in compound-forming liquid metallic alloys, there is still a lack of information, due to the limited number of systems where all the partial correlation functions could be determined. Mainly two different views of the nature of the CSRO in these alloys have been considered: A frequently adopted picture is based on the existence of chemical complexes, often called associates, with well-defined stoichiometry AliBy, embedded in a matrix where the A atoms and the B atoms are statistically distributed. This point of view
10-
i
c:
A
amorphous
+4
§, 5-
£
-
\J
V
liquid
00
0.5
1.0
1.5
R (nm)
Figure 4-49. Liquid Fe 83 B 17 and amorphous Fe 80 B 20 : Total F - Z pair correlation functions (Nold et al., 1983).
274
4 Structure of Amorphous and Molten Alloys
was stimulated by the observation that in most cases the degree of CSRO is highest at the composition of intermetallic phases in the system. The liquid system is described as a ternary alloy characterized by the chemical equilibrium: fi A 4- v B <-> A^ Bv The degree of CSRO is determined by the number of complexes in the liquid. Within the complexes it is maximum and in the statistical matrix it is zero by definition. Accordingly, variation of the short-range order parameter with the composition, as well as with the temperature, is attributed to the number of complexes. Theoretical models have been developed, using the amount of the complexes in the system as one of the adjustable parameters, to fit experimental structural quantities and thermodynamic functions (see e.g., Steeb and Buhner, 1970; Predel and Oehme, 1974 and 1976; Bhatia and Ratti, 1975; Boos etal., 1977; Sommer, 1989; Bhatia and Singh, 1982). Hoshino (1984) applied a molecular model for the CSRO in liquid Li 4 Pb. The alloy was considered as a ternary system consisting of Li and Pb atoms and Li 4 Pb molecules. The total neutron S (Q) was calculated using the PercusYevick hard-sphere model for a mixture of three different hard spheres. The concentration of Li 4 Pb molecules was taken as a free parameter. In the case of an Li 4 Pb content of 0.7 the result agreed quite well with the experimental curve of Ruppersberg and Reiter (1982) (Fig. 4-45 b). From the viewpoint of chemical complexes in compound-forming alloys the value of any experimentally observed quantity, such as the coordination number or the density, is explained as the average of this quantity in the complex and in the remaining disordered matrix. The experimental values often deviate very strongly
from the values expected for a statistical alloy. The value of this quantity for the complex itself has to deviate even more from randomness than the measured average value, and then a high degree of inhomogeneity would be expected in the alloy. It has often been noted that the simultaneous existence of two phases in dynamical equilibrium, having very different structures, should also be observable by smallangle scattering. This was never possible however for these compound-forming metallic alloys. An alternative point of view is related to solid solutions with a certain degree of short-range order, i.e., a certain degree of hetero-coordination. From this viewpoint the molten alloy is in principle homogeneous, but statistical concentration fluctuations exist with preference for unlike atomic pairs. The degree of this preference is characterized by the negative value of the short-range order parameter a of the n-ih coordination sphere, where n = 1,3,5 etc., which gradually approaches zero with increasing distance from any chosen reference atom. The spatial extension of the chemical ordering, i.e., the correlation length Rc, is given by the range where GCC(R) shows oscillations. An estimation of Rc from the width of the main peak of S cc (6) (Eq. (4-12)) yields i ? c % l n m for liquid Mg 30 Bi 70 . Note that this range involves more than a hundred atoms. A snapshot of such a system would show statistical fluctuations from atomic site to atomic site with a preference for hetero-coordination, but no well-defined chemical complexes. Copstake etal. (1984) calculated the Bhatia-Thornton partial structure factors of liquid Li 4 Pb from pair potential functions. They adopted two theoretical relations provided by liquid state theory, namely the mean spherical approximation
4.5 Structure of Molten Metallic Alloys
(MSA) and the hypernetted chain approximation (HNC). Based on the idea of a charge transfer from Li to Pb, the interionic forces were described as screened Coulombic interactions, being attractive between Li and Pb. The short-range forces were taken as hard-sphere interaction in the MSA model, and as a softer repulsion in the HNC model. The effective charge, used as a parameter to fit the experimental SCC(Q) function, corresponded to the ions Li 0>5+ and Pb 2 ". The decrease in the CSRO with temperature was explained by a decreasing degree of charge transfer. Comparison of the calculated S cc (Q) functions with the experimental one in Fig. 4-45 c shows that the hard-core MSA model reproduces the range of the main peak of SCC(Q) reasonably well, whereas the soft-core HNC model accounts better for the flat region between 25 and 35 nm" 1 . Of course, in reality all the intermediate possibilities between both views of the short-range order in compound-forming alloys may exist. This will strongly depend on the type of bonding, which may be quite different in real systems, ranging in the triangle of metallic, ionic and covalent bonding. In those cases where a charge transfer causes mainly ionic bonding a non-directional, rather long-ranged ordering is to be expected, thus tending more to the second view. On the other hand, a covalent contribution to the bonding would be correlated with spatially restricted structures, thus defining a specific associate. An example of the case of two types of bonding occurring at the same time is the concept of Zintl ions forming structural entities in equiatomic liquid alkali group IVB alloys, such as K - P b and Na-Sn (Geertsma et al., 1984; Van der Lugt and Geertsma, 1984). It was proposed that the liquid consists of poly-anion clusters, such
275
as (Pb 4 ) 4 tetrahedra, formed by covalent bonding, which are separated by alkali ions. This concept is based on the occurrence of such units in the corresponding crystalline alloys and on the fact that they are iso-electronic with tetrahedral molecules formed by the group VB elements P and As in the gas phase. Neutron diffraction curves of equiatomic liquid alkali Pb (Reijers et al., 1989 a, b) and alkali Sn (Reijers et al., 1990 a) alloys have been simulated using calculated models where Pb or Sn tetrahedra, surrounded by the alkali ions, were presupposed, and good agreement has been stated. In a molecular dynamics simulation study of liquid alkali Pb alloys (Pb4)4~ ions have been incorporated, and the distribution of the alkali ions around the tetrahedra has been analyzed (Reijers et al., 1990 b). 4.5.2.2 Segregation Alloys
In segregation alloys there is preferred coordination between like atoms compared with a statistical distribution. The value of the Warren-Cowley short-range order parameter a1 is positive in such alloys and their enthalpy of mixing is also positive. The liquidus lines in the phase diagrams often show an inflection point or even a miscibility gap in the liquid state. The correlated concentration fluctuations cause a rise of the partial BhatiaThornton structure factor SCC(Q) towards Q = 0. Provided the weighting factor of S cc (Q) in the total structure factor S (Q) is large enough [Eq. (4-25)], a small-angle scattering effect is observed with segregation alloys. In the wide angle regime a splitting of the main peak of S (Q), more or less pronounced, is typical for these alloys, where the atomic diameters of both constituents are sufficiently different. This is due to the
276
4 Structure of Amorphous and Molten Alloys
fact that the contribution of the correlation function between unlike atoms to the total function is small, which causes the contributions of the like atomic pairs to appear more distinctly separated. As an example, Fig. 4-50 shows the total A-L neutron structure factors of liquid Cu-Pb alloys (Lamparter and Steeb, 1980). The Cu-Pb system has a miscibility gap in the liquid state with a critical point near 65 at.% Cu. There is a tendency to preserve the structural properties of the pure constituents in the alloys: The main peak occurs at the same position as the predominant component has its maximum in the pure liquid state, and a shoulder occurs at the position where the minor component has its maximum. The dashed curves in Fig. 4-50 show a simple, but quite illustrative model for this behavior. It is the sum of three contributions: the two structure factors of pure Pb and Cu, weighted by the short-range order parameter a1, and a third structure factor calculated using
at % Pb
7 (°C) 603
960
+5
1050
+4
1050
+3
1170
+2
1025
+1
the Percus-Yevick hard-sphere model for a statistical Cu-Pb alloy, weighted by (1 — a1). Best fits to the experimental curves were obtained using the values of a1 listed in Table 4-6, where the larger value is obtained for the alloy with the critical composition Cu 6 5 Pb 3 5 . The most significant structural feature of segregation alloys is the small-angle scattering effect. Figure 4-51 shows the total A-L X-ray structure factor of Cd 5O Ga 5O . This alloy has the critical composition of the miscibility gap in the liquid state. Splitting of the main peak is missing in this case, but a strong small-angle scattering is observed (Hermann et al., 1980 a). The concentration, as well as the temperature dependence of the small-angle scattering of the Cd-Ga system, has been investigated in detail by Hermann et al. (1980 b). Using Eq. (4-34) within the range of small Q and known values of the compressibility as well as of the dilatation factor, S^c (Q) could be obtained from the measured structure factor S (<2). It was shown that the small-angle scattering [after subtraction of the compressibility term in Eq. (4-34)] follows the Ornstein-Zernike theory of critical scattering, given by Eqs. (4-35, 4-36). From the Ornstein-Zernike plot (Fig. 4-52 for
1120
40 60 Q (nrrr1)
80
100
Figure 4-50. Liquid Cu-Pb alloys: Total A-L structure factors (Lamparter and Steeb, 1980). ( ) from neutron diffraction, ( ) segregation model (see text).
Figure 4-51. Liquid Cd5OGa5O: Total A-L structure factor (T = 296 °C) displaying a small angle scattering effect (Hermann et al., 1980 a).
277
4.5 Structure of Molten Metallic Alloys
Table 4-6. Liquid segregation alloys: Structural parameters. T = temperature. RG = Guinier radius. Rc = correlation length, a1 = Warren-Cowley short-range order parameter. v,y = critical exponents, s = reduced temperature.
Alloy Ref., Method
T[°C]
Al loo _ x Sn x (x = 3-30) Hezel and Steeb (1970), SAXS
near Tm
0.16-0.4
A hoo-xlnx (x = 4.7-90) Hohler and Steeb (1975), SAXS
near Tm
0.33-0.4 0.18-0.27
Ga-Pb Wignall and Engelstaff (1968), SANS
RG [nm] Rc [nm]
800-1080
0.21-0.22 0.14-0.16
Li 61 Na 39 Ruppersberg and Knoll (1977), SANS
317 452
Kc = 1.7 £ c = 0.37
Li
307-316
0.05-0.21
a1 = 0.5 a1 = 0.3
296-440
near Tm (x = 35,65) Lamparter and Steeb (1980), N
Cd5OGa5O), i.e., 1/7(0 versus Q2, the two parameters ScC(0) and RC are obtained from the axis intercept and the slope of the straight lines for each temperature [see Eq. (4-35)]. The results for the Cd-Ga system are shown in Fig. 4-53 a and b. ScC(0) in Fig. 4-53 a represents the concentration fluctuations in the thermodynamic limit and is related to the Darken excess stability function according to Eqs. (4-31, 4-33). It attains values much larger than for the case
0.57 1.1 0.655 1.296
2.0-50.7
Wu and Brumberger (1975), SAXS Cd1Oo_xGa^ (x = 20-80) Hermann et al. (1980b), SAXS
V
y
0.6 -
Biioo-xCu, (x = 10-90) Zaiss et al. (1976), SANS
66Na34
oc1
£>8-10~3 0.51 1.06 £<8-10~3 0.615 1.185
for Rc see Fig. 4-52 b 0.4-4
x=35 of = 0.58 x= 0.65 <xl = 0.46
of an ideal mixture, where S^c = cA cB. The temperature dependence of the scattering is described by power laws of the reduced temperature according to Eqs. (4-37, 4-38). Classical theories of critical phenomena as the mean field theory predict critical exponents v = 0.5 and y = 1.0, while according to a lattice gas model v = 0.63 and y = 1.24. At temperatures not too close to the critical temperature the critical exponents for liquid Cd-Ga alloys are close to the classical mean-field values (Table 4-6). The
278
4 Structure of Amorphous and Molten Alloys
3, O
50 Ga concentration (at.%)
14
Figure 4-52. Liquid Cd5OGa5O: Ornstein-Zernike plot of the small angle scattering intensity (Hermann et al, 1980 b). (eu: electron units.)
42U-
(b)
/*^v
400-
Rc (nm) o 0.4 °\
/ 380-
correlation length Rc, as presented in Fig. 4-53 b, is a measure of the range where concentration fluctuations are spatially correlated. At the critical composition the correlation length is largest, extending up to 4nm. The critical behavior of the segregating Li-Na system was investigated by Ruppersberg and Knoll (1977). Using the isotope 7Li the S cc (<2) function of the "zero alloy", Li 61 Na 39 « / > = 0) very close to the critical composition with 64 at. % Li, was measured by neutron diffraction in the small-angle and in the wide-angle scattering regime (Fig. 4-54). Beyond the smallangle regime the oscillations are very small, but nevertheless present. The corresponding Fourier transform 4nR2 QCC (R) in Fig. 4-55 is positive and approaches zero at quite large distances, beyond 10 nm at 317 °C. The short-range order parameter
/
2> 340-
1
320" K
/
^ 0.5 \
/ O 360-
100
/
• 0.6
*1 °8
^\ \
/ h/^ \ \ A
\ \
\
• °
\
n 1.2
| 300280-
X ^ x 4.0
260P40-
\ 20
40 60 80 Ga concentration (at.%)
100
Figure 4-53. Liquid Cd-Ga alloys: (a) 5cC(0) from extrapolation of the curves in Fig. 4-52 to Q = 0. (b) Phase diagram and lines of constant correlation length Rc.
a1, as calculated according to Eq. (4-29), is positive which is also the case for the CSRO parameters at larger atomic distances. Note, that in the case of segregation alloys the extension of the correlations between concentration fluctuations may exceed by
4.5 Structure of Molten Metallic Alloys
40
20 Q
80
5 10 R (nm)
279
15
(nirr 1 )
Figure 4-54. Liquid Li 61 Na 39 : Partial B-T structure factor SCC(Q) (Ruppersberg and Knoll, 1977). Scale of ordinate is for the full line. The dotted curve is enhanced by a factor of 10.
Figure 4-55. Liquid Li 61 Na 39 : Partial B-T correlation function 4 n R 2 QCc(RY ( ) Experimental, ( ) Ornstein-Zernike function. Parts (a) and (b) show the same functions on different R-scales.
far the range of the topological ordering. This is in contrast to the compound-forming alloys. At large distances (Fig. 4-55 b) the function gcc is determined practically only by the small-angle scattering part, and therefore follows the smooth theoretical Fourier transform of the OrnsteinZernike SQQ(Q), as given in Eq. (4-36). At smaller distances, however, the atomicscale structure, connected mainly with the wide-angle scattering range, causes oscillations around the O-Z function (Fig. 4-55 a). Estimated values of the short-range order parameter for the first coordination shell a1 (Table 4-6) show a decrease at higher temperatures. In early studies of metallic segregation alloys the small-angle scattering behavior has been interpreted in terms of segregated clusters, formed by one of the constituents which exist isolated in a statistical matrix. This view may be used for the case of very small correlation lengths covering only the nearest neighbor distances, i.e., at temperatures well above any critical temperature. Guinier radii RG have been calculated from the scattering function of these alloys (Table 4-6). In fact, with molten Bi-Cu alloys, investigated by Zaiss et al. (1976)
(Fig. 4-56), the scattering could be better described using the Guinier law, and a Guinier radius of the order of 0.2 nm, almost independent of the temperature, was obtained (Table 4-6). Treating the OrnsteinZernike scattering law with the Guinier approximation yields the theoretical relationship RG = (^/3/2)Rc. However, at large
Figure 4-56. Liquid Bi5OCu5O: Total A-L structure factor. (•) Neutron diffraction, ( ) Guinier fit, ( ) Ornstein-Zernike fit (Zaiss et al., 1976).
280
4 Structure of Amorphous and Molten Alloys
correlation lengths Rc the idea of isolated clusters no longer makes sense and is therefore not acceptable. 4.5.2.3 Metal-Nonmetal Transition
Certain alloys, where the constituents are metals in their pure liquid state, undergo a transition from the metallic state to a non-metallic state near certain stoichiometric compositions. Hereby drastic structural changes occur, and these changes cannot be described by a more or less pronounced deviation from the hypothetic case of a statistical distribution of the constituents. It is more appropriate to regard these alloys at specific compositions as completely new alloys. Note that the metallic compound-forming alloys described above showed already a more or less distinctive tendency towards nonmetallic behavior at certain compositions, and hence they are not completely different from the examples discussed in the present section. A complete change to a non-metallic state is due to a change from metallic to another type of bonding, either more ionic or more covalent. The resulting atomic structures may be rather complex and quite specific, including a variety of saltlike, molecular, and semiconducting liquids. In the following, two examples are described. In the system Au-Cs a transition from the pure metallic constituents to a molten salt Au5OCs5O occurs (Fig. 4-57). At 50 at.% the electrical conductivity crel drops from values larger than 103 Q" x cm" 1 , typical for metals, down to 3Q~ 1 cm~ 1 , which is typical for molten salts (Fig. 4-57 a). The large difference in the electronegativities of Cs (0.7) and Au (2.4) causes a complete charge transfer at the stoichiometric composition and the
I
3^
I 2^ O
0 60 4020(b)
0
1000-
800-
E
£400-
200-
20
40 60 80 Au concentration (at.%)
100
Figure 4-57. Liquid Cs-Au alloys: (a) (-•-) electrical conductivity
melt is composed of the ions Cs + and Au . This is accompanied by a large negative excess volume of up to 50% (Fig. 4-57 b; Martin et al, 1980d). The structure of Au-Cs melts was studied by Martin et al. (1980 a, b, c) using neutron diffraction (Figs. 4-58, 4-59). The prepeak at Q = 12 nm" 2 in S(Q) of the Au5OCs5O alloy reflects the ordering effect in this alloy. It
281
4.5 Structure of Molten Metallic Alloys
at.% Cs
T (°C)
100
600
o.u-
/ " \
h +5
100
80
7|7^^-^ 80
420
T/ 500
—
+2
+4 1.5
70
+2.5 •
2.5-
+3
\
70
^
+ 1.5
] A ^ —
-v
+1
10-
+2
0.5
\ J / I \ / I ^-y,
50
+0.5
V V
+1
0 -0.5 -1.0-1.5
V
0.2
0.4
0.6 R (nm)
0.8
1.0
1.2
Figure 4-58. Liquid Cs-Au alloys: Total F-Z structure factors. ( ) from neutron diffraction (Martin etal., 1980 b); dashed curve for Au: Percus-Yevick hard-sphere model; dashed curve for 50 at.% Cs: MSA model (Gopala and Satpathy, 1989).
Figure 4-59. Liquid Cs-Au alloys: Total F-Z pair correlation functions calculated from the experimental S (Q) in Fig. 4-58. The G(£)-function of molten Au is characterized by pronounced oscillations.
appears quite small due to the small weighting factor of the S cc (Q) function in this case, but the value of the corresponding amplitude of SCC(Q) estimated from this hump is SCC(Q = 12 nm" 1 ) = 4, which is of the same order as found in molten alkali chlorides. The main peak at 0.358 nm in the correlation function G (R) of this alloy occurs at a distance much smaller than the average of the diameters of metallic Cs and Au, and corresponds to the sum of the ionic radii of Cs + (0.167 nm) and Au" (0.190 nm). The mean spherical approximation theory (MSA) relates the structure functions S (Q) and G (R) to the pair potential functions of charged hard spheres and has frequently been applied to the description of molten salts (Waisman and Lebowitz, 1970 and 1972; Blum, 1975; Hiroike, 1977). Based
on the concept that molten AuCs is fully ionic, the partial structure factors of this liquid have been calculated from theory by Evans and Telo da Gama (1980) and by Gopala and Satpathy (1989). Comparison of the calculated total S (Q) with the experimental one shown in Fig. 4-58 proves the existence of a molten salt AuCs. The intermediate compositions between Cs and stoichiometric CsAu show a smallangle scattering effect and thus are segregating alloys. It is interesting to note that they cannot be classified into one of the groups in Sec. 4.5.2.1 and 4.5.2.2 because in this case the segregation involves pure Cs and the compound-forming alloy CsAu. The G(R) in the intermediate concentration range show a main maximum with two peaks at those positions where Cs and CsAu have their main maximum (Fig. 4-59).
282
4 Structure of Amorphous and Molten Alloys
Analysis of the small-angle scattering effect, including its temperature dependence, revealed that the strongest segregation tendency occurred at the equimolar composition Cs5O(AuCs)5O. The correlation length Rc of the concentration fluctuations is of the order of 0.5 nm (Fig. 4-57 a), i.e., the microscopic segregation is restricted to the range of only a few coordination shells. The temperature dependence of the scattering follows the classical mean field behavior, with the parameters v = 0.5 and y = 1.0. Liquid Cs-Sb alloys represent another example of a system showing a metal-nonmetal transition. The conductivity drops drastically within a very narrow concentration range around the stoichiometric composition Cs3Sb, but also shows a min-
0
20 Sb
40
60
80
100
concentration (at.%)
Figure 4-60. Liquid C s - S b alloys: (a) electrical conductivity <7el at 750 °C (Redslob et al., 1982), (b) phase diagram.
imum at CsSb (Fig. 4-60 a). This indicates complicated structural behavior in the molten alloys. Furthermore, liquid Sb in the unalloyed state already has a more complicated structure than simple metals. In contrast to the Au-Cs system, the phase diagram of Cs-Sb is complex (Fig. 4-60b). From the electronic properties it was concluded that liquid Cs-Sb alloys are not predominantly ionic in character, but are determined by an ionic-covalent mixed bonding (Redslob et al., 1982). Figures 4-61 and 4-62 show structural results from neutron diffraction (Lamparter et al., 1983). Within the intermediate composition range a small peak in front of the main maximum of S(Q) occurs at Q = 10 nm" 1 , which, however, in this case should not be referred to as a prepeak. The neutron scattering lengths of Cs (0.55 • 10" 1 2 cm) and Sb (0.56 • 10" 1 2 cm) are very close together, and hence the contribution of S cc (Q) to the total S{Q) is negligible [Eq. (4-25)]. Consequently, the peak has to be attributed to the SNN(g) function, i.e., it is caused by a special topological arrangement in the melts. It was concluded that structural complexes exist with an intermolecular distance of about 0.8 nm and, as illustrated by the sharpness of the peak, with a quite extended mutual distance correlation. As confirmation, there is a fairly pronounced maximum in G(R) at 0.8 nm for the concentrations 50, 65 and 75 at.% Cs (Fig. 4-62). In addition, the coordination number of 8 atoms at a distance 0.40 nm in liquid Cs 75 Sb 25 is the same as in the solid semiconductor Cs3Sb, suggesting quite similar short range order in both states. Approaching the composition Cs5OSb5O, in G(R) an additional distance with a rather sharp bond length develops at R = 0.284 nm with a corresponding coordination number Z = 2. This compares
4.6 Conclusions
283
tems would, of course, require determination of the complete set of partial structure factors. The examples given here were selected to illustrate that liquid alloys outside the range of metallic alloys may have rather complex atomic arrangements which renders their classification into specific groups difficult.
4.6 Conclusions 50
100 Q
(nnr 1 )
Figure 4-61. Liquid Cs-Sb alloys: Total F - Z structure factors from neutron diffraction (Lamparter et al., 1983).
0.5
1.0 R
(nm)
Figure 4-62. Liquid Cs-Sb alloys: Total F - Z pair correlation functions calculated from S(Q) in Fig. 4-61.
well with the covalent Sb-Sb bond length (0.285 nm) in spiral Sb chains occurring in the solid semiconductor CsSb. A more detailed characterization of the structure of such complicated liquid sys-
The structure of amorphous and liquid metals has been investigated extensively in the past, beginning in the 1930s. The most widely used were the diffraction methods, where substantial advances have occurred in the last twenty years by improvement of experimental techniques. More intense radiation sources have become available and more efficient diffractometers have been designed. In particular, neutron diffraction in combination with the isotopic substitution method yielded detailed information on the structure. The increasing bulk, as well as the reliability of experimental data, stimulated the development of theoretical models for the structure of non-crystalline metallic alloys. Much work has been devoted to the simulation of the structure by constructing computer models in accordance with the experimental atomic distribution functions. These models provide deeper insight into the structure because they give information on the three-dimensional arrangement of the atoms, in contrast to the experimental pair distribution functions which represent a one-dimensional description of the structure. Starting from dense random packing of hard spheres, the models became more and more sophisticated. The increasing power of computers has enabled rapid calculation of larger model clusters.
284
4 Structure of Amorphous and Molten Alloys
The field of research into amorphous metals, so-called metallic glasses, has developed very rapidly during the last two decades. Due to their special mechanical, electrical, and magnetic properties, metallic glasses are used in various technical devices. On the other hand, they drew attention to basic scientific studies in metal physics. In the meantime, for many binary amorphous alloys the structural parameters, i.e., the three partial pair distribution functions, the atomic distances, and the coordination numbers, are well established. A certain degree of chemical ordering, characterized by a preference for unlike nearest neighbors, is a common structural feature of metallic glasses. The chemical interaction between the constituents favors the formation of the metastable glassy state. Structural changes in this state during relaxation annealing below the crystallization temperature are usually extremely small, but they cause often substantial changes in the properties. Any similarity in the nearest neighbor distribution in an amorphous alloy and its crystalline counterpart, if found at all, is usually restricted to one of the constituents. For some metallic glasses phase separation into two amorphous phases has been found on a length scale of some ten angstroms. Most of the glasses also show some structural inhomogeneity on very extended length scales of the order of some thousand angstroms. However, further research of this feature remains to be done. In the future, further progress in the elucidation of the structure of amorphous metals is expected from the enhanced combination of complementary experimental techniques applied to the same alloy, such as X-ray absorption spectroscopy, Mossbauer spectroscopy, nuclear magnetic resonance, and electron microscopy. In particular, investigation of the X-ray absorption near-edge structure will
play an important role because it is one of the few techniques which is sensitive to higher-order distribution functions. Certainly, the range of amorphous alloys to be studied in future will expand continuously from the metallic alloys to the non-metallic alloys. Further important subjects will be the study of hydrogen in amorphous alloys and the dependence of the structure on the different production methods of amorphous alloys. The molten metallic alloys have lagged behind during recent years, since the novel amorphous metals took the attention of the relevant investigators in many parts of the world. However, this field of research will remain important. Bearing in mind the multiplicity of ordering effects, observed up to now in molten alloys, the bulk of the experimental information is still relatively restricted. The structure of most of the molten alloys deviates more or less from a random distribution of the constituents. It is determined either by a preference for unlike nearest neighbors, i.e., a compoundforming tendency, or by a preference for like neighbors, i.e., a segregation tendency. Due to the different possible types of chemical interaction (metallic, ionic, and covalent bonding) a variety of liquid structures exist. Liquid alloys range continuously from metallic to non-metallic alloys. In several alloy systems a change in the type of bonding was found involving a metalnonmetal transition within a certain concentration range. Basic scientific studies have been devoted to the calculation of atomic distribution functions from the principles of the theories of the liquid state as well as from the simulation of the structure by computer models. From such calculations information about interatomic potentials, being the fundamental input to theoretical models, could be derived. In future, further detailed experimental investi-
4.7 References
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Wildermuth, A., Lamparter, P., Steeb, S. (1985), Z. Naturforsch. 40 a, 191. Wong, J. (1981), in: Glassy Metals I, Topics in Physics, Vol. 46: Guntherodt, H.-J., Beck, H. (Eds.). Berlin: Springer, pp. 45-77. Wong, J., Liebermann, H. H. (1984), Phys. Rev. B29, 651. Wright, J. G. (1977), Inst. Phys. Conf. Ser. No. 30, Chap. 2, Parti, pp. 251-267. Wu, E. S., Brumberger, H. (1975), Phys. Letters 53 A, 475. Yu, Z., Rongheng, D., Gongxian, H., Zhongyi, H. (1988), J. Mater. Sci. Lett. 7, 555. Zaiss, W, Bauer, G. S., Steeb, S. (1976), Phys. Chem. Liq. 6, 21.
General Reading Bhatia, A. B., Thornton, D. E. (1970), Phys. Rev. B2, 3004. Chieux, P., Ruppersberg, H. (1980), /. Phys. (Paris) C8, 145. Faber, T. E. (1972), Introduction to the Theory of Liquid Metals. Cambridge: Cambridge University Press. Faber, T. E., Ziman, J. M. (1965), Philos. Mag. 11, 153. Gaskell, P. H. (1983), in: Glassy Metals II, Topics in Applied Physics, Vol. 53: Beck, H., Guntherodt, H.-J. (Eds.). Berlin: Springer, pp. 5-49. Gurman, S. J. (1981), in: Extended X-Ray Absorption Fine Structure: Joyner, R. W. (Ed.). New York: Plenum, Chap. 4. Huijben, M. J. (1978), Thesis, University of Groningen. Janot, C. (1983), in: Les Amorphes Metalliques. Les Ulis: Les Editions de Physique, pp. 81-167. Lamparter, P., Steeb, S. (1988), in: Proc. 4th Int. Conf on the Structure of Non-Crystalline Materials, NCM 4 (Oxnard, California, 1988), Wagner, C. N. I, Wright, A. C. (Eds.). /. Non-Cryst. Solids 106, 137. Sadoc, J. R, Wagner, C. N. J. (1983), in: Glassy Metals II, Topics in Applied Physics, Vol. 53: Beck, H., Guntherodt, H.-J. (Eds.). Berlin: Springer, pp. 51-92. Sauerwald, R (1950), Z. Metallkd. 41, 97, 214. Steeb, S. (1968), Fortschr. chem. Forsch. 10, 473. Wagner, C. N. J. (1978), J. Non-Cryst. Solids 31, 1. Waseda, Y. (1980), The Structure of Non-Crystalline Materials. New York: McGraw-Hill.
5 Lattice Vibrations Herbert R. Schober x and Winfried Petry 2 1 2
Institut fur Festkorperforschung, Forschungszentrum Julich, Jiilich, Germany Fakultat fur Physik, Technische Universitat Miinchen, Garching, Federal Republic of Germany
List of 5.1 5.2 5.2.1 5.2.2 5.2.3 5.2.3.1 5.2.3.2 5.2.3.3 5.2.3.4 5.2.3.5 5.2.3.6 5.2.3.7 5.2.4 5.2.4.1 5.2.4.2 5.2.4.3 5.2.4.4 5.3 5.3.1 5.3.2 5.3.3 5.4 5.4.1 5.4.2 5.4.3 5.4.4 5.4.5 5.4.6 5.5 5.5.1 5.5.2 5.5.3 5.5.4 5.5.5
Symbols and Abbreviations Introduction Elements of Lattice Dynamics Equations of Motion Harmonic Approximation Phenomenological Models of Lattice Dynamics General Born-von Karman Model (Tensor Force Constant Model) Axially Symmetric Born-von Karman Model Valence Force Models Nearly Free Electron Models Rigid Ion Model Shell Model Bond Charge Model First Principle Methods Frozen Phonon Method Force Constant Method Linear Response Method Molecular Dynamics Simulations Properties of the Ideal Harmonic Crystal Frequency Spectrum and Related Properties Green's Functions and Correlation Functions Long Wavelength Limit Anharmonicity Quasi-Harmonic Approximation Anharmonic Perturbation Theory Self-Consistent Phonon Theory Computer Simulation Electron-Phonon Interaction Phonon Lifetimes Imperfect Crystals and Disordered Solids Single Defect Dynamics Finite Defect Concentration Extended Defects Amorphous Materials and Glasses Surfaces
Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.
291 294 295 295 297 304 304 305 306 306 307 308 308 309 309 310 310 311 311 311 316 318 320 320 324 326 327 327 328 329 329 333 336 337 340
290
5.6 5.6.1 5.6.1.1 5.6.1.2 5.6.1.3 5.6.1.4 5.6.1.5 5.6.2 5.6.3 5.7 5.8
5 Lattice Vibrations
Experimental Methods Inelastic Neutron Scattering Inelastic Scattering Cross-Section Selection Rules and Different Brillouin Zones The Triple Axis Spectrometer (TAS) The Incoherent Inelastic Scattering Cross-Section Time-of-Flight Spectroscopy X-Ray Backscattering Inelastic He Scattering Outlook References
341 343 344 346 347 348 349 350 351 352 353
List of Symbols and Abbreviations
List of Symbols and Abbreviations ai,a2,a3 a
J
a b1,b2,b3 b~j~,bj Cj(q/q) c, c C
al3,yd'CaP
Cafi yS'CaP
Cy,Cp D ej e (q,j) E frJt F F
G g(w) 9(co2) G(co) h h
Hn(x) k kB K L m
M,M n
i
n{co) Nj
N P q Q r R s s
basis vectors of the direct lattice vibrational amplitude of mode j lattice constant coherent and incoherent scattering length, respectively basis vectors of reciprocal lattice creation and annihilation operator of mode j sound velocity of mode j in direction q/q longitudinal, transverse sound velocity Huang tensor elastic constant constant volume and constant pressure specific heat dynamical matrix eigenvector of mode j polarization vector of mode j energy/atom radial, transversal force constant free energy/atom force structure factor vibrational spectrum spectrum of squared frequencies Green's function Planck's constant/2 n difference of cell index vectors (m — n) Hamilton operator Hermite polynomial wave vector Boltzmann constant Kanzaki force combined matrix — Mco2 +
cell index vector atomic mass, atomic mass matrix mode occupation number Bose factor occupation number operator total number of atoms in lattice dipole force tensor wave vector scattering vector position vector instantaneous atomic position number of atoms in unit cell static displacement
291
292
5 Lattice Vibrations
S t T U u v aj3 Vc X Z
entropy/atom time temperature potential energy atomic displacement strain tensor unit cell volume atomic equilibrium position partition function
/? y F 8Uj 5 (co) s0 f 00 x^ vA G 1 T T <£ O W ij/j co
l/(kBT), inverse temperature Griineisen constant damping of mode or phonon Kronecker symbol 5-function electrical field constant reduced wave vector Debye temperature displacement of sublattice \i with respect to unit cell origin Debye cutoff frequency cross section self-energy reciprocal lattice vector phonon lifetime many-body potential coupling constant matrix total vibrational wave function wave function of n o r m a l m o d e frequency
ATA b.c.c. BS B.v.K. CPA EELS f.c.c. FIR h.c.p. HFR ILL INS IR LA LO
average T-matrix approximation body-centered cubic Brillouin scattering Born-von Karman coherent potential approximation electron energy loss spectroscopy face-centered cubic far-infrared (spectroscopy) hexagonal close-packed high flux reactor Institute Laue-Langevin inelastic neutron scattering infrared (spectroscopy) longitudinal acoustic (phonons) longitudinal optical (phonons)
List of Symbols and Abbreviations
PG RS STA TA TAS TO TOF VCA
293
pyrolytic graphite Raman spectroscopy single T-matrix approximation transverse acoustic (phonons) triple axis spectrometer transverse optical (phonons) time-of-flight spectrometer virtual-crystal approximation
Indices D def E h H j / L m,n,... a, /?, y /I, }i, v 0
Debye defect Einstein difference of cell index vectors (n — m) harmonic mode number atom number combined atomic and Cartesian coordinate index cell index vector Cartesian coordinates sublattice ideal harmonic crystal
conversion factors between phonon frequency units 13 1 1 v = l T H z o co = 0.6283 x 10 rads" ohco = 4.136 rneVo 33.36 cm- <>T = 48.00 K
294
5 Lattice Vibrations
5.1 Introduction The atoms in a solid will oscillate around their equilibrium configuration at any temperature. At absolute zero temperature atoms oscillate due to their zero point motion. Oscillation of the atoms is the major heat bath in solids and is responsible for important properties such as heat capacity, thermal conductivity, lattice expansion, displacive phase transitions etc. The theory of these lattice vibrations is often referred to as lattice dynamics. In its present form it is now eighty years old, originating in papers by Born and von Karman in 1912. Lattice dynamics is closely related to the thermodynamics of solids and elasticity theory which had both been studied much before that date. The present chapter is intended to give a short outline of the present status of lattice dynamics. Since this is a well established field there are a number of textbooks available where one can find more detailed accounts (see list of references). Vibrational properties of specific classes of substances are also often discussed in books and reviews on these substances. We hope that this chapter can serve as a source of first information and a guide to more detailed references. To achieve this goal, rather than quoting the original publications, we have tried to give newer and more detailed references. We also had to limit ourselves to only a few examples, thus only typical examples have been chosen. Lattice dynamics is infamous for the multitude of indices: Cartesian coordinates, atom numbers, unit cell numbers, sublattice numbers etc. In different areas different notations are useful. An ideal crystal is best described in terms of cell and sublattice numbers, whereas such a numbering scheme makes no sense for a glass. Never-
theless, many basic formulae are valid for both. In order not to have to repeat the equations we have adopted, where appropriate, the more general scheme of indexing. In order to avoid confusion different letters are used for the different indices: a,/?,...: Cartesian coordinates; /: atom number; m, n,...: cell index vector; \i, v,...: sublattice number and L: combined atomic and Cartesian index. As far as possible we have indicated summations explicitly and have resorted to a matrix notation only where otherwise this would have complicated the formulae beyond all reason. In the following section first the basic equations of lattice dynamics are introduced, starting from Newton's equation of motion of the atoms. These are used to derive the harmonic approximation, and the notion of phonons and how these can be derived form phenomenological or first principle models for ideal lattices. The third section gives a survey of the poperties of the ideal harmonic crystal. In harmonic approximation the contributions of the single phonons are often additive and can therefore be expressed by a spectrum function. The scattering of particles by the lattice, on the other hand, depends on the "structures" of the phonons and can be desribed by correlation and Green's functions. These are also convenient tools in the study of anharmonic effects and defect lattices discussed in later sections. For long wavelength acoustic phonons, lattice dynamics turns into elasticity theory. The section on anharmonic properties and other effects limiting the phonon lifetimes had to be restricted to just a few remarks on basic concepts. A full treatment involves rather complicated formulae. In practice the quasi-harmonic approximation is mostly encountered, which only accounts for the shift of phonon energies but not for the finite lifetimes. This lowest or-
5.2 Elements of Lattice Dynamics
der anharmonic approximation is treated in more detail. Real solids are normally not ideal crystals, and a section is devoted to the concepts introduced by imperfections. Emphasis has been given to point imperfections, both single defect properties and their effect on the host lattices. Other areas such as the dynamics of surfaces are beyond the scope of this article and therefore only a small taste of this expanding field can be given. Finally, a section is devoted to the experimental methods available for the investigation of phonons. The most important method there is, of course, neutron scattering.
5.2 Elements of Lattice Dynamics 5.2.1 Equations of Motion
Lattice dynamics is conventionally based on the adiabatic or Born-Oppenheimer approximation which eliminates the loosely bound electrons from the equations of motion of the nuclei. The basic idea behind this approximation is that typically the velocities of the electrons are much higher than those of the nuclei ( « 106 ms" 1 and « 103 ms" 1 respectively). One can therefore assume that, on the timescale of the motions of the nuclei, the electronic configuration adjusts itself instantaneously to the momentaneous configuration of the nuclei. The electrons always remain in their respective ground-state configurations. The energy of the electronic ground state changes with the position of the nuclei and thus acts, in addition to the direct interaction, as an effective potential between the nuclei. We will not give a detailed derivation and justification of this approximation, but only a few of the basic equations. A more detailed discussion can be found in
295
Born and Huang (1954) and in many textbooks. There is no general criterion for the applicability of the adiabatic approximation, i.e., for the neglect of the interaction between lattice vibrations and excitations of the electronic system (electron-phonon coupling). One would expect the approximation to be valid in insulators, where the gap in the electronic excitation spectrum is of order 1 eV, much larger than the 10 meV typical for a vibrational excitation. It has, however, been shown that the approximation is quite accurate as regards the vibrational energies, even for metals where there is no gap in the electronic spectrum (Brovman and Kagan, 1974). The lifetimes of the vibrational states, on the other hand, are strongly influenced by the electron-phonon coupling which can at low temperatures be the dominant effect. Electron-phonon coupling effects are important for many phenomena such as superconductivity and are generally discussed in those contexts. After elimination of the electronic degrees of freedom the dynamics of the atoms in the solid are determined by a general many-body potential $, depending on the instantaneous atomic positions. Obviously no exact solution of the resulting equation of motion is possible. We are here interested in vibrations of the atoms around their equilibrium positions. The amplitudes of these vibrations will in general be small compared to the interatomic distances. We expand, therefore, the potential energy,
(5-1)
where X denotes the equilibrium position and ul the displacement. The Taylor expan-
296
5 Lattice Vibrations
sion of
2
£ <\l2A\<\
Z
To avoid the large number of indices we will often use a matrix notation. In this notation Eq. (5-3) reads Mu=
(5-2)
Here the constant term
=
_
2 ..h
y
_
(5-3)
where Ml denotes the atomic mass of atom /. The coefficients $la\%,..., &la\';;!;n are called second, ..., nth order force constants or coupling constants. They are defined as derivatives of the many-body interatomic potential with respect to atomic displacements from the equilibrium position
(5-3 a)
-
The force constants have to obey a number of symmetry requirements. First, it follows immediately from their definition Eq. (5-4) that the force constants are invariant against permutations of the index pairs (lj9 <Xj). Second, the invariance of the potential energy # and its derivatives against translations (ul -• ul + t) of the solid as a whole requires (5-5) and additional conditions are imposed by rotational invariance. Third, in addition to these relations due to rigid displacements of the solid as a whole, a given form of the interaction potential can introduce additional relations between the elements of the force constants (Sec. 5.2.3). The conditions discussed so far are valid for any collection of atoms. For crystals additional restrictions are imposed by the structural symmetries which have been discussed in Chap. 1. In an infinite ideal crystal the equilibrium positions of the atoms can be given as X11 =m1a1+
m2 a2 + m3 a3
(5-6)
where m denotes the triple of integers mi9 the at are primitive translation vectors of d
5.2 Elements of Lattice Dynamics
venient to use, instead of the translation vectors a, a more symmetry adapted set. E.g., in an f.c.c. lattice the atomic sites are normally given by X™ = (a/2) ma where a is the lattice constant and the ma are restricted to integers with an even cross sum. Equation (5-6) can now be used to replace the arbitrary atom indices lt in the coupling parameters by the more systematic enumeration by ™. In these new indices the coupling parameters exhibit the invariance of the lattice against symmetry operations from its space group. These symmetry operations do not only leave the potential energy and its derivatives invariant but also the undistorted lattice. Under these symmetry operations, therefore, not only the numerical value of the potential energy but also the form of the expression for it, Eq. (5-2), is invariant. Most important is the invariance of the lattice against translation by a basis vector a{. This invariance implies that the coupling parameters of the ideal lattice do not depend on the absolute positions of the primitive cells but only on their distance vector. For the harmonic force constants one has On-m
(5-7)
where we have used the standard notation of one uppermost vector index h = n — m. The number of independent nonzero elements of the force constants is further reduced by symmetry axes, inversion symmetry and more complicated symmetry elements such as screw axes and glide planes. Under such a symmetry operation a lattice point ™ is transformed into an equivalent one ™. As an example let us consider the nearest neighbor coupling in a simple cubic lattice. The six equivalent positions are given by ± a (1,0,0), + a (0,1,0) and ±tf(0,0,1) and the coupling matrix between nearest neighbors has only two independent elements
297
(5-8)
The offdiagonal elements vanish due to symmetry against the inversions y -> — y and z -+ — z while the y y- and z z-elements are identical because of the fourfold symmetry around the x-axis. The coupling to the neighbors at a distance a (3,2,1) has in general the maximum number of 6 independent elements, since this distance vector cannot be transformed into itself by a symmetry element. The couplings to equivalent neighbors are related by symmetry operations. The symmetry conditions of the coupling parameters are important technically because they reduce the calculational effort. In any model starting from a reasonable form of the potential they are automatically fulfilled. If, however, the coupling constants are used as fit-parameters, as is commonly done, the symmetries have to be built in to safeguard against unphysical results. 5.2.2 Harmonic Approximation Taking only the harmonic terms the equation of motion (5-3) can be solved exactly for any assembly of atoms. For simplicity we combine the indices (
) of
Eq. (5-2) or I \i I of Eq. (5-7) to give a single
W
index L. The equation of motion can then be written in symmetrized form as UL
=
-
where for N atoms the index L takes 3N values, ML = Mx and 1
(5-10)
298
5 Lattice Vibrations
The 3 JV x 3 N matrix D is called the dynamical matrix. It is real and symmetric. It should be noted that for spatially periodic systems the term dynamic matrix is also used for the spatial Fourier transform of DLL, (see below). For the displacements uL (t) we make an ansatz y/MLuL{t)
icot
= aeLQ
(eJ)* = (e-J)9
aj = a-j9
COJ = CO_J (5-16)
The energy of the harmonic system is
(5-11)
where a stands for an arbitrary amplitude. Eq. (5-9) is thus reduced to an eigenvalue problem for the matrix D: .eL.
the normal modes are plane waves which are mostly characterized by eigenvectors of the form exp{i &•/?}. The reality of the atomic displacements then requires the conditions
(5-12)
There are 3 AT solutions with eigenvalues, co?, and eigenvectors ej to this problem. Since the dynamical matrix is real and symmetric the eigenvalues are real. The harmonic system is stable if all eigenvalues are positive. An eigenmode with frequency cOj and eigenvector ej is called they'th normal mode of the system. The eigenvectors can be chosen to be real and orthogonal to each other (5-13) and satisfy the closure condition
L
1 2L 1
\ Uriij + - y
Li
LI
f\
*mm^
LILI
Li
LI
^ LU
(5-17)
So far we have treated the lattice vibrations with classical mechanics. For many applications, however, quantum properties are needed. To obtain these we interpret the energy function as the Hamiltonian, Jtf, of our system and can then derive the vibrational wavefunctions by the standard methods for the harmonic oscillator. The quantum mechanics of the lattice vibrations is commonly treated by introducing phonon creation and annihilation operators, bj~ andfc;-,respectively:
(5-14) The 3N amplitudes, aj9 of the normal modes obey independent harmonic oscillator equations dj=-tfaj
(5-15)
The actual atomic amplitudes in the normal modes are gained from Eq. (5-11) where the weighting with yJM expresses the fact that, assuming similar force constants, the amplitudes of heavy atoms are smaller than the ones of light atoms. It is sometimes convenient to use, instead of real eigenvectors, complex ones. E.g., in a translationally invariant lattice
(5-18) The operators b and b + obey the commutation rules for bosons [bj,bf] = 8jr, [bj,br] = [bf,bt,] = 0 (5-19) With these new operators the Hamiltonian takes a very simple form /<- ^
where the hermitian operator, Nj = bf bj is called the particle number operator.
5.2 Elements of Lattice Dynamics
Its eigenvalues are nonnegative integers (0,1,2,...), the numbers of vibrational quanta in the eigenstate. The term \ h cOj is the zero point vibrational energy. In this socalled particle number representation a state | rij} is characterized by the number of vibrational excitation quanta, np in each mode, j . The vibrational excitation quanta are in this representation quasi-particles, the phonons. An additional phonon is created by applying bf to the state, while one is deleted by applying by Any state can be created by successive applications offcj1"to the ground state |0>. Defining as ground state the state without phonons and using the normalization condition <w_y| w^ = 1 we get the following relations =0
(5-21) with (5-22)
K-> = - r = ; W
The eigenstates are in this representation the product of the eigenstates for the single normal modes 2...
... |n3JV> = 3N
3JV
7=1
7=1
and the energy is the sum of the normal mode energies
299
chanical calculations. The same results can also be derived in real space from the Schrodinger equation J^W = EW, where Jf is the Hamiltonian corresponding to Eq. (5-17) with the standard equivalences pL=
—ih(d/dXL)
= MLuL.
In the normal
mode description the many particle wavefunction is given by the product of the wavefunction for the single modes (5-25) 7=1
exp
- 2h
(5-26)
where we have used a real representation of the normal modes and Hn. is the rijth Hermite polynomial. The relations in Eq. (5-21) follow from the elementary properties of the Hermite polynomials. With the help of Eq. (5-11) the normal mode wavefunctions can be expressed in terms of the single atom displacements uL. The diagonalization of the dynamical matrix can be done at present for systems of a few thousand atoms numerically. For larger systems without atomic order it is not feasible. For an infinite periodic system, such as an ideal crystal, translational symmetry can be used to blockdiagonalize the dynamical matrix by a Fourier transformation with respect to the cell vectors A™, introduced in Eq. (5-6). Due to the periodicity, the eigenvectors have the form
3N
(5-24)
(5-27)
Equations (5-23) and (5-24) reflect the independence of the normal modes, the phonons, in the harmonic approximation. The introduction of creation and annihilation operators simplifies the quantum me-
where Nc is the number of cells and the vectors q are restricted to one unit cell of the reciprocal lattice (see Chap. 1). It is usual to use a symmetric unit cell, the first Brillouin zone of the reciprocal lattice; q is
300
5 Lattice Vibrations
then the wavevector of the vibration. X = 2 n/q is the wavelength of the vibrational wave in the lattice which travels in direction q/q; e (qj) is the polarization vector, which determines the direction of the atomic vibrations. For lattices with more than one atom in the unit cell e (qj) determines, together with the phase factor exp (i q • x"), the relative displacements of the atoms in the unit cell. The polarization index j takes 3 s values with s the number of atoms in the unit cell, see below. The polarization vectors depend in general on q. The choice of the first Brillouin zone as the unit cell of the reciprocal lattice has the advantage that it is invariant under the spacegroup of the crystal. Figure 5-1 shows the Brillouin zone of an f.c.c. crystal with the standard notation of symmetry points and directions. At least for the simpler structures the polarization vectors in the main symmetry directions are determined purely by symmetry, independent of q. Similarly, degeneracies in the normal modes can also be determined. These symmetry arguments are very important for
the determination of the phonons from experiments. The unit vectors, bt, of the reciprocal lattice are derived from the ones of the direct lattice by aiAj = 27i8 ij
(5-28)
which implies that the eigenstate, Eq. (5-27), is not changed if one adds a reciprocal lattice vector T to q. The reciprocal lattice vectors are defined as in Eq. (5-6) for the direct lattice by = m1b1+ m2 b2
m3 b3
(5-29)
This invariance against addition of reciprocal lattice vectors can be utilized to expand the different eigenstates belonging to the same q in an extended zone scheme as in electronic band structure theory. An important consequence is that conservation laws in the wavevector are always modulo T. In the previous derivations we have always assumed a finite number of atoms. For an infinite periodic lattice the allowed values of q become dense, q is a continuous variable and summations over q have to be replaced by integrations T/
r
q
(2 7 r ) 3 j - *
(5-30)
with Vc the cell volume. Inserting the eigenfunction, Eq. (5-27), into the equation of motion Eq. (5-9), the problem of calculating the phonons is reduced to an eigenvalue problem of dimension 3 s for each q, where s is the number of atoms in the unit cell coj (?) e£ (qj) = Z D% (q) e} (qj)
(5-31)
where j denotes the 3 s possible polarizations and D is a new dynamical matrix Figure 5-1. First Brillouin zone of the f.c.c. lattice. The symmetry points are in units of 2 it/a; F: (0,0,0);
(5-32)
5.2 Elements of Lattice Dynamics
The dynamical matrix, D, is hermitian and, due to Eq. (5-16), has the properties Dii (q) = (D}i (q))* = [D£J ( - q)]
only on q/q, and the frequencies are q
(5-33)
and hence
If two q values in the Brillouin zone can be mapped onto each other by a symmetry operation of the lattice the phonon frequencies for both q values are identical with the polarization vectors transformed correspondingly. Thus, for example, for a cubic lattice all phonons can be calculated from the q values of l/48th of the Brillouin zone. In the long wavelength limit, q -»0, three of the 3 s phonon branches become sound waves, as they are known in continuum theory. In lattice dynamics these phonon branches are called acoustic phonons. In these acoustic phonons the atoms in the unit cell vibrate in phase with each other. The polarization vectors of these branches depend in the limit q -• 0
THz
1.0
0.8
0.6
-
0.4
1
(5-35)
where Cj is the (phase) sound velocity which depends, in a lattice, on direction and polarization (see also Sec. 5.3.3). In the limit q = 0 the acoustic phonons become rigid translations of the lattice as a whole. The frequencies of the other 3 5-3 phonon branches go to a finite limit for q -• 0 and the atoms of the unit cell vibrate against each other. If the atoms carry a charge, as in ionic crystals, such a vibration causes a macroscopic oscillation of the electric dipol moment which can interact with electromagnetic radiation. These modes are therefore called optic modes. In Figs. 5-2 to 5-4 the phonon dispersion curves, i.e., the dependence of the frequency on the value of q, in the main symmetry directions are shown for three typical materials. Such curves normally have a characteristic shape depending on the lattice structure, e.g., f.c.c, and on the type of
<(q,j) = (* ( - q,j))* and coj (q) = o>? ( - q)
1.0 0
301
0.2
0
0.5
Figure 5-2. Phonon dispersion curves of Cu at T = 49 K plotted against the reduced wavevector, J = a • q/2n. The continuous lines represent a sixth neighbor axially symmetric Born-von Karman fit to the experimental points (after Nicklow et al., 1967). The diagram follows a path A from F to X (see Fig. 5-1) then path Z via W to X of the neighboring Brillouin zone. From there it follows path Z via K back to the origin f, and finally reaches along path A the point L.
302
5 Lattice Vibrations
Figure 5-3. Phonon dispersion of NaCl at T = 80 K. The continuous lines represent an eleven parameter shell model fit (Raunio and Rolandson, 1970).
0
0.4 0.2
10-jO) Z
0
0.2 0.4
0.2 0.4 0.4 0.2 0 0.2 0.4 _ _ ( ± ± 0 ) _ _ (000) _ * . [OO1] 1 l u 2 2 U) V 6 Y R C T O A Reduced wavevector J
- (000) — A
T
(^
Figure 5-4. Phonon dispersion curves for deuterated naphthalene at T = 6 K plotted against the reduced wavevector. The labels JJ, Cf and At denote different representations. The lines are guides to the eye only (Natkaniec et al., 1980).
5.2 Elements of Lattice Dynamics
material, e.g., metallic, ionic or molecular crystal. The absolute magnitude of the frequencies depends on the specific material, e.g., it scales with >/M according to Eq. (5-32). The dispersion curves of two different materials with the same electronic structure will be particularly similar, and one can define a homology relation between the frequencies co1 and co2 of the two
303
not a center of inversion symmetry. Continuing in the Z direction one reaches the X-point of a neighboring zone which in turn explains the degeneracy of the [0,1,1] / and [0,1,1] t x phonons which are identical to the [0,0,1]£ phonon. This apparent contradiction between longitudinal and transversal character stems from the continuation of the Z branch into a second materials cojco2 = \/M2a\ {~jM±a\)~x Brillouin zone where one has e\\(q + T) where a stands for the lattice constant. which in general does not imply e\\q. Such a homology relation is obeyed for The phonon dispersion of the ionic crysexample between Na and K with an avertal NaCl, shown in Fig. 5-3, is more comage deviation of only 3%. We will now plicated. There are two atoms in the Bradiscuss some of the basic properties of the vais cell and, therefore, three optic and phonon dispersion curves using the three three acoustic branches. The acoustic examples shown. branches have a structure similar to the The phonon dispersion of Cu, Fig. 5-2, is one of Cu due to the f.c.c. unit cell. Comrepresentative of an f.c.c. metal with short pared with the sine-like shape of the acousrange forces. The fit to the experimental tic branches the optic ones are much flatvalue shown is by a sixth neighbor interacter. In a number of places optic and tion model with 12 parameters, but a acoustic branches cross each other. Denearest neighbor model would also give a pending on the symmetries the branches reasonable fit. Since Cu has a Bravais latcan intersect or hybridize with each other, tice there are three phonon branches (poe.g., [0,0,1]/ branch. Symmetry again imlarizations) for each q value. A number of plies various degeneracies. properties follows directly from the symFinally, as an example of a molecular metries of the dynamical matrix which are crystal, Fig. 5-4 shows the dispersion curve a consequence of the symmetries of the of deuterated naphthalene (C 10 D 8 ) which Brillouin zone. In the main symmetry dicrystallizes in a monoclinic structure with rections (A, I, A, Fig. 5-1) the polarization two molecules in the unit cell. In a molecuvectors are completely determined by symlar crystal vibrations which deform the metry and one can distinguish between molecule, intramolecular vibrations, here longitudinal (/), e\\q, and transversal (t) of the two coupled rings of C atoms, are modes, e JL q. The latter ones are degenerusually at much higher frequencies than ate in A and A directions. This distinction vibrations of the molecules against each of longitudinal and transversal modes is other (intermolecular vibrations). There not possible for general directions but, nevare six degrees of freedom for each moleertheless, often holds approximately. In the cule which do not change its shape (3 X and L points the dispersion curves have translations and 3 rotations). There are horizontal shapes since these points are therefore 2 x 6 = 12 intermolecular vibracenters of inversion symmetry between the tions in our example giving 3 acoustic and first Brillouin zone and the adjoining ones 9 optic branches. The latter can have pre[co (q) = co (q + r)]. The K-point lies on the dominantly translational or rotational boundary of the first Brillouin zone but is character. The "optic" branches above
304
5 Lattice Vibrations
5 THz are intramolecular vibrations. Their small q dependence testifies to the small influence of the intermolecular interaction on these vibrations. Phonon dispersion curves have been measured for most elemental crystals and for many compounds. Compilations can be found for elemental metals (Schober and Dederichs, 1981; Kress, 1987) for alloys (Kress, 1983) and for insulators (Bilz and Kress, 1979). The harmonic theory discussed so far does not allow for a variation of the frequencies with temperature and therefore does not provide for the lattice expansion, both in contrast to experimental results. Measured phonon dispersion curves at some given temperature have to be understood in the framework of the quasi-harmonic approximation, discussed later, where the expansion of the energy around the equilibrium configuration is done at the given temperature. This way the phonon frequencies become temperature dependent and so, strictly speaking, do empirical models fitted to them. These effects become important near phase transitions. 5.2.3 Phenomenological Models of Lattice Dynamics In applications one is normally not interested in a specific phonon, but wants to know the influence of all phonons or of a group of phonons on some property of the lattice, e.g., specific heat, heat conductivity, thermal atomic displacements or electron scattering. What is needed, therefore, is a fast means to calculate the dynamical matrix. First principle calculations are at present much too time consuming, thus it is necessary to resort to phenomenological models. The parameters of these models are fitted to experimentally determined phonon frequencies, elastic constants and
electric polarizabilities. The models serve to extrapolate from these properties to the whole dispersion curves. This extrapolation will be more reliable the more information on the nature of the material is built into it. Consequently there is an ever increasing multitude of models ranging from simple Born-von Karman models to nearly first principle calculations. The compilation of Schober and Dederichs (1981) already gives some sixty models for the phonon dispersion of Cu, and the number keeps increasing. In the following subsections the main classes of models will be briefly discussed. Depending on the kind of interatomic bonding, preference is given to models with short range interaction (Born-von Karman model variants), screened electronic interactions for metals (nearly free electron models), and models taking Coulomb and electric dipole interactions explicitly into account (rigid ion model, shell model, bond charge model). 5.2.3.1 General Born-von Karman Model (Tensor Force Constant Model) In principle the Born-Oppenheimer approximation allows the dynamic matrix of any system to be expressed by force constants which only have to obey the symmetry requirements discussed before. In practice this can be done only for very short range interactions in lattices of high symmetry or when the analytic form of the interaction is known. To describe a general nearest neighbor interaction in an f.c.c. lattice 3 independent force constants are needed. An extension to twice the range (4th neighbors) already needs 12 parameters. If there are two nonequivalent atoms in the unit cell (e.g., h.c.p.) these numbers are doubled. On the other hand, the shape of the dispersion curves often necessitates fits by long range interactions, giving high
5.2 Elements of Lattice Dynamics
Fourier coefficients. In order to reduce the number of parameters additional restrictions are introduced, e.g., axial symmetry, which can often be justified from first principle calculations. Fits with models of different range show that in general only the first few force constants are reasonably stable against the cutoff. Force constants for the interaction with the more distant neighbor fluctuate in sign and magnitude and are purely fit parameters without any physical meaning. The parameters are mostly fitted to phonons in symmetry directions and this can cause inaccuracies for the off-symmetry phonons (Stedman et al, 1967). Born-von Karman fits to the phonons alone reproduce the elastic constants only very badly. These are often, therefore, included in the fit as constraints. In q-space localized anomalies, e.g., Kohn anomalies, cannot of course be reproduced by short range real space models. Bornvon Karman (B. v. K.) models cannot be used to extrapolate from one structure to another without additional assumptions on the underlying interaction. A determination of the real physical couplings involves not only an exact knowledge of the phonon frequencies but also needs information on the polarizations of off-symmetry phonons (Leigh et al., 1971). Particularly when there is more than one atom in the unit cell, equally good fits to the phonon frequencies can be obtained with different models, resulting in qualitatively different polarization vectors, i.e., different amplitude patterns [see for example Chesser and Axe (1974) for h.c.p. Zn]. 5.2.3.2 Axially Symmetric Born-von Karman Model In the axially symmetric model, used to obtain the fit in Fig. 5-2, one assumes that between a pair of atoms there is only one
305
radial (longitudinal) and one transversal force constant, fT and ft respectively. Such a model can be thought of as derived from a central pair potential U (R) and the two force constants are related to the derivatives of U: _d2U(R)
1 dU{R)
The coupling matrix then takes the form **B =
-(fr-fd:
{Xmn)2
mn 2
(Xmn))
(5-37)
where 5a/? is the Kronecker symbol. In case the interaction is only via a pair potential the transversal force constants between different neighbors are restricted by the equilibrium conditions of the lattice. The force exerted by atom m on atom n is ft (R™ — R"). In equilibrium the sum of all the forces has to vanish for each atom. For inversion symmetric infinite lattices this condition is automatically satisfied. Additional relations between the transversal force constants result from the condition that, in equilibrium, the lattice energy has to be minimal with respect to homogeneous deformations, i.e., changes of the unit vectors a{ Eq. (5-6) and hence dE/daia = 0. For the transversal force constants this means
X f.h(xha + <
+ < - x}) = o
l
This condition implies the Cauchy relation for the elastic constants, see Sec. 5.3.3. It is violated if many-body forces, or volume forces are present. For hexagonal crystals a modified axially symmetric model is often employed, where different paramters are used in Eq. (5-37) for directions in the basal plane and perpendicular to it, respectively.
306
5 Lattice Vibrations
5.2.3.3 Valence Force Models In covalently bonded materials the interatomic potential is often expressed in terms of bond lengths and angles between the bonds. In the limit of small displacements, relevant for lattice dynamics, such a model amounts to a set of relations between the parameters of a short range B.v.K. model. In the notation of Musgrave and Pople (1962) the first terms of the valence force potential for the diamondzincblende type structure are
+ ...
(5-39)
where the indices i, j , k denote the atoms; brtj is the change of bond length from its equilibrium value r 0 ; and 80ijk is the corresponding change of the angles between the bonds from atom j to atoms i and k, respectively.fcris then the nearest neighbor bond stretching constant, whereas krr, k0 and kr@ describe the interaction between the bonds on atom j . An often used simplified form of Eq. (5-39) was introduced by Keating (1966) who parameterized 0 in terms of the scalar variations 5(ro- • rjk). 5.2.3.4 Nearly Free Electron Models For simple metals the electronic structure can be calculated by perturbation theory starting from plane waves for the conduction electrons and eliminating the tightly bound core electrons by a pseudopotential. These theories provide a well founded framework which is treated in many textbooks and articles, e.g., Cohen
and Heine (1970). First principle pseudopotentials are nonlocal electron-ion potentials. In actual calculations a specific form, local or nonlocal, is usually assumed, with fitted parameters. For a local pseudopotential the dynamical matrix takes a particularly simple form which makes these models popular for the description of phonons. In a Bravais lattice the dynamical matrix can be written as DaAq) = T;?t(T + q)ax
(5-40)
Here the Coulomb term Fc (Ewald term) stems from the direct Coulomb interaction of the ions with a compensating uniform background of electrons Fc(q) =
Z2e2 q2
(5-41)
whereas the bandstructure term represents the indirect interaction of the ions via the response of the conduction electrons to the ionic movement in the adiabatic approximation (5-42) with wo(q) the Fourier transform of the local pseudopotential and e (q) the electric permittivity of a homogeneous electron gas of the average density of the conduction electrons. Through the permittivity the dynamical matrix depends explicitly on the cell volume of the crystal. In this description also the constant term in the energy expansion Eq. (5-2) depends explicitly on the volume. Due to these volume dependencies these models do not have the correct limit for all long wavelength phonons. This deficiency can be corrected
5.2 Elements of Lattice Dynamics
for by including higher order terms in the pseudopotential. The model can also be formulated in real space, resulting in a long range oscillating pair potential. The equilibrium condition again involves explicit volume dependencies, therefore giving a long range axial symmetric model without the restriction of the Cauchy relations. With these models the phonon dispersion of simple metals can be fitted with very few (in some cases only two) parameters. This does not, however, guarantee their physical significance. It has been found, for example, that a good fit to second order, such as Eq. (5-40) can be destroyed by higher order terms. The same parameters need not be appropriate for different structures (Heiroth et al., 1986). To simulate the repulsion of the ionic cores expression, Eq. (5-40), is often augmented by nearest neighbor B.v.K. parameters. Such models can then also be applied to fit the phonons in transition metals. There are a large number of models using simplified expressions for the formfactor, Eq. (5-42), plus short range B.v.K. models, e.g., Krebs (1964). These models can be quite successful as far as the quality of the fit is concerned. Physical rigor, however, is sacrificed for computational simplicity. Even further simplified approaches omit all terms with T ^ O and thus violate the periodicity condition of the phonons. 5.2.3.5 Rigid Ion Model
In the simplest approximation for ionic crystals the ions are not polarizable and carry a point charge Ze, where e is the electronic charge. The ions are kept apart by an additional repulsive overlap energy, approximated for example by a BornMayer potential V(R) = AQ~BK. The force constants consist of two axially symmetric parts, a short range one from the repulsive
307
potential and the long range Coulomb part: (5-43) 2
= zmzn
Ane
{xmnmn\5 y
Due to the long range of
=
(5-44)
where B(CD = 0) and 8^ are the static and high frequency electric permittivities respectively. In this simple form the relation is valid for optically isotropic diatomic crystals. Generalizations to crystals with arbitrary geometry were given by Cochran and Cowley (1962). Since at high frequencies the ions cannot follow the electric field, B^/BQ should be one, whereas the observed values for the alkali halides are between 2 and 3. To correct this deficiency the polarizability of the electron wave function has to be taken into account.
308
5 Lattice Vibrations
5.2.3.6 Shell Model
The most popular way to account for the electronic polarizability is the shell model developed by Dick and Overhauser (1958) and by Cochran (1959). In this model the ions consist of a non-polarizabile core with charge X surrounded by a shell of charge Y In the simplest version, in addition to the Coulomb force constants, O cc , the shells are coupled by harmonic coupling constants kt to their respective cores, Ocs, and by nearest neighbor force constants (/ r ,/ t ) to each other,
ud>css + ~sd>sss
(5-45)
where u stands for the ionic displacements and 5 for those of the shells. In the adia-
batic approximation the shell displacements adjust instantaneously to the core displacements 0 = u CS + O ss s
(5-46)
and we get an adiabatic potential energy 1
d> ,. ,
- 1
<
. = — it <(bcc —
adiabatic
~
I
V
/
,
1
(5-47) Here the shell-shell interaction enters via the inverse of the coupling matrix which is no longer just a sum of long range Coulomb and short range repulsive forces. Even without the Coulomb forces the shell-shell interaction transmits long range effective core-core interactions. The finite shell core coupling together with the shell charge Y fixes the optic permittivity s^ and in consequence allows a better fit of the optic phonons. To improve the fits for special materials a number of extensions of the shell model are in use. An additional "breathing" deformability of the shell was introduced by Schroder (1966), and a quadrupolar deformability by Fischer et al. (1972). To describe the large drops in the dispersion curves of TaC and related compounds Weber et al. (1972) proposed a double shell model for two sets of valence electrons. The shell model is intuitively appealing for ionic crystals. It is also widely used for semiconductors where its applicability is not so clear and where fits often need many parameters. 5.2.3.7 Bond Charge Model
Figure 5-5. Simple shell model for ionic crystals. The ions are represented by cores with charges Xi plus electronic shells with charges Yt coupled to their cores with coupling constants /c;. Shells are coupled by axially symmetric force constants ft and ft to each other.
In covalent crystals like Si the charge is accumulated between the atoms near the centres of the bond. This electronic distribution is simulated in the bond charge model by point ions and point bond charges. As in the shell model the latter is eliminated from the equations of motion
5.2 Elements of Lattice Dynamics
by the adiabatic approximation resulting in an expression similar to Eq. (5-47). In the original model the bond charge was fixed to the centre of the bond (Phillips, 1968; Martin, 1969). Weber (1974, 1977) extended this model to allow the bond charges to move adiabatically. Using this model the phonons in diamond type semiconductors can be fitted with only four or five constants. In particular the flattening of the transverse acoustic branch can be reproduced, which needs many more parameters in a valence force model. 5.2.4 First Principle Methods
The phenomenological models of lattice dynamics, discussed in the previous section, are an efficient way to calculate the phonon dispersion over the whole Brillouin zone. The fit parameters do not, however, necessarily have any physical meaning. Often the same lattice is even described by conceptually different models, e.g., semiconductors by valence force, shell or bond charge models. Furthermore, the phenomenological models need not be transferable from one structure to another. To get an insight into the origin of the differences of the lattice dynamics of different materials and their different structures, ab initio calculations starting from the electronic structure are needed. Such calculations often involve substantial computing with high accuracy requirements, since phonon energies are typically 10 meV whereas electron ones are typically 10 eV. In recent years much progress has been achieved and accuracies of a few percent are now standard. There are two main approaches: (i) the direct approach where one calculates directly the energy change or the force due to the atomic displacements (frozen phonon and force constant methods) and (ii) the
309
linear response method. In recent years molecular dynamics methods have gained importance, particularly for calculating anharmonic properties and for phonons in disordered systems. 5.2.4.1 Frozen Phonon Method
Displacing the atoms of a periodic lattice according to the displacement pattern of some phonon with an arbitrarily chosen small amplitude, a new priodic lattice is obtained, with a much larger unit cell. The size of the unit cell depends on the symmetry of the phonon. For phonons in the main symmetry directions and q values which are low rationals of the reciprocal lattice vectors, one obtains managable unit cells (super cells). From the energy difference between the displaced and equilibrium structures the phonon frequencies can be calculated. This approach requires both energies to be calculated by the same method and with high accuracy. The number of phonons which can be calculated is limited by the size of the super cell which can be adequately treated. The idea of the method is some twenty years old, but for most materials reliable results only become available during the last decade, e.g., Si: Yin and Cohen (1982); Ge: Kunc and Gomes Dacosta (1985); GaAs: Kunc (1985); GaP: Rodriguez and Kunc (1988); simple metals: Lam and Cohen (1982); and transition metals: Ho et al. (1984). The calculations mostly employ the local density functional approximation and the core electrons are eliminated by an ab initio pseudopotential. For reviews see Kunc and Martin (1983) and Kunc (1985). By varying the displacement amplitude anharmonic effects can be studied. Comparing the energies for different displacement patterns with the same q value, eigenvectors can also be obtained where they are not already given by symmetry.
310
5 Lattice Vibrations
For long wavelength longitudinal optical phonons in polar crystals the procedure has to be modified to account for the induced macroscopic electric field which causes the Lyddane-Sachs-Teller relation, Eq. (5-44). The periodic boundary conditions needed in the energy calculation imply that this field vanishes, which is equivalent to an added constant depolarizing field. This field can be calculated from first principles and its effect subtracted. Long wavelength acoustic phonons can be calculated from homogeneous deformations of the unit cell. 5.2.4.2 Force Constant Method
To extend the frozen phonon method from selected single phonons to whole branches it is necessary to extract force constants from the calculation. This is done by proceeding as in the frozen phonon method and using the HelmanFeynman theorem to obtain the forces from the self-consistent charge distribution with the phonon frozen in. A powerful variant of this method is the planar force constant method by Kunc and Martin (1982), see also Kunc (1985). There the lattice is decomposed into planes perpendicular to the chosen q vector. In harmonic lattice theory such planes are connected by planar force constants which are linear combinations of the conventional interatomic force constants. Displacing the planes rigidly against each other transverse and longitudinal interplanar force constants are calculated by the HelmanFeynman theorem, and from these the frequencies (Oj(q) of the phonon branches. For the long wavelength optic modes it is again necessary to correct for the macroscopic field.
5.2.4.3 Linear Response Method
In the linear response method the displacements of the atoms by the phonon are treated as perturbations of the equilibrium lattice, and the response of the electron distribution is accounted for by an electric permittivity. This change in electron energy acts as an additional effective interatomic potential. The simplest form of such models was given in the previous subsection for simple metals in the approximation of local pseudopotentials, Eq. (5-40). Using first principle (nonlocal) pseudopotentials this method can be easily adopted to more rigorous calculations for simple metals. In these materials the conduction electrons are in plane-wave like states up to the Fermi surface. The electric permittivity matrix, s(q) has a singularity for q = 2 kF where k¥ is the vector which describes the Fermi surface. These singularities cause small kinks, Kohn anomalies, in the phonon dispersion of metals, e.g., Brovman and Kagan (1974). This simple procedure breaks down for materials other than the simple s/p bounded metals, e.g., if electronic d bands come into play. The permittivity s then no longer depends only on q but becomes a matrix s (q, q'). Equation (5-42) and its equivalents in rigorous theories involve, however, the inversion of s (q, q') and this cannot be done without additional approximations (see Devreese et al., 1985). These difficulties in a #-space representation can be circumvented by adopting real space techniques. Varma and Weber (1977) introduced a tight binding method for transition metals and their compounds (Weber, 1984). This approach uses the idea that the tight binding orbitals move with the displaced cores; it provides an intuitive understanding of the origin of phonon anomalies in these substances.
5.3 Properties of the Ideal Harmonic Crystal
Wang and Overhauser (1987) and Rakel et al. (1988) decompose the charge density of the metal into atomic charge densities to calculate the phonon dispersion in a number of metals. 5.2.4.4 Molecular Dynamics Simulations
Molecular dynamics involves the study of the motion of the atoms according to Newton's equation of motions for some given interatomic potential. Simulations have been used for many years for the determination of anharmonic properties at high temperatures, e.g., Dickey and Paskin (1969). The advent of the Car-Parinello (1985) method, where electronic and atomic configurations are calculated simultaneously, has made molecular dynamics into an ab initio method, e.g., Buda et al. (1989). Willaime and Massobrio (1991) fit an interatomic potential with four adjustable parameters to a tight binding calculation for Zr and then calculate the parameters from experimental data of the low temperature h.c.p. phase. This potential is then used for the high temperature b.c.c. phase. They find that the transverse (Ti) AT-point phonon in that phase is unstable in harmonic approximation and only stabilized
311
by anharmonic effects. Experimentally this phonon, which is important for the martensitic h.c.p.-b.c.c. phase transition, shows a large anharmonicity extending even to zero frequency. Future years will certainly bring a large increase in molecular dynamics applications both for anharmonic properties and for disordered systems.
5.3 Properties of the Ideal Harmonic Crystal 5.3.1 Frequency Spectrum and Related Properties
In the harmonic approximation the normal modes, i.e., the phonons, are independent. The total wavefunction is the product of the single mode wavefunctions and the total energy and the thermodynamic functions are sums of the single mode contributions. These sums can be expressed as averages over the frequency spectrum, g(co\ defined such that g (co) dco is the fraction of eigenfrequencies in the interval [co, co + dco]: (5-48)
and 0.5
THz- 1
(U
1
00
Cu -T:29 8 K -T=
,0 3
0.1
/!
k9 K
1 1 N f y i
\
0
Figure 5-6. Frequency spectra of Cu at T = 49 K and 298 K. The spectra were calculated from a sixth neighbor axially symmetric Born-von Karman fit, as used in Fig. 5-2 (Schober and Dederichs, 1981).
$ dcog (co) =
(5-49)
0
Figures 5-6 and 5-7 show the frequency spectra of Cu and NaCl, respectively, corresponding to the dispersion curves shown in Figs. 5-2 and 5-3. At low frequencies the spectrum increases as co2 which is a direct consequence of the linear dispersion of the long wavelength acoustic modes Eq. (5-35). The spectrum of Cu shows, typical for f.c.c. metals, two maxima: one related to the short wavelength transverse phonons, the
5 Lattice Vibrations
co (10
rad sec )
Figure 5-7. Frequency spectrum of NaCl at T = 80 K obtained from the shell model fit of Fig. 5-3 (after Raunio and Rolandson, 1970).
other one to the longitudinal ones. It should be remembered that the number of phonons for a given magnitude of q increases as q2. The NaCl spectrum also shows a separation into two parts predominantly stemming from acoustic and optic modes respectively. Due to the overlap between these modes in that particular crystal the separation is not complete, as it would be for example in NaBr or Nal. The spectra have a number of singularities (van Hove singularities) where the derivative dg (co)/dco is discontinuous. These singularities originate from extrema of the dispersion curves [Vq co (q) = 0] which are always present due to the periodicity. Near a maximum in the dispersion curves, say com, the singular contribution to the spectrum is of the form: (5-50) for co In one and two dimensional lattices not only the derivative but also the spectrum itself becomes discontinuous or diverges.
A numerical calculation of the frequency spectrum requires an integration over the whole Brillouin zone or over its irreducible part if symmetry is used. The straightforward root sampling method requires a large number of q points. An improved method is due to Gilat and Raubenheimer (1966). There the frequencies coj(q) and their gradients Vq co} (q) are calculated at a smaller number of points and the q dependence of Wj (q) is included by linear extrapolation. In another popular method the volume is filled by tetrahedra and the frequencies within these tetrahedra are estimated by linear interpolation (Jepsen and Anderson, 1971; Lehman and Taut, 1972). Care has to be taken near points of degeneracy to correlate the phonon branches correctly. The thermodynamic functions can be derived from the partition function z — tr(e
K
)
P-M)
where tr stands for trace, Jf is the Hamiltonian and /? = l/(kB T) with kB the Boltzmann constant and T the absolute temperature. In the harmonic approximation the trace can be evaluated for each mode separately, resulting in products of sums over occupation numbers giving exp
(5-52)
The Helmholtz free energy per atom (N: number of atoms) F=--knTlnZ = N
(5-53)
n {2 sinh [h coj
From this the internal energy £, the specific heat at constant volume Cv and the vibra-
313
5.3 Properties of the Ideal Harmonic Crystal
tional entropy S of the crystal, all per atom, can be calculated by standard thermodynamic relations
CQ!
Cu
K mole 6
A
°
(5-54)
*
/
: = 3 - j do; coth [h co/(2 kB T)] co g (co)
c = 3 kB J dco
ha>
(5-55)
2k^f
*•--(**
= 3 kB j dco
(5-56) coth [ft co 1(2 kB T)] -
-ln{2sinh[ftG)/(2feBT)]}lflf(©) These expressions are strictly limited to the harmonic approximation. If the phonon frequencies depend on the temperature, as in the quasi-harmonic approximation, the spectrum g(co) becomes temperature dependent g (co, T) and Eqs. (5-54) to (5-56) have to be augmented by additional terms involving dg(co, T)/dT (see Sec. 5.4). Since
j
- c
200
400 T
^p * p I BXpt. ?xpt.
C
c , B VK E
h
600
800
K
1000
Figure 5-8. Lattice specific heat in Cu (dashed line: harmonic specific heat; full line: quasi-harmonic specific heat; symbols: caloric experiments by various authors) (after Miller and Brockhouse, 1971).
there is no lattice expansion, in the harmonic approximation Cp and Cv are equal. Figure 5-8 shows as a typical example the vibrational part of the specific heat of Cu. Up to about 200 K the harmonic theory works very well; at higher temperatures anharmonic corrections become important. The high and low temperature limits of the thermodynamic functions are given in Table 5-1. The specific heat and entropy obey a T 3 law at low temperatures. In one and two dimensions the corresponding dependencies would be Cv ~ T and Cv ~ T 2 , respectively. This T 3 dependence is typical for the ordered lattice. In
Table 5-1. Low and high temperature limits of the thermodynamic functions per atom in the harmonic approximation. The constant 0D is the (zero temperature caloric) Debye temperature. Function
T->0
T-^oo 3 kB T$ In (Phco)g(co) dco
31 \~hcog(co)6.co H
Cv
15
314
5 Lattice Vibrations
amorphous solids a leading term ~ T is found which is attributed to tunneling centres. In metals an additional term Ce ~ T is due to the specific heat of the electronic system. For high temperatures the harmonic specific heat becomes constant at « 3 kB per atom. The increase in the actually observed specific heat is due to the softening of the phonons with temperature and can be explained within the quasi-harmonic approximation. The low temperature behavior can be approximated very well by a simplified form of the spectrum, the Debye spectrum. At low temperatures the thermodynamics of the crystal will be dominated by the low frequency phonons. The spectrum for low frequencies, on the other hand, is always proportional to co2, stemming from the linear dispersion for small q. In the Debye approximation one assumes a purely quadratic behavior of g (co) up to a cutoff given by the normalization condition, Eq. (5-49), (5-57) 0
CO > COD
Since the specific heat is temperature dependent, a temperature dependent Debye frequency coD(T) and Debye temperature 0D(T) are thus obtained. At low temperatures this coD(T) will be a true representation of the actual g (co\ whereas with increasing temperature it is a somewhat complicated function of the spectrum which quite strongly depends on the fitted quantity, e.g., Cv. The expression for the harmonic specific heat can be rewritten for the Debye spectrum as
Cy(T)=
(5-59)
- " " •
fern))
Equating this result with the specific heat, calculated from the true spectrum or measured experimentally, fixes the caloric Debye temperature 0D(T). Figure 5-9 shows the comparison of the Debye temperature of Cu determined experimentally to the ones calculated from three different Bornvon Karman models. There is a variation of about 10% of the Debye temperature with temperature. Spectra containing parts
The Debye frequency coD is often expressed in terms of a Debye temperature by 350
kB0D
= h coD
(5-58)
In essence the Debye approximation is a low temperature approximation, and the only parameter in gD{co), the Debye frequency, should be determined from the low frequency part of the true spectrum g (co) or equivalently from the sound velocity. It is, however, popular to use the Debye spectrum as a simple reference system to describe some physical property of the lattice, most often the specific heat. The parameter coD is fixed by the condition that the Debye spectrum should give the same value of the specific heat as the true spectrum does.
340
Cu
\
o expt. • expt.
330
A
exp t.
a? 320
~T5
6
•
310
3
°°0
50
100
150
200
250
K
300
T
Figure 5-9. Comparison of the caloric Debye temperature 0D(T) of Cu calculated from the phonon spectra with caloric experiments. Curves A and B refer to the spectra for 49 K and 298 K (Fig. 5-6). Curve C represents a general Born-von Karman model at 80 K (after Nilsson and Rolandson, 1973).
315
5.3 Properties of the Ideal Harmonic Crystal
at very high frequency would give stronger variations. Another approximation is gained if the vibrations of the single atoms are calculated with the rest of the lattice fixed to their equilibrium positions. For each atom three frequencies are thus obtained, called Einstein frequencies. If the average of all these atomic Einstein frequencies are taken, the Einstein frequency of the lattice, coE is obtained. In a cubic Bravais lattice the atomic Einstein frequencies are all identical and equal to coE. In general they are the eigenvalues of the "self interaction" part of the real space dynamical matrix, D^ 2 , for lt = \2. The average Einstein frequency coE is given by the trace of the total dynamical matrix, and due to the invariance of the trace under unitary transformations can be expressed by the spectrum col = — tr D = J co2 g (co) dco
(5-60)
It can be seen immediately that co2 = -\/5/3 coE. The cutoff frequency for n = — 3 can be defined by equating the divergent parts in Eq. (5-62), and is thus equal to the Debye frequency defined above (5-64) _^ = cor For n = 0 where Eq. (5-62) becomes undefined, coo can be defined for the limit n -> 0 00
"1
(5-65)
coo = exp I - + J In (co) g (co) dco
In Fig. 5-10 the cutoff frequencies determined from the spectrum are compared with data gained from calorimetric experiments. Often it is not the average spectrum of the crystal which is of interest, but the partial spectrum, say of the vibrations of atom / in direction a. This can be defined in terms of the eigenvectors, Eq. (5-12), of the dynamical matrix as
In the Einstein approximation the continuum of phonon frequencies is replaced by the Einstein frequency gE (co) = 5 (co — coE)
(5-61) 12
This very simple model can be used to estimate the lattice properties for temperatures T>0E = hcoE/kB, but it breaks down for T -• 0. Experimental data can often be used to calculate moments of the spectrum. Since the actual moments cover many orders of magnitude it is convenient to relate the moments to Debye spectra and so define "Debye cutoff frequencies" con.
-
7.1 — .
17n(4
7.0
"
1.6.9 a
E6.8
/
1
1 \
6.7 \l \
6.6
I
\
(On
dco (5-62)
(5-63) Yn + 3 I /" of = ——
V.'
6.5 6
and the Debye cutoff frequencies are 1
T
t
"-10
-5
0
5
10 n
15
20
25
30
Figure 5-10. Debye cutoff frequencies, vn = con/(2 n), for Cu obtained from the frequency spectra of Fig. 5-6 compared to values obtained from caloric experiments (Nicklow et al, 1967).
316
5 Lattice Vibrations
also called local spectrum of atom / in direction a. Each mode is weighted by the projection of its amplitude on atom / in direction a. In the case in which all gla(co) are equal, this definition is identical to the one of the average spectrum Eq. (5-48). These local spectra can be used to calculate the temperature dependent mean square displacement of an atom h
i
i
(5 67)
-
This is the simplest form of the equal time displacement correlation functions discussed in the next subsection. Expanding Eq. (5-67) one obtains for low temperatures (5-68) and for high temperatures
M
(5-69)
co
At low temperatures the mean square displacement has a constant value determined by the zero point motion. For high temperatures it increases linearly with temperature, with its magnitude given by the co"2
moment of the local spectrum. We will see that this is the static displacement Green's function. In Fig. 5-11 the mean square displacement in Cu in the harmonic approximation is depicted. Also shown is the deviation at higher temperatures due to the softening of the phonons. The atomic mean square displacement determines the Debye-Waller factor observed in scattering experiments. 5.3.2 Green's Functions and Correlation Functions
Whereas it is sufficient to know the frequency spectrum to calculate the thermodynamic functions, more information is needed to calculate the response of the lattice to some perturbation, e.g., the scattering of neutrons or X-rays. This lattice response can be expressed most easily in terms of the two time Green's function, or equivalently in terms of two time correlation functions. Detailed derivations are given in Leibfried and Breuer (1978) and Dederichs and Zeller (1980) which we mainly follow here. Under the influence of an outside perturbation which we write as a force Fl on the atoms, the equation of
0.03
0.02 ~
0.01 -
200
400 600 TEMPERATURE [K]
800
1000
Figure 5-11. Atomic mean square displacement in Cu (dashed line: harmonic values; solid line: quasiharmonic values).
5.3 Properties of the Ideal Harmonic Crystal
motion (5-3) is in the harmonic approximation Mhuh
+ y
317
Often a Green's function for the eigenmode j is introduced by
(5-70 a)
1
(5-75)
l2<*2
or in matrix notation (5-70 b) A partial solution of this equation of motion can be written in terms of the Green's function Gla\li2(t) <\{t)= Z I2&2
? dt'G[%{t-t')Flal(t')
(5-71)
~ °o
where the retarded Green's function is the response to a 5 force
= 8« ia2 8 llJ2 5(0
(5-72 a)
where Mj is the appropriately averaged atomic mass and the Green's function is then given as the sum over all G; (co) multiplied by the appropriate element of their eigenvectors. Apart from factors, Gj(co) is the phonon propagator at T = 0 which is used in the theory of anharmonic effects for example. For the ideal lattice the eigenvectors are given by the Fourier term and the polarization vector, Eq. (5-27), and j is replaced by (qj). Using the standard relation 1 x — if]
or (5-72 b) with the retardation condition G (t) = 0 for t < 0. This condition ensures that the displacement at a time t does not depend on forces later than t. Taking the Fourier transform with respect to time, Eq. (5-11), G (co) can be written formally as
rj
x + r\L + 1-x + Y\
1 X
the Green's function can be split into its real and imaginary parts which are connected by the Kramers -Kronig relation 1 °° 1 Re {G (co)} = - da/ —= 7T 0
G(co) =
1
> i/-0
(5-73)
The infinitesimal quantity Y\ guarantees the retardation. We have chosen the sign of G so that the static Green's function G (co = 0) is positive. In quantum theory often the opposite sign is used in the definition of G. The Green's function G (co) is the inverse of
j 0)j -{co + irj
CO
, Im {G (co')\
—CO
(5-76) which reduces the task of calculating G to calculating the imaginary part
2
-vj)
(5-77)
The diagonal elements are, apart from factors, the local spectra, Eq. (5-66). The physical meaning of the real and imaginary part of G can be understood by evaluating Eq. (5-70) for a point force. The real part then describes the displacements in phase with the force and the imaginary part describes displacements lagging behind in phase by
318
5 Lattice Vibrations
sublattice will vary slowly in space. In this limit the acoustic phonons where the atoms move in phase become sound waves, which are usually defined by differential equations for an elastic continuum instead of the difference equations in lattice dynamics. In the optic modes, on the other hand, the atoms in the different sublattices will vibrate against each other; there is a strong variation of the displacements in one unit cell, i.e., on an atomic scale. These intracell degrees of freedom have to be taken into account explicitly and only the intercell motion can be treated by continuum theory. If the crystal is ionic, the long wavelength limit is not only determined by the short range force constants, 0, but also where the Hamilton operator was given in by the ionic charges and the electric polarEqs. (5-17) and (5-20) and the angular izability which enters through the Lydbrackets < >av imply thermal averaging. By dane-Sachs-Teller relation, Eq. (5-44). an easy manipulation of the creation and Ionic crystals without inversion symmetry annihilation operators, defined before, the are normally piezoelectric. The resulting correlation function formulae are given in the text books, e.g., Maradudin et al. (1971). — j dco ^ 7T — oo We will restrict ourselves here to sound waves in the absence of electric charges. Even in this case the derivation of the h ° sound velocities, or equivalently the elastic = - f dco {2 [n (co) + 1] cos co t — no constants, can be quite tedious since it necessitates the elimination of the internal m{G^a22(co)} (5-79) degrees of the unit cell. The relative motion is obtained, with the Bose factor of the atoms in the unit cell can have drasUo 1 n(co) = (e!" -l)(5-80) tic effects on the sound velocity. We will restrict ourselves here to the simplest case Similarly, the momentum correlation funcof a Bravais lattice without electric polariztion is given as ability. In this case we can treat u™ (t) = u (r = X"1, t) as a smooth, continuous and = -1 dco{[2n (co) + 1] cos co t - (5-81) a differentiable function of X. Additionally, 71 o we have to assume that the driving force h h 2 l l - i sin co t} M M co Im {G a\ £2 (co)} F varies slowly in time and space. We expand the equation of motion (5-3) in pow5.3.3 Long Wavelength Limit ers of Xn-Xm = Xn-m = Xh and keep In the long wavelength limit, q -> 0, the only the lowest nonvanishing term. The displacements of the atoms in the same first two terms vanish because of transla7c/2. The latter are connected with the energy dissipation into the lattice. The larger the spectral density and hence Im{G(co)} is for a given frequency, the more energy can dissipate into the lattice. The Green's function can be used to express the time dependent correlation functions. We replace the classical displacements by their operator counterparts. In the Heisenberg picture these obey the same equations as the classical coordinates and therefore the same applies to the time dependent correlation functions (5-78)
5.3 Properties of the Ideal Harmonic Crystal
tional invariance and inversion symmetry, X"1 = — X~m, and the equation of motion becomes, with Q = M/Vc,
319
length 2n/q to be much larger than the range of the couplings. The dynamical matrix becomes, in the limit, (5-86) Q yd
and the frequencies are given as
h,pyd
j
(5-82) 1
V,
0,7,8
where C is the Huang tensor, which is invariant under the exchange of a with /? or of y with 3. To convert the Huang tensor to the elastic (stiffness) constants, Cafityd9 invariance against interchange of (a /?) with (yd) is also needed, (C^yd = CydtaP). For general coupling constants this is an additive restriction on the force constants (Huang condition). In cubic crystals this condition is automatically fulfilled. The Huang tensor is then related to the elastic constants by a linear relation Coty,lid
=
Cocp,yd + ^yp,aS ~ ^ya,(id
(5-83)
For cubic crystals one has the simple expressions (using the pairing convention for indices, e.g., 11 —>-1) (5-84) C l 2 = -
C1 2 ,
'12
With these relations Eq. (5-82) can be written in the usual form known from continuum elasticity theory
+ —Fa(r,t)
(5-85)
c
A completely equivalent derivation is gained by expanding the spatial Fourier transform of the equation of motion for long wavelengths [exp (i q X"1) = 1 + {(qX™) + ...]. This requires the wave-
j
(5-87)
where Cj (q/q) is the (phase) sound velocity in the direction of q for polarization j . The sound velocities for arbitrary directions can be calculated analytically only for hexagonal crystals. For cubic crystals they can be calculated in symmetry directions or for isotropy (Leibfried and Breuer, 1978). In the isotropic case there is one longitudinal mode with sound velocity cx = -S/C11/Q, and two transverse modes with velocities c = /c4jg9 for each q, independent of t
direction, and the isotropy condition c n — c i2 = 2c 4 4 holds. Two important points concerning force constant models should be mentioned. The first is the Cauchy relation which reduces the number of independent elastic constants from 21 to 15, in the general case, or in the cubic lattice, from 3 to 2 (c12 = c44). This condition holds if (i) each atom is a centre of inversion, which is always the case in Bravais lattices, and (ii) the atoms interact only by central forces. The Cauchy relation is then a consequence of the equilibrium conditions of the infinite lattice. The Cauchy relations hold in halides within a few percent, e.g., in NaCl, C C ii/ 4.4. = 0-98, whereas they are strongly violated in metals, e.g., in Cu, c 12 /c 44 = 1.62. The second point concerns potentials which explicitly depend on the unit cell volume; for example, the pair potentials derived by perturbation theory for metals (see Sec. 5.2.3) where the average conduction electron density enters explicitly. For these interaction models the equilibrium
320
5 Lattice Vibrations
condition mixes the orders of the perturbation expansion. This, in turn, means that the long wavelength limit of the phonons involving lattice compression becomes inaccurate (Brovman and Kagan, 1970; Pethick, 1970). To be correct to second order, fourth order perturbation theory would be needed, which would destroy the simplicity of the models. To obtain the compression modulus one has to calculate the volume derivative of the energy rather than the long wavelength limit.
5.4 Anharmonicity The harmonic approximation discussed in the previous section gives a satisfactory description of the vibrational properties of most solids at low temperatures when the vibrational amplitudes are small. The neglect of the anharmonic terms in Eq. (5-2) has, however, a number of consequences. Notably there is no thermal expansion, and thus the phonon frequencies, force constants and elastic constants are also independent of temperature. Additionally, the specific heat becomes constant at high temperatures, and the constant volume and constant pressure specific heats are equal, Cf = C^. As eigenstates of the harmonic Hamiltonian, in harmonic approximation the phonons have infinite lifetimes, their linewidths are zero, their mean free paths are infinite and hence the thermal conductivity of a perfect harmonic crystal is infinite. To correct these shortcomings, anharmonicity has to be taken account of. This can be done on different levels. Effects of the temperature variation on the phonon frequencies can be treated adequately and relatively easily by the quasiharmonic approximation, where the lifetime of the phonons is still infinite. To give an adequate description of the phonon life-
time, phonon-phonon interaction effects must be treated. As long as these effects are not too large perturbation theory in the anharmonic terms of Eq. (5-2) will be sufficient. For temperatures near the melting point, the convergence of the perturbation expansion becomes very slow and a selfconsistent phonon description is more adequate. The analytic difficulties can be circumvented by numerical calculations for model substances using molecular dynamics or Monte Carlo methods. Finally it should be kept in mind that the phonon lifetimes are not only limited by the anharmonicity of the ideal infinite lattice, but also by surface and defect scattering and by electronic excitations, neglected in the adiabatic approximation where the atomic interaction is given by an effective interatomic potential. The latter effect, the electron-phonon interaction, is important in metals and also in semiconductors. 5.4.1 Quasi-Harmonic Approximation
In the harmonic approximation the equilibrium positions of the atoms were defined by the minimum of the potential energy, and were thus independent of temperature. In the quasiharmonic approximation, the equilibrium positions are taken as parameters which are determined by the minimum of the temperature dependent free energy. The expansion is then, as in the harmonic theory, limited to quadratic terms in the displacements from these minimum positions. The free energy Fqh takes the same form as in the harmonic approximation, potential energy $ 0 + vibrational free energy [Eqs. (5-2) and (5-53)] (5-88) + feB T E In {2 sinh [h coj (q, T)/(2 kB T)]}
Since the expansion is no longer from a minimum of the potential energy there will
5.4 Anharmonicity
be a linear term
(5-89)
or by the thermal strains, their values at T = 0 Xs =
, relative to = 0)
The equilibrium positions are now obtained by taking, for constant T, the derivatives of the free energy with respect to the unit vectors or, equivalently, to the thermal strains. If there are atoms in the unit cell whose relative position, x11 (T), is not determined by symmetry, the relaxation of the internal positions in the cell has to be allowed for by additional minimization. 8F>h th
Q
8$o
+
,.th
V
±] (5-91)
ith
321
The thermal strain is then given by vih« =
aP
3NVr (5-94)
where hcoj(q){n(wj(q)) + |} is the mean thermal energy of mode (q,j) at temperature T, and where for simplicity we have restricted ourselves to cubic crystals with compressibility x. From Eq. (5-94) the thermal expansion coefficient can be calculated as a£p = dy^/dT. At high temperatures it becomes independent of temperature, but proportional to ^qjy(qj). The lattice constant increases linearly with temperature. At low temperatures the expansion coefficient goes to zero, in accordance with Nernst's theorem, but the lattice constant differs slightly from the harmonic one due to the zero point motion included in Eq. (5-88) but not in Eq. (5-2). To determine the mode Griineisen constants, Eq. (5-93), measurements or calculations of the phonon dispersion at different temperatures are needed. Normally neither is available. Therefore, mostly average Griineisen constants y are used. One such definition is via the Debye temperature, Eq. (5-58), which can be gained from the change of specific heat with temperature,
V
Here the temperature enters via the Bose factor n(co), Eq. (5-80). The strain derivative of the potential energy can be related to the harmonic elastic constants, Eq. (5-83) w' h
9w"
8, yd "yd
(5-92)
The strain derivatives of the phonon frequencies are usually expressed by (mode) Griineisen parameters (5-93)
y(co)=
-
81n0 D 61nK
(5-95)
The experimental values of y typically vary between 1 and 2. In this approximation the expansion coefficient is proportional to the specific heat and should therefore vanish with T 3 for T -> 0. This normal behavior is violated in a number of substances where with increasing temperature the lattice constant first decreases at lower temperatures before it increases at higher temperatures (e.g., ice, Si, Ge, ZnS). Such behavior can occur if some groups of phonons have
322
5 Lattice Vibrations
gitudinal frequencies soften over the whole temperature range by about 10% and the transverse ones by about 15%. This smooth variation is also reflected in the shifts of the corresponding frequency spectra, Fig. 5-13. Even in simple structures the behavior is not always so simple. For example, in the high temperature phase of Zr (b.c.c. (3-Zr) there are two groups of phonons. The majority of the phonons soften with increasing temperature, as expected for a normal metal. The phonons in certain regions of the Brillouin zone, however, show the opposite behavior; they harden considerably. In Fig. 5-14 this effect shows up as a shift of the low frequency peak of the spectrum to higher frequencies. The frequency of some phonons increases by about 50% in a temperature range of 700 K. This inverse temperature behavior
negative Griineisen constants whereas others have positive ones. In Si, for example, the transverse acoustic phonon at the Xpoint has y = — 1.4 whereas the corresponding optical phonon has y = + 0.9 (Weinstein and Piermarini, 1975). At low temperatures the expansion coefficient is determined by the low frequency modes, and the negative y values of the low-lying branches are dominant. At higher temperatures, on the other hand, the expansion coefficient is proportional to the sum over all mode Griineisen coefficients, which is always positive. In simple lattices a description of the phonon softening by an average Griineisen constant works quite well. As an example Fig. 5-12 shows the dispersion curves of Cu measured at temperatures ranging from 49 K to 1336 K, i.e., 20 K below the melting point. The lon-
Cu
THz
IOO£]
7
r n
6 --
5h
L -
3 -
°T = 296 K •T = 673 K ^ T = 973 K = 1336 K • T -- 296 K vT -- />9 K
n (f SI r
2 -
If
f 0
0.2
i
i
0.^
0.6
J
-~
i
0.8
1.0 0
0.2
i
^m i
0.6 —^*-
r—
i
0.8
1.0
\
I
I
I
1
0.8
0.6
0.4
0.2
^ 1
0
E
Figure 5-12. Temperature dependence of the phonon dispersion in Cu. The lines are guides to the eye (Larose and Brockhouse, 1976).
323
5.4 Anharmonicity
stant pressure is obtained; Cp(T) = 3fcB| dcox
hco V f . , / hco sinh ; \ 2kBT xg(qi)
7THz 8
Figure 5-13. Frequency spectra of Cu at different temperatures obtained from Born-von Karman constants (Schober and Dederichs, 1981).
is related to the h.c.p.-b.c.c. martensitic phase transition of Zr at 1138 K, and is an example of the connection between phase transitions and phonon anharmonicity. The above mentioned negative y value of the transverse acoustic X-point phonon in Si correlates with a pressure induced phase transition (Soma, 1978). The vibrational entropy in the quasiharmonic approximation has the harmonic form, Eq. (5-56), with the quasi-harmonic frequencies instead of the harmonic ones (Werthamer, 1969; Hui and Allen, 1975). By differentiation, the specific heat at con-
0.5
bcc Zr
0.4 0.3-
0.1 10 15 ENERGY t i w (meV)
20
Figure 5-14. Frequency spectra of b.c.c. Zr at different temperatures obtained from Born-von Karman constants (Heiming et al, 1991).
81nco(T)1
'J
-i
(5-96)
In addition to the harmonic part, which is modified by the change of spectrum with temperature, there is an anharmonic correction term, 8 In co{T)/d In T. The high temperature limit of the spectrum exceeds the Dulong-Petit value of 3 kB due to the general softening of the phonons (see Fig. 5-8). To estimate the effect, a usual approximation is to replace the logarithmic correction for each phonon by an average value (Hui and Allen, 1975)
324
5 Lattice Vibrations
lation of the high temperature isothermal constant to T = 0. The isothermal constants for T = 0 differ by a few percent due to the anharmonic effects of the zero point motion. The isothermal elastic constants are observed in static strain and generally also in the long-wavelength limit of neutron and X-ray experiments. In sound waves, on the other hand, the adiabatic constants are observed. The two sets of constants are related by 9 TCV
(5-98)
The transition from one set of elastic constants or sound velocities to another is a complicated problem, beyond the scope of this chapter. For details the reader is referred to Beck (1975) or Gurevich (1986). Due to its simplicity, the quasi-harmonic approximation is usually assumed in interpreting temperature dependent phonon data. The measured frequency shift of the phonons, however, not only includes the effect of the thermal lattice expansion, but also additional anharmonic effects stemming from the corrections to the quasi-harmonic free energy. As long as the observed phonon line width is small, these corrections will also be small. In extreme cases it is possible that the quasi-harmonic phonon does not exist, and the vibration is stabilized by the anharmonic terms. In such a situation the interpretation of the experiment in terms of the quasi-harmonic approximation would give a positive frequency. Such a case was found by Willaime and Massobrio (1991) in a calculation for b.c.c. Zr. 5.4.2 Anharmonic Perturbation Theory
In the harmonic and quasi-harmonic approximations the phonons are noninteracting quasiparticles with energy ha)j(q)
and quasimomentum h q. In a real crystal the phonons are no longer independent of each other. There is a phonon-phonon interaction, caused by the anharmonic terms of the potential energy, Eq. (5-2). This anharmonicity becomes more important as the temperature increases, and the phonons have finite lifetimes and linewidths due to it. Additional effects such as electronphonon interaction, phonon scattering by defects or finite size effects will be discussed later. In perturbation theory the total Hamiltonian of the anharmonic lattice is written as
+
anh
(5-99)
where J^° is the harmonic Hamiltonian (5-20) and j ^ a n h = A
5.4 Anharmonicity
materials without large anharmonic effects, such as the alkali metals, low order perturbation theory agrees well with experiment (Vaks et al, 1980). The actual calculations can be done by any of the usual techniques of perturbation theory. The starting point is always the exact solution of the harmonic problem. The anharmonic terms are then expressed in terms of phonon creation and annihilation operators, b+(qj) and b(#,;), introduced in Eq. (5-18) (since we are considering periodic lattices the general index j is replaced by the pair qj). The first anharmonic terms then describe all processes involving just three phonons. For example, term involving b + (q3 J3) b+ (q2j2) b (qx j x ) describes the decay of the phonon (#x j ^ ) into two phonons {q2j2) and (q3j3). Energy conservation requires (5-101) -hco(q2,j2)
-hco(q3j3)
=0
and quasimomentum conservation hql-hq2-hq3
= tiT
(5-102)
where T is a reciprocal lattice vector. If T = 0 this is a normal (N-) process, while processes with t # 0 are referred to as "Umklapp" (U-) processes. To calculate the lifetime of a particular phonon one can now calculate, e.g., by Fermi's golden rule, the probability that the phonon is destroyed or created in such a process. The lifetime is then the reciprocal value of the total destruction probability minus the creation probability. These lifetimes are temperature dependent since all phonon-phonon processes depend on the mean thermal occupation of the phonons involved. Even though such a calculation is simple in principle it involves substantial computations due to the many summations involved. Systematic derivations of anharmonic properties can be given by many-body per-
325
turbation theory methods, see Barron and Klein (1974), Kwok (1967). These methods can be utilized to calculate the anharmonic Green's function and the related correlation functions which are measured in scattering experiments. In Eq. (5-75) the Green's function for a single phonon at T = 0 was given in the harmonic approximation as Gj(q9co) = [Mjcoj(q) - Mj(co + i*/)2]"1. This function had poles on the real co-axis for co = (Oj (q). The anharmonic phonon propagator can be similarly written with an additional term, the self energy Z, calculated to some order of perturbation theory. This self energy is a complex matrix, and is only diagonal in the polarization j of the harmonic phonons if all polarizations belong to different symmetries. Let us consider the simplest case, in which I is diagonal in j and varies slowly in co. We replace co by the harmonic frequency coj(q) in S, which is now a complex number, and we obtain in analogy to Eq. (5-75) the phonon Green's function G,. (q, co) = {Mj co] (q) + Ij [q9 coj (q)] -Mjco2}'1
(5-103)
If the phonon damping is not too large the propagator will have poles near the real axis at the shifted phonon frequency
and the damping or inverse lifetime of the anharmonic phonon is ranh ,„, ^
I m
{Zj [9><»j(9)]}
(5.1O5)
The line shape of the phonon in this approximation is Lorentzian. The exact line shape can be more complicated. In Fig. 5-15 experimental values for the frequency shifts and damping in Li are shown. These values
326
5 Lattice Vibrations
Figure 5-15. Phonon frequency shifts and half widths in 7Li determined by neutron scattering experiments (dashed line: T = 293 K; full line: T = 385 K; dash-dotted line: T = 424 K) (Beg and Nielsen, 1976).
can be reproduced by second order perturbation theory (Vaks et al., 1980). Analytically, the self energy can only be evaluated in some limiting cases. One such case is the Landau-Rumer result that the damping (attenuation) of transverse sound waves is proportional to co T 4 in isotropic solids. This strong temperature dependence is a consequence of the linear dispersion of the sound waves which causes a quadratic frequency dependence of the frequency spectrum, g(co)ccoj2. The number of phonon states which can be excited thermally at low temperatures increases as T 3 . More generally, anharmonic damping is at low temperatures proportional to cov T 5 ~ v , with v usually between 2 and 4 depending on symmetry (Herring, 1954). For more details see Kwok (1967) and the book by Srivastava (1990). The knowledge of phonon damping is a necessary ingredient for the understanding of heat conduction, which is beyond the scope of this chapter. Second order perturbation theory calculations of the mean square displacement in an b.c.c. lattice have shown that the quasi-harmonic approximation accounts
for about 90% of the anharmonic corrections and only 10% is intrinsically anharmonic (Shukla and Mountain, 1982). 5.4.3 Self-Consistent Phonon Theory
Perturbation theoretic treatments are limited to small anharmonicities. They become unreliable near the melting point and break down if the harmonic phonon is not stable [(Oj(q)2 < 0]. In the self-consistent phonon approximation the force constants are not calculated from the equilibrium distances of the atoms, but averaged using the pair correlation function of these atoms, for which the harmonic expression Eq. (5-78), is used for the lowest order. Since the pair correlation itself depends on the frequencies and, therefore, on the force constants, iteration is needed for self consistency. This first approximation still breaks down at high temperatures. An essential improvement is gained by taking the core repulsion at short distances into account (Horner, 1974; Horton and Cowley, 1987). With this method Cowley and Horton (1987) were able to reproduce the
5.4 Anharmonicity
lattice dynamical properties of Ar to within 3 K of the melting temperature. 5.4.4 Computer Simulation
Apart from the simple quasi-harmonic approximation all analytic approaches to anharmonicity are limited to simple systems or to discussions of qualitative behavior in certain limits. This restriction to simple systems is much less severe in molecular dynamics simulations, where the motion of the atoms is studied by solving Newton's equations for a given interatomic interaction. Molecular dynamics is thus strictly limited to classical mechanics and, therefore, best suited for high temperatures where quantum effects are not important. That is, of course, just that region where analytic methods have their greatest difficulties. The accuracy of molecular dynamics methods is determined by the number of atoms which can be included which excludes long wavelength phonons and their effects. Also, finite size effects cannot be fully avoided. With the usual periodic boundary condition, each atom sees itself through its interaction with the other atoms. The constraints of available computer time makes molecular dynamics well suited to short time phenomena, it cannot fully sample the phase space. A limitation that computer simulation has in common with analytic methods is the difficulty in providing accurate interaction potentials. There the Car-Parinello (1985) method, where the electronic configuration is relaxed simultaneously with the atomic one, has brought great progress. As long as its limitations are kept in mind, computer simulation is to date the most powerful method in calculations of more complex structures. A recent review of its application to lattice dynamics was given by Lewis and Klein (1990).
327
Thermodynamic quantities can also be calculated by Monte Carlo techniques which are, in principle, exact in the classical limit. In this method the partition function, Eq. (5-52), and its derivatives is calculated by random sampling over instantaneous configurations (Cowley, 1983). Convergence can be accelerated by suitable sampling algorithms (Day and Hardy, 1985). 5.4.5 Electron -Phonon Interaction
So far we have considered lattice dynamics in the adiabatic or Born-Oppenheimer approximation where the electrons arrange themselves instantaneously. If one is interested in anharmonic phonon properties, and in particular in phonon damping, excitations of the electronic states have to be included. A phonon can decay by exciting an electron hole pair. This mechanism is of great importance in metals where the finite electronic density of states at the Fermi level provides a large reservoir of possible excitations. Due to this finite density of states at the Fermi level the phonon damping by their interaction with the electrons stays finite at zero temperature. An accurate calculation is rather involved, since now one has to deal with the calculational difficulties of both the electron and the phonon systems. A rough estimate of the magnitude to be expected can be derived from a jellium approximation (Pines, 1963). The ratio of the damping F to the phonon frequency is
r
nfZmV12
(5-106)
where Z is the number of conduction electrons per atom and m and M are the electronic and atomic mases, respectively. For Al this estimate gives about 5 x 10~3. The
328
5 Lattice Vibrations
1.0
0.8
0.6
(U
0.2
0 0
0.1
0.2
results of a more rigorous calculation for Nb are shown in Fig. 5-16. 5.4.6 Phonon Lifetimes
The lifetimes T of the phonons are limited by a large number of processes. In addition to the strongly temperature dependent damping by anharmonic effects (x ~x = T) and the electron phonon interaction, a large number off additional processes limit the phonon lifetimes. These will only be mentioned without going into details. The finite size of the crystal will cause the phonon to be scattered at the surfaces. The lifetime between two such boundary scattering events will be roughly L c
Tbs = —
(5-107)
where c is the speed of the phonon and L the typical length of the sample. Further
0.3
0.4
Figure 5-16. Calculated phonon linewidth due to electron phonon interaction in Nb (full width at half maximum; solid histogram: longitudinal modes; dashed and dotted histograms: transverse modes) (Butler et al, 1977).
0.5
scattering occurs due to imperfections. The scattering at point defects will be discussed in the next section. Even a pure crystal will in general contain different isotopes giving rise to mass defect scattering which can be estimated as (Pomeranchuk, 1942; Klemens, 1955) 4 TIC
3
AMV 1A)
CO
(5-108)
where <... > denotes the configurational average and AM the deviation from the average atomic mass M. In real crystals additional scattering occurs by extended defects such as dislocations, stacking faults or grain boundaries. These will give additional contributions to T~~ 1 proportional to co and co2 (Klemens, 1958). All these estimates are only valid in the long wavelength limit.
5.5 Imperfect Crystals and Disordered Solids
5.5 Imperfect Crystals and Disordered Solids The perfect periodic lattice treated in the previous sections is an idealization of the real solid. In real crystals periodicity is destroyed by a number of imperfections: point defects (substitutional impurities, vacancies, interstitial atoms and small defect clusters) or extended defects such as dislocations, stacking faults, grain boundaries and surfaces. The local dynamics of these impurities can differ considerably from that of the host material. There are often local vibrational modes at the impurity: resonant and localized modes of point defects or surface waves. Even a small concentration of defects can strongly affect the dynamics of a crystal. With an increasing number of substitutional impurities the periodic crystal will become a disordered alloy where there is still an underlying periodic lattice, with distortions which are not too large. If the impurities have a discernible orientation, e.g., CN molecules, an orientational glass is obtained. On the other hand, if the number of structural defects increases then spatial order is destroyed and the solid becomes amorphous, or a structural glass. Obviously there is a huge multitude of possible combinations of imperfections and the following discussion has to be restricted to some highlights. We will discuss the dynamics of vibrations around the equilibrium positions of the atoms in the harmonic approximation of the defect crystal or glass. The static equilibrium configuration, on the other hand, must in general be calculated from the full potential energy, and the harmonic force constants in the defect crystal need not be the same as in the ideal host crystal.
329
5.5.1 Single Defect Dynamics In the harmonic theory of the vibrations in the defect lattice the vibrations of the crystal atoms, in particular the defect atoms, around their equilibrium positions are studied. These are either calculated from the full potential, or are fitted to experimental data such as volume expansion. Detailed reviews are given, e.g., in the book by Maradudin et al. (1971) and by Dederichs and Zeller (1980). A second order expansion of the potential energy with respect to the dynamic displacements u from their static equilibrium positions R = X + s gives, for substitutional defects, the equation of motion: (M + AM) «
A
(5-109)
Here and in the following discussion indices have been dropped and matrix notation is adopted, with M = MSmndap the mass matrix of the ideal host crystal, AM the matrix of mass changes, O the force constant matrix of the ideal crystal and ACE> the matrix of force constant changes. AO contains not only the direct couplings due to the defect but also contributions from the anharmonic host potential. After Fourier transformation with respect to time Eq. (5-109) can be written as (L + AL)n=/
(5-110)
where the ideal host lattice terms have been combined to L = — M co2 + and the perturbation terms to AL = - AM<x>2 + Ad>. Equation (5-109) can formally be solved by a Green's function G = (L + A L ) " 1
(5-111)
As for the ideal crystal (Sec. 5.3.2), experimentally observable quantities can be conveniently expressed by this Green's function, e.g., the local frequency spectrum g[ (co) of atom / for vibrations in direction a, Eq. (5-66).
330
5 Lattice Vibrations
Various ways to calculate G are currently in use: (i) The most simple one is to do a straightforward diagonalization of Eq. (5-111) for a finite crystallite. On modern computers this can be done for about a thousand atoms in a couple of minutes. The resultant frequencies and eigenvectors will be accurate for localized and short wavelength vibrations. Long wavelength acoustic vibrations can be estimated from their static limit, i.e., from the elastic constants. (ii) In recent years the recursion technique, proposed by Haydock et al. (1972) for tight binding electronic calculations, is frequently used for lattice dynamical calculations of point defects and disordered systems. This method is based on the fact that it is possible to construct for a quadratic Hamiltonian a set of basis states which are coupled by a three point symmetric recursion relation. In this basis the inversion Eq. (5-111) can be done by an infinite continued fraction. Dependent on the size of the problem the continued fraction is terminated at a given level by some appropriate closure procedure. This causes the discrete spectrum of a finite fraction to be replaced by a smooth continuous one. Usually ten to twenty levels are sufficient for convergence comparable to experimental resolution (Punc and Hafner, 1985). (iii) Equation (5-111) can be expanded in terms of locators where the locator is the one particle Green's function in the Einstein approximation for this particle (Czachor, 1980). This method is restricted to defects which are weakly coupled to the lattice so that the Einstein approximation is a good starting point. (iv) An exact solution of Eq. (5-111) for the single defect case can be given by T-matrix techniques where G is expressed by the ideal host lattice Green's function 6 and
the perturbation AL. Denoting the Green's function of the ideal lattice, Eq. (5-74), by d(co) = L(co)"1, Eq. (5-111) can be written for a substitutional defect as G = 6-6ALG = 6 - 6 t 6 (5-112) with the single defect T-matrix t = AL(l + 6 A L ) " 1 = = AL -
1
&
(5-113)
The matrix t has the same dimension as the defect matrix AL, thus it can be fairly easily evaluated for "small" defects, whereas for "large" defects mostly approximate methods are preferred. The point symmetry of the defect subspace can be utilized to speed up the inversion and to gain extra insight into the physics behind G. Complex and interstitial defects can be dealt with by a projection technique (Krumhansl and Mathews, 1968) where the degrees of freedom are divided into those of the ideal crystal on one hand and the additional ones, e.g., the internal vibration modes of a substitutional molecule, on the other. From Eqs. (5-112) and (5-113) one sees that large values of G have to be expected if det 11 + AL 61 ~ 0. The determinant can only vanish exactly if the real and imaginary part vanish simultaneously. The imaginary part stems from the imaginary part of the host's Green's function 6 and vanishes therefore for all frequencies outside the bands of ideal lattice frequencies. For those frequencies where the real part also vanishes, 5-shaped contributions to the spectrum result. These vibrational modes are spatially localized and thus are called localized modes. The localization is a consequence of the asymptotic behavior of the Green's function. Such modes are to be expected for defects with either small mass (e.g., H in metals) or very strong couplings (e.g., intramolecular vibrations of substitutional molecules). If the real part of
5.5 Imperfect Crystals and Disordered Solids
the determinant vanishes while the imaginary part is small then a resonant mode is obtained. Such a mode is not as strongly localized around the defect, thus it is also called a quasilocalized mode, and its frequency is broadened, with the width given by the imaginary part. Since Im 6 (co) ~ co for small co such resonant vibrations occur at low frequencies and therefore for either heavy defects (e.g., Ag in Al with M A g « 4MA1) or for weak couplings. Figure 5-17 depicts the local spectra for light substitutional mass defects for a nearest neighbor model of an f.c.c. crystal. The spectrum consists of a part where the defect vibrates with the lattice modes and the sharp localized line. With decreasing mass of the defect, the localized mode frequency increases and agrees more and more with the defect's Einstein frequency (coE oc Mdef); the intensity of the localized mode line increases as M/(M — Mdef). Considering the normalization of the spectrum this implies that the vibration gets more localized on the defect. The corresponding local spectra for heavy mass defects are shown in Fig. 5-18. The behavior is now opposite to that of the light defects. Another feature is the narrowing of the resonance part of the spectrum. The half width of the resonance decreases approximately as r oc wr2es oc (1/Mdef). Many defects show neither of these pronounced effects. As a typical example Fig. 5-19 shows the spectrum of a neighbor to a vacancy in Cu. As expected the spectrum in the vacancy case shows a general softening, due to the missing couplings, which is however not strong enough to cause a resonant mode. A much more interesting behavior is shown in Fig. 5-20 for the local spectrum of the self-interstitial atom in f.c.c. Cu which is discussed in Chap. 6, Sec. 6.4.2.3. Localized and resonant modes occur simulta-
M/2
.•••••"
; M A /
331
MA
•-.
M/2
it
Figure 5-17. Local frequency spectra of light isotopes (having masses M/2 or M/4 embedded in a matrix of isotopes with mass M) in a nearest neighbor model of an ideal f.c.c. crystal. The localized modes are indicated by vertical lines labeled with the defect mass. The dotted line indicates the ideal spectrum. The arrows indicate the values of the Einstein frequencies of the different isotopes (Dederichs and Zeller, 1980).
neously. For our purposes it can be envisaged as a substitutional Cu 2 molecule where the two atoms are strongly coupled causing a localized bond stretching vibration (Alg). At the same time the restoring forces against libration (£g) or translation (A2u) of the molecule as a whole are weak,
Figure 5-18. Local frequency spectra of heavy isotopes in a nearest neighbor model of an ideal f.c.c. crystal. As in Fig. 5-17, the dotted line indicates the ideal spectrum. The arrows indicate the values of the Einstein frequencies of the different isotopes (Dederichs and Zeller, 1980).
332
5 Lattice Vibrations
Figure 5-19. Local frequency spectra of a neighbor to a vacancy in an f.c.c. lattice. The broken line indicates the ideal spectrum (Dederichs and Zeller, 1980).
causing resonant vibrations. This example shows the importance of the tensor character of the coupling. The simultaneous occurrence of localized and resonant modes is typical for "large" defects which compress the lattice locally and thus cause a tendency to instability in some directions. This example shows that not only weak coupled defects can have resonant vibrations but that a defect which would be nor-
mally considered as strongly coupled to the host lattice can have low frequency resonant modes due to the local atomic arrangement. The defect configuration can even become instable and is then often split into configurations of lower symmetry. The configuration of the self-interstitial atom can be envisaged as the result of an instability of the octahedral configuration which would have higher symmetry. Defect vibrations, especially localized ones, are often used to identify the defect. An example is the localized modes of H in Nb where the 1:2 intensity distribution between the two localized modes shows that H occupies tetrahedral and not octahedral sites in Nb, see Fig. 5-21. Another such investigation was carried out to identify the configurations of Ag2 in noble gas matrices (Bechthold et al., 1986). There exist one, two or three distinct such configurations for matrices of Ar, Kr and Xe respectively. These could be identified by a comparison of experiment with the results of structure calculations followed by a calculation of the Ag2 vibrations. Figure 5-22 shows the
5 arb. units
a-N ^Ho.CK
4
n
v/ \V /
1.0
0.5
1.5
oo/u) m
Figure 5-20. Local frequency spectra of a self-interstitial atom in the f.c.c. Cu lattice. The localized modes are indicated by vertical lines. The modes are labelled according to their symmetry (A2u,AH, Eg, Eu), which is also shown schematically. The broken line indicates the ideal spectrum (Dederichs and Zeller, 1980).
Figure 5-21. Localized frequency spectra for H in Nb according to time of flight experiments (Verdan et al., 1968).
5.5 Imperfect Crystals and Disordered Solids
333
Figure 5-22. Resonance Raman spectrum of Ag2 in Kr, demonstrating the simultaneous occupation of two trapping sites and the correlation of modes. The internal Ag2 bond-stretching modes are at 194 and 203 cm" 1 , respectively. The species with the 203 cm" i mode also has a broad in-band contribution at 40 cm ~x and a combination band at 234 cm" 1 . The other configuration has a localized mode at 57 cm" 1 with a combination band at 260 c m 1 (Bechthold et al., 1986). 400
15
(cm"1)
spectra seen by resonance Raman scattering on Ag2 trapped in Kr. A compilation of localized modes observed in semiconductors is given in the book by Srivastava (1990). 5.5.2 Finite Defect Concentration With increasing concentration the defects will affect the average behavior of the crystal. The more the local dynamics of the defect differs from that of the host lattice the greater the effects can be expected to be. If the constituents of the defect crystal differ only slightly one expects only a slight change of the phonon dispersion of the ideal lattice with a small broadening of the phonons. Such a situation can be described by average force constants (virtual crystal approximation). o) = ?,ciLi
(5-114)
i
where ct are the concentrations of the different constituents and L was defined in Eq. (5-110). Localized modes of single defects will merge to optic bands. An example is the optic phonon band in the metal hy-
drides, Fig. 5-23, which can be described by an additional atom in the unit cell and the virtual crystal approximation. The effects on the host phonons of defects having resonant modes can normally not be dealt with by such simple methods. A more accurate description is in terms of average Green's function methods which have been reviewed in detail by Elliot et al. (1974) and by Dederichs and Zeller (1980). We consider first a defect concentration sufficiently small that overlap effects between the defects are negligible. The total perturbation is then the sum of the perturbations of the single defects, and as in the case of a single defect the Green's function can again be calculated exactly in terms of a T-matrix, Eq. (5-113). This formal solution, however, is not very useful. Macroscopic properties such as phonon dispersion curves will not depend on the exact positions of all defects but only on their configurational or volume averages. We are therefore interested in an average Green's function,
334
5 Lattice Vibrations I
I
I
i
!
i
i
i
?
1
1
To-" 8 "!
o
25.0 o8
o
Q Q
P-NbD 0.75
LA |
6.0
77 5.0 x i—
?>
\
/
4.0
*
\
\
\
,LA
\V \
^ ^
\ 3.0
2.0
1.0
" ; // / / -/ / '
\
Hi
/T T A i //
\\\ \ \\\ \\\
/ / /) // i
w
III 1
r -
1/1
\/ ' 0
TA, r
N
i
i
0.2
i
i
1
0.6
1
1
0.8
1
0.6
1
i
0.4
i
V Y i
, ,1
0.2
toog]
trix, L, as for the lattice anharmonicity, Eq. (5-103). Elco)]- 1
(5-115)
with
(5-116) £ (co) =
X(co)~cZ
(5-117)
i
with Nd the number of defects. In general the configuration average will include averaging over equivalent defect orientations. For not too large perturbations of the phonons the matrix element of the self energy with a particular phonon determines the frequency shift and damping of that phonon, Eqs. (5-104) and (5-105) respectively. Especially large effects are ex-
0
i
0.2
i
i
0.4
Figure 5-23. Phonon dispersion curves of (3-NbD0 75 at room temperature. The symbols represent the experimental data. The full line is the result of a shell model calculation. The dashed line is the dispersion curve of pure Nb. To allow comparison with Fig. 5-21 the optic frequency has to be scaled by the mass ratio (MD/MH)1/2 = y/2 (Lottner et al., 1978).
pected when t(l) has contributions from resonant modes of the single defects. Whether or not a particular resonance couples to a particular phonon is determined by symmetry. For very low concentrations Eq. (5-115) will still have well defined phonon solutions which are somewhat deformed and damped near the resonance frequency. By comparing these results with the single defect Green's function it can be seen that the damping of the phonon is inversely proportional to the damping of the defect mode. For well defined defect levels the concept of a well defined phonon at the resonance frequency, therefore, breaks down even at low concentrations and a description in terms of hybridized phonon and resonance states is more adequate, Fig. 5-24. Both kinds of behavior have been observed in experiment. Figure 5-25 shows the hybridization of a phonon line in Al with the resonance level of substitutional Ag. Due to the large mass ratio (MAg/MAl « 4) and a small difference in
5.5 Imperfect Crystals and Disordered Solids
335
(a)
2Vc
Figure 5-24. Effects of resonance vibrations on the phonon dispersion curves, (a) Distorted phonon line, shifted by Aco (qj) and broadened by F (qj) (valid if 2 F (qj) <| r res ). (b) Hybridization of the ideal phonon line with the resonance mode resulting in a split of the dispersion curves (valid if 2 F (q,j) > r res ).
coupling strength, this is an example of a mass defect. By expanding the phonon wavevector for small q (long wavelengths) one can show that only resonance modes with even symmetry couple to long wavelength phonons to order q2, whereas "uneven" resonances couple only to order qA. Therefore only the "even" resonances contribute appreciably to the change of elastic constants. This so-called diaelastic change can be very large. For example, for the selfinterstitial in Cu the Eg resonant modes shown in Fig. 5-20 causes a softening of the c 44 shear module by Ac 44 « — 16 c 44 times defect concentration (Dederichs and Zeller, 1980). The "single T-matrix approximation" (STA), Eq. (5-117), is only valid in the low concentration limit. It also does not give a finite width to the spectrum of localized vibrations, as has to be expected from disorder. For diagonal (pure isotope) disorder one can improve the STA by an "average T-matrix" approximation (ATA) which uses the virtual crystal approximation as reference system and all atoms are "defects". This approximation is symmetric in the high and low concentration limits, c -> 0 and (1 — c) -• 0. A further improvement,
the self consistent "coherent potential approximation" (CPA), is gained by determining the reference lattice self-consistently, requiring that the average T-matrix becomes zero. In this approximation also the localized modes have finite widths. With off-diagonal (force constant) disorder these approximations are, due to the translational invariance condition, only possible with restrictions on the force constants, e.g., <£
336
5 Lattice Vibrations -i
counts
200
1
A!-77.Ag T-820K Transverse (00£)
100
1
1
1
r
0.6
0.7
0.8
u n NbD 0.85 T=220 K
—-v(THz) 1
0
1
l£C£l LA
200 100 0 200 100 0
1.0
4.0
2.0
200 100 0 100
1.0 s
50 0 1 1.0 t 200 -counts
2.0
3.0
100 0
3.0
1.0
-
[THz]
-
Al-7% Ag T=820K
40
0
0.1
02
0.3
(U
0.5
0 9
1.0
Figure 5-26. Equal intensity contours of P-NbD0 85 (Shapiro et al., 1981).
Q2
Q3 along (001)
Figure 5-25. Constant q scans and dispersion curves at T = 820 K for Al0 965 Ag 0 0 3 5 . Transverse phonons with q = (2n/a)(00Q (Zinken et al., 1977).
near the crossing of the resonant like H-vibration with the phonon line in NbD 0 85 . The broad distribution of intensities can be seen, and also how a flat phonon mode in the ordered structure evolves from the single defect resonant mode. A study of such diffuse intensities could provide information on partial ordering effects etc.
Apart from the diaelastic change discussed above there is often a paraelastic one which stems from reorientations of defects under strain. Since this latter effect needs a thermal activation it is strongly temperature dependent, contrary to the diaelastic one (Leibfried and Breuer, 1978; Robrock, 1990). 5.5.3 Extended Defects Besides point defects there are a number of extended defects in crystalline solids: dislocations, stacking faults, grain boundaries, precipitates etc. All these will scatter the host phonons. For long wavelength phonons their effects have been estimated from continuum models, e.g., Klemens
337
5.5 Imperfect Crystals and Disordered Solids
(1958). For phonons with shorter wavelengths the atomistic structures of these defects are important and the effects are strongly dependent on the material and details of the defect. A few such defects have been studied by computer simulations of model crystallites, e.g., Aharon and Brokman (1991). 5.5.4 Amorphous Materials and Glasses
Amorphous materials and structural glasses are solids with no long range order and no sharp range order. The structure can be visualized roughly as that of a frozen-in liquid. We will not distinguish between amorphous solids and glasses. Glasses are produced normally by quenching from the liquid state. Under heavy irradiation with neutrons a crystal will become amorphous. Many glassy properties show up before the amorphous state is reached (Laermans, 1987). Although the typical glassy properties are universal, they depend quantitatively on the material and even on its production history. Sound waves, the long wavelengths limit of the phonons, still exist in glasses. The sound velocities and the density of states are given by the elastic constants of the glass. With decreasing wavelength the sound waves will be increasingly scattered by the inhomogeneity of the glass on an atomic scale. Damping of phonons by disorder is always proportional to some power of co. When the wavelength approaches the atomic scale a description of the atomic vibrations in terms of phonons, taken here as meaning plane waves, is no longer sensible. The lifetime of such a phonon would be comparable to its inverse frequency. Nevertheless there are well defined atomic vibrations, but with rather complicated eigenvectors (structure factors). The density of state of these vibra-
tions is similar to their counterpart in crystals, Fig. 5-27. In particular the maxima present for the crystalline form are also found in the amorphous material. The maxima in the density of state of crystals stem from the zone boundary phonons, i.e., the short wavelength phonons. These phonons probe the short range order which is more or less preserved in the amorphous phase. By the same reasoning the optical phonons and intramolecular vibrations in crystals will still be discernable in the density of states of the amorphous state, e.g., Kamitakahara et al. (1984). Disorder will of course broaden all features. At low temperatures the properties of glasses differ significantly from those of crystals (Philips, 1981). Best known is the anomalous low temperature behavior of the lattice specific heat, Fig. 5-28. In crystals Cp oc T 3 and the proportionality constant is determined by the sound velocities (Sec. 5.3.1). In glasses there are additional contributions to the specific heat. Below T « 2 K the specific heat increases approximately linearly with T. At T % 2 K there is a crossover to a T5 dependence. The linear part in the specific heat is attributed to
60
80
fioo (meV)
Figure 5-27. Experimental density of states of amorphous Si (a). The density of states of the crystalline phase, calculated from a bond charge model is given for comparison (b) (Kamitakahara et al., 1984).
338
5 Lattice Vibrations 1
1 50
V 20
3-
10
CD
k
5
I—
Q O
H
2
-
Suprasil W
m
j Suprasil
o
Heralux
I
-
r*
c-7"
/
V \
- Debye
-
0.1
Spectrosil
•
1
-
1
0.5
*
A
\
~
I
i
I
0.2
0.5
/
I
C T
~
-
1
10
Figure 5-28. Specific heat (Debye contribution subtracted) over temperature cubed versus temperature in a log-log plot for different vitreous silica samples (Buchenau et al., 1991).
T (K)
two-level systems; certain groups of atoms can be envisaged as tunneling between two minimum configurations (Anderson et al., 1972; Philips, 1972). In some suitably chosen reaction coordinate this can be desribed in terms of double minimum potentials. The Cp oc T1 dependence is then the consequence of the distribution of the tunnel splittings and the asymmetries in this potential. The concentration of these tunneling centers in glasses is typically about 10 ~6 per atom. Buchenau et al. (1986) have shown that the anomaly in the specific heat above 2 K can be attributed to localized nearly harmonic soft vibrations, Fig. 5-29. A common description of the two-level systems and the soft vibrations was proposed in the soft potential model
1
U.UD
•
i
i
Neutrons Specific heat
0.0A
0.03
-
Debye /
0.02 / 0.01
if •7 /
/
/
^
/
/
' i
n
0
1
2 Frequency v (Thz)
i
3/,
Figure 5-29. Density of state g (v) in vitreous silica derived (full line) from the inelastic scattering intensities (vibrational modes) and (dotted line) from the heat capacity (Buchenau et al., 1986).
339
5.5 Imperfect Crystals and Disordered Solids
(Karpov et al., 1983). In this model two and one-well potentials are described, causing two-level systems and soft vibrations, respectively, by a common distribution. The model reproduces the results of the tunneling model for low temperatures. The T 5 dependence of the contribution of the soft modes to the specific heat is a consequence of a singularity of the distribution of the harmonic frequencies in that model. A potential with harmonic frequency = 0 (zero curvature in the minimum) is highly susceptible to perturbations which shift the minimum so that it has finite curvature. This "sea-gull" singularity is universal to glasses. A fit of the potential to the experimental data of three different glassy materials gives numbers of 20 to 80 atoms participating in the tunnelling and the soft vibrations (Buchenau et al., 1991). Similar numbers were found in a computer simulation of a model glass (Schober and Laird, 1991). There it was found that the soft vibrations occur around atoms whose local environment differs from the average one in the glass. The tunneling centers and soft vibrations are responsible for the sound attenuation at low temperatures in glasses (Hunklinger and Raychaudhuri, 1986). Calculations for glasses are mainly done by computer simulation, see Lewis and Klein (1990) for a review. Analytically the effective medium approximation, similar to the coherent potential approximation (CPA) but with a distribution of force constants, has been employed (Schirmacher and Wagener, 1989). Such a model of force constant disorder is expected to be useful to describe the bulk of the atomic vibrations, but not the soft local modes. A separate class of amorphous materials are the aerogels. Aerogels are highly porous materials whose density can be reduced by about two orders of magnitude from that of the "normal" amorphous one.
In the case of a broad distribution of pore sizes the solid is described in terms of a fractal structure. Theory then predicts for the density of vibrations, whose typical wavelengths are of the order of the pore sizes, a spectrum g (co) oc
.^-1
(5-118)
where d < 3 is the spectral dimension of the fractal conjectured to lie at 4/3 (Alexander and Orbach, 1972). The lattice vibrations in this frequency region are called fractons. For very low frequencies, when the wavelengths of the phonons are larger than the pore sizes, sound waves with a velocity appropriate to the reduced density should again be found. Recent experiments reviewed by Buchenau etal. (1992) have found a crossover from normal phonons at low frequencies to fractons, Fig. 5-30. Fracton models have also been used to explain the "normal" amorphous materials. Evidence to support a fracton concept in these materials is not convincing.
i
phonons
i
fractons
i
/
aerogel —/
\
/
10" -
I io- 3 vitreous / silica •—y /
10"4
i
10"3
i/
i
10 "2
i
10"1
1
frequency
v(THz)
Figure 5-30. Vibrational density of states of an aerogel sample compared to that of vitreous silica (Buchenau et al., 1992).
340
5 Lattice Vibrations
waves are called Rayleigh waves, after their discoverer. Their penetration depth is of the order of their wavelength. Their frequency is a linear function of their wavevector cos = cRq; they are acoustic waves. Waves with polarization perpendicular to the sagittal plane are not localized to the surface and are therefore termed surface skimming bulk transverse waves. By a slight variation of the properties these waves can also become surface waves. For a discussion of long wavelength surface waves on real surfaces see Maradudin (1987). On an atomistic scale the situation will be much more intricate. The atomic configuration near the surface can differ from the one in the bulk due to surface reconstruction, and the distance between planes of atoms can be different due to surface relaxations. In addition, all sorts of defects can be found on surfaces. To give just one example we show in Fig. 5-31 the dispersion curves for a clean surface of Cu (Zeppenfeld et al., 1988). The experimental points were obtained by He-scattering, whereas
5.5.5 Surfaces
All real crystals are bounded by surfaces. The study of these surfaces is of great technological interest and has attracted much attention. The knowledge about phonons on surfaces is a rapidly developing field. Atomic scattering techniques and electron energy loss spectroscopy are providing a wealth of experimental data. At the same time there is a rapid increase in theoretical results using both first principle methods and empirical models. Recent reviews are given in the book edited by Kress and De Wette (1991) and in Vol. 6 of the series edited by Horton and Maradudin (1990). The existence of waves on the surface of a solid was established by Lord Rayleigh in 1885. He solved the continuum equations of motion for an isotropic elastic medium with a free surface. In this model there are a pair of degenerate surface waves. These are acoustic waves with polarization in the sagittal plane, i.e., the plane spanned by the direction of propagation of the wave and the normal to the surface. These surface
0
0.2
0.4
0.6
0.8
1.0
0
0.2
0.4
0.6
08
10
Figure 5-31. Phonon dispersion curves of a 30-layer (110) slab crystal in [111] and [001] directions for a nearest neighbor central force constant model with modified constants near the surface. The thick lines show modes with considerable amplitudes of the topmost layer in the sagittal plane. The open circles are experimental data from He scattering experiments (Zeppenfeld et al., 1988).
341
5.6 Experimental Methods
the lines are gained from the dynamical matrix of a slab of 30 layers interacting via nearest neihbor central forces. The force constants were slightly modified near the surface. The modes denoted by S{ are surface modes, while M S denotes surface resonances, i.e., modes with enhanced amplitudes at the surface. Modes of the upmost layer in the sagittal plane with a strong amplitude are indicated by thick lines. From the dimension and the linear dispersion for small q it follows for the density of states of the surface waves gs (co) oc co for small co
(5-119)
As for the bulk, the low temperature thermodynamics can be described by a surface Debye model. The contribution of the surface to the specific heat is then Csa:ST2
(5-120)
ultrasonic waves, etc.). The energies are given by (5-121) / F 1 ^photon ^phonon
^ coj{q)<* / F
\ ^particle
\E \
h2k2
2m = hck
ultrasonic
where c stands for the velocity of light and sound, respectively and k = 2 n/1 is the length of the wave vector, with X being the wavelength. The matching of the energyto-momentum distribution of these particles with the one to be investigated, namely the phonon dispersion, is illustrated in Fig. 5-32. The cloud in the center of the plot represents roughly 90% of the typical energy-momentum range which is of interest for phonon studies. Important exceptions are optical phonons, which are often dispersionless and therefore can be
It is proportional to the surface area S and the square of the absolute temperature, as opposed to T 3 for the bulk. For large surface to volume ratios it can dominate at low temperatures.
5.6 Experimental Methods So far the properties of the collective excitations or phonons in solids have been treated in a system that is in equilibrium and in isolation from the "rest of the world". To see these vibrations, this splendid isolation has to be disturbed; phonons have to be excited or annihilated by their interaction with matter. Suitable probes are those particles whose energy and momentum distribution fit to that of the phonons (photons, electrons, neutrons and helium atoms). Also suitable are externally produced phonons (ultra sound shocks or
Figure 5-32. Energy-momentum distribution for various particle waves used as phonon probes. A typical velocity of externally produced ultrasound is 3 x 103 ms" 1 . Shown is the cloud of phonon dispersions, typical for solids, with two important extensions: dispersionless optical phonons and the long wavelength limit of acoustical phonons. The abbreviations for the spectroscopic methods are: FIR: far infrared, IR: infrared.
342
5 Lattice Vibrations
studied at small momentum transfer, and acoustic phonons in the long wavelength limit, where the dispersion reduces to Wj {q) = Cj (q/q) q with the sound velocity cj9 Eq. (5-87). All methods using light as the interacting "particle", such as far-infrared (FIR) and infrared spectroscopy (IR), Raman spectroscopy (RS), Brillouin scattering (BS) and X-ray scattering have the drawback that either momentum transfer or their energy transfer is several orders beyond the main interest of phonon spectroscopy. Despite this they do, however, often yield precise and complementary information. For instance optical modes in the low q region are better studied by infrared or Raman spectroscopy. Fig. 5-22 shows an example of Raman scattering by a localized mode. Brillouin scattering differs from Raman scattering only in that it focuses on lower energy transfers and therefore interacts with the acoustic phonons in the continuum regime. The observed initial slopes of the phonon dispersion then allow the determination of the elastic constants. Xrays, i.e., photons with energies of some tenth of a keV, have wavelengths comparable to those of phonons; however, energy transfers of some meV due to phonon scattering are hard to measure. Despite this large difference between incident energy and energy transfer, indirect information about phonons has been extracted from thermal diffuse X-ray scattering experiments (Colella and Batterman, 1970). Recently an energy resolution AE/E « 10" 6 with variable scattering vector Q was achieved by means of backscattering techniques, and allowed the direct measurements of phonons by X-ray spectroscopy (Dorner et al., 1989). Obviously the best method to study phonons is the use of particle waves with a particle mass of some nucleons, because
their dispersion falls into the relevant momentum-energy range. For thermal neutrons, helium and molecular beams like H 2 , D 2 , Ne, the de Broglie wavelength [resulting from the corresponding /c-value in Eq. (5-121)] matches typical phonon wavelengths in an energy range also comparable to that of the phonons. Due to their limited penetration in dense matter, helium and molecular beams are excellent probes for surface phonons. Slow neutrons have the unique advantage of penetrating dense matter - for example, only 1.4% of a thermal neutron beam (velocity: 2200 ms" 1 ) is absorbed when passing through a 1 cm thick Al plate - and inelastic neutron scattering (INS) is the workhorse for the exploration of phonons. There are a number of additional methods with specific applications. Ultrasonic measurements probe directly the sound velocity and yield the most precise data for elastic constants. Point contact spectroscopy measures the current-voltage characteristics of metal-insulator-metal tunnel junctions at liquid He temperatures. Phonons are excited via the electronphonon coupling and the second derivative of the voltage with respect of the current, d 2 F/d/ 2 , is proportional to g{co\ the phonon density of state, times a term reflecting electron-phonon coupling. Electron energy loss spectroscopy (EELS) applies the scheme of a classical scattering experiment. Free electrons with incident energies of « 10 eV are backscattered at the surface of a crystal and analyzed for their momentum and energy transfers (resolution of some meV). Since electrons penetrate at most a few layers of the crystal, information on surface excitations is thus obtained. Only three of these methods will be presented in more detail: inelastic neutron scattering, as the most versatile method in
5.6 Experimental Methods
studying phonons; X-ray backscattering spectroscopy, which will gain in importance with the increasing availability of synchrotron sources; and He scattering, as a very successful method for studying surface phonons. A fuller discussion of other methods can be found in Bruesch's book (1986). 5.6.1 Inelastic Neutron Scattering
Thermal neutron scattering is so successful in examining collective excitations in solids for a number of reasons. First of all, the momentum and energy transfer of thermal neutrons matches exactly the expected momentum and energy range of typical phonons. Neutron Brillouin scattering (Suck, 1991) reaches down to low momentum transfers o f 2 ~ 3 x l O ~ 2 A ~ 1 . At a momentum transfer in the order of a Brillouin zone dimension, energy transfers of as little as 1 jieV up to some 100 meV are measurable. Secondly, neutron scattering observes directly the particle correlation function in space and time. The word "directly" has to be taken with reservation insofar as the scattering experiment observes the space and time Fourier transform of the correlation. Thirdly, the neutron has no charge and easily penetrates several centimetres into most solids. Therefore, in general, surface effects can be neglected and true bulk properties are detected. All kinds of sophisticated sample environments can also be realized, such as high pressures (up to lOOkbar), low or high temperatures (JIK to some 103 K). Finally, the scattering cross-section of neutrons has no general trend with the different elements, and is large enough to enable the study of all elements, whereas X-ray scattering favours heavy atoms due to the Z 2 dependence of its cross-section. It is especially important that neutrons scatter at
343
the nuclei, i.e., large differences in the crosssection may occur for different isotopes of the same element. The strength of the neutron-nucleus interaction is described by the scattering length b, which in general is a complex quantity, where the imaginary part gives the absorption (not discussed here) and the real part the elastic scattering by the nucleus. It strongly varies between the isotopes and, if the spin of the nucleus is not zero, depends on whether the spin of the neutron and the nucleus are parallel or antiparallel. Even for a one-element sample, nuclei of different scattering lengths are randomly distributed over the scattering volume, and therefore the neutron sees a sample volume in which the scattering power varies from one point to another. Only that part of the scattering length of each individual scattering center which is common to all scatterers can contribute to interference effects of the incident neutron plane wave. This part is called the coherent scattering length b and corresponds to the average value of all b's. The incoherent scattering length, on the other hand, is the mean square deviation of the individual fo's
&inc = J\b-b\2
= Jb2 - (b)2
(5-122)
By definition the incoherently scattered part of the wavefunction contains no information on the correlation between different scattering centers. It is convenient to define cross-sections crcoh = 4 n (b)2 and crinc = 4 n bfnc. Within the wavelength range of slow neutrons coherent and incoherent scattering cross-sections are in the first approximation independent of wavelength. Absorption cross-sections, which account for the neutron capture are in general small and increase in the energy range of interest linearly with the neutron wavelength. Exceptions are isotopes which show a resonance absorption at low neutron energies,
344
5 Lattice Vibrations
with the famous example of Cd with a resonance at 175 meV. We conclude with a comparison to Xrays. For X-rays, the quantity analogous to the scattering length is the atomic scattering factor. Unlike the scattering length, it depends on the scattering vector or scattering angle because the dimension of the electron cloud which scatters the X-rays and the X-ray wavelength are comparable. Whereas b varies randomly from isotope to isotope and may even change its sign, the X-ray atomic scattering factor increases monotonically with the number of electrons in the atom. For practical purposes it is helpful to repeat some of the basic quantities of neutron waves. The energy of a neutron with wave vector k and neutron mass m = 9.396 • 108 eV/c2 is given by the de Broglie relation (5-121). It is sometimes useful to rewrite this relation in terms of neutron wavelength 81.8055
(5-123)
Neutrons in thermal equilibrium at room temperature (300 K) have E « 25 meV, X = 1.8 A or v = 2200 m/s and consequently neutrons coming from a thermal moderator will be peaked around that energy. Neutron spectroscopy has one crucial drawback: neutron sources are large and expensive experimental facilities and their flux is low. Nuclear reactors and spallation sources are the two varieties of neutron sources available nowadays. Whereas nuclear reactors provide a continuous neutron flux, spallation sources deliver neutron pulses given by the pulsed structure of the flux of the accelerators driving them. Since its inauguration in 1971 the high flux reactor (HFR) of the Institute LaueLangevin (ILL) at Grenoble, with approxi-
mately 1.2 xlO 1 5 thermal neutrons/cm2 s in its moderator, is the world's leading neutron source (Yellow Book, 1988). More recently spallation sources have gained in importance. ISIS, at the Rutherford Appleton Laboratory, England, with a design intensity of 4xlO 1 6 fast neutrons/cm2 s, a repetition frequency of 50 Hz and a pulse width varying from 1 to 100 (is is the leading facility. 5.6.1.1 Inelastic Scattering Cross-Section In a scattering experiment the incident and scattered neutron waves are characterized by the wave vectors k{ and &f, respectively. The change in momentum or scattering vector Q and transferred energy hco = Ex — Ef are then Q=
(5-124)
ki-k[
and (5-125) respectively (Fig. 5-33). We adopt here the convention of positive energy transfer for energy gain of the sample. For this reason Q also has the opposite sign as that defined
real space
reciprocal space Q
source
sample K
detector
Figure 5-33. Scattering vector in real and reciprocal space, left and right, respectively.
5.6 Experimental
345
Methods
for most diffraction experiments. The number of scattered neutrons in a given volume and energy element are measured and normalized by the incident neutron flux. The scattered intensity is then described by the double differential cross-section neutron counts scattered per second into volume element dQ |_ and energy range between Ef and £ f + d£ [incident neutron flux in energy range (Ef, £ f 4- dE)]
d2 a dQ d£ f
For inelastic coherent scattering and in harmonic approximation this reads (Lovesey, 1987) inel
dQdE(
coh
k{2nh (Q)]
(5-127)
— oo
*l')tQ
•e"i(
l
'(t)>
_
l]ei(Otdt
The prefactor contains a kinematic term (/cf//Ci). The function in the integral describes the structural [e"1^*1"*1')] and dynamical [
1 phonon
(5-130) coh
00
• f
and by Eqs. (5-11) and (5-27), describing the displacements in terms of a planar wave, the coherent cross-section becomes d2e
fcf 1
(5-126)
(5-128)
If we expand the exponential containing the displacement correlation function, the first nonzero term describes the onephonon scattering
• 2 S,6(6 ± q - i)\G(Q,q,j)\2 Fj(q,ta) qjj
where the structure factor (not to be mistaken with the Green's function defined earlier) unit cell
G(Q,qJ)= '
• eL
e
(5-131)
and the dynamical response function
-8[co±(Oj(q)]
(5-132)
The phonon occupation is given by n((o) and the upper and lower signs account for phonon annihilation and phonon creation by the scattered neutron wave. In the limit of T -* 0 only phonon creation is possible. The 8 functions guarantee momentum and energy conservation. Anharmonic and other effects limiting the phonon lifetime can be taken into account by introducing a damping constant in the response function, see Eq. (5-105). The 5 function is then in the lowest approximation replaced by a Lorentzian line shape. For details see Lovesey (1987).
346
5 Lattice Vibrations
5.6.1.2 Selection Rules and Different Brillouin Zones
The conservation of momentum and energy [8 functions in Eqs. (5-130) and (5-132)] and the scalar product Q • & {qj) define where in the 4-dimensional Q-oo space a phonon of a particular polarization can be observed. Consider a selected q value. According to the dispersion cDj{q), only discrete energies are possible; in the simplest case for a monoatomic lattice only two transverse (T) and one longitudinal (L) excitation are allowed. But those are only measurable where the scalar product Q ' & (
(110) reciprocal plane
In practice one tries to have Q and q as perpendicular as possible for transverse phonons in order to rule out contributions from other polarizations. Conditions to measure longitudinal phonons are less restrictive; Q and q have to be parallel because q || elong. For this case it is not necessary to know the full orientation of the single crystal with respect to the scattering plane. Because Q always starts at the origin, it is sufficient to have the direction of the desired longitudinal phonon within the scattering plane. Figure 5-34 shows the example of the longitudinal L[((0] branch accessible in the (110), (001) as well as in all other planes containing the <110> direction. Because polarization vectors are only well defined along main symmetry directions like [001], [110] and [111], coj{q) is often measured only along these propagation directions, and the dispersion in the full reciprocal space is deduced by a model description of cOj (q). Besides measuring the frequency of a given mode cOj(q), the polarization vector or the direction of atomic displacements of a mode can also be determined. For this purpose the intensity of a given phonon (Oj(q) has to be measured at different Qs, but always with the same propagation vector q. The components of the eigenvector e(qj) can then be evaluated by fitting the theoretical intensities - the structure factor
(001) reciprocal plane (220)
(T10 ) _
(130)
(040)
(020)
(130)
(110)
(220)
f / / t (000)
Figure 5-34. Reciprocal lattice planes for a b.c.c. lattice. Conditions for the observation of transversal (T) and longitudinal (L) phonons in directions of high symmetry are shown. {k{,kf: incident and final neutron wavevectors; Q: scattering vector; and q = Q — T: the phonon wave vector in the reduced zone scheme.)
5.6 Experimental Methods
(5-131) - to the observed intensities. In cases of polarization mixing this time-consuming procedure - phonon intensities are at least 3 orders of magnitude weaker than Bragg intensities - may be the unique way to determine polarization vectors unambiguously. 5.6.1.3 The Triple Axis Spectrometer (TAS)
In order to measure the double differential neutron cross-section a monochromatic neutron beam scattered from the (monocrystalline) sample has to be analyzed for its energy and momentum transfer. Directions are determined by the positions of sample and detector in real space (see Fig. 5-33); energies or wavelengths of the incident and scattered beam can be determined by two methods: (i) by defining the velocity of the neutron by time-of-flight technique (TOF) or (ii) by selecting a wavelength X making use of a Bragg reflection at a (different) single crystal nl = 2ds'm&
(5-133)
One notes that a direct determination of the neutron energy by the detector itself is not practicable because the detection process is always based on a nuclear reaction involving energies in the 104 — 106 eV range which are too large to separate phonon energies in the 10~ 3 eV regime. Triple axis spectrometers, which are based on the second method are the most versatile instruments to measure discrete phonons in a single crystal. The time-offlight technique is more suited to measuring integral quantities such as the phonon density of state or excitations in disordered systems. Since the first TAS was built by Brockhouse (1961) the basic principle has not
347
from reactor concrete monochromator changer, movable in translation and rotation
analyzer table
/ diaphragm / sample table detector
Figure 5-35. Schematic view of the thermal triple axis spectrometer IN 8 at the high flux reactor in Grenoble.
changed and is outlined in Fig. 5-35. Neutrons pass along a beam tube through the shielding of the source. A wavelength Xt is selected by a single crystal set at the Bragg angle 0m and hits the sample at 2 0m. Shielding material of at least 1 m thickness surrounds the zone of the primary spectrometer in order to suppress fast neutrons and y-radiation from the reactor. The incidenct monochromized neutron flux is monitored by a transparent counter in front of the sample. To facilitate orientation of the single crystal sample the latter is mounted on a goniometer table which allows rotation in the scattering plane and tilts in the two directions perpendicular to the rotation axis. Thus, the crystal can be oriented with an angle \j/ with respect to kt (see Fig. 5-33). The scattered neutrons arrive at an angle 0 in the secondary spectrometer. The secondary spectrometer analyses the energy by a suitable set-up of
348
5 Lattice Vibrations
a Bragg crystal set at the angle 0a with the detector at the angle 2 0 a. Nowadays, "black" 3 He detectors are used with an efficiency close to unity for a large wavelength band of slow neutrons. Horizontal collimators along the path of the neutron serve to control the energy and Q resolution. At the expense of an increase in divergence, neutrons can be focussed on the sample and/or the detector using bent monochromators and analyzers. Vertical focussing is used for most of the TAS monochromators. The efficiency of a TAS depends crucially on the proper choice of the monochromator and analyzer crystals. Sufficient intensities at the sample and in the detector demand crystals of high reflectivity, sufficiently large mosaic spread and negligible primary extinction effects (Freund and Forsyth, 1979). Increasing intensity is usually at the expense of resolution in energy and 0-space. The energy resolution depends on the Ad/d of the monochromator and on the beam divergence A0 which is the dominating term at small 0
fixed and the 4-dimensional Q-h co space is scanned in 2-dimensional slices at constant Q and variable hco, or vice versa. The choice of which of the two scan variables, Q or h co is kept fixed is made by resolution considerations. The resolution function is oval shaped and often inclined in the Q-h co space. Depending on how the dispersion cuts this resolution, ellipsoid intensity is focused or defocused into the detector. Nowadays simulation programs for the resolution function are part of standard TAS and should be consulted to optimize measuring strategies. Further details are given in Dorner's book (1982).
5.6.1.4 The Incoherent Inelastic Scattering Cross-Section
In analogy to the coherent case (Eq. (5-130)) the one-phonon incoherent crosssection for a Bravais lattice is given by d2a
1 phonon
h
(5-134) The most commonly used monochromator is pyrolytic graphite (PG) with a layered structure, high reflectivity, large mosaic spread and a particularly large lattice spacing c = 6.71 A. Copper is also frequently used since it can be produced with tailored mosaic spreads (Freund, 1975). The (111) reflection of germanium or silicon is sometimes used because the (222) reflection has a vanishing structure factor, and therefore 1/2 contaminations are completely suppressed. The number of parameters characterizing the TAS (ki9kf9
incoh unit cell
~ kx f
( in
M" (5-135)
•[n((oJ(q))+l/2+l/2]d(w±OJJ(q)) As in the coherent case energy conservation in gain (lower sign) and loss scattering (upper sign) is given by the 5 function, but no momentum conservation is associated with Eq. (5-135). Hence the incoherent scattering only gives information on the number of phonon modes as a function of energy, but not on the dispersion relation itself. In general it is not possible to estimate \Qe*(q,j)\2 and e[~2>rM(G)] for the different species and therefore their relative contributions cannot be separated.
5.6 Experimental
For Bravais lattices Eq. (5-135) simplifies to 1 phonon
dQdE{
K
i incoh
4
M (5436)
the (g-averaged spectrum, is a valuable quantity in determining the vibrational behavior. Such disordered systems are generally multicomponent systems, and in harmonic approximation Eq. (5-137) simplifies to
(5-138) 1 phonon
• [n(a>j(q)) + 1/2 + 1/2]8(© ±
The summation can be expressed in terms of the phonon density of states g (co% Eq. (5-48), and the scalar product \Qe(qJ)\2 is replaced by its average for a given energy. A measurement of the incoherent one-phonon scattering cross-section can therefore be used to determine directly the phonon density of states g (co) of a Bravais lattice d2a
dQdEi
1 phonon incoh
2M fc:
(5-137)
349
Methods
Q2
components
dQdE{ incoh CD
ji now stands for the different components of the amorphous system with their differing incoherent cross-sections. In the absence of translational symmetry q is no longer defined and has to be replaced by Q, and the summation over the scalar product \Qe(qJ)\2 is replaced by an average Q/3 and a weighted density of state Q^co) = Z\e?(q,j)\ 2 5(co ± co,)
(5-139)
j
The practical use of this equation is limited to a few cases. The incoherent cross-section has to be large in comparison to the coherent one to be sure that the incoherent scattering is really the dominating process in the observed scattering law. Systems containing hydrogen are very favourable because (jincoh/(Tcoh = 79.7/1.76 = 45.3. Also favorable are metals like V, Co, Ni, which all have a crincoh « 5 barn, but only V is of practical use because the competing crcoh « 0 barn. In principle it is possible to separate the incoherently scattered intensity by neutron spin analysis, but this is at the expense of neutron intensities and, therefore, rarely used. Incoherent inelastic scattering is of real importance in determining the vibrational spectra in strongly disordered systems like glasses. Translational symmetry is absent or at least strongly disturbed and the incoherent inelastic intensity distribution, i.e.,
for component fi. From the experimental point of view the problem remains that most amorphous systems are not incoherent but coherent scatterers. An approximation often made for these low energy excitations is then to average the coherent signal over a large range in scattering angle and to interpret it in terms of an incoherent scattering, i.e., Eq. (5-138) is applied (see Lovesey, 1987). A different, but also approximate approach, is to factorize the low frequency coherent scattering from a disordered system into energy- and (^-dependent parts (Buchenau, 1985; Dianoux, 1989). 5.6.1.5 Time-of-Flight Spectroscopy
Time-of-flight (TOF) spectrometers are most efficient for measuring the inelastic incoherent scattering from disordered systems. The energy of the scattered neutrons is here determined from the time-of-flight
350
5 Lattice Vibrations
of the neutron package for a known distance. This method can be used for both the primary and secondary spectrometer. Alternative TOF and Bragg scattering are combined in a spectrometer. In a TOF spectrometer detectors are spread over a large range in the scattering plane at fixed positions. Because neutrons are counted at a fixed scattering angle
hco/E.y/2cos(f)-hco/E{ = Q2/k2
(5-140)
which generates a family of curves of possible energy transfers as a function of Q for
a given incident energy E{. Figure 5-36 shows examples for cold, thermal and hot neutrons of E{ = 5, 25, 100 meV, respectively. Cold neutrons are used to study low energy excitations in the range of 10 ~2 to 10 meV but as Fig. 5-36 indicates this is possible only in energy loss because the incident neutrons are already too cold to create phonons. In the positive energy transfer region mainly thermal or hot neutrons have to be used at the expense of energy resolution. Usually neutrons are counted in the detector in constant time intervals, i.e., measuring d2o/(dQdt). Using the relation d2a dQdtm
d%
(5-141)
where d0 is the flight distance and (t — t0) the flight time; the measured intensity can be converted into an intensity proportional to the scattering law. 5.6.2 X-Ray Backscattering
As mentioned before, X-rays have a momentum transfer in the order of Brillouin zone dimensions but with an incident energy of some 10 keV, which makes it difficult to measure typical energy transfers due to phonons in the order of some meV. However, recently energy resolutions AE/E « 10 ~6 have been achieved using the backscattering technique (Dorner et al, 1989; Burkel, 1991). This technique makes use of the fact that - according to Eq. (5139) - for a scattering angle & = 90°, AE/E is solely determined by the crystal quality Figure 5-36. Loci of energy transfer h co and scatterAd/d, whatever the beam divergence A0 is. ing vector Q accessible to neutrons of incident enerWith an energy selection of AE/E « 10" 6 gies Et in the range of cold neutrons (5 meV), thermal such a backscattering spectrometer is best neutrons (25 meV) and hot neutrons (100 meV). Curves refer to different scattering angles <j> for the placed at a synchrotron source; stanindividual £•. dard X-ray tubes are too weak in intensity.
5.6 Experimental Methods curved and heated analyzer
351
crystal
double crystal premonochromator
synchrotron
-43 m-
Figure 5-37. Schematic view of the X-ray backscattering spectrometer INELAX at Hasylab in Hamburg. Diaphragms are used to reduce background radiation and to avoid broadening of the image (Dorner et al., 1989).
An outline of the scheme is presented in Fig. 5-37. A double crystal monochromator preselects a small wavelength band to reduce heat load on the spherically curved monochromator of dislocation free Si or Ge. The analyzer, which can be turned in the scattering plane, consists of a similar curved crystal focussing the analyzed beam into the detector. Geometrical constraints do not allow compliance with the backscattering condition. Large distances are chosen to make this deviation from backscattering as small as possible. Phonon groups are scanned by varying the analyzer temperature and scattering geometry. By changing the analyzer temperature, the lattice parameter and thereby the analyzed wavelength is changed. Whereas its energy resolution is still inferior to that in neutron spectroscopy, Xray backscattering may fill a gap in Q-hco space at high energy transfers and low Q. 5.6.3 Inelastic He Scattering
He atoms are the ideal probe to investigate surface phonons (Feuerbacher, 1980; Toennis, 1991). As for neutrons, momen-
tum transfers are in the range of the dimension of the Brillouin zone, and the energy of a Helium beam is in the order of surface phonon energies. Figure 5-38 shows the schematic set-up of a He time-of-flight spectrometer. A "free-jet beam" source ejects, through a system of nozzles from a pressure cell ( « 200 At), a highly monochromatic He beam with energy spreads in the range of AE/E = 0.7% - 2%. As the energy of the beam depends only on temperature, limits are given by the temperatures achievable. Next, the monochromatic beam passes a fast-turning chopper providing a pulse length in the jasec range. The beam at the sample surface is analyzed for its energy by time-of-flight, triggered by the chopper. To provide energy resolutions in the order of the primary spectrometer ( « 0.2 meV), a flight path in the order of 1 to 2 m is needed. Conventional mass spectrometers are used as detectors, allowing - in comparison with other spectroscopic methods - a large counting range. Intensities between 10 counts per second and 107 counts per second are routinely measured. It is apparent that low background counting
352
5 Lattice Vibrations
magnet monochromized He-beam chopper
ionizer
detector target
Figure 5-38. Left: schematic diagram illustrating the conservation of momentum for surface scattering. For a planar scattering geometry the projections of k{ and k{ on the surface are arranged in parallel. Right: layout of a He spectrometer.
and the need for very clean sample surfaces require excellent differential pumping systems. This and the large time-of-flight distance leads, for practical reasons, to system geometries with a fixed scattering angle of typically 90°. Access to different locations in reciprocal space is then possible by a tilt parallel to the y axis and by turning around the surface normal axis, z. Usually, a planar scattering geometry, where the scattered beam is in the plane formed by the incident beam and surface normal is chosen (see Fig. 5-38).
5.7 Outlook Although lattice dynamics is by now an old research subject, steady progress is expected over the coming years. The phonons have by now been measured for most simple crystals, at least in their room temperature phases. Theoretical calculations on the other hand, are lagging behind. Even for simple crystals the anharmonic prop-
erties are, in most cases, not known quantitatively. Anharmonicity is an essential ingredient for the understanding of phase transitions. Both theoretical and experimental data are needed. Increasing effort is directed towards more complicated structures. A typical example is the large number of investigations of vibrational properties of high Tc superconductors. There the phonons provide important information on structural properties and are, of course, a crucial ingredient in all explanations based on the electron-phonon interaction. Another class of substances currently at the centre of interest are amorphous solids and glasses. In the future also, phonon investigations will certainly play a major role in the study of "new" materials, e.g. fullerenes. The development of new techniques opens new avenues to the study of vibrational properties. In situ crystal growth techniques enable one to investigate new phases. Scattering of molecular beams has triggered an ever increasing number of investigations of surfaces. The new synchro-
5.8 References
tron sources open windows into hitherto inaccessible regions. An impression of present trends can be gained from the proceedings of the last phonon conferences, listed at the end of the references.
5.8
References
Aharon, S., Brokman, A. (1991), Acta Metall Mater. 39, 2489. Alexander, S., Orbach, J. (1972), J. Phys. (Paris) 43, 1. Anderson, P. W., Halperin, B. I., Varma, C. M. (1972), Philos. Mag. 25, 1. Barron, T. H. K., Klein, M. L. (1974), in: Dynamical Properties of Solids 1: Horton, G. K., Maradudin, A. A. (Eds.). Amsterdam: North Holland, p. 205. Bechthold, P. S., Kettler, U., Schober, H. R., Krasser, W. (1986), Z. Phys. D: At. Mol. Clusters 3, 263. Beck, H. (1975), in: Dynamical Properties of Solids 2: Horton, G. K., Maradudin, A. A. (Eds.). Amsterdam: North Holland, p. 205. Beg, M. M , Nielsen, M. (1976), Phys. Rev. B14, 4266. Bilz, H., Kess, W. (1979), Phonon Dispersion Relations in Insulators. Berlin: Springer Verlag. Born, M., Huang, K. (1954), Dynamical Theory of Crystal Lattices. Oxford: Clarendon Press. Brockhouse, B. N. (1961), in: Inelastic Scattering of Neutrons in Solids and Liquids. Vienna: IAEA, p. 113. Brovman, E. G., Kagan, Y. M. (1970), Sov. Phys.JETP 30, 883. Brovman, E. G., Kagan, Y. M. (1974), in: Dynamical Properties of Solids, Vol. 1: Horton, G. K., Maradudin, A. A. (Eds.). Amsterdam: North Holland, p. 191. Briiesch, P. (1982, 1986, 1987), Phonons: Theory and Experiments I, II, III. Berlin: Springer-Verlag. Buchenau, U. (1985), Z. Phys. B58, 181. Buchenau, U., Prager, M., Niicker, N., Dianoux, A. I, Ahmad, N., Phillips, W. A. (1986), Phys. Rev. B34, 5665. Buchenau, U., Galperin, Yu. M., Gurevich, V. L., Schober, H. R. (1991), Phys. Rev. B43, 5039. Buchenau, U., Monkenbusch, M., Reichenauer, G., Frick, B. (1992), J. Non-Cryst. Solids (in print). Buda, R, Chiarotti, G. L., Car, R., Parinello, M. (1989), Phys. Rev. Lett. 63, 294. Burkel, E. (1991), Inelastic Scattering of X-rays with very High Energy Resolution, Springer Tracts in Modern Physics, Vol. 125. Berlin: Springer. Butler, W. H., Smith, H. G., Wakabayashi, N. (1977), Phys. Rev. Lett. 39, 1004.
353
Car, R., Parinello, M. (1985), Phys. Rev. Lett. 55, 2471. Chesser, N. I, Axe, I D. (1974), Phys. Rev. B 9,4060. Cochran, W. (1959), Proc. R. Soc. London A 253, 260. Cochran, W, Cowley, R. A. (1962), J. Phys. Chem. Solids 23, 447. Cohen, M. L., Heine, V. (1970), in: Solid State Phys.: Seitz, F. D., Turnbull, D., Ehrenreich, H. (Eds.). 24, 38. Colella, R., Batterman, B. W. (1970), Phys. Rev. B 1, 3913. Cowley, E. R. (1983), Phys. Rev. B28, 3160. Cowley, E. R., Horton, G. K. (1987), Phys. Rev. Lett. 58, 789. Czachor, A. (1980), Phys. Rev. B21, 4458. Day, M. A., Hardy, J. R. (1985), /. Phys. Chem. Solids 46, 487. Dederichs, P. H., Zeller, R. (1980), Point Defects in Metals II, Springer Tracts in Modern Physics 87. Berlin: Springer. Devreese, J. T., Van Camp, P. E., Van Doren, V. E. (1985), in: Electronic Structure Dynamics and Quantum Structural Properties of Condensed Matter: Devreese, J. T., Van Camp, P. V. (Eds.). New York: Plenum. Dianoux, J. (1989), Philos. Mag. B59, 17. Dick, B. G., Overhauser, A. W. (1958), Phys. Rev. 112, 90. Dickey, J. M., Paskin, A. (1969), Phys. Rev. 188, 1407. Diehl, H. W, Leath, P. L., Kaplan, T. (1979), Phys. Rev. B 19, 5044. Dorner, B. (1982), Coherent Inelastic Neutron Scattering in Lattice Dynamics, Springer Tracts in Modern Physics 93. Berlin: Springer. Dorner, B., Burkel, E., Illini, T., Peisl, J. (1989), in: Phonons '89: Hunklinger, S., Ludwig, W, Weiss, G. (Eds.). Singapore: World Scientific, p. 1405. Elliot, R. I, Krumhansl, J. A., Leath, P. L. (1974), Rev. Mod. Phys. 46, 465. Feuerbacher, B. (1980), in: Vibrational Spectroscopy of Adsorbates, Springer Series in Chemical Physics 15. Berlin: Springer. Fischer, K., Bilz, H., Haberkorn, R., Weber, W (1972), Phys. Status Solidi (b) 54, 295. Freund, A. (1975), Nucl. Instruments & Methods 124, 93. Freund, A., Forsyth, J. B. (1979), in: Treatise on Materials Science and Technology, Vol. 15, Neutron Scattering: Kostorz, G. (Ed.). London: Academic Press, p. 461. Gilat, G., Raubenheimer, L. J. (1966), Phys. Rev. 144, 390. Gurevich, V. L. (1986), Transport in Phonon Systems. Amsterdam: North Holland. Haydock, R., Heine, V, Kelly, M. J. (1972), J. Phys. C5, 2845. Heiming, A., Petry, W, Trampenau, J., Alba, M., Herzig, C , Schober, H. R., Vogl, G. (1991), Phys. Rev. B43, 10948.
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Heiroth, M., Buchenau, U., Schober, H. R., Evers, J. (1986), Phys. Rev. B 34, 6681. Herring, C. (1954), Phys. Rev. 95, 954. Ho, K. M., Fu, C. L., Harmon, B. N. (1984), Phys. Rev. B29, 1575. Horner, H. (1974), in: Dynamical Properties of Solids 1: Horton, G. K., Maradudin, A. A. (Eds.). Amsterdam: North Holland, p. 451. Horton, G. K., Cowley, E. R. (1987), in: Physics of Phonons: Paskiewicz, T. (Ed.). Berlin: Springer, p. 50. Horton, G. K., Maradudin, A. A. (Eds.) (1974-91), Dynamical Properties of Solids, Vols. 1-6. Amsterdam: North Holland. Hui, J. C. K., Allen, N. R. (1975), /. Phys. C 8, 2923. Hunklinger, S., Raychaudhuri, A. K. (1986), in: Progress in Low Temperature Physics, Vol. IX: Brewer, D. F. (Ed.). Amsterdam: Elsevier, p. 265. Jepsen, O., Anderson, O. K. (1971), Solid State Common. 9, 1763. Kamitakahara, W. A., Shanks, H. R., McClelland, J. K, Buchenau, U., Gompf, R, Pintschovius, L. (1984), Phys. Rev. Lett. 52, 644. Karpov, V. G., Klinger, M. I., Ignat'ev, F. N. (1983), Sov. Phys.-JETP 57, 439. Keating, P. N. (1966), Phys. Rev. 145, 637 and 149, 674. Klemens, P. G. (1955), Proc. R. Soc. London A 68, 1113. Klemens, P. G. (1958), in: Solid State Physics: Seitz, R, Turnbull, D. (Eds.). New York: Academic Press. Krebs, K. (1964), Phys. Lett. 10, 12. Kress, W. (1983), in: Landolt-Bornstein, Vol. III/13b: Hellwege, K. H., Olsen, J. L. (Eds.). Berlin: Springer Verlag. Kress, W. (1987), Phonon Dispersion Curves up to 1985, Physics Data, Karlsruhe: Fachinformationszentrum Karlsruhe. Kress, W, de Wette, F. W. (Eds.) (1991), Surface Phonons. Berlin: Springer Verlag. Krumhansl, J. A., Mathews, J. A. D. (1968), Phys. Rev. 166, 856. Kunc, K. (1985), in: Electronic Structure, Dynamics and Quantum Structural Properties of Condensed Matter: Devreese, J. T., v. Camp, P. (Eds.). New York: Plenum, p. 227. Kunc, K., Gomes Dacosta, P. (1985), Phys. Rev. B32, 2010. Kunc, K., Martin, R. M. (1982), Phys. Rev. Lett. 48, 406. Kunc, K., Martin, R. M. (1983), in: Ab initio Calculation of Phonon Spectra: Devreese, J. T., v. Doren, V. E., v. Champ, P. E. (Eds.) New York: Plenum, p. 65. Kwok, P. C. K. (1967), in: Solid State Physics 20: Seitz, R, Turnbull, D., Ehrenreich, H. (Eds.). New York: Academic Press, p. 213. Laermans, C. (1987), Diffusion and Defect Data 5354, 451.
Lam, P.K., Cohen, M. L. (1982), Phys. Rev. B25, 6139. Larose, A., Brockhouse, B. N. (1976), Can. J. Phys. 54, 1990. Lehmann, G., Taut, M. (1972), Phys. Status Solidi (b) 54, 469. Leibfried, G., Breuer, N. (1978), Point Defects in Metals, Springer Tracts in Modern Physics 81. Berlin: Springer. Leigh, R. S., Szigetti, B., Tewary, V. K. (1971), Proc. R. Soc. London A 320, 505. Lewis, L. I, Klein, M. L. (1990), in: Dynamical Properties of Solids 6: Horton, G. K., Maradudin, A. A. (Eds.). Amsterdam: North Holland. Lottner, V, Kollmar, A., Springer, T, Kress, W., Bilz, H. (1978), in: Lattice Dynamics: Balkanski, M. (Ed.). Paris: Flammarion, p. 247. Lovesey, S. W. (1987), Theory of Neutron Scattering from Condensed Matter, 3rd ed. Oxford: Oxford Science Publications. Ludwig, W. (1967), Springer Tracts in Modern Physics 43, Berlin: Springer. Maradudin, A. A. (1987), in: Physics of Phonons: Paskiewicz, T. (Ed.), Lecture Notes in Physics, Vol. 285. Berlin: Springer, p. 82. Maradudin, A. A., Montroll, E. W, Weiss, G. H., Ipatova, I. P. (1971), Theory of Lattice Dynamics in the Harmonic Approximation, Solid State Physics, Supplement 3. New York: Academic Press. Martin, R. M. (1969), Phys. Rev. 186, 871. Miller, A. P., Brockhouse, B. N. (1971), Can. J. Phys. 49, 704. Natkaniec, I., Bokhenkov, E. L., Dorner, B., Kalus, J., Mackenzie, G. A., Pawley, G. S., Schmelzer, U., Sheka, E. F. (1980), J. Phys. C13, 4265. Nicklow, R. M., Gilat, G., Smith, H. G., Raubenheimer, L. J., Wilkinson, M. K. (1967), Phys. Rev. 164, 922. Nilsson, G., Rolandson, S. (1973), Phys. Rev. B 7, 2393. Pethick, C. J. (1970), Phys. Rev. B 2, 1789. Phillips, J. C. (1968), Phys. Rev. 166, 832. Phillips, W. A. (1972), /. Low Temp. Phys. 7, 351. Philips, W. A. (Ed.) (1981), Amorphous Solids, Low Temperature Properties. Berlin: Springer. Pines, D. (1963), Elementary Excitations in Solids. New York: Benjamin. Plakida, N. M. (1987), in: Physics of Phonons: Paskiewicz, T. (Ed.). Berlin: Springer. Pomeranchuk, I. (1942), /. Phys. (Moscow) 6, 237. Punc, G., Hafner, J. (1985), Z. Phys. B: Condens. Matter 61, 231. Rakel, H., Falter, C , Ludwig, W. (1988), /. Phys. F: Met. Phys. 18, 2181. Raunio, G., Rolandson, S. (1970), Phys. Rev. B2, 2098. Robrock, K.-H. (1990), Mechanical Relaxation, Springer Tracts in Modern Physics 118. Berlin: Springer.
5.8 References
Rodriguez, C O . , Kunc, K. (1988), /. Phys. C21, 5933. Schirmacher, W., Wagener, M. (1989), Spring. Proc. Phys. 57,231. Schober, H. R., Dederichs, P. H. (1981), in: LandoltBornstein, Vol. III/13a: Hellwege, K.-H., Olsen, J. L. (Eds.). Berlin: Springer Verlag. Schober, H. R., Laird, B. B. (1991), Phys. Rev. B 44, 6746.
Schroder, U. (1966), Solid State Commun. 4, 347. Shapiro, S. M., Richter, D., Noda, Y., Birnbaum, H. (1981), Phys. Rev. B23, 1594. Shukla, R. C , Mountain, R. D. (1982), Phys. Rev. B25, 3649. Shukla, R. C , Shanes, F. (1985), Phys. Rev. B31, 372. Soma, T. (1978), J. Phys. Soc. Jpn. 44, 469. Srivastava, G. P. (1990), The Physics of Phonons. Bristol: Adam Hilger. Suck, J.-B. (1991), J. Phys.: Condens. Matter 3, F73. Stedman, R.. Almquist, L., Nilsson, G. (1967), Phys. Rev. 162, 549. Toennis, J. P. (1991), in: Surface Phonons: Kress, W, de Wette, F. W. (Eds.). Berlin: Springer, p. 111. Vaks, V. G., Kravchuk, S. P., Trefilov, A. V. (1980), /. Phys. F: Met. Phys. 10, 2105. Varma, C. M., Weber, W. (1977), Phys. Rev. Lett. 39, 1094. Verdan, G., Rubin, R., Kley, W (1968), in: Neutron Inelastic Scattering, Vol. 1. Vienna: IAEA, p. 223. Wang, Y. R., Overhauser, A. W. (1987), Phys. Rev. B 35, 497. Weber, W. (1974), Phys. Rev. Lett. 33, 371. Weber, W. (1977), Phys. Rev. B 15, 4789. Phys. Rev. Lett. 29, 373. Weber, W. (1984), in: Electronic Structure of Complex Systems: Phariseau, P., Temmerman, W. (Eds.). New York: Plenum. Weber, W, Bilz, H., Schroder, U. (1972), Phys. Rev. Lett. 29, 373. Weinstein, B. A., Piermarini, G. J. (1975), Phys. Rev. B12, 1172. Werthamer, N. R. (1969), Phys. Rev. B 1, 572. Willaime, F, Massobrio, C. (1991), Phys. Rev. B43, 11653.
355
Yellow book, Neutron Research Facilities at the ILL High Flux Reactor (1988): Blank, H., Maier, B. (Eds.). Grenoble: ILL (available on request at the ILL). Yin, M. T., Cohen, M. L. (1982), Phys. Rev. B26, 3259. Zeppenfeld, P., Kern, K., David, R., Kuhnke, K., Comsa, G. (1988), Phys. Rev. 38, 12329. Zinken, A., Buchenau, U., Fenzl, H. I , Schober, H. R. (1977), Solid State Commun. 22, 693.
General Reading Bottger, H. (1983), Principles of the Theory of Lattice Dynamics. Berlin: Akademie-Verlag. Born, M., Huang, K. (1954), Dynamical Theory of Crystal Lattices. Oxford: Clarendon Press. Bruesch, P. (1982, 1986, 1987), Phonons: Theory and Experiments I, II, HI. Berlin: Springer Verlag. Horton, G. K., Maradudin, A. A. (Eds.) (1974-91), Dynamical Properties of Solids, Vols. 1-6. Amsterdam: North Holland. Kress, W, de Wette, F. W (Eds.) (1991), Surface Phonons. Berlin: Springer Verlag. Maradudin, A. A., Montroll, E. W, Weiss, G. H., Ipatova, I. P. (1971), Theory of Lattice Dynamics in the Harmonic Approximation, Solid State Physics, Supplement 3. New York: Academic Press. Srivastava, G. P. (1990), The Physics of Phonons. Bristol: Adam Hilger.
Conference Proceedings Hunklinger, S., Ludwig, W, Weiss, G. (Eds.) (1990), Phonons '89. Singapore: World Scientific. Kollar, X, Kroo, N., Menyhard, N., Siklos, T. (Eds.) (1985), Phonon Physics. Singapore: World Scientific.
6 Point Defects in Crystals Heinrich J. Wollenberger Hahn-Meitner-Institut Berlin GmbH, Berlin, Federal Republic of Germany
List of 6.1 6.2 6.3 6.3.1 6.3.2 6.4 6.4.1 6.4.1.1 6.4.1.2 6.4.1.3 6.4.1.4 6.4.1.5 6.4.2 6.4.2.1 6.4.2.2 6.4.2.3 6.4.2.4 6.4.2.5 6.4.3 6.4.4 6.4.5 6.5 6.5.1 6.5.1.1 6.5.1.2 6.5.1.3 6.5.1.4 6.5.1.5 6.5.2 6.5.2.1 6.5.2.2 6.5.2.3 6.5.2.4 6.6 6.7
Symbols and Abbreviations Introduction Thermodynamics Diffusion-Controlled Reaction Kinetics The Rate Equation Approach Diffusion of Charged Defects Point Defects in Metals Vacancies Enthalpy and Entropy of Formation Vacancy Diffusion Structure Agglomerates Interaction with Other Defects - Vacancies in Solid Solutions Self-Interstitials Formation Structure of Self-Interstitials Dynamic Properties of Self-Interstitials Agglomerates of Self-Interstitials Interaction of Self-Interstitials with Solutes Self-Organization of Defect Agglomerates Atomic Redistribution by Persistent Defect Fluxes Defect Features for Ordered Alloys Point Defects in Ionic Crystals Halides General Remarks The F-Center Impurity-Related F-Centers Photolytic Damage Impurities Oxides General Remarks Alkaline-Earth Oxides Transition-Metal Oxides Other Oxides Acknowledgements References
Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.
358 360 361 364 364 365 365 365 365 368 370 371 371 373 373 379 382 385 387 390 392 393 395 395 395 395 400 401 403 403 403 404 405 406 407 407
358
6 Point Defects in Crystals
List of Symbols and Abbreviations a0, a b0 C, C c c a ,c b cs c° c i? c v c 2v c vp D D\,DB D\, DB Do E Eg,AH,A2u /r g2y G Gflv G\y HO,HT,HC AH^sa AH*,AH% AHmd AH{ Af/^v AH[p AH™ ^A,B,C,D,E
Ji9Jy K kB kiw /cis,/cvs L M, m n p(T)
lattice parameter jump distance of the vacancy elastic moduli atomic defect concentration atomic concentrations of reacting defects atomic concentration of point defect sinks vacancy concentration in thermal equilibrium atomic concentration of interstitials and vacancies, respectively atomic concentrations of divacancies atomic concentration of vacancy pairs in ionic crystals diffusion coefficient diffusion coefficient of alloy constituents A and B due to vacancy diffusion, respectively diffusion coefficients of alloy constituents A and B due to interstitial diffusion, respectively temperature independent prefactor of the diffusion coefficient incident energy of irradiating particles, transition energy of an F-center lattice vibration modes resonance frequency geometry factor Gibbs free energy Gibbs free energy change for vacancy formation Gibbs free energy change for divacancy binding interstitial configurations in the f.c.c. lattice change of enthalpy for vacancy solute atom binding change of enthalpy for vacancy formation in the a and (3 sublattices, respectively change of enthalpy for mixed dumbbell motion change of enthalpy for vacancy formation change of enthalpy for divacancy binding change of enthalpy for vacancy pair formation change of enthalpy for vacancy migration resistivity recovery stages fluxes of interstitials a n d vacancies, respectively production rate of freely migrating defects Boltzmann constant rate constant for the recombination of vacancies a n d interstitials rate constant for annihilation of vacancies a n d interstitials, respectively, at sinks thickness of a foil sample masses of target nucleus and neutron, respectively defect density in ionic crystals displacement probability
List of Symbols and Abbreviations
359
q rah Ro r0 AS f ASl t T Td ^dam T max Tq AV™\AVJcl AF2rf AFvf
electrical charge reaction radius governing the reaction between defects a and b equilibrium atomic distance shift of interatomic potential to simulate solute-solvent interaction change of entropy for defect formation change of entropy for vacancy formation time absolute temperature, recoil energy threshold energy for atomic displacement recoil energy diminished by the electronic energy loss m a x i m u m transferred energy quenching temperature relaxation volume of interstitials and vacancies, respectively relaxation volume of divacancies volume of vacancy formation
x X Q¥ @v Agq o G0 od
elastic modulus wavelength of concentration fluctuations resistivity contribution per unit of atomic concentration of Frenkel defects resistivity contribution per unit of atomic concentration of vacancies quenched-in resistivity increment electrical conductivity temperature independent prefactor of the electrical conductivity displacement cross section time-integrated flux density of irradiating particles atomic volume atomic vibration frequency
b.c.c. DD DXS ENDOR EPR ESR f.c.c. h.c.p. KKR nnd PAS TDPAC
body-centered cubic differential dilatometry diffuse X-ray scattering electron nuclear double resonance electron paramagnetic resonance electron spin resonance face-centered cubic hexagonal close-packed Korringa-Kohn-Rostocker nearest neighbor distance positron annihilation spectroscopy time differentiated perturbed angular correlation
360
6 Point Defects in Crystals
6.1 Introduction The perfect lattice structure of crystals is a theoretical conception. Nature prefers imperfect lattices. Frenkel (1926) first pointed out that, at any finite temperature, the atomic arrangement of a crystal will not be that of a perfect lattice. The mixing entropy gained by the large numer of possible configurations of the defective crystal will always ensure a benefit of the free energy, however high the formation energy per defect might be. This benefit is largest for the point defects with their zero-dimensionality. Less entropy is gained by the formation of linearly extended dislocations or planar interfaces. Lattice imperfections (distortions) around point defects are confined to a few atomic shells. Important physical properties of materials often result from properties of the imperfections or defects. Examples are the mechanical properties of steels, the electronic properties of impurity doped semiconductors and the optical properties of solid state lasers. Point defects may be of intrinsic nature like vacancies and self-interstitials and extrinsic like impurity atoms. Self-interstitial atoms situated at the interstice among regular lattice sites cause, in general, lattice distortions which are significantly larger than those arising from vacancies and substitutional impurities. While small impurities are found in the octahedral or tetrahedral gap, self-interstitials tend to form dumbbell-like configurations or crowdions. In the first case, two atoms share one lattice site while in the second one an additional atom is squeezed into a close-packed lattice row. For metals the vacancy is the natural point defect. Vacancies migrating by site exchange with nearest neighbor atoms provide atomic transport. Because of this role, the migrational properties of the vacancy
have certainly attracted more attention than any other vacancy property. The selfinterstitial is only found in crystals irradiated by energetic particles. It may be characterized by its extremely large relaxation volume and exciting dynamic properties. Properties of the self-interstitial determine significantly the damage structure of metallic materials after ion implantation (for purposes of property improvement, e.g. surface hardening) or nuclear reactor irradiation. For semiconductors and insulators electrical neutrality must be obeyed even for the imperfect crystal. The charge of the missing atom forming the vacancy must be compensated by charge redistribution. Ionic crystals are characterized by their Coulombic binding forces. They are very good electrical insulators with a wide energy gap (9-12 eV for the alkali halides). Point defects in ionic crystals frequently show several different states of charge. At the same time, they cause significant lattice relaxation accompanied by strong electron-lattice interaction. The charged defects may efficiently trap electrons or holes, thus controlling the lifetime of free carriers. They also act as scattering centers, thus controlling the mobility of the carriers. The energy gap of about 10 eV allows for electric transitions between the various levels of a defect center which give rise to absorption and luminescence bands within the visible optical spectrum. The defects involved are called color centers. Because of the very high electrical resistivity of the ionic crystals, magnetic resonance techniques have fruitfully been applied for studies of the defect structure. Nevertheless, quite a number of questions on structural details for the more complex defects remain unresolved. A great number of complex defects are observed in the oxide materials. The complexity arises from the fact that for these materials ionic as well as covalent bonding
6.2 Thermodynamics
is responsible for the lattice structure with more influence of the one or the other for the different species of oxides. Correspondingly, the properties of the defect are determined more by ionic or covalent bonding features in one or the other case. Despite of the complexity, quantitative knowledge about the properties of such defects is urgently needed as the application of oxide materials cover a broad and quite differentiated field of modern technology. Examples are materials for lasers and integrated optics, for high Tc superconductors and solid electrolytes of fuel cells. In most of the cases, point defects or defect agglomerates determine the specific properties used for application. In the present chapter the fundamental properties of vacancies and self-interstitials in metals are described in detail. The properties of these defects are less complex than those of defects in nonmetallic crystals. Quantitative understanding of their properties may be regarded as being almost complete. A similar detailed description of the wealth of different point defects in ionic crystals would exceed the framework of this chapter. Instead, the F-center and the related impurity centers in alkali halides are described to some detail in order to provide a guideline for understanding of the many other defects occurring in ionic crystals. For the oxides only general features of the point defect properties are described. Here, the detailed knowledge seems to be less complete than that for the defects of alkali halides. The point defects in semiconductors are not treated in this chapter. One whole chapter in Vol. 4 of this series (Watkins, Chap. 3) is devoted to this topic. Moreover, another chapter in that volume (Gosele and Tan, Chap. 5) is devoted to the diffusional properties of the point defects in semiconductors.
361
6.2 Thermodynamics The concepts of equilibrium thermodynamics explain the occurrence of intrinsic point defects by first principles. The Gibbs free energy change SG caused by changing the atomic defect concentration c by 5c at the temperature T reads for c < 1: 8G = (AHf - TASf + kB Tine)8c
(6-1)
where AH{ is the activation enthalpy for defect formation, kB In c is the ideal entropy of mixing and ASf is the excess entropy caused by the defect formation. The change of entropy ASf is dominated by the contribution arising from the perturbation of the phonon spectrum. The harmonic approximation for high temperatures yields for vacancies CO:
(6-2)
Oj H-
where coi are the atomic eigenfrequencies and 5 ^ the changes caused by the vacancy. The value of AS/ may be estimated by using the most simple model. The nearest neighbor interaction is replaced by spring forces with the force constant /. We calculate the change of the Einstein eigenfrequencies of the vacancy's nearest neighbor atoms. For each of the nearest neighbors in the f.c.c. lattice one of the twelve coupling springs is removed. As a result the frequencies are changed from co| = 4 f/M to (coE + 5coE)2 = 3 f/M parallel to the removed springs. The frequencies perpendicular to these remain unchanged. Hence, we obtain
Improvements of the calculation must take into account the static atomic relaxations around the vacancy (Dederichs and Zeller,
362
6 Point Defects in Crystals
1980). These change the force constants for many more atoms than the nearest neighbors. The influence was followed up to 19 atomic shells around the vacancy for Cu (Hatcher et al., 1979). Depending on the used potentials, values between 2.3 kB and 1.6 kB were obtained for AS/. The higher value corresponds to a cut-off Morse potential which, when used in computer modelling, gives reasonable agreement with experimental data for lattice constants, bulk modulus, enthalpies for vacancy formation and migration. The uncertainty related to the specific potential used for the calculation of vacancy properties becomes quite obvious for calculations of the relaxation volume AFvrel. It is the volume change of the crystal caused by removing one atom from the interior and withdrawing it from the crystal. The quantity is measured by the change of the lattice constant. With the two potentials as used for calculation of the above quoted two AS/ values, the AVyrel values -0.02(3 and -0.47(2 (Q: atomic volume) were obtained, respectively (Dederichs et al., 1978). The experimental volume is —0.2(2 (see Sec. 6.4.1.3). The two potentials yield AH{ to be + 1.29 eV and — 0.41 eV, respectively. Only the former fits reasonably well to experimental data. For ab initio calculations of vacancy properties the treatment of the electronic contributions provides considerable difficulties (Heald, 1977; Stott, 1978). More promising than the various approximations applied in the past seems to be the combination of self-consistent cluster calculations (quantum-chemical approach) with lattice defect calculations (for example, see Adams and Foiles, 1990). This method fails when perturbations of electron states govern the defect properties as it holds true for insulators and semiconductors. An equally good consideration of
lattice distortions and electron redistributions seems to be a very complex problem. Sometimes the opposite approach is adopted, i.e. neglect of the lattice distortion. Despite the serious drawbacks, the electron redistribution among electron levels in semiconductors has commonly been calculated this way (Catlow et al., 1980). For ionic crystals, however, both contributions are equally important in most of the cases. The concentrations of vacancies observed in experiments may considerably differ from those valid for thermodynamic equilibrium. The quenching experiments (Sec. 6.4.1.1) make intentionally use of vacancy concentrations being significantly enhanced above equilibrium concentrations at the temperature of measurement. At low temperatures, the sluggish vacancy migration may prevent the system from reaching thermodynamic equilibrium. At temperatures near the melting point, however, equilibrium is attained easily. Thermodynamics is, indeed, an appropriate tool for describing point defect properties and reactions as long as such processes occurring far from equilibrium may be excluded. Examples of the latter are treated in Sees. 6.4.3 and 6.4.4. From Eq. (6-1) one derives the equilibrium concentration of vacancies for 8G = 0: AS/ AHA 0 (6-3) exp c" = exp kBTj Before discussing methods of deriving enthalpy and entropy of vacancy formation from measurements of c®(T) according to Eq. (6-3) we consider the formation of vacancy agglomerates. For conditions of equilibrium, higher aggregates than the divacancy may be neglected. In Eq. (6-3) we have to replace c° by c°v + 2 c%y, where the indices 1 v and 2 v stand for single and divancancies, respectively. In thermal equi-
363
6.2 Thermodynamics
librium the divacancy concentration c\y is related to c°u by , „ C
2v = Qiv(civ)2 e x P I
H^~ I e x P
where AHb2v = 2 AH{V - AH{, is the socalled binding enthalpy, AS\y the corresponding binding entropy and g2y a geometry factor which takes a value of 6 for the two vacancies at nearest neighbor sites in the f.c.c. structure. The experimentally derived enthalpy for vacancy formation would then represent A
f
AH3 =
ion vacancies as shown in Fig. 6-1. For most cases it is energetically favorable to have adequate numbers of both species to ensure electrostatical neutrality of the crystal. For AB type crystals pairs of cation and anion vacancies assure electrostatical neutrality of the crystal on a local scale. The ideal entropy of mixing then contains the squared number of possible arrangements or c^p with cvp the atomic concentration of vacancy pairs [for comparison see Eq. (6-1)]. Correspondingly,
3 In cv°(T) A r r f ^ (AH{V - A ^ v ) e x p [ ( G ^ v - G Q / ( / c B r ) ] = AH lvi + 2 a-> 8[l/(/cBT)] ^ 2v In ^ 2v exp[(G> v -G{ v )/(/c B T)]
where G\y is the Gibbs free binding enthalpy defined corresponding to AH\V. The Eq. (6-5) introduces a temperature dependence of AH{ which must be considered whenever a deviation from the Arrhenius behavior is determined for an extended temperature range. For b.c.c. and h.c.p. structures different divacancy configurations may exist simultaneously. Then Eq. (6-4) will become more complicated (Seeger, 1973). New theoretical calculations yield binding enthalpies for the divacancy in Cu, Ni, Ag and Pd of 0.08, 0.07, 0.08 and 0.11 eV, respectively (Klemradt etal., 1991). Such values would not allow for measurable deviations from the Arrhenius behavior. For metals the formation enthalpy of self-interstitials is generally larger than that of vacancies by a factor of about 3 to 5, and the diffusivity is much larger than that of vacancies. Therefore, the concentrations of interstitials are almost always negligibly small for conditions of thermal equilibrium. Their occurrence in irradiated crystals always imply a reaction system far from equilibrium (Sees. 6.4.3 and 6.4.4). In ionic crystals vacancies may occur as positive ion vacancies as well as negative
(6-5)
the equilibrium concentration is given by c v p *exp[-AH v y(2/c B T)], with AH{p the formation enthalpy of the pair. The latter amounts to 1.3-2.7 eV for the different alkali halides. In thermodynamic equilibrium cation or anion vacancies are formed as Schottky defects in alkali halides and as Frenkel defects (Fig. 6-1) in silver halides. Anion vacancies may simply be produced by heating an alkali halide within excess va-
©
o © Vacancy
pair
o ©
G 1 ©o 0 ©
©
o
nO o ©
©,
• ©
©
o ©
p
Schottky defect
Frenkel defect
o
o © o ©o © Figure 6-1. Fundamental point defects in the rocksalt structure. The Frenkel defect is shown as a cation defect. Anion defects occur as well.
364
6 Point Defects in Crystals
por of the alkali metal. Cation vacancies are obtained when divalent cations (Ca + + , for example) are introduced into an ionic crystal containing monovalent cations (KC1, for example).
6.3 Diffusion-Controlled Reaction Kinetics 6.3.1 The Rate Equation Approach
Migrating defects react with each other and with localized defects. This reaction system is currently described by means of the chemical rate equation approach. The term chemical refers to spatially homogeneous reactions for which the reaction rate is assumed to only depend on the concentrations Cj of the reactants (law of mass action) and not on the local distributions Cj (r). In solid state reactions the rate-determining step for the reaction between defect species a and b is the transport of at least one species to the other. It has been shown that for a diffusion-limited reaction the reaction rate is proportional to ca cb as long as the mean initial distance between a and b exceeds, at least by a few times the reaction radius rab for which distance the reaction occurs. The size of the reaction radius has been calculated for lattice diffusion, for two- and three-dimensional reactants and various topologies (e.g., edge dislocations, stacking fault tetrahedra or spherical voids as sinks for the mobile point defect) and for the influence of long-range interaction potentials (review by Schroeder, 1980). Systems in which the point defects interact with an extended localized inexhaustable sink like an edge dislocation are spatially inhomogeneous. Such a system is characterized by a net flow J of mobile defects towards the closest sink sites. The divergence of J = VD Vc is equivalent to a
reaction rate like rab D ca ch/Q, where D is the diffusion coefficient of the mobile defect and Q the atomic volume. Chemical kinetics would require Vc = 0. This condition will be fulfilled for a foil sample of thickness L with its surfaces acting as sinks if ca < bl/L2, where ca is the concentration of the migrating defect a and b0 the elementary jump distance. Then, the annihilation rate will be proportional to ca cs with cs, the sink density given by cs &n2 Q/(4nrsaL2), where rsa is the reaction radius for the annihilation of a at the sink. Reaction systems involving dislocation loops as point defect sinks are often treated accordingly. For ca > bl/L2 locally valid rate equations are commonly used. For an irradiated metallic crystal in which freely migrating vacancies and interstitials are produced with the rate K the equations would read 9c - ^ - K - kiy ^ cv - fcvscvcs
•Vcv) (6-6)
8c, ^ = K - kivc{cv - kisc{cs + V(D{ Vq) dt
The second term on the right hand side describes the recombination of vacancies and interstitials, the third term the annihilation of vacancies or interstitials at sinks, and the last term the divergence rate. This term makes the otherwise nonlinear differential equations nonlinear partial ones with spatially dependent concentrations. Solution of them requires specified boundary conditions in addition to the initial local concentrations [Cj(r)]r=0 of the mobile species j (interstitials or vacancies). This type of rate equation system is frequently used for modelling the defect reactions in ion beam irradiated thin film samples. Here, the sample surface acts as a significant extended sink s. The rate equation systems rapidly increase when reactions like cluster forma-
6.4 Point Defects in Metals
tion of the migrating defects and trapping at localized traps are included (review by Beeler, 1983). In metals clustering of vacancies and interstitials as well as trapping of them at solutes must not be neglected for most of the irradiation conditions.
6.3.2 Diffusion of Charged Defects
In ionic crystals mass transport is, in general, coupled with charge transport. In an electric field the charged defects perform a drift diffusion and because of the NernstEinstein relationship o/D = n q2/{kB T) with o being the conductivity, D the diffusion coefficient, n the density of charged defects and q the charge of them, diffusion coefficients may be determined by means of electrical conductivity measurements. In alkali halides cation vacancies migrate faster than anion vacancies. They therefore determine the electrical conductivity of the material. Ionic transport has attracted great attention for the solid electrolytes (superionic conductors). Such materials are not close-packed but contain in their structure passage-ways which allow a rapid transport of ions. Such passage-ways are formed by connecting void sites of a given ion sublattice. Since for different substances completely different sublattices may provide such tunnels, both, cationic and anionic superionic conductors have been found. The former are given by betaalumina compounds while the latter by fluorite-like oxides. Diffusivity and conductivity are characterized by high absolute values and small temperature dependence. The temperature independent prefactors Do and a0 are small. Successful interpretation of the phenomenon seems to be possible on the base of order-disorder transformations in the conducting sublattice (Lechner, 1983).
365
6.4 Point Defects in Metals 6.4.1 Vacancies 6.4.1.1 Enthalpy and Entropy of Formation
In metals vacancies are formed by thermal activation to concentrations which are measurable with satisfying accuracy. The classic method of measurement is the differential dilatometry (DD) as proposed by Wagner and Beyer (1936) and successfully applied to Al, Cu, Ag and Au by Simmons and Balluffi (I960,1963). The method measures the difference between the relative change of the macroscopic sample volume represented by the length change AL/L and that of the microscopic sample volume as represented by the lattice constant Aa/a upon the introduction of vacancies of the concentration cv. As the missing atoms in the interior have to be added to the surface of the sample the difference 3 (AL/L — Aa/a) measures the relative volume change by these added atomic volumes, i.e., the atomic concentration of vacancies is given by 4
(6-7)
In order to obtain enthalpy and entropy of formation according to Eq. (6-3) c° has to be determined according to Eq. (6-7) as function of temperature T. Near the melting point c° was found to be around 10" 4 . This small difference between the relative changes of the sample length and the lattice constant must be measured according to Eq. (6-7) on top of a large background of AL/L and Aa/a. For Al, for example, AL/L changes by about 10" 2 when going from 400 °C to the melting point. These figures indicate the extreme efforts required in order to attain the necessary sensitivity and precision of measurement. In addition, one realizes that for an enthalpy of formation of around 1 eV the vacancy concentration
366
6 Point Defects in Crystals
already falls below the detection limit of the DD method at 100 to 150 °C below the melting point. The DD method has nowadays been supplemented by the positron annihilation spectroscopy (PAS) which detects vacancies of much lower concentrations (review by Hautojarvi, 1987). High energy positrons injected into metal crystals are rapidly thermallized by electron-hole excitations and interactions with phonons. The thermallized positron diffuses through the lattice and terminates its life by annihilation with an electron. The lifetime depends on the total electron density occurring along the diffusional path of the positron. Vacancies trap positrons in a bound state and, because of the missing core electrons at the vacant lattice site, the local electron density is significantly reduced. This condition causes the lifetime of trapped positrons to be enhanced by 20 to 80% as compared to that of the free positrons in the perfect lattice. As the probability for trapping is proportional to the vacancy concentration the total lifetime is a measure for the vacancy concentration. Critical discussion on nonthermal trapping was given by Kluin and Hehenkamp (1991). Lifetime measurements are possible as y quanta are emitted with the creation of a positron as well as with its decay. Fortunately, thermallization happens within about 1 x 10" 1 2 s whereas the average lifetime in the metal crystal is of the order of 2 x l O " 1 0 s . The PAS can be applied for vacancy concentration measurements only when vacancies provide bound states for the positrons. For a large number of metals the existence of such states can be concluded from the measurements and have been predicted theoretically. For other metals like Li, Na, Ga, Sb, Hg and Bi such states could not be found although theoretical calculations predicted them.
The PAS is not only performed by means of positron lifetime measurements but also by angular correlation measurements which refer to the angular correlation between the directions of the emitted annihilation y quanta and by measurements of the Doppler broadening of the y momenta. These quantities yield information on the net momenta of the annihilating electron positron pairs. They allow a distinction to be made between annihilations with the higher momentum core electrons and the lower momentum valence or conduction electrons. Vacancy concentrations are determined by means of lifetime measurements as well as momentum techniques. The former avoids additional assumptions on positron annihilation parameters but requires high resolution measuring techniques and expanded data deconvolution. Although the deconvolution of momentum distribution data requires critical assumptions on the temperature dependence of positron annihilation parameters these methods have become more usual for AH{ determinations. Vacancy concentrations derived from angular correlation measurements are shown in Fig. 6-2 for Cu and Au. One realizes that the PAS data extend to about two orders of magnitude smaller vacancy concentrations than the DD data. According to the calculated values for the binding enthalpy of divacancies (Klemradt et al., 1991) the temperature range covered by the PAS method and the obtained concentrations of vacancies ensure that the data represent the monovacancy concentrations. Hence, PAS studies play a fundamental role for the determination of equilibrium vacancy concentrations. The data in Fig. 6-2 yield the activation enthalpies for vacancy formation for Au to be 0.97 ± 0.01 eV and for Cu 1.29 ± 0.02 eV It is obvious from Fig. 6-2 that DD data can never reach uncertainties
6.4 Point Defects in Metals
1000
800
77 °C 600
500
400
measurement. Also, theory is not yet in the position to give values within reliable and tolerable error limits. Nevertheless, by measuring A^q (Tq) where Tq is the quenching temperature the enthalpy of vacancy formation would be obtained according to Eq. (6-3). By taking one c° value at a certain temperature T = Tq from a separate D D measurement,
8
9
10
11
12
13
14
15
Figure 6-2. Arrhenius plot of the vacancy concentration as derived from positron annihilation spectroscopy for gold (full symbols) and copper (open symbols) according to Triftshauser and McGervey (1975). The left hand side of the arrows represents the range covered by the differential dilatometry.
as small as derived from the PAS data. On the other hand, determination of the entropy of formation from PAS measurements is difficult as absolute values of the vacancy concentration cannot yet be derived from first principles. Considerable efforts have been made to replace the difficult measurements of vacancy concentrations at high temperatures by quenching experiments. The aim was the quenching of equilibrium vacancy concentrations down to temperatures at which the vacancies are immobile. Measurement of physical properties like the residual electrical resistivity which are proportional to the vacancy concentration, for quenched and well-annealed samples would then allow to derive from the resistivity increment A@q the vacancy concentration at the quenching temperature, provided that the specific resistivity contribution of vacancies QV is known. Actually, this quantity is not known from any kind of independent
367
Qv
= AQq(Tq)/c°{T - Tq)
may be obtained. The Qy data in the literature have indeed been derived by this kind of conclusion. The great number of quenching studies revealed significant problems involved in the quenching process (review by Balluffi, 1978). During the quenching the vacancies are still highly mobile within a substantial part of the total temperature interval passed during quenching. The migrating vacancies are able to react with each other or with other defects. Important processes are clustering and annihilation at sinks, i.e., dislocations and interfaces. The annihilation causes the quenched vacancy concentration to be smaller than the equilibrium concentration at the quenching temperature. The clustering causes repartitioning of the cluster size distribution existing at the quenching temperature. Larger clusters are favored with respect to the equilibrium distribution at the quenching temperature. Considerable effort was expended in modelling reaction schemes which were assumed to describe the quenching process. Within 20 years the experimental research lead to higher and higher quenching rates by using thinner and thinner foil samples and improving the heat transfer from the samples to the quenching media. After about 20 years of development it was shown that the successful way is provided by quenching of comparatively thick (3 mm diameter) single crystals with dislocation densities being by four orders of magnitude smaller than those found in the
368
6 Point Defects in Crystals
earlier used thin foil samples (Lengeler, 1976). The quenched-in vacancy concentrations were substantially higher than in the earlier experiments. The obtained entropies of formation (2.2 kB for Cu) agreed well with those from theoretical calculations (see Sec. 6.2). 6A.I.2 Vacancy Diffusion
The migrational step of the single vacancy is the jump of the nearest neighbor atom into the vacant lattice site. This problem has been treated by analytical methods (Flynn, 1968) as well as by computer simulation (Beeler, 1983). The model developed by Flynn (1968) predicts activation enthalpies for the vacancy migration AH™ in remarkably good agreement with experimental data for quite a number of metals. The model relates AH™ to the elastic moduli. According to the author, the jumping atom passes the saddle point by means of a fluctuation of the kinetic energy which was picked up near the atom's equilibrium position where the kinetic energy takes maximum values. There, however, the atomic movement can be described by the harmonic approximation. Experimental information on vacancy migration is currently derived from annealing studies of quenched or irradiated samples. Such studies aim at the measurement of diffusion-controlled annihilation of the excess vacancies at the annihilation temperature (Balluffi, 1978). The residual electrical resistivity has most frequently been used to measure the vacancy concentration. This property can conveniently be measured and the method is of satisfying sensitivity with respect to the interesting vacancy concentrations. Unfortunately, resistivity results can be interpreted unambiguously only if no secondary defect reactions occur besides the vacancy annihila-
tion. Secondary reactions are the formation of divacancies or vacancy impurity complexes. With the electrical resistivity, which is a transport property, different electron scattering contributions cannot be separated. A breakthrough was achieved when methods measuring the hyperfine interaction of probe atoms with their neighborhood allowed to detect the arrival of point defects like vacancies at the probe atoms intentionally implanted into the samples. One of the nuclear methods is the measurement of the perturbed y —y angular correlation. Here, the crystal is doped with radioactive probe atoms which decay by emitting two y quanta separated by the lifetime of a nuclear metastable state. The probability for the emission of y1 with the momentum kx depends on the orientation of the nuclear spin with respect to k1. The probability for the emission of y2 with k2 depends on the angle between k± and k2. The hyperfine interaction of the nuclear magnetic moment with the electric or magnetic crystal field causes the Lamor precession of the nuclear momentum which, in turn, causes the angular rotation of the y momenta kx and k2. This effect is measured by two detectors watching the sample from different but fixed angles and counting coincident events. The counting rate shows the time modulation given by the Lamor precession superimposed onto the experimental decay of the metastable state of the nucleus. Positioning of one or more point defects (vacancies or interstitials) at nearest neighbor positions of the probe atom changes the electric gradient of the crystal field to an extent which can easily be measured by the time differentiated perturbed angular correlation (TDPAC). There exists an interaction between vacancies and solutes which in many cases is attractive such that the solute traps the randomly walking
6.4 Point Defects in Metals
3 CO
0.10 0.05 200
300
f/ns Figure 6-3. Time dependence of the normalized coincidence rate R showing the perturbed angular correlation spectrum for Au-In quenched from 1280 K and annealed at 257 K (Wichert, 1982).
300 200 of/Mrad/s Figure 6-4. Angular frequency spectrum obtained by Fourier analysis (a) of the spectrum in Fig. 6-3 and (b) after electron irradiation at 257 K (Wichert, 1982).
100
369
vacancy whenever it encounters a nearest neighbor site of this solute. An important example is the solute In in Au for which the coincidence rate R (t) is shown in Fig. 6-3 as function of the observation time according to Wichert (1982). The sample was prequenched from 1280 K and then annealed at 257 K. The Fourier analysis of this spectrum with respect to the frequency of the rotating y momenta is shown in Fig. 6-4 a. For comparison the result for a sample preirradiated with electrons at low temperatures and then annealed at 257 K is shown in Fig. 6-4 b. It is obvious that the same hyperfine interaction of the In atom with its neighborhood is produced by the two different treatments. From the tensor properties of the hyperfine field gradient it has been concluded that this signal is caused by the monovacancy at a nearest neighbor site of the In atom. The sensitivity of the method for differentiating between the various defect configurations is illustrated in Fig. 6-5. The signal showing the annealing behavior as given by the solid lines was identified as resulting from
Au - ions
o
protons
20
electrons
""5 DC
quenching
200
300
400
500
Annealing temperature/K
Figure 6-5. Fraction of In atoms in gold which emit perturbed y — y angular correlation signals upon isochromal annealing after quenching and low temperature irradiation with the particles indicated. The different kind of lines indicate different signals which are interpreted as follows: solid lines - monovacancy at nearest neighbor position, dotted and dashed lines - multiple vacancies differently arranged around the probe atom, dash-dotted lines planar <111) vacancy loop around the probe atom (Wichert, 1982).
370
6 Point Defects in Crystals
the monovacancy at a nearest neighbor site of the probe atom. The dotted and the dashed lines show the annealing behavior of multiple vacancies in different arrangements around the probe atom and the dash-dotted lines give the behavior of a planar <111 > vacancy loop nucleated at the probe atom. From Fig. 6-5 the complexity of the defect reactions occurring within the stage III recovery (around Q = 0.7 eV in Fig. 6-15) becomes obvious. In the period before the application of PAC this recovery stage was almost always interpreted in terms of monovacancy or monointerstitial reactions. 6.4.1.3 Structure
For a vacancy, the missing atom induces relaxation of the neighboring atoms to new equilibrium positions. For a tight packing of spheres like the f.c.c. lattice, one realizes that the shift of the nearest neighbor atoms towards the vacancy pushes the next nearest neighbors away from it. This effect substantially reduces the total relaxation volume. The experimental determination of AFvrel is difficult and subject of substantial uncertainty. Direct methods are the differential dilatometry (DD) and the measurement of the pressure dependence of the vacancy concentration. Combination of diffuse Xray scattering (DXS) with lattice parameter changes also allow the determination of AFvrel. Once c° has been determined according to Eq. (6-7) the relaxation volume of the vacancy may be calculated according to AVyrel/Q = 3Aa/(ac°) which relationship was derived by Eshelby (1956) for point dilatation centers of dilute concentration in the elastic continuum. The DXS cross section is given by the product of cv and the square of the scattering amplitude caused per vacancy (Dederichs, 1973) for low vacancy concentrations and random
spatial distribution. The scattering amplitude is essentially given by the elastic moduli and the dipole force tensor which describes the atomic displacements resulting from the insertion of the vacancy. As AFvrel is determined by the same quantities and can be derived from Aa/a measurements as shown above if cv is known, the combination of DXS and Aa/a measurements allows the evaluation of both, cv and AFvrel (Ehrhart et al., 1979). The DXS-Aa/a measurements yielded AFvrel - ( - 0.19 ± 30%) £2 for gold when measured at 4 K. The DD method yielded — 0.5 Q instead (see review by Wollenberger, 1983). Ehrhart et al. (1979) trace this difference, also observed for the results of the pressure dependence method, back to a potential temperature dependence of AFvrel. For other metals like Al, Cu and Ni, the DXS-Aa/a method yielded - 0.05 Q, - 0.2 Q, and - 0.22 Q, respectively. The third method determines the volume of vacancy formation AV/ = AFvrel + Q by measuring the quenched resistivity increment (as measure for c° at the quenching temperature Tq) as function of temperature and hydrostatic pressure. The order of magnitude of the pressure dependence is such that for gold 6 kbar pressure increase corresponds to a temperature decrease of about 30 K around 900 K with respect to c°
The way of determining AFvrel from (Aa/a)q and Agq measurements on quenched samples must be based upon well known QW values. As these are obtained by use of absolute c° data from DD measurements this method goes back essentially to DD data but does involve all quenching problems. For the b.c.c. metals AFvrel values do not exist because of difficulties in determining absolute c° values (Schultz, 1991). For h.c.p. metals similar values have been reported as for the above quoted f.c.c. metals.
6.4 Point Defects in Metals
6.4.1.4 Agglomerates
The significance of the divacancy was extensively discussed in the literature in relation to observed deviations from the Arrhenius behavior of c°(T) according to Eqs. (6-3) and (6-5). Values of 0.2 eV up to 0.5 eV were derived for the binding enthalpies of the divacancy for the f.c.c. metals Al, Cu, Ag, Au, and Ni from the nonlinearity of the Arrhenius plot. Such evaluations, however, completely neglected any possible temperature dependence of the formation enthalpy of the single vacancy. As already mentioned in Sec. 6.2 latest theoretical calculations obtain binding enthalpy values around 0.1 eV for the same metals and therefore leave no possibility for this kind of interpretation of the observed nonlinearity. The existence and migration of divacancies have often been invoked for the explanation of complex recovery features which did not fit to simple reaction kinetics as they were expected for single vacancy annihilation. Such interpretations have to be reassessed in view of the new theoretical results for the binding enthalpy. Information on AF^ 1 is lacking. The same holds true for multiple vacancies up to cluster sizes near the resolution limit of the electron microscopy. The vacancy clustering effect per se has been observed for Au by Huang scattering measurements (Ehrhard etal, 1979) from 3 to 4 up to about 20 vacancies per cluster during stepwise annealing of quenched samples between - 2 5 ° C and +80°C. Also PAS studies show the cluster formation by characteristic changes of the line shape parameter. This fact was used to demonstrate the vacancy mobility within annealing stage III (Mantl and Triftshauser, 1978). Larger vacancy agglomerates have been studied by means of electron microscopy
371
(review by Ruhle and Wilkens, 1983). The topology of configurations involves dislocation loops, stacking-fault tetrahedra and voids. With the resolution of conventional transmission electron microscopy of 1 2 nm the observable agglomerates generally contain more than 10 vacancies. About the same resolution limit holds for diffuse X-ray scattering. Field ion microscopy does allow imaging of agglomerates consisting of less than 10 vacancies (review by Wagner, 1982). It has, however, not been applied yet to problems like shape and size distribution of small vacancy agglomerates as formed by the encounter of migrating vacancies. A brief survey on the field is given by the respective papers in the conference proceedings edited by Abromeit and Wollenberger (1987). 6.4.1.5 Interaction with Other Defects Vacancies in Solid Solutions
Vacancies interact with defects like interstitials, dislocations, surfaces, solutes, Bloch walls, etc. The interaction with interstitials governs the Frenkel pair recombination (Sec. 6.3). The interaction with dislocations, interfaces, and surfaces controls the vacancy annihilation in quenching experiments (Sec. 6.4.1.1). In irradiated samples the annihilation at such sinks competes with the Frenkel pair recombination (Sees. 6.3 and 6.4.3). The interaction with solutes governs the solute diffusion and correspondingly the solvent diffusion. While the first types of interaction are treated briefly in the quoted sections of this chapter, the vacancy solute interaction is treated in the present section. In theory, a number of different approaches have been followed in the past in order to obtain quantitative information about the vacancy solute interaction (review by Doyama, 1978). Nowadays ab ini-
372
6 Point Defects in Crystals
tio calculations are performed by means of the KKR Greens' function method based upon the theory of local density functions in the local spin density approximation (Klemradt et al., 1991). The method allows to calculate the nearest neighbor interaction of vacancies with 3 d and 4 sp solute atoms in Cu and Ni as well as with 4 d and 5 sp solute atoms in Ag and Pd. As a result the data shown in Fig. 6-6 were obtained. Positive sign of the interaction energy means repulsion and negative sign attraction of vacancy and solute. For the sp solute atoms in Ag and Pd, the attraction is approximately proportional to the valence difference. Such proportionality was discussed for a long time in the literature for
7s Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb
-0.6 C a S c T i V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se
Figure 6-6. Calculated interaction energies of a vacancy with a solute atom of the element given by the abscissa for the solvents silver ( ) and palladium ( ) in (a) and copper ( ) and nickel ( ) in (b) according to Klemradt et al. (1991).
experimental data and was often compared with a proportionality to the lattice parameter according to a different approach of modelling (brief review of this discussion by Benedek, 1978). The same behavior is seen for the sp solute atoms in Cu and Ni. The significantly different behavior of the 3 d solute atoms in Cu and Ni as compared to that of the 4 d solute atoms in Ag and Pd is due to their magnetic moment. The magnetic exchange energy reduces the repulsive energy to very small values. Klemradt etal. (1991) find remarkably good agreement of their data with experimental ones as obtained from solute diffusion measurements (see below). The authors also report on first results for binding energies of a second solute atom with a vacancy solute pair. This configuration and high order clusters are of great interest for the understanding of the dependence of solvent and solute diffusion on solute concentration. The direct experimental method of determining AH*sa for vacancy solute atom pairs is the comparison of DD measurements for dilute alloys and the pure solvent metal. In the dilute alloy c° is enhanced by a term being proportional to C s a exp[-(Atf/-A# v b s a )/(/c B T)]. Further methods are as described in Sec. 6.4.1.1 equilibrium PAS measurements and resistivity measurements after quenching, both comparing dilute alloys and pure solvent metals (Doyama, 1978). A careful comparison of DD measurements and PAS measurements in Cu and dilute Cu-Ge alloys performed by Kluin and Hehenkamp (1991) yields good agreement for the results from both methods after a reassessment of the PAS models applied for the evaluation ofcv°. For f.c.c. metals with substitutional solutes the formation of vacancy solute complexes influences solute and solvent diffusion as well. Consequently, diffusion coeffi-
6.4 Point Defects in Metals
cients for solutes and solvent atoms in dilute alloys were always traced back to the pure solvent self-diffusion coefficient by taking into account approximate binding enthalpies. While the effect of AH*sa on c° is easily taken into account as has been shown above, the effect on the diffusion coefficients for solute and solvent atom is more complex. A number of models describing the atomic jump mechanisms near a solute have been applied in the past (Doyama, 1978; Le Claire, 1978; Faupel and Hehenkamp, 1987). Deeper insights into the effect of more than one solute atom bound to one vacancy on the diffusion coefficients are obtained from the dependence of these diffusion coefficients on the solute concentration (see e.g., Faupel et al., 1988). A data collection of diffusion coefficients was edited by Mehrer (1990). The structure of the complex consisting of one vacancy and one solute is simply that of a pair of nearest neighbor lattice positions. Details of the atomic displacements or the relaxation volume of the complex are not yet known, neither theoretically nor experimentally. For divacancy and trivacancy complexes with special solute atoms, however, detailed structural information comes from TDPAC measurements (Sec. 6.4.1.2). The In solute atom in Al and Cu relaxes from its nearest neighbor place of a divacancy to the center of the three resulting vacancies. Similarly, it relaxes to the center of the resulting vacancy tetrahedron from its former nearest neighbor position of {111} planar trivacancy. The same kind of information was obtained by ion-channeling (Wiechert, 1987). 6.4.2 Self-Interstitials 6.4.2.1 Formation The formation enthalpy for self-interstitials amounts to a few electron volts. Ther-
373
mally activated production therefore remains negligible as compared to that of vacancies. Energies of this order of magnitude and even more are transferred to the nuclei of lattice atoms by energetic particle irradiation. The head-on collision of an electron of 400 keV kinetic energy with a Cu nucleus results in a recoil energy of about 20 eV. Neutrons of 2 MeV kinetic energy emitted due to nuclear fission of uranium transfers 125 keV recoil energy to Cu nuclei by head-on collision (for more details see Corbett, 1966; Seitz and Koehler, 1956 and Norgett et al, 1974; Schilling and Ullmaier, Chap. 8, Vol. 10 of this series). Radiation damage in materials was studied since the beginning of the nuclear reactor technology with the Manhattan Project. The basic problems stimulated research on the interaction of energetic particles with lattice atoms as well as the behavior of the Frenkel defects which arise from the former interaction. In the present section, we look at the production process for Frenkel defects as it occurs for different kinds of irradiating particles, electrons, ions and neutrons. At this occasion we sketch the quantities which are involved when the defect concentration cd is to be determined for a given time-integrated flux density
374
6 Point Defects in Crystals
ing. The electron-electron interaction also scatters the incident electrons out of the incident beam direction. This multiple scattering must be taken into account when the flux density is to be determined. The electron-nucleus interaction may lead to a high fractional energy loss for the colliding electron but the probability of such collisions is small and such is the average nuclear energy loss (nuclear stopping power) of electrons. Collisions with recoil energies T larger than the threshold Td for permanent displacement lead to the production of a Frenkel defect. The probability of an electron-nucleus collision with a recoil energy in the interval (7^ T + dT) is determined by the differential cross section for the scattering of a relativistic electron by a point nucleus do/AT (Mott, 1932; Corbett, 1966). The total cross section of a nucleus to get knocked-on by an electron of the energy E and the recoil energy falling into the interval (Td, Tmax) is given by
da(E,T) dT dT
(6-8)
where Tmax is the maximum transferred energy (head-on collision). Creation of a stable Frenkel defect by atom displacement requires an atomic collision process which separates vacancy and interstitial far enough to prevent the defects from spontaneous recombination. The term spontaneous refers to the absence of thermal activation. Spontaneous recombination is the result of mechanically unstable Frenkel defects (review by Wollenberger, 1970). The recoil energy of the primary knocked-on atom is consumed by collisions with its neighboring atoms. To achieve the necessary minimum distance between vacancy and interstitial then
means that the formation energy of the interstitial is to be left at least after that sequence of atomic collisions which transported the energy above this minimum distance. Computer simulations have shown that the optimum way to fulfill this condition is verified by a focussed replacement collision sequence along the closest packed <110> row in the f.c.c. lattice. The primary knocked-on atom replaces its nearest neighbor at its lattice position, the displaced one replaces its nearest heighbor along the same row and soforth until the recoil energy is consumed down to the minimum required for one further replacement collision. The focussing effect diminishes deviations from the impact direction with respect to <110> and minimizes the energy release perpendicular to <110> (Leibfried, 1965). The finally displaced atom ends up interstitially. Eventually, it forms a dumbbell together with its nearest neighbor atom (Gibson et al., 1960; King et al., 1981, Sec. 6.4.2.2). It is obvious that a primary impact deviating significantly from the principal lattice directions will lead to manybody collisions which split the available recoil energy into a number of small fractions. Accordingly, an interstitial can be created merely in close vicinity to the vacancy and might not be stable there. By this way the majority of the recoil energy is released to the phonon system. The influence of the lattice structure on the way the primary recoil energy is split among many atoms and the required minimum separation of vacancy and interstitial causes the threshold Td to depend on the impact orientation with respect to the lattice. In Fig. 6-7, this dependence is shown for Cu as derived from resistivity measurements on samples irradiated by the electron beam of a high voltage electron microscope according to King et al. (1981). The irradiation has been performed
6.4 Point Defects in Metals
at temperatures below 10 K. The defect production rate has been monitored by the rate of residual resistivity increase which was measured in samples of about 400 nm thickness and 0.1 x 0.1 nm 2 irradiated area. Data were collected for 6 different electron energies for each of 35 different crystal orientations. One recognizes minima of Td around <110> and <100> in accordance with the theoretical predictions. Irradiation of textureless polycrystals results in randomly distributed recoil impact orientations with respect to the lattice directions. When calculating the total cross section of the nuclei to recoil such that a Frenkel pair is produced, Eq. (6-8) is not appropriate. The anisotropy of Td is commonly taken into account by a displacement probability p(T):
(6-9)
Now Td min is the minimum threshold within the angular dependence. The displacement probability corresponding to the angular dependence in Fig. 6-7 is shown in Fig. 6-8. This displacement probability is commonly replaced by a single step function (i.e., isotropic threshold) at some higher threshold value, at 30 eV, for instance. The so-called effective threshold is used for total cross section calculations. By using the residual electrical resistivity as measure of the Frenkel defect concentration one obtains (besides the anisotropic threshold energy as shown in Fig. 6-7) also @F, the specific resisitivity contribution of the unit concentration of Frenkel defects because of (6-10)
375 [111]
/
\ 61 \ 24 \ 26 \
28 I 45 \ 25 \ 20 \ 22 \
/Z2
As
23
/27
30
26
24
23
20 j 20 I
23
25
25
29
23
21
23
25
29 / 23 J 21
20
20 I 22 /
J [100]
[110]
Figure 6-7. Anisotropy of the threshold energy Td for copper as derived from resistivity measurements according to King et al. (1981). The figures give the value of Td in eV. For recoil momenta pointing in the respective direction.
S
0.0 20
40 60 Recoil Energy / eV
80
Figure 6-8. Displacement probability versus recoil energy T for copper resulting from the threshold energy anisotropy as shown in Fig. 6-7. The error bars indicate the uncertainty resulting from the fit of Eq. (6-8) to the measured resistivity data.
376
6 Point Defects in Crystals
where 0 is the time-integrated electron flux density. In the quoted study QF = 2.85 x 10~~4Q cm was obtained (reviews by Wollenberger, 1983; Ehrhart et al., 1986). An entirely different method of monitoring the defect production rate was applied by Urban and Yoshida (1981). They irradiated various metals at temperatures above 50 K and measured nucleation and growth rates for interstitial type dislocation loops which they observed in situ by means of the high voltage electron beam of the microscope. This method only counts such interstitials which have migrated long distances in order to agglomerate in loops. The obtained threshold energy is therefore valid for the production of long-range migrating interstitials. For 50 K the results agree with those described before within the experimental uncertainty. Urban et al. (1982) observed a significant decrease of Td with further increasing irradiation temperature (e.g., 10 eV for Cu at 550 K compared to 17 eV at 80 K). The minimum separation of vacancy and interstitial required in order to prevent spontaneous recombination defines the spontaneous recombination volume around the vacany within which the interstitial would not be stable. This volume of spontaneous recombination was found by computer simulations to be anisotropic as well. The spontaneous recombination reduces the production rate according to Eq. (6-9) to an effective one which decreases approximately linearly with increasing defect concentration. The total volume of spontaneous recombination produced by the Frenkel defects already present in the sample must be subtracted from the sample volume available for defect production. From the shape of the linearly decreasing damage rate the volume of spontaneous recombination was determined to be about 100 Q (Wollenberger, 1970).
Defect Production by Neutron and Ion Irradiation The nuclear reactor technology raised the problem of the radiation damage produced by energetic neutrons. Neutrons are heavy projectiles as compared to electrons. The maximum recoil energy of knocked-on target nuclei via head-on collision is given by the classic formula Tmax = 4MmE/(M + mf with E being the incident neutron energy, m and M the masses of neutron and target nucleus. For Cu we have Tmax « £/16 and the average recoil energy
6.4 Point Defects in Metals
range around 107 eV are used when the envisaged measurements require samples of 10 to 100 |im thickness (Jung and Ullmaier, 1990; Gavillet et al, 1988). Lattice atoms which obtained recoil energies as large as 105 eV act as projectiles in the lattice. They initiate a whole cascade of collisions which all involve recoil energies well above the displacement threshold. With decreasing kinetic energy, the cross section for the collisions increases. Correspondingly, the spatial density of collisions and of produced defects strongly increases towards the end of the cascade. For the range of high recoil energies, the collisions can be treated as binary collisions and may thus be described by a linear Boltzmann type transport equation (Kinchin and Pease, 1955). These authors obtained n = T/(2 Td) for the number n of displaced atoms in the cascade initiated by a primary recoiling nucleus with energy T and counting all displaced atoms with recoil energies above the displacement threshold Td. This type of model breaks down with the binary collision approximation for energies below 102 eV. The multiple collision events in the low energy region form so-called displacement spikes. This displacement spike yet provides considerable difficulties for theoretical treatment. Significant decrease of the atomic density in the center of the cascade (Diaz de la Rubia et al., 1990) prevents a reliable application of the manybody potentials valid for the perfect crystal to this region. On the other hand, the atom transport within this region is of great importance for the quantitative understanding of the so-called atomic mixing. There is still a significant discrepancy between measured mixing efficiencies and the atomic transport calculated by binary collision codes (Wollenberger, 1990).
377
The total cross section od as introduced in Eq. (6-8) must now be reformulated by taking into account the number n of defects produced in the cascade:
da(E,T) n(T)dT AT I'd,
(6-11)
eff
Now, da/dTis the differential cross section of the target atoms for neutron-nucleus or for ion-nucleus collisions, respectively. From computer simulations the function n(T) reads n = 0.8 Tdam/(2 Tdteff), where Tdam is the total recoil energy of the primary knocked-on atom diminished by the electronic energy losses released during the sequence of the cascade collisions. In Ni, for example, these losses become significant above 104 eV and amount to about 60% of the total energy at 106 eV. The threshold Td eff was found by computer simulations to be about twice the minimum threshold determined by electron irradiation. For energies near the threshold, n is defined as follows: n = 1 for Td < T < 2.5 Td and n = 0 for T
378
6 Point Defects in Crystals
mental cross section is generally smaller than the calculated one. The efficiency £ = (7d (meas)/crd (calc) is shown for Ni as function of n in Fig. 6-9. An efficiency being smaller than unity is to be expected as the spontaneous recombination is not taken into account when using the differential cross section dcr/dT in Eq. (6-11) and probably not sufficiently considered in the computer models used for calculating n. The data indicate that the loss of defects by recombination increases drastically with increasing cascade size. When the irradiation is performed at elevated temperatures at which the defects are mobile the recombining fraction is even larger. For the radiation-enhanced diffusion, i.e., long distance mass transport under irradiation, the fraction of those defects is relevant which escape recombination or clustering with defects of the parent cascade. This percentage is shown in Fig. 6-10. Information on the structure and volume of defect cascades comes from computer simulation as well as from experiments. Figure 6-11 gives an example of simulation results. The figure shows that the defect density within one cascade fluctuates strongly. It also shows that the vacancies tend to be accumulated whereas the interstitials are mostly arranged as single defects. Real cascades have been imaged by electron microscopy (Jenkins and Wilkens, 1976; Kiritani, 1982, 1987) and field ion microscopy (Seidman, 1978). While field ion microscopy does resolve single point defects and thus images the defect arrangement directly, electron microscopy can only image contrasts produced by defect agglomerates or disordered zones. In the latter case long-range ordered alloys have been irradiated with ions and the zones disordered by the replacement collisions within the cascades became visible in a
1
0.8 0.6 0.4 0.2
t
1
I -
H 1.0
1
•
1.2
He
: 1u I 1
c JNO "T Ne A
.Fe^AgBL -
i
i
i
i
6 8 10 n
20
i
40
Figure 6-9. Efficiency £ of defect production at 4.2 K for copper as function of the average number of defects n produced per cascade. The irradiations have been performed with the ion species as indicated at the data points. The data scatter is shown for He, Li, C and O (Averback et al., 1978).
1 MeV H + 2 MeV He + 2 MeV Li+
10.0-
3 MeV Ni + •
• 300keVNi 4
2 keV 0+
3.25 MeV Kr+.
0.1 1
10 Number of Defects per Cascade
100
Figure 6-10. Percentage of freely migrating defects (FMD) as function of the average number of defects n produced per cascade for nickel and the irradiating ions as indicated. Squares indicate self-diffusion measurements (Naundorf and Wollenberger, 1990) and circles impurity diffusion (Wiedersich, 1990).
6.4 Point Defects in Metals
Figure 6-11. Computer simulation of a displacement cascade with T = 2 x 105 eV for copper. The primary knock-on happened in the lower left. The larger dots are vacancies and the smaller dots are interstitials. Cube edge measures 125 nearest neighbor distances (Heinisch, 1981).
dark field image using a superlattice reflection of the ordered lattice. More information on the structure of defect cascades was obtained from measurements of the Huang X-ray and small angle synchroton radiation scattering (Peisl et al., 1991; Rauch etal., 1990). According to these studies interstitial and vacancy type loops are formed in Cu by reactor neutron irradiation at 4.2 K. The loop radii amount to about 1 nm. For Fe and the same irradiation conditions agglomerates with 1.5 interstitials per cluster in the average and 60% divacancies besides 40% single vacancies were derived. The cascade radii amounted to 1.5 nm for interstitials and 2.2 nm for vacancies. All these values are for about 10 keV mean damage energy. For the damage production, as it has been treated here, the electronic energy loss of the travelling ion has been considered just as a loss term diminishing the
379
available damage energy. The ionisation losses have been taken as heat release to the sample environment. This treatment is correct as long as the ionisation density (stopping power) remains within the order of magnitudes valid for the incident ion energies smaller than a few MeV. A complete new regime of damage production occurs for heavy ion energies of the order of GeV. For such ions the electronic stopping power reaches the order 10 keV/nm and is two orders of magnitude larger than the nuclear stopping power. This kind of irradiation produces tubularly damaged zones called tracks which are known for a long time for nonmetallic materials. Such tracks have now been observed for Ni-Zr alloys. The radiation-induced resistivity increase, observed after such kind of irradiation for pure metals, has successfully been interpreted by Frenkel defects arranged in tubular geometry along the ion's path-way. A review on this topic is given by Dunlop and Lesueur (1992). 6.4.2.2 Structure of Self-Interstitials
For the f.c.c. lattice, the self-interstitial configurations shown in Fig. 6-12 have been taken into account. Theoretical calculations of the enthalpy of formation yielded values between 2 and 4 eV for copper and 4 eV for nickel. Despite the largely differing values from different authors, the Ho configuration was universally found to be the stable configuration. The formation enthalpies for the different configurations to not deviate by more than 15% from the minimum value and for 2 or 3 different configurations, the deviations remain below 5%. For the b.c.c. lattice, again the dumbbell configuration was found to be the most stable one. Here, its orientation is along <110> instead of <100> for f.c.c. For the h.c.p. magnesium, configurations like
380
6 Point Defects in Crystals
Figure 6-12. Self-interstitial configurations in the f.c.c. lattice. O, T and C stand for octahedral, tetrahedral and crowdion. Ho, HT, Hc denote the dumbbell interstitials with axes along <100>, (111) and <110>, respectively.
odumbbell, octahedral, tetrahedral, and trigonal gap and crowdions along a and c direction have been found by different authors to be most stable (for more details see Wollenberger, 1983 and Ehrhart et al., 1986). Experimental information on interstitial configurations comes from diffuse X-ray scattering and mechanical or magnetic relaxation experiments. The analysis of diffuse X-ray scattering makes use of the information included in the intensity profile outside the Bragg reflections. This intensity is due to the imperfections of the crystal. For point defects, the atomic displacements around them are responsible for the diffuse scattering. This intensity occurs on a background of other scattering contributions which are shown in Fig. 6-13. As the point defect relevant intensity remains one to two orders of magnitude below that of the Compton scattering, only very careful comparison of the intensities from interstitial-free and interstitial-containing crystals allows reliable determination of the defect relevant contribution. The diffuse intensity profile contains information on the symmetry of the atomic
CO
10
10"
K/nrrf Figure 6-13. Diffuse X-ray scattering cross sections versus wave vector K parallel to [100] (X = 0.154 nm) for aluminum according to Ehrhart et al. (1974). Line presentations: Compton scattering, thermal diffuse scattering at 4.2 K, contribution by <100>-split interstitial and by the vacancy, both being present at the concentration 5 x 10 ~4.
381
6.4 Point Defects in Metals
distortions and the distortion strength as well. The unknown configuration is determined by comparing calculated intensity profiles with the measured ones. Figure 6-14 demonstrates such a comparison for aluminum. The aggreement for the <100> dumbbell configuration is convincing. The calculations are for 0.85 nearest neighbor distance (nnd) separation of the two dumbbell atoms. The four nearest neighbors of the dumbbell had to be displaced outward by about 0.07 nnd in order to attain this agreement. The long-range atomic displacement field around lattice defects determines the scattering at small values of the scattering vector. Since the anisotropy of this displacement field reflects the symmetry of the interstitial configuration, intensity measurements close to the Bragg peaks (Huang scattering) can therefore resolve the symmetry of interstitials, too. Although high resolution techniques must be applied in order to obtain reliable results, the overall experimental expenditure and the requirements for sample quality are less pronounced than for diffuse X-ray measurements far away from the Bragg peaks. This advantage is due to the much higher defect-induced scattering intensity near the Bragg peaks. Huang scattering has in particular been applied for point defect cluster studies (Ehrhart et al, 1986). As the X-ray scattering yields the atomic distortions around the self-interstitial they are also used for determining the relaxation volume AFirel. It is that volume, by which a crystal expands when an additional atom is introduced into this crystal and takes its part in the interstitial configuration. Experimental data are given in Table 6-1. One realizes that for the f.c.c. lattice AV{rel amounts to about two atomic volumes. The part exceeding the one atomic volume given by the introduced
< 100 > - Split
3
c
o
15°
30°
45° (a)
60° 15° 30° scattering angle
45°
60°
(b) Tetrahedral
15°
30°
45° (c)
60° 15° 30° scattering angle
45°
60°
(d)
Figure 6-14. Comparison of the diffuse X-ray scattering intensity measured for aluminum electron irradiated at 4.2 K (a) with curves calculated (b) to (d) for the interstitial configurations Ho, O, and T in Fig. 6-12. The numbers 1 to 4 indicate the circles of the Ewald sphere shown above (Ehrhart et al., 1974). Table 6-1. Relaxation volume of self-interstitialsa. Metal AV^/Q a
Al Co Cu Fe Mo Ni Pt Zn Zr 1.9 1.5 1.6 1.1 1.1 1.8 1.8 3.5 0.6
For references see Wollenberger (1983) and Ehrhart etal. (1986).
382
6 Point Defects in Crystals
atom is spent because the regular closest packing of atoms is perturbed by the interstitial. Perturbation of the f.c.c. structure obviously requires more additional volume than the more open b.c.c. structure. Determination of the interstitial configuration by mechanical or magnetic relaxation measurements essentially test the symmetry of dynamic interstitial properties. They are treated in the following section. 6.4.2.3 Dynamic Properties of Self-Interstitials The dynamic behavior of the self-interstitial appears to be rather unusual (see also Chap. 5 of this volume by Schober and Petry). Actually, it reflects the influence of the large lattice distortion more clearly than the static properties. Already from the first resistivity recovery measurements after low temperature particle irradiation it was commonly concluded that the interstitial migrates freely around 30 K in metals like Al, Cu and Ni and the activation enthalpy for this migration measures about 0.1 eV (Corbett et al., 1959). It is one order of magnitude smaller than the activation enthalpy for vacancy migration (see Sec. 6.4.1.2 and Fig. 6-15). This low value fits to the insensitivity of the enthalpies of interstitial formation to its configuration as has been discussed in the foregoing section. The large lattice relaxation around the interstitial which determines most of the formation enthalpy is enhanced by a few percent only when the dumbbell is lead across the saddle point. Details of the migrational step have been investigated by computer simulation (Scholz and Lehmann, 1972; Imafuku et al., 1982). The most probable path obtained is shown in Fig. 6-16 for Cu and for the b.c.c. oc-Fe. In the f.c.c. structure it consists of a translational motion of the center of gravity by
100
(XL
I 50
& 2
1 •K i
.
i
i -i
t\
"A 5
1
\
i 1 I
i • -
0.5
^
1
.
\
1.5
2.5
O/eV Figure 6-15. Electrical resistivity recovery spectrum for noble metals (except gold) as function of the activation energy for thermally activated annihilation processes according to Van Bueren (1961). The pretreatments are irradiation with electrons (—) and neutrons ( ), quenching ( ), and cold working
one atomic distance and the rotational motion of the dumbbell axis by 90°. The 90° axis rotation with a fixed center of gravity was found to require an activation enthalpy being a factor of four larger than that for the migrational step (Dederichs et al., 1978; Lam et al., 1981). For the b.c.c. structure we have a translational step of one atomic distance but rotation of the axis by 60°. The activation enthalpy was found to be 0.21 eV. Here, rotation by 90° without translation requires only 0.04 eV energy more than the migrational step. The computer simulations furthermore revealed vibration modes of the <100>
(a) (b) Figure 6-16. Migrational steps of the <100>-split interstitial in the f.c.c. lattice (a) and the <110>-split in the b.c.c. lattice (b) according to computer simulation results.
6.4 Point Defects in Metals
\ a.u
-
Eg
^
A
-^ "§* N -
<
\l\
,.-<
:\
1^
,.A
0.0
\
1.0
0.5 €O/(O
Alg \
max
Figure 6-17. Local frequency spectrum of the <100>split interstitial in the f.c.c. lattice (averaged over all lattice directions) for a modified Morse potential representing the behavior of copper (Dederichs et al., 1978). Dashed curve for the perfect lattice. Localized modes: w > comax, resonant modes: w < comax.
dumbbell in the f.c.c. structure of low frequency (resonant modes) and of high frequency (localized modes) (see Fig. 6-17). For the dumbbell atoms vibrating along <100> in opposite directions as shown in Fig. 6-18 a, the small equilibrium separation between the two atoms (0.77 nearest neighbor distances) leads to a very high force constant for the coupling spring. Such coupling yields a localized mode
383
which lies well above the maximum lattice frequency comax (see Alg in Fig. 6-17). For the displacement directions as shown in Fig. 6-18 c, the strongly compressed spring between the two dumbbell atoms causes a negative bending spring component acting perpendicular to the dumbbell axis. The resulting force constant is small producing a low frequency resonant mode (see Eg on the left hand side of Fig. 6-17). Another resonant mode (A2u) is excited with the atomic motion shown in Fig. 6-18 b. The low frequency resonances lead to large thermal displacements of the dumbbell atoms when compared to those of regular lattice atoms. At 50 K, the average squared displacement <s2> is about twice and at 150 K about 3.5 times as large as that of the atoms at regular lattice sites. As the resonance mode shown in Fig. 6-18 c leads the dumbbell atoms towards the saddle point configuration for the migrational step and the thermal population of this mode starts increasing at 20 K already, the on-set of interstitial migration around 30 K (Figs. 6-15 and 6-19) remains no longer mysterious. The question whether the interstitial migration may follow an Arrhenius behavior at all was positively an-
(0,1,1),
(1.0,1)
(a)
(b)
(c)
Figure 6-18. Localized and resonance vibrational modes of the <100>-split interstitial in the f.c.c. lattice: (a) localized mode Alg (see Fig. 6-17), (b) resonance and localized mode A2u, (c) resonance and localized mode E . The arrows give the directions of vibration.
384
6 Point Defects in Crystals
swered by Flynn (1975). The explanation relies on the classic excitation of the resonant mode. The flatness of the energy contour along the migrational path is the direct consequence of the negative bending spring action. The high thermal mobility of the interstitial with respect to that of other defects can clearly be seen in Fig. 6-15 drawn according to Van Bueren (1961). The resistivity recovery with the activation enthalpy around 0.1 eV results from the annihilation of migrating interstitials. More details on the migrational properties of the interstitial can be seen in Fig. 6-19. It shows the temperature differentiated resistivity recovery curves for electron-irradiated platinum. The stages JA, / B and Ic are due to close Frenkel pair annihilation. This first order reaction (peak temperature and shape independent of the initial resistivity, i.e., defect concentration) is due to the strong attractive interaction of vacancy and interstitial. The stages / D and 7E are due to correlated and uncorrelated recombination, respectively, of freely migrating interstitials
20
15
r/K
25
with immobile vacancies (Sonnenberg et al., 1972). The negative bending spring action of the dumbbell interstitial is also responsible for the high elastic polarizability of the dumbbell interstitial. A {100} <001> shear stress application will excite just the liberational resonance mode shown in Fig. 6-18 c. Computer simulation shows that the rotation angle of the dumbbell axis is by a factor of 20 larger than the macroscopic shear angle due to the bending spring action. The large negative modulus change was first detected by measurements in a-particle irradiated polycrystalline Cu samples (Konig etal., 1964; review by Wenzl, 1970). The first measurements in single crystals containing randomly distributed single Frenkel defects have been reported by Holder et al. (1974) for thermal neutron irradiated Cu (thermal neutrons produce recoil energies close to the displacement threshold similar to 1 MeV electron irradiation) and by Rehn and Robrock (1977) for electron irradiated Cu. The data obtained by Holder et al. (1974) are shown in Fig. 6-20 demonstrating the largest effect for C = c 44 . The
30
Figure 6-19. Temperature-differentiated isochronal recovery curves for platinum electron-irradiated to six different resistivity increments (defect concentrations) according to Sonnenberg et al. (1972). The resistivity increments are 1.93, 7.90, 21.7, 56.2, 104.6, and 227.0 nQ cm for the curves from the bottom upwards, respectively.
6.4 Point Defects in Metals 0 0
1
2 .
3 + 2c 1 2 .}f_ 11 3 -^-*. c
-1 /
D
N,
\
E
x
C -
C
11-c
2
o-2
5
-
\ -3 •
\
-4
i
10
20 30 Irradiation Time / h
40
50
Figure 6-20. Elastic moduli decrease for copper single crystals during electron irradiation at 4.2 K (Holder et al.f 1974).
data were derived from ultrasonic attenuation measurements. Rehn and Robrock (1977) found the same results by means of internal friction measurements using a torsion pendulum. The interstitial concentrations amounted to 10" 7 for the former measurements and 10 ~ 4 for the latter ones. The relative changes of the moduli are given in Table 6-2. Quenched-in vacancies were found to produce much smaller Table 6-2. Normalized relative modulus changes by the presence of interstitialsa. Metal 1 AC 1 AC 1 Ax C;
a
Al
Cu
Mo
27 + 2
17 ± 2
5.5 + 2
16 + 2
16 + 2
14+2
2±2
X
Data according to: Robrock and Schilling (1976) for Al; Rehn and Robrock (1977) for Cu; Jacques (1982) for Mo.
385
changes of the elastic constants. In fact, the figures are valid for interstitials. If the interstitial concentration could be enhanced up to 3.7 at.% for Al (only hypothetical because spontaneous recombination limits cd to about 0.5 at.%) and the softening proceeded linearly up to this concentration, the crystal would be unstable against a {100}<001> shear stress. The figures for Mo in Table 6-2 with the maximum negative change for the modulus C indicate the <110> orientation of the split interstitial in the b.c.c. structure. Explanation of the negative modulus change caused by the presence of interstitials was a long-standing problem and could not be solved before the resonance mode was detected in computer experiments. Handwaving arguments starting from highly stressed coupling springs with their enhanced force constants always arrived at positive modulus changes. Considerable efforts have been undertaken to prove the existence of the soft modes shown in Fig. 6-17 by inelastic neutron scattering measurements. The proof turned out to be very difficult because of the small effect. The only positive result was reported by Nicklow et al. (1979). Resonance-like perturbation of the phonon dispersion curves were observerd in accordance with the expected behavior. Quantitative evaluation leads to a resonance frequency fT for the dumbbell of 0.8xl0 1 2 Hz when the modulus data are taken into account. The quantitative relationship between fr and the elastic modulus change has been derived by Dederichs and Zeller (1980). 6.4.2.4 Agglomerates of Self-Interstitials Computer calculations have shown that the binding energies for interstitials in the f.c.c. lattice amount to about 1 eV (Ingle et al., 1981). The most stable di-interstitial
386
6 Point Defects in Crystals
measurements of the diffuse X-ray scattering, Huang scattering and mechanical relaxation measurements. In Fig. 6-21 diffuse X-ray data obtained for electron irradiated Al after 40 K annealing are shown and compared with the calculated scattering curves for two different di-interstitial configurations. For the Huang scattering intensity, clusters containing n point defects is n times that for n single interstitials. The cluster formation can easily be monitored as the wave vector dependence of the Huang intensity according to K~2 is replaced by a K~A dependence for higher K values. From the boundary of the K~ 4 validity the cluster radius can be estimated. For Al and the same experimental conditions as in Fig. 6-21, N = 2 was found (Fig. 6-22) in agreement with the conclusion drawn from the diffuse scattering results. For Cu small loop shaped clusters with 5 to 10 interstitials have been found at the end of recovery stage I. From the lack of clusters smaller than 5 interstitials it was
configuration in the f.c.c. lattice consists of dumbbells with parallel axes and their centers of gravity at two nearest neighbor sites situated in a plane perpendicular to the dumbbell axes. The relaxation volume decreases by about 10% per interstitial entering the dimer configuration. The cluster formation was studied up to 36 interstitials. Above about 10 interstitials, two-dimensional (111) stacking fault loops were found to be more stable than the three-dimensional clusters. The binding energies increased from 1 eV to more than 2 eV per interstitial for larger clusters. The relaxation volume per interstitial dropped by about 25% from the single interstitial to a loop of 36 interstitials. The migration enthalpy for the di-interstitial was found to be about the same as that of the single interstitial. For the tri-interstitial it was found to be twice of that for the single defect. The existence of small multiple interstitial arrangements was concluded from
13
erf —
I
15°
45°
30
(a)
60°
15
30° 45° scattering angle e
(b)
60° (c)
Figure 6-21. Comparison of the diffuse X-ray scattering intensity measured for aluminum electron irradiated at 4.2 K and subsequently heated to 40 K (a) with calculated curves for two different di-interstitial configurations (b) and (c). Numbers 1 to 4 as in Fig. 6-14 (Ehrhart et al., 1974).
6.4 Point Defects in Metals 1
•
i
1
i
50 30 A20
Loop formation
V 10
Jf
/ 5
End of stage I
3 2 -
10
! L 50
1
100
200
T/K Figure 6-22. Average number of clustered interstitials versus annealing temperature for Au ( ), Cu (••••), Al ( ) and Ni ( ) and initial defect concentrations of about 300 ppm (Ehrhart et al., 1986).
concluded that in Cu di- and tri-interstitials are mobile in the same temperature region as the single interstitial. Small interstitial clusters were found by Huang scattering in Au, Al, Cd, Co, Mg, Nb, Zn and Zr (Erhard and Schonfeld, 1982) as well. The strong interstitial clustering found for Au already at 10 K (Fig. 6-22) is in accordance with the interstitial mobility found already at 0.3 K (Birtcher et al., 1975). For the b.c.c. lattice computer calculations yielded two dumbbells with parallel axes along <110> and their centers of gravity at nearest neighbor sites situated in a plane perpendicular to the axes as the most stable configuration. The binding enthalpy was found to be 0.9 eV and the migration enthalpy approximately equal to that for the single interstitial.
387
6.4.2.5 Interaction of Self-Interstitials with Solutes
From recovery measurements of irradiated metals it became obvious that solutes trap migrating interstitials in the temperature range of stage I up to temperatures of stage II to stage III depending on the species. An example is shown in Fig. 6-23. The kinetics of trapping and detrapping has been studied by means of electrical resistivity measurements in great detail (Schilling et al., 1970; Takamura et al, 1987). As the trapping happens upon interstitial migration the capture rate is diffusion-controlled. For three-dimensionally migrating defects one obtains the capture radius rt as the rate controlling quantity being specific for the solute. This quantity was determined by resistivity damage rate measurements for irradiation temperatures above stage I (review by Wollenberger, 1978). The radii found for solutes in Cu amounted to 1 to 2 nnd. From the dissociation temperatures (100 K in Fig. 6-23) one derives the enthalpy of dissociation and finds for a number of solutes in copper values between 0.1 and 0.4 eV. For a number of solutes like Ni and Cu no trapping was observed by the resistivity measurements and for Be in Cu the enthalpy of dissociation
0.3
Cu + 112ppm Au
0.2
0.1
Cu
80 120 160 T/K Figure 6-23. Isochronal resistivity recovery of pure copper and a dilute copper-gold alloy electron irradiated at 4.2 K (Cannon and Sosin, 1975). 40
388
6 Point Defects in Crystals
was found to measure more than 1 eV. It is worth mentioning that trapping as concluded from resistivity measurements means immobilization in the trap. If solute and self-interstitial were mobile as a complex at the temperature of formation, resistivity measurements would not indicate any complex formation. For the solute Be in Cu its interstitial transport was observed with the enthalpy of migration amounting to about 0.6 eV (Gudladt et al, 1983). This value fits well to the theoretical predictions for the migrating mixed dumbbell (see Fig. 6-25). The volume misfit of the solute Be amounts to about — 30%. For Ni in Cu the misfit is almost zero. Trapping stages are not observed in recovery measurements. The observation of low temperature short-range ordering in concentrated CuNi alloys suggests mixed dumbbell transport (Poerschke and Wollenberger, 1980). For the solute Si in Ni similar conclusions may be drawn from radiation-induced segregation measurements. Computer simulations of the interstitial-solute interaction are difficult because of the lack of knowledge on the details of the atomic interaction potential for a sol-
(a)
(c)
ute in the host. Dederichs et al. (1978) simulated a solute in Cu by simply shifting the Cu-Cu interaction potential by a certain distance r 0 . Enlarging the equilibrium distance of the neighbor simulates an oversized solute, diminishing it simulates an undersized one. The simulation yielded a number of interesting results. For undersized solutes it shows the formation of a mixed dumbbell. Solute and one solvent atom share one lattice position with the dumbbell axis parallel <100>. For oversized solutes the self-interstitial dumbbell was found trapped at different neighboring lattice positions to the solute with different binding energies in the order of 0.1 eV. Studies of the mixed dumbbell dynamics discovered three different kinds of jumps shown in Fig. 6-24. Corresponding enthalpies for dissociation, rotation and the cage motion of the mixed dumbbell are shown in Fig. 6-25. As the enthalpy for self-interstitial migration AH™ amounts to 0.1 eV one realizes that the model predicts very small enthalpies for cage motion. Rotation should be possible for larger size misfits. For solutes in Al interionic potentials derived from first principles have been
(b)
1 i
looping
dissociation
(d)
Figure 6-24. Elementary jumps of a mixed dumbbell in the f.c.c. lattice: (a) jump of the solute, (b) six mixed dumbbells are formed consecutively by the cage motion of the solute around the octrahedral position, (c) jump sequence of solvent atoms and the solute which enables long range transport of the solute, (d) mixed dumbbell rotation (Dederichs etal., 1978).
6.4 Point Defects in Metals
-ro/Ro/
%
Figure 6-25. Normalized enthalpies AHmd/AH™ of mixed dumbbell dissociation, rotation and cage motion according to computer simulation for copper versus size misfit of the solute (Dederichs et al., 1978). The denominator AHj" is the activation enthalpy for interstitial migration. The ratio ro/Ro of the potential shift r0 divided by the equilibrium atom distance Ro must be multiplied by about 6 in order to obtain the volume size misfit comparable with those of solutes in copper.
used for computer simulation (Lam et al., 1981). For the solute Zn mixed dumbbell formation was found with a binding energy of 0.38 eV. The saddle point energy for mixed dumbbell migration was found to be much larger than the dissociation energy. The occurrence of the cage motion was confirmed for Co in Al and in Ag by Mossbauer studies (Petry et al., 1982). Dilute alloys with the Mossbauer isotope 57 Co were irradiated at temperatures below stage II and then subjected to Mossbauer spectroscopy. Besides the wellknown Mossbauer line resulting from the solute transmutating from 57 Co to 57 Fe at substitutional positions (unirradiated case) a second line was observed for which the isomer shift indicated a smaller distance to the nearest neighbor host than for the substitutional position and the Debye-Waller factor shows an unusual temperature de-
389
pendence. According to Fig. 6-26 the intensity of this Mossbauer line drops down between 15 and 20 K by about 70%. This decrease can be measured reversibly as long as the sample is not heated into recovery stage III, i.e., the trapped interstitials are not annihilated. This decrease has been explained quantitatively by a cage motion of the Mossbauer isotope. The motion of the Mossbauer isotope within the cage starts at about 15 K and takes many different positions within the lifetime of the Y quantum of 10" 7 s inducing by this means self-interference of the emitted Y wave bunch. This results in the drop of the Debye-Waller factor. Performing such measurements in single crystal samples allows the determination of the jump directions within the cage with respect to the lattice directions (Vogl et al., 1982).
Figure 6-26. Temperature dependence of the intensity of the Mossbauer lines for iron impurities in aluminum (solid symbols - unirradiated sample, i.e., Fe atoms in substitutional positions, open symbols - behavior of the additional line after electron irradiation at 4.2 K arising from Fe atoms at interstitial sites) according to Petry et al. (1982).
390
6 Point Defects in Crystals
The cage motion has also been shown by internal friction measurements (Rehn et al., 1978) and by ultrasonic attenuation measurements (Granato et al., 1982; Granato, 1982) for dilute alloys of Fe in Al. For the internal friction measurements with 80 Hz frequency the damping maximum occurred around 8 K whereas for the ultrasonic attenuation measurements with 10 MHz frequency it occurred around 18 K. From this observation one derives the activation enthalpy for the cage motion to be 0.014 eV. For more results on this topic see Granato (1982). Configurations like the mixed dumbbell have been searched extensively also by the ion-channeling method (review by Howe and Swanson, 1982). Backscattered He ions are measured as function of the outcoming beam direction with respect to the crystal orientation. Like for Rutherford backscattering the energy of the outcoming He ions reflects scattering processes at solutes and at host atoms in a resolvable way. If the outcoming beam direction coincides with channel directions of the lattice the yield of the reflected ions contains the information whether solutes are displaced from regular lattice sites towards the center of the channel or not. For mixed dumbbells the solute does not occupy a regular lattice site and its position in the interstice of the lattice can be measured, in principle. In fortunate cases, the position can clearly be specified by investigating a number of different channels. Like Rutherford backscattering this method is sensitive only for heavy solutes in light solvents. Therefore, this method has been applied predominantly to dilute Al alloys. The existence of <100> mixed dumbbells has been concluded for the investigated alloys with undersized solutes and no solute displacement was found for oversized ones. By using proton-induced X-ray emission as an
indicator for the presence of impurities in interstitial positions the mass difference problem is avoided (Ecker, 1982). A serious limitation for the application of this method to our problem is an effect that channeling measurements can unambiguously identify the solute position only when the solute does not occupy more than one type of interstitial position. Besides the different configurations possible for the complex formed by one interstitial and the solute, the trapping of more than one interstitial per solute enhances the number of specific solute positions. This problem could only be solved by studying the solute position as function of the defect concentration. Detailed considerations show that the resolution limit of the method allows the investigation of the case of only one interstitial per solute in only a few cases. Data on self-interstitial solute interaction in a series of metals are reviewed by Robrock (1983, 1989). 6.4.3 Self-Organization of Defect Agglomerates Irradiation of a crystal with heavy particles produces single vacancies and interstitials as well as defect agglomerates, preferentially dislocation loops. At irradiation temperatures such that vacancies and interstitials are mobile, dislocation loops act as sinks for the point defects. For reasons of simplicity we consider vacancy loops as the only sinks for the point defects. The elastic interaction of the migrating vacancies and interstitials with dislocations is different in strength because of the different relaxation volumes of the two defects. As a result one obtains a bias for the sink strengths of a given dislocation for interstitials on one hand and vacancies on the other. The sink strength for interstitials is larger than that for vacancies. Let us con-
6.4 Point Defects in Metals
sider an homogeneous production of freely migrating vacancies and interstitials and randomly distributed dislocation loops. The randomly migrating point defects annihilate at the sinks as soon as they cross the capture radius. Annihilation of the vacancies enlarge the loop length and, hence, the total sink strength. Annihilating ion interstitials diminishes this quantity. Such a reaction system already fulfils the conditions for spatial self-organization of the sink structure under persisting irradiation (Abromeit and Wollenberger, 1988). The quantitative treatment of the process uses an appropriate rate equation system and investigates the stability of the differential equation system with respect to small spatial fluctuations in terms of the linear stability analysis (Haken, 1977). Despite the fact that self-organization is always the result of nonlinear relationships among the reacting agents it is worth trying to understand the physical origin of the pattern formation. The spatial fluctuation of the sink density 8cs/
391
x/X Figure 6-27. Sketch of the spatial fluctuation of the concentrations of sinks (s), vacancies (v) and interstitials (i).
The amplification is at maximum for some finite value of k. The structure evolves a lattice characterized by this wave length. A self-organized system of dislocations observed after proton irradiation is shown in Fig. 6-28. One realizes that the broken symmetry of the defect annihilation process at the dislocation loop (vacancies enhancing the sink density, interstitials diminishing it) and the annihilation bias are the essentials for the ordering process. Another example for broken symmetry would be the three-dimensionally migrating vacancy and a two- or one-dimensionally migrating interstitial. Such kind of diffusion is supposed to occur for the crowdion (onedimensionally) shown in Fig. 6-12 or interstitials in h.c.p. and tetragonal lattices (two-dimensionally). With this kind of asymmetry the occurrence of void lattices has been explained as well (Evans, 1990; Hahner and Frank, 1990). Coupling of the persistent defect fluxes induced by an ordered sink structure with mass transport may lead to chemically ordered alloys. Coupling mechanisms for defect and solute fluxes are described in the following section. Theory and experimental results on radiation-induced ordered
392
6 Point Defects in Crystals
Figure 6-28. Periodic arrays of planar {001} walls of dislocations and stacking-fault tetrahedra in copper irradiated with 3.4 MeV protons around 100 °C to 2 dpa (displacements per atom) fluence. Imaged in <100> projection (Jager et al., 1990).
structures are reviewed by Abromeit (1989). 6.4.4 Atomic Redistribution by Persistent Defect Fluxes
Irradiation of a crystal causes the supersaturation of vacancies and interstitials in the lattice. At irradiation temperatures such that both defect species are mobile the defects will annihilate by recombination and annihilation at sinks. For very low sink densities the annihilation rate will be controlled by the recombination reaction. As the defects migrate randomly recombinations will occur homogeneously in space. For descriptions in terms of the rate equa-
tion approach the defect annihilation is considered by the "lossy medium". In case of high sink densities the defect annihilation can generally not be described by the lossy medium. Unsaturable sinks like surfaces or nonsessile dislocations cause persistent defect fluxes under stationary circumstances (Sec. 6.3). The situation in front of such a sink is sketched in Fig. 6-29 for a solid solution consisting of A and B atoms. If the partial diffusion coefficients D\ and D^ of the constituents A and B of a binary alloy due to vacancy migration are different, the vacancy flux Jv to the sink will cause different atom fluxes JA and J B with the result of a depletion of the faster transported constituent near the sink as shown in Fig. 6-29 b. The process is the inverse Kirkendall effect (Smilgalskas and Kirkendall, 1947). If the interstitial flux J{ provides atom fluxes as shown in Fig. 629 c constituent A will be enriched as shown in Fig. 6-29 d. For steady state conditions and rvs = ris the concentration gradient VcA was derived by Wiedersich et al. (1979) to be proportional to [{D\/DD Atomic redistribution of this kind also occurs if one of the defect species forms a mobile tightly bound complex with one of the constituent atoms and annihilates at the sink. In this case, the complex forming constituent will be enriched at the sink. One realizes that such atomic redistributions may lead to the formation of new phases. The most simple case is the irradiation of an undersaturated solid solution of B in A. When the redistribution amplitude is large enough, the solubility limit will be exceeded near or at the sink and an intermediate phase will be formed whenever nucleation conditions allow it. At sufficiently high irradiation temperatures the process may be reversed by the action of thermal vacancies after stopping the irradiation.
6.4 Point Defects in Metals
(a)
(c)
393
(b)
(d)
Theoretical description of the radiationinduced segregation has been reviewed by Wiedersich and Lam (1983). For studies of precipitation near internal sinks high voltage electron microscopy has proved to be a very valuable tool. Electron energies from about 0.5 MeV upwards are sufficient to produce Frenkel defects during the imaging. The beam intensity is extremely high such that the number of atomic displacements produced by irradiation in nuclear reactors can be exceeded by orders of magnitude within minutes. Radiation-induced formation of ordered phases like Ni3Si can thus be imaged easily by means of the dark field technique. For Cu-Be the formation of radiation-induced Guinier- Preston zones (Kell and Wollenberger, 1989) has been studied by means of field ion microscopy and that of the long range ordered y-phase CuBe by transmission electron microscopy (Koch et al., 1981). The experimental material on radiation-induced segregation was reviewed by Rehn and Okamoto (1983). Radiation driven atomic redistribution is not necessarily bound to sink annihila-
Figure 6-29. Inverse Kirkendall effect in an A-B alloy induced by fluxes of vacancies (a) and (b) and interstitials (c) and (d) to a sink. The faster moving atom is A and CK its concentration.
tion of the defects. If the rate of Frenkel pair recombination depends on the alloy composition, which might be the case for a strong influence of the chemical potential, the recombination rate would become inhomogeneous in space. This again would create persistent defect fluxes which could amplify existing concentration fluctuations (Martin, 1980). The field of phase transformations in irradiated materials has been reviewed by Russell (1984).
6.4.5 Defect Features for Ordered Alloys
Long-range atomic order is a common phenomenon for intermediate phases, for solid solutions as well as for intermetallic compounds. Examples are Cu3Au for the solid solutions and CoGa for the intermetallic compounds. For many intermetallic compounds the critical temperatures for the order-disorder transformation exceed the melting temperature which indicates high ordering energies. For such alloys the point defects show additional features to those known from pure metals or disor-
394
6 Point Defects in Crystals
dered solid solutions. Let us inspect the already quoted CoGa which belongs to the group of (3-brass electron compounds and crystallizes in the CsCl structure. This structure is made up of two cubic primitive sublattices a and P occupied by A (Co) and B (Ga) atoms, respectively. The latter atoms are positioned at the body-centered sites of the first lattice. Vacant lattice sites in the two sublattices will most probably not be energetically equivalent due to their different nearest neighbor atomic shells. The vacancy fractions in the two sublattices may be quite different which can be inferred from the following rather naive picture. The vacancy in the oc-sublattice is surrounded by only B atoms. This arrangement might suggest for the enthalpies of formation AH* = AH^ where the right hand side describes the pure B metal case. This assumption is indeed confirmed for a number of group VHI-group IIIA compounds. For Ga we have AH^ ^ 0.5 eV leading to vacancy fractions in the Co sublattice of about 10% at 900 °C. The more detailed treatment by Miedema (1979) yields AH* = 0.48 eV while equilibrium measurements yielded 0.23 + 0.06 eV. The same reasoning leads to Aif J « 1.4 eV and, hence, vacancy fractions in the (3-sublattice being many orders of magnitudes smaller than that in the a-sublattice. Certainly, such large total amounts of vacancies will significantly determine macroscopic properties of the material. How are the excess vacancies in the asublattices being formed? An excess vacancy (excess with respect to the vacancies in the P-sublattices) can be formed only by transferring an A atom into the p-sublattice. Such an atom forms an antisite defect (also called antistructure atom), i.e., a point defect specific for ordered alloys. Because of the requirement of equal numbers of sites per sublattice the one antisite defect
must be accompanied by two vacancies in the oc-sublattice. We have a triple defect. The creation of antisite defects introduces the ordering energy as a controlling parameter for concentration and mobility of the vacancies on the a-sublattice. The usual nearest neighbor jump distance « l l l > / 2 ) must be replaced by the next nearest neighbor distance «100». In particular, the vacancies are retained from annihilation during quenching or even slow cooling. By this means, vacancy concentrations of the 1 % order of magnitude can easily be observed at room temperature for a number of intermetallics. On the other hand, the degree of order does not only depend on the ordering energy but also on the formation enthalpy of the a-vacancies. This condition leads to a curved Arrhenius plot for the degree of order as being observed. For some of the intermetallics significant repulsive interaction of the vacancies has been observed and explained by the electrical charge which may be as high as one electron per vacancy (for FeAl, Koch and Koenig, 1986). Repulsive interaction leads to dispersed vacancy distribution as observed for FeAl whereas formation of large voids (50-100 nm in diameter) is observed for NiAl. For NiAl and for CoAl the vacancy concentration is significantly smaller than for FeAl and CoGa. The latter compounds show a higher degree of intrinsic disorder than the former ones. Long-range ordering of vacancies has been observed for a number of more complex intermetallics (Liu Ping and Dunlop, 1988). An important feature of many intermetallic compounds is their stability with respect to deviation from stoichiometry. The compound CoGa is found to be stable from 45 to 65 at.% Co for slowly cooled specimens. The vacancy concentration in the Co-sublattice amounts to about 10%
6.5 Point Defects in Ionic Crystals
for 45 at. % Co and falls below 0.1% at 65 at.% Co. At the stoichiometric composition it amounts to about 2.5%. At first sight, these vacancies might be taken as structural ones, i.e., formed to assure the lattice stability for the respective composition. On the other hand, the above-mentioned sluggish approach towards the thermal equilibrium concentration of vacancies does substantially aggravate the discrimination between thermal and structural vacancies. For CsCl structure compounds the existence of structural vacancies is still discussed controversially in the literature (Kim, 1986; Wever, 1992). For the intermetallic compounds of more complex structures the above-treated questions are studied to even lesser extent by systematic means which is mainly due to the great difficulties with the preparation of reliable sample material. Nevertheless, for a few cases, most interesting properties have been studied in great detail. One example is the Zintl phase (3-LiAl which is of great interest as an anode material for lithium sulfur batteries. At room temperature this material contains 0.4% and 7% Li vacancies for 47 at.% Al and 52 at.% Al, respectively. The activation energy for Li self-diffusion is of the order of 0.1 eV. The Li vacancies are long-range ordered at 80 K (Kim, 1986). The intermetallic compounds Nb 3 Sn, V3Ga, and Nb 3 Ge (A 15 structure) are of great interest because of their excellent superconducting properties. These properties are closely related to a specific feature of the A15 structure which contains linear chains of the transition metal atoms. The transition temperature for the superconducting to the normal state was found to depend strongly on the quenching temperature for V3Ga. This dependence could successfully be interpreted in terms of the antisite defects (Ga atoms within the V
395
atom chains) produced by thermal disordering at the quenching temperatures (Bakker, 1987). Diffusion properties of intermetallic compounds are reviewed by Wever (1992).
6.5 Point Defects in Ionic Crystals 6.5.1 Halides 6.5.1.1 General Remarks The point defects in alkali halides and silver halides have certainly induced more scientific work than those in metallic materials. The attraction is due to the great variety of defects and defect properties which occur and at the same time due to the many highly specific and sensitive methods of investigation which can be applied. The following sections are confined to the properties of the F-center and related defects in alkali halides. In addition, the photolytic damage process in these materials is briefly described. Important topics like the properties of hole centers, e.g., H and Fk, of the self-trapped exciton and vibronic states have been omitted here, the defect properties and the photograph process in silver halides as well. For reviews on these topics, the reader is referred to the books by Agullo-Lopez et al. (1988), by Tuchkevich and Shvarts (1981), and Crawford and Slifkin (1972). 6.5.1.2 The F-Center In ionic crystals vacancies may occur in each of the two sublattices. The possible configurations are shown in Fig. 6-1. For ionic crystals formation of a vacancy must induce a redistribution of electric charge in order to ensure the electrical neutrality for the insulator. The most simple and thus best understood defect is the halide (anion)
396
6 Point Defects in Crystals
vacancy having trapped one electron. Heating of a alkali halide in the vapor of excess alkali metal introduces excess alkali ions into the crystal. Conservation of the lattice structure requires the formation of an equal number of vacancies in the halide sublattice. The valence electron of the alkali atoms is not bound to the atom. It may migrate through the lattice and eventually become bound at a halide vacancy which in the perfect ionic lattice acts as a positive (missing negative) charge. In fact, the electron is distributed mainly on the positive metal ions neighboring the vacancy. It is held there by the electrostatic forces of the remainder of the crystal. Essentially, the defect represents a valence electron without nucleus. Such electrons may assume a number of energy states. By optical absorption it is transferred from its ground state to the first excited one. For each alkali halide there is a specific absorption band originating from this defect. Frequently it occurs in the visible part of the spectrum making the crystal to appear colored. This effect was already studied to some detail in the second decade of the century by Pohl and co-workers in Gottingen, Germany. Pohl called the defect an F-center according to the German word for color, Farbe. The quantum energies associated with the Fcenters in alkali halides range from 1.8 eV for RbBr to 5.0 eV for LiF. Most of the values lie between 2 and 3 eV. The above model of the F-center has been verified by surprisingly simple experiments. It was found that the F-band absorption is specific for the sample crystal and does not depend on the alkali metal used in the vapor. Furthermore, the colored crystals were observed to be less dense than the uncolored ones which confirms the assumption that the effect is due to a vacancy type defect. And finally, the
integrated spectral absorption corresponds quantitatively to the total number of vacancies as derived from chemical analysis of the excess alkali atoms (about 1016 to 1019 per cm3). Details of the probability of residence of the bound electron are derived from the width of the electron spin resonance line of the electron which is essentially determined by its hyperfine interaction with the nuclear magnetic moments of the adjacent alkali atoms. The method works as follows. An F-center is paramagnetic because of the unpaired electron spin. In a magnetic field, the ground state of this electron is split into two levels corresponding to the orientation of the electron spin which may be either parallel or antiparallel to the magnetic field. The electron can be transferred between these states by coupling it to an ac magnetic field of the proper frequency and oriented at a right angle to the dc field. The condition of resonance is given by hv = A£ = gfiBH, where v is the frequency of the ac magnetic field, usually in the microwave region, g is the Lande factor, fiB is the Bohr magneton, and H is the dc magnetic field. Besides the externally applied dc field the F-center electron experiences the magnetic field generated by the nuclei of the adjacent alkali atoms. This modulation of the total magnetic field causes a corresponding modulation of the resonance frequency (hyperfine interaction). The K + ions in KC1 have a nuclear magnetic moment of 3/2. The Fcenter of this crystal has 6 nearest neighbors K + which lead to a total spin varying from -I- 9 to — 9 in unit steps, leading to 19 different transitions for the electron. A few of the alkali halides like LiF, NaF, RbCl, CsCl and NaH do show such a resolved hyperfine structure in the electron spin resonance spectra. In other cases, the interaction of the electron with the nuclei of ions at greater distances smears out details of
6.5 Point Defects in Ionic Crystals
the spectrum with the result of a structureless broad absorption. Further complications arise due to the presence of isotopes with different nuclear magnetic properties. Information on the isotropic nature of the F-center can be obtained by measuring the resonance spectrum for different orientations of the crystal in the magnetic field. More detailed information on the electronic structure of the F-center was obtained by measurements of the electron nuclear double resonance (ENDOR). The method takes advantage of the possibility of saturating the electron spin resonance by working at low temperatures and applying high power inputs. For the condition of a saturated electron spin resonance signal, a radio frequency is applied which excites nuclear resonances of such nuclei which are coupled to the F-center electron. The frequency of this interaction depends on the electron density at the position of the resonating nucleus. Detection of the nuclear resonance is made by observing changes in the saturated F-center electron spin resonance. By this way, the interaction of the F-center electron with nuclei as far away as 8 shells as well as the isotopes of these ions can well be resolved. Results of
A \
20 K
I
i //
i
Q.
o
50K
.J 2.5
20 K
>78 K
\
\
1.5
J
%
1.0
Energy I eV Absorption
•
|
\\V \\\ 2.0
this method are still important data for the development of the complete theory of the F-center (Agullo-Lopez et al., 1988). As already mentioned, the absorption energies of the F-center lie between 5 eV for LiF and 1.7 eV for Rbl. The luminescence energies are smaller by about 1 eV or even 2 eV for NaF and NaCl. The width amounts to 5 to 10% for the absorption spectrum and to 10 to 15% for the luminescence spectrum. As an example, the spectra for KBr are shown in Fig. 6-30. Even at 20 K there is a considerable width. The large shift in energy between absorption and luminescence is the so-called Stokes shift, specific for optical active centers in ionic crystals. The theory of the F-center begins with the simple case of the electron in a box. Correspondingly, one obtains results for the electrons strongly localized within the vacancy. The model predicts for the transition energies Eoca^ 2 , where ax deviates from the lattice parameter by a factor of the order of one. The so-called MollowIvey law describes the experimental data by E = 57 a~177, where a is the lattice parameter in A and E in eV. The ENDOR measurements show that only 90% of the
r
r
B to
397
Emission
•
Figure 6-30. The F-center absorption and emission bands in KBr with the temperature of measurement as parameter (Gebhardt and Kuhnert, 1964).
398
6 Point Defects in Crystals
ground state wave function is confined within the vacancy. Nevertheless, higher state transitions as well as more complex defects can be treated successfully with this simple model. Data for F, F 2 and F 3 (see Table 6-3) are shown in Fig. 6-31. For higher state transitions conduction band states can be occupied leading to photoconductivity. The width of the absorption and luminescence lines as well as the Stokes shift are explained by the electron-lattice interaction which has to be calculated by means of many-body Hamiltonians. The physics of the processes can be understood qualitatively on the base of the single configuration coordinate model, derived from the Born-Oppenheimer approximation and
Table 6-3. Electron centers in alkali halides. Center
Structure
F+
A halogen ion vacancy. A lattice excitation near the F + -center produces the oc band. A halogen ion vacancy that has trapped an electron. A lattice excitation near an F-center produces the p-band. Absorption produced by transitions of the F-center to higher states give the K, L1? L2 and L 3 absorption bands. F-centers bound to one or two neighboring alkali-metal impurities. A variant of the F-center with a divalent neighboring metal ion. A halogen ion vacancy that has trapped two electrons, F' absorption band. A pair of adjacent F-centers aligned along <100>, M absorption band. A triangular array of three F-centers. Although possible arrangements are a linear chain of three vacancies, an L-shaped set in the (100) plane or a triangular set in the (111) plane, the model currently believed is the latter (111) arrangement, R absorption band. Four F-centers linked together. Two possible configurations are a parallelogram of vacancies in the (111) plane and a tetrahedron of vacancies, N absorption band. F 2 , F 3 , F 4 centers having trapped an additional electron, corresponding absorption bands are M', R' and N'.
F
F A ,F B Fz F~ F2 F3
Peak wavelength / nm
0.4
8
8
CVJ
CO
8 8 8 8 8 8 8 a "^
IO
(D
S
CO
Iflr
Flip
F4
-NaF
F^",FJ F4
-LiCI
Hole centers in alkali halides
E0.5
8 8
-LiBr -RbF NaCI
VK H
0.6 -NaBr
"CsF -KCI -Nal -Rb Cl K Br
0.7
-RbBr -Kl -Rbl
Figure 6-31. The Mollwo-Ivey plot for the vacancy defects F, F 2 and F 3 in alkali halides (Agullo-Lopez etal, 1988).
V4 HA'HB I
Self-trapped hole on a pair of halogen ions. Hole trapped along four halogen ions in three adjacent lattice sites aligned along <110>. This is a neutral crowdion interstitial. A halogen di-interstitial, probably aligned along <111>. Variants of H-centers adjacent to alkali impurity ions. Interstitial halogen ion.
6.5 Point Defects in Ionic Crystals
applied commonly in molecular physics. This model treats the energy broadening which results from configurational changes of the ions interacting with the electron of the center. Configurational changes are due to thermal vibrations and to lattice relaxations arising from electronic transitions. The influence of the thermal vibrations on both, the ground and the excited state, explains the width of the spectra. The Stokes shift arises from the fact that the absorbed photon brings the electron into an excited state which can only be a metastable one because the interacting ions relax due to the change of the electron distribution. The consumed relaxation energy is no longer available for subsequent transition to the ground state with its photoemission. At low temperatures the zeropoint vibration together with the influence of the configurational energy leads to still wide lines at low temperatures as can be seen from Fig. 6-30. The F-center levels are shown in the band scheme in Fig. 6-32. The lifetime of the excited state is of the order of 10~~6 s and becomes strongly temperature dependent above a certain temperature as can be seen in Fig. 6-33. The long lifetime arises from the fact that in the excited state 2 p and 2 s levels mix in local electric fields. The partial s character is essentially responsible for the long lifetime. The radiative decay and the thermal excitation contribute to the lifetime. The forConduction Band Emission
Absorption
Valence Band Figure 6-32. Band scheme of the F-center.
399
1CTV
o "o JZ Q_
SO
100
150
Temperature/K
Figure 6-33. Lifetime of luminesence (o) and photoconductivity (•) for the F-center in KC1 versus temperature of measurement (Swank and Brown, 1963). (Courtesy of Academic Press Ltd., London.)
mer is temperature independent while the latter causes the observed temperature dependence. From the F-center one may easily proceed to the F +-center (empty anion vacancy) and to the F~-center (two trapped electrons in the halide vacancy). The F-center has been treated here in some detail although even more properties like those of the higher excited states, the excitons etc., might be of interest. For more, the reader is referred to the review by Klick (1972), and the book edited by Fowler (1968). The present description, however, may serve as an introduction into the wealth of intrinsic and extrinsic point defects in ionic crystals with their manyfold properties. In Table 6-3 further defects are named and briefly characterized. The fundamental ones are shown in Fig. 6-34 and the band scheme is given in Fig. 6-35. The absorption lines are shown in Fig. 6-36. Among these defects are the aggregates of F-centers F 2 , F 3 , and F 4 . Higher order aggregates are called N-centers. The F 2 center absorbs light polarized parallel to <110>. The same is true for luminescence emission. The F2-center can be described
400
6 Point Defects in Crystals
Figure 6-34. Atomic configurations of a number of color centers in the alkali halides. The small black dots marked "e" indicate trapped electrons. The dashed cation stands for an impurity atom. The black ion symbols indicate defects with trapped holes.
to first approximation similar to the hydrogene molecule embedded in a dielectric continuum. Impurity-related F2-centers have been found as well. 6.5.1.3 Impurity-Related F-Centers
The migration enthalpies for vacancies amount to 0.3 to 1.5 eV depending on the Conduction Band
y/////////////////. Jbands
Valence Band
Figure 6-35. Band scheme for alkali halides, showing the ground and excited states for the A- and the Fcenter and the ground states for the F' and the V-center.
ionic crystal material. In general, the migration enthalpies of the alkali vacancies are smaller than those of the halide vacancies. For NaCl we have 0.69 and 0.77 eV, for example. Upon migration the vacancies encounter impurity atoms and become trapped here. F-centers are found to be bound to monovalent alkali metals (FA-centers) or to divalent cation impurities (Fz-centers). The FA-centers have an alkali metal impurity as a nearest neighbor (Figs. 6-34 and 6-37) and seem to only exist for impurities with a negative size misfit. Most of the information comes from ENDOR measurements. In contrast to the cubic F-center the FA-center has tetragonal symmetry which leads to splitting of the F-center band into two polarized components. With regard to luminescence two groups of FA-centers designated as FA(J) and F A (//) are observed which behave quite differently. Whereas type / behaves similar to F, type / / shows a very marked Stokes shift and a much shorter lifetime than that of the F-center. For Li in KC1 type / / has been explained by a very strong relaxation of the excited state such that the nearest neighbor Cl~ is shifted to an interstitial position at the saddle point between two adjacent anion vacancies. The electron wave function is then spread as shown in Fig. 6-37 over the two symmetric partial vacancies. Because of the low symmetry of the FA-centers reorientation can be studied by measurements of the dichroic properties of the absorption. A very low activation energy of about 0.1 eV has been found for the reorientation of F A (/) in the excited states in contrast to the ground state where the activation energy is about 0.6 eV. For FA(/7) no thermal activation of reorientation has been observed. The activation enthalpies for dissociation have been found to be 0.99 eV for Li in KC1 (Klick, 1972).
6.5 Point Defects in Ionic Crystals
401
Figure 6-36. Relative positions of the major optical absorption bands for defect containing alkali halides. Perfect crystals do not show absorption between the infrared reststrahl absorption IR and the absorption edge Eg. (Courtesy of Academic Press Ltd., London.) Photon energy
A number of F A (//(-centers have already been applied for laser emission. In particular for tunable lasers monovalent cations other than alkali metals like Ag + , Ga + , Tl + and In + have been studied. 6.5.1.4 Photolytic Damage
Figure 6-37. The FA(/)-center in KC1 with an impurity Na + and the F A (//)-center with Li + . The F A (//)center is shown in its excited state according to Liity (1968).
In alkali halides vacancies and interstitials can be produced by electromagnetic radiation like X-rays and ultraviolet light with energies close to or even less than the band gap value (5-10 eV). As this kind of radiation can directly affect only the electronic system of the solid, there must exist a mode of energy transfer for electron excitations to the lattice which ends up in the production of point defects. By ESR studies, it was found for LiF, KC1 and KBr that the radiation-induced defect is given by a dihalide molecular ion, occupying one lattice site and interacting with two additional halide ions at nearest-neighbor sites in the <110> direction (Kanzig and Woodruff, 1958). This defect, the H center, is shown in Fig. 6-34 and its absorption line in Fig. 6-36. At the same time, the absorption band of the F-center was observed in the irradiated crystals. While the interstitial and vacancy were identified here by means of electronic properties, the Frenkel defect was directly identified by compara-
402
6 Point Defects in Crystals
tive measurement of the crystal volume expansion and lattice constant increase (Simmons-Balluffi method, see Sec. 6.4.1.1) upon irradiation. The result (Balzer et al, 1968) is shown in Fig. 6-38. The relative changes of the sample length and that of the lattice constant are equal within the experimental uncertainty. This result excludes the production of Schottky defects as observed in metals upon thermal activation (Sect. 6.4.1.1) and can only be understood for Frenkel defects (see Fig. 6-1). Hence, the production of Frenkel defects of formation energies around 3 to 4 eV by electronic excitations between 5 and 10 eV is to be explained. Double or multiple excitations of the halide ions have been investigated as potential energy accumulators which then become electrostatically ejected into interstitial positions. Capture of an electron would then produce the F-centerH-center pairs. It was found that the short
lifetimes of the multiply excited states do not explain the observed production efficiencies. The defect production must, therefore, be explained by single ionization. The now widely accepted explanation starts from a self-trapped exciton which equals a Vk center (X^ in Fig. 6-34) plus an electron. The two halide ions of the Vk center are displaced considerably from their regular lattice sites towards each other. The recombination of the electron-hole pair provides 7 eV electrostatic energy which is released to the Fk ion pair. The resulting momentum points along <110>. As we know from Sec. 6.4.2.1, replacement collision sequences propagate along this close packed direction with minimum transversal energy loss. Hence, formation of a wellseparated Frenkel defect (larger distance than the radius of the spontaneous recombination volume) is likely to occur. Important for the distance achieved are the in-
•
y • m•
f •
• •
• •
0
V
/
10-A///
Figure 6-38. Comparison of the relative changes of lattice parameter a and sample length / for X-ray irradiated alkali halides (• KBr, • NaF, • LiF, o KC1), according to Balzer et al. (1968).
6.5 Point Defects in Ionic Crystals
stantaneous charge state of the moving ion (neutral would be the optimum) and the ratio of the ionic distance to the ionic diameter (focussing parameter). The instantaneous charge state controls the transversal energy release of the moving ion. The influence of the focussing parameter has been studied by radiation-induced F-center production in various alkali halides. Within a critical range of the focussing parameter an increase of less than the factor two reduces the total damage energy elapsed per F-center by about four orders of magnitude (Rabin-Klick diagram, see Agullo-Lopez et al., 1988).
6.5.1.5 Impurities
Impurities are known as cation impurities as well as anion impurities. Alkali, alkaline-earth and transition metal ions are positioned substitutionally in the alkali sublattice. Characteristic optical absorption and luminescence bands are observed. Of special interest are the monovalent cations with larger negative size misfits like Li + , Cu + and Ag + . Their equilibrium positions are off-center substitutional. For Li + in KC1 the off-center position is at a <111) axis which leads to eight equivalent positions around the substitutional site. The impurity may tunnel between these sites, thereby producing multiply split vibration levels. Furthermore, the defect shows elastic as well as electric dipole moments, which may be reoriented by extrinsic elastic and electric fields, thus contributing to the respective susceptibility. Also localized modes in the vibrational spectra are observed similar to those described in Sec. 6.4.2.3. For Li in KBr even the isotope effect of 6Li and 7Li on the localized mode has been measured (review by Nowick, 1972).
403
Another type of defect is produced by divalent cation impurities. The excess charge with respect to the host cation generates a neutralizing cation vacancy which by Coulomb attraction is located at the first or second neighbor cation site of the impurity. This leads to impurity-vacancy dipole formation along <110> for the former location of the vacancy and along <100> for the latter one. Direct evidence for both types of dipoles were found by electron paramagnetic resonance measurements for the impurities Mn 2 + and Eu2 + in several alkali halides. As the vacancy concentration in these crystals is entirely determined by the concentration of the divalent impurities, the thermally activated escape of vacancies can easily be followed by means of the electrical conductivity provided by the free vacancies. The Arrhenius behavior of the conductivity yields binding energies between 0.4 eV and 0.8 eV for different host-impurity combinations. Anion impurities are halogen ions, H~, H°, OH", O2~, CN" and NO 2 , for example. Most studied are the H~ and the OH~ impurities as they occur as unavoidable contaminants when growing crystals in the presence of alkali hydride or in air. The H " substitutes for a host halogen ion forming the U center which shows a specific ultraviolet absorption band. By illumination at this band, the conversion from substitutional H~ to interstitial H~ is induced. A localized vibrational mode of the U-center gives rise to an infrared absorption in the 10-30 |um range. 6.5.2 Oxides 6.5.2.1 General Remarks
Oxide materials may be characterized by their mixed bonding nature. Properties of point defects are determined mainly by
6 Point Defects in Crystals
ionic bonding for some of the oxides and by covalent bonding for others. Examples are MgO for the first species and quartz (SiO2) for the latter one. The relevance and application of oxide materials is broadly spread all over the branches of modern technology. Corrosion products of metallic materials, materials for lasers and integrated optics, high Tc superconductors, piezo- and ferroelectric actuators and fuel cells do characterize the multiplicity of interesting properties which in most of the cases are governed by point defect properties. Such have been studied earlier for MgO, SiO2 and A12O3 (sapphire) while later more attention has been paid to materials like LiNbO 3 , BaTiO 3 and Bi 4 Ge 3 O 12 . Reviews on defects in oxides are given in the book edited by S0rensen (1981). 6.5.2.2 Alkaline-Earth Oxides For MgO Schottky disorder is assumed to be energetically favorable with the enthalpy of formation amounting to about 4 eV as derived from conductivity data. Reported enthalpies of vacancy migration ange from 1.6 to 2.13 eV for the Mg vacancy and from 1.7 to 3.8 eV for the O-vacancy. As for the alkali halides, a number of centers have been characterized by paramagnetic resonance and optical spectroscopy. The paramagnetic F +-center consists of an oxygen vacancy having one trapped electron. Despite of the positive charge it corresponds to the F-center in alkali halides with respect to its main properties. Two electrons trapped at the oxygen vacancy form the F-center being neutral and diamagnetic. The absorption and luminescence bands are positioned around peak energies of about 5 eV and 3 eV, respectively, for the F +-center and around 5 eV and 2.4 eV, respectively, for the F-cen-
ter. For other binary oxides like CaO, SrO and BaO, these energies are somewhat smaller. For a-Al 2 O 3 they are even larger than for MgO. The optical bands are markedly asymmetric as can be seen in Fig. 6-39. The curves also show resolved phonon resonances. The peak situated between the emission and the absorption bands is the zero-phonon line which results from a pure electronic transition. The band shape has been explained by dynamic Jahn-Teller coupling with noncubic vibration modes. The effective charge of the F + center leads to a more localized ground state wave function than that of the F-center in alkali halides. The F +-center also causes larger lattice distortions. The nuclear quadrupole hyperfine effects are more pronounced. The overlap of the electron wave function with orbitals of the adjacent ions causes larger shifts of the Lande factor g according to electron paramagnetic resonance (EPR) spectroscopy. A number of V-centers have been identified by means of EPR spectroscopy. For the V"-center, the hole is trapped at an alkaline earth vacancy which results in an O~ ion at an adjacent anion site. For MgO, this center exhibits an optical ab-
13
ZPl
404
03 0)
/ /
c
£
/
L
\j... i
3.0
3.5
4.0
Energy/eV Figure 6-39. Absorption ( ) and emission ( ) features for the F + -center in CaO x (ZPL-zerophonon line), according to Henderson et al. (1972).
6.5 Point Defects in Ionic Crystals
405
sorption band in the ultraviolet with 2.3 eV peak energy. The V°-center is the cation vacancy with two trapped holes showing an optical absorption band with about the same peak energy as given above. More hole centers are formed with OH~ or an impurity ion adjacent to the cation vacancy. Review on defects in alkaline-earth oxides is given by Henderson and Wertz (1968).
6.5.2.3 Transition-Metal Oxides
This group of oxides is characterized by a still higher complexity of existing defect species and their structures. In addition, nonstoichiometry creates the structural vacancies which may occur to very high concentrations with specific configurations. From the oxides TiO, VO, MnO, FeO, CoO and NiO, all of the rock-salt structure, the first two are metallic and all others are paramagnetic semiconductors. The range of nonstoichiometry is remarkably broad for FeO ranging from 5% deficiency of Fe to 16% surplus. From the iron deficient range, the structure shown in Fig. 6-40 was found to be in accordance with the experimental and theoretical findings (Catlow etal., 1977). The so-called 4:1 cluster consists of a tetrahedral arrangement of 4Fe 2 + vacancies with a Fe 3 + interstitial at the center. Additional Fe 3 + interstitials at neighboring positions provide the necessary compensation of charge. Also shown in this figure is the 6:2 cluster resulting from 4:1 cluster aggregation. Higher aggregates have also been discussed. Substantial clustering is expected for FeO and MnO above 1% cation deficiency. The presence of high concentrations of intrinsic defects is not confined to nonstoichiometric compositions. For TiO and VO about 15% vacancies distributed to the two sub-
Figure 6-40. Defect clusters in MnO type oxides. Lower part: four M 2 + vacancies (O) in tetrahedral configuration with a M 3 + ion (®) at the central interstitial position (4:1 cluster). Upper part: two 4:1 clusters may form the 6:2 cluster. (Courtesy of Academic Press Ltd., London.)
lattices are observed for the exactly stoichiometric composition. The noncubic binary transition-metal oxides respond to nonstoichiometry by forming extended planar defect arrangements, the so-called crystallographic shear planes. They are formed by a rearrangement of the MO 6 octahedra (M stands for the respective metal ion) building up the lattice structure. Examples are WO 3 and MoO 3 . The crystallographic orientation of the shear planes depends on the deviation from stoichiometry. Often, the planes form long-range ordered arrays. Oxides showing high electrical conductivity are of the type MO 2 with M being a tetravalent transition metal or rare-earth cation. They crystallize in the fluorite structure. High conductivity occurs after doping such crystals with divalent M 2 + cations appearing as trivalent M 3 + cations. Electrical neutrality is achieved by the creation of oxygen vacancies. Vacancy formation has been concluded from den-
406
6 Point Defects in Crystals
sity and X-ray measurements. The impurity vacancy pairs have been identified by dielectric and inelastic relaxation studies. The migration enthalpy for the oxygen vacancies has been found to be about 1 eV and that of the cation vacancies between 4 eV and 5 eV. At high impurity concentrations short- and long-range defect order was observed. The dominant electronic defect of most of the transition metal oxides is the selftrapped electron (small polaron, Holstein, 1959). It migrates by thermally activated hopping among equivalent trapping sites. Mobilities are of the order of 10-4m2V-1s-1. No color centers have been reported for these oxides.
6.5.2.4 Other Oxides Further oxides of great practical importance are sapphire (a-Al2O3), the (3-alumina ( M ^ A l ? t - x i i P l i w i t h 0 < x < 0.3 and M + = Na + , Li + , K + , Ag+ and more), quartz and silica (crystalline and amorphous SiO2) and the ternary oxides (perovskites or related structures). Sapphire has been an ideal substance for color center studies for more than 30 years. It has a broad band gap, it can easily be doped with impurities in a well-defined way, i.e., with a low level of natural impurities and has a rhombohedral structure. Impurity doping creates color centers which, because of the lattice structure, exhibit anisotropic absorption and emission features. Such behavior eases detailed checks of color center models. The P-alumina are known for their superionic conducting property. In the nonstoichiometric compositions the M + ions are highly mobile within definite lattice planes which leads to a distinct anisotropy
of the electrical conductivity (review by Geller, 1977). Quartz, like sapphire, has a large band gap and has been studied for about 30 years with respect to defect properties. No less than 15 different centers have been classified (Abu-Hassan, 1987). Synthetic quartz can be produced with very high purity and very low dislocation densities leading to quite an ideal reference material. Similar defect properties like in quartz are found in amorphous SiO 2 . This similarity might be due to the similarity in the shortrange atomic structure. Silica shows a significant short-range order of the SiO4 tetrahedra. A common impurity of silicon is the OH radical. One differentiates between dry or waterfree and wet silica depending on the OH concentration level. The latter one would contain around 1000 ppm OH. The presence of this radical may be responsible for radiolytic damage production at room temperature. For quartz such effect is observed only at temperatures below 100 K. The ternary oxides crystallize in the perovskite or related structures. Examples are LiNbO 3 , SrTiO3 and BaTiO 3 . They are of great practical importance because of their pyroelectric and electrooptic properties. Photochromic and photorefractive properties are produced by electron and hole photoionization and specific trapping capabilities. Again, oxygen vacancies seem to be the most important defects although cation vacancies may be present under respective nonstoichiometric conditions. By impurity doping a large variety of centers may be produced particularly because transition metal impurities may appear as multiply charged ions with charges from + 2 e to + 5 e.
6.7 References
6.6 Acknowledgements Ms. Heike Krzyzagorski and Ms. Ulrike Wollmann kindly phonotyped the manuscript. Ms. Dagmar Kopnick and Ms. Katharina Macht took care of the figure preparation. Dr. Christian Abromeit was always ready to help clarifying difficult problems with the matter treated here. My wife Hanna was patient once again during the weekend-time-consuming preparation of this chapter.
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Lam, N. Q., Doan, N. V, Dagens, L., Adda, Y (1981),./ Phys. Fll, 2231. Lechner, R. (1983), in: Mass Transport in Solids: Beniere, F., Catlow, C. R. A. (Eds.). New York: Plenum Press, pp. 169-226. Le Claire, A. D. (1978), /. Nucl. Mater. 69 & 70, 70-96. Leibfried, G. (1965), Bestrahlungseffekte in Festkorpern. Stuttgart: Teubner. Lengeler, B. (1976), Phil. Mag. 34, 259. Liu Ping, Dunlop, G. L. (1988), /. Mater. Sci. 23, 1419-1424. Liity, F. (1968), in: Physics of Color Centers, Fowler, W. B. (Ed.). New York: Academic Press, pp. 182242. Mantl, S., Triftshauser, W (1978), Phys. Rev. B17, 1645. Martin, G. (1980), Phys. Rev. B21, 2122. Mehrer, H., (Ed.) (1990), Diffusion in Solid State Metals and Alloys, Landolt-Bornstein (New Series), Group III, Vol. 26. Heidelberg, Berlin: SpringerVerlag. Miedema, A. R. (1979), Z. Metallkunde 70, 345-353. Mott, N. F (1932), Proc. Roy. Soc. A135, 429. Naundorf, V, Wollenberger, H. (1990), /. Nucl. Mater. 174, 141. Nicklow, R. M., Crummett, W P., Williams, X M. (1979), Phys. Rev. B20, 5034.
6.7 References
Norgett, M. X, Robinson, M. T., Torrens, I. M. (1974), Nucl. Eng. Design 33, 50. Nowick, A. S. (1972), General and Ionic Crystals, Vol. 1 of Point Defects in Solids. New York: Plenum Press, pp. 151-200. Peisl, X, Franz, H., Schmalzbauer, A., Wallner, G. (1991), in: Defects in Materials, Materials Research Society Symposium Proceedings, Vol. 209: Bristowe, P.D., Epperson, J.E., Griffith, J.E., Liliental-Weber, Z. (Eds.). Pittsburgh, PA: Mat. Res. Soc, pp. 271-282. Petry, W., Vogl, G., Mansel, W. (1982), Z. Phys. B46, 319. Poerschke, R., Wollenberger, H. (1980), Radiation Effects 49, 225-232. Rauch, R., Peisl, X, Schmalzbauer, A., Wallner, G. (1990), J. Phys.: Condens. Matter 2, 9009-9017. Rehn, L. E., Robrock, K. H. (1977), J. Phys. F7, 1107. Rehn, L. E., Okamoto, P. R. (1983), in: Phase Transformations During Irradiation. London: Applied Science Publishers, pp. 247-290. Rehn, L. E., Robrock, K. H., Jaques, H. (1978), /. Phys. F8, 1835. Robrock, K.-H. (1983), in: Phase Transformations During Irradiation. London: Applied Science Publishers, pp. 115-146. Robrock, K.-H. (1989), Mechanical Relaxation of Interstitials in Irradiated Metals, in: Springer Tracts in Modern Physics, Vol. 118. Berlin: Springer-Verlag. Robrock, K.-H., Schilling, W. (1976), /. Phys. F6, 303. Ruhle, M., Wilkens, M. (1983), in: Physical Metallurgy: Cahn, R. W, Haasen, P. (Eds.). Amsterdam: North-Holland, pp. 788-792. Russell, K. C. (1984), Prog. Mater. Sci. 28, 229. Schilling, W, Burger, G., Isebeck, K., Wenzl, H. (1970), Annealing Stages in the Electrical Resistivity of Irradiated fee. Metals, in: Vacancies and Interstitials in Metals: Seeger, A., Schumacher, D., Schilling, W, Diehl, X (Eds.). Amsterdam: NorthHolland, p. 255. Scholz, A., Lehmann, C. (1972), Phys. Rev. B6, 813. Schroeder, K. (1980), Theory of Diffusion Reactions of Point Defects in Metals, in: Springer Tracts in Modern Physics, Vol. 87. Berlin: Springer-Verlag, pp. 171-262. Schultz, H. (1991), Mater. Sci. Eng. A 141, 149-167. Seeger, A. (1973), /. Phys. F3, 248. Seidman, D. N. (1978), Surf. Sci. 70, 532. Seitz, E, Koehler, X S. (1956), Displacement of Atoms During Irradiation, in: Solid State Physics, Vol. 2: Seitz, E, Turnbull, D. (Eds.). New York: Academic Press, p. 305. Simmons, R. O., Balluffi, R. W. (1960), Phys. Rev. 117, 52. Simmons, R. O., Balluffi, R. W. (1963), Phys. Rev. 129, 1533.
409
Smigalskas, A. D., Kirkendall, E. O. (1947), Trans. AIME 171, 130. Sorensen, O. T. (Ed.) (1981), Nonstoichiometric Oxides. New York: Academic Press. Sonnenberg, K., Schilling, W, Mika, K., Dettmann, K. (1972), Radiation Effects 16, 65. Stott, M. X (1978), /. Nucl. Mater. 69/70, 157. Swank, R. K., Brown, F. C. (1963), Phys. Rev. 130, 34. Swanson, M. L., Howe, L. M. (1983), Nucl. Instr. Methods 218, 613. Takamura, X Shirai, Y, Furukawa, K., Nakamura, F. (1987), Mater. Sci. Forum 15-18, 809-828. Trifthauser, W, McGervey, X D. (1975), Appl. Phys. 6, 171. Tuchkevich, V M., Shvarts, K. K. (Eds.) (1981), Proc. Intern. Conf. on Defects in Insulating Crystals, Riga. Berlin: Springer-Verlag. Urban, K., Yoshida, N. (1981), Phil. Mag. A 44,1193. Urban, K., Saile, B., Yoshida, N., Zag, W. (1982), in: Point Defects and Defect Interactions in Metals, Takamura, X, Doyama, M., Kiritani, X (Eds.). Tokyo: Univ. of Tokyo Press, pp. 783-788. Van Bueren, H. G. (1961), Interperfections in Crystals. Amsterdam: North-Holland, p. 300. Vogl, G., Petry, W, Sassa, K., Mantl, S. (1982), in: Point Defects and Defect Interactions in Metals. Tokyo: Univ. of Tokyo Press, pp. 33-40. Wagner, C , Beyer, X (1936), Z. Phys. Chem. B32, 113. Wagner, R. (1982), Field-Ion Microscopy, in: Crystals, Vol. 6. Berlin: Springer-Verlag. Wenzl, H. (1970), Physical Properties of Point Defects in Cubic Metals, in: Vacancies and Interstitials in Metals: Seeger, A., Schumacher, D., Schilling, W, Diehl, X (Eds.). Amsterdam: North-Holland, p. 363. Wever, H. (1992), in: Diffusion in Solids - Unsolved Problems: Murch, G. E. (Ed.). Aedermannsdorf (Switzerland): Trans. Tech. Public, pp. 55-72. Wichert, Th. (1982), in: Point Defects and Defect Interactions in Metals: Takamura, XL, Doyama, M., Kiritani, M. (Eds.). Tokyo: Univ. of Tokyo Press, pp. 19-26. Wichert, T. (1987), Mater. Sci. Forum 15118, 829848. Wiedersich, H., Okamoto, P. R., Lam, N. Q. (1979), J. Nucl. Mater. 83, 98. Wiedersich, H., Lam, N. Q. (1983), in: Phase Transformations During Irradiation. London: Applied Science Publishers, pp. 1 -46. Wiedersich, H. (1990), in: Radiation Effects and Defects in Solids 113, 97-108. Wollenberger, H. X (1983), in: Physical Metallurgy, Vol. 2: Cahn, R. W, Haasen, P. (Eds.). Amsterdam: Elsevier Science Publishers, pp. 1140-1221. Wollenberger, H. (1990), Nucl. Instr. and Methods in Phys. Res. £¥#,493-498.
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6 Point Defects in Crystals
Wollenberger, H. (1970), Production of Frenkel Defects during Low-Temperature Irradiations, in: Vacancies and Interstitials in Metals: Seeger, A., Schumacher, D., Schilling, W, Diehl, J. (Eds.). Amsterdam: North-Holland, p. 215. Wollenberger, H. (1978), 1 Nucl. Mater. 69/70, 362371.
General Reading Abromeit, C , Wollenberger, H. (Eds.) (1987), Mater. Sci. Forum 15-18, 1-1426. Agullo-Lopez, K, Catlow, C. R. A., Townsend, P. D. (1988), Point Defects in Materials, London: Academic Press. Crawford, J. H., Jr., Slifkin, L. M. (Eds.) (1972), General and Ionic Crystals, Vol. 1 of Point Defects in Solids. New York: Plenum Press. Dederichs, P. H., Zeller, R. (1980), Dynamical Properties of Point Defects in Metals, in: Springer Tracts
in Modern Physics, Vol. 87: Hohler, G., Niekisch, E.A. (Eds.). Berlin: Springer-Verlag. Ehrhart, P., Robrock, K.H., Schober, H.R. (1986), in: Physics of Radiation Effects in Crystals: Johnson, R. A., Orlov, A. N. (Eds.). Amsterdam: Elsevier Science Publishers, pp. 3-115. Robrock, K.-H. (1989), Mechanical Relaxation of Interstitials in Irradiated Metals, in: Springer Tracts in Modern Physics, Vol. 118. Berlin: Springer-Verlag. Schroeder, K. (1980), Theory of Diffusion Reactions of Point Defects in Metals, in: Springer Tracts in Modern Physics, Vol. 87. Berlin: Springer-Verlag, pp. 171-262. Sorensen, O.T. (Ed.) (1981), Nonstoichiometric Oxides. New York: Academic Press. Wollenberger, H.J. (1983), in: Physical Metallurgy, Vol.2: Cahn, R.W., Haasen, P. (Eds.). Amsterdam: Elsevier Science Publishers, pp. 1140-1221. Ullmaier, H. (Ed.) (1991), Atomare Fehlstellen in Metallen, Landolt-Bornstein (New Series), Group III, Vol. 25. Berlin: Springer-Verlag.
7 Dislocations in Crystals David J. Bacon Department of Materials Science and Engineering, University of Liverpool, Liverpool, U.K.
List of 7.1 7.1.1 7.1.2 7.1.3 7.1.4 7.1.5 7.2 7.2.1 7.2.2 7.2.3 7.2.4 7.2.5 7.2.6 7.2.7 7.3 7.3.1 7.3.2 7.3.3 7.3.4 7.3.5 7.3.6 7.3.7 7.4 7.4.1 7.4.2 7.4.3 7.5 7.5.1 7.5.2 7.6 7.6.1 7.6.2 7.6.2.1 7.6.2.2
Symbols and Abbreviations Introduction Background to the Study of Dislocations Geometry of Dislocations The Burgers Circuit Dislocation Core Structure Observations of Dislocations Movement of Dislocations Crystal Plasticity Glide and Slip Cross Slip The Peierls Barrier Velocity of Dislocations Climb Plastic Strain due to Dislocation Movement Dislocations in Elastic Media Elements of Elasticity Theory Elastic Field and Energy of a Straight Dislocation Forces on Dislocations Interaction Between Dislocations Image Forces Interaction Between Point Defects and Dislocations General Line Shapes and Anisotropic Media Generation of Dislocations Thermodynamic Considerations Nucleation at Stress Concentrations Multiplication Sources Dislocations 4En Masse' Intersection of Dislocations Dislocation Pile-Ups Dislocations in Particular Crystal Structures Introduction The fee. Metals Dislocations and Slip Other Dislocations
Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. Allrightsreserved.
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7.6.3 7.6.4 7.6.5 7.6.6 7.6.7 7.7 7.8
7 Dislocations in Crystals
The h.c.p. Metals The b.c.c. Metals Ionic Crystals Ordered Alloys Covalent Crystals Concluding Remarks References
464 468 471 474 477 479 480
List of Symbols and Abbreviations
List of Symbols and Abbreviations a A b c, c 0 c/a Ct d e etj E E core Ecl Ef, Em Eint EY JBj £P /, F G* kB K / n N p Ptj q ra r0 R S t T u v V Vh, Fs F mis w
lattice parameter, atomic spacing, interplanar spacing anisotropy ratio Burgers vector concentration, equilibrium concentration (of vacancies) lattice parameter ratio for h.c.p. metals transverse sound velocity spacing of partial dislocations electronic charge component of strain tensor dislocation line energy (per unit length) dislocation core energy (per unit length) dislocation elastic energy (per unit length) formation, migration energy (of point defects) dislocation-dislocation interaction energy dislocation-point defect interaction energy j o g energy Peierls barrier energy chemical, mechanical force on a dislocation Gibbs free energy of activation Boltzmann's constant prelogarithmic energy factor of a straight dislocation vector segment of line unit vector normal to a surface number of dislocations pressure dipole moment tensor electric charge radius of point defect dislocation core radius radius, radius of curvature surface time temperature displacement vector dislocation velocity volume hole, sphere volume (of point defects) misfit volume dislocation core width
y F d
stacking-fault energy line tension of a dislocation misfit parameter
413
414
7 Dislocations in Crystals
dij e £ijk e p , 8p 0D X \i v Q, Qm a,
misfit strain misfit parameter or strain permutation tensor plastic strain, plastic strain rate Debye temperature Lame constant or core parameter in dislocation energy shear modulus Poisson's ratio density, mobile density (of dislocations) stress, normal component of stress component of stress tensor stress factor for a straight dislocation resolved shear stress, critical and maximum values Peierls stress vibration frequency of dislocation volume per atom
APB b.c.c. CRSS CSF f.c.c. h.c.p. RSS SISF TEM
antiphase boundary body-centred cubic critical resolved shear stress complex stacking fault face-centred cubic hexagonal close packed resolved shear stress superlattice intrinsic stacking fault transmission electron microscopy
7.1 Introduction
7.1 Introduction 7.1.1 Background to the Study of Dislocations
In the 50 to 60 years since dislocations were "discovered", they have become the most intensely investigated of all the defects in crystals. The reason for this is not hard to find, for either individually or collectively, dislocations influence the chemical, mechanical and physical behavior of crystalline solids, and an understanding of their properties is essential for the successful development of models and theories of this behavior. To say that dislocations were "discovered" in the 1930s is somewhat misleading, for their existence in observations made in the nineteenth and early twentieth centuries is now clearly recognized (Hirth and Lothe, 1982; Nabarro, 1967,1984), and some of the underlying theory of dislocation properties was formulated long before their discovery (Volterra, 1907; Love, 1927). However, it was not until the work of Orowan (1934), Polanyi (1934), Taylor (1934) and Burgers (1939) that these geometrical defects on the atomic scale were found to influence the macroscopic properties of materials. The background to this development was investigation of the response of metal crystals to stresses above the elastic limit, i.e. their plasticity, and the prime stimulus to subsequent research on dislocations has come from the need to explain and improve the mechanical properties of crystalline solids. As noted above, however, dislocations can have a profound influence on other properties, and it is important to bear this in mind when studying their characteristics. The key to appreciating their wide-ranging significance is their geometry at the atomic level in the crystal in question. Once this is understood, the effect of dislocations on each other and on
415
other crystal defects, and their role in phenomena such as mass and charge transport can be developed by more detailed exposition. This is the approach developed in this chapter, for it provides an introductory review of dislocation properties in general and their structure in common crystal systems. It offers a background to the more specialized descriptions of dislocation-related processes given in other chapters of this series. We first provide a brief introduction to what dislocations actually are and how they may be observed. Their specific properties are explained in later sections. 7.1.2 Geometry of Dislocations
A dislocation is a line defect in a crystal, by which we mean that it is rather akin to a "tube", usually only a few atom spacings across, threading its way through the crystal. Within the tube, atoms may be displaced with respect to each other so that their coordination is quite distinct from that in the perfect crystal, whereas outside the coordination is approximately perfect. As we shall see, there is no clear distinction between the interior and exterior of the tube, but rather a gradual transition: nor need the tube be circular in section. This somewhat ill-defined region is called the dislocation core (see Sec. 7.1.4). The simplest geometry is that of the edge dislocation, first proposed independently by Orowan, Polanyi and Taylor in 1934. An example in a simple cubic crystal only three atomic planes thick is shown in Fig. 7-1, where the line is straight and perpendicular to the page. It can be seen that the distortion is concentrated within the core region at the centre of the diagram. Furthermore, this distortion is that which would arise if a half-plane of atoms is inserted into an otherwise perfect region of
416
7 Dislocations in Crystals
Figure 7-1. An edge dislocation in a simple cubic crystal. (From Hirth and Lothe, 1982; reprinted with permission of John Wiley and Sons, Inc.)
crystal. The dislocation runs along the bottom of this extra half-plane in Fig. 7-1. For simplicity, this edge dislocation is often denoted by the symbol _L, the vertical stroke showing the orientation of the extra halfplane. The other elementary geometry to consider is that of the screw dislocation. This is depicted in a simple cubic crystal in Fig. 7-2. Here, the line runs from top-to-bottom of the block, and is seen to convert the
Figure 7-2. A screw dislocation in a simple cubic crystal. (From Hirth and Lothe, 1982; reprinted with permission of John Wiley and Sons, Inc.)
planes parallel to the top and bottom faces into a single helicoidal surface. Again, the crystal is almost perfect outside the highly distorted core. It may be noted that the screw dislocation in Fig. 7-2 results in a helicoidal surface with a right-handed thread. A lefthanded screw is equally possible, and it is easy to envisage that if these two dislocations were to coincide, they would mutually annihilate and leave a perfect crystal. Similarly, the edge dislocation in Fig. 7-1, call it "positive", has a "negative" counterpart with the extra half-plane orientated thus: T. These edge dislocations are physical opposites and mutually annihilate if brought into coincidence. It is clear, therefore, that the labels "edge" and "screw" are insufficient to uniquely define the physical nature of a dislocation. We shall now show that there exists a vector which, together with the direction of the dislocation line, enables a unique description to be achieved. 7.1.3 The Burgers Circuit
The Burgers vector b is defined by the Burgers circuit construction (after Frank, 1951). A Burgers circuit is a closed, atomto-atom path taken in a crystal containing dislocations. Such a path is illustrated in Figs. 7-3 a and c for edge and screw dislocations, i.e. MNOPQ. It is essential that the circuit in the real crystal passes entirely through almost perfect material. If the same atom-to-atom sequence is made in a dislocation-free crystal and the circuit does not close, then the first circuit must enclose one or more dislocations. The vector required to complete the circuit is called the Burgers vector. This is demonstrated in Figs. 7-3 b and d, from where it is seen that the closure failure QM, i.e. Burgers vector, is perpendicular to the dislocation for the
7.1 Introduction
417
(a)
(b)
p Burgers
o
N
Figure 7-3. Burgers circuit round (a) an edge and (c) a screw dislocation, (b) and (d): the same circuit in perfect crystals. (From Hull and Bacon, 1984; reprinted with permission of Pergamon Press, Ltd.)
edge line and parallel to it for the screw. This result provides the definition of edge and screw dislocations. In the most general situation, dislocations may not be exactly parallel or perpendicular to 6, in which case the line is mixed, having a combined edge and screw character, as given by the vector sum of these two components (see Fig. 7-4). However, the total Burgers vector of a single dislocation has fixed crystallographic length and direction, and is independent of the position and orientation of the dislocation line.
Dislocation
Figure 7-4. The Burgers vector of a mixed dislocation line resolved into its edge and screw components bp = b sin a and b. = b cos a.
418
7 Dislocations in Crystals
Burgers circuits taken around other defects, such as vacancies and interstitials, do not lead to closure failure. A convention is implied by the Burgers circuit construction used above. First, when looking along the dislocation line, which defines the positive line sense /, the circuit is taken in a clockwise fashion (see Fig. 7-5). Second, the Burgers vector is taken to run from the finish to the start point of the reference circuit in the perfect crystal. This has become known as the FS/RH convention (Bilby et al., 1955). Mathematically, b for a continuum is given by the line integral of the elastic displacement u taken around a circuit C about the dislocation (Fig. 7-5) b = § dw c
(7-1)
It is readily shown by use of sketches similar to those of Fig. 7-3 that reversing the line sense reverses the direction of the Burgers vector for a given dislocation. Furthermore, dislocations with the same line sense but opposite Burgers vectors (or alternatively with opposite line senses and the same Burgers vector) are physical opposites, in that if one is a positive edge, the other is a negative edge, and if one is a right-handed screw, the other is lefthanded. The Burgers vectors defined in the simple cubic crystals above are the shortest lattice translation vectors which join two points in the lattice. A dislocation whose Burgers vector is a lattice translation vector is known as a perfect or unit dislocation. Dislocation lines can end at the surface of a crystal and at grain boundaries in polycrystals, but not in an otherwise perfect region of crystal. Thus, dislocations must either form closed loops or branch into other dislocations. When three or more dislocations meet at a node, it is a necessary condition that the Burgers vector is con-
Dislocation Figure 7-5. The clockwise circuit for a positive line direction /.
served. Consider the dislocation bx (Fig. 7-6) which branches into two dislocations with Burgers vectors b2 and b3. A Burgers circuit has been drawn round each according to the line senses indicated, and it follows from the diagram that b± = b2 + b3
(7-2)
The large circuit on the right-hand side of the diagram encloses two dislocations but since it passes through the same good material as the bx circuit on the left-hand side, it is equivalent to it and the Burgers vector must be the same, i.e. b x. It is easy to see that if all line senses are taken as positive pointing away from the node, then for N dislocations N
I* ;
=0
(7-3)
The dislocation density Q is a measure of the total dislocation content of a crystal,
Line sense
b2 + b3 =
Line
Line sense
Figure 7-6. A dislocation node.
7.1 Introduction
and is defined as the total length of dislocation line per unit volume. Thus for a volume V containing line length L, Q = L/V. An alternative definition, which is sometimes more convenient to use, is the number of dislocations intersecting a unit area. If all the dislocations are parallel, the two density values are the same, but for a completely random arrangement the volume density is twice the surface density. In wellannealed metal crystals Q is usually between 106 and 108 cm" 2 , but it increases rapidly with plastic deformation, and a typical value for a heavily cold-rolled metal is about 1010 to 1011 cm" 2 , Q is usually lower in non-metallic crystals and values down to 10cm~ 2 can be obtained in carefully grown semiconductor crystals. 7.1.4 Dislocation Core Structure It was noted earlier that most of the disregistry of atoms caused by a dislocation is concentrated within a core region, which is somewhat ill-defined. It has proved useful to have a means of quantifying this disregistry, and we now describe one means of doing this. Again, we consider for simplicity a simple cubic crystal. When an extra half-plane of atoms is inserted by the creation of an edge dislocation of Burgers vector A, the atoms in planes above (A) and below (B) the slip plane (see Sec. 7.2.2) are displaced by w, as illustrated in Fig. 7-7 a. To accommodate the dislocation, there is a disregistry of atomic coordination across the slip plane which is defined more fully in Sec. 7.2.2. This disregistry is defined by the displacement difference Aw between two atoms on adjacent sites above and below the slip plane i.e. Aw = w(B) — w(A). The form of Aw, in units of b, versus x is shown in Fig. 7-7 b. The width of the dislocation w is defined as the distance over which the magnitude of the disregistry is greater than
419
(a)
*- A
Slip ( plane
Au/b
(b)
+ 0.5
Figure 7-7. (a) Open and full circles represent the atom positions before and after the extra half-plane is inserted. The points P and P' mark a shift of the atoms at x = 0 as will be discussed later on in Sec. 7.2.4. (b) The displacement difference AM across the slip plane for the dislocation in (a). (From Hull and Bacon, 1984; reprinted with permission of Pergamon Press, Ltd.)
one-half of its maximum value, i.e. over which — b/4 < Aw < b/4. This parameter is shown in Fig. 7-7 b. The width provides a measure of the size of the dislocation core, i.e. the region within which the displacements and strains are unlikely to be close to the values of elasticity theory (see Sec. 7.3.2). Typical forms of Aw curves for an edge dislocation in a simple cubic crystal are shown schematically in Fig. 7-8: in these cases, Aw has been increased by b when negative in order to produce continuous curves. The core widths of undissociated dislocations (see Sec. 7.6) in Figs. 7-8 a and b are usually found by computer simulation to be between b and 5 b, and to depend on the interatomic potential and crystal
420
7 Dislocations in Crystals
tttffl w
Narrow
Wide
Dissociated
~w*
Au b
n X
f Areab/2
n
Area b/2
A (c)
Figure 7-8. Atomic positions, disregistry Aw, and Burgers vector distribution / for (a) wide, (b) narrow and (c) dissociated edge dislocations in a simple cubic crystal. (From Hull and Bacon, 1984; reprinted with permission of Pergamon Press, Ltd.)
structure. Dissociation into two partial dislocations (see Sec. 7.6) as illustrated in Fig. 7-8 c can only occur when a stable stacking fault exists on the glide plane (e.g., in the f.c.c. lattice, see Sec. 7.6.2). The spacing d of the partials may be large and is determined principally by their elastic interaction and the stacking fault energy (Sec. 7.6.2), whereas the width w of the partial cores is determined by inelastic atomatom interactions. Another useful representation of core structure can be obtained simply from the derivative of the disregistry curve: d(Au) dx
(7-4)
The form of f(x) for the three core structures of Fig. 7-8 is shown by the lower curve in each case. This function is known as the distribution of Burgers vector because the area under an f(x) curve equals b. As can be seen from Fig. 7-8, the distribution curve shows clearly where the disregistry is concentrated: it is particularly
useful when a dislocation consists of two or more closely-spaced partials. For non-planar cores in which the disregistry and Burgers vector are not distributed mainly on one plane - a situation sometimes found for screw dislocations - a two-dimensional plot in which the displacement difference is shown by the length of arrows can be illuminating. (This procedure is illustrated later in Sec. 7.6 for the core of screw dislocations in h.c.p. and b.c.c. metals in Figs. 7-49 and 7-52). The displacement difference and Burgers vector distribution functions have proved to be particularly useful for presentation of details of the atomic structure of dislocation cores obtained by computer simulation. 7.1.5 Observations of Dislocations The arrangement, density and Burgers vectors of dislocations have been studied by a variety of techniques. (For an introductory review, see Hull and Bacon (1984).) Some, such as surface methods, in
7.2 Movement of Dislocations
which the point of emergence of individual dislocations at a surface is revealed by etching, and decoration techniques, in which dislocations in transparent crystals are decorated with precipitates to reveal their position, are not widely applicable, and do not readily provide detailed information on dislocation character. Similarly, X-ray topography is only generally used when distributions of dislocations (of low density) are of interest. By far the most extensively-used method for investigating dislocation properties and processes has been transmission electron microscopy (TEM). Chapter 1, Volume 2 A of this Series contains a detailed exposition of TEM. Dislocations observed by this technique are in specially-prepared thin foils that are transparent to electrons with energy in the range 100keV-l MeV, i.e. thickness from < lOOnm to about 1000 nm. In most studies, dislocations are imaged by strain contrast. This is achieved because the elastic distortion produced by a dislocation in the surrounding crystal (see Sec. 7.3.2) bends atomic planes, and these can diffract electrons away from the main electron beam if at the Bragg angle (see Fig. 7-9). By imaging a dislocation with different diffracting planes, it is possible to determine not only its direction but also its Burgers vector. By its very nature, the dislocation image produced by strain contrast is wide - from a few nm to (more typically) tens of nm - in comparison with atomic spacings, and does not provide direct information on the atomic arrangements within the dislocation core. Such information can be obtained by TEM using lattice imaging (see Volume 2 A, Chapter 1 of this Series). This technique is more restrictive in terms of material and sample thickness, but has proved invaluable in many problem areas, e.g. dislocation structure in interfaces. It is important to note, however, that much of
421
Incident
Transmitted
Diffracted
Figure 7-9. Planes near an edge dislocation bent into the orientation for diffraction (schematical).
the interpretation carried out using TEM methods would not be possible without a firm understanding of basic dislocation properties, and we now turn to this aspect.
7.2 Movement of Dislocations 7.2.1 Crystal Plasticity As noted earlier, the discovery of dislocations came from those who were attempting to understand plastic flow in crystalline solids. Plastic deformation in a single crystal under an axial tensile stress o is shown schematically in Fig. 7-10. At normal temperatures, the deformation occurs by slip (or glide) of parts of the crystal over each other, the slip planes and slip directions often having particular crystallographic form. This demonstrates the shearlike, constant-volume character of plastic deformation. Experiments have shown that slip in this situation occurs when the resolved shear stress (RSS), T, on the slip
422
7 Dislocations in Crystals
Slip plane normal
Slip direction
Slip plane
Figure 7-10. The geometry of slip in a single crystal under uniaxial stress o.
7.2.2 Glide and Slip
plane in the slip direction X = G COS(j) COS X
is the shear modulus of elasticity, yet experimentally TC is much smaller than this, and typically of the order of JLL/106 to fi/103. The discrepancy between theory and experiment was resolved with the recognition that dislocation movement is responsible for the low value of r c . We now know that the macroscopic slip bands observed in crystals are the result of slip of many dislocations. An alternative to glide occurs when a dislocation moves out of its glide plane. This process of climb requires a mobile population of point defects, and as such is particularly important at elevated temperatures. We now consider in turn the glide and climb of individual dislocations in greater detail.
(7-5)
reaches a critical value TC , called the critical resolved shear stress (CRSS). The slip that occurs at x = TC is not a rigid sliding of the two parts of a crystal with respect to each other, for the RSS required to slide two planes of atoms rigidly past each other is of the order of /z/10 (Frenkel, 1926), where \i
Glide occurs when a dislocation moves in its glide plane, which is the surface that contains both the dislocation line and its Burgers vector. (Strictly, "glide" refers to one dislocation and "slip" to many, though the two terms are now often synonymous.) Consider the edge dislocation in Fig. 7-1, and imagine for the moment that it is formed in the crystal by the procedure il-
gers vector (b) ia) Figure 7-11. Formation of a pure edge dislocation FE. (From Hull and Bacon, 1984; reprinted with permission of Pergamon Press, Ltd.)
7.2 Movement of Dislocations
lustrated in Fig. 7-11. Here, a cut is made over the surface AEFD in the crystal, and the upper surface of the cut is displaced by b - the shortest lattice vector in Fig. 7-1 relative to the lower surface. An extra halfplane EFGH and a dislocation line FE are formed and the distortion produced is identical to that of Fig. 7-1. Although dislocations are not formed in this way in practice, this approach demonstrates that the dislocation can be defined as the boundary between the slipped and unslipped parts of the crystal. It should be emphasized that as far as the distortion around a dislocation is concerned, the imaginary cut-and-displace procedure is not unique. A relative displacement of b given to atoms across any cut surface terminating at the line would suffice, e.g. EFGH in Fig. 7-11. This should be clear from Eq. (7-1) and the description of the Burgers circuit, where no specific reference is made to a cut surface. The distortion in Fig. 7-11 b arises from giving the same relative displacement b to atoms on opposite sides of the cut. This creates a Volterra dislocation (Volterra, 1907), and although more general forms arising from variable displacements are possible, they are not treated here.
x y
(a)
423
Only a relatively small RSS is required to move the dislocation along the slip plane, as demonstrated in Fig. 7-12. Within the dislocation core, atomic coordination is far from perfect, and small relative changes in position of only a few atoms are required for the dislocation to move. For example, a small shift of atom 1 relative to atoms 2 and 3 effectively moves the extra half-plane from x to y and this process is repeated as the dislocation continues to glide. The applied stress required to overcome the lattice resistance to the movement of the dislocation is the PeierlsNabarro stress (see Sec. 7.2.4) and is much smaller than the Frenkel theoretical shear stress of a perfect lattice. Figure 7-12 demonstrates why the Burgers vector is the most important parameter of a dislocation. Two neighboring atoms (say 1 and 3) on sites adjacent across the slip plane are displaced relative to each other by the Burgers vector when the dislocation glides past. Thus, the slip direction (see Fig. 7-10) is necessarily always parallel to the Burgers vector of the dislocation responsible for slip. The movement of one dislocation across the slip plane to the surface of the crystal produces a surface slip step defined by the Burgers vector. The
x y
(b)
(c)
(d)
Figure 7-12. Glide of an edge dislocation: the arrows indicate the applied shear stress. (From Hull and Bacon, 1984; reprinted with permission of Pergamon Press, Ltd.)
424
7 Dislocations in Crystals
plastic shear strain in the slip direction resulting from dislocation movement is derived in Sec. 7.2.7. Since the Burgers vector of an edge dislocation is perpendicular to the dislocation line, the glide surface (given by both directions) is unique. For the screw dislocation, however, b is parallel to the line, and thus any plane that contains the line can be its glide plane, at least in principle. This is clear in Fig. 7-2, where the dislocation can be imagined to result from cutting the material over the vertical plane terminating at
(a)
(c)
the line and displacing the two sides of the cut parallel to the line. Glide of the dislocation by small movements of atoms in the core would extend the resulting surface step in either the same plane as the cut or any other vertical glide plane the dislocation moves on. The direction in which a dislocation glides under stress can be determined by physical reasoning. Consider material under an applied resolved shear stress (Fig. 7-13 a) so that it deforms plastically by glide in the manner indicated in Fig. 7-13 b.
(b)
(d)
-A/ (e)
(f)
Figure 7-13. Plastic deformation by glide of edge and screw dislocations under the applied shear stress shown. (From Hull and Bacon, 1984; reprinted with permission of Pergamon Press, Ltd.)
7.2 Movement of Dislocations
From the description above, a dislocation responsible for this deformation must have its Burgers vector in the direction shown. It is seen from Figs. 7-13 c and d that a positive edge dislocation glides to the left in order that the extra half-plane produces a step on the left-hand face as indicated, whereas a negative edge dislocation glides to the right. A right-handed screw glides towards the front in order to extend the surface step in the required manner (Fig. 7-13 e), whereas a left-handed screw glides towards the back (Fig. 7-13 f). These observations demonstrate that (a) dislocations of opposite sign glide in opposite directions under the same stress, as expected of physical opposites, and (b) for dislocation glide a shear stress must act on the slip plane in the direction of the Burgers vector, irrespective of the direction of the dislocation line. In the examples illustrated above it has been assumed that the moving dislocation remains straight. However, dislocations are generally bent and irregular, particularly after plastic deformation. A more general shape of a dislocation is shown in Fig. 7-14. The boundary separating the slipped and unslipped regions of the crystal is curved, i.e. the dislocation is curved, but the Burgers vector is the same all along its length. It follows that at point E the dislocation line is pure edge and at S is pure screw. The remainder of the dislocation has a mixed character, with b resolvable into edge and screw components, as illustrated in Fig. 7-4. 7.2.3 Cross Slip Despite the observation above that any plane which contains a screw dislocation is a possible glide plane, it is found that screw dislocations tend to move only in particular crystallographic planes in real materials (Sec. 7.6). This is a consequence of the influ-
425
./^\.S .S*/*S*/*/ /
S
jf
S_ ^
^
^
7~
tfl
Mi
r
Burgers vector
Figure 7-14. The curved dislocation SME is pure edge at E and pure screw at S and mixed elsewhere (M). (From Hull and Bacon, 1984; reprinted with permission of Pergamon Press, Ltd.)
ence of the crystal structure and atomic bonding on the atomic arrangement in the core. For example, in the face-centred cubic (f.c.c.) metals, dislocations glide on the close-packed {111} planes, but screw dislocations can switch from one {111} plane to another. This process of cross slip is illustrated in Fig. 7-15. In Fig. 7-15 a, a dislocation with b = 1/2 [101] is gliding in the (111) plane under a RSS. The only other {111} plane containing b is (111), and if at some point the local stress field changes so that glide is preferred on that plane, then, as shown in Figs. 7-15 c and d, the screw segment can cross slip onto the (111) plane and further movement of the line occurs on it. The process can be reversed, and double cross slip is illustrated in Fig. 7-15 d. Cross slip provides an important mechanism for dislocations to by-pass obstacles on their glide plane. Also, it results in wavy slip lines in materials where there is not a strong preference for one type of glide plane. 7.2.4 The Peierls Barrier The atomic disregistry in the core region results in a dislocation core energy and re-
426
7 Dislocations in Crystals
[To>]
(d)
sistance to movement. In the first estimates of the lattice resistance (Peierls, 1940; Nabarro, 1947), it was assumed that the atom planes A and B (Fig. 7-7) interact by a simple sinusoidal force and that in equilibrium the resulting disregistry forces on A and B are balanced by the elastic stresses from the two half-crystals above and below these planes. This condition provided an analytical solution for Aw, from which w was found to be a/(l — v) for an edge dislocation and a for a screw dislocation, where a is the interplanar spacing and v is Poisson's ration; the core is therefore "narrow". The dislocation energy was also found by combining the disregistry energy, calculated from Aw and the sinusoidal forces, with the elastic energy stored in the two half-crystals. (It is similar in form to Eq. (7-28), with r0 replaced by approximately w/3.) When the dislocation in Fig. 7-7 moves to the right to PP', say, the atoms in planes A and B cease to satisfy the equilibrium distribution of Aw, and the disregistry energy increases. Peierls and Nabarro calculated the dislocation energy per unit length as a function of dislocation position, and found that it oscillates with period b
Figure 7-15. Cross slip in a facecentred cubic metal. The Burgers vector is parallel to the [101] direction which is common to the (111) and (111) glide planes. (From Hull and Bacon, 1984; reprinted with permission of Pergamon Press, Ltd.)
and maximum fluctuation (known as the Peierls energy) given by — 2TTMA p
exp jc(l-v)
(7-6)
The maximum slope of the periodic energy function is the critical force per unit length required to move the dislocation through the crystal. Dividing this by b gives the Peierls stress 2n Tp =
— 2nw exp (1-V)
(7-7)
This simple model successfully predicts that TP is orders of magnitude smaller than the theoretical shear strength estimated by Frenkel (see Sec. 7.2.1). Although the Peierls method has now been superseded by more realistic approaches using computer simulation, it has qualitative features generally recognized to be valid. Slip usually occurs on the mostwidely spaced planes, and a wide, planar core tends to produce low values of TP . For this reason, edge dislocations are generally more mobile than screws. Also, the Peierls energy can be very anisotropic, and dislocations will then tend to lie along the most
7.2 Movement of Dislocations
closely-packed directions, for which TP is a maximum. It should be noted, however, that the magnitudes of EP and TP depend sensitively on the nature of the interatomic forces. TP is low (<10~ 6 to 10~5/i) for the face-centred cubic and basal-slip hexagonal metals, in which dislocations dissociate, but is high (~10~ 3 /i) for covalent crystals such as silicon and diamond, in which dislocations have a preference for the <110> orientations either parallel or at 60° to their l/2<110> Burgers vector. The body-centred cubic metals, for which the <111 > screw does not have the planar form of the Peierls model, and the hexagonal metals, for which slip on the prism planes fall between these extremes. These aspects are discussed further in Sec. 7.6. The energy of a dislocation as a function of position in the slip plane is illustrated in Fig. 7-16 for a dislocation lying predominantly parallel to the z direction, which is a direction of low line energy. The energy Eo per unit length fluctuates by EP (usually < Eo) due to the Peierls energy, with a period a given by the repeat distance of the lattice in the x direction. If the dislocation is unable to lie entirely in one energy minimum, it contains kinks where it moves from one minimum to the next. The shape and length m of a kink depend on the value of £ P and are a balance between the dislocation tending to lie as much as possible in the lines of minimum Peierls energy, which would give m = 0, and wanting to reduce its energy by being as short as possible, which would give a straight line with m |> a. Thus, a high £ P produces low m and vice-versa. The RSS required to move a kink laterally along the line, and thus move the line from one energy minimum to the next, is less than TP, which is the stress required to move a long straight line rigidly over the energy hump EP. Thus, at a low applied stress, pre-existing kinks can move later-
427
X
Maxima I of potential Minima[ energy E(x)
-/Kinks
Dislocation lineatO°K
Potential energy A Dislocation at of dislocation E(x) finite temperature
Figure 7-16. Schematic illustration of the potential energy surface of a dislocation line due to the Peierls energy EP. (From Seeger et al., 1957; reprinted with permission of McGraw-Hill, Inc.)
ally until reaching the nodes at the ends of the dislocation segments. The resulting plastic strain (preyield microplasticity) is small and leaves long segments of line lying along energy minima. At 0 K, a stress of at least TP is therefore required for further (macroscopic) plastic flow. As the temperature is raised, however, there is an increasing probability that atomic vibrations resulting from thermal energy may enable the core to bulge from one minimum to the next over only part of the line (Fig. 7-16) and thus reduce the flow stress. This process of double-kink nucleation increases the line length and energy, however. Also, the two kinks are of opposite sign and therefore tend to attract and annihilate each other. As a result, the double kink is not stable under an applied stress unless the length of the bulge is sufficiently large, typ-
428
7 Dislocations in Crystals
ically ~20b and ^>m. Double-kink nucleation therefore has an activation energy which is a function of EP. The effect on the applied stress required to maintain plastic flow is as illustrated schematically in Fig. 7-17. The flow stress for a given applied strain rate e decreases with increasing temperature T up to Tc as thermal activation becomes increasingly significant, and z consists of a thermal component T* plus an athermal component T^, which is almost independent of T. Stress and strain rate at a given temperature are related by (Hull and Bacon, 1984)
s=
QmAexp(-AG*/kBT)
(7-8)
where gm is the mobile dislocation density, A = baco, and co is the vibration frequency of the dislocation (< atomic vibration frequency). The stress-dependence of s arises from the stress-dependence of the Gibbs free energy of activation AG* given by the empirical equation (Kocks et al, 1975) (7-9) with T* = z — TM (T* depending mainly on T but in a minor way also on e), 0 < p < 1 and i
T*(O)
Figure 7-17. Variation of the flow stress with temperature for the Peierls mechanism.
7.2.5 Velocity of Dislocations Dislocations move by glide at velocities which depend on the applied shear stress, purity of crystal, temperature and type of dislocation. A direct method of measuring dislocation velocity was developed by Johnston and Gilman (1959) using etch pits to reveal the position of dislocations at different stages of deformation in lithium fluoride. Dislocation velocity v was determined from the positions of the dislocations before and after a stress pulse. This method and several others have been applied to many materials (Vreeland, 1968). An illustration of the range of velocity data as a function of applied RSS in several materials is shown in Fig. 7-18. It can be seen that v is approximately linear in z at very high and low stresses, and strongly dependent on z in between. The ranges of z for a given material depend on the CRSS, and the high-velocity stage may not be observed when the CRSS is large. There has been a good deal of experimental and theoretical study of the mechanisms that control v in a given situation (see, for example, Mason (1968) and Hirth and Lothe (1982)), and of the influence of lattice phonons, conduction electrons and the Peierls barrier, in particular. Also, it is known that at
7.2 Movement of Dislocations io4
LiF(b) IO"
1
10' IO2 10° Shear stress(MN/m )
IO3
Figure 7-18. The dependence of dislocation velocity on applied shear stress. The data are for 20 °C except Ge (450 °C) and Si (850 °C). (After Haasen, 1978; reprinted with permission of Cambridge University Press.)
high velocities comparable with the transverse sound velocity, C t , v is limited by shock-wave radiation. Eshelby (1949) showed that this effect is negligible for i><0.5Ct, but above this range a dislocation moves supersonically. In pure metals, where the Peierls barrier is small (Sec. 7.2.4), dislocations move relatively freely at stresses above the CRSS and the velocity in the range 10" 3
= Bv/b
(7-10)
where B is the viscous drag coefficient. The two main contributions to B arise from phonon and electron scattering (Leibfried, 1950; Alshits and Indenbom, 1975; Kaganov et al., 1974). The latter is approximately independent of temperature, T, and is the larger at T < 0.1 0 D , where 0 D is the Debye temperature. Above this temperature, however, phonon scattering increases
429
approximately linearly with T and is totally dominant, and dislocation velocity increases with decreasing temperature. In Fig. 7-18 the dislocation velocities are mainly given for values v/C{< 10" 3 , where they are very sensitive to stress, i.e. v is proportional to xm and the slope m of the curves can reach values 20 to 50. The reduction of m above this range indicates that this is the regime where viscous damping becomes important (for LiF this is for D>4mms~ 1 ). The mechanisms in the lower region are particularly dependent on the nature of the material, i.e. the height of the Peierls barrier and the energy to nucleate and move kinks (Sec. 7.2.4): purity and microstructure are also important (Seeger and Engelke, 1968; No wick and Berry, 1972; de Batiste, 1972). A more detailed discussion of these effects in ceramics and semiconductors can also be found in Volume 11, Chapter 7 and Volume 4, Chapter 6 of this Series. 7.2.6 Climb
In conditions where diffusion is difficult, the movement of dislocations is restricted almost entirely to glide but at higher temperatures, or in the presence of a supersaturation of mobile point defects, a dislocation can move out of its slip plane by climb. Consider the edge dislocation in Fig. 7-1. If the row of atoms at the bottom of the extra half-plane is removed, the dislocation line moves up one atom spacing from its original slip plane; this is called positive climb. Similarly, if an additional row of atoms is introduced below the extra half-plane the dislocation line moves down one atom spacing, negative climb. Positive climb can occur by either diffusion of vacancies to the core or the formation of self-interstitial atoms in the core and their diffusion away. Conversely, negative climb can occur ei-
430
7 Dislocations in Crystals
ther by interstitial absorption or vacancy emission. More generally, if a small segment d/ of line undergoes a small displacement ds (Fig. 7-19), the local change in volume is
dV = b - (dlxds) = (bx d/) • ds
(7-11)
d#
Figure 7-19. Displacement of a segment of line d/ by ds.
Figure 7-20. Jogs on an edge dislocation.
for the two sides of the area element d/ x ds are displaced by b relative to each other. The glide plane of the element is by definition perpendicular to b x d/, and so when either ds is perpendicular to bxdl or b x d/= 0, which means the element is pure screw, dV is zero. This is the condition for glide discussed in Sec. 7.2.2. For other cases, volume is not conserved {dV^ 0) and the motion is climb, the number of point defects required being dV/Q, where Q is the volume per atom. The mass transport involved occurs by diffusion and therefore climb requires thermal activation. The vacancy is the dominant species of intrinsic point defect in most crystals in equilibrium, and so climb processes usually involve the diffusion of vacancies. It is statistically most unlikely that a complete atomic row is removed or added simultaneously, and in practice point defects are involved individually. The effect of this is illustrated in Fig. 7-20 which shows climb of a short section of a dislocation line resulting in the formation of two jogs. Climb proceeds by the nucleation and motion of jogs. Conversely, jogs are sources and sinks for point defects. Jogs are steps on the dislocation which move it from one atomic slip plane to an-
other, in distinction to kinks (Sec. 7.2.4) which displace it on the same slip plane. The two are distinguished in Fig. 7-21. Jogs and kinks have the same Burgers vector as the line on which they lie, and thus the kink, having the same slip plane as the dislocation line, does not impede glide of the line. (In fact, it may assist it (Sec. 7.2.4).) Similarly, the jog on an edge dislocation (Fig. 7-21 c) does not affect glide. The jog on a screw dislocation (Fig. 7-21 d) has edge character, however, and can only glide along the line; movement at right angles to the Burgers vector requires climb. This impedes glide of the screw and results in point defect production during slip (Sec. 7.5.1). The jogs described have a height of one atomic-plane spacing and a formation energy E} ~ 1 eV resulting from the increase in dislocation line length (Sec. 7.3.2). They are produced extensively during plastic deformation by the intersection of dislocations (Sec. 7.5.1), and exist even in wellannealed crystals, for there is a thermodynamic equilibrium number of thermal jogs per unit length of dislocation given by nj = n o exp(-E j /fc B T)
(742)
where n0 is the number of atom sites per
7.2 Movement of Dislocations
431
y
y*\ 1
w
y
(a)
(b) Burgers vector b
Figure 7-21. (a, b) Kinks in edge and screw dislocations, (c, d) Jogs in edge and screw dislocations. The slip planes are shaded. (From Hull and Bacon, 1984; reprinted with permission of Pergamon Press, Ltd.) (c)
(d)
unit length of dislocation. The climb of dislocations by jog formation and migration is analogous to crystal growth by surface step transport. There are two possible mechanisms. In one, a pre-existing single jog migrates along the line by vacancy emission or absorption. This involves no change in dislocation length and the activation energy for jog diffusion is (£fY + E^) = Ed, where Efv and E Y are the vacancy formation and migration energy, respectively, and Ed the activation energy for selfdiffusion. In the other, thermal jogs are nucleated on an otherwise straight line and the jog migration energy is (£ d + £j). In most situations the first process dominates. The effective activation energy in some circumstances can be less than one half the value of Ed measured in the bulk, for the crystal is distorted at atom sites close to the dislocation line itself and Ed is smaller there. The process of mass transport occurring along the line is known as pipe diffusion. Although jog diffusion in thermal
equilibrium occurs at a rate proportional to exp( — Ed/kB T), the actual rate of climb of the dislocation depends also on the forces acting on it, as discussed in Sec. 7.3.3. Pure screw dislocations have no extra half-plane and in principle cannot climb, i.e. d F = 0 in Eq. (7-11). However, a small edge component or a jog on a screw dislocation will provide a site for the start of climb. The production of helical dislocations is a good example. (See Amelinckx etal, 1957; Hirth and Lothe, 1982; Hull and Bacon, 1984 for more details.) Glide and climb are sometimes referred to as conservative and non-conservative motion, because of the conditions d F = 0 and dV ^0 discussed above. However, conservative climb is possible if climb of one part of the line leads to climb of another part with dV equal in magnitude but opposite in sign. This process, which involves no net flux of point defects to or from the line, can occur by pipe diffusion. It has
432
7 Dislocations in Crystals
been observed in the case of movement of dislocation loops, for instance (Kroupa and Price, 1961). 7.2.7 Plastic Strain due to Dislocation Movement As noted in Sec. 7.2.1, the movement of a dislocation, usually in response to an externally-applied load, results in plastic strain. This is an addition to the elastic strain, which is simply related to the external stress by Hooke's law (Sec. 7.3.1). The relation between plastic strain and the applied stress is more complicated and depends on factors such as temperature, applied strain rate and, in particular, the microstructure of the material. There is, however, a simple relationship between the plastic strain and the dislocation density defined as in Sec. 7.1.3. It is based on the fact that when a dislocation moves, two atoms on sites adjacent across the plane of motion are displaced relative to each other by the Burgers vector b. The relationship for slip is derived first. Consider a crystal of volume hid containing, for simplicity, straight edge dislocations (Fig. 7-22 a). Under a high enough applied RSS, the dislocations will glide, as shown, positive ones to the right, negative ones to the left. The top surface of the sample will therefore be displaced plastically by D relative to the bottom surface as demonstrated in Fig. 7-22 b. If a dislocation moves completely across the slip plane through the distance d, it contributes b to D. Since b is small in comparison to d and /z, the contribution made by a dislocation i which moves a distance xt may be taken as the fraction {xjd) ofb. Thus, if the number of dislocations which move is N, the total displacement is
5
d i=t
(7-13)
(a)
y
/
/
r
\7
i i ii
/\
i
i
\ \ \ \ (b)
Figure 7-22. Edge dislocations gliding in a crystal subjected to a RSS. Dislocation i has moved a distance xt, as shown. (From Hull and Bacon, 1984; reprinted with permission of Pergamon Press, Ltd.)
and the macroscopic plastic shear strain ep is given by D/h. Equation (7-13) can be simplified by writing Nx for the sum, where x is the average distance moved by a dislocation. Since the density of mobile dislocations £m is (Nl/hld) we arrive at an equation for the plastic shear strain (7-14) The strain rate is therefore (7-15)
7.3 Dislocations in Elastic Media
where v is the average dislocation velocity. The same relationships hold for screw and mixed dislocations. Also, since the average area of slip plane swept by a dislocation A equals /x, an alternative to Eq. (7-14) is s = bnA, where n is the number of lines per unit volume. It is emphasized that gm appearing in the above equations is the mobile dislocation density, for dislocations which do not move do not contribute to the plastic strain. Climb under an external tensile load is shown schematically in Fig. 7-23. When an edge dislocation climbs, an extra plane of thickness b is inserted into, or removed from the crystal in the area over which the line moves. As in the analysis for slip, if a dislocation moves through distance xi9 it contributes b(xjd) to the plastic displacement H of the external surface. It is therefore easy to show that the total plastic tensile strain (H/h) parallel to the Burgers vector and the strain rate are given by Eqs. (7-14) and (7-15) respectively. The same relations also hold for mixed dislocations, except that b is then the magnitude of the edge component of the Burgers vector.
(b)
Figure 7-23. Edge dislocations, denoted by their extra half planes, climbing in a crystal under a tensile load.
433
7.3 Dislocations in Elastic Media 7.3.1 Elements of Elasticity Theory The atoms in a crystal containing a dislocation are displaced from their perfect lattice sites, and the dislocation is therefore a source of internal stress in the crystal. For example, the region above the slip plane of the edge dislocation in Fig. 7-1 accommodates the extra half-plane and is in compression: the region below is in tension. The stresses and strains have a range much larger than the interatomic spacing, and in the bulk of the crystal are sufficiently small for conventional elasticity theory to be applied to obtain them. This approach only ceases to be valid at positions within the dislocation core. Although most crystalline solids are elastically anisotropic, i.e. their elastic properties are different in different crystallographic directions, it is much simpler to use isotropic elasticity theory. This still results in a good approximation in most cases as discussed in Sec. 7.3.7. From a knowledge of the elastic field, the energy of a dislocation and its interaction with other defects can be obtained. The elastic field produced by a dislocation is not affected by the application of stress from external sources: the total stress on an element of the body is the superposition of the internal and external stresses. The early mathematical treatments of the elastic distortions due to dislocation by Volterra (1907), Love (1927), Burgers (1939) and others are fully reviewed in texts such as de Wit (1960), Hirth and Lothe (1982), Nabarro (1952, 1967), Mura (1982) and Teodosiu (1982), and more elementary analyses are given in Cottrell (1953), Weertman and Weertman (1964) and Hull and Bacon (1984). Here, we simply define the terms required for the elementary theory presented in the following sections.
434
7 Dislocations in Crystals
The displacement of a point from its position in the unstrained state is represented by the vector u = [ux,uy,uz], where each component is in general a function of position (x, y, z). The nine components of the strain tensor etj (with ij = x, y or z) are obtained from the first derivatives of the components of u in linear elasticity e.g.
The internal energy of a body is increased by strain. The strain energy per unit volume is one-half the product of stress times strain for each component. Thus, for an element of volume dF, the elastic strain energy is
i = x,y,
z j = x,y,
z
etc. 2\dy
du, dx
(7-16) etc.
The three normal components with i —j are measures of stretching and compressive deformation, and the six components with i ^j define shear strains. The relationship between stress and strain in linear elasticity is Hooke's Law, in which each stress component is linearly proportional to each strain. For isotropic solids, only two proportionality constants are required: X {exx + eyy + ezz),
etc. (7-17)
etc.
X and \x are the Lame constants, but \i is more commonly known as the shear modulus. Other elastic constants are frequently used, the most useful being Young's modulus, E, and Poisson's ratio, v. Under uniaxial loading in the longitudinal direction, E is the ratio of longitudinal stress to longitudinal strain and v is minus the ratio of lateral strain to longitudinal strain. Since only two material parameters are required in Hooke's law, these constants are interrelated. For example, = 2fjL(l+v), and V =
(7-18)
Typical values of E and v for metallic and ceramic solids lie in the ranges 4 0 600 GPa and 0.2-0.45 respectively.
7.3.2 Elastic Field and Energy of a Straight Dislocation
The elastic distortion created by a straight, infinitely-long dislocation is relatively easy to analyze in comparison with that of curved and polygonal dislocations, as discussed in Sec. 7.3.7. Fortunately, much insight can be gained with the results for the simple geometry. Consider first the screw dislocation shown in Fig. 7-2. The elastic cylinder in Fig. 7-24 a has been deformed to produce a similar distortion by making a radial cut LMNO and displacing the cut surfaces rigidly with respect to each other by b in the z-direction. The distortion is one of anti-plane strain, and the displacements are seen by direct inspection to be ux = u = 0 ,
uz =
b9
(7-20)
The non-zero strains are readily found to be (Hull and Bacon, 1984) b sinO p
•=• p
P
=
=
b cos 9 yz
P
(7-21)
=
zy
4 7i
and the stresses can be obtained using Eq. (7-17). These shear components take their simplest form in cylindrical polar coordinates: 2nr
(7-22)
7.3 Dislocations in Elastic Media
435
(b)
Figure 7-24. Elastic distortion of a cylinder following the creation of (a) a screw and (b) an edge dislocation along its axis. (From Hull and Bacon, 1984; reprinted with permission of Pergamon Press, Ltd.)
Thus, the elastic distortion is one of pure shear and exhibits complete radial symmetry since the cut LMNO can be made on any radial plane. For a dislocation of opposite sign, i.e. a left-handed screw, the signs of all the field components are reversed. The stresses and strains are proportional to 1/r and diverge to infinity as r->0, and so the cylinder in Fig. 7-24 is shown as hollow with a hole of radius r0. This is the core radius (see Sec. 7.1.4) in the elastic model, because as the centre of a dislocation in a crystal is approached, elasticity theory cases to be valid and a non-linear, atomistic model must be used. From Eq. (7-22) it is seen that the stress reaches the theoretical limit (Sec. 7.2.1) and the strain exceeds about 10% when rzzb, and so a reasonable value for r0 therefore lies in the range b to 4 b, i.e. r o < 1 nm in most cases. The elastic field of the edge dislocation is more complex than that of a screw because it is one of plane strain. With reference to Fig. 7-24 b, it is generated by displacing the two surfaces of cut LMNO rigidly past each other by b in the direction x perpendicular to the dislocation lying along the z-axis, as in Fig. 7-11. The displacement
and strains in the z-direction are zero and the non-zero stresses are found to be (e.g. Nabarro, 1967)
= Dy
0"^
(x2-y2) (x2+y2)2 (x2-y2)
(7-23)
=
where D=
Lib 2TC(1-V)
The stress field therefore has both dilational and shear components. The hydrostatic pressure on a volume element is P= -3 3
-y2)
(7-24)
It is compressive above the slip plane and tensile below, as implied qualitatively by the distortion illustrated in Figs. 7-1 and 7-11. As in the case of the screw, the signs of the components are reversed for a dislo-
436
7 Dislocations in Crystals
cation of opposite sign, and again the elastic solution has an inverse dependence on distance from the line axis and breaks down when x and y tend to zero. It is valid only outside a core of radius r 0 . The elastic field produced by a mixed dislocation (Fig. 7-4) is obtained from the above equations by adding the fields of the edge and screw constituents. The two sets are independent of each other in isotropic elasticity. The distortion created by a dislocation has an associated strain energy which may be divided into two parts F —F ^
— ^core
+ F ' ^el
(1-7 $\ \'
Fig. 7-24 a and b) 1 d£ el (screw) = -azyb dA 2 (7-27) 1 d£el(edge) =-oxyb&A 2 with the stress evaluated on y = 0. The factor 1/2 enters because the stresses build up from zero to the final value given by Eqs. (7-22) and (7-23) during the displacement process. The element of area is a strip of width dx parallel to the z axis, and so the total strain energy per unit length of dislocation is nb2Rcdx lib21 / £ N
ZJ
)
The elastic part, Eel9 stored outside the core may be determined by integration of the energy of each small element of volume. This is a simple calculation for the screw dislocation, for from the symmetry the appropriate volume element is a cylindrical shell of radius r and thickness dr. From Eq. (7-19), the elastic energy stored in the cylinder of Fig. 7-24 a per unit length of dislocation is
This analysis based on the volume integration of Eq. (7-19) is much more complicated for other dislocations having less symmetric fields. Formally, the volume integral of the products of stress times strain may be converted to a surface integral of stress times displacement taken over the surface enclosing the volume by use of Gauss' theorem (e.g., de Wit, 1960), as discussed in Sec. 7.3.7. This much more tractable approach has a simple physical interpretation in that we consider £ el as the work done in displacing the faces of the cut LMNO by b against the resisting internal stresses. For an infinitesimal element of area &A of LMNO, the work done is (see
471
£el(edge) =
4TT
4TC(1-VU x
»"2 4TT(1-V)
(7-28)
.,/* VO
The screw result is the same as Eq. (7-26). Strictly, Eqs. (7-28) neglect small contributions from the work done against the tractions on the core surface r = r0 of the cylinder (Bullough and Foreman, 1964; Hirth and Lothe, 1982) but they are adequate for most requirements. We note that £el(edge) is greater than £ el (screw) by 1/(1—v), or about 3/2, and Eel (mixed) falls inbetween. The energy of a mixed dislocation (Fig. 7-4) in isotropic elasticity theory is given by the addition of the two energies in Eq. (7-28), with b in the two cases replaced by focosa and bsinoc, respectively. If we denote the prelogarithmic factor in Eqs. (7-28) by K then K=
jub2 (1 - v cos2 a) 4TT(1-V)
(7-29)
In crystals containing many dislocations, the dislocations tend to form in configurations in which the superimposed longrange elastic fields cancel. The energy per
7.3 Dislocations in Elastic Media
dislocation is thereby reduced and an appropriate value of R is approximately half the average spacing of the dislocations arranged at random. Early estimates of £ core in Eq. (7-25) made before the advent of atomic-scale computer modelling suggested that it was within the range of 10 to 25% of the total energy for straight lines (e.g., Cottrell, 1953). To consider this further, we can use Eqs. (7-28) to rewrite the total energy per unit length as R
(7-30)
n
where K is the prelogarithmic energy factor, Eq. (7-29). The parameter X is chosen so that r0 may be replaced by b and E includes £ core . Calculations for a range of materials using realistic interatomic potentials suggest A~l/4 (Hirth and Lothe, 1982; Gao and Bacon, 1992), lending firm support to the magnitude of the Ecore contribution suggested above. (Note that 2 ~ l / 4 only applies to the use of Eq. (7-30) for the total energy. It does not imply that the radius r0 of the non-linear core region as defined in preceding sections is only b/4.) From the expressions above it is clear that the energy per unit length is relatively insensitive to the character of the dislocation and also to the values of R and r 0 .
437
Taking realistic values for R and r 0 , all the equations can be written approximately as E = kfib2
(7-31)
where fc^0.5 to 1.0. This makes apparent a simple rule known as Frank's rule (Frank, 1949) for determining whether or not it is energetically feasible for two dislocations to react and combine to form another. Consider the two dislocations in Fig. 7-25 with Burgers vectors bx and b2 given by the Burgers circuit construction (Sec. 7.1.3). Allow them to combine to form a new dislocation with Burgers vector b3 as indicated. From Eq. (7-31), the energy per unit length of the dislocations is proportional to b\, b22 and b\, respectively. Thus, if (b\ + bl) > bl, the reaction is favorable for it results in a reduction in energy. If (bl + bl) < bl the reaction is unfavorable and the dislocation with Burgers vector b3 is liable to dissociate into the other two. If (bl + bl) = bl there is no energy change. These three conditions correspond to the angle 0 in Fig. 7-25 satisfying n/2<
\
Linei
Line 2
Line 3
b1+b2=b3
Figure 7-25. Reaction of two dislocations to form a third.
438
7 Dislocations in Crystals
Frank's rule is used to consider the feasibility of various dislocation reactions in Sec. 7.6. 7.3.3 Forces on Dislocations A dislocation can respond to a stress applied externally to a crystal as though it actually experiences mechanical forces for glide and climb. These are not real forces applied to the atoms in the vicinity of the line, but virtual forces which describe the energy changes that accompany dislocation motion. These mechanical forces on a line can also arise from sources of internal stress in the crystal, such as other dislocations, point defects and even the dislocation itself (self-forces). Additionally, there may be chemical forces which, like the mechanical ones, are a convenient way of describing the way a dislocation responds to thermodynamic changes. We consider the mechanical forces first (Peach and Koehler, 1950). Consider an element d/ undergoing a small displacement ds, as shown in Fig. 7-19. Atoms on opposite sites of the shaded area will be displaced by b relative to each other as this movement occurs. For glide (dF = 0 in Eq. (7-11)), the appropriate applied stress is the RSS T, and the average shear displacement of the crystal surface produced by glide of d/ is b (ds dl/A), where A is the area of the slip plane. The external force on this area is i i so that the work done when the element of slip occurs is (7-32)
For climb (dK^O in Eq. (7-11)), the relevant applied stress is the normal component of stress, an, acting perpendicular to the shaded area. The normal displacement of the crystal surface produced by climb of d/ is again b (ds dl/A), and the work done is
given by Eq. (7-32) with T replaced by <7n. In both cases, the virtual force on d/ is F =dW/ds, and so the glide and climb forces Fg and Fc per unit length of dislocation line are (7-33)
F=Tb,
respectively. The force F acts normal to the dislocation at every point along its length, irrespective of the line direction. The positive sense of the force is given by simple physical reasoning, as in Sec. 7.2.2. Equations (7-33) for the Peach-Koehler forces contain the resolved shear and normal stresses T and on explicitly. A more general expression, valid for all possible combinations of line direction /, Burgers vector b and stress tensor a is (7-34) i= x k= x m= x
where j = x, y or z, and the permutation tensor sijk = +1 if ij k is an even permutation of xyz; = — 1 if ij k is an odd permutation of xyz; — 0 if any two subscripts are the same. This gives the correct magnitude and direction for the three components of the force vector [Fx, Fy, Fz] when b is signed according to the FS/RH convention (Sec. 7.1.3). (An alternative form of Eq. (7-34) is given by Eq. (7-66) in Sec. 7.3.7.) For example, for the straight edge dislocation running along the z axis in Fig. 7-24 b, / = [0,0,1] and the FS/RH rule gives b = [fo,0,0], and Eq. (7-34) reduces to X""1
J7
(7-35)
i= x
and so 77
—
p
rr
/•) -
(7-36 a)
7.3 Dislocations in Elastic Media
(7-36 b) (7-36 c)
F=0
Eqs. (7-36 a) and (7-36 b) are the vectorized form of Eqs. (7-33), with the correct signs, and Eq. (7-36 c) simply reflects the fact that the straight line cannot experience a force parallel to itself. For the right-handed screw dislocation in Fig. 7-24 a with 1= [0,0,1], the FS/RH convention gives 6 = [0,0,b] and it is readily shown that the two non-zero components of force are =
-ozxb
(7-37)
These, correctly, are both glide components, since the straight screw dislocation can only move by glide (Sec. 7.2.2). The mechanical Peach-Koehler forces described above also apply when the source of stress is within the crystal itself (see Sec. 7.3.4 on the interaction between dislocations). A difficulty arises, however, when the force on a dislocation due to its own stress field is required. Analysis of this self-force is complicated by the fact that the stress diverges to ± oo when evaluated at the centre of the dislocation core (Sec. 7.3.2). There are several ways in which this problem can be handled. The simplest is the line tension approximation. This treats the dislocation as an elastic string with a tension, which arises because, as outlined in the previous section, the strain energy of a dislocation is proportional to its length and an increase in length results in an increase in energy. The line tension F has units of energy per unit length. From the approximation used in Eq. (7-31), the line tension, which may be defined as the increase in energy per unit increase in the length of a dislocation line, will be F = E, see Eq.(7-31). Consider the curved dislocation in Fig. 7-26. The line tension will produce forces tending to straighten the line and the direc-
439
Dislocation
Figure 7-26. Curved element of dislocation under line tension forces.
tion of the net force is perpendicular to the dislocation and towards the centre of curvature. The line will only remain curved if there is a force in the opposite sense. For glide equilibrium, this is x b per unit length of line (Eqs. (7-33)), and the stress needed to maintain a radius of curvature R is found in the following way. The angle subtended at the centre of curvature is dd = dl/R, assumed to be <^ 1. The outward force along OA due to the RSS T is x b d/ and the opposing inward force along OA due to the line tension F at the ends of the element is 2Tsin(d0/2) which is equal to FdO for small values of d6. The line will be in equilibrium in this curved shape when the forces balance, i.e. F
(7-38)
Substituting for F from F = E, Eq. (7-31) k/ib X =
R
(7-39)
Equation (7-38) gives the stress required to bend a dislocation to a radius R in the line tension approximation, and Eq. (7-39) assumes from F = E (see Eq. (7-31)) that edge, screw and mixed segments have the same energy per unit length. The curved dislocation is therefore the arc of a circle in the constant line tension approximation but this is strictly valid only if Poisson's
440
7 Dislocations in Crystals
ratio v equals zero. In all other cases, the line experiences a torque tending to rotate it towards the screw orientation where its energy per unit length is lower. The true line tension of a mixed segment is (de Wit and Koehler, 1959)
r=E(a)
d2E(a) da 2
(7-40)
where a is the angle between the segment direction and b and E(a) is given by the analysis of Sec. 7.3.2 below, Eqs. (7-28) and (7-29). As a result -)
(7-41)
is obtained. The line tension F for a screw segment is four times that of an edge when v = 1/3, and thus the radius of curvature at any point is still given by Eq. (7-38), but the overall line shape is approximately elliptical with major axis parallel to the Burgers vector; the axial ratio is approximately 1/(1—v). For many calculations, however, Eq. (7-39) is an adequate approximation. The force calculated in the line tension model depends only on the local curvature of the line, not on its more distant shape. This is a reasonably valid approach for regular line shapes, such as circles and ellipses with R^>b, but is invalid when this is not the case or when other parts of the line approach the segment in question to within distances small compared with R. In these situations, equilibrium between the self and other forces requires that the former be obtained from all the parts of the line that contribute to the self-force, and calculation of this requires in turn a definition of self-stress that avoids the mathematical divergence in the core. Brown (1964) showed that the self-force can be obtained from the average of the two values of the appropriate stress component
evaluated at + r0 with respect to the centre of the core. Inconsistencies in his definition were later removed by refinement of the analysis (Gavazza and Barnett, 1976; Scattergood, 1980; Kirchner, 1981), and selfforce methods have been used to compute equilibrium shapes of dislocations under stress in a wide variety of problems (for review see Bacon et al, 1978). Fortunately, the leading term in the self-force is the line tension, and so the line-tension model provides a good description except in the situations just referred to. The mechanical force for climb (Eqs. (7-33) and (7-36b)) results from the resistance or assistance of the normal component of stress to the introduction of the extra half-plane of the dislocation. The stress can arise from external or internal sources, and can include the line itself, in which case the line tension will tend to reduce the line length in the extra halfplane. However, since either the creation or annihilation of point defects is involved in climb, chemical forces due to defect concentration changes must be taken into account in addition to these mechanical forces. It was seen in Sec. 7.2.6 that when an element of d/ of dislocation is displaced through ds, the local volume change is (b x d/) • ds. Consider a segment length d/ of a positive edge dislocation climbing upwards through distance ds in response to a mechanical force F per unit length. The work done is Fdlds and the number of vacancies absorbed is b dl ds/Q, where Q is the volume per atom. The vacancy formation energy is therefore changed by F Q/b. Thus, the equilibrium vacancy concentration at temperature T in the presence of the dislocation is reduced to c = exp [- (£fv + FQ/b)/kB T] = = c0Qxp(-FQ/bkBT)
(7-42)
441
7.3 Dislocations in Elastic Media
where c0 is the equilibrium concentration in a stress-free crystal. For negative climb involving vacancy emission (F < 0) the sign of the chemical potential is changed so that c> c0. Thus, the vacancy concentration deviates from c 0 , building up a chemical force per unit length on the line /=
In (c/c0)
II
(7-43)
until / balances F in equilibrium. Conversely, in the presence of a supersaturation c/c0 of vacancies, the dislocation climbs up under the chemical force / until compensated by, say, external stresses or line tension. The latter is used in the analysis for a dislocation climb source in Sec. 7.4.3. By substituting reasonable values of T and Q in Eq. (7-43), it is easy to show that even moderate supersaturations of vacancies can produce forces much greater than those arising from external stresses. The rate of climb of a dislocation in practice depends on (a) the direction and magnitude of the mechanical and chemical forces, F and / , (b) the mobility of jogs (Sec. 7.2.6) and (c) the rate of migration of point defects through the lattice to or from the dislocation. 7.3.4 Interaction Between Dislocations The Peach-Koehler force discussed in the preceding section applies when the source of stress is another dislocation. This is readily demonstrated by deriving the force from the additional work done in introducing the second dislocation into a crystal which already contains the first. The derivation can be complicated (Sec. 7.3.7), but for simplicity consider two straight dislocations lying parallel to the z axis as in Fig. 7-27, shown for convenience as edge dislocations. The total energy consists of the self-energy of dislocation I, the
Figure 7-27. Interaction between two dislocations.
self-energy of dislocation II, and the elastic interaction energy Eint between I and II. £ int is the work done in displacing the faces of the cut which creates II in the presence of the stress field of I. The displacements across the cut are bx, by, bz, the components of the Burgers vector b of II. By visualizing the cut parallel to either the x or y axes, two alternative expressions for Eint per unit length of II are im = I (K x
ayy
bz
GZV)
dx (7-44)
00
mt = ~ f (
ayx + bz
GZX)
dy
where the stress components are those due to I. The interaction force on II is obtained simply by differentiation of these expressions, i.e. Fx = — 9£ int /9x and Fy = -9E i n t /9y: FX = t , GXy + by Gyy ^ ^ , G
^ ^
For the two parallel edge dislocations with parallel Burgers vectors shown in Fig. 7-27, by = bz = O and bx = b, and the components of the force per unit length acting on II are therefore Fx = Gxyby
Fy = -Gxxb
(7-46)
where Gxy and GXX are the stresses of I evaluated at position (x, y) of II. The forces are
442
7 Dislocations in Crystals
reversed if II is a negative edge. Equal and opposite forces act on I. Equations (7-46) are seen to agree with the glide and climb components given in Eqs. (7-36). Substituting from Eq. (7-23) gives x
y
fib2 x(x2-y2) 2 n ( l - v ) {x2 + y2)2 fib2 y(3x2-y2) 2 n ( l - v ) (x2 + j 2 ) 2
(7-47)
Since the glide planes of the edge dislocations in Fig. 7-27 are perpendicular to the y axis, the component of force which is important in determining the behavior in the absence of climb is Fx. Thus, for x > 0, Fx is negative (attractive) when x
edges of the same sign and ± y if they have the opposite sign. It follows that an array of edge dislocations of the same sign is most stable when the dislocations lie vertically above one another as in Fig. 7-29 a. This is the arrangement of dislocations in a small-angle pure tilt boundary. On the other hand, edge dislocations of opposite sign gliding past each other on parallel slip planes tend to form stable dipole pairs as indicated in Fig. 7-29 b when the applied stress is removed. Equations (7-44) for £ int are generally valid, and so if dislocations I and II in Fig. 7-27 are replaced by parallel screws (b = bz\ it is easily seen that Eqs. (7-45) are valid for the interaction force. This can be extended to cylindrical polar coordinates by noting that oze = azy and azr = azx for y = 0. Hence, the radial and tangential components of force on II are (see Eq. (7-22)) Fr = ozeb =
jib2/2nr
(7-48)
o-3r
0-2 -,
S o 0)
-0
-0-2 -J
-0-3L
Figure 7-28. Force between parallel edge dislocations with parallel Burgers vectors in units of fib2/[2n(l —v)y]. Curve A is for like dislocations and curve B for unlike dislocations. (After Cottrell, 1953)
7.3 Dislocations in Elastic Media
443
490°
(a)
Figure 7-29. Stable positions for two edge dislocations of (a) the same sign and (b) opposite sign.
(b)
The force is much simpler in form than that between two edge dislocations because of the radial symmetry of the screw field. Fr is repulsive for screws of the same sign and attractive for screws of opposite sign. It is readily shown from Eqs. (7-45) that no forces act between a pair of parallel edge and screw dislocations in isotropic elasticity, as expected from the lack of mixing of their stress components.
tions are satisfied. When evaluated at the dislocation line, as in Sec. 7.3.3, they result in a force. The analysis for infinite, straight dislocation lines parallel to the surface is relatively straight-forward. Consider screw and edge dislocations parallel to, and distance d from, a surface x = 0 (Fig. 7-30); the edge dislocation has Burgers vector b in the x direction. For a free surface, the tractions aXX9 ayx and ozx
must be zero on the plane x = 0. Consideration of Eqs. (7-17) and (7-21) shows that these boundary conditions are met for the screw if the infinite-body result is modified by adding to it the stress field of an imaginary screw dislocation of opposite sign at x = — d (Fig. 7-30). The required solution for the stress in the body (x > 0) is therefore
7.3.5 Image Forces
A dislocation near a surface experiences forces not encountered in the bulk of a crystal. It is attracted towards a free surface because the material is effectively more compliant there and the dislocation energy is lower: conversely, it is repelled by a rigid surface layer. To treat this mathematically, extra terms must be added to the infinitebody stress components given in Sec. 7.3.2 in order that the required surface condi-
Ay
Ay 2
(xl+y ) (xl+y2) Ax + Ax_ 2 (xl+y ) (x\+y2)
(7-49)
Figure 7-30. (a) A screw y y y y
y
yz y y y
Imagee rl
^
*• y y y,
(a)
Image
Screw X
T
O
1 i 1
(b)
Edge
rl
and (b) an edge dislocation at a distance d from a surface x = 0. The image dislocations are in space a distance d from the surface. (From Hull and Bacon, 1984; reprinted with permission of Pergamon Press, Ltd.)
444
7 Dislocations in Crystals
where x_ = (x — d), x+ = (x + d) and A = fi 6/2 7i. The force per unit length in the x direction Fx = azyb (see Eqs. (7-37)) induced by the surface is obtained from the second term in ozy evaluated at x = d, y = 0. It is F ' = •
And
(7-50)
and is simply the force due to the image dislocation at x = — d. For the edge dislocation (Fig. 7-30 b), superposing the field of an imaginary edge dislocation of opposite sign at x = — d annuls the stress oxx on x = 0, but not Gyx. When the extra terms are included to fully match the boundary conditions, the shear stress in the body is found to be (Head, 1953) (x2+-y2) (xl+y2)2 (x\+y2)2 2Dd[x_ x% - 6xx+y2 + y4]
(7-51)
where D =A/(l—v). The first term is the stress in the absence of the surface, the second is the stress appropriate to an image dislocation at x = — d, and the third is that required to make (ryx = 0 when x = 0. The force per unit length Fx( =
4n(l-v)d
that a second dislocation near the surface would experience a force due to its own image and the surface terms in the field of the first. The interaction of dipoles, loops and curved dislocations with surfaces is therefore complicated, and only given approximately by images.
(7-52)
and is again equivalent to the force due to the image dislocation. The image forces decrease slowly with increasing d and are capable of removing dislocations from near-surface regions. They are important, for example, in specimens for transmission electron microscopy (Sec. 7.1.5) when the slip planes are inclined steeply to the surface. It should be noted
7.3.6 Interaction Between Point Defects and Dislocations
The most important contribution to the interaction between a point defect and a dislocation is usually that due to the distortion the point defect produces in the surrounding crystal. The distortion may cause an interaction with the stress field of the dislocation to raise or lower the elastic strain energy of the crystal. This change is the interaction energy Ex. If the defect occupies a site where EY is large and negative, a situation that may be met either when a dislocation glides through a crystal containing a random distribution of defects or when defects are able to diffuse to such favored positions, work |£J will be required to separate the dislocation from it. The simplest model of a point defect is an elastic sphere of natural radius ra (1 + 5) and volume F s , which is inserted into a spherical hole of radius ra and volume Vh in an elastic matrix (Fig. 7-31 a). The sphere and matrix are isotropic with the same elastic constants. The difference between the defect and hole volumes is the misfit volume F mis .
Km, = VS-Vb*4nr?8
(if8< 1)
(7-53)
The misfit parameter 5 is positive for oversized defects and negative for undersized ones. On inserting the sphere in the hole, Vh changes by AVh to leave a final defect radius r a (l -ha). The change AVh is given by (7-54)
7.3 Dislocations in Elastic Media Volume AVS
445
Volume y< Defect jo
(a)
(b)
Parameter s is determined by the condition that in the final state the inward and outward pressures developed on the sphere and the hole surfaces are equal. The result is (Eshelby, 1956, 1957) 8 =
(1+v) 3(1-v)'
Figure 7-31. (a) Model for a defect of natural radius ra(l +5) inserted in a hole of radius ra. The final radius is ra (1 + s). (b) Geometry for the interaction with a dislocation.
(7-55)
The total volume change experienced by an infinite matrix is AKh, for the strain in the matrix is pure shear with no dilational part. In a finite body, however, the requirement that the outer surface be stress-free results in a total volume change given by (Eshelby, 1956, 1957) (7-56) which, from Eqs. (7-53) to (7-55), equals "mis'
If the material is subjected to a pressure /?, the strain energy is changed by the presence of the point defect by (7-57)
When a dislocation is the source of p, it is evaluated at the site of the point defect (Cottrell, 1948; Bilby, 1950). For a screw dislocation, p = 0 (Sec. 7.3.2) and thus El = 0. However, for an edge dislocation lying along the z axis (Fig. 7-31 b), Eqs.
(7-24), (7-53) and (7-56) give 3
y
(7-58)
or in cylindrical coordinates El
=
4(1+v)fibred sinO 3(1-v)
r
(7-59)
In terms of s (Eq. (7-55)), £, simplifies to (7-60) but note that this equation would not be valid if the defect were, say, incompressible, for then Eq. (7-55) would be replaced by e = <5. For an oversized defect (d > 0) El is positive for sites above the slip plane (0 < 9 < n) and negative below (n<6<2n). This is because the edge dislocation produces compression in the region of the extra halfplane and tension below (Sec. 7.3.2). The positions of attraction and repulsion are reversed for an undersized defect (5<0). The value of ra in Eq. (7-59) can be estimated from the fact that Vh is approximately equal to the volume per atom Q for substitutional defects and vacancies, and is smaller for interstitial defects. A given species of defect produces strain in the lattice proportional to 6, and this parameter can be determined by measurement of the lat-
446
7 Dislocations in Crystals
tice parameter as a function of defect concentration. It is found to range from about -0.1 to 0 for vacancies, -0.15 to +0.15 for substitutional solutes, and 0.1 to 1.0 for interstitial atoms. For a given distance r the sites of minimum El9 i.e. maximum binding energy of the defect to the dislocation, are at 0 = n/2 or 3 TI/2, depending on the sign of 3. The strongest binding occurs within the dislocation core at r ~ b , and from Eq. (7-59) the energy ranges from about 3131 eV for the close-packed metals to 20131 eV for silicon and germanium. Although the use of this equation of linear elasticity within the dislocation core is questionable, these energy values may be considered upper limits and are within an order of magnitude of estimates obtained from experiment and computer simulation. The interaction of a dislocation with a spherically symmetric defect is a special case of a more-general size-effect interaction. Many defects occupy sites that have lower symmetry than the host crystal and thereby produce asymmetric distortions. Solute interstitial atoms, such as carbon in a-iron, and self-interstitial atom dumbbells are good examples. Unlike the spherical defect, the asymmetric defect can also interact with screw dislocations. In the elasticity approximation, the defect can be treated as three orthogonal force dipoles parallel to the x, y, z axes and of moment Pxx, Pyy and Pzz (e.g., Bacon, 1969; Bacon et al., 1978). The interaction energy with a dislocation is then = - (Pxx exx
Pyy eyy
Pzz ezz)
(7-61)
where etj is the strain field of the dislocation at the defect site. (For the tetragonal distortions produced by the carbon interstitial in a-iron or the <100> dumbbell self-interstitial in f.c.c. metals, for example, Pxx = Pyy 7^ Pzz.) Alternatively, if the misfit strains produced at the surface of the defect
by these dipoles are dxx, 3yy, 3ZZ (Cochardt et al., 1955; Hirth and Lothe, 1982), then by analogy with Eq. (7-57) 4 (7-62) where axx, oyy and ozz are the stresses produced by the dislocation. The dislocation shear field may enter Ex when the dipole axes are transformed to those of the dislocation coordinate system, and the magnitude of El for a screw dislocation may then be comparable with that for the edge. In both cases Ex is proportional to 1/r, as in Eq. (7-59), but it has a more complicated orientation dependence than sin#. A second form of elastic interaction arises if the defect is considered to have different elastic constants from the surrounding matrix. A vacancy, for example, is a soft region of zero modulus. Both hard and soft defects induce a change in the stress field of a dislocation, and this produces the inhomogeneity interaction. The interaction is always attractive for soft defects, because they are attracted to regions of high elastic energy density. (This is analogous to the attraction of dislocations to a free surface (Sec. 7.3.5).) Hard defects are repelled. The dislocation strain energy over the defect volume is proportional to 1/r2 for both edges and screws (see Eqs. (7-19), (7-21) and (7-22), for instance), and the interaction energy therefore decreases as 1/r2. Although it is of second order in comparison with the 1/r size-effect interaction, it can be important for substitutional atoms and vacancies when the misfit parameter 3 is small. It can also be important when an external stress is applied. In this situation, EY for the inhomogeneity effect depends on the orientation of the line and b with respect to the principal axes of stress. This results in a stronger interaction of point defects with some dislocations than others, and can lead to a stress-
7.3 Dislocations in Elastic Media
induced preferred absorption at these dislocations when the point defects are mobile. This is an important mechanism in irradiation creep in metals (e.g., Bullough, 1985). There are other sources of interaction between point defects and dislocations. For instance, when a perfect dislocation dissociates into partial dislocations (Sec. 7.6), the ribbon of stacking fault formed changes the crystal structure locally. The solute solubility in the fault region may be different from that in the surrounding matrix, and the resulting change in chemical potential can cause solute atoms to diffuse to the fault if the temperature is sufficiently high. This chemical (or Suzuki) effect is not a longrange interaction, but it may present a barrier to the motion of dissociated dislocations (Suzuki, 1962). There are several effects that can give rise to an electrical interaction. In metals, the conduction electron density tends to increase in the dilated region below the half plane of the edge dislocation and to decrease in the compressed region. The resulting electric dipole could, in principle, interact with a solute atom of different valency from the solvent atoms, but free-electron screening probably reduces the interaction to negligible proportions in comparison with the size and inhomogeneity effects of the elastic model. In ionic crystals, lack of screening leads to a more significant interaction, but of more importance is the presence of charged jogs. The electric charge associated with jogs (Sec. 7.6.5) produces a 1/r Coulomb electrostatic interaction with vacancies and charged impurity ions, and this probably dominates the interaction energy in such materials (see Volume 11, Chapter 7 of this Series). In covalent crystals, the relative importance of the possible effects is less clearly understood. In addition to the elastic interaction and the electric-dipole interaction with impurities of different valency, an elec-
447
trostatic interaction can occur if conduction electrons are captured by the dangling bonds which may exist in some dislocation cores (Sec. 7.6.7 and Volume 4, Chapter 6 of this Series). In a crystal containing a concentration c0 of point defects in solution, the energy of a defect in the vicinity of a dislocation is changed by Ex from its value in the matrix. The equilibrium defect concentration c at a position near the dislocation therefore changes from c0 to
= c0Qxp[-El/kBT]
(7-63)
This formula assumes that the defects do not interact with each other. Defects therefore tend to concentrate in core regions where E{ is large and negative, and dense atmospheres of defects can form in weak solutions with small values of c0. The condition for these Cottrell atmospheres is that the temperature is sufficiently high for defect migration to occur but not so high that the entropy contribution to the free energy causes the atmosphere to disperse into the solvent matrix. It is readily shown from Eq. (7-63) that even in dilute solutions with, say, c0 = 0.001, dense atmospheres with c>0.5 may be expected at r = 0.5Tm in regions where — EY>3kBTm, where Tm is the melting point. This corresponds to a typical defect-dislocation binding energy of 0.2 to 0.5 eV for metals. Solute segregation to the core of a dislocation is important because an extra applied stress is required to move the dislocation away from the concentrated solute region. The dislocation is said to be locked by the solutes. Once a sufficiently high stress has been applied to separate a dislocation from the defects, its movement is unaffected by locking, but subsequent heat treatment can allow the defects to diffuse back to the dislocations and re-establish locking: this is the basis of strain ageing.
448
7 Dislocations in Crystals
Also, at sufficiently high deformation temperatures and low strain rates, point defect mobility may enable solutes to repeatedly lock dislocations during dislocation motion, giving a repeated yielding process known as dynamic strain ageing or serrated yielding and characterized by a serrated stress-strain curve. (These aspects are discussed in more detail in, for example, Hirth and Lothe (1982) and Hull and Bacon (1984).) 7.3.7 General Line Shapes and Anisotropic Media
The expressions derived in the preceding sections for dislocation stress and energy, and the interaction forces between dislocations, are based on the results for a straight, infinitely-long line. They are often adequate and can provide good insight into dislocation properties. There are problems, however, for which the true line shape needs to be analyzed, and this invariably adds complexity. The general equation for the elastic strain energy of a dislocation is (de Wit, 1960) =
iffy * > . AJ
i=x^
(7-64) j —x
where A is the cut surface bounded by the line (on which u = b) and the core surface, Gtj are the stress components produced by the line, and n is the unit vector normal to any element &A of A, with positive sense for n given by the right-hand rule with respect to the positive line direction (Sec. 7.1.3). This surface integral is the generalized version of Eqs. (7-27) and (7-28). Similarly, the interaction energy of dislocations I and II is (7-65) A
i= x j = x
where altj is the stress field of I and b11 is the Burgers vector of line II, which bounds surface A. This is the generalized version of Eq. (7-44). By considering the variation in £ int as line II is displaced, the Peach-Koehler force on II in a plane with normal n is found to be (Bacon et al., 1978; Hirth and Lothe, 1982)
F= Z Z '
(7-66)
i=x j = x
which is an alternative to Eq. (7-34). Whether the integrations can be performed easily, or the force F evaluated readily, depends on the form of atj. It is clearly simple for a straight line, but more complex for curved and polygonal lines. Nevertheless, it has an analytic form for some geometries, and this lends itself to tractable treatment. For anisotropic elasticity, otj almost invariably requires numerical evaluation, even for straight lines, and so Eqs. (7-64) to (7-66) cannot be dealt with in a simple way. We therefore close this section with a brief account of one approach that has proved useful in this context. The stress field of a general curvilinear dislocation can be expressed as an integral taken around the line (Mura, 1968; Bacon et al., 1978). The integrand of the line integral is a function of products of the Burgers vector, the elastic constants and the first derivative of the Green's function of elasticity. The latter can be reduced to either a single integral or the eigenvectors of a sixdimensional eigenvalue equation, and has to be obtained numerically for anisotropic elastic media in general. However, Brown (1967) developed a formula for planar geometries which reduces the problem to manageable proportions. The geometry relevant to Brown's formula is shown in Fig. 7-32. The dislocation is a planar curvilinear line L and xf is the position vector from a fixed origin to an
7.3 Dislocations in Elastic Media
449
and can therefore be evaluated in terms of the field factors for infinite straight lines having directions AP and BP, i.e. <> / = ! reference and 0 = 2 > respectively. Bacon et al. (1978) have shown how Eq. (7-68) may also be applied to three-dimensional problems. tangent For anisotropic materials the field facLine L tors Ztj as well as the energy factor K, Eq. Figure 7-32. The geometry used for the elastic field at (7-30) require numerical evaluation, but P due to dislocation line L. can be expressed analytically for isotropy, as in Eqs. (7-22), (7-23) and (7-29). Both factors have been computed for many element d/ on L. Brown showed that the metals with a wide range of anisotropies, stress at point P (with position vector x) in and tabulated in convenient forms (e.g., the plane of the line is Bacon and Scattergood, 1974). However, it , , 1 has been found that the isotropic approxi•61 mation is generally good provided appro\x — x (7-67) priate values of \i and v are used in the where angles a and <j> are measured from isotropic formula. These effective isotropic an arbitrary reference direction, which is constants can be defined uniquely by notusually chosen for convenience in two-diing that for isotropic media, the energy facmensional problems as the direction of b. tors for screws (a = 0) and edges (a = 7t/2) Itj is a function of <> / and is the angular part are (see Eq. (7-29)) of the stress field of an infinite straight disr location with direction x — x (not direction (7-69) = ^ and K (JC/2) = d/) and the same b as L. This means that 4n(l— v) the stress at distance r from an infinite straight of orientation
(7-68)
Figure 7-33. The geometry for defining the field at P due to a straight segment AB.
450
7 Dislocations in Crystals
Table 7-1. The effective shear modulus fi (in GPa) and Poisson's ratio v for dislocations in some common cubic metals. Each column heading indicates the direction of b and the plane of the dislocation, which is not always the slip plane. Starred entries are values of fi/(l — v). The anisotropy ratio A is a measure of the deviation of the metal from its value A = l in an isotropic solid (Hirth and Lothe, 1982). (From Bacon, 1985) f.c.c.
A
Ag
3.01
[110] (111)
[110] (111)
[112] (111)
[111] (111)
26.6 0.449
4.35 0.918
27.8 0.434
48.7*
v =
\i =
Al
1.21
\i = V ==
25.9 0.360
22.6 0.447
25.9 0.359
39.7*
Au
2.90
n=
24.7 0.498
6.70 0.872
25.9 0.484
48.9*
42.1 0.431
10.1 0.864
44.2 0.413
69.0*
78.6 0.363
64.4 0.526
80.7 0.351
126.3*
[Til] (110)
[111] (110)
[100] (110)
[110] (110)
[111] (211)
128.4 0.102
120.8 0.187
114.6 0.260
150.4*
128.4 0.103
62.5 0.473
80.5 0.341
98.1 0.060
115.9*
62.5 0.485
130.1 0.248
122.7 0.291
116.2 0.358
175.6*
130.1 0.249
44.3 0.270
37.4 0.391
34.6 0.474
63.0*
44.3 0.278
61.2 0.428
68.3 0.365
74.9 0.254
104.9*
61.2 0.432
50.1 0.313
47.5 0.355
45.8 0.397
74.5*
50.1 0.314
Cu Ni
3.21 2.51
V
==
\k
=
V
=
IX = V =
b.c.c. Cr
A 0.69
\i
=
V =
Fe
2.35
11 = V =
Mo
0.77
\i
=
V =
Nb Ta V
0.51 1.56 0.78
fl
=
V
=
fl
=
V
=
fi = V
==
metals. They enable the anisotropic field and energy of an infinite straight dislocation to be approximated to within a percent or two, except for the starred columns. (For those, the line is pure edge in all orientations and the stress is independent of angle, whereas the anisotropic field can very. The biggest error here is 12% (for iron)). Note that the effective constants for a given metal vary from plane to plane, depending on the degree of anisotropy of the material. They have been demonstrated to give a good representation of
anisotropic results in several complex dislocation problems: for a review see Bacon etal. (1978).
7.4 Generation of Dislocations 7.4.1 Thermodynamic Considerations
The contribution the entropy of a dislocation makes to the free energy of a crystal is about —2kBT/b per unit length (Cottrell, 1953). This compares with the strain energy contribution of about jub2 (Eq.
7.4 Generation of Dislocations
(7-31)). Since jib3 is typically ~5eV and kB T at 300 K is 1/40 eV, the net free energy change due to dislocations is positive, and so the equilibrium density of dislocations in a stress-free crystal is zero. Nevertheless, apart from carefully prepared crystals of materials like silicon (Volume 4, Chapter 6 of this Series) dislocations occur in all crystals. There is a variety of sources from which dislocations may have been generated in freshly grown crystals. These include (i) defect development from the 'seed' or from the mould surface; (ii) internal stresses due to differential thermal contraction at impurity particles or in regions with strong temperature gradients (Sec. 7.4.2); (iii) impingement of different parts of the solidification interface; (iv) misfit of the lattice spacing between materials in epitaxial contact (Sec. 7.4.2); and (v) nucleation and growth of dislocation loops in supersaturations of either vacancies or self-interstitials, which occur in rapidly quenched or irradiated materials. It should be noted that homogeneous nucleation is not included in this list. Simple estimates (Cottrell, 1953; Hull and Bacon, 1984) demonstrate that a stress of the order of the theoretical shear strength ~/i/10 (Sec. 7.2.1) is required to nucleate a dislocation: conversely, at the yield stress, the activation energy for nucleation is several keV. Thus, dislocations cannot be nucleated homogeneously in a crystal in thermodynamic equilibrium at the yield stress. The large increase in dislocation density that accompanies plastic flow (Sec. 7.1.3) results from multiplication of pre-existing dislocations (see Sec. 7.4.3).
451
cients be al and oc2. Suppose that at some temperature the natural stress-free size of the inclusion (radius rx) is the same as the hole it occupies in the matrix. Then if the material undergoes a temperature change AT, the inclusion becomes a misfitting sphere of radius ri(l + e), in analogy with Sec. 7.3.6. The elastic displacement field generated in the matrix is purely radial and decreases as r~2, and since it equals srx at r = r x , it is ur — tr1/r
(7-70)
The strain field in the matrix is one of pure shear, and the maximum shear stress rmax acting in a radial direction on a cylindrical surface with radial axis occurs at the interface between the inclusion and matrix on a cylinder of diameter y/2r1, as illustrated in Fig. 7-34. It is W = 3e/*
(7-71) 6
1
Thus, if, say, a1 = 3xlO~ K~ , a2 = 30 x l O ^ K " 1 and AT = 600K, then e = 0.016 and rmax = fi/20, which is at the level required for dislocation nucleation (Sec. 7.4.1). The prismatic loops (i.e., loops where the slip plane is a cylinder surface containing the dislocation line and b) produced by this process of prismatic punching glide on their glide cylinder away from the inclusion to regions of lower stress, and rows of such loops may form from a single
Cylindrical surface with max. resolved shear stress in direction 0 A.
7.4.2 Nucleation at Stress Concentrations
As an example of this, consider a spherical inclusion embedded in a matrix, and let their respective thermal expansion coeffi-
Figure 7-34. The geometry for dislocation nucleation at a misfitting particle. (After Hull and Bacon, 1984; reprinted with permission of Pergamon Press, Ltd.)
452
7 Dislocations in Crystals
particle. They may be either interstitial in nature (if s > 0) or vacancy (if e < 0) depending on the physical need to move material either away from or towards the inclusion, respectively. It is not necessary for material inhomogeneity to be present in order that dislocations are generated by differential thermal expansion. Thermal gradients of quite modest proportions can readily produce stresses that exceed the CRSS, and hence cause dislocations to move and multiply. This is a process that produces unwanted dislocations in GaAs crystals grown by the Czochralsky technique (Jordan et al, 1980; Meduoye et al., 1991). There is another source of misfit or coherency strain that can result in dislocations. It may arise when two crystals bond across an interface with different atomicplane spacings, e.g. different crystal structure and/or different composition. This is illustrated schematically in Fig. 7-35 for two structures with interplanar spacing a1 and a2. In the incoherent interface in (a), there is only irregular matching in the continuity of planes across the interface, and both crystals are stress-free. In (b), \a1 — a2\ is small and coherency of the planes is achieved, albeit at the expense of elastic coherency strain of order \a1 — a2\/a1 in each crystal. The elastic energy stored
is therefore proportional to \a1—a2\2. Clearly, if \a1 — a2\ is not small and/or the material is thick enough, it is energetically favorable to nucleate dislocations to reduce the strain. This results in a semicoherent interface containing misfit dislocations of spacing ax a2/\a1—a2\9 as shown in Fig. 7-35 c. The criteria that result in unwanted dislocations in epitaxially-grown semiconductor device materials have been the subject of much study (Volume 4, Chapter 6 of this Series; Gosling et al., 1992). 7.4.3 Multiplication Sources
The plastic strain that occurs at yield (Sec. 7.2.7) is accompanied by an increase in the density of mobile dislocations. The processes of regenerative dislocation multiplication discussed in the literature are similar to the mechanism proposed by Frank and Read (1950). It is shown schematically in Fig. 7-36, where a dislocation segment DD' of length L experiences a glide force T b (per unit length) in its glide plane. The remainder of the line may be pinned or lie on other planes. As T is increased from zero, the line will bow to a radius of curvature R which balances the line tension (Eq. (7-38)). The minimum R, corresponding to a semicircle in the constant line-tension approximation (Eq. (7-39)), therefore occurs at a stress ?max-/^/£
/
>
a->
(a)
(b)
3;
(c)
Figure 7-35. Representation of incoherent, coherent and semi-coherent interfaces. (From Hull and Bacon, 1984; reprinted with permission of Pergamon Press, Ltd.)
(7-72)
This is a maximum or critical stress for the source, in the sense that for stresses beyond this, R increases and Eq. (7-39) cannot be satisfied. The unstable line forms a complete loop, because the segments m and n in Fig. 7-36 d, which move in opposite directions under T, are of opposite sign and annihilate. The original segment DD' is seen to regenerate itself as the large outer loop is created, and the process can repeat.
7.5 Dislocations 'En Masse'
xb
xb xb 4 4 • !!
(a)
L(b)
D' (c)
of the paper. By equating the climb force under a supersaturation of vacancies (Eq. (7-43)) to the line-tension force (Eq. (7-39)) at the critical radius R = L/2, we find that the critical supersaturation is given by ikixbQ
KTL
Figure 7-36. Representation of the operation of the Frank-Read source. Slip has occurred in the shaded area. (From Read, 1953; reprinted with permission of McGraw-Hill, Inc.)
This mechanism is believed to be important in the production of wide slip bands. In this case, a dislocation that has experienced double cross-slip (Fig. 7-15) can act as a source on the plane parallel to the first if the parts of it in the inclined cross-slip plane are relatively immobile. Multiple cross-glide can result in the macroscopic slip bands frequently observed in deformed materials. When one of the pinning points DD' in Fig. 7-36 is a node of dislocations, and the other dislocations have a b component perpendicular to the glide plane of the source segment, operation of the source results in the formation of a conical helix by a pole mechanism at the node (Fig. 7-37). As the source winds round the pole, it produces a helix with pitch equal to the screw component of the pole dislocation. Dislocations can also multiply by a regenerative process involving climb rather than slide. The Bardeen-Herring source (Bardeen and Herring, 1952) is similar to the Frank-Read source in Fig. 7-36, except that the segment DD' is pure edge in character with the extra half-plane in the plane
453
(7-73)
where /c~0.5. By inserting typical values on the right-hand side of this equation, it is readily found that climb sources can operate at supersaturations as low as a few percent. Further discussion of sources can be found in Hirth and Lothe (1982), Chaps. 17 and 20.
7.5 Dislocations 'En Masse' 7.5.1 Intersection of Dislocations Since even well-annealed crystals contain a non-zero density of dislocations in a network, a single dislocation gliding in its slip plane will either interact with other dislocations because of their stress field (see Sec, 7.3.4) or cut through the forest dislocations that intercept its slip plane. The latter intersection process can influence the ease of glide and is discussed here. When a dislocation of Burgers vector b moves over its glide plane, two atoms ini-
Figure7-37. The pole mechanism for a dislocation with a Burgers vector bl. (a) At the node, b2 and b3 have a component perpendicular to the glide plane for bl. (b) The helix advances by this component on each turn. (From Hirth and Lothe, 1982; reprinted with permission of John Wiley and Sons, Inc.)
454
7 Dislocations in Crystals
tially adjacent to each other across the plane experience a relative displacement of b (Sec. 7.2.2). Thus, when two dislocations intersect by the glide of one through another, each acquires a step equal to the Burgers vector of the other. The step may be a jog or kink (Sees. 7.2.4 and 7.2.6). This is illustrated in Fig. 7-38 for (a) orthogonal edges with orthogonal Burgers vectors, (b) orthogonal edges with parallel Burgers vectors, (c) orthogonal edge and screw, and (d) orthogonal screws. The direction of movement is shown by an arrow, and the glide plane for the edge is indicated as appropriate. Whether the jog/kink affects subsequent dislocation movement or not depends on the glide geometry. One jog is formed in (a) and, since it can glide with line AB, does not affect its motion. The same is true for the edge line AB in (c). Both steps in (b) are kinks and do not impede glide of the lines. However, all the steps in the screw lines of (c) and (d) have edge character, and can only glide on one plane. If the screw dislocation glides on the same plane, the step is a kink and offers no resistance to the line. However, if the screw glides on a different plane, the step is a jog and can only move with the line by climb. This requires pointdefect generation, and is illustrated in Fig. 7-39 for a screw line gliding upwards, but restricted by several jogs of spacing x. When a single point defect of formation energy Ef is created, the line sweeps an area of approximately b x and the work done by the load is T b2 x. Thus the stress required is T = Ef/b2 x
After
Before
Y*tb
(d)
Figure 7-38. Insertion of dislocations AB and XY, showing the situation before and after intersection. The glide planes of the edge dislocations are labeled PAB and PXY. (After Read, 1953)
(7-74)
£ f is usually lower for vacancies than selfinterstitials, and they are the dominant species produced in plastic deformation. At temperatures above OK, thermal activation assists defect formation and reduces z in Eq. (7-74). The phenomenon of thermal
Figure 7-39. Glide of a jogged screw dislocation producing trails of point defects. (After Hull and Bacon, 1984)
7.5 Dislocations 'En Masse'
activation of dislocation glide is discussed in more detail in Volume 6, Chapter 5 of this Series. The processes outlined here are relatively simple, in the sense that the dislocations cut through each other and simply leave steps on the lines, and these steps are only b in height. In reality, the Burgers vectors may be orientated favorably according to Frank's rule (Sec. 7.3.2) and then an attractive junction segment can form along the line of intersection of the two glide planes. Also, the dislocations may be dissociated (Sec. 7.6) and this can influence the nature of the interactions and products formed. Finally, the jogs may not be of 'unit' height, for superjogs many b high can form during plastic deformation. Their behavior under stress is not the same as that of the unit jogs described above. A review of these aspects is to be found in Hull and Bacon (1984), Chap. 7. 7.5.2 Dislocation Pile-Ups Dislocations moving as a group can have a synergetic effect, as typified by the pile-up. Consider a dislocation source which emits a series of dislocations all lying in the same slip plane (Fig. 7-40). When the leading dislocation meets a barrier such as a grain boundary or sessile dislocation configuration, further movement is pre-
vented and the dislocations then pile up. Being of the same sign, they do not combine. They interact elastically and their spacing, which depends on the applied shear stress and the type of dislocation, decreases towards the front of the pile up. The stress T1 experienced by the leading dislocation of a pile-up can be deduced as follows. For n dislocations, the leading dislocation experiences a forward force due to the applied stress T and the other (n — 1) dislocations, and a backward force due to the internal stress xob produced by the obstacle. If the leading dislocation moves forward by a small distance dx, so do the others, and the applied stress does work per unit length of dislocation equal to n b T dx. The increase in the interaction energy between the leading dislocation and iOb is bxOhdx. In equilibrium, these energies are equal and T1 = rOb, so that (7-75) = nx Thus, the stress at the head of the pile-up is magnified to n times the applied stress. The pile-up exerts a back-stress i b on the source, which can only continue to generate dislocations provided (T — tb) is greater than the critical stress for source operation. Eshelby et al. (1951) found that for a singleended pile-up spread over the region 0<x
Pile-up / \ Barrier
Figure 7-40. Dislocations piled-up against barriers under an applied shear stress T. (After Hull and Bacon, 1984)
455
(7-76)
where D is fib/n for screw dislocations ixb/[n(l— v)] for edges. The shear stress outside the pile-up well away from the first and last dislocations is approximately the same as that produced by a single superdislocation of Burgers vector n b at the centre of gravity of the pile-up {x = 3 L/4). Dislocation pile-ups therefore produce large, long-range stresses. At grain boundaries, this can nucleate either yielding in the adjacent grains or boundary cracks. It can
456
7 Dislocations in Crystals
also assist in the cross slip of screw dislocations held-up at obstacles such as precipitates or dislocation locks. More general aspects of collective behavior and effects of multiple dislocations are described in Chapter 4, Volume 6 of this Series.
7.6 Dislocations in Particular Crystal Structures
views by Amelinckx (1979) and Hirth and Lothe (1982)): here, we simply consider a few. One of the most widely studied is that of the f.c.c. metals, such as Ag, Al, Au, Cu and Ni, and they illustrate many of the basic concepts required for understanding other classes. We therefore start with these relatively simple metals, and then consider other common materials. 7.6.2 The f.c.c. Metals 7.6.2.1 Dislocations and Slip
7.6.1 Introduction The preceding sections of this chapter have taken little account of the effects of crystal structure on dislocation properties. It has been noted that the slip vector b of a perfect dislocation is usually the shortest translation vector of the lattice - 'shortest' because the energy is proportional to b2 (Sec. 7.3.2) and 'translation vector' so that the dislocation leaves perfect atomic coordination in its wake - and that the glide planes tend to be widely spaced - because this lowers the Peierls stress (Sec. 7.2.4), which is, of course, a result of crystal structure. Nevertheless, most of the discussion so far has been based on a model in which the crystalline solid is treated as an elastic continuum. This is fully justified for dislocation properties that have a much larger range than the interatomic spacing, e.g. the elastic field, but is inadequate for effects within the core region and for mechanisms that are specific to a crystal structure. These effects are important for understanding phenomena related to dislocations in real materials, and are the subject of this section. It will be seen that they depend not only on the lattice structure of the crystal but also on the nature of the atomic bonding. There are therefore many possible classes of material to treat (see the extensive re-
Slip in these metals is almost always observed to occur on the {111} planes and in the <110> directions. The Burgers vectors of the perfect dislocations responsible for slip are of the form 1/2 <110>, the shortest vector of the lattice. However, the core structure and behavior of these dislocations is dominated by the fact that they can decompose into a more stable arrangement of dislocations. This occurs because stable stacking faults can exist in the f.c.c. structure. These are planar defects in which atomic coordination across the plane is not perfect. A stacking fault can be envisaged as being created by a dislocation moving across the plane, but with b not equal to a lattice translation vector. Thus, stacking faults are bounded by imperfect, or partial dislocations. The atoms in the close-packed metals can be approximated as hard spheres and the structure can be regarded as closepacked planes of atoms stacked one above the other in a particular sequence. In the f.c.c. case, the planes are of the {111} type and the sequence is 3-fold, i.e. if one layer is labelled A, it and the others follow the sequence ABCABCA... A projection of this sequence is shown in Fig. 7-41 a, where only the atoms in A sites are shown as spheres. The perpendicular vector from one A plane to another is <111>. Within
7.6 Dislocations in Particular Crystal Structures
457
(a)
(b)
Figure 7-41. (a) Projected stacking positions of atoms in the {111} planes of the f.c.c. metals, (b) Important vectors in the (111) plane. (After Hull and Bacon, 1984)
any one layer, the vector from one site to a nearest neighbor is of the type 1/2 <110> and that from one site to the projected position of a nearest neighbor from an adjacent layer is 1/6<112>, as shown for the particular example of the (111) plane in Fig. 7-41 b. If the crystal energy is computed, and then a B layer, say, and all the layers above are rigidly translated relative to the A layer and all those below, the en-
ergy will increase, except for translations of the type bx = 1/2 <110> in Fig. 7-41 a which restore perfect stacking. However, it is clear from the symmetry and hard-sphere arrangement that a translation of the type b2 or 6 3 ( = 1/6<112» leads to a local minimum in the energy. This is demonstrated by the computed variation of the change in energy with fault vector for aluminium in Fig. 7-42. Thus, a stable stack-
Figure 7-42. The y surface computed for aluminium, showing the variation of the energy with fault vector on the (111) plane, y is zero, i.e. the crystal is perfect, for translations of the form 1/2<110>, and a metastable fault is seen for translations of the form 1/6<112> from the perfect stacking pattern. The scale in the x direction is 1/2 <112> and in the y direction 1/2 <110>. (From Vitek, 1968; reprinted with permission of Taylor and Francis, Ltd.)
458
7 Dislocations in Crystals
ing fault exists with the stacking sequence ...ABCACABC... The additional energy per unit area of fault is known as the stacking fault energy, y. Typical values of y in mJm~ 2 for pure metals are in the range 16 (Ag) - 32 (Au) - 4 5 (Cu) - 166 (Al) (Hirth and Lothe, 1982). As may be surmised by its slip-like nature, this intrinsic fault can be created by the glide of a dislocation with b = 1/6<112>, known as a Shockley partial dislocation. This glissile dislocation therefore moves on a {111} plane, and can be edge, mixed or screw in character, but, even when pure screw, cannot cross-slip (Sec. 7.2.3) because the 1/6<112> Burgers vector lies in only one {111} glide plane. The connection between these partial dislocations and perfect dislocations is clear from Fig. 7-41 a. As the glide plane of the perfect dislocation with Burgers vector b1 is traversed, there is a transition in the dislocation core between the region where the B atoms are displaced relative to the A layer atoms by bt and the region where no displacement has occurred. Unless the core width is narrow (see Sec. 7.1.4), B atoms within the core are translated only partway along b1, and therefore relax towards the C sites to achieve greater stability on the y energy surface (Fig. 7-42). Thus, the core of the perfect dislocation dissociates, or extends, into two dislocations according to the reaction * i = * 2 + *3,
(7-77)
i.e., 1/2 [T10] = 1/6 [211] + 1/6 [T2T] and consists of a ribbon of intrinsic fault bounded by two Shockley partials. This is energetically feasible according to Frank's rule (Sec. 7.3.2) since b\ is greater than {b\ + b\\ i.e. a2/2 c.f. a 2 /3, where a is the lattice parameter. However, the partials cannot separate without limit since the fault has an energy y per unit area which partly offsets the reduction in dislocation
elastic energy. Equilibrium is achieved when the total energy is minimized, i.e. the repulsive force between the partials per unit length equals y. Elasticity theory (Sec. 7.3.4) gives a good approximation to the force (Duesbery and Vitek, 1985; Gao and Bacon, 1992), and the equilibrium spacing, d, between the centre of the cores of the Shockley partials is found to vary from D (2 — 3 v)/y for the dissociated screw dislocation to D(2 + v)/y for the edge, where D = iia2/[24n(l-v)]. Taking the values for \i and v for the <112> {111} system in Cu from Table 7-1 and the value 45mJm~ 2 for y quoted above, we find that d varies between 2.2 and 7.0 nm for these cases. A projection of atomic positions found by computer simulation for the 60° mixed dislocation in Cu is shown in Fig. 7-43 a. The separation d is about 12 a for this orientation. The partial spacings are about twice as big in Ag and about one quarter as big in Al, mainly as a result of the small and large values of y9 respectively. Thus, the core of a l/2<110> dislocation in the f.c.c. metals is a planar configuration extended on the {111} glide plane. As it glides, the leading Shockley partial shifts atoms adjacent to the glide plane by 1/6 <112> into the stable intrinsic-fault position, and the trailing partial adds another shift to complete the l/2<110> translation and leave perfect stacking. The intrinsic fault results in a low core energy and Peierls stress (Sec. 7.2.4), and the CRSS for glide is low. This explains the predominance of {111} slip in these metals. The planar, dissociated core structure has further implications for plastic flow in the f.c.c. metals. Cross slip (see Sec. 7.2.3) is difficult to achieve because a 1/6<112> vector lies in only one {111} plane and so an individual Shockley partial cannot cross slip. An extended dislocation is therefore constrained to glide in the {111}
7.6 Dislocations in Particular Crystal Structures
459
(a)
[HI] 'W\A/U\/\i\/U\/\/\/\j\/\|\j\J\j\f
1 '•'
= 11.9a
(b)
[111] 1
' - A .
—m [11?]
.
a
V
A
• *
- *
- / /
Cu : dt = 8.2a , d= 2.4a Figure 7-43. Projection of the positions of atoms in two adjacent (110) planes for (a) the dissociated perfect 60° dislocation and (b) the Lomer-Cottrell dislocation in copper. The dislocations are perpendicular to the paper. In (a), the Burgers vectors of the left- and right-hand partials are at 30° and 90° to the dislocation line, respectively. In (b), the two pure-edge Shockley partials lie on (111) and (111) with b = 1/6[112] and 1/6[112], respectively. The stair-rod has b = 1/6 [110]. The arrows denote the difference in displacement of the pairs of atoms at the end of each arrow: constant differences occur across a stacking fault. [See Sec. 7.1.4 and Vitek (1974), Duesbery (1989).] They denote the pure-edge [112] component in (a) and are the superposition of the [112] and [112] components in (b). (From Gao and Bacon, 1992; reprinted with permission of Taylor and Francis, Ltd.)
460
7 Dislocations in Crystals
plane of its fault. Although a dissociated screw dislocation cannot cross slip it is possible for a short segment to constrict and form an unstable, unextended screw core, and this can then dissociate and glide on the other {111} plane that contains its l/2<110> slip vector. The sequence of events envisaged during the cross-slip process is illustrated in Fig. 7-44. In this example, a b = 1/2 [lTO] dislocation, extended in the (111) slip plane, has constricted along a short length parallel to the [lTO] direction, and so the constricted dislocation has a pure screw orientation. This segment has then dissociated on the (111) plane and eventually the whole dislocation has crossslipped onto this plane. Since the constriction is unstable, its formation is likely to be assisted by stress, such as when the dislocation is held up at an obstacle, and high temperature, since thermal activation increases the probability of forming a constricted core. The probability of constriction is also higher when d is small i.e. y is high. Thus, the ease with which dislocations in the f.c.c. metals can overcome obstacles by cross slip is a sensitive function of stress, temperature and y. Another consequence of the dissociation of these dislocations is their ability to interact and form barriers to further dislocation movement during plastic deformation. This leads to progressive strain-hardening. Several possible reaction products exist,
but the most important are the homer and Lomer-Cottrell dislocations. Consider first the interaction of perfect dislocations gliding on two intersecting {111} planes. There are three possible l/2<110> Burgers vectors in each (Fig. 7-41) so that, including reversals, there are 36 interactions to consider. Of these, two have antiparallel Burgers vectors and, according to Frank's rule (Sec. 7.3.2), are very attractive, and two have parallel Burgers vectors and are very repulsive. Of the remainder, four are particularly interesting because they are attractive, yet result in a product dislocation that cannot glide on either of the two planes in question (Lomer, 1951). For example, a dislocation on (111) with A = l/2[IlO] can interact with another on (Til) with b = 1/2[101] (see Fig. 7-45a) to form a product dislocation with b = 1/2[Oil]. The elastic energy is reduced by 50%, but this Lomer dislocation cannot glide on either of the (111) and (111) planes. Furthermore, it can decompose (Cottrell, 1952) into two ribbons of intrinsic fault on the two planes by the following reaction: 1/2 [011] = 1/6 [211] + 1/6 [011] + 1/6 [211] (7-78) This is favorable according to Frank's rule. The resulting arrangement is depicted in Fig. 7-45 c, where it is shown that the dislocation with b = 1/6 [011] resides at the junction of the two faults. This is (descriptively)
111) plane
(111) plane
(c)
Figure 7-44. Four stages (a)-(d) in the cross-slip of a dissociated screw dislocation by the formation of a constriction. A constriction has occurred at stage (b) and cross slip has started at (c).
7.6 Dislocations in Particular Crystal Structures (111)
(T11)
Stacking fault 1/6 1112]
Lomer - Cottrell sessile dislocation
461
two Shockley partials according to Frank's rule, and these three partial dislocations form a stable, sessile arrangement. It acts as a strong barrier to the glide of further dislocations on the two {111} planes, and is known as a Lomer-Cottrell lock. Although the two ribbons of stacking fault in Fig. 7-45 c are depicted as having the same width, elasticity theory and computer simulations show that the lock is actually asymmetric, with the two widths having a ratio of about 3.8 (Korner et al., 1979; Bonneville and Douin, 1990; Gao and Bacon, 1992). The computed structure for Cu is plotted in Fig. 7-43 b. It should also be noted that the change in energy between the Lomer and Lomer-Cottrell configurations may not always be as negative as implied by Frank's rule (Saada and Douin, 1991). 7.6.2.2 Other Dislocations
1/6 [211]
Figure 7-45. Formation of a Lomer-Cottrell sessile dislocation. (From Hull and Bacon, 1984; reprinted with permission of Pergamon Press, Ltd.)
called a stair-rod partial. Figure 7-45 b also demonstrates that the same structure can be formed even if the two perfect dislocations in (a) dissociate before interacting, because the two leading Shockley partials have a very favorable reaction:
The Burgers vector of the stair-rod partial is perpendicular to the dislocation line and does not lie in either of the two {111} planes of the adjacent faults. Thus, it cannot glide on these planes, and it cannot glide on its own (100) slip plane without producing a high-energy fault. It is said to be sessile. It exerts a repulsive force on the
The intrinsic fault of the preceding section is produced by glide on a {111} plane of a Shockley partial dislocation. An identical fault can also be created if an atomic plane is missing from the stacking sequence ... ABC ABC... of {111} planes. For example, the sequence ... ABCACABC... is obtained by removal of a B layer from the perfect arrangement and displacement of the surrounding crystal into the space created. If this fault exists over only part of a {111} plane, it must be surrounded by perfect crystal in which the layer of atoms in B sites is present. This layer is equivalent to the extra half-plane of the edge dislocation (Fig. 7-1), and so the fault is bounded by an edge dislocation with A = l/3<111> perpendicular to the fault plane. This is a Frank partial dislocation, and is sessile because its Burgers vector is such that it would produce an unstable stacking fault of high energy if it were to glide.
462
7 Dislocations in Crystals
A different fault is created if an extra {111} plane of atoms is inserted in the perfect sequence. For example, if the layer is inserted between A and B layers, its atoms must occupy C positions, giving the stacking sequence ... ABCACBCAB..., which is distinct from the intrinsic fault. The energy of this extrinsic fault arising from the addition of extra material is similar to that of the intrinsic fault, because in both cases two layers are only two planes apart rather than three. Where the extrinsic fault terminates in perfect crystal, the inserted plane is equivalent to the extra half-plane of an edge dislocation with b = 1/3 <111>, which is again a Frank partial dislocation. Although the Frank partial dislocation is sessile, it can climb by the absorption/emission of point defects. A closed loop of a Frank partial dislocation enclosing a stacking fault can be pro-
duced by the collapse of a platelet of vacancies: this may arise from the local supersaturation of vacancies produced by rapid quenching or in the displacement cascades formed by irradiation with energetic particles. By convention this is called a negative Frank dislocation. A positive Frank dislocation may be formed by the precipitation of a close-packed platelet of interstitial atoms as produced by irradiation damage. The stacking fault can be removed in both cases by a dislocation reaction as follows. Consider the vacancy loop on (111) shown schematically in Fig. 7-46 a, formed by vacancies 'removing' an area of plane C: it has b = 1/3 [111]. The fault will be removed if the atoms above the fault are displaced relative to those below so that A->C, B^A, etc. This 1/6<112> displacement is achieved by the movement of a Shockley partial dislocation across the fault. The
[111] [111]- PLANES
tin]
"H Figure 7-46. (a, c) Atomic structure through vacancy and interstitial Frank loops on the (111) plane of a face-centred cubic metal, and (b, d) the perfect loops formed by unfaulting reactions. (From Ullmaier and Schilling, 1980)
7.6 Dislocations in Particular Crystal Structures
Shockley partial may have one of the three 1/6 <112> vectors lying in the fault plane and will react with the Frank partial dislocation to produce a perfect dislocation according to a reaction of the type
The resulting perfect loop is illustrated in Fig. 7-46 b. For the interstitial loop, two Shockley partials are required to remove the extrinsic fault. With reference to Fig. 7-46 c one partial glides below the inserted layer transforming B->A, A->C, etc., leaving an intrinsic fault and the other sweeps above the layer with the same result as in the vacancy case. A possible reaction in the (111) plane is 1/6 [121] + 1/6 [211] + 1/3 [111] = 1/2 [110] (7-81) The perfect interstitial loop is shown in Fig. 7-46 d. The prismatic loops of perfect dislocation thus formed can slip on their cylindrical glide surfaces to adopt new positions and orientations. Reactions in Eqs. (7-80) and (7-81) will occur only when the stacking fault energy is sufficiently high, for unfaulting depends on whether the thermodynamic conditions result in the nucleation of a Shockley partial dislocation and its spread across the stacking fault. A necessary condition is that the energy of the Frank loop with its associated stacking fault is greater than the energy of the perfect dislocation loop, i.e., there is a reduction in energy when the stacking fault is removed. Simple calculations using elasticity theory for the energy of a circular loop of radius JR and core radius r0 show that the unfaulting reaction will be energetically favorable if (Hull and Bacon, 1984) y • 24TTK\1-V
(7-82)
463
Taking a = 0.35 nm, R = 10 nm, \i = 40 MPa, ro = 0.5nm and v = 0.33, the critical stacking fault energy is about 60mJm~ 2 . The value ,R = 10nm is close to the minimum size for resolving loops in the electron microscope, and so faulted loops are rarely observed in metals with y values larger than this. Condition (7-82) based on initial and final energy values may not be a sufficient condition for unfaulting, for it neglects the fact that there is an energy barrier for nucleation of the Shockley partials. As a loop grows by the absorption of point defects, it becomes increasingly less stable, but if the Shockley nucleation energy is independent of R, the equilibrium unfaulted state may not be achieved. The probability of nucleation is increased by increasing temperature and by the presence of external and internal sources of stress, but there is no simple rule governing loop unfaulting. Another dislocation arrangement that has been observed in quenched and irradiated metals consists of a tetrahedron of intrinsic stacking faults on the four {111} planes with 1/6 (110> type stair-rod dislocations along the edges of the tetrahedron. According to the preceding discussion, when a platelet of vacancies collapses to form a loop of Frank partial dislocation, it will be stable if the fault energy is sufficiently low. The Frank partial loop on (111), say, may dissociate into a low-energy stair-rod dislocation and a Shockley partial on the intersecting (111) plane according to the reaction 1/3 [111] = 1/6 [101] + 1/6 [121]
(7-83)
and similarly on the other two planes (111) and (111). Discounting the energy of the stacking fault, the reaction is energetically favorable. It is found that the partials attract each other in pairs on the three other {111} planes and form another set of stair
464
7 Dislocations in Crystals
rods along their three intersecting lines, as described in the preceding section (Eq. (7-79)). Thus, a complete stacking-fault tetrahedron can be created with {111} faces and <110> edges. Tetrahedra are mainly found in metals of low stacking fault energy such as Ag and Au. The increase in energy due to the formation of stacking faults places a limit on the size of the tetrahedron that can be formed. If the fault energy is relatively high, the Frank loop is stable, or it may only partly dissociate, thereby forming a truncated tetrahedron. Stacking-fault tetrahedra based on interstitial loops have not been identified (Jenkins etal., 1987). A way in which the relationship between the geometry of a regular tetrahedron and the Burgers vectors and stacking faults can be exploited to simplify the analysis of dislocation reactions in the f.c.c. structure was proposed by Thompson (1953). The Thompson tetrahedron notation is now widely used (Nabarro, 1967; Hirth and Lothe, 1982; Hull and Bacon, 1984). 7.6.3 The h.c.p. Metals The atoms in the (0001) basal plane (see Fig. 7-47) have the same hard-sphere arrangement as the {111} planes of the f.c.c. metals, and these planes are stacked in the 2-fold sequence ... AB AB... consistent with the hexagonal variant of close-packing, but none of the metals have the ideal lattice parameter ratio c/a = (8/3)1/2 = 1.633 for close-packing of hard spheres. Magnesium and cobalt are close to this, but most have c/a smaller - e.g. titanium: 1.587; beryllium: 1.568 - or considerably larger e.g. zinc: 1.856; cadmium: 1.886. This results from the directional nature of the bonding, a factor that also affects dislocation properties. The shortest lattice vectors are 1/3 <1120>, and it may be anticipated,
(0001)
{10T0>
{1011}
{1122}
Figure 7-47. The hexagonal unit cell of the h.c.p. structure, showing some important planes and directions.
therefore, that dislocation glide occurs in the basal plane with this Burgers vector. This slip system is frequently observed and is the favored one in some h.c.p. metals, e.g. Be, Mg, Co, Zn and Cd. However, other metals, such as Ti and Zr, slip most easily with b = 1/3 <1120> on the first-order prism planes {10T0}. In polycrystalline metals, basal and prism slip do not supply sufficient slip modes to satisfy von Mises' criterion that every grain should be able to plastically deform to accommodate the shape changes imposed by its neighbors. This requires five independent slip systems, whereas basal and prism slip provide only two each. Consequently, twinning and occasionally other slip systems play an important role in the plasticity of these metals (Yoo, 1981).
7.6 Dislocations in Particular Crystal Structures
The complexity of the slip behavior suggests that the core structure of the perfect dislocations with A = l/3<1120> varies in form from ones that are extended on the basal plane to others where it is widest on the prism planes. The former can be explained on the same basis as the dissociated cores in the f.c.c. metals (Sec. 7.6.2). According to the hard-sphere model, three basal-plane faults exist which do not affect nearest-neighbor arrangements of the perfect stacking sequence. Two are intrinsic and conventionally called Ix and I 2 . Fault Ix is formed by removal of a basal layer, which produces a very high energy fault, followed by slip of 1/3 <1010> of the crystal above this fault to reduce the energy. The stacking sequence in these stages is: ABAB ABABA... -* AB ABB ABA... -> ABABCBCB...(I1) (7-84) Fault I 2 results from slip of 1/3 <1010> in a perfect crystal: ABABABAB... -+ ABABCACA... (I2) (7-85) and is illustrated by the y energy surface calculated for Mg in Fig. 7-48 a. In this plot the fault vector is 1/6 [1120] + 1/6 [1100]
800 n
465
= 1/3 [1010] with respect to the origin. The extrinsic fault (E) is produced by inserting an extra plane: ABABABAB... -> ABABCABAB... (E) (7-86) These faults introduce into the crystal a thin layer of f.c.c. metal stacking (ABC) and the main contribution to the stacking-fault energy y arises from changes in the secondneighbor sequences of the planes. Consequently, yE& 3/2 y l 2 ^ 3yu. There are few firm experimental estimates of y for the basal plane, but the tight-binding calculations of Legrand (1984) suggest values in the range 15-900mJm~ 2 . There is no direct experimental evidence of stable faults on the prism planes, but tight-binding estimates and the results of computer simulations using pair and many-body potentials (Bacon and Liang, 1986; Vitek and Igarashi, 1991) indicate that a metastable fault may exist in some metals with a fault vector l/6<112x>, where x~0.1 to 0.9. This is demonstrated by the calculated y surface of Mg in Fig. 7-48 b. Thus, slip by the system 1/3 <1120> (0001) in those metals in which it is favored is similar to 1/2
800
Figure 7-48. The calculated shape of the y surface for (a) the basal plane and (b) the first-order prism plane in magnesium. (From Vitek and Igarashi, 1991; reprinted with permission of Taylor and Francis, Ltd.)
466
7 Dislocations in Crystals
face-centred cubic metals, in that the critical resolved shear stress is low ( < 1 MPa) and the perfect dislocation dissociates into two Shockley partials bounding a ribbon of I 2 stacking fault. The Burgers vector reaction is of the form (7-87) The geometry is the same as that of the f.c.c. metals, in that the partial vectors lie at ±30° to the perfect vector and the fractional reduction in dislocation energy given by b2 is 1/3. The basal-slip metals can also be induced to slip on the 1/3 <1120> {1100} system, but the CRSS is one to two orders of magnitude higher. This can be explained by a combination of the need for constriction (see Sec. 7.6.2.1) to occur if a dislocation already dissociated on (0001) is to cross slip onto {10T0}, and the absence of a low-energy stacking fault on the prism planes. More precisely, it is the ratio of the energy of the metastable prism-plane fault to that of the intrinsic I 2 fault on the basal plane that matters. When it is large, the dislocations with 6 = l/3<1120> have a wide planar core in the (0001) plane, and the CRSS for prism slip is high because of the need for constriction to occur repeatedly to prevent the screw dislocation from dissociating on the basal plane. Magnesium is an example where this occurs. When the ratio is close to unity, or less, dissociation on the prism planes according to the reactions of the type = l/6<112x> + l/6<112x>
(7-88)
becomes more favorable, where x ~ 0.7-0.9 (see above). Computer simulations (Vitek and Igarashi, 1991) indicate that even when the core dissociates on the {lOTO}
planes, the screw may still spread to a limited extend on (0001), leading to the high CRSS (>10MPa) observed in the prismslip metals. This structure is illustrated by the core structure of the <1120> screw in titanium found by computer simulation in Fig. 7-49. More subtle points concerning the roles of the core structure and crossslip mechanisms in <1120> glide in the h.c.p. metals are to be found in Naka et al. (1991) and Couret et al. (1991). Slip with Burgers vector 1/3 <1123> (see Fig. 7-47) has been widely reported, although only under conditions of high stress and orientations in which more favored slip vectors cannot operate. The glide planes are {1011} and {1122}. The magnitude of the Burgers vector is large and the core structure is complex. Computer simulations (e.g., Minonishi et al.,
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[11001 Figure 7-49. The core structure of the 1/3 [1120] screw dislocation extended on the prism plane in titanium. The dislocation has a direction perpendicular to the paper and is extended on the prism plane (T100). The arrows denote the screw component (which is normal to the paper) of the displacement difference of pairs of atoms marked by the connecting arrows. (From Vitek and Igarashi, 1991; reprinted with permission of Taylor and Francis, Ltd.)
7.6 Dislocations in Particular Crystal Structures
1982a,b; Liang and Bacon, 1986) suggest that this dislocation can extend on several planes, and that the movement under stress may initiate microtwins. Thus, there may be a close connection between <1123> dislocations and the important process of deformation twinning. Dislocations are also responsible for the propagation of twins once they have been nucleated. Twinning dislocations are steps on the matrix-twin interface which move at the CRSS for twinning. They are not perfect dislocations in the sense defined above, but, depending on the step height and twin mode, can have very wide cores and can glide at a low applied stress (Serra et al., 1991). As in the f.c.c. metals (Sec. 7.6.2.2), vacancies and interstitials in excess of the equilibrium concentration can precipitate as platelets to form dislocation loops. The situation is more complicated, however, because although the basal plane is closepacked, the relative density of atoms in the different crystallographic planes varies with c/a ratio. Also, the stability of stacking faults varies, as discussed above. The simplest geometries are considered here, starting with basal-plane loops. Condensation of vacancies in a single basal plane, say an A layer, results in B over B stacking, and this unstable situation of high energy is avoided in one of two ways. In one, the stacking of one layer adjacent to the fault is changed; for example B to C as in Fig. 7-50 a. This is equivalent to the glide below the layer of one Shockley partial with Burgers vector 1/3 <1010> followed by glide above of a second Shockley of opposite sign. The Burgers vector reaction is therefore 1/2 [0001] + 1/3 [1100] + 1/3 [1100] = = 1/2 [0001] (7-89) for example. The resultant sessile Frank partial surrounds the extrinsic (E) stacking
467
fault described above. In the alternative mechanism, a single Shockley partial sweeps over the vacancy platelet, displacing the atoms above by 1/3 <1010> relative to those below. The Burgers vector reaction is, for example, 1/3 [T100] + 1/2 [0001] = 1/6 [2203]
(7-90)
This sessile Frank partial surrounds the type Ix intrinsic fault (Fig. 7-50 b). The Etype loop of reaction (7-89) can transform to the Ix form by reaction (7-90), and since yE is expected to be approximately three times yh, the 1/6 <2023> loops may be expected to dominate. The dislocation energy is proportional to fc2, however, and the total energy change accompanying reaction (7-90) is dependent on the y values and loop size, in a similar manner to the unfaulting of loops in f.c.c. metals. There is therefore a critical loop size for the reaction, which may be influenced by factors such as stress, temperature and impurity content. Experimentally, both forms are observed in quenched or irradiated metals (Eyre et al., 1977; Griffiths, 1991). Precipitation of a basal layer of interstitials as in Fig. 7-50 c produces an E-type fault surround by a Frank partial loop with Burgers vector 1/2 [0001]. Again, provided the loop is large enough, this can transform to an I x loop by the nucleation and sweep of a Shockley partial according to reaction (7-90). The stacking sequence is shown in Fig. 7-50 d. Interstitial loops with Burgers vectors 1/2 [0001] and l/6<2023> have been seen in irradiated magnesium, cadmium and zinc (Eyre et al., 1977; Griffiths, 1991). In the latter two metals, perfect loops with Burgers vector [0001] are also observed. They result from a double layer of interstitials. The atom density of the basal planes is only greater than that of the corrugated {10T0} prism planes when c/a>^/3, sug-
468
7 Dislocations in Crystals High energy fault
Low energy fault
(d)
ABAB-
-B -A -B
Figure 7-50. Illustration of the basal-plane stacking sequence after the two possible partial-unfaulting reactions for (a, b) vacancy loops and (c, d) interstitial loops on the basal planes of the h.c.p. metals. (After Berghezan et al., 1961)
gesting that the existence of basal stacking faults aids the stability of the basal vacancy and interstitial loops in magnesium. These faults have relatively higher energy in metals such as titanium and zirconium, however, and large basal-plane loops are not expected. It is surprising, therefore, that vacancy basal loops are produced by ligh doses of radiation damage in zirconium, for reasons that are not understood Griffiths, 1991). Nevertheless, most interstitial and vacancy loops in irradiated titanium and zirconium do grow on the prism planes and have the perfect Burgers vector 1/3 <1120> [see, for example, Griffiths (1991), Phythian et al. (1991)]. They lie on planes of the [0001] zone at angles up to 30° from the pure-edge {1120} orientation. The {1120} planes are neither widelyspaced nor densely-packed, and it is believed that the point defects precipitate initially as single-layer loops on the {10T0} prism planes. The resulting stacking fault has a high energy and is removed by shear when the loops are small by a Burgers vec-
tor reaction of the type 1/2 <10T0> + 1/6
(7-91)
The glissile loops thus produced can adopt the variety of orientations observed in practice. Many papers concerned with dislocations and other defects in h.c.p. metals are to be found in Bacon (1991). 7.6.4 The b.c.c. Metals
The body-centred cubic metals (such as iron, molybdenum, tantalum, vanadium, chromium, tungsten) slip in close-packed <111 > directions with Burgers vector of the type 1/2 <111>. The crystallographic slip planes are {110}, {112} and {123}. Three {110}, three {112} and six {123} planes intersect along the same <111 > direction, and it is possible for screw dislocations to move on combinations of planes favored by the applied stress. Thus, slip lines are often wavy and ill-defined, and the apparent
7.6 Dislocations in Particular Crystal Structures
slip plane varies with composition, crystal orientation, temperature and strain rate (Christian, 1983). It is found that the slip plane of a single crystal deformed in uniaxial compression may be different from that which operates in tension for the same crystal orientation. In other words, there is an asymmetry of slip in which the CRSS in a slip plane in one direction is not the same as that in the opposite direction in the same plane. Slip is easier when the applied stress is such that a dislocation would move in the twinning sense on {112} planes rather than the anti-twinning sense, even when the actual slip plane is not {112}. Electron microscopy of metals deformed at low temperatures reveals long screw dislocations, implying that non-screw dislocation are more mobile and that, as in the h.c.p. case, screw dislocations dictate the slip characteristics. Stacking faults have not been observed experimentally in the b.c.c. metals, and the ease of cross slip suggests that faults are at best of very high energy. This is confirmed by computer calculations of the y surface (Vitek, 1974; Harder and Bacon, 1986), which reveal no metastable faults. The computed y surfaces for the {110} and {112} planes in molybdenum are shown by way of example in Fig. 7-51. Consequently, simple dissociation models of dislocations are inadequate, and much of our knowledge about core structure and behavior has been obtained from computer simulation. This shows that the screw dislocation with Burgers vector 1/2 <111> has a core with non-planar character. An example of two equivalent atomic configurations for a 1/2[111] screw are shown in Fig. 7-52a and b. The atom positions are projected on the (111) plane and the orientation of the traces of the {110} and {112} planes are shown in Fig. 7-52 c. The only atomic displacements which are not negligible are
469
(a)
[1TO]
Figure 7-51. The calculated shape of the y surface for (a) the (110) plane and (b) the (112) plane of molybdenum. The energy units are eV A~2 (=1.6Jm~ 2 ). (From Harder and Bacon, 1986; reprinted with permission of Taylor and Francis, Ltd.)
parallel to the dislocation line [111]. For the isotropic elastic solution Eq. (7-20), the arrows would exhibit complete radial symmetry. In the atomic model, the displacements are concentrated on the three intersecting {110} planes, each of which contains an unstable fault produced by a 1/6[111] displacement. Although the 1/2[111] dislocation spreads into three 1/6 [111] cores, these fractional dislocations do not bound stable stacking faults, unlike Shockley partial dislocations. Close exami-
470
7 Dislocations in Crystals (6)
•
12]
nation of the displacements (Vitek, 1974) reveals that the fractional cores also spread asymmetrically, but on three {112} planes in the twinning sense. This accounts for the slip asymmetry referred to above. By applying stress to the model crystals, it has been found that the core structure changes before slip occurs (e.g., Vitek and Yamaguchi, 1981). For example, under an increasing shear stress on (T01) tending to move the screw dislocation in Fig. 7-52 a to the left, the fractional dislocation on (T01) extends the core to the left and the two others constrict towards the core centre. As the stress is increased, the fractional dislocation on (Oil) disappears to be replaced by another on (110) before glide of the whole core occurs on (T01). Movement of
•-—* - •
Figure 7-52. (a) Atomic positions on a (111) plane and displacement differences (shown by arrows) for a screw dislocation with Burgers vector 1/2[111] in a b.c.c. metal. The displacement differences are in the direction normal to the paper, (b) Alternative to (a), (c) Orientation of the {110} and {112} planes of the [111] zone. (After Vitek, 1974)
the core to the right under reverse stress takes a different form. Although the detailed core changes are dependent on the interatomic potentials, it is found that under pure shear stress slip occurs on the {110} planes with the asymmetry described above. Furthermore, computer simulation has shown that the screw core responds differently to stresses with different nonshear components, in good agreement with the effects of compression and tension found in experiment (Duesbery, 1984 and 1989). Similar studies of non-screw dislocations, on the other hand, show they have cores which are planar in form on either {110} or {112} but do not contain stable stacking faults. Like their f.c.c. and h.c.p.
7.6 Dislocations in Particular Crystal Structures
counterparts, they are not sensitive to the application of non-shear stresses, and they glide at much lower RSS values than the screw dislocation. Another set of perfect dislocations in the b.c.c. metals are those with Burgers vector <001>. They are occasionally observed in dislocation networks and are believed to occur from the reaction of two perfect 1/2 < 111 > dislocations, e.g. 1/2[111]+ 1/2[111]-[100]
(7-92)
Computer modelling of the edge dislocation with this large Burgers vector has shown that large tensile forces in the core region just below the extra half-plane lead to severing of atomic bonds and the formation of a microcrack there. This dislocation is therefore unlikely to take part in plastic deformation, but since the main cleavage planes are {100}, it may play a role in crack nucleation. Dislocation loops formed by condensation of interstitials and vacancies are believed to nucleate on the most denselypacked {110} planes with Burgers vector l/2<110>. These partial dislocations are associated with an unstable, high-energy fault, and so the planes above and below the fault shear at an early stage of growth to become perfect by one of two reactions (Eyre and Bullough, 1965). 1/2
(7-93)
471
7.6.5 Ionic Crystals
These are an important class of solid because they are the foundation for many of the ceramic materials that are finding increasing use in engineering. The distinctive feature of dislocations in ionic crystals, as compared to metals, is that electric charge effects are associated with them. This is inherent in the ionic nature of the bonding, and the influence it has on plasticity in ceramics is discussed in detail in Volume 11, Chapter 7, Section 7.4 of this Series. Here, therefore, we simply introduce some essential characteristics of dislocations. We consider, by way of example, NaCl, which has the rocksalt structure illustrated in Fig. 7-53 a. The Bravais lattice is f.c.c. and the crystal has an anion-cation pair at each lattice site, one at 0,0,0 and the other at 1/2,0,0. Each anion has six cation nearestneighbors, and vice versa. The Burgers vector of the dislocations responsible for slip is 1/2 <110>, the shortest lattice vector. The principal slip planes are {110}, although slip occurs (occasionally) on the {111} and {112} planes at high stresses and/or high temperatures. Cross slip of screw dislocations can occur only by glide on planes other than {110}, for only one <110> direction lies in a particular {110} plane. A schematic illustration of a pure edge dislocation with a 1/2 [110] Burgers vector and (110) slip plane emerging on the (001)
(7-94)
The resultant dislocation in reaction (7-93) has the lower energy (i.e., b2) and loops with this Burgers vector have been observed in many metals. In oc-iron and its alloys, however, a high proportion of loops are of <100> type: this somewhat surprising effect is as yet unexplained, and may contribute to the void-swelling resistance of ferritic steels (Little et al., 1980).
• Sodium ion o Chlorine ion
Figure 7-53. (a) The NaCl structure and (b) an AB3 alloy with ordered Ll 2 structure.
472
7 Dislocations in Crystals [1101
( nib]
0
© TO © 0 0© (a)
surface is shown in Fig. 7-54. The {110} planes have a 2-fold stacking sequence, and the extra half-plane actually consists of two supplementary half-planes. The ions in the planes below the surface alternate between those shown in Figs. 7-54 a and b. The figure serves to illustrate that there is an effective charge associated with the point of emergence of the dislocation on the (001) surface, and if it is — q in (a) it must be + q in (b). It is readily shown that q = e/4, where e is the electronic charge, as follows. In any cube of perfect crystal in which the corner ions are of the same type, as in Fig. 7-53 a, the ions of that sign exceed in number those of opposite sign by one. This excess charge of + e may be considered as an effective charge ± e/S associated with each of the eight corners. In a cube with equal numbers of anions and cations, the positive and negative corners neutralize each other. Consider the dislocation of Fig. 7-54 a to be in a block of crystal ABCD bounded by {100} faces. The effective charge of the four corners A,B,C,D is (-\-e/S-3 e/S) = -e/4, and so the net effective charge at the (001) surface is ( — e/4 — q). If a single layer of ions is removed to expose the new face A ' B ' C D ' (Fig. 7-54 b), the net effective charge changes to (— e/S + 3 e/S + q) by removal of the sixteen anions and fifteen cations, i.e. —e. Since the initial charge
Figure 7-54. Edge dislocation in the sodium chloride structure, with the Na + cations represented by + and the Cl~ anions by —. (a) Configuration of ions on the (001) surface, (b) Configuration after removal of one surface layer. (From Amelinckx, 1979)
must equal the final charge plus the charge removed. e e - - - q = - + q-e (7-95) and so q = e/4. The same result holds for emergence on {110} planes. When an edge dislocation glides on its (lTO) slip plane, there is no atomic displacement parallel to the line, and the effective charge of the emergent point does not change sign. For the same reason, kinks in edge dislocations bear no effective charge. If the dislocation climbs by, say, removal of the anion at the bottom of the extra halfplane in Fig. 7-54 a, the configuration changes to the mirror image of that in Fig. 7-54 b, and so the effective charge at the emergent point changes sign. Consequently, jogs carry effective charge, as demonstrated by the illustration in Fig. 7-55 of two unit jogs of one atom height. The bottom row of ions has an excess charge —e, which is effectively carried by the two jogs. Thus, depending on the sign of the end ion of the incomplete row, a jog has a charge ± e/2, and cannot be neutralized by point defects of integer charge. Charged jogs attract or repel each other electrostatically, and only jogs of an even number times the unit height are neutral. Of course, in divalent crystals such as
7.6 Dislocations in Particular Crystal Structures
MgO, the effective charges are twice those discussed here. Anion and cation vacancies have different formation energy, in general, and this results in a higher probability of a jog being adjacent to a vacant site of the lower energy. Dislocations thus have an effective charge per unit length in thermal equilibrium, although this is neutralized by an excess concentration of vacancies of opposite sign in the remainder of the crystal. The charge effects associated with the 1/2 <110) screw dislocation are more complicated. The ions in any particular <110> row are of the same sign (see Fig. 7-53 a), and since atomic displacements are parallel to the Burgers vector, motion of the screw results in displacement of charge parallel to the line. Consequently, both kinks and jogs on screw dislocations are charged, the effective charge being + e/4 in each case. The effective charge of the point of emergence on {110} and {100} surfaces is ±e/8. The mechanisms that lead to the preference for {110} slip are unclear. It has long been considered that the glide system is determined by the strengths of the electrostatic interactions within the dislocation core. This is partly supported by computer
Figure 7-55. Extra half-planes of the edge dislocation in Fig. 7-54 with jogs denoted by the squares. (From Amelinckx, 1958)
473
simulations of the edge dislocation (Puls and So, 1980), for although stable stacking faults do not exist, the core spreads on the {110} planes to a width of about 6 b, and thus consist of two fractional dislocations of Burgers vector 1/4 <110> bounding an unstable fault. Dissociation on the {100} planes does not occur and is small on {111}. It is also possible, however, that since the ions of the row at the bottom of the extra half-plane of the 1/2 <110> edge dislocation all have the same sign for {100} and {111} slip, but alternate in sign for {110} slip, interaction between edge dislocations and charged impurities may be an important factor in crystals that are not ultrapure. The CRSS shows a strong temperature dependence, as illustrated schematically in Fig. 7-17. In high-purity crystals, this is due to dislocation movement being controlled at low temperatures by double-kink nucleation over the Peierls barrier, as discussed in Sec. 7.2.4. At high temperature, this barrier is not significant and line tension (Sec. 7.3.2) determines the bowing of dislocations. When impurities of different valency from the host ions are present, they can control the CRSS at intermediate temperatures. Two mechanisms exist for this, both based on the charged nature of such impurities (Kalman et al, 1982, 1984). A divalent metal impurity (charge + 2 e) is associated with a vacancy on a Na site (effective charge —e) to provide charge compensation. This charge dipole pair creates tetragonal distortion in the crystal and interacts elastically with dislocations (see Sec. 7.3.6). The dipoles therefore impede dislocation motion, either by being static in the lattice (at low temperatures) or by migrating and exerting a drag on dislocations (at intermediate temperatures). The other interaction is electrostatic, as outlined above, and has different strength for different slip planes.
474
7 Dislocations in Crystals
A detailed discussion of these factors is to be found in Haasen (1985), and a comprehensive description of dislocations structures in ceramics has been given by Mitchell et al. (1985). 7.6.6 Ordered Alloys
Alloys in which the atoms occupy sites in an ordered structure, as opposed to the arrangement in a random solid solution, have been the subject of much attention in recent years. They have properties, particularly with regard to mechanical behavior at high temperature, that make them attractive in many applications. (See Chapter 6, Volume 6 of this Series.) These special properties are now known to arise from the detailed atomic structure in the dislocation core. As in the previous section, it is beyond the scope of this introductory survey to detail the current state of knowledge of numerous materials. Instead, we consider the LI 2 superlattice structure, which is important and illustrates a range of core characteristics. The unit cell of the L l 2 structure for an alloy of composition AB3 is shown in Fig. 7-53 b: it is based on the f.c.c. structure of the solid solution. Examples include Ni3Al, Ni3Si and Cu3Au. At low temperatures, these materials generally behave like the f.c.c. metals in that the CRSS is almost independent of temperature, but as the temperature is raised, the CRSS increases - the yield stress anomaly - and if the order-disorder transition temperature is high enough, the CRSS reaches a peak (at between 800 and 1000 K for Ni3Al). The peak temperature and CRSS are different for tension and compression, and depend on the crystallographic orientation of the load axis. Slip can occur on both the <110> {111} and <110> {100} systems, the former being seen below the peak and the latter, which tends
to be wavy, above (Pope and Ezz, 1984; Stoloff, 1984). To understand the connection between this and dislocation properties, it needs to be recognized that antiphase boundaries (APBs) can exist in the ordered crystal. These fault surfaces separate domains that have the same crystal structure and orientation, but atoms on one side of the APB have neighbors on the other of the wrong type. These conditions are less restrictive than those for the formation of stable stacking faults (Sec. 7.6.2), where atoms have the wrong number of neighbors, and APBs can occur on several crystallographic planes. They frequently arise when domains nucleated during the disorder-order transition grow and meet "out of phase". Further, they can be created by glide of dislocations that have a perfect Burgers vector of the disordered structure, i.e. b = 1/2 <110) in a f.c.c. lattice (Sec. 7.6.2). Since this vector is not a translation vector of the Ll 2 lattice (Fig. 7-53 b), glide of the dislocation leaves a surface of APB in its wake. This is illustrated for the {111} and {100} planes in Figs. 7-56 a and b, respectively, where the atomic arrangement in two adjacent planes is shown. In (a), displacement of one layer by 1/2 <110> shifts X to Y and creates an APB in which A atoms are in nearest-neighbor sites. Order may be restored by a second shift of 1/2 <110>, completing the lattice translation X to Z. For the {100} plane in (b), the shift X to Y leaves nearest-neighbor chemical bonds across the APB unchanged, and the second-neighbor changes are the major contribution to the APB energy. Again, the second shift Y to Z restores order. Thus, a perfect dislocation moving on either the {111} or {100} planes of the Ll 2 structure can consist of two l/2<110> dislocations joined by an APB. This superdislocation of the superlattice has A = <110>. Thus, the
7.6 Dislocations in Particular Crystal Structures
475
1/2 [110]
(a)
9
A in plane of diagram
O B in plane of diagram ® A in adjacent plane © B in adjacent plane
£
A in plane of diagram
O
B in plane of diagram
(5) B in adjacent plane Figure 7-56. Arrangement of atoms in two adjacent atomic layers for (a) (111) planes and (b) (100) planes in an AB3 superlattice.
superdislocation can glide as a pair of l/2<110> superpartial dislocations joined by a ribbon of APB, with a spacing given in equilibrium by the balance between the elastic repulsive force per unit length between the dislocations and the APB energy, which is typically ~ 10 to lOOmJm" 2 .
The simple geometrical picture of APBs above suggests that the APB energy on {100} is lower than that on {111}. Whilst the relative energy and stability of these boundaries will vary from alloy to alloy, this difference is likely to be a general feature. The question arises, then, as to why {111} <110> slip dominates at low to moderate temperatures? Further consideration has to be given to possible core structures of the <110> superdislocation on {111}, for two other metastable faults can exist on these planes, besides the APB. Their translation vectors are denoted by c and s in Fig. 7-56a. Vector c is of the type 1/6<112>, and corresponds to the Shockley partial vector of the disordered f.c.c. metals (Sec. 7.6.2): it creates a complex stacking fault (CSF) in the superlattice. Vector s is of the type 1/3 <112> and creates a superlattice intrinsic stacking fault (SISF). Both faults may be stable, depending on the alloy system. Thus, there are four basic forms for the <110> dislocation on {111} planes. Three are illustrated in Fig. 7-57. One (Fig. 7-57 a) consists of two 1/2 <110> superpartials bounding a stable APB, and the second, Fig. 7-57 b, shows each superpartial dissociated into 1/6 <112> Shockley partials bounding a stable CSF. The third, Fig. 7-57 c, has two 1/3 <112> superpartials sep-
476
7 Dislocations in Crystals
t\
1\
/
\
APB
CSF A P B C S F
SISF
(a)
(b)
(C)
Figure 7-57. Three dislocation modes for the <110> screw superdislocation on a {111} plane in the Ll 2 structure. The arrows denote Burgers vectors.
arated by a ribbon of stable SISF. The forth, not shown, would arise when the three faults are unstable, and consists simply of a single <110> superdislocation. The three dislocations in Fig. 7-57 are expected to have a low Peierls stress, i.e. high mobility, on the {111} planes. However, the actual structure of a <110> superdislocation depends also on the stability and energy of the APB. As in the h.c.p. and b.c.c. metals discussed in Sees. 7.6.3 and 7.6.4, the screw is the most important dislocation, and much insight into its form has been gained from computer simulation. Some possible core configurations of one of the 1/2 <110> screw superpartials in an alloy with a stable low-energy APB on {100} are depicted in Figs. 7-58 a-c, where
(111)
(11D (010)'
\1T1) (010)" (a)
»
\111) (b)
APB on
(111)
(010)x
\(111) (c)
{111}
APB on {100}
Figure 7-58. Block representation of superpartial dislocation cores and their orientation with respect to the APB in the case of low APB energy on {100} planes in the Ll 2 structure. (After Duesbery, 1989)
the superpartial is linked by the APB on (010) to its partner (not shown). In (a) and (b), the CSF is stable and the l/2<110> core has dissociated on either (111) or (111) into two Shockley partials. In (c), the CSF is unstable and the core simply extends mainly on the two {111} planes. In all these cases, the superpartial core is not confined to the (010) plane of the APB and is sessile. This dislocation can only be expected to move at high stresses when the core can constrict to allow glide on (010) or, more likely, when alternate movements on (010)/ (111) or (010)/(lIl) can occur by APB dragging. In conclusion, when the APB (or the SISF) is stable on {111}, the <110> superdislocation is stabilized on these planes by one of the structures of Fig. 7-57. This gives rise to a low CRSS at low to moderate temperatures. As the temperature is increased, thermally-activated constriction and cross-slip of the superpartials onto the {100} planes produces sessile dislocations and increases the CRSS. The restriction of {111} glide by the thermally-activated cross-slip on {100} is the basis of the KearWilsdorf lock proposed by Kear and Wilsdorf (1962), although the detailed core mechanisms have only recently been understood (Hirsch, 1992). Most Ll 2 alloys behave in this way. When there is no stable APB on {111}, the <110> screw is sessile (Fig. 7-58), and can only be induced to move at high stresses, with thermal assistance. The CRSS therefore decreases with increasing temperature in this situation. More comprehensive reviews of these mechanisms are in Duesbery (1989), Umakoshi and Yamaguchi (1990), and Volume 6, Chapter 6 of this Series. Other classes of ordered alloys, such as the cubic B2 structure, which includes CsCl, NiAl and FeAl, and the hexagonal D0 19 structure, which includes Ti3Al and
7.6 Dislocations in Particular Crystal Structures
Mg3Cd, are of importance and exhibit interesting mechanical properties. Again, the core structures of their dislocations play the key roles, but they are not as well understood as those in the Ll 2 alloys, and the reviews cited in the preceding paragraph should be consulted for more detail. 7.6.7 Covalent Crystals
Although the directionality of atomic bonding is a significant factor in some metallic materials, such as the transition metals, it finds greatest importance in the covalent solids. The nature of the covalent bond has a strong influence on dislocation properties, and this has received much attention from the sectors concerned with these materials, e.g. microelectronics and advanced ceramics. As in previous sections, we concentrate here on one structure, which is the most representative, is important and has been studied widely. It is the diamond-cubic structure of diamond, silicon and germanium shown in Fig. 7-59. The space lattice is face-centred cubic with two atoms per lattice site, one at 0,0,0 and the other at 1/4,1/4,1/4. In solids such as GaAs with the sphalerite structure,
477
the two atoms of the basis are of different type. Each atom is tetrahedrally bonded to four nearest-neighbors, and the shortest lattice vector 1/2 <110> links a secondneighbor pair. The closely-packed {111} planes have a six-fold stacking sequence AaBbCcAaBb... as shown in Fig. 7-60. Atoms of adjacent layers of the same letter such as Aa lie directly over each other, and planar stacking faults arising from insertion or removal of such pairs do not change the tetrahedral bonding. By reference to the f.c.c. metals (Sec. 7.6.2), the intrinsic fault has stacking sequence AaBbAaBbCc... and the extrinsic fault has AaBbAaCcAaBb..., and the intrinsic fault can also be formed by a translation of the form 1/6<112> between layers of a different letter, say b and C. These faults restore tetrahedral bonding to each atom and have modest energy (~60mJm~ 2 in Si and Ge). In contrast, faults formed between adjacent layers of the same letter do not restore tetrahedral bonding and are expected to be unstable. Perfect dislocations have Burgers vector l/2<110> and slip on {111} planes. They usually lie along <110> directions at 0° or 60° to the Burgers vector, as a result of
[ml
[011]
(a)
(b)
Figure 7-59. (a) Diamond-cubic unit cell, (b) Illustration of the structure showing the tetrahedral bonds and important crystallographic directions. (From Hull and Bacon, 1984; reprinted with permission of Pergamon Press, Ltd.)
478
7 Dislocations in Crystals
[111] — 'Shuffle - - 'Glide' (011)
Figure 7-60. (Oil) projection of the diamond-cubic lattice showing the stacking sequence of the (111) planes and the shuffle and glide planes defined in the text. (From Hull and Bacon, 1984; reprinted with permission of Pergamon Press, Ltd.)
relatively low core energy in those orientations. From consideration of dislocations formed by the cutting operations of Sec. 7.1.2, two dislocation types may be distinguished. The cut may be made between layers of either different letters, e.g. aB, or the same letter, e.g. bB. Following Hirth and Lothe (1968) (see Hirth and Lothe, 1982) the dislocations produced belong to either the glide set or the shuffle set, as denoted in Fig. 7-60. Diagrammatic illustrations of the two sets are shown in Fig. 7-61. The dangling bonds formed by the free bond per atom along the core are apparent. Dislocations of the glide set can readily dissociate and the core is glissile. The perfect dislocation dissociates to two 1/6 <112> Shockley partials separated by the intrinsic stacking fault, as in the f.c.c. metals (Sec. 7.6.2). This dissociated state has now been observed by TEM in a wide range of elemental and compound semiconductors. Dissociation of the shuffle dislocation is not so simple because of the absence of low-energy shuffle faults. It would occur by the nucleation and glide of a Shockley partial of the glide type between an adjacent pair of {111} layers. This results in a fault of the glide set bounded on one side by a Shockley partial and on the
other by, depending on whether the glide fault is above or below the shuffle plane, a row of interstitials or vacancies. This dislocation is less mobile than the glide-set dislocation because movement of the row of point defects within the core can only occur during slip by shuffling. Climb, which involves point defect absorption or emission, transforms shuffle-set dislocations to glide-set dislocations, and vice-versa. It is probable that the glide-set dislocations dominate, but that segments of the shuffle set can occur when circumstances are favorable, e.g. in the presence of point defects or high stress. Computer calculations of the core structure and energy of dislocations in model crystals suggest that the energy is reduced by bond reconstruction, a process in which dangling bonds reform with others so that all atoms retain approximately tetrahedral coordination. This is shown schematically
Figure 7-61. Perfect 60° dislocations of (a) the glide set and (b) the shuffle set. (After Hirth and Lothe, 1982)
7.7 Concluding Remarks
(a)
in Fig. 7-62 for the glide-set partial with b at 30° to the line direction. Reconstruction occurs by the dangling bonds at CCCC rebonding in pairs along the core. Such reconstruction appears less favorable for shuffle-set cores because the orbitals of the unpaired electrons exhibit less overlap. Since an atom can form one of these bonds with either of its neighbors along the core, a single dangling bond is created when a series of reconstructed bonds in one sense meet a series of the opposite sense. This is a soliton (Heggie and Jones, 1982) or antiphase defect (Hirsch, 1980). Kinks in dislocations can consist of either a reconstructed or an unreconstructed step. One of these is converted to the other by absorbing an antiphase defect. The situation is more complex for the compound semiconductors. For GaAs, for instance, all atoms in planes designated with small letters (Fig. 7-60) are of one type and those in planes labelled with capitals are of the other type. Thus, reconstruction along the core of glide-set dislocations cannot be crucial for stabilizing these dislocations in GaAs, since it would require the bonding of C-C, C-C, ... in Fig. 7-62 to occur between atoms of the same kind. Nevertheless, dissociated dislocations of the glide set are commonly seen. An addi-
(b)
479
Figure 7-62. Two possible core configurations for the 30° partial of the glide set in silicon, (a) Dangling bonds occur at atoms CCCC. (b) Reconstructed core. (From Marklund, 1979)
tional feature is that a dislocation of given orientation and Burgers vector can have one of two forms, for the extra half-plane can terminate in the core with either anions or cations. This information must be furnished if dislocations are to be fully specified since it affects their dissociation. As may be inferred from the descriptions of core structure in this section, the nature of the bonds within the core affects not only the mechanical response of these materials, but also their electronic properties. These aspects are reviewed in detail in Volume 4, Chapter 6 of this Series, and in Hirsch (1985) and Alexander (1986). Numerous recent research papers in this area are also to be found in Roberts et al. (1989).
7.7 Concluding Remarks As was noted several times in this chapter, the general aim here has been to provide an introductory description of dislocations that will serve as background information for the other chapters on materials science and technology in this series. More advanced treatments are available and certainly more mathematical ones as far as the theory of dislocations is concerned - but they can miss the needs of
480
7 Dislocations in Crystals
engineers and scientists who do not intend to become expert in the field. The references cited here should provide an adequate resource for those who wish to go further. The first two-thirds of this review cover much that is general in the science of dislocations, and we have tried to establish there the principles that govern dislocation properties. In the last part (Sec. 7.6) we have considered specific materials classes, which hopefully are the ones of major interest, and, by being selective, have attempted to demonstrate how dislocations behave in real materials. It will be clear from that section that each material must be considered in its own right. Whilst the crystal structure and nature of atomic bonding give a broad indication of behavior, it is the actual atomic structure of the core that matters most, and an understanding of that falls outside the scope of the continuum approach of Sees. 7.3 to 7.5. The presence of stable, planar faults is a dominant influence on core structure, and whilst the symmetry and bonding of a particular crystal can be used to show that faults may exist, they cannot be used to show that they actually do. Recent advances in atomic-scale computer simulation have been of real benefit in this context, and this topic has been reviewed in depth by Duesbery (1989). Finally, no coverage has been offered here of the dislocation content of interfaces in crystalline solids and the interaction of dislocations with interfaces. This is a large and important topic, and is reviewed in Chapter 9 of this Volume.
7.8 References Alexander, H. (1986), in: Dislocations in Solids, Vol. 7: Nabarro, F. R. N. (Ed.). Amsterdam: North-Holland, Chap. 35.
Alshits, V. L, Indenbom, V. L. (1975), Sov. Phys. Usp. 18, 1. Amelinckx, S. (1958), Nuovo Cimento 7, 569. Amelinckx, S. (1979), in: Dislocations in Solids, Vol. 2: Nabarro, F. R. N. (Ed.). Amsterdam: North-Holland, p. 66. Amelinckx, S., Bontinck, W., Dekeyser, W., Seitz, F. (1957), Phil Mag. 2, 355. Bacon, D. J. (1969), Scripta Metall. 3, 735. Bacon, D. J. (1985), in: Fundamentals of Deformation and Fracture: Bilby, B. A., Miller, K. J., Willis, J. R. (Eds.). Cambridge: Cambridge University Press, p. 401. Bacon, D. J. (Ed.) (1991), "Defects in H.C.P. Metals", Phil. Mag. A 63 (5). Bacon, D. X, Liang, M. H. (1986), Phil. Mag. A 53, 163. Bacon, D. X, Scattergood, R. O. (1974), Phys. Stat. Sol. (a) 25, 395. Bacon, D. X, Barnett, D. M., Scattergood, R. O. (1978), Prog. Mater. Sci. 23, 51. Bardeen, X, Herring, C. (1952), in: Imperfections in Nearly Perfect Crystals. New York: Wiley, p. 261. Berghezan, A., Fordeux, A., Amelinckx, S. (1961), Acta Metall. 9, 464. Bilby, B. A. (1950), Proc. Phys. Soc. A63, 191. Bilby, B. A., Bullough, R., Smith, E. (1955), Proc. Roy. Soc. A 231, 263. Bonneville, X, Douin, X (1990), Phil. Mag. A62, 247. Brown, L. M. (1964), Phil. Mag. 10, 441. Brown, L. M. (1967), Phil. Mag. 15, 363. Bullough, R. (1985), in: Dislocations and Properties of Real Materials. London: Inst. of Metals, p. 283. Bullough, R., Foreman, A. X E. (1964), Phil. Mag. 9, 315. Burgers, X M. (1939), Proc. Ron. Ned. Akad. Wetenschap. 42, 293, 378. Christian, X W. (1983), Metall. Trans. A14, 1233. Cochardt, A. W., Schoeck, G., Wiedersich, H. (1955), Acta Metall. 3, 533. Cottrell, A. H. (1948), in: Report of a Conference on the Strength of Solids. London: The Physical Society, p. 30. Cottrell, A. H. (1952), Phil. Mag. 43, 645. Cottrell, A. H. (1953), Dislocations and Plastic Flow in Crystals. Oxford: Oxford University Press. Cottrell, A. H. (1975), An Introduction to Metallurgy, 2nd ed. London: Edward Arnold. Couret, A., Caillard, D., Puschl, W., Schoeck, G. (1991), Phil. Mag. A63, 1045. de Batiste, R. (1972), Internal Friction of Structural Defects in Solids. Amsterdam: North-Holland, de Wit, G., Koehler, X S. (1959), Phys. Rev. 116,1113. de Wit, R. (1960), Solid State Phys. 10, 249. Duesbery, M. S. (1984), Proc. Roy. Soc. A 392, 145 and 175. Duesbery, M. S. (1989), in: Dislocations in Solids, Vol. 8: Nabarro, F. R. N. (Ed.). Amsterdam: North-Holland, Chap. 39.
7.8 References
Duesbery, M. S., Vitek, V. (1985), Mater. Sci. Eng. 72, 199. Eshelby, J. D. (1949), Proc. Phys. Soc. 62 A, 307. Eshelby, I D. (1956), Sol. State Phys. 3, 79. Eshelby, J. D. (1957), Proc. Roy. Soc. A 241, 376. Eshelby, J. D., Frank, F. C , Nabarro, F. R. N. (1951), Phil Mag. 42,351. Eyre, B. L., Bullough, R. (1965), Phil Mag. 12,31. Eyre, B. L., Loretto, M. A., Smallman, R. E. (1977), in: Vacancies '76. London: The Metals Society, 63. Frank, F. C. (1949), Physica 15, 131. Frank, F. C. (1951), Phil. Mag. 42, 809. Frank, F. C , Read, W. T. (1950), in: Sympos. on Plastic Deformation of Crystalline Solids. Pittsburgh: Carnegie Inst. of Tech., p. 44. Frenkel, J. (1926), Z. Phys. 37, 572. Frost, H. X, Ashby, M. F. (1982), Deformation Mechanism Maps. Oxford: Pergamon Press. Gao, F , Bacon, D. J. (1992), Phil. Mag. A66, 839. Gavazza, S. D., Barnett, D. J. (1976), /. Mech. Phys. Sol. 24, 111. Gosling, T. J., Jain, S. C , Willis, J. R., Atkinson, A., Bullough, R. (1992), Phil. Mag. A66, 119. Griffiths, M. (1991), Phil. Mag. A63, 835. Haasen, P. (1978), Physical Metallurgy. Cambridge: Cambridge University Press. Haasen, P. (1985), in: Dislocations and Properties of Real Materials. London: Inst. of Metals, p. 312. Harder, J. M., Bacon, D. J. (1986), Phil. Mag. A 54, 651. Head, A. K. (1953), Proc. Phys. Soc. 66B, 793. Heggie, M., Jones, R. (1982), /. Physique (Paris) 43, Cl-45. Hirsch, P. B. (1980), /. Microscopy 118, 3. Hirsch, P. B. (1985), in: Dislocations and Properties of Real Materials. London: Inst. of Metals, p. 333. Hirsch, P. B. (1992), Phil. Mag. A 65, 569. Hirth, J. P., Lothe, J. (1982), Theory of Dislocations, 2nd ed. New York: John Wiley. Hull, D., Bacon, D. J. (1984), Introduction to Dislocations, 3rd ed. Oxford: Pergamon Press. Jenkins, M. X, Hardy, G. X, Kirk, M. A. (1987), Mater. Sci. Forum 15-18, 901. Jesser, W. A., van der Merwe, X H. (1989), in: Dislocations in Solids, Vol. 8: Nabarro, F. R. N. (Ed.). Amsterdam: Elsevier, Chap. 41. Johnson, W. G., Gilman, X X (1959), J. Appl. Phys. 30, 129. Jordan, A. S., Caruso, R., Von Neida, A. R. (1980), Bell System Tech. J. 59, 539. Kaganov, M. I., Kravchenko, V. Y, Natsik, V D. (1974), Sol. Phys. Usp. 16, 878. Kalman, P., Toth, A., Keszthelyi, T., Sarkozi, X (1982), Mater. Sci. Eng. 54, 85. Kalman, P., Toth, A., Keszthelyi, T, Sarkozi, X (1984), Mater. Sci. Eng. 64, 223. Kear, B., Wilsdorf, H. G. F. (1962), Trans. AIME 224, 382.
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Kirchner, H. O. K. (1981), Phil. Mag. A43, 1393. Kocks, U. E, Argon, A. S., Ashby, M. F. (1975), Prog. Mater. Sci. 19. Korner, A., Schmid, H., Prinz, F. (1979), Phys. Stat. Sol. (a) 51, 613. Kroupa, F., Price, P. B. (1961), Phil. Mag. 6, 243. Legrand, B. (1984), Phil. Mag. B49, 111. Legrand, B. (1985), Phil. Mag. A52, 83. Leibfried, G. (1950), Z. Phys. 127, 344. Liang, M. H., Bacon, D. X (1986), Phil. Mag. A53, 181 and 205. Little, E. A., Bullough, R., Wood, M. H. (1980), Proc. Roy. Soc. A 372, 365. Lomer, W. M. (1951), Phil. Mag. 42, 1327. Loretto, M. H., Smallman, R. E. (1975), Defect Analysis in Electron Microscopy. London: Chapman and Hall. Love, A. E. H. (1927), The Mathematical Theory of Elasticity. Cambridge: Cambridge University Press. Marklund, S. (1979), Phys. Stat. Sol. (b) 92, 83. Mason, W. P. (1968): in: Dislocation Dynamics: Rosenfield, A. R., Hahn, G. T., Clement, A. L., Jaffee, R. I. (Eds.). New York: McGraw-Hill, p. 487. Meduoye, G. O., Bacon, D. X, Evans, K. E. (1991), /. Crystal Growth 108, 627. Minonishi, Y, Ishioka, S., Koiwa, M., Yamaguchi, M. (1982a), Phil. Mag. A 45, 835. Minonoshi, Y, Ishioka, S., Koiwa, M., Yamaguchi, M. (1982b), Phil. Mag. A 46, 761. Mitchell, T. E., Lagerlof, K. P. D., Hener, A. H. (1985), in: Dislocations and Properties of Real Materials. London: Inst. of Metals, p. 349. Mura, T. (1968), Adv. Mater. Res. 3, 1. Mura, T. (1982), Micromechanics of Defects in Solids. Amsterdam: Martinus Nijhoff. Nabarro, F. R. N. (1947), Proc. Phys. Soc. 59, 256. Nabarro, F. R. N. (1952), Adv. Phys. 1, 269. Nabarro, F. R. N. (1967), The Theory of Crystal Dislocations. Oxford: Oxford University Press. Nabarro, F. R. N. (1984), in: Dislocations in Solids: Suzuki, H., Ninomiya, T., Sumino, K., Takeuchi, S. (Eds.). Tokyo: Univ. Tokyo Press, p. 3. Naka, S., Kubin, K. L., Perrier, C. (1991), Phil. Mag. A63, 1035. Nowick, A. S., Berry, B. S. (1972), Anelastic Relaxation in Crystalline Solids. New York: Academic Press. Orowan, E. (1934), Z. Phys. 89, 605, 634. Peach, M., Koehler, X S. (1950), Phys. Rev. 80, 436. Peierls, R. (1940), Proc. Phys. Soc. 52, 34. Phythian, W. X, English, C. A., Yellen, D. H., Bacon, D. X (1991), Phil. Mag. A 63, 821. Polanyi, M. (1934), Z. Phys. 89, 660. Pope, D. P., Ezz, S. S. (1984), Int. Metall. Rev. 29, 136. Puls, M. P., So, C. B. (1980), Phys. Stat. Sol. (b) 98, 87.
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Read, W. T. (1953), Dislocations in Crystals. New York: McGraw-Hill. Roberts, S. G., Holt, D. B., Wilshaw, P. R. (Eds.) (1989), Structure and Properties of Dislocations in Semiconductors. Bristol: Inst. Phys., Conf. Ser. 104. Saada, G., Douin, J. (1991), Phil. Mag. Lett. 64, 67. Scattergood, R. O. (1980), Acta Metall. 28, 1703. Seeger, A., Engelke, H. (1968), in: Dislocation Dynamics: Rosenfield, A. R., Hahn, G. T., Clement, A. L., Jaffee, R. I. (Eds.). New York: McGrawHill, p. 623. Seeger, A., Donth, H., Pfaff, F. (1957), Disc. Faraday Soc. 23, 19. Serra, A., Pond, R. C , Bacon, D. J. (1991), Acta Metall. Mater. 39, 1469. Stoloff, N. S. (1984), Int. Metall. Rev. 29, 123. Suzuki, H. (1962), J. Phys. Soc. Japan 17, 322. Taylor, G. I. (1934), Proc. Roy. Soc. A145, 362. Teodosiu, C. (1982), Elastic Models of Crystal Defects. Berlin: Springer. Thompson, N. (1953), Proc. Phys. Soc. B66, 481. Ullmaier, H., Schilling, W. (1980), in: Physics of Modern Materials. Vienna: IAEA, 301. Umakoshi, Y, Yamaguchi, M. (1990), Prog. Mater. Sci. 34, 1. Vitek, V. (1968), Phil. Mag. 18, 113. Vitek, V. (1974), Crystal Lattice Defects 5, 1.
Vitek, Y, Igarashi, M. (1991), Phil. Mag. A63, 1059. Vitek, V, Yamaguchi, M. (1981), in: Interatomic Potentials and Crystalline Defects: Lee, J. K. (Ed.). New York: Metall. Soc. AIME, 223. Volterra, V. (1907), Ann. Ecole Norm. Super. 24, 400. Vreeland, T. (1968), in: Techniques of Materials Research, Vol. 2: Bunshaw, R. F. (Ed.). New York: Wiley, p. 341. Weertman, I , Weertman, J. R. (1964), Elementary Dislocation Theory. New York: Macmillan. Yoo, M. H. (1981), Metall. Trans. A 12, 409.
General Reading Hirth, J. P., Lothe, J. (1982), Theory of Dislocations, 2nd ed. New York: Wiley. Hull, D., Bacon, D. J. (1984), Introduction to Dislocations, 3rd ed. Oxford: Pergamon Press. Loretto, M. H. (Ed.) (1985), Dislocations and Properties of Real Materials. London: Institute of Metals. Nabarro, F. R. N. (1967), The Theory of Crystal Dislocations. Oxford: Oxford University Press. Nabarro, F. R. N. (Ed.), Dislocations in Solids. Amsterdam: North-Holland (several volumes from 1979 on).
8 Crystal Surfaces Michel A. Van Hove Center for Advanced Materials, Materials Sciences Division, Lawrence Berkeley Laboratory, Berkeley, U.S.A.
List of 8.1 8.2 8.2.1 8.2.2 8.2.3 8.2.4 8.2.5 8.2.6 8.3 8.3.1 8.3.2 8.3.2.1 8.3.2.2 8.4 8.4.1 8.4.2 8.4.3 8.4.4 8.4.5 8.5 8.5.1 8.5.2 8.5.3 8.5.3.1 8.5.3.2 8.5.4 8.5.4.1 8.5.4.2 8.5.4.3 8.5.4.4 8.5.5 8.5.6 8.5.7 8.5.7.1
Symbols and Abbreviations Introduction Experimental and Theoretical Techniques Diffraction Techniques Interference Techniques Scattering Techniques Microscopic and Topographic Techniques Complementary Techniques Theory Two-Dimensional Ordering and Nomenclature Ordering Principles at Surfaces Nomenclature Miller Indices Superlattices: Wood and Matrix Notations Clean Surfaces Bulk-Like Lattice Termination Stepped Surfaces Relaxations Reconstruction Surface Segregation Adsorbate-Covered Surfaces Physisorption Atomic Chemisorption Sites and Bond Lengths Atomic Multilayers Metallic Adsorption on Metals Multilayer Growth on Semiconductors Molecular Adsorption Molecular Adsorption Sites and Ordering CO and NO Adsorption Straight-Chain Hydrocarbon Adsorption Benzene Adsorption Coadsorption Adsorbate-Induced Relaxations Adsorbate-Induced Reconstructions Displacive Local Reconstruction Induced by Adsorption
Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. Allrightsreserved.
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8.5.7.2 8.5.7.3 8.5.7.4 8.5.8 8.6 8.7 8.7.1 8.7.2 8.8 8.9 8.10
8 Crystal Surfaces
Removal of Reconstruction by Adsorption Creation of Reconstruction by Adsorption Change of Reconstruction by Adsorption Compound Formation and Surface Segregation Disordered Surfaces Mechanisms Bond Length Changes Restructuring Outlook Acknowledgements References
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List of Symbols and Abbreviations
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List of Symbols and Abbreviations a, b a\b' B5 hkl ra, n M m
tj
two-dimensional unit cell vectors of the substrate two-dimensional unit cell vectors of the superlattice a substitutional site Miller indices of a surface; h, k, I = 0,1 independent stretch factors in different surface directions in Wood notation matrix of the matrix notation for unit cells matrix elements of M; ij = 1,2 "adatom": adding atom on top position with a total of 4 bonding partners (reconstruction mechanism) atomic weight fraction parameter
a
rotation angle in the Wood notation for unit cells
AES ALICISS ARPEFS ARUPS ARXPD b.c.c. c CI DLEED EAM EHT EMT ESDIAD f.c.c. FIM GVB HEIS HF, HFS HREELS h.c.p. IRAS LCAO LEED LEEM LEIS MEIS MO NEXAFS NMR NPD
Auger electron spectroscopy alkali-ion impact collision ion scattering spectroscopy angle-resolved photoelectron emission fine structure angle-resolved ultraviolet photoelectron spectroscopy angle-resolved X-ray photoelectron diffraction body-centered cubic centered (optional prefix in Wood notation) configuration interaction diffuse LEED embedded atom method extended Hiickel theory effective-medium theory electron-stimulated desorption ion angular dependence face-centered cubic field-ion microscopy generalized valence bond high-energy ion scattering Hartree - Fock, Hartree - Fock - Slater high-resolution electron energy loss spectroscopy hexagonal close-packed infrared absorption spectroscopy linear combination of atomic orbitals low-energy electron diffraction low-energy electron microscopy low-energy ion scattering medium-energy ion scattering molecular orbital near-edge X-ray absorption fine structure nuclear magnetic resonance normal photoelectron diffraction
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p PEM PhD, PD R SAM SCF-Xa-SW SCF-HF SEXAFS STM TDS VB XANES
primitive (optional prefix in Wood notation) photoelectron microscopy photoelectron diffraction implies rotation in the Wood notation scanning Auger microscopy self-consistent-field-Xa-scattered-wave self-consistent-field-Hartree-Fock surface extended X-ray absorption fine structure scanning tunneling microscopy thermal desorption spectroscopy valence bond X-ray absorption near-edge structure
8.1 Introduction
8.1 Introduction The atomic-scale structure and properties of the solid/vacuum and solid/gas interfaces have been extensively investigated in the last two decades (see the references to general reading). Recently, notable progress has also been achieved at the solid/ solid interface (see Chap. 9 and Vol. 4, Chap. 8). Rather less is known so far about the solid/liquid interface, especially regarding structure (Lipkowski and Ross, 1991). Our emphasis will be on the solid/ vacuum and solid/gas interfaces, which we will collectively label "surfaces". Thanks to the availability of ultra-high vacuum techniques, it has become possible to control the composition and condition of interfaces at the atomic level, and to determine their structure by using electrons, photons, ions and other probes. This is accomplished most effectively by starting with free surfaces of single crystals. Atomically clean surfaces can be prepared, while foreign matter can be added at will in submonolayer to multilayer amounts. Interfaces, and in particular solid crystalline surfaces, serve as models for the understanding of many phenomena of technological importance, such as those occurring in semiconductor devices, in heterogeneous catalysis, oxidation and corrosion, electrochemistry, friction and wear. For the purposes of this discussion, we shall define a surface as a region of space limited to a few atomic diameters on either side of the interface plane. We shall treat both the clean surface, before deposition of foreign matter, and the adsorbate-covered surface obtained after deposition of amounts ranging from less than a monolayer to several monolayers. The substrate can be a metal, an alloy, a semiconductor (whether elemental or com-
487
pound), an insulator, or any other substance that crystallizes. The clean or adsorbate-covered substrate surface can "reconstruct" into a lattice that is quite different from the three-dimensional bulk lattice, a very characteristic phenomenon of surfaces. Many types of adsorbate can be deposited on a substrate. Typically, one deposits molecules, resulting frequently in atomic adsorbates due to molecular decomposition, or resulting in adsorbed molecular species directly related to or different from the initial molecule. Atoms or clusters can also be deposited, sometimes in ionic form. Chemisorption occurs when strong substrate-adsorbate bonds form. Otherwise physisorption can occur (at low enough temperatures). Submonolayers often do not order into a lattice, or else they may create two-dimensionally ordered superlattices, or also generate close-packed islands. Multilayers grow into thin films that may be epitaxial (i.e., grow in some registry with the substrate lattice) or that may be totally independent of the substrate. One can identify several levels of detail in the description of the structure of singlecrystal surfaces. One level of detail is the two-dimensional unit cell of the surface structure, as given typically by a diffraction pattern (which is for example provided by the commonly available technique of low-energy electron diffraction). This first of all indicates the presence or absence of ordering on the surface. It also defines, for an ordered surface, its periodicity in relation to the periodicity of the bulk material. Such information is very valuable for adsorbates. A second level of detail identifies the chemical composition of the surface, which can be different from the bulk composition by segregation to the surface of compounds, or by the deposition of foreign matter. A third level of detail defines
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the crystallographic bonding structure: here bond lengths and bond angles are the important quantities. We shall discuss all three levels of detail for various classes of structures. In Sec. 8.2 we shall describe the experimental and theoretical techniques used to study surfaces. The reader who is more interested in results can pass over this section. Since many single-crystal surfaces are ordered, the causes of such ordering will be introduced in Sec. 8.3, while notation for such ordering will be introduced there as well. The structures of clean and adsorbate-covered surfaces will then be presented in Sees. 8.4 and 8.5, respectively. Disordered surface layers will be addressed in Sec. 8.6. Mechanisms for explaining the origins of surface structures will be explored in Sec. 8.7. And finally, an outlook into future developments will be offered in Sec. 8.8.
8.2 Experimental and Theoretical Techniques In the following, we give an overview of the experimental and theoretical methods used to study the structure of surfaces. Some aspects of the theoretical underpinnings of experimental techniques are included, because more or less complicated calculations are usually required to extract the geometrical information from the measured data. Only the main techniques will be discussed: many more are available, but have contributed less to our topic. A crucial requirement for experimental techniques for surface structure determination is surface sensitivity: it is necessary to obtain structural information about only a few atomic layers of material, which contain at most a few times 1015 atoms on the typical surface area of 1 cm2.
Sample preparation (Somorjai, 1981) is a critical step in surface experiments, since few surfaces exist in the natural state with the required purity or crystallinity. Most surfaces in contact with air are naturally covered with layers of oxides and other compounds, which in turn are covered with a film of organic deposit. Exceptions are such inert surfaces as the basal planes of graphite and layer compounds (e.g., molybdenum disulfide), which can be relatively easily cleaned of weakly-adsorbed organic deposits. Ultra-high vacuum in the 10" 1 0 Torr range is a prerequisite for the atomic-scale control of surfaces. Such a vacuum allows a surface to remain in the same condition of cleanliness and crystallinity for hours, for the duration of the experiment. It also allows the deposition of foreign material in well-controlled amounts. Surfaces are commonly prepared from single-crystal rods by mechanical or spark cutting in air, thereby exposing a surface with a preselected crystallographic orientation. The sample is then introduced into the vacuum, where it can be cleaned of impurities, surface oxides, etc. and finally smoothed by annealing. One cleans by case-dependent methods and combinations of methods, which include: annealing (i.e., long heating cycles) to drive impurities from the subsurface bulk to the surface for subsequent elimination; flashing (short heating cycles) to evaporate impurities off the surface; ion bombardment (also called ion milling) to kick off tenacious impurities like oxides; chemical treatment to reactively and selectively eliminate certain elements. Some surfaces (especially some semiconductor surfaces) are obtained directly in vacuum by cleavage along cleavage planes; annealing is still useful to smooth them on the atomic scale.
8.2 Experimental and Theoretical Techniques
Atoms and molecules are usually deposited onto a clean surface from the gas phase in small amounts that generate controllable "coverages" on the surface, i.e. controllable surface densities in the range from a fraction of a monolayer to a few monolayers. The coverage is determined by the exposure time to the gas. Deposition can also be accomplished with evaporation sources aimed at the surface (this is commonly done with metallic deposition). Several techniques are available to monitor both the chemical composition and the crystallinity of a surface. The composition can be monitored with, among other techniques (Ertl and Kiippers, 1979; Somorjai, 1981): Auger electron spectroscopy, which identifies non-destructively and semiquantitatively the chemical elements present on the surface; thermal desorption mass spectroscopy (also called temperature-programmed desorption), which quantitatively detects species evaporated from the heated surface; and high-resolution electron energy loss spectroscopy, which identifies adsorbed species by their vibrational frequency spectrum. 8.2.1 Diffraction Techniques
Diffraction has been a most successful approach to atomic-scale structure determination of bulk materials and surfaces. Diffraction can deliver atomic coordinates at surfaces with precisions in the range of 1 to 10 pm. Electron and X-ray diffraction are the main diffraction techniques which have been applied for surface structure determination. The majority of known surface structures has been studied with low-energy electron diffraction (LEED) (see general reading). LEED uses as probes elastically diffracted electrons with energies in the 20-300 eV range, which corresponds to
489
electron wavelengths in the 0.05 to 0.2 nm range. Mono-energetic electrons are beamed at a surface, from which they are diffracted (only elastically scattered electrons are normally recorded). Inelastic scattering processes severely limit the penetration depth of such electrons into a surface to about 0.5 to 1 nm, giving a surface sensitivity of only a few atomic layers. The diffraction pattern is easily displayed on a fluorescent screen and serves as a convenient monitor of the surface condition. In particular, it tells about the long-range ordering of the surface, showing the two-dimensional periodicity parallel to the surface. Interferences between different scattering paths pick up the local surface structure information in the form of modulations of diffracted electron beam currents. Elastic interactions are strong enough that multiple scattering of electron from one atom to another is important in this energy range. This complicates the analysis of experimental diffraction data, but the necessary theoretical methods have been very successful in obtaining bond lengths and angles at surfaces of almost any chemical composition. The methods employed to analyze LEED measurements simulate the entire multiple scattering process in a way similar to the calculation of electronic band structures in the bulk and at the surface. A variety of methods has been devised to most efficiently perform these simulations for complex surface structures. For two decades, the only way to solve structures by LEED was to simulate the diffraction for all plausible surface geometries and choose the geometry which gave the best fit to the experiment. This requires great efficiency of calculation for each trial structure. Only very recently have automated search methods been developed that
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can much more efficiently solve a surface structure (Cowell etal., 1986; Kleinle et al., 1989; Rous et al., 1990). The LEED technique has recently also been extended from ordered layers to the case of disordered layers of adsorbates on single-crystal substrates: in this case, the diffraction occurs in all directions, leading to the name of diffuse LEED. The diffusely scattered intensity is measured and then analyzed in terms of the local bonding geometry: adsorption site, bond lengths, bond angles, etc. (Saldin et al., 1985; Heinz etal., 1985b; Wander etal., 1991). X-ray diffraction, with its inherent conceptual simplicity, has been an obvious choice for surface crystallography (Feidenhans'l, 1989). However, long mean free paths of X-rays in solids permit the desired surface sensitivity only when grazing incidence and/or emergence are used: angles within a fraction of a degree from the surface plane are required. This demands extremely flat surfaces and strict control of diffraction angles, both challenging experimental tasks. Also a sufficient photon flux is required, which is often sought at synchrotron radiation facilities. At surfaces, X-ray diffraction has been used in first instance to study clean or adsorbate-induced surface reconstructions. It is particularly suitable to investigate disordering phenomena (like surface roughness and surface phase transitions). X-ray diffraction, because of its grazing angles, has better sensitivity to distances parallel to the surface than perpendicular to the surface. As a result, a good number of X-ray surface structures provide the lateral atomic positions without the perpendicular distances. A variant of X-ray diffraction has been applied to obtain interlayer spacings between adsorbates and bulk atomic planes. These bulk atomic planes include not only the planes parallel to the surface, but also
any crystallographic planes inclined to the surface. Then "triangulation" allows the adsorption site to be determined. This approach uses X-ray standing waves due to reflection from atomic planes in the crystal bulk (Cowan et al., 1980). This method is used in conjunction with detection of the fluorescence that is unique to the adsorbate. The fluorescence is sensitive to interference between the directly reflected beams and beams reflected from the bulk planes; hence it can tell the interplanar separation. Atom diffraction has been used to obtain information about surface corrugation, i.e. the shape of the outer electron density contours of the surface, some 0.5 to 1 nm away from the atomic nuclei (Engel and Rieder, 1982). The corrugation often allows the nature of the surface structure to be determined. However, it is difficult to model the surface-atom interaction potential and to relate the corrugation information to nuclear positions. This has limited the usefulness of atom diffraction for detailed structural studies. On the other hand, the technique is very valuable for studying the long-range order and disorder of surfaces.
8.2.2 Interference Techniques
A number of techniques exploit interference effects to determine surface structures, without using diffraction from the periodic surface lattice. Prominent among these interference techniques, as we shall call them here, are the various forms of photoelectron diffraction and the finestructure techniques, as well as forward focusing. The main difference between the interference and diffraction techniques lies in the origin of the scattering particles. In the
8.2 Experimental and Theoretical Techniques
diffraction case, beams are incident onto the surface from outside and can be represented by plane waves. In the interference case, by contrast, electrons are used which are excited locally in the surface and can be represented by spherical waves. These excited electrons are generated by another incident probe, such as a photon or another electron. Even the same electron can be used as probe: it is allowed to inelastically scatter, thereby losing its coherent planewave properties. These excited electrons then scatter from nearby atoms. The interference between different scattering paths picks up structural information in the form of modulations of the emitted electron current. In the case of photoelectron diffraction (Fadley, 1990), electrons are photoemitted from particular electronic orbitals, such as core levels in individual surface atoms. Often synchrotron radiation is used as a source of photons. Those electrons scattered from nearby atoms toward the detector interfere with electrons travelling directly from the emitting atom to the detector, in a way that depends on the local geometry. The electrons are emitted with kinetic energies in the 400-1000 eV energy range, where single-scattering events dominate to give a qualitatively simple scattering picture (but multiple scattering must be taken into account for accuracy). The method has been applied primarily to the study of atomic chemisorption on singlecrystal surfaces. Several names are in use for this method, emphasizing different modes of operation, including: ARUPS (angle-resolved ultraviolet photoelectron spectroscopy), ARXPD (angle-resolved Xray photoelectron diffraction), PhD or PD (photoelectron diffraction), NPD (normal photoelectron diffraction), ARPEFS (angle-resolved photoelectron emission fine structure).
491
Particularly well known among the finestructure techniques is surface extended Xray absorption fine structure (SEXAFS) (Citrin, 1986; Rowe, 1990). Again, photoelectrons are excited and allowed to scatter from nearby surface atoms. However, in this case the electrons return to the emitting atom and modulate (by wave interference) the emission process itself. This modulation is again interpreted in terms of the local geometry. Here, 1000 eV is a typical electron kinetic energy, and a single-scattering model is often adequate to interpret the experimental data. Any emitted particle can be chosen for detection, including photons, electrons and ions. Like photoelectron diffraction, this method has been applied primarily to the study of atomic adsorbates on single-crystal surfaces. Related to the extended fine-structure techniques are the near-edge techniques, such as near-edge X-ray absorption fine structure (NEXAFS, also called XANES, for X-ray absorption near-edge structure) (Outka and Stohr, 1988). NEXAFS is SEXAFS conducted at much lower electron kinetic energies, where multiple scattering is strong. This technique is nowadays primarily used to monitor excitations among valence electrons, from which structural information like molecular orientation and bond lengths is accessible. Indeed, the electronic excitation energies depend on the bond lengths in characteristic ways that permit the application of simple empirical relationships to the determination of surface structure. And the polarization of the incident photons can be used to detect the orientation of bonds and thus the orientation of molecules relative to the surface plane. A recently developed technique that also uses electron interference has become very promising for a variety of applications. It
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can be called "forward focusing" or "forward scattering" (Egelhoff, 1990; Fadley, 1990; Xu and Van Hove, 1989). Photoelectrons and other secondary electrons (such as Auger or inelastic electrons) are generated at an atomic site in a surface with kinetic energies on the order of 5002000 eV. These are forward scattered by other nearby atoms, i.e. they are focussed in the forward direction as if by a converging electrostatic lens. As a result, they produce peaks of emission in directions corresponding to bond directions around the emitting atom. By detecting such peaks, one can in effect "triangulate" the local structure around the emitting atom. It is possible, for example, to determine the location of foreign atoms within a solid surface, such as the site of interstitial atoms below a surface, or also the orientation of molecules on surfaces. Another very recent approach uses "electron holography" (Barton, 1988; Saldin and de Andres, 1990). The angular distribution of an emitted electron (e.g., a photoelectron) is a hologram, due to interference between the directly emitted wave and the same wave scattered from nearby atoms. From this hologram, one can computationally reconstruct an image of the neighborhood of the emitting atom. Current work aims at improving the resolution of the reconstructed image, which at present is only on the order of 0.1 nm. 8.2.3 Scattering Techniques
A number of techniques rely on scattering, as opposed to interference, to obtain geometrical information from surfaces. Of particular value has been ion scattering both at low energies (around 1 keV) and at high energies (100-1000 keV). Low-energy ions cast wide shadows behind surface atoms. These shadows obscure further atoms if they lie within the
shadow cone. By varying the incidence direction, the shadow cone can be swept through the surface and expose or hide individual atoms. It is possible to monitor the disappearance and emergence of atoms in the shadow cone, thereby obtaining structural information. This information tends to be restricted to the top one or two atomic layers, since the shadows obscure all deeper layers. The technique is generally called LEIS (low-energy ion scattering) (Aono etal., 1982). Alkali ions provide particularly good structural sensitivity when monitored near the 180° scattering direction: this feature is used in ALICISS (alkali-ion impact collision ion scattering spectroscopy) (Niehus and Comsa, 1984). High-energy ions (helium nuclei and protons are commonly used) are directed along bulk crystal axes of the surface material. The ions can channel relatively deeply into the crystal between rows of atoms, because the shadow cones are in this case very narrow. But if surface atoms deviate from the ideal bulk lattice positions and block the channels through which the ions move, the ions will scatter strongly back out of the surface. This conceptually simple approach has been used successfully to obtain detailed structural information for a number of clean and adatom-covered surfaces. It is uniquely suited to study buried solid-solid interfaces (i.e., interfaces that lie deep in the bulk below a surface), since the ions can be made to penetrate relatively deeply into a surface. The technique still requires ultra-high vacuum for its operation, because the ion beams can only be formed in such a vacuum (van der Veen, 1985). Depending on the energy range used, and other experimental choices, the technique is known under the names of medium-energy ion scattering (MEIS) or high-energy ion scattering (HEIS).
8.2 Experimental and Theoretical Techniques
Another directionally-sensitive scattering technique is electron-stimulated desorption ion angular dependence (ESDIAD) (Madey, 1985). Here, incident electrons break an atom off from the surface. Such an atom is often ionized in the process, which makes it easy to detect. The ion is found to travel in a direction close to that of the broken bond. For instance, assume that an H 2 O molecule stands on a surface with the oxygen atom down and the hydrogen atoms pointing at an angle from the surface. Then, an electron can easily break an O-H bond and will emit a hydrogen ion in the direction of the O-H bond. This technique thus provides direct information on molecular bond orientations (as well as vibrational amplitudes). 8.2.4 Microscopic and Topographic Techniques
A number of powerful techniques have been developed that study surfaces in a microscopic sense: they image directly individual microscopic parts of a surface rather than structure as averaged over macroscopic distances. Some, like field-ion microscopy (FIM) (Miiller and Tsong, 1969; Ehrlich, 1985) and scanning tunneling microscopy (STM) (Kumar Wickramasinghe, 1989) can image individual atoms. Near-atomic resolution can also be obtained with electron microscopy (Takayanagi, 1989), which can image end-on individual rows of atoms. Other techniques, like scanning Auger microscopy (SAM) (Briggs and Seah, 1983), low-energy electron microscopy (LEEM) (Telieps and Bauer, 1988) and photoelectron microscopy (PEM) (Margaritondo and Cerrina, 1990), image larger regions with micron or submicron resolution. Often these techniques provide chemical specificity; for ex-
493
ample, one atomic element can be imaged while all others are ignored. None of these microscopic techniques readily provides complete information about bond lengths or other bonding details. In special conditions, distances parallel to the surface can be obtained with some accuracy, but mostly these techniques are used to map out surface topography or composition, down to atomic resolution in the case of STM and FIM. A particular advantage of imaging is that defects of many types can be analyzed. For example, steps in an otherwise flat surface can be mapped out in detail, or domain boundaries can be imaged between substrate crystallites and between domains of ordered overlayer structures. The effects of non-ideal coverages of overlayers stand out easily as defects or interstitials, etc. Especially STM (and its derivative techniques like atomic force microscopy) has gained wide acceptance as a powerful tool for visualizing surface topography, as well as for mapping out electronic structure across a surface. 8.2.5 Complementary Techniques
A number of other techniques have been extensively applied to study surfaces in a variety of useful ways. The electronic structure of surfaces has been studied in great detail with the help of photoelectron spectroscopy (Plummer and Eberhardt, 1982). Two common well-established techniques for the analysis of surface composition are Auger electron spectroscopy (AES) (Briggs and Seah, 1983) and thermal desorption spectroscopy (TDS) (Petermann, 1972), mentioned earlier. Several kinds of optical spectroscopy are available to study electronic excitations and surface vibrations, in particular infrared absorption spectroscopy (IRAS) (Hoffmann,
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1983). For vibrational analysis, high-resolution electron energy loss spectroscopy (HREELS) (Ibach and Mills, 1982) and atomic scattering (Lahee et al., 1986) have also become particularly effective. 8.2.6 Theory Theory has made great progress in modelling surfaces on the atomic scale. Understandably, the most accurate results are obtained for the simplest surfaces, using the most sophisticated theories. These can give accurate electronic and geometrical structure. Conversely, less detailed simulations are available for the more complex surface structures, and for the description of more complex processes like diffusion, reactions, etc. Accurate results have been obtained for clean metal surfaces and atomic adsorbates thereon in terms of surface energies and relaxations, binding geometries and energies, as well as electronic structure and vibrational frequencies. Properties of clean surfaces are generally well accounted for, including reconstructions. Calculations for atomic absorption are now often quite successful, but the case of adatom-induced reconstructions is still in its infancy. Good results are also available for the smallest adsorbed molecules, such as carbon monoxide and acetylene (C 2 H 2 ) and a very few larger ones (e.g., C 6 H 6 ). However, the more complex structures, such as reconstructions and larger molecules, require more severe approximations in the calculations if any appreciable parameter variation is attempted, in particular to optimize surface geometries. Many theoretical methods have been devised for surface calculations. They include a wide variety of quantum chemical approaches based on using finite-size clusters to represent the extended surface. A series
of solid-state methods incorporating the two-dimensional periodicity of the surface have also been introduced. On the whole, it appears that the extended-surface approach is most successful, and several quantum chemical methods are being extended to include this feature. Several overviews are available in the literature (Louie and Cohen, 1984; Hamann, 1988; Sauer, 1989; Van Hove etal., 1989), and we shall here outline only major developments and methods. Among the quantum chemical approaches, one may first cite several semiempirical methods. The extended Hiickel theory (EHT), pioneered by Hoffmann, has been very effective in reproducing trends in geometrical and energetic surface properties. It has been implemented for extended periodic surfaces (Hoffmann, 1987; Hoffmann, 1988). EHT has been used to study a variety of molecular adsorption systems on metal surfaces, including straigth-chain and aromatic hydrocarbons. The self-consistent-field-Xa-scattered-wave (SCF-Xoc-SW) method (Johnson, 1966) has been extensively used to model the adsorption of carbon monoxide and small hydrocarbon molecules (e.g., ethylene) on metal surfaces. The HartreeFock-Slater (HFS) method (Baerends et al., 1973; Rosen et al., 1976) has been used to study, among other systems, carbon monoxide and atomic adsorption. A number of ab initio quantum chemical calculations have been applied to surfaces, including metal and semiconductor surfaces, as well as atomic and molecular adsorption. They employ either the molecular orbital (MO) or the valence bond (VB) schemes. The most commonly used ab inito method is the self-consistent-fieldHartree-Fock (SCF-HF) scheme with or without configuration interaction (CI) (Bagus, 1981). One CI approach incorpo-
8.2 Experimental and Theoretical Techniques
rates a method to embed a finite cluster in an extended surface (Whitten and Pakkanen, 1980). Worthy of mention is also the generalized valence bond (GVB) method (Hunt et al., 1972), which improves on some of the HF procedures. The numerous applications of these approaches have included the adsorption of atoms and small molecules on metal and semiconductor surfaces. An important development is the introduction of the density-functional formalism, which can be derived from first principles, and which enables more complex structures to be handled. A vast literature also exists that is based on solid-state methods which use the twodimensional periodicity of the surface. Early methods obtained only the electronic structure of a surface. Later methods also calculated the total energy of the surface. Other approaches produce only the cohesive energy due to the bonding in the surface. Among the many early methods applied to surfaces we mention a few prominent approaches that have remained in use. One method matches wave functions of the solid and of the vacuum using a Gaussian expansion of core orbitals and interstitial charge densities (Appelbaum and Hamann, 1978). Supercells containing slabs with two back-to-back surfaces have been used within a self-consistent pseudopotential formalism (Schliiter et al., 1975). A self-consistent local orbital method has been developed specifically for d-band metals (Smith et al., 1981). And a self-consistent numerical basis set method which generalized the discrete-variational LCAO (linear combination of atomic orbitals) method has been particularly successful (Wang and Freeman, 1979). Those approaches which are based on pseudopotentials allow the core electrons to be frozen and treated in a simplified and
495
efficient manner. Total-energy calculations have allowed optimization of the geometrical structure. Accuracy is combined with efficiency in the mixed-basis scheme, in which both plane waves and atomic or Gaussian orbitals are used (Louie et al., 1974). One very effective method to optimize a surface geometry uses the Hellmann-Feynman forces on the atoms to determine in which direction they need to be moved toward equilibrium (Ihm et al., 1981). Many of these schemes are based on the local-density-functional approximation for the exchange potential (Hohenberg and Kohn, 1964; Kohn and Sham, 1965). Tight-binding methods, in which semiempirical parameters describe the necessary interaction potentials, have been developed and have allowed the calculation of surface structures with greater complexity. Particularly successful have been applications to surface reconstructions of semiconductors (Chadi, 1979) and metals (Tomanek and Bennemann, 1985). Another direction which has proven quite profitable is represented by the effective-medium theory (EMT) and the embedded atom method (EAM). Both methods are often coupled with Monte Carlo and molecular dynamics techniques to explore not only energy minimization but also the time evolution of surfaces, such as with atomic diffusion along the surface or melting. They were designed for metallic surfaces, but extension of EAM to semiconductors is possible (Baskes et al., 1989). In EMT (Norskov and Lang, 1980; Norskov, 1982) the individual atoms are embedded in a matrix of homogeneous electron gas, and their binding energy can be calculated once and for all within the local-density approximation. The embedding energy can be obtained from first principles, or also by fitting to bulk materi-
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al properties. The method has been extended to molecular reactions at surfaces, for example to model the ammonia synthesis over iron surfaces (Stoltze and Norskov, 1985). The EAM approach (Daw and Baskes, 1984) generalizes EMT by viewing the cohesive energy of a metal as comprised of the embedding energy and electrostatic interactions. Atoms near a defect such as a surface are embedded into an electron gas of different profile than in the bulk. The interaction potential, which is fitted to bulk and surface properties, includes many-body effects, while the calculations are sufficiently fast that large numbers of independent atoms can be handled in molecular dynamics simulations.
8.3 Two-Dimensional Ordering and Nomenclature Deposition of adsorbates on a singlecrystal substrate can produce quite different two-dimensional periodicities than the clean surface has. And the clean surface may have a different two-dimensional periodicity than one would expect from simple truncation of the three-dimensional bulk lattice. We shall in this section first address the question of how ordering takes place at surfaces, with emphasis on the case of adsorbates. Then we shall introduce the nomenclature that is used in surface science to describe ordered surface structures, whether due to adsorbates or due to reconstructions. 8.3.1 Ordering Principles at Surfaces
A large number of ordered surface structures can be produced experimentally on single-crystal surfaces, especially with adsorbates (Ohtani etal., 1987). Ordering can manifest itself both as commensurate
and as incommensurate structures. There are also many disordered surfaces. For selected surfaces, order-order and order-disorder phase transitions have been explored in considerable detail both experimentally and theoretically (Van Hove et al., 1986a). We shall adopt the following definitions of the terms coverage and monolayer. The surface coverage will be unity when each two-dimensional surface unit cell of the unreconstructed substrate is occupied by one adsorbate (the adsorbate may be an atom or a molecule). A coverage of 1/2 per cell thus corresponds to filling every other equivalent adsorption site. The term monolayer will indicate a saturated single adsorbate layer with a thickness equal to the dimension of the adsorbate perpendicular to the surface. Thus, deposition after this coverage can only be achieved by starting a second monolayer growing on top of the first monolayer. The driving force for surface ordering originates, analogous to three-dimensional crystal formation, in the interactions between atoms, ions, or molecules in the surface region. The physical origin of the forces is of various types (covalent, ionic, van der Waals), and the spatial dependence of these interaction forces is often complex. For adsorbates, an important distinction must be made between adsorbatesubstrate and adsorbate-adsorbate interactions. The dominant adsorbate-substrate interaction is due to strong covalent or ionic chemical forces in the case of chemisorption, or two weak van der Waals forces in the case of physisorption. Adsorbate-adsorbate interactions may also be of different kinds: they may be strong covalent bonding interactions (as with dense metallic layers), weaker orbital-overlapping interactions or electrostatic interactions (e.g., dipole-dipole interactions), or
8.3 Two-Dimensional Ordering and Nomenclature
weak van der Waals interactions, etc. These are many-body interactions that may be attractive or repulsive depending on the system. Frequently, an adsorbate lattice is formed that is simply related to the substrate lattice. In the ordered case this yields commensurate superlattices. The most common of these are simple superlattices with one adsorbate per superlattice unit cell. They occur for adsorbate coverages of 1/4, 1/3, or 1/2 per cell, for example. An incommensurate relationship exists when there is no common periodicity between an overlayer and the substrate. Such a structure is dominated by adsorbate-adsorbate interactions rather than by adsorbate-substrate interactions. The classic example is that of rare-gas monolayers physisorbed (weakly adsorbed) on almost any substrate.
8.3.2 Nomenclature 8.3.2.1 Miller Indices Single-crystal surfaces are characterized by a set of Miller indices that indicate the particular crystallographic orientation of the surface plane relative to the bulk lattice. Thus, surfaces are labelled in the same way that atomic planes are labelled in X-ray crystallography. For example, a P t ( l l l ) surface exposes a hexagonally close-packed layer of atoms, given that platinum has a face-centered cubic bulk lattice. For reference, such a surface is often additionally labelled ( l x l ) , thus Pt(lll)-(1 x 1): this notation indicates that the surface is not reconstructed or otherwise modified into a periodicity different from that expected from simple truncation of the bulk lattice. Further examples are shown in Fig. 8-1.
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8.3.2.2 Superlattices: Wood and Matrix Notations Most surfaces exhibit a different two-dimensional periodicity than expected from the bulk lattice, as is most readily seen in diffraction patterns: often additional diffraction features appear which are indicative of a "superlattice". This corresponds to the formation of a new two-dimensional lattice on the surface, usually with some simple relationship to the expected "ideal" ( l x l ) lattice. For instance a layer of adsorbate atoms may occupy only every other equivalent adsorption site on the surface, in both surface dimensions. Such a lattice can be labelled (2x2): in both surface dimensions the repeat distance is doubled relative to the ideal substrate. In general, the (2 x 2) notation can take the form p(mxn) Ra° (where the prefix p is optional) or c(mxn)R(x°. This form is called the Wood notation (Wood, 1964). The prefixes p and c indicate "primitive" and "centered" lattices, respectively. (Note that this notation, as well as the matrix notation below, describes the translation lattice, and does not indicate the number of atoms in the basis.) Here the numbers m and n are two independent stretch factors in different surface directions. These numbers need not be integers: irrational values yield incommensurate lattices, while rational values, expressible as a ratio of integers, correspond to commensurate lattices. In addition, this stretched unit cell can be rotated by any angle a° about the surface normal. Thus, the Wood notation allows the ( l x l ) unit cell to be stretched and rotated; however, it conserves the angle between the two unit cell vectors in the plane of the surface, disallowing "sheared" unit cells. Various examples are shown in Fig. 8-2. For instance, the square c(2 x 2) lattice on f.c.c. and b.c.c. (100) surfaces
498
A
B
8 Crystal Surfaces
C
A
B
C
A
(a) fee (111)
(e)bcc(IOO)
A
B
A
B
A
(b)hcp(OOOI)
(c) bec (110)
(f) fee (110)
(g)bcc(lll)
(d) fee (100)
(h) fee (311)
Figure 8-1. Surface structures of low-Miller-index unreconstructed metal surfaces. In each panel a top view (above) and a side view (below) are shown. Dotted circles in the side views (e) to (h) indicate the unrelaxed ideal bulk positions, whereas the corresponding solid circles represent the relaxed positions in the surface (displacements are drawn to scale). For cases (a) to (d) the effect is too small to draw (see Sec. 8.4.3).
can also be described by the equivalent Wood notation Ql x y2)R45°. A more general notation is available for all unit cells, including those that are sheared, so that the superlattice unit cell can take on any shape, size and orientation. It is the matrix notation, defined as follows (Van Hove et al., 1986 a). We connect the unit cell vectors a' and b' of the superlattice to the unit cell vectors a and b of the substrate by the general relations: a! = mxl a + m12b b = m21 a + m22b
(8-1)
The coefficients m 1 m22 define the matrix 11 M = m,
m 12 m, m
21
m 12, (8-2)
22/
which serves to denote the superlattice. The ( l x l ) substrate lattice and the (2 x 2) superlattice are then denoted by the matrices 1 M= and 0 1 2 M = 0
2
8.4 Clean Surfaces
p(2x2) (v/3xy3)R30° fcc(111), hcp(0001)
p(2x1) bcc(11O)
p(2x2) c (2x2) fcc(001)t bcc(OOI)
p(2xi)
c (2x2) fee (110)
respectively. Similarly, the c (2 x 2) lattice on f.c.c. and b.c.c. (100) can be denoted by the matrix M=
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1 -1 1 1
Very useful is the fact that the determinant of the matrix M is simply the ratio of the unit cell areas of the superlattice and the substrate lattice.
8.4 Clean Surfaces Once a clean surface has been prepared, it is often found to have the two-dimensional periodicity which one would expect from simple ideal truncation of the bulk lattice parallel to the surface plane. However, there are many exceptions: they are called reconstructions and we shall define them to be those clean structures that do not have a (1 x 1) unit cell: they thus have a two-dimensional periodicity which is not expected from the bulk structure. In all cases, whether reconstructed or not, there is the possibility that bond
Figure 8-2. Commonly occurring superlattices on low-Miller-index metal surfaces, including their Wood notation.
lengths and interlayer spacings near the surface can differ from those in the bulk. 8.4.1 Bulk-Like Lattice Termination
A fair number of clean surfaces exhibit a bulk termination (with perhaps some minor relaxations which we will discuss in Sec. 8.4.3) (Ohtani etal., 1987). They are then denoted as having a ( l x l ) surface lattice. This occurs most prominently with many pure metal surfaces that have low Miller indices, such as f.c.c.(Ill), f.c.c. (100), f.c.c.(110), b.c.c.(110), b.c.c.(Ill), h.c.p.(OOOl) and h.c.p.(lOTO). Among the few oxide surfaces which have been studied, the low-index surfaces derived from the bulk NaCl lattice also exhibit a (1 x 1) lattice, e.g. NiO(100). A number of alloy surfaces (again, few have been studied) have also been found to have the bulk termination. Some semiconductor compounds also have the ( l x l ) termination, such as the (110) surface of many III-V and II-VI compounds (GaAs, AlAs, A1P, GaP, GaSb, InAs, InP, InSb, CdTe, ZnS,
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ZnSe and ZnTe), although they can involve large atomic displacements. Surface termination is more ambiguous in the case of lattices with more than one atom per primitive unit cell, such as the h.c.p. metals and any compounds. Many of their crystallographic surface orientations imply a choice between two or more inequivalent terminations, depending on the exact depth at which the surface is cut. The h.c.p. (lOTO) surface offers this choice; the choice is resolved in favor of the least corrugated of the two possible bulk truncations, at least for Ti(10T0) (Mischenko and Watson, 1989), Co (1120) (Welz et al., 1983) and Re(10T0) (Davis and Zehner, 1980). As an example for compounds, NiAl (100) consists of alternating layers of pure Ni and pure Al: so the surface could either have pure Ni or pure Al in its outermost layer [a non-bulk-like mixture and a reconstruction are excluded by the observation of a ( l x l ) unit cell]. Nature appears to favor an Al termination (Davis and Noonan, 1988). By contrast, it appears that the (111) surface of the same NiAl compound adopts both terminations simultaneously on different parts of the surface (Noonan and Davis, 1988). The different possible terminations are expectsd to have different surface energies in general, and therefore one will be favored over the others, but not necessarily to the complete exclusion of the other at a finite temperature. There are cases where the bulk material is so close to a phase transition that one might expect the surface to have undergone the transition before the bulk material does. This possibility was explored with Co, which undergoes a phase transition at 723 K between the low-temperature h.c.p. phase and the high-temperature f.c.c. phase. At 300 K the surface was found to be h.c.p.-like, and at 730 K it was found to
be f.c.c.-like, i.e. no deviation from the bulk phase was found (Lee et al., 1978). By contrast, the phenomenon of surface premelting has been well documented, at least for certain Pb surfaces, particularly the (110) surface of this f.c.c. metal (van der Veen et al., 1988). Premelting within a few outermost surface layers is observed already some 100 K below the bulk melting temperature of about 600 K. 8.4.2 Stepped Surfaces It is found that on well-annealed clean f.c.c. and b.c.c. metal surfaces, steps between adjacent flat terraces are of monoatomic height (Somorjai, 1981). This is partly due to the fact that on ideal f.c.c. and b.c.c. surfaces, successive steps are structurally equivalent and multiheight steps are less favorable. On h.c.p. metal surfaces, however, steps are often of double height. The difference is that on most h.c.p. surfaces mono-atomic height steps alternate among two inequivalent structures and can compose a more favorable double-height step. Similarly, steps on many semiconductor surfaces have a twoatom height. Little is known about step structures at other surfaces. 8.4.3 Relaxations Surface atoms have a highly asymmetrical environment: they have neighbors toward the bulk and in the surface plane, but none outside the surface. This anisotropic environment forces the atoms into new equilibrium positions, relative to the bulk, as illustrated in Fig. 8-1. For clean unreconstructed surfaces, there is generally a contraction of bond lengths between atoms in the top layer and in the second layer under the surface, relative to the bond length in the bulk: the contraction is on the order of a few percent (MacLaren
8.4 Clean Surfaces
etal., 1987; Van Hove etal., 1989). This relaxation in the topmost interlayer spacing is larger the more open (or rougher) is the surface, i.e. the fewer neighbors the surface atom has (Jona and Marcus, 1988). The closest packed surfaces, such as f.c.c.(lll) and f.c.c.(lOO), show almost no relaxation; there may even be a very slight expansion for Pd and P t ( l l l ) . Relatively large inward relaxations occur by contrast at surfaces like f.c.c. (110), with interlayer spacings contracted by about 10%. The contractions are material dependent, Pb(110) showing a particularly large interlayer spacing contraction of 16%, corresponding to a bond length contraction of 3.7% (Frenken et al., 1987). It should be recalled that interatomic distances for diatomic molecules are much shorter than bond distances in solids where atoms have many more nearest neighbors. The surface atoms may be viewed as having a chemical environment that is between a diatomic molecule and the bulk of the solid, and the bond lengths indeed fit in between those limits. They are closer to the bulk value when the number of nearest neighbors is closer to that in the bulk, and vice versa. The trends found at surfaces satisfy the well-known behavior of bond lengths with number of neighbors in other situations, as described already by Pauling (Pauling, 1960). Relaxations of interlayer spacings occur also deeper than the second layer (Jona and Marcus, 1988). The amplitudes of these relaxations decay approximately exponentially with depth. The decay appears to be related more to physical depth than to interlayer spacing, since higher-Millerindex surfaces with smaller interlayer spacings show relaxations that propagate more layers down, but not deeper in distance. The deeper relaxations are by no means all contractions. At least in metals, it is more
501
common to have alternating contractions and expansions in the interlayer spacings. These oscillations have a wavelength that can be close to twice the interlayer spacing (but are not systematically related to this spacing). Thus, one could have, in penetrating the surface, first a contraction, then an expansion, followed by another expansion, and then again a contraction. At stepped surfaces the atoms at the step edges are more exposed and are expected to exhibit relatively large relaxations, thereby causing a partial smoothing at these rough surface sites. The relaxations at the step edges are similar to those found at f.c.c. (110) surfaces, for example (Jona and Marcus, 1988). Relaxations parallel to the surface are also expected and observed for atoms at step edges (Jona and Marcus, 1988): they tend to relax sideways toward the upper terrace of which they are a part. The amplitude of the displacement is probably similar to the relaxations observed perpendicular to rough surfaces. Such sideways relaxations can also propagate deeper below the surface, again with an exponential decay and an oscillatory character. Semiconductor surfaces often present larger relaxation effects than metal surfaces, because there is more room for bond angle changes in the less close-packed semiconductors (Duke, 1988; Duke, 1991). The bond lengths also appear to change more than they do in metals. For example, in GaAs (110) and a number of similar unreconstructed surfaces of III-V and II-VI compounds, large rotational relaxations occur, cf. Fig. 8-3. Whereas the bulk has tetrahedral angles of 109.5°, some of the bond angles at surface atoms are reduced to 90 ±4°, while others are increased to 120 + 4°. These changes vary from layer to layer, and decay to the bulk value within a few atomic layers. In these examples, bond
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Figure 8-3. Perspective side view of the surface structure of GaAs (110): the bonds between the uppermost Ga and As atoms have rotated from being parallel to the surface to a tilt angle of about 27° from the surface plane.
lengths change by up to 9%, but more typically by about 5%; both contractions and elongations occur in the same structure. Such effects are due to the rehybridization of atomic orbitals around the surface atoms. At (110) surfaces of III-V compounds, the group III element (anion) rehybridizes toward sp2 and a planar neighborhood with 120° bond angles, whereas the group V element (cation) adopts a distorted p 3 hybridization that favors 90° bond angles. The metastable unreconstructed Si(lll)-(1 x 1) surface (which is obtained by laser annealing and thus rapid quenching) exhibits similar relaxations: bond angles ranging from 105 to 114°, and bond lengths changed by - 2 . 5 % and + 5 . 1 % between first and second layers, and between second and third atomic layers, respectively (Jona et al., 1986). Relaxations are observed at the surface of certain alloys, in which the surface layer is composed of more than one type of atom. The different atoms can relax by different amounts perpendicular to the surface. A good example is provided by NiAl(llO) (Davis and Noonan, 1988). All bulk (110) layers are equal mixtures of
coplanar Ni and Al atoms. The surface layer is buckled by about 0.02 nm, due to the fact that the larger Al atoms relax outward by 0.01 nm, while the Ni atoms relax inward by 0.01 nm. Not enough information is available for insulating surfaces, like metal oxides, to establish trends. But it is clear that the (100) surfaces of NiO, MgO, CaO and CoO exhibit little, if any, relaxation (MacLaren e t a l , 1987). The relatively open CoO (111) surface appears to have a 17% first-spacing contraction between the outermost O atoms and the following Co layer (Ignatiev et al., 1977), while the (100) surface of SrTiO3 shows similar relaxations, in addition to a buckling of the topmost compound layer, two different surface terminations being observed (Bickel etal., 1989). 8.4.4 Reconstruction
Among the clean metal surfaces, nearly a dozen are known to reconstruct. Over 40 clean semiconductor reconstructions have been reported. Numerous reconstructions have also been found for oxides and other compounds. Depending on preparation methods, some of these surfaces can present different superlattices, some of which are metastable. Thus, Si (111) reconstructs readily into a (7 x 7) structure, but can also be stabilized into a (2x1) structure, a (yfe x ^/3) R30° structure and even an unreconstructed ( l x l ) form. Similarly, Ir(100) normally reconstructs into a (1 x 5) lattice, but can be prepared in a metastable ( l x l ) structure. Several types of clean-surface reconstruction can be distinguished. First, one finds displacive reconstructions (King, 1989), in which atoms are displaced slightly from their ideal bulk-like positions in different directions which break the ideal
8.4 Clean Surfaces
periodicity and create a superlattice. Generally, no bonds are broken or created in this type of reconstruction, but bond lengths and angles are changed. Mo and W(100) are good examples. The case of W(100) is easily visualized, see Fig. 8-4: opposite lateral displacements by about 0.03 nm occur on alternating atoms, giving rise to a c (2 x 2) periodicity. Next are the missing-row reconstructions, exemplified by Ir, Pt and Au(110) (Sowa et al., 1988 b). In this case, rows of atoms are missing from the ideally-truncated substrate, cf. Fig. 8-5. The resulting troughs consist of 3-atom wide inclined facets (composed of adjacent white, gray and black rows of atoms in Fig. 8-5), which have the (111) crystallographic orientation and are therefore more closepacked than the ideal (110) surface. The most common missing-row reconstruction produces a (2x1) unit cell with facets 2 atoms wide. Facets of 3-atom width can also form, in a (3 x 1) unit cell, and wider facets are possible, but rarely observed. In fact, it is probable that these wider facets are impurity-stabilized. Relaxations occur in addition to the reconstruction, as with unreconstructed surfaces. Another type of reconstruction seen on metal surfaces is the formation of a closerpacked top layer. This can be explained by the tendency for bond lengths to decrease as the bonding coordination decreases. Such a reconstruction occurs for Ir, Pt and Au(100), as well as A u ( l l l ) (Van Hove et al., 1981). In these cases, the interatomic distance within the topmost layer shrinks by a few percent parallel to the surface. It then becomes more favorable for this layer to collapse into a denser layer that is nearly hexagonally close packed rather than maintain the square or perfectly hexagonal lattice of the underlying layers. The dense monolayer does not fit well on the underly-
503
I>-A
W(100)
-c(2x2)
Figure 8-4. Top view of the reconstructed W(100)c (2 x 2) surface. Light circles represent top-layer W atoms displaced from a square array into zigzag rows. Gray and black circles show atoms in the second and third layers, respectively.
ing layers and can adopt various positions and orientations, depending on the metal. In-plane distortion and out-of-plane buckling are expected and found in such layers. Semiconductors often exhibit bond breaking and creation in one or more surface layers, relative to the ideal truncation (Duke, 1988; Duke, 1991). This effect is due to the directionality of the bonding in these materials. The "dangling" bonds broken by the creation of the surface are
fee (110) - (1 x 2) missing row
Figure 8-5. Perspective side view of a f.c.c. (110) surface with a (1 x 2) missing-row reconstruction, looking along the channels due to the missing rows.
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8 Crystal Surfaces
energetically unfavorable. An important driving mechanism for semiconductor reconstruction is the minimization of the number of such dangling bonds. This is accompanied by more or less drastic rearrangements of the surface lattice. A relatively simple case occurs with Si (100), where atom pairing satisfies half of the dangling bonds, forming a (2 x 1) superlattice, cf. Fig. 8-6. A more extensive rearrangement is found in the (2x1) reconstruction of Si (111) and diamond C ( l l l ) : here atoms bond in zigzag chains along the surface, while the 6-membered rings of the bulk are replaced by 5- and 7-membered rings next to the surface, cf. Fig. 8-7.
Si (100) - (2 x 1) buckled dimer
Figure 8-6. Perspective side view of the Si (100)(2x1) reconstruction, showing tilted dimers.
Si (111) - (2 x 1) n-bonded chain
Figure 8-7. Perspective side view of the Si (111)(2x1) reconstruction, showing 7-atom rings that incorporate zigzag chains, and 5-atom rings.
Another type of reconstruction is the expulsion of a fraction of the surface atoms. This is observed with the recently discovered metastable Si(lll)-( v / 3 x > /3)R30 o reconstruction (Fan etal., 1989). In this structure, one third of the Si atoms in the top half of the topmost bilayer is missing. The remaining atoms relax around the resulting vacancies in a manner that is reminiscent of the relaxations of III-V and I I VI compounds. The topmost bilayer is nearly flattened, giving bond angles within the layer of 119° and bond angles to the second bilayer of close to 90°, rather than the original tetrahedral angle of 109.5°. Also, bond lengths within the top bilayer are considerably expanded to 0.262 nm (from the bulk value of 0.235 nm). This reflects a rehybridization of the top-layer Si atoms towards sp2 from sp 3 . A closely-related type of reconstruction results from changes in surface stoichiometry. This happens in GaAs and GaP(lll)(2x2), for example (Tong etal., 1984; Tong etal., 1985): one quarter of the Ga atoms in the surface is missing, allowing a more optimal rehybridization of the p 3 type about the remaining atoms, as illustrated in Fig. 8-8. These remaining atoms relax without further bond breaking or creation to generate bond angles similar to those of GaAs and GaP(llO). A converse reconstruction mechanism is the addition of atoms on top of the ideal truncation. The Ge(lll)-c(8 x 2) reconstruction is of this type (Tong et al., 1990): two so-called T 4 adatoms per unit cell are located over 3-fold symmetrical shallow hollow sites, in contact with 3 Ge atoms in the top half of the topmost bilayer and in a top position over 1 Ge atom in the bottom half of the topmost bilayer (this top position with a total of 4 bonding partners gives the T 4 notation). The T 4 site is detailed in Fig. 8-9. This again removes a
8.4 Clean Surfaces GaAs/GaP(111)-(2x2)
Figure 8-8. Perspective view of the GaAs or GaP(lll)-(2 x 2) reconstruction. Darker atoms and bonds lie in the reconstructed top bilayer. Missing Ga positions lie at the centers of the 3-pointed stars consisting of three arcs of five atoms each. Ga and As atoms are drawn large and small, respectively. Si(111)T4 cluster
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stretched bond to the atom directly below it (0.253 nm, compared to the bulk value of 0.235 nm), which is itself depressed by about 0.065 nm, compressing the Si-Si bond from the first to the second bilayer down to 0.215 nm. This compression propagates into the second bilayer, which is flattened to some extent. Other bond lengths around the T 4 site appear to be close to the bulk value. The Si(lll)-(7x7) surface is the most complex reconstruction solved to date (Takayanagi, 1985). It incorporates many of the abovementioned effects, and some additional ones. One half of the unit cell presents a stacking fault between the first and second bilayers, as if the top bilayer had been rotated by 180° about the surface normal. This stacking fault joins the nonfaulted half of the unit cell at a seam that consists of paired atoms. Six such seams meet at large and deep holes (one hole per unit cell), which expose the second bilayer. The combination of adatoms, dimers and stacking fault reduces the number of dangling bonds per (7 x 7) unit cell from 49 on the ideal unreconstructed surface to 19 on the reconstructed surface.
8.4.5 Surface Segregation Figure 8-9. Geometry of a T 4 "adatom" site, in perspective view: the adatom is drawn dark, sitting on top of a central atom, dark gray, around which four atoms are tetrahedrally positioned (three in the surface plane, light gray and one in the second bilayer, white).
good fraction of the dangling bonds. The Si(lll)-(7 x 7) reconstruction includes the same T 4 adatom site, of which there are 12 in each (7x7) unit cell (Tong et al., 1988). The adatom has a highly non-tetrahedral bonding environment. It includes a
A number of bulk compounds like oxides, carbides, sulfides and semiconductors maintain their bulk composition at the surface. Examples include NiO(100), TaC(100), GaAs (110) and other related compounds. Often the bulk lattice is retained as well, but with possible changes in bond lengths and bond angles. There are, however, numerous exceptions that exhibit a deviating surface composition. An example is the case of the (111) faces of GaAs, GaP, and other such semiconductors, where a deficiency of Ga
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8 Crystal Surfaces
leads to a (2 x 2) reconstruction due to Ga vacancies (see Sec. 8.4.4). The clean alloys and intermetallic phases fall into two main categories: those for which the bulk alloy is ordered and those for which it is disordered. It appears that the surface structures of ordered bulk alloys are generally also ordered and maintain the bulk concentration. NiAl (Davis and Noonan, 1988) and Ni3Al (Sondericker etal., 1986) in particular have been extensively studied and found to satisfy this principle. With disordered bulk alloys, the surface is most often also disordered, but surface segregation can be very marked and can be strongly layer-dependent, with the possibility of an oscillating layer-by-layer concentration. For instance, different crystallographic faces of the Pt^-Ni^^ bulk alloy have been found to exhibit very different segregation behavior as well as a strong layer dependence. Thus, the (111) surface with a bulk Pt concentration of 50% has first-, second- and third-layer Pt concentrations of 88%, 9% and 65%, respectively (Gauthier, 1988). Other alloys, exemplified by Cu-rich CuAl (Baird et al., 1986), are disordered in the bulk, but ordered at some faces for certain bulk compositions. Thus the (111) face of the f.c.c. ocCu84Al16 exhibits a (y/3 x >v/3)R30o surface periodicity [relative to the ( l x l ) surface lattice of pure Cu (111)]. The other low-Miller-index faces of this alloy do not order.
8.5 Adsorbate-Covered Surfaces A large number of atomic and molecular adsorbates have been studied on singlecrystal surfaces over the last two decades (Ohtani et al., 1987). Very different structures are found when physisorption is
compared with chemisorption, or when comparing atomic with molecular adsorption, or when mixing in a second type of adsorbate. Such differences will be addressed in this section. We shall also discuss multilayer growth, relaxations, reconstructions, compound formation and surface segregation of surfaces upon adsorption. 8.5.1 Physisorption
At low enough temperatures most gasphase species will physisorb on any surface. Particularly with inert gases and with saturated hydrocarbons, physisorption is commonplace and stable on many types of substrate. These substrates include metals, semiconductors and insulators as well as inert surfaces such as the graphite basal plane. Also, more reactive species such as O 2 , CO, N 2 , NO and S2 have been physisorbed stably on the graphite basal plane. Physisorption usually involves chemically stable species. One can therefore examine large parts of the phase diagrams of these adsorption systems as a function of coverage and temperature. Many phases have been observed in physisorption, and new classes of phases continue to be discovered. There are commensurate and incommensurate phases, disordered latticegas and fluid or liquid phases. There are out-of-phase domain structures, including striped-domain phases, pinwheel and herringbone structures, and modulated hexatic reentrant fluid phases, among others (Nielsen et al., 1980). The simpler among the observed LEED patterns for physisorbed species can often be easily interpreted in terms of structural models. The known van der Waals sizes of the species (as defined by van der Waals radii about each atom) lead to satisfactory structures which are more or less close-
8.5 Adsorbate-Covered Surfaces
packed. This is especially straightforward with inert gases. Thus, with Xe in an incommensurate overlayer on Ag(lll), the Xe-Xe and Xe-Ag interatomic distances correspond closely to van der Waals distances (Stoner et al., 1978). With molecules, the best structural models usually involve flat-lying species, which are arranged in a close-packed superlattice (Firment and Somorjai, 1977). Physisorption allows the formation of multilayers, which normally grow with their own lattice constant on any substrate. For example, xenon films exposing a Xe(lll) surface have been grown on an Ir(100) substrate and analyzed by LEED to show that the bulk f.c.c. Xe structure is maintained (Ignatiev et al., 1971). 8.5.2 Atomic Chemisorption Sites and Bond Lengths
Thousands of atomic chemisorption systems have been studied in terms of their ordering characteristics (Ohtani et al., 1987). It is very frequent that chemisorbed atoms order well on surfaces, particularly at specific coverages like 0.25, 0.5, 0.75, etc. per cell, where regular superlattices can develop. In many cases, order-disorder transitions are observed as the temperature is raised. We first address the issue of the adsorption site of chemisorbed atoms. The simple atomic adsorption structures on metal surfaces are generally characterized by the occupancy of high-coordination sites, as illustrated in Fig. 8-10. Thus, Na, S, and Cl overwhelmingly adsorb over "hollows" of the metal surface, bonding to as many metal atoms as possible (Van Hove et al., 1989). The situation is slightly more complicated with the smaller adsorbates, such as H, C, N, and O, on metal surfaces. And all
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adsorbates appear to behave in a more complex manner on semiconductor surfaces. By contrast, little crystallography has been accomplished on atomic adsorption on other types of substrates, such as insulating compounds and alloys. With the more complex atomic adsorption, there still remains a preference for high-coordination sites. However, the atoms often penetrate deeper within or even below the first substrate layer. The penetration can be interstitial (as occurs with small atoms on metals) or substitutional (as is relatively more frequent on semiconductors and compounds). In either case the surface can reconstruct as a result, especially at higher coverages. For instance, a monolayer of N penetrates into interstitial octahedral sites between the first two layers of Ti(0001) with minimal distortion of the Ti lattice (Shih et al., 1976). Both C and N burrow themselves within the hollow sites of the Ni(100) surfaces so as to be almost coplanar with the topmost Ni atoms (Onuferko et al., 1979; Wenzel et al., 1988). The nearest Ni atoms are also pushed sideways by perhaps 0.04 nm, a good example of adsorbate-induced reconstruction (see Sec. 8.5). Perhaps the only structure of an adatom at steps solved so far is that of O on Cu(410), analyzed with X-ray photoelectron diffraction (Thompson and Fadley, 1984). The Cu(410) surface consists of (100) terraces, 3 atoms wide, on which the O adatoms can arrange themselves in a c(2 x 2) array at hollow sites, cf. Fig. 8-11. Oxygen atoms bond within the step edge between adjacent Cu step atoms: the bonding arrangement is just like the 4-fold hollow site, except that one of the four surrounding Cu atoms is missing. Atomic adsorption on semiconductors has been a controversial matter: many contradictory structural solutions have been
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(a) fcc(11l), hcp(OOOI): hollow site
(d)bcc(100): hollow site
(b)bcc(UO): 3-fold site
(e)fcc(110); center long-and-short bridge sites
(c) fee(100)' hollow site
(f) hep (0001) : underlayer
Figure 8-10. Top and side views (above and below in each panel) of common atomic adsorption geometries on various metal surfaces. Adatoms are shown as hatched circles. The dotted circles represent metal atoms before adsorption, while the corresponding slightly displaced solid circles indicate their relaxed positions after adsorption (displacements are drawn to scale). (100) terrace
(010) step
Cu (410) - (1 x 1) - 20
Figure 8-11. Perspective view of oxygen (small black circles) adsorbed on Cu(410), the step edges of which are marked as shaded atoms. The coverage is near 1/2 per unit cell in the terrace.
proposed and it will still take time before the overall picture becomes clear. However, one discerns three emerging major trends in the adsorption sites: low-coordination adatoms, high-coordination adatoms and substitutional atoms. Some atoms adsorbed on the (111) face of C (diamond), Si and Ge cap the dangling bonds of the ideally truncated surface. For instance, H, Cl, Br and I choose capping sites on Si (111), forming bonds
8.5 Adsorbate-Covered Surfaces
through single coordination to Si atoms (MacLaren et al., 1987). These adatoms in effect continue the bulk Si lattice outward, removing any clean-surface reconstruction when the adatom coverage is large enough. The second trend is illustrated by several adsorbed atoms on Si (111) and G e ( l l l ) which at low coverage appear to prefer the T 4 adatom site (the same that Si and Ge adatoms occupy in the clean-surface reconstructions, cf. Fig. 8-9). The adatoms are thereby bonded to 4 substrate atoms of the top bilayer. Examples are Pb on G e ( l l l ) (Huang et al., 1989), and both Al (Huang etal., 1990 b) and Ga on Si (111) (Kawazu and Sakama, 1988). It appears from these results that the larger adatoms induce larger distortions in the substrate. The third trend involves substitutional penetration of the adsorbate into the substrate. One example is boron on Si (111): instead of becoming a T 4 adatom, B interchanges its position with the Si atom immediately below the T 4 site, i.e. with the central atom in Fig. 8-9 (this substitutional site is called B 5 site) (Huang et al., 1990a). Another example is Al on GaAs(llO), in which Al substitutionally replaces Ga atoms, largely retaining the relaxations of clean GaAs(llO) (Kahn etal., 1981). Next we consider bond lengths between adsorbate and substrate atoms. The observed bond lengths generally fall well within 0.01 nm of corresponding bond lengths measured in bulk compounds and molecules. In a few cases the accuracy is sufficient to detect chemically significant variations in bond lengths. As a dramatic example, when the surface coverage of Cs atoms is varied from 1/3 to 2/3 per cell on Ag(lll), the Ag-Cs bond length changes from 0.320 to 0.350 nm (Lamble etal., 1988). In this case the charging state of the adsorbate changes with coverage (as observed through work function changes),
509
with a concomitant effect on bond lengths. This also illustrates an expected effect of mutual interactions between adsorbates: the denser the adsorbate layer, the weaker the individual adatom-substrate bonds. 8.5.3 Atomic Multilayers Atomic multilayer growth has been studied most frequently for metal deposition on metal surfaces, and for semiconductor or metal deposition on semiconductors (van der Merwe, 1984; Vook, 1982; Ludeke, 1984). Also, the growth of oxides and other compounds has been studied, but rarely in structural detail. Two aspects are of particular interest: (1) the growth mode, whether layer-by-layer and/or epitaxial or as three-dimensional crystallites; and (2) the interface structure between the substrate and the growing film. For metals, the growth mode tends to attract the most attention, while the interface structure is of particular interest for growth on semiconductors. 8.5.3.1 Metallic Adsorption on Metals More than 400 surface structures of metal layers deposited on metal surfaces have been reported so far (Ohtani et al., 1987). At low coverages, most of the metallic adsorbates form commensurate ordered overlayers: the overlayer unit cells are closely related to the substrate unit cells. Furthermore, in many cases a (1 x 1) LEED pattern is observed. This suggests that these adsorbed metal atoms attract each other to form two-dimensional closepacked islands. On the other hand, a disordered LEED pattern is observed when the adsorbed metal atoms repel each other. This is found for example in the case of alkali metal adsorption on a transition metal, since the charged adatoms undergo repulsive interactions.
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8 Crystal Surfaces
Some metals undergo layer-by-layer growth, while others form three-dimensional crystallites ("balls"). Many cases fall between these two extremes. Comparison of the surface tension of the adsorbate metal and of the substrate metal has failed to explain these phenomena, and up to now there is no simple rule to predict which metal film growth mechanism applies in any particular case. The limit of layer-by-layer growth has been studied in structural detail, thanks to the frequent formation of simple ( l x l ) overlayer unit cells. Striking is the growth of metastable films with lattices that are not favored in the bulk. Fe grown on Ni(100) (Lu etal., 1989) and Cu(100) (Clarke et al., 1987) has received considerable attention: the Fe film can be made to grow with an f.c.c.-like lattice (which continues the f.c.c. lattice of Cu), rather than with its b.c.c. bulk lattice. At room temperature, a (1 x 1) pattern is formed, which is probably accompanied by some interdiffusion between the substrate and overlayer. The spacing between overlayers varies both from layer to layer and, for a given layer, as the film thickness grows, so that the Fe film does not adopt a perfectly cubic lattice; but the Fe is 12-fold coordinated, as in the f.c.c. lattice. Such effects are particularly interesting in view of the magnetic properties of thin metallic films, which appear to depend strongly on the growth geometry. When a metal undergoes a ( l x l ) epitaxial growth despite a substrate lattice constant that differs from its own bulk lattice constant, the overlayer metal can be considerably strained. This is the case with the metastable Fe films grown on Ni and Cu(100). Therefore, the epitaxial growth must at some point be accompanied by a lattice constant change. The change can occur after only a few layers or after many
layers, depending on the combination of metals. Such a change is probably accompanied by dislocations occurring within a dozen or so layers from the interface. 8.5.3.2 Multilayer Growth on Semiconductors
Multilayer growth on semiconductor substrates is of great importance to the semiconductor industry: it is highly relevant to the formation and electrical properties of semiconductor-metal contacts, of semiconductor-semiconductor heterojunctions and of "superlattices" (here understood to mean the stacking of thin films of alternating composition). Nevertheless, relatively little structural information on the subnanometer level is available. On a more qualitative level, many of the features described above for metal-metal interfaces are thought to apply here as well (Ludeke, 1984). One reason for the scarcity of structural information of such interfaces is the difficulty in studying deeply buried interfaces. Even with interfaces buried only a few atomic layers below a solid-vacuum surface, few experimental techniques are capable of sampling the buried structure. The technique which has been most successful in this respect has been high-energy ion scattering (HEIS), while LEED, SEXAFS and LEIS have also contributed by studying shallow buried interfaces. A few interfaces between Si and a metal silicide have been investigated in detail. Examples are the Si(11 l)-NiSi 2 (111) interface (Vlieg et al., 1986) and the similar interface produced with Co instead of Ni (Fischer etal., 1987). In both cases, the two materials match very closely in lattice constant parallel to the interface, forming a (1 x 1) surface lattice. Despite the identical lattices of NiSi2 and CoSi2, the bond-
8.5 Adsorbate-Covered Surfaces Rh(111)-(2x2)-C2H3
Figure 8-12. Perspective view of the (2 x 2) ordered structure of ethylidyne (CCH3) on Rh(lll). Hydrogen and carbon atoms are drawn as small and medium-size circles.
ing arrangement across the silicon-silicide interface is topologically slightly different between the two cases. 8.5.4 Molecular Adsorption Over 400 ordered LEED patterns have been reported for the adsorption of molecules (Ohtani et ah, 1987). By far the most frequently studied substrates are metals. Platinum substrates have been most extensively used, due no doubt to their importance in heterogeneous catalysis. The most common adsorbates are C 2 H 2 (acetylene), C 2 H 4 (ethylene), C 6 H 6 (benzene), C 2 H 6 (ethane), HCOOH (formic acid), and CH 3 OH (methanol). Ordered LEED patterns for organic adsorption are frequent at lower temperatures. They can often be interpreted in terms of close-packed layers of molecules, consistent with known van der Waals sizes and shapes. These ordered structures usually are commensurate with the substrate lattice, indicating strong chemisorption in preferred sites. It appears that many hydrocarbons lie flat on the surface, using
511
unsaturated 7i-orbitals to bond to the surface. By contrast, non-hydrocarbon molecules form patterns that indicate a variety of bonding orientations. Thus CO and NO are found to strongly prefer an upright orientation. However, upon heating, unsaturated hydrocarbon adsorbates evolve hydrogen and new species may be formed which bond through the missing hydrogen positions, often in upright positions. An example is ethylidyne, CCH 3 , which can be formed from ethylene, C 2 H 4 , upon heating. Ethylidyne has the ethane geometry, but three hydrogens at one end are replaced by three substrate atoms, cf. Fig. 8.12. 8.5.4.1 Molecular Adsorption Sites and Ordering
When the adsorbate-substrate bond is strong and localized, the molecule presents clear preferences for particular adsorption sites and it orders well. Thus, ethylidyne (CCH3) bonds through one carbon atom to a three-fold coordinated hollow site on many f.c.c. (Ill) surfaces, and typically orders as a (2 x 2) overlayer (Koestner et ah, 1982; Koestner etah, 1983), cf. Fig. 8-12. When the molecular species is large and bonds to many metal atoms simultaneously, as is the case with benzene lying flat on a surface, there is less preference for particular sites, which then depend on the metal and can easily be affected by coadsorbed species (e.g., acceptors like CO). For instance, benzene will shift its center from a bridge site to a hollow site when coadsorbed with CO on R h ( l l l ) . Ordering is relatively weak under such conditions. Thus, benzene does not order at room temperature on Pd and P t ( l l l ) surfaces, and only weakly on R h ( l l l ) . (But coadsorption with CO produces stable ordering through strong interactions, see Sec. 8.5.5.)
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In the case of weaker chemisorption, such as when CO or NO adsorb without dissociation, there is also relatively little site preference and ordering is less pronounced as well: such molecules choose sites that depend on the metal and on the coverage, as well as on coadsorbates, while low order-disorder transition temperatures are found.
metal and the crystallographic face: it is a bridge site on Pd(100) (cf. Fig. 8-14) and a 3-fold hollow site on P d ( l l l ) (cf. Fig. 815). At higher coverages the coordination generally increases, towards two-fold bridge sites and three-fold hollow sites (but apparently never four-fold hollow sites). Pd( 100)-(2/2x/2)R45°-2CO
8.5.4.2 CO and NO Adsorption Detailed structural studies of adsorbed carbon monoxide and nitric oxide have been performed for a dozen surface structures, primarily on close-packed metal surfaces (Ohtani etal., 1987; Ohtani etal., 1988 b). They have confirmed the site assignments based on vibrational frequencies, as originally derived for metal-carbonyl and similar complexes (Albert and Yates, 1987). On many metals, CO prefers low-coordination sites at low coverages, e.g. linear coordination at top sites for CO on R h ( l l l ) , cf. Fig. 8-13. However, the low-coverage site depends strongly on the
)-(/3x/3)R30°-CO
Figure 8-13. Perspective view of carbon monoxide on Rh(lll), adsorbed in top sites (linear metal-C-O bonding). CO molecules are perpendicular to the surface.
Figure 8-14. Perspective view of carbon monoxide on Pd(100), adsorbed in bridge sites (2-fold coordinated metal-C-O bonding). CO molecules are perpendicular to the surface, with two possible bridge azimuths.
Pd(111)-(/3x/3)R30'-CO
Figure 8-15. Perspective view of carbon monoxide on Pd(l 11), adsorbed in hollow sites (3-fold coordinated metal-C-O bonding). CO molecules are perpendicular to the surface.
8.5 Adsorbate-Covered Surfaces
The metal-C bond length has been found to increase strongly with coordination, and the C-O bond length increases slightly at the same time (Ohtani e t a l , 1988 b). This is again in agreement with the case of metal-carbonyl complexes, and confirms the C-O bond weakening implied by the decreasing vibration frequency. On some metals, CO is more prone to dissociation and one might expect to see that in the adsorption structure. (We do not discuss here the complete dissociation into separate C and O adatoms, a common occurrence on metals to the left of the periodic table: the dissociated atoms then behave just like adsorbed atoms discussed in Sec. 8.5.2.) Iron is a boundary case between strongly reactive and less reactive metal surfaces for CO. Using forward focusing, CO on Fe(100) was indeed found to be anomalous. CO is tilted by about 55° from the surface normal on this surface, possibly leaning against the side of a 4-fold hollow site (Saiki et al., 1989). At high coverages, crowding occurs and part of the CO and NO molecules have to settle for lower-symmetry sites. For instance, at a coverage of 3/4 per cell on Rh(lll), one third of the adsorbed CO molecules occupy bridge sites, while the remainder are pushed off the top sites by about 0.05 nm (Van Hove et al., 1983). NO in the same circumstances does the same thing, but the displacement from the top site is half as large, owing to the smaller packing diameter of this molecule (Kao et al., 1989). There is no clear indication for CO tilting away from the surface normal in these structures, but tilting by about 10° has been observed by angle-resolved photoemission and other techniques in several other close-packed structures. A LEED study gave a 17° tilt angle for the C-O axis in the close-packed Ni(110)-(2 x 1)-2CO
513
Ni(110)-p(2x1)-2CO
Figure 8-16. Perspective view of carbon monoxide on Ni(110). CO molecules are tilted by 17° from the surface normal in two opposite directions due to steric crowding. Therefore two CO molecules populate each (2x1) superlattice unit cell.
structure (Hannaman and Passler, 1988), cf. Fig. 8-16. Coadsorption of CO or NO with other adsorbates affects the adsorption site markedly and can lead to dissociation. It is apparent that CO and NO are unusually sensitive monitors of the surface condition. They react strongly to changes in substrate identity, coverage, and coadsorbates. The changes are easily measured, especially through vibrational analysis (HREELS and IRAS). C-O and N - O bond lengths in adsorbed CO and NO are generally close to the gas-phase value of 0.115 nm, with some expansion to about 0.117 nm in the presence of coadsorbates (Ohtani et al., 1988 b). The metal-C and metal-N bond lengths vary strongly with adsorption site and to a lesser extent with metal. For instance, R h - C bond lengths are 0.195 nm for a top site (one-fold C coordination to Rh), 0.202 nm for a bridge site and 0.216 nm for a 3-fold hollow site (the latter induced by coadsorption).
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8.5.4.3 Straight-Chain Hydrocarbon Adsorption
Acetylene and ethylene have been studied on several transition metal surfaces and are generally believed to lie flat on the surface, based on results from a variety of techniques. The probable geometry of acetylene on several f.c.c.(lll) surfaces is shown in Fig. 8-17. Molecular distortions away from the linear or planar gas-phase configuration are expected from theory (Gavezzotti and Simonetta, 1980; Kang and Anderson, 1985) and found with HREELS (Ibach and Lehwald, 1981; Steininger etal., 1982). A lengthening of the C-C bond is observed experimentally with NEXAFS (Stohr etal., 1984). The observed C-C bond lengths are about 0.145 and 0.149 nm for acetylene and ethylene, respectively (elongated from 0.120 and 0.133 nm, resp., in the gas phase; for comparison, the ethane single C-C bond length is 0.154 nm in the gas phase). Similar elongations have been observed by NMR for acetylene on Pt clusters (Wang et al., 1985). These effects are due to strong rehybridization, as the carbons change to-
ward sp3 hybridization to form strong new bonds to the metal atoms. This causes C - C - H bond angles to tend toward the tetrahedral angle. The lengthening of C-C bonds points to a weakening within the molecule. The detailed ethylene adsorption structure is not known at present, other than its parallelism with the surface and its C-C bond expansion to nearly a single-bond length. Ethylidyne can easily be produced from either acetylene or ethylene, by addition or subtraction of one hydrogen atom per molecule (Koestner etal., 1982; Koestner et al., 1983). This is a very stable species on many metal surfaces, at least on those which expose 3-fold hollow sites (Fig. 812). Ethylidyne can also form on 4-fold hollow sites, such as on an f.c.c.(lOO) surface (Slavin etal., 1988). On the other hand, some metals, such as Ni, tend to break the C-C bond of acetylene or ethylene, and thus do not give rise to stable ethylidyne species. The C-C bond of ethylidyne orients itself perpendicularly to the f.c.c.(lll) surface of several transition metals. The bonding to the metal occurs at three-fold coordinated hollow sites, cf. Fig. 8-12. The resulting bond lengths are in close agreement with values found in organometallic complexes. In particular, the C-C bond is intermediate in length between a single and a double bond, ranging from 0.145 to 0.155 nm. The metal-C bonds range from 0.200 to 0.212 nm. 8.5.4.4 Benzene Adsorption
Figure 8-17. Perspective view of acetylene on f.c.c.(lll). Hydrogen atoms are predicted by theory to bend away from the surface (as shown), relative to the straight gas-phase geometry of HCCH.
Benzene adsorbs parallel to f.c.c. (100), f.c.c.(lll) and h.c.p.(OOOl) surfaces, and probably also on other close-packed surfaces (Ohtani etal., 1988a). Benzene does not order easily on close-packed metal sur-
8.5 Adsorbate-Covered Surfaces
515
Figure 8-18. Top and side views of disordered benzene on Pt(lll), including distortions away from the gasphase benzene geometry (H positions are guessed). The side view (at top) shows the out-of-plane buckling of the carbon ring, as well as selected C-metal bond lengths. The top view (at bottom) shows C-C bond lengths and ring radii.
faces, compared to CO, NO and especially atomic adsorbates. At room temperature, benzene does not order at all on Pt and Pd(lll), while it weakly orders on R h ( l l l ) (a short exposure to the LEED electron beam is sufficient to destroy the ordered structure). The adsorption site of benzene is variable, depending on metal, crystallographic face and coadsorbates. It has so far only been determined on f.c.c.(Ill) surfaces (Ohtani et al., 1988b). On P t ( l l l ) , the molecule centers itself over a bridge site, whether benzene is mixed with CO or not (Ogletree et al., 1987; Wander et al., 1991),
cf. Fig. 8-18. On Rh(111), the same site is found for a pure benzene layer, but a 3fold hollow site emerges in the presence of coadsorbed CO (Van Hove et al., 1986b; Lin et al., 1987). On P d ( l l l ) , in the presence of CO, the 3-fold site is also found (Ohtani et al., 1988 a), while the site is not known for the pure benzene layer. Since the 3-fold hollow site does not exist on other crystal faces, this already implies a change of site in some of these cases. The height of the benzene carbon ring over the metal surface varies with the metal, as determined by LEED for CO-coadsorbed structures on Pd, Rh and P t ( l l l )
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surfaces (Ohtani et al., 1988 b): it is largest on P d ( l l l ) , smallest on P t ( l l l ) and intermediate on R h ( l l l ) . Another, more remarkable, trend is an expansion of the carbon ring radius: the radius is close to the gas-phase value (0.140 nm) on P d ( l l l ) , larger by about 0.035 nm on P t ( l l l ) and intermediate on R h ( l l l ) (these values again apply to CO coadsorption). Pure benzene on P t ( l l l ) has an average C-C expansion of about 0.012 nm, illustrating again that one adsorbate can significantly affect the structure of others nearby. In addition, LEED studies of chemisorbed benzene exhibit a reduction of the rotational ring symmetry from the 6-fold symmetry in the gas phase. This shows up as long and short C-C bonds in the same ring. The symmetry is three-fold (with three mirror planes, as in the Kekule distortion) when adsorption takes place over a hollow site possessing that symmetry, while it becomes two-fold (with two orthogonal mirror planes) over a bridge site (Fig. 8-18). 8.5.5 Coadsorption Over 150 ordered surface structures have been formed upon coadsorption of two or more different atomic or molecular adsorbates (Ohtani et al., 1987). In general, coadsorbed surface structures may be classified in two categories: cooperative adsorption and competitive adsorption (Kao et al., 1988). In cooperative adsorption, the two kinds of adsorbate mix well together and interpenetrate. In the competitive case the adsorbates segregate to form separate non-mixed domains. It has emerged that, generally, cooperative adsorption is found whenever the two species have opposite donor/acceptor character, i.e. when one adsorbate is an electron donor while the other is an electron acceptor
toward the metal substrate. Competitive adsorption occurs mostly when the two adsorbates are of similar type, i.e. both donors or both acceptors. However, some adsorbates, like CO, are ambivalent: the same adsorbate may sometimes be a donor and sometimes an acceptor. For example, Na (a donor) and S (an acceptor) coadsorbed on Ni(100) mix and order well. As a function of the two component coverages, different structures can be prepared. But common to these structures is the fact that each adsorbate surrounds itself as much as possible with the other kind of adsorbate: thus Na is surrounded by S and vice versa (Andersson and Pendry, 1976). This indicates an attractive interaction between a donor adsorbate and an acceptor adsorbate. A good analogy is the bulk structure of ionic crystals, such as NaCl: here positive ions are surrounded by negative ions and vice versa. The situation is not quite so simple for adsorption at a metal surface, because the metal electrons induce image charges of opposite sign below the surface. These produce in effect dipoles, each consisting of an adsorbate and an image charge; the dipole is reversed ("antidipole") when changing a donor to an acceptor. We thus have several kinds of dipole-dipole interactions: dipole-antidipole interactions are attractive and thus favorable, while dipole-dipole and antidipole-antidipole interactions are repulsive and thus unfavorable. However, even the dipolar interaction model in coadsorption is oversimplified and neglects various through-substrate and dynamic interactions (Mate et al., 1988). Coadsorption structures involving molecules have been extensively examined on P d ( l l l ) , R h ( l l l ) and P t ( l l l ) , using various pairs of adsorbates from the set C 2 H 2 , C 2 H 3 (ethylidyne), C 6 H 6 , Na, CO,
8.5 Adsorbate-Covered Surfaces
and NO (Mate et al., 1988). Among these, the hydrocarbons and Na transfer electrons to R h ( l l l ) when adsorbed (as measured by work function changes): they are donors. CO and NO have the opposite electron-transfer character, and are therefore acceptors, at least in these cases. The combination of ethylidyne and CO (or NO) on R h ( l l l ) , illustrated in Fig. 819, is particularly striking. It produces a molecular monolayer that is nearly chemically inert: this monolayer can resist air at room pressure for minutes (instead of oxidizing instantaneously, as do most overlayer structures on metals).
hep hollow
517
A further illustration of coadsorbate-induced ordering is the case of benzene mixed with CO on Pd(lll), R h ( l l l ) and P t ( l l l ) (Ohtani et al., 1988b). As described above, benzene alone adsorbs in a disordered or weakly ordered manner at room temperature. However, addition of CO to these disordered overlayers produces ordered surface structures, as shown in Fig. 8-20. The resulting structures are very stable, indicating again a strong attractive interaction between the coadsorbates. Note again how the donors (benzene) are surrounded by the acceptors (CO), and vice versa.
fee hollow
c(4X2) NO + Ethylidyne(CCH3)
Figure 8-19. Side view (above) and nearly perpendicular top view (below) of a mixed NO + ethylidyne layer on Rh(lll), showing selected interlayer spacings and bond lengths (hydrogen positions are guessed). Medium-size and large circles represent NO and metal atoms, respectively.
Figure 8-20. Side and top views (above and below) of benzene coadsorbed with CO on Rh(lll) in a 1:1 ratio, forming a unit cell which is outlined. Van der Waals radii are assumed for overlayer atoms. CO molecules are shaded. Large dots indicate C and O positions, connected small dots indicate guessed hydrogen positions and small dots represent metal atoms in the second metal layer.
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Several mutual effects between coadsorbates have been seen. One effect is a site change. With CO, ethylidyne and benzene, site changes have been induced by coadsorption. For instance, CO is relocated by donor coadsorbates toward higher-coordination sites. As an other example, pure ethylidyne on R h ( l l l ) selects only one of the two possible three-fold hollow sites (it is the so-called h.c.p. site). But after CO coadsorption, the ethylidyne site is changed to the f.c.c. hollow site (which differs in the absence of a second-layer metal atom directly below the site). Nevertheless, the calculated binding energies of ethylidyne at the two sites are virtually identical. It appears then that some effect due to the coadsorption itself must be active. Another effect is a bond length change. This has been noticed, for example, with benzene due to CO coadsorption, as mentioned in Sec. 8.5.4.4 (Ohtani et al., 1988 b). A bond length change is also found when another adsorbate is mixed into a CO layer. Donor adsorbates induce a modest lengthening of the C-O bond by about 2 pm (this seems to occur at the same time as the site change towards higher coordination and a charge transfer). Connected with this effect is a large change in vibrational frequencies, again well illustrated with CO: as the bond is lengthened, i.e. weakened, by coadsorption, the C-O stretch frequency decreases (in some cases by up to 25%) (Ogletree etal., 1986). 8.5.6 Adsorbate-Induced Relaxations
Chemisorption on a surface modifies the chemical environment of the surface atoms and therefore affects the structure. In particular, upon adsorption, any clean-surface relaxation is generally reduced as the surface atoms of the substrate move back
towards the ideal bulk-like position or even beyond (Somorjai and Van Hove, 1989; King, 1989). Relaxations of deeper interlayer spacings are also usually reduced upon adsorption. In addition, it is becoming increasingly clear that small local distortions on the scale of 0.01 nm are induced around each adsorption site in directions other than the surface normal. Good examples of the outward relaxation of interlayer spacings are provided by atomic adsorption on the (110) surfaces of nickel and other f.c.c. metals (Somorjai and Van Hove, 1989). The clean (110) surfaces typically exhibit contractions by about 10% (0.01 to 0.015 nm) in the topmost interlayer spacing relative to the bulk value. Upon adsorption these contractions are reduced to less than 3 to 4% (3 to 5 pm). Adsorbates can also induce relaxations while maintaining an existing clean-surface reconstruction. This occurs with S on Ir(110): the missing-row structure is maintained, while the interatomic distances revert to nearly their bulk value (Chan and Van Hove, 1987). One also expects similar effects on other kinds of substrate, such as semiconductor surfaces. Thus, H on Si (111), relative to the metastable clean ( l x l ) surface, relaxes the top Si layers to very near their ideal bulk positions (Jepsen et al., 1980). Also, Sb deposited on GaAs(llO) removes the large rotation angles of the clean surface, reestablishing the bulk structure to within 0.01 nm (Duke etal., 1982). But adsorption is more often accompanied by a change in reconstruction, which will be discussed below. 8.5.7 Adsorbate-Induced Reconstructions
Adatoms can induce a restructuring of a surface in a variety of ways (Somorjai and
8.5 Adsorbate-Covered Surfaces
Van Hove, 1989). A mild form of reconstruction occurs when substrate atoms are displaced by small amounts in different directions, thereby changing the unit cell of the substrate (displacive reconstruction). An opposite situation is the removal of a clean-surface reconstruction by an adatom. Also possible is the change from one reconstruction to another. Adatoms can furthermore give rise to new compound formation, or can change surface segregation in an existing compound. On a larger scale, adatoms have also been found to cause macroscopic reshaping of surfaces. The energy needed for surface restructuring is paid for by the increased bond energies between the adsorbed atom and the substrate. Therefore, such surface restructuring is expected only upon chemisorption where the adsorbate-substrate bond energies are similar to or larger than the bond energies between the atoms in the substrate. This is clearly the case for the adsorption of carbon, oxygen, and sulfur on many transition metals. 8.5.7.1 Displacive Local Induced by Adsorption
Reconstruction
519
which it expands by pushing the four neighboring metal atoms outward from the site, parallel to the surface. This allows the adatom to penetrate deeper into the metal surface and to bond not only to the four first-layer metal atoms but also to a metal atom in the next layer. The surrounding metal lattice cannot accept a corresponding compression at a coverage of 0.5 per cell and instead forces a rotation of the square of four metal atoms about the surface normal. Thereby, the average metal density in the top layer is kept constant, while accommodating the additional foreign atoms. Another example that involves a rotation to better accommodate the adsorbate is offered by O on N i ( l l l ) (Vu Grimsby etal., 1990). Here O atoms adsorb over three-fold hollow sites and primarily cause the three surrounding Ni atoms to rotate about the surface normal. Oxygen on Ni(100), as well as other such adsorbate structures, also induce substrate relaxations that can be called reconstruction. In this case, the relaxations are purely perpendicular to the surface. In the c(2x2) and p (2 x 2) structures, a buckling appears Ni(100)-p(2x2)-2C
The adsorption of atoms may displace substrate atoms to provide better adsorbate-substrate bonding, in such a way that a new unit cell results in the substrate (King, 1989). This is a generalization of the adsorbate-induced relaxations discussed above. The local displacements are of the same order of magnitude, at most a few hundredths of nm. The effect is well illustrated with the structure induced by carbon or nitrogen adsorbed on Ni(100) (Onuferko et al., 1979; Wenzel et al., 1988), also seen with O on Rh(100) (Oed et al., 1988), cf. Fig. 8-21. The adatom occupies a four-fold site,
Figure 8-21. Top view of the structure of Ni(100)(2 x 2)-2C, with C shown as small dark circles.
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in the second Ni layer, while the first Ni layer moves slightly outward relative to the clean surface. The second-layer Ni atom directly below an oxygen atom (which is centered over a hollow site) is pushed down away from the oxygen atom, unlike the other second-layer Ni atoms, thereby creating the buckling (Oed etal., 1989; Oed etal., 1990). Another example of adsorbate-induced surface restructuring through lateral displacements is provided by sulfur on Fe(110) (Shih etal., 1981). The clean Fe(110) surface offers two-fold and threefold coordination sites for adsorption. Sulfur maximizes its coordination number to nearly four by distorting the two-fold coordination site into a nearly square "hollow" site. Row pairing is another kind of adsorbate-induced reconstruction. It occurs on f.c.c.(HO) surfaces after adsorption of hydrogen, in particular (Kleinle et al., 1987). On Ni and Pd(110), a suitable coverage of H [1.5 adatom per (1 x 1) cell] causes a (1 x 2) reconstruction of the substrate, in which adjacent close-packed metallic rows move closer together by 0.02 nm, thereby causing an alternation of narrow and wide troughs between rows. This creates a narrower channel which may allow the hydrogen atoms to bond to more metal atoms at the same time (however, the H positions in these structures are not fully determined yet). 8.5.7.2 Removal of Reconstruction by Adsorption
The chemisorption of atoms frequently removes surface reconstruction and produces a more bulk-like surface structure. Examples of this effect are offered by the removal with hydrogen of the reconstruction of clean Si (100), Si (111) and diamond
C (111), and the removal with carbon, oxygen or CO of the reconstructions of the (100) and (110) faces of Ir and Pt. Electron acceptors, like O and S, are particularly effective at removing reconstructions. Sometimes small amounts of adsorbate suffice to remove a reconstruction, but more frequently amounts comparable to a monolayer are required. Hydrogen has to be adsorbed to a coverage of 2 per cell to remove the W(100)-c(2 x 2) reconstruction (Passler et al., 1985). Removal of the adsorbate (such as by thermal desorption) reverses the process: the clean-surface reconstruction reappears again. But kinetic effects may trap a metastable state of the surface, as has been observed with small amounts of adsorbates on Ir and Pt(100) (Heinz and Besold, 1983). Clean diamond C (111) favors the (2x1) Pandey reconstruction (Sowa et al., 1988 a), cf. Fig. 8-7. One hydrogen per ( l x l ) unit cell restores the bulk lattice termination, which is found to be ideally terminated within the LEED accuracy (Yang et al., 1982). The H atoms are thought to cap the dangling bonds of this ideal truncation. Hydrogen also removes the reconstructions of clean Si (111) in a similar way (Jepsen et al., 1980). The clean Ir(110) has the (2x1) missing-row reconstruction: a dense layer of oxygen can remove this reconstruction and restore the ideal lattice termination. In the process the Ir-Ir bond lengths nearly regain their bulk value (Chan etal., 1978). 8.5.7.3 Creation of Reconstruction by Adsorption
Adsorbates have frequently been found to induce new reconstructions on surfaces that were not reconstructed in the clean state (Somorjai and Van Hove, 1989). Of-
8.5 Adsorbate-Covered Surfaces
ten a small fraction of a monolayer suffices to make the entire surface reconstruct. Electron donors, like alkali metals, are particularly well known to induce reconstructions on metal surfaces. For example, a small coverage (below 0.1 per cell) of disordered alkali adatoms is sufficient to cause reconstruction of the Ni (Behm etal., 1987), Cu (Copel etal., 1985), Pd (Barnes etal., 1985), and Ag(110) (Hayden et al., 1983) surfaces, which transform to the (1 x 2) missing-row structure (Fig. 8-5). Cs adsorbed on the (1 x 2) reconstruction of Au(110) causes a (1 x 3) structure, with only about 5% coverage per ( l x l ) cell (Haeberle and Gustafsson, 1989). This structure is also of the missingrow type, but with deeper troughs than the clean (1x2) structure. A likely reason for this is that large alkali atoms bond more strongly (thanks to more near neighbors) within the deep troughs of the missing rows than in the shallow troughs of the ideal (110) surface (Behm et al., 1987; Jacobsen and Norskov, 1988). More generally, electron-donating adsorbates tend to stabilize metal reconstructions. Stabilization is exemplified by alkali adsorption on the hexagonal reconstruction of Ir(100) (Heinz et al., 1985 a). There the clean-surface reconstruction is maintained in the presence of alkali atoms. Electron acceptors, however, can also induce reconstructions, as often happens in compound formation (see Sec. 8.5.8). An example is the case of oxygen deposited on Cu(100), which system has produced much controversy. While initially a c (2 x 2) pattern was thought to be formed at 0.5 atoms per ( l x l ) cell, in fact a c (4 x 2) pattern is now believed to occur. And, whereas at first simple hollow-site adsorption was suspected, a missing-row reconstruction is now more probable (Zeng et al., 1989). The missing rows form
521
in effect alternating up and down steps, which are identical to the steps of Cu(410). As discussed in Sec. 8.5.2, oxygen on Cu(410) chooses hollow sites at the terrace edges, cf. Fig. 8-11 (Saiki etal., 1989): it emerges that O on Cu(100) selects the same adsorption sites. Other examples are the systems Ni (110)( 2 x l ) - O (Kleinle etal., 1990), Cu(110)(2 x 1)-O (Parkin et al., 1990) and Fe(211)(2 x 1)-O (Sokolov et al., 1986). The clean surfaces consist of troughs between ridges of close-packed metal atoms. Oxygen replaces every other ridge atom, forming ridges of alternating metal and O atoms, cf. Fig. 8-22. 8.5.7.4 Change of Reconstruction by Adsorption
It stands to reason that surfaces which are already reconstructed when clean are particularly prone to further reconstruction in the presence of adsorbates. This is specially apparent with semiconductor surfaces. Cu(110)-(2x1)-O
Figure 8-22. Perspective view of the structure of Ni(110)-(2 x 1)-O, with O shown as small dark circles substituting for Ni atoms in the light-gray Ni ridges.
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Numerous examples exist where adatoms change the reconstruction of semiconductor surfaces. However, the number of resulting structures which have been solved is relatively small. They concern mostly metal adatoms, such as Al, Ga, and Pb deposited on Si and G e ( l l l ) (Kawazu etal., 1988; Huang etal., 1989; Huang etal., 1990b). Their adsorbate location has been discussed in Sec. 8.5.2 (T 4 adatom site occupation, Fig. 8-9). The substrate lattice relaxes noticeably around the adsorption site, with displacements up to about 0.03 nm. Larger adatoms induce larger displacements. The relaxation is noticeable down to the second double layer, which is strongly buckled. Hydrogen on W (100) and Mo (100) provides examples of this process on metal surfaces (Estrup, 1979; Hinch et al., 1983). Clean W(100) and Mo (100) at low temperatures are characterized by a reconstructed surface with long zigzag chains of surface metal atoms. When hydrogen is adsorbed at half-saturation coverage, the tendency is to break up these chains into individual W - H - W or M o - H - M o trimers. Each H atom bridges a pair of W or Mo atoms, replacing the zigzag chain geometry (King, 1989; Willis, 1979). Oxygen and nitrogen also change the reconstruction of W (100). Oxygen, as studied at lower coverages (Rous et al., 1986), settles deep in hollow sites surrounded by four W atoms that are drawn in radially toward the oxygen by about 0.02 nm. Nitrogen, studied at higher coverages, also settles deep into similar sites, but because of its higher coverage of about 0.4 atoms per cell, it causes islands of the top W layer to shrink into contracted rafts of average size (4x4) (Griffiths etal., 1982). This is an example where one might speak of compound formation, at the monolayer level, since the adatoms are relatively close to
being coplanar with the metal layer (with interlayer spacings of 0.059 and 0.049 nm for O and N, respectively). 8.5.8 Compound Formation and Surface Segregation
In compound formation from adsorption, a reconstruction occurs that resembles a bulk compound. Continued addition of adsorbate atoms may enable the formation of a thicker film with the threedimensional lattice of a bulk compound. Such behavior is characteristic of oxidation, nitridation, carbide formation and alloying of metal surfaces. Questions of interest include whether the compound is ordered and, if so, which is the crystallographic orientation of the growing compound. Also the question of lattice matching is important: most growing compounds have a lattice which is mismatched to the substrate (Somorjai and Van Hove, 1989). The initial oxidation step of a metal is illustrated in Fig. 8-23. Ta(100) is shown after uptake of a submonolayer amount of oxygen, which takes interstitial positions between the first and second metal layers (Titov and Jagodzinski, 1985). An intermediate nitridation step which has been observed consists of the penetration of one monolayer's worth of N atoms between the first and second metal layers of Ti(0001) (Shih et al., 1976); the same happens for N adsorbed on Zr(0001) (Wong and Mitchell, 1987). In Sec. 8.5.7.4, we gave another example of monolayer compound formation: N deposited on W(100) produces rafts of twodimensional W 2 N compound with a lattice constant parallel to the surface different from that of the substrate. S on Ni (111) has been observed to form a compound monolayer of composition
8.6 Disordered Surfaces
523
Figure 8-23. Perspective side view of the structure ofTa(100)-(lx3)-O, with O shown as small gray circles in near-tetrahedral subsurface positions. Ta(100) - (1 x 3) - 0
Ni2S (Kitajima et al, 1989). It is suggested to have a square lattice of Ni atoms, with every other hollow site occupied by S atoms in a c (2 x 2) array. Metal silicide compounds are commonly formed after adsorption of metal atoms onto silicon surfaces. Thus, upon Ni deposition on Si (111), NiSi2 grows with its (111) surface interfaced to the substrate (Vlieg et al., 1986). Cobalt (Fischer et al., 1987) and other transition metals behave similarly. For example, Al forms a substitutional GaAsAl compound after deposition on GaAs(llO) (Kahn e t a l , 1981). Adsorbates may induce large changes of surface composition in multicomponent systems, i.e. surface segregation (Somorjai and Van Hove, 1989). Such changes involve atomic diffusion perpendicular to the surface, and thus bond breaking and rebonding. This occurs particularly when the chemisorption bond energies between the alloy components are very different. One example is the behavior of the AgPd alloy (Bouwman et al., 1972). The clean surface of a Ag-Pd alloy is enriched in silver at any bulk composition because of the lower surface energy of Ag as compared to Pd. Upon adsorption of CO, the surface composition changes rapidly. Because of the greater strength of the Pd-CO bond as compared to the Ag-CO bond, the Pd atoms move to the surface and the alloy surface becomes enriched in Pd. Upon heating CO desorbs and the surface excess of Ag is reestablished.
8.6 Disordered Surfaces Above a certain critical temperature that depends on coverage, many ordered atomic and molecular overlayers become disordered. If the substrate maintains its crystallinity and the adsorbate maintains a preference for certain adsorption sites, we have a lattice gas. Thus the disordered overlayer consists of adsorbates at one or a few preferred sites, but without longrange order. Clean surfaces that reconstruct may also undergo such an order-disorder phase transition, for instance W(100)-c(2 x 2) and Si(lll)-(7 x 7); above the critical point, a disordered version of the reconstruction exists. Surfaces with disorder and defects are more common in experiments than are perfectly ordered surfaces. They are more realistic models of practical surfaces used in various technologies. This is especially true if one considers defects, which for the most part are disordered. So far, however, almost no structures have been determined at defect sites (steps, vacancies, dislocations, etc.). Diffraction techniques are excellent for studying the long-range nature of the disorder, in particular the phase diagram, the critical exponents of the phase transition and the pair-correlation function. But diffraction is inherently poor for obtaining short-range information in disordered systems.
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Nevertheless, LEED has been developed to provide just that local bonding information from the diffuse LEED intensities caused by a disordered layer (Saldin et al., 1985; Heinz et al., 1985 b; Van Hove, 1988; Wander et al., 1991). This is possible mainly because of the strong multiple scattering that occurs in LEED and which delivers information about the relative positions of adsorbates and substrates. SEXAFS has been applied to many disordered surface structures. These include especially atomic adsorbates on Si (111) surfaces (Citrin, 1986; Rowe, 1990). In many cases, the adsorbate changes or removes the clean-surface reconstruction (as seen by the LEED pattern). Adsorbate-Si bond lengths and coordination numbers are determined, and, through the light-polarization dependence of SEXAFS, the orientation of bonds are to some extent determined. From that information, models for the local bonding site and geometry are deduced. For instance, Cl (Thornton etal., 1989) and I (Citrin etal., 1982) on Si (111) are found to cap dangling bonds of the Si (111) surface, forming a Si-adsorbate bond perpendicular to the surface. Since the (7 x 7) LEED pattern is maintained, the relationship between the local bonding geometry and the (7 x 7) structure cannot be ascertained. Complete disordered local geometries have been obtained with DLEED (diffuse LEED) for a few atomic and molecular adsorption systems on metal surfaces. The first result described the location of oxygen on W(100) at a low coverage [about 0.1 atom per (1 x 1) cell] (Rous etal., 1986). Not only is the adsorption site found to be the 4-fold hollow site, but in addition a relaxation of the metal surface is detected: the oxygen atoms attract the four neighboring W atoms to form a contracted W square with better W - O bond lengths. CO 2
disordered on Ni(110) is found to probably occupy two different sites centered on the ridges of the substrate (Illing et al., 1988). CO on P t ( l l l ) , at a coverage just above a third of a molecule per ( l x l ) cell, occupies primarily top sites and partly bridge sites (in agreement with HREELS site assignments) (Blackman etal., 1988). The bond lengths and bond angles also match those of known ordered structures of CO on Pt(lll). And the disordered benzene adsorption structure on P t ( l l l ) has been determined by the same approach, cf. Fig. 8-18 (Wander et al., 1991).
8.7 Mechanisms We shall here briefly summarize the main mechanisms responsible for the observed surface structures. More examples can be found in the earlier sections. 8.7.1 Bond Length Changes
Bond lengths at surfaces generally match corresponding bond lengths found in molecules and solids to well within 0.01 nm. They also satisfy Pauling's principle which relates them to bond order (Pauling, 1960). In essence, the number of bonding neighbors strongly affects the bond lengths, due to rehybridization of atomic orbitals involved in the chemical bond. Some exceptions are observed, but it is likely that these are actually incorrectly solved structures, i.e. cases when the final structure has not yet been found. In fact, one may suggest that bond lengths which do not fit the established principles often indicate an incorrect structure. Mitchell and co-workers (Mitchell et al., 1986) have analyzed in detail the surface bond lengths obtained for atomic adsorbates in terms of a modernized version of
8.7 Mechanisms
the Pauling rule. They exhibit the adequacy of such a description. But their work also highlights the limitations of such an empirical approach. In particular, multilayer relaxations, reconstructions and molecular adsorbates would be difficult to incorporate into their model. Some caution must be exercized when considering unusual bonding geometries. For instance, the T 4 adatom geometry and other features of the Si(lll)-(7 x 7) and Ge(lll)-c(2 x 8) reconstructions provide unique bond lengths and angles. They include 5-fold coordinated Si and Ge atoms, perhaps only known otherwise in small clusters of these materials (Tomanek and Schluter, 1987). Under these conditions, unusually long and short bond lengths are found, cf. Sec. 8.4.4. In addition, it should come as no surprise to find very long bond lengths in chemisorbed molecules which are close to decomposition, as in benzene on some metal surfaces. 8.7.2 Restructuring
Reconstruction in general is the result of optimizing the surface chemical bond. However, different detailed mechanisms are responsible for the variety of reconstructions that are observed. In the case of clean surfaces, restructuring can be due to the following causes, among others: (a) Bond shortening: Surface atoms that have lost near neighbors due to the surface formation, have shorter bond lengths to the remaining near neighbors. As a result, surface layers may contract parallel to the surface into a different lattice, as in the hexagonal reconstruction of Ir, Pt and Au(100) (Van Hove etal., 1981). (b) Jahn-Teller-like pairing: In certain cases, such as with Mo (100) and W(100), a clean metal surface with half-filled d-
525
bands has a high density of states near the Fermi level. Then the total energy can be reduced by splitting this density of states by superlattice formation. This can be accomplished by a rebonding arrangement in which zigzag rows of atoms are formed, as happens with the Mo and W (100) surfaces (Inglesfield, 1985). (c) Rehybridization: The absence of near-neighbor atoms can give rise to substantial rehybridization of orbitals around surface atoms, in particular at semiconductor surfaces. Then different bond angles become favored, often yielding radically different bonding configurations. This applies to most semiconductor surfaces, including the large bond angle distortions on (110) surfaces of III-V compounds (Chadi, 1978; Tong et al., 1990), and missing atoms on the (111) surfaces of some of these compounds (Chadi, 1984; Tong etal., 1990). (d) Reduction of the number of dangling bonds: Solids with strong orientational preference in their bonding have difficulty reconciling that preference with a surface. The dangling bonds formed there by removal of atoms to form the surface can to some extent be compensated by new bonding geometries different from those in the bulk. This mechanism often goes hand in hand with the rehybridization described above. The minimization of dangling bonds operates in many elemental semiconductor surfaces, including especially Si(100)-(2 x 1) (Yin and Cohen, 1981) and Si(lll)-(7 x 7) (Tong et al., 1990). (e) Small-facet formation: Since closepacked crystallographic faces have lower surface energy, one can expect such facets to form on more open surfaces. This occurs with the missing-row reconstruction of Ir, Pt and Au(110), where small (111) facets are formed (Tomanek and Bennemann, 1985).
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8 Crystal Surfaces
An adsorbate can induce or hinder clean-surface reconstruction by changing the relative importance of the various reconstruction mechanisms. It can thus induce a new or different reconstruction or remove a clean-surface reconstruction by giving precedence to a different mechanism. An adsorbate can also be responsible for new mechanisms of reconstruction. New reconstruction mechanisms acting in the presence of adsorbates include the following: (a) Local distortions due to the adsorbate-substrate bond: Distortions of the substrate may occur that improve the coordination number, adsorbate-substrate bond lengths and bond angles relative to the undistorted structure. Many examples of this displacive reconstruction mechanism are known for adatoms on metal surfaces. (b) Adsorbate preference for certain sites, attainable only through reconstruction, facetting or recrystallization: If an adsorbate requires, for example, a certain coordination number for bonding to a substrate, and if no suitable sites are available on a given substrate, then the substrate may restructure in such a way as to provide the correct type of site. (c) Strong adsorbate-substrate bonds leading to compound formation or to modified surface composition of an alloy: When adsorbate-substrate bonds are much stronger than adsorbate-adsorbate or substrate-substrate bonds, the surface atoms will rearrange in order to maximize the number of adsorbate-substrate bonds.
8,8 Outlook Surface crystallography started about 20 years ago and has reached a stage where
many simple structures have been solved in detail. The results have shown a number of important phenomena, such as: clean-surface and adsorbate-induced relaxations and reconstructions; surface segregation at the monolayer level in alloys, influenced by adsorbates; atomic adsorption at highcoordination sites with bond lengths close to those expected from known covalent radii; molecular species with geometries similar to those also found in analogous organometallic compounds; strongly stretched and/or distorted chemisorbed molecular species; coadsorbate-induced ordering and mutual distortions. In recent years, the phenomenon of adsorbate-induced restructuring has been observed with increasing frequency. Not only are interlayer spacings within the substrate changed by the adsorbate, but also lateral relaxations parallel to the surface and buckling of substrate layers are observed. This effect is likely to be more general than previously expected and many of the earlier structure determinations will have to be repeated with greater attention to such details. The observation of adsorbate-induced relaxations is an example of the increase in accuracy achieved in surface crystallography over the years. Another simultaneous improvement has been the ability to analyze increasingly complex and interesting structures. This has allowed the structure determination of complex surface reconstructions and coadsorbed molecular layers, for example. Recently, also, disordered overlayers have been made accessible to detailed local structure determination. It is safe to say that there are more disordered surface structures than ordered ones. Therefore, it is of great importance to extend this capability: one should expect a rapid increase in the investigation of such structures.
8.10 References
A new direction being developed at this time is that of stepped surfaces. Only a very limited class of stepped surfaces has been studied in any detail, namely clean surfaces with narrow terraces (i.e., with not too high Miller indices). In view of the importance of steps as a primary type of surface defect where catalytic reactions can take place, it is imperative that adsorption structures at steps be investigated. Metallic multilayers on metal substrates have yielded new metastable structures with novel properties. By contrast, nonmetallic multilayers have hardly been investigated so far. Metallic and non-metallic multilayers grown on a variety of substrates are directly relevant to interfacial structures (e.g., heterojunctions, Schottky barriers and grain boundaries), to mechanical bonding of different materials and epitaxial growth of one crystal on another. Much work lies ahead in this direction, as well. Relatively few compound surfaces have been studied to date, whether prepared from bulk compounds or by formation of thin compound layers on pure substrates. Here also, numerous technologically interesting structures invite analysis.
8.9 Acknowledgements This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Materials Sciences Division of the U.S. Department of Energy under Contract No. DE-AC0376SF00098.
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General Reading Clarke, L.X (1985), Surface Crystallography: An Introduction to Low-Energy Electron Diffraction. Chichester: Wiley-Interscience. Ertl, G., Kuppers, X (1979), Low Energy Electrons and Surface Chemistry. Weinheim: VCH. Feidenhans'l, R. (1989), Surf. Sci. Rep. 10 (X-ray, diffraction), 105. Heinz, K. (1988), Progr. Surf. Sci. 27, 239. Hull, R., Ourmazd, A., Tung, R. T. (1992), in: Series Materials Science and Technology, Vol. 4: Schroter, W. (Ed.). Weinheim: VCH publishers, in press. Ibach, H., Mills, D.L. (1982), Electron Energy Loss Spectroscopy and Surface Vibrations. New York: Academic.
8.10 References
Inglesfield, J. (1985), Progr. Surf. Sci. 20, 105. Lipkowski, J., Ross, P.N. (Eds.) (1991), Structure of Electrified Interfaces. Weinheim: VCH publishers, in press. MacLaren, J.M., Pendry, I B . , Rous, P.I, Saldin, D.K., Somorjai, G.A., Van Hove, M.A., Vvedensky, D.D. (1987), Surface Crystallographic Information Service: A Handbook of Surf ace Structures. Dordrecht: D. Reidel. Ohtani, H., Kao, C.-T., Van Hove, M.A., Somorjai, G.A. (1987), Progr. Surf. Sci. 23, 155. Pauling, L. (1960), The Nature of the Chemical Bond, 3rd ed. Ithaca: Cornell University Press. Pendry, J. B. (1974), Low-Energy Electron Diffraction: The Theory and its Application to Determination of Surface Structure. London: Academic Press. Somorjai, G.A. (1972), Principles of Surf ace Chemistry. Englewood Cliffs: Prentice-Hall. Somorjai, G.A. (1981), Chemistry in Two Dimensions. Ithaca: Cornell University Press. Somorjai, G.A., Van Hove, M.A. (1979), Adsorbed Monolayers on Solid Surfaces, Structure and Bonding, Vol. 38. Heidelberg: Springer-Verlag, p. 1. Somorjai, G.A., Van Hove, M.A. (1989), Progr. Surf. Sci. 30,201. Tong, S.Y., Van Hove, M.A., Takayanagi, K., Xie, X.D. (Eds.) (1991), Proceedings: The Structure of Surfaces III. Heidelberg: Springer-Verlag.
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Van der Veen, J.F., Van Hove, M.A. (Eds.) (1988), Proceedings: The Structure of Surfaces II. Heidelberg: Springer-Verlag. Van Hove, M.A. (1988), in: Chemistry and Physics of Solid Surfaces VII: Vanselow, R., Howe, R. (Eds.). Heidelberg: Springer-Verlag, p. 513. Van Hove, M.A. (1991), in: Surface Physics and Related Topics: Yang, J.-F. (Ed.). Singapore: World Sci. Publ. Co., in press. Van Hove, M. A., Tong, S. Y. (1979), Surface Crystallography by Low Energy Electron Diffraction: Theory, Computation and Structural Results. Heidelberg: Springer-Verlag. Van Hove, M. A., Tong, S.Y. (Eds.) (1985), Proceedings: The Structure of Surfaces. Heidelberg: Springer-Verlag. Van Hove, M.A., Weinberg, W.H., Chan, C.-M. (1986), Low-Energy Electron Diffraction: Experiment, Theory, and Structural Determination. Heidelberg: Springer-Verlag. Van Hove, M.A., Wang, S.W., Ogletree, D.F., Somorjai, G.A. (1989), Adv. in Quantum Chem. Vol. 20, p. 2. Watson, P.R. (1987), J. Phys. Chem. Ref Data 16, 953. Watson, P. R. (1989), /. Phys. Chem. Ref Data 19, 85. Zangwill, A. (1988), Physics at Surfaces. Cambridge: Harvard Univ. Press.
9 Structures of Interfaces in Crystalline Solids Michael W. Finnis and Manfred Riihle Max-Planck-Institut fur Metallforschung, Institut fur Werkstoffwissenschaft, Stuttgart, Federal Republic of Germany
List of 9.1 9.2 9.2.1 9.2.1.1 9.2.1.2 9.2.1.3 9.2.2 9.2.3 9.3 9.4 9.4.1 9.4.2 9.4.3 9.4.4 9.5 9.5.1 9.5.2 9.6 9.6.1 9.6.1.1 9.6.1.2 9.6.1.3 9.6.1.4 9.6.1.5 9.6.2 9.6.2.1 9.6.2.2 9.6.2.3 9.7
Symbols and Abbreviations Introduction Homophase Boundaries - Geometrical Aspects Crystallography Coincidence Site Lattice O-Lattice DSC Lattice Small Angle Boundaries Dislocations in Large Angle Boundaries Heterophase Boundaries - Geometrical Aspects Simulation of Homophase Boundary Structures Interatomic Potentials Simulation Techniques Results Structural Unit Models Simulation of Heterophase Boundary Structures Interatomic Potentials Results Experimental Studies of Interface Structures by X-Ray Techniques X-Ray Diffraction Introduction General Form of the Boundary Diffraction Experimental Difficulties with Measuring the Boundary Diffraction Strategies for Determining the Boundary Structure Some Current Results Structure Determination by Grazing Incidence X-Ray Scattering The Technique Experimental Studies Experimental Results Experimental Studies of Interface Structure by High-Resolution Electron Microscopy 9.7.1 Introduction 9.7.2 Comments on Direct Lattice Imaging of Distorted Materials 9.7.2.1 Resolution 9.7.2.2 Determination of the Atom Column Locations Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.
535 537 538 539 539 540 542 543 545 548 549 549 552 554 558 560 561 561 563 563 563 563 565 567 569 571 571 572 572 574 574 575 576 578
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9.8 9.8.1 9.8.2 9.8.3 9.8.3.1 9.8.3.2 9.8.4 9.8.4.1
9 Structures of Interfaces in Crystalline Solids
Experimental Results and Comparison to Results of Simulation Structures of Grain Boundaries in Semiconductors Structures of Grain Boundaries in Metals Structures of Grain Boundaries in Ceramics Grain Boundaries in Simple Oxides Grain Boundaries in Polyphase Ceramics Structures of Metal/Ceramic Interfaces Structure of the Interface Between Sapphire (a-Al2O3) and a Single Crystalline Nb Film Grown by Molecular Beam Epitaxy (MBE) . . . 9.8.4.2 Structure and Bonding at the Ag/MgO Interfaces 9.9 Concluding Remarks 9.10 Acknowledgements 9.11 References
578 579 580 584 584 585 586 587 595 599 600 600
List of Symbols and Abbreviations a1? a2
b D d0
d2 E F Fj
A/ A/s g k L
q(x,y) Q (w, v) R T
5 e 9 0 vf, Qt
I
ARM b.c.c. BDL CSL CTF DDL DSC EAM ELNES
unit vectors of periodicity in grain boundary plane unit vectors of periodicity in boundary diffraction lattice Burgers vector planar density of coincidence sites minimum distance between neighboring columns of atoms imaged by HREM point-to-point resolution in Scherzer condition in HREM information limit of HREM energy at zero temperature glue function structure factors defocusing distance in HREM optimum defocusing distance in HREM (Scherzer defocus) diffraction vector vector in reciprocal space 3 x 3 matrix which linearly transforms lattice vectors transmission function of an object Fourier transform of transmission function reliability factor translation vector of two crystals adjacent to the interfaces coordinates in diffraction plane weighting factors coordinates lattice mismatch strain boundary misorientation twist angle types of atoms occupying sites i,j local density ratio of unit cell volumes for boundary classification density function (of a free electron) amplitude in the image plane atomic resolution microscopy body-centered cubic boundary diffraction lattice coincidence site lattice contrast transfer function double diffraction lattice complete pattern shift lattice; the letters D S C are not historically an acronym for anything (see Bollmann, 1970) embedded atom model energy loss near edge structure
536
EXAFS f.c.c. FP-LMTO FLAPW GIXS h.c.p. HREM MBE MC MD OR SAD TEM
9 Structures of Interfaces in Crystalline Solids
extended X-ray absorption fine structure face-centered cubic full potential linear combination of muffin tin orbitals full potential linear augmented plane wave grazing incidence X-ray scattering hexagonal close packed high-resolution electron microscopy molecular beam epitaxy Monte Carlo method molecular dynamics orientation relationship selected area diffraction transmission electron microscopy
9.1 Introduction
9.1 Introduction Most materials applied in materials science are used in polycrystalline form. This is not only true for metals but also for ceramics and polymers. Only semiconductors are quite often used in single-crystal form. Boundaries between the different grains or, in general, between different phases play an important and controlling role in determining the properties of different materials. For example, the toughness and strength of steels was drastically improved in the early 1960s by rigorous control of grain size, of the chemistry of grain boundaries and of non-metallic inclusions in the metal. Since internal interfaces play this important role, a large amount of research has been conducted over the last five decades and a rich body of literature is available. Balluffi and Sutton (1993) summarize all scientific issues with respect to structure and properties of internal interfaces. In the present article the structural aspects of internal interfaces will be discussed. It is helpful to distinguish between two different groups of boundaries (Fig. 9-1) (Cahn and Kalonji, 1982). Homophase boundaries are interfaces between grains of identical crystal structure and identical
537
composition. They include grain boundaries, twin boundaries, domain boundaries and stacking faults. Heterophase boundaries are interfaces between regions of different crystal structure which may also vary in their chemical composition. Examples of the first kind include boundaries between coexisting allotropic modifications, for instance between grains of the tetragonal and monoclinic phase of ZrO 2 . Heterophase boundaries of the second kind are present in all technical alloys and ceramics, e.g. boundaries between austenitic and ferritic steels in a duplex stainless steel or the surfaces of the non-metallic inclusions mentioned above (Fischmeister, 1985). Another group of materials which contain the latter boundaries are composites (Ishida, 1987; Suresh and Needleman, 1988). In all these cases, the regions adjoining the interfaces belong to different phases in the classical thermodynamic sense, therefore, the simpler term phase boundary is also often used in the literature. For low-angle grain boundaries a dislocation model was proposed by Taylor (1934). Its verification by edge pit observations as well as the prediction of the influence of misorientation upon interfacial energy (Read and Shockley, 1950), and
homophase boundaries grain boundaries twins domain boundaries stacking faults heterophase boundaries ZrO2 (t) Metal / Metal / Metal /
/ ZrO2 (c) Ceramic Semiconductor Polymer
Figure 9-1. Classification for different internal interfaces (Cahn and Kalonji, 1982).
538
9 Structures of Interfaces in Crystalline Solids
boundary diffusivity (Turnbull and Hofmann, 1954) were early triumphs of the evolving understanding of real crystal properties in terms of lattice dislocations. In contrast, for high-angle boundaries the development of structural models has been much slower and more difficult. Theory and experimental techniques for testing detailed models have only recently become available. New techniques, especially X-ray scattering and high resolution electron microscopy resulted in exciting developments which were the theme of many recent review articles and conference proceedings (Aucouturier, 1984, 1989; Balluffi, 1979; Balluffi and Sutton, 1993; Chadwick and Smith, 1976; Gleiter and Chalmers, 1972; Ishida, 1986, 1987; Murr, 1975). In contrast to grain boundaries, heterophase boundaries are more complicated. Only quite recently, first steps have been taken towards an understanding of the bonding and structure, particularly of metal/ceramic heterophase boundaries (Fischmeister, 1985; Riihle et al., 1990, 1992). In contrast to grain boundaries, the engineering control of phase boundaries has yet to be developed. The mechanical properties of advanced materials are not presently understood at an atomic scale. The present article will describe the crystallographic aspects of homophase and heterophase boundaries (Sees. 9.2 and 9.3), the simulation of interfaces (Sees. 9.4 and 9.5), and the experimental techniques X-ray diffraction and high resolution electron microscopy (Sees. 9.6 and 9.7). Results are summarized in Sec. 9.8, and conclusions drawn in Sec. 9.9.
9.2 Homophase Boundaries Geometrical Aspects In this section we discuss planar boundaries between two crystals A and B. In general, nine parameters are required to describe the macroscopic geometry of such a boundary (Balluffi et al., 1981). Three parameters are required to describe the relative orientation of the two crystals, another three to describe the location of the boundary plane and a further three to describe the rigid body translation of A with respect to B. At a microscopic level, the boundary structure is by no means specified by these nine parameters as each atom has its own three degrees of freedom. We shall return to the microscopic description of grain boundary structure in Sec. 9.4, but note here that for randomly oriented boundaries there is little one can predict about the microscopic structure and properties. Fortunately there are many special boundaries, across which there is some symmetry relation between the two crystals and in which the atomic structure displays periodicity. The periodicity often reduces the number of degrees of freedom to manageable proportions. In the following we describe some general crystallographic concepts which, although developed to describe homophase boundaries, are helpful in the description and classification of both kinds of boundaries. For this reason, we shall describe the relative orientation of the crystals in terms of a linear transformation bringing one into coincidence with the other; in the case of a homophase boundary this can always be thought of as a rotation, whereas for heterophase boundaries a deformation of one of the lattices is required.
9.2 Homophase Boundaries - Geometrical Aspects
9.2.1 Crystallography 9.2.1.1 Coincidence Site Lattice The coincidence site lattice (CSL) has proved a useful concept in the crystallography of interfaces and in the description of dislocations in interfaces. It is defined as follows (see Fig. 9-2). Two crystal lattices A and B which meet at an interface are imagined to be extended through each other in both directions. This interpenetration of the lattices is a purely mathematical device. Crystal A or B is then translated so that a lattice point in A coincides with one in B. This point which we label O is desig-
nated as the origin of coordinates (Fig. 9-3). Now it is possible that no other lattice points of A and B coincide, in which case O is the only common lattice point. However for many special orientations of the crystals there will be other coincidences. The coinciding lattice points if there are any form a regular lattice, which is known as the CSL. Fortes (1979) has shown how CSL rotations may be generated. In hexagonal crystals, there can be no three-dimensional CSL in general because the c/a ratio is irrational. A CSL can only be found if (c/a)2 is a rational fraction. An approximate CSL for the real hexagonal system
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Figure 9-2. Coincidence site lattices (Z5) formed by interpenetration of (a) like crystals, and (b) unlike crystals. The unit cell of the CSL is outlined to the left, and the linear transformation relating the two lattices is shown on the right in each case.
Figure 9-3. An (001) projection of the O-lattice between two simple cubic lattices with different lattice parameters, (a) The atoms of each crystal are represented by dots and crosses and the O-points are circled, (b) Showing the lines midway between the Opoints which can be geometrically regarded as the cores of dislocations. After Smith and Pond (1976).
540
9 Structures of Interfaces in Crystalline Solids
can nevertheless be defined as the CSL of the nearest reference structure obtained by stretching or compressing the real structure until its (c/a)2 is rational. For example in Zr, with c/a = 1.595, a convenient assumption is (c/a)2 = 5/2, c/a =1.581. Similar considerations apply to tetragonal and rhombohedral systems. The construction of CSL lattices in hexagonal, tetragonal and rhombohedral structures is discussed with further references by Warrington (1975), Chen and King (1988), Gertsman (1989), Grimmer (1990), King and Shin (1990). Real boundaries often occur in an orientation which can be associated with a CSL. The simplest examples are perhaps symmetric tilt boundaries in a cubic crystal, a special case of which is the (111) twin in face-centered cubic (f.c.c.) crystals. The stacking sequence ABCABC... at such a twin is reversed on passing through the boundary. In the real twin, there is a degree of translation of the two halves normal to the boundary, because the spacing between (111) planes near the boundary does not exactly keep its bulk value. After removing this translation normal to the boundary, if the crystals are imagined to interpenetrate, every third (111) plane in one crystal coincides with a (111) plane in the other. The CSL in this case has lattice points on every third (111) plane of atoms. Boundaries are classified according to the ratio of the volume of a unit cell of the CSL to the volume of a unit cell of A or B. This value is denoted by I. Thus a low value of I implies a high frequency of coincidences of the interpenetrating lattices and at the other extreme I = oo implies a completely incommensurate or random orientation. Boundaries with a relatively low (i.e. unambiguously measurable) value of I are referred to as coincidence boundaries and are sometimes associated with
special physical properties such as low interfacial free energy or high mobility (for reviews see Chadwick and Smith, 1976; Balluffi, 1977, 1979). As an example, Fig. 9-2 a shows a I 5 CSL for simple cubic crystals, where both crystal lattices are rotated about the cubic axis normal to the paper. If the boundary is parallel to the drawing plane it is a twist boundary with a twist angle of 36.9°. Figure 9-2b represents a similar CSL for two unlike crystals which have a special ratio of lattice constants. The (111) twin in f.c.c. mentioned above has a I value of 3 and is referred to as a S3 boundary. A boundary plane which is also a plane of the CSL has a certain planar density of coincidence sites D per unit area, which may also be relevant to structural properties (Pond, 1974). The area \/D is geometrically important. For CSL boundary planes, periodically repeating unit cells of the bicrystal can be defined (unit cells of the CSL) whose faces pave the boundary. The area of one of these faces is \/D. An early idea was that boundaries with a high D would have a low free energy because the atoms occupying coincidence sites are in their bulk equilibrium positions referred to either lattice. This is not tenable as a general rule, because in a real boundary the geometrical coincidences are destroyed by relative translations of the grains or by atomic rearrangements (relaxation) at the interface. Low I values and other purely geometrical criteria cannot be correlated directly with measured physical properties of boundaries, as Sutton and Balluffi (1987) have discussed. 9.2.1.2 O-Lattice A generalization of the CSL can be made called the O-lattice which is sometimes a helpful concept. Start with two ide-
9.2 Homophase Boundaries - Geometrical Aspects
al interpenetrating lattices A and B coinciding at one point which we take as the origin. We remark in advance that the definition of the O-lattice, like the CSL is independent of the location of any interface, although in the end we shall only be concerned with lattice points lying in the boundary. Consider a linear homogeneous mapping which brings A into complete coincidence with B at every site. The mapping may be a pure rotation or a shear or some other operation which can be represented by a matrix multiplying a lattice vector referred to the chosen origin. The coincidence sites are completely equivalent to each other, so any one of them could be regarded as the origin of the transformation. In general, there are even more points which could be taken as the origin of the mapping and all such points together form a set called the set of O-points (O for Origin) (Bollmann, 1970). There may be complete lines of O-points, referred to as Olines, as on the axis of a rotation, or planes of O-points, referred to as O-planes, as on the invariant plane of a shear transformation. The CSL is a subset of the O-points. O-points are located wherever the interior coordinates of a point in a unit cell of lattice A referred to the origin and axes of that unit cell are the same as the interior coordinates of the same point referred to the origin and axes of a unit cell in lattice B. A conceptual advantage of the O-lattice over the CSL is that the set of O-points moves continuously as crystal A is rotated or deformed with respect to crystal B, whereas CSL points disappear and appear abruptly. There are O-points even when there are no coincidences. A consequent feature of the O-lattice concept is that it applies to phase boundaries in which the two crystals are incommensurate; their lattices need only to be related by a matrix of strain and rotation. A very simple example
541
is the case of two cubic lattices in the same orientation with slightly different lattice parameters. One is transformed into the other by a homogeneous dilation. If their (100) faces are in contact, a square array of O-points is formed where the O-lattice intersects the boundary. Lines bisecting the array of O-points in the boundary can be geometrically regarded as the cores of dislocations which accommodate the misfit. There is thus a close relationship between the O-lattice and the geometrical theory of interfacial dislocations (Bollmann, 1974) (Fig. 9-3). Such geometrical dislocations are a mathematical device, and do not necessarily correspond to physical dislocations, which are observable atomic structures. The principle utility of an O-lattice construction is that it enables the geometrical location and Burgers vectors of interfacial dislocations to be discussed. A weakness in the concept of an O-lattice stems from the fact that for a given orientation of A and B, the transformation which brings A into coincidence with B is not representable by a unique matrix. Since the O-lattice is by definition a property of the transformation and not of the orientations it is perhaps no surprise that it is also not unique. A simple example serves to illustrate this point. Consider a symmetrical tilt boundary in a cubic crystal, in which lattice A is generated from lattice B by a certain rotation about the tilt axis. This axis of rotation is clearly an Oline, and there is a lattice of such lines intersecting the CSL-points. An alternative transformation from A to B would be a shear on the boundary plane. In this case we have an O-lattice which is a stack of O-planes, consisting of the plane of the boundary itself and all the similar parallel planes which intersect CSL-points. This non-uniqueness is why one must be cautious about attaching any physical signifi-
542
9 Structures of Interfaces in Crystalline Solids
cance in terms of observable dislocations to the mathematical dislocations defined by means of the Olattice. To determine the location of O-lattice points it is straighforward to show that one has to solve a simple linear matrix equation, namely (9-1) xAxB where xA is any lattice point in crystal A, xB is any lattice point in crystal B, 0 p is an O-point, xA0P is the vector joining xA to 0 p , xAxB is the vector joining xA to x B , L is a 3 x 3 linear transformation matrix which operating about O (or about any of the O-points such as 0 p to be determined) brings crystal A into coincidence with crystal B and 1 is the unit matrix (see Fig. 9-4). A complete set of O-points is given by the solutions to this equation for all xA and xB. In fact it is sufficient to choose a single xB and range over xA in order to generate the complete set. If a coincidence site is present at x B , then for the particular choice xA = xB Eq. (9-1) for an O-point is satisfied: the right-hand side is zero in this case, so that a solution is Op = xA. Further more detailed discussions of the O-lattice are \ - l
Figure 9-4. Illustration of the definition of an Opoint Op [see Eq. (9-1)]. If a lattice point xA is transformed to a lattice point xB by the linear transformation L acting about O then O is an O-point.
given by Goux (1974), Pond (1974) and Sutton (1985). In order to see the difference between the CSL lattice and the O-lattice we refer to Fig. 9-5, which represents a cube-oncube interface in four degrees of mismatch. The two lattice constants aA and aB are related by aB = laA with / ranging from 1/2 to 4/9 in this figure. The lattice constant of the O-lattice is determined by the condition ao = (n — \)aA = naB, where n is a real number, but not necessarily an integer. The solution for n is n = (1 — /) ~1, with the result that ao = (1 — I)"1 aB. The relation to the more general Eq. (9-1) is obvious. In the cases depicted in Fig. 9-5, the CSL and the O-lattice are identical only in Fig. 9-5 a. This is the case n — 29 and we can see that it is only for integer values of n that the CSL and O-lattices are identical. In Figs. 9-5 b and 9-5 c only the O-point in the upper left corner belongs to the CSL. In Fig. 9-5 d all four corners are also CSL points. 9.2.1.3 DSC Lattice Another lattice which is of importance for the discussion of isolated dislocations and steps at boundaries is called the DSC lattice sometimes referred to as Displacement Shift Complete although not historically an acronym at all (Bollmann, 1970) (see Fig. 9-6). This is defined as the coarsest lattice which includes the lattices of A and B (in an orientation for which there is some coincidence) as sublattices. Any vector joining a lattice point of crystal A to a lattice point of crystals A or B, and vice versa, is also a vector of the DSC lattice. Thus if either lattice is translated by a DSC vector, the complete pattern of the interpenetrating lattices and the CSL is either invariant or is simply displaced by the same vector. The DSC lattice is unique to the given orientations of A and B and is
9.2 Homophase Boundaries - Geometrical Aspects O+CSL
O+CSL
•
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543
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Figure 9-5. The O-lattice: example of a cube-on-cube interface with four degrees of mismatch. The cubic lattice of black atoms is progressively dilated with respect to the lattice of white atoms. The smaller black spots [visible in (b) and (c)] denote the O-lattice sites corresponding to this transformation. Unit cells of the black and white lattices are drawn dashed and solid respectively. The transformation centered on the O-point which is within both of these cells (labeled O) maps the white onto the black lattice. The top left hand site is also an O-point which remains so in each stage of dilation (a)-(d). As the black lattice expands, the O-lattice contracts, (a) A E 8 coincidence boundary occurs when the black lattice parameter is exactly twice that of the reference (white lattice). The O-points are also all CSL-points and occur at each black site, (b) The dilation of the black lattice is about 4%. The O-lattice has contracted somewhat. Only the O-point at the top left of the figure remains a CSL-point within that part of the crystal which is illustrated, (c) The dilation of the black lattice is about 8%. The O-lattice has contracted further, (d) The dilation of the black lattice is exactly 12.5%. This is now a Z729 coincidence boundary, and the O-points at each corner of the figure are also CSL points.
useful for predicting or explaining the observable (physical) dislocations in boundaries, as we describe in Sec. 9.2.3. Further discussion of CSL and DSC properties including a method of constructing them for cubic crystals is given by Grimmer et al. (1974). 9.2.2 Small Angle Boundaries
A small angle boundary is a boundary between two identical crystals that are nearly in complete coincidence. In reality it
appears as a planar array of dislocations in a single crystal, and it can be described geometrically as such. The dislocations localize the misfit, leaving areas of perfect matching crystal between them. This idea seems to have been first introduced by Taylor (1934), Burgers (1940) and Bragg (1940). A mathematical description of grain boundaries in terms of dislocations has been developed, notably by Frank (1950), Christian (1975) and Bilby etal. (1955) which can also be applied with caution to heterophase boundaries and large
544
9 Structures of Interfaces in Crystalline Solids
(a)
LOQ
Figure 9-6. The DSC lattice corresponding to the interpenetrating lattices illustrated in Fig. 9-2 a. The DSC lattice points are the points of the fine grid.
angle grain boundaries. The basic equation in this context relates the orientation of the crystals A and B to an equivalent planar dislocation distribution in A or B. As in the discussion of the CSL, the half crystal lattices are imagined to be extended through each other, and the actual atomic positions are irrelevant; the discussion refers to the crystals as if the atoms remained on their perfect lattice sites. Consider a point Q in the boundary and an origin O, also in the boundary (Fig. 9-7). We now ask the question: what is the dislocation density at the interface equivalent to the misorientation, or what is the total Burgers vector crossing the boundary between O and Ql The answer is obtained by considering a closed Burgers circuit in the system [Frank's method, see Hirth and Lothe (1982)]. Starting from Q we take a closed path through B to O then from O through A back to Q. Now consider the linear transformation L that maps A onto B, or the inverse transformation L"1 that maps B onto A. It is not necessary at this
(b)
(c)
1
1
1
•Q
Figure 9-7. (a) Illustration of the Frank-Bilby-Christian formula, Eq. (9-2). The transformation L maps crystal A onto crystal B. Before the transformation, a path of the Burgers circuit starts at Q, goes through B and A and returns to Q. The Burgers vector b is the closure failure of the path after the transformation [Eq. (9-2)]. (b) Illustration of the same transformation referred to the median lattice, which gives the Frank formula relating the Burgers vector per length OQ to the boundary misorientation [Eq. (9-3)]. (c) A possible distribution of discrete dislocations which would give the total Burgers vector of Fig. 9-7 b above.
point to think of the atomic structure of A or B, but their lattices must be brought into coincidence by the transformation. For the discussion of a low angle boundary, the transformation is a pure rotation. However, the following derivation is valid for completely general linear transforma-
9.2 Homophase Boundaries - Geometrical Aspects
tions. In the general case, for example for a phase boundary, crystal A may be in a state of strain after the forward transformation L. Now apply the transformation L to crystal A, allowing the point Q to move with A to the position L * (Tg. The Burgers vector b is defined just as for bulk dislocations by the closure failure 0 Q — L * O g, thus (9-2) We refer to this result as the Frank-BilbyChristian formula (see Fig. 9-7). Note that it is not symmetric with respect to crystals A and B, since the crystal B was held fixed as the reference system while A was transformed. It is equally valid to take A as the reference system and transform B, in which case L is simply replaced by L"1. Another choice of reference system which is useful for rotations is the midpoint of the transformation, the so-called median lattice (Frank, 1950) into which A and B are transformed by equal and opposite rotations of 9/2 about an axis u. The resulting b is easily shown to be given by Frank's formula for a boundary misorientation 9:
= 2sin(9/2)(OQxu)
(9-3)
This is the central result for obtaining the dislocation representation of low angle grain boundaries (Fig. 9-7). The total b crossing (TQ may be expressed in terms of sets of discrete lattice dislocations (bulk dislocations). In general each set consists of parallel equidistant dislocation lines each with the same Burgers vector, and up to three sets are required with linearly independent Burgers vectors. However, this representation of b is not unique, and these purely crystallographic arguments cannot uniquely predict which sets of dislocations will actually occur in natural low angle boundaries, only what is geometrically possible. In the simplest cases of symmet-
545
ric low angle tilt boundaries this nonuniqueness causes no difficulties; only one set of dislocations appears. Their Burgers vector is predictable from the lattice structure hence their spacing is derivable from Eq. (9-3). It is simplyfe/[2sin(0/2)].In more complicated cases, particularly for higher angles of tilt or twist, there may be no useful solutions to the Frank-Bilby-Christian equation in the form of observable discrete dislocations. 9.2.3 Dislocations in Large Angle Boundaries
If the misorientation between crystals A and B is large, they are said to be separated by a large angle boundary. The formal theory of representing large angle boundaries in terms of dislocations is no different to that for small angle boundaries, but it merits some special discussion as it has been a source of controversy. There is no question that in the case of low angle boundaries, the discrete dislocations actually have physical significance. As predicted by Frank's formula, they are geometrically equivalent to the boundary between two crystals. Their individual strain contrasts in regions of otherwise perfect crystal can be observed in the electron microscope. The situation is different for large angle boundaries. Although large angle boundaries can be represented formally as dislocation arrays, no physically identifiable discrete dislocations can be predicted by this representation. In general, we refer to dislocations which formally satisfy geometrical requirements without necessarily being observable as geometrical, mathematical or crystallographic dislocations. The Frank-Bilby-Christian formula, Eq. (9-2), is closely related to the O-lattice introduced previously. In the Frank-Bilby-Christian formula, b connects a partic-
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9 Structures of Interfaces in Crystalline Solids
ular point we designate as being in crystal B (the point to which Q is transformed by L) to a point we designate as being in crystal A (the original point Q). O is by definition an O-point, because it is the origin of the transformation. If we now impose the condition that Q is also an O-point, then the position of Q within the unit cell of B before the transformation is the same as its position in the unit cell of A before the transformation; that is the definition of an O-point. After the transformation Q has in general been carried to a different unit cell of B, which now coincides with the particular unit cell of A within which Q was carried by the transformation. Since its position in the unit cell of A is not changed by a linear transformation, its new position in the unit cell of B is the same as its untransformed position in the unit cell of B. In other words, we have proved that b is a lattice vector of B. By working this argument backwards the converse is also easily proved, that if A is a lattice vector of B then Q as well as O must be an O-point. This result is also expressed by the O-lattice Eq. (9-1), in which xA corresponds to Q, Op to O and xAxB to b. With this substitution it is seen that the O-lattice equation is a special case of the Frank-Bilby-Christian equation. The O-lattice approach and the Frank-Bilby-Christian approach are equivalent ways of describing the mathematical structure of crystalline interfaces in terms of dislocations (Knowles, 1982). These are sometimes referred to as primary dislocations to distinguish them from additional dislocations in the boundary which are referred to as secondary. In the previous discussion of the O-lattice we noted that the transformation L is not unique. Consequently according to the Frank-Bilby-Christian equation the dislocation representation b of a boundary is not unique. This non-uniqueness is in ad-
dition to the non-uniqueness of the discrete representation of b referred to previously. If crystallographic dislocations are added to a perfect boundary between A and B these secondary dislocations may or may not be observable entities. If the crystallographic dislocations exist as a regular array of secondary dislocations they serve to alter the original orientation of the boundary. One can define new primary dislocations corresponding to the reoriented boundary, in which case the secondary dislocations have been defined away. Thus the observability of regularly spaced secondary dislocations cannot be established simply by crystallography, it depends on the atomistic relaxation at the interface. A thorough discussion is in Sutton (1984). So far our discussion has been concerned with arrays of dislocations as a description of the misorientation of two crystals. The situation is rather different if we consider the geometry of a single dislocation in an otherwise perfect boundary. The crystallography of this situation, and that of the more complex case of a single partial dislocation in a boundary, in which the dislocation separates regions of the boundary with different structures, have been thoroughly analyzed by Pond and coworkers (Pond, 1977,1983; Pond and Bollmann, 1979; Pond and Vlachavas, 1983). The dislocation at an initially perfect boundary is defined by a Volterra-like process (see Fig. 9-8). As in the previous theories we have described, the definition of a dislocation refers to perfect unrelaxed lattices. However, the dislocation so defined is a topological feature which does not alter its character (the Burgers vector) when the atoms relax locally. The process is as follows: In the continuum theory an infinite straight hollow cylinder of vanishingly small radius is created with its axis in
9.2 Homophase Boundaries - Geometrical Aspects
Figure 9-8. Schematic diagram of the Volterra process for creating a dislocation in a boundary.
the boundary. This will be the dislocation core. This cylinder is unnecessary if discrete lattices are explicitly considered. Now a planar cut is made in the boundary up to the core and the material above and below the cut is translated by tA and tB respectively. Material is then added or removed to restore the match across the plane and the boundary is thereby repaired. The Burgers vector of the secondary dislocation so defined is tA — tB. The translational symmetry of the crystals requires that tA — tB is a vector of the DSC lattice. Such dislocations are called ReadShockley dislocations after the authors who first discussed them. They differ from lattice dislocations in crystals A or B whose Burgers vectors are tA or tB respectively. Read-Shockley dislocations are associated with a step equal in height to the component of tA — tB normal to the boundary. They can also exist and have been observed at interphase boundaries. There are other kinds of dislocations whose Burgers vector is not a vector of the DSC lattice, called supplementary displacement dislocations. They are possible when at least one of the crystals is not symmor-
547
phic, such as for example a crystal of the diamond structure. They arise if one of the translations tA, tB is not a lattice vector but a glide or screw displacement (such as the displacement a [111]/4 in silicon; see also Sec. 1.5.2 of Chap. 1). It is necessary that the two crystals have the same point group operation associated with different displacements. Then a supplementary displacement dislocation would separate domains of equivalent structures. Besides the crystal dislocations, ReadShockley dislocations and supplementary displacement dislocations already described, other less common kinds of line defect can exist at crystalline interfaces. We mention them only and refer readers to the papers of Pond and co-workers for detailed descriptions (Pond, 1977, 1983; Pond and Vlachavas, 1983). Firstly there is the possible case which arises if, in the Volterra process, instead of combining displacements we combine a point group operation on the lattice of A with one on B. The resulting defect is called an interfacial disclination. If in addition at least one of the lattices is non-symmorphic, point group operations can be combined with translations in the Volterra process and the result is called an interfacial dispiration. Finally we note that the foregoing geometrical description of dislocations at interfaces is incomplete as a description of real interfaces unless atomic relaxations are taken into account. Although atomic relaxation cannot alter the total Burgers vector of an interfacial dislocation, it can induce dissociations, reorientation of the boundary or redistribution of the Burgers vector from the presumed core, all of which profoundly modify the structure of the boundary. For example, consider the secondary dislocation network predicted in order to relieve the mismatch from a CSL orientation. The dislocations actually
548
9 Structures of Interfaces in Crystalline Solids
occurring in nature, or in a relaxed atomistic model, may have a slightly different Burgers vector to the prediction of DSC theory. The naturally occurring Burgers vector can nevertheless satisfy the crystallographic constraints by means of inequalities in the spacing of the dislocations. Furthermore, new types of dislocations can be introduced by relaxation. For example, the unrelaxed bicrystal may have some special symmetry, such as a mirror in the boundary, which is broken by atomistic relaxation. In that case two degenerate, symmetry related structures in the boundary are possible. Special dislocations, called relaxation diplacement dislocations, would separate domains of such structures. The atomistic simulation of boundaries will be discussed in Sec. 9.4.
9.3 Heterophase Boundaries Geometrical Aspects Heterophase boundaries can be described geometrically in the same terms as homophase boundaries. Although there is no exact CSL unless the lattices of the two phases happen by chance or design to contain vectors in a rational ratio to each other, an O-lattice can be constructed as described in Sec. 9.2.1. Interfaces typically contain a lattice of O-points, bisected by lines of geometrical dislocations. One often deals with crystals of identical lattice structure but different lattice constants, in the same geometrical orientation, perhaps the simplest being two cubic crystals in the so-called cube-on-cube orientation, which means that their crystal axes are parallel. A particularly simple boundary of cube-oncube orientation is one parallel to {100} (Fig. 9-3). The geometrical dislocations lie on a square lattice parallel to the cube ax-
es, the spacing of which varies inversely with the misfit. Consider a real but perfect interface in which the atoms are relaxed from their perfect crystal sites. The geometrical dislocation lines may be associated with real dislocation cores, at which the misfit across the interface is localized and between which there are regions of the interface across which the lattices match well. Another possibility is that the dislocations dissociate into partials, separating regions of perfect, but not identical, interface. The dislocations whether whole or partial whose role it is to localize the misfit are called misfit dislocations. One refers to interfaces as coherent, semicoherent or incoherent. These terms mean that the interface is perfectly matched, that it consists of regions of perfect matching separated by misfit dislocations, or that there are no perfectly matched regions respectively. Misfit dislocations are of great interest in electronics, because of the importance of lattice-matched III-V compound semiconductor junctions in high-performance devices. The performance of the devices is very sensitive to the structure of the interfaces. Jain et al. (1990) have reviewed the theoretical and experimental work on the structure of Ge^S^..^ strained layers and superlattices and they include a very extensive bibliography. A key question is: if a layer is grown epitaxially, to what thickness can it be grown until the coherent boundary becomes unstable to the formation of misfit dislocations? This is called the critical thickness. A theoretical answer was provided by the model of Frank and van der Merwe (Frank and van der Merwe, 1949; van der Merwe and Ball, 1975). The idea is that the introduction of a misfit dislocation is energetically favorable when the energy of such a dislocation is
9.4 Simulation of Homophase Boundary Structures
less than the relief of elastic strain energy which it provides. There have been several refinements of the original model, as discussed by Jain et al. (1990), but it is generally recognized that the observed critical thicknesses are larger than those predicted by energy balance considerations, and that kinetic effects or the availability of dislocations play an essential role.
9.4 Simulation of Homophase Boundary Structures 9.4.1 Interatomic Potentials We now discuss the theoretical methods which have been used to predict the positions of atoms at an interface. Although the geometrical orientation of the two surfaces which meet imposes some constraints on the structure of the interface, the freedom of the local atomic coordinates still allows a huge variety of structures. The equilibrium atomic structure adopted in reality is that which minimizes its free energy and in general this bears no resemblance to a simple juxtaposition of perfectly terminated free surfaces. Even an educated guess at the structure, based on an understanding of interfacial crystallography, can be far from the one observed by microscopy. Detailed prediction of boundary structures, given the orientation of the two half-crystals, is the province of the computational science known as atomistic simulation. The theoretical prediction of a boundary structure by atomistic simulation can never be totally reliable. For one thing, the number of degrees of freedom is generally too large for them all to be explored. However, at least some plausible alternatives for the structure can be derived by the techniques to be described. Another basic
549
problem is the uncertainty in the calculation of the free energy associated with any given structure. A critical input to all types of calculation is the description of the energy and interatomic forces, which we now discuss. It is generally accepted that the electronic coordinates are not independent variables in the calculation of total energy. This is the essence of the Born-Oppenheimer or adiabatic approximation, which says that once the positions of the atomic nuclei are specified, the wave functions of the electrons and hence the electronic charge density are uniquely determined. Hence the total potential energy of the solid is a unique function of the atomic (i.e. nuclear) coordinates. If one knew what the corresponding electronic charge density was one could calculate the force on each nucleus purely by electrostatics (the Hellmann-Feynman theorem). Until very recently this has only been attempted in systems with few atoms or for simple metals (to be discussed later); for systems involving transition metals and thousands of independent atomic coordinates a variety of more empirical methods have been used to model the interatomic forces. The most common approach for all metals and insulators has been to assume that the atoms are classical point objects interacting via pairwise potentials. A number of ways have been followed to derive these potentials. Firstly, for simple s-p bonded metals, e.g. Li, Na, K, Mg, Al, such interatomic potentials can be derived rigorously from the electrostatics of interacting spherical screening clouds of electrons. This is because the ion cores in such systems are a weak perturbation to the conduction electrons which screen them in a linear additive way. From the point of view of the conduction electrons, the ion cores may be represented by a weak poten-
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9 Structures of Interfaces in Crystalline Solids
tial with no core states, called a pseudopotential (see for example Harrison, 1966). The resulting electron density is a superposition of spherical charge densities, hence the interatomic forces are pairwise additive. The resulting pairwise potentials may be completely ab initio, in the sense that they are derived without fitting any parameters to experimental data. Recent ab initio potentials of this type have been published by Walker and Taylor (1990 a, b). Further theoretical background to pair potentials in simple metals is given in the extensive articles by Heine etal. (1990), Pettifor (1987) and Hafner (1987). The presence of d-electrons at or near the Fermi level spoils this simple linear screening picture. For the transition and noble metals therefore ab initio pair potentials cannot be derived. The traditional approach has been empirical. Pairwise potentials have been generated by fitting more or less arbitrary functions to bulk properties such as cohesive energy, lattice parameter, elastic constants and vacancy formation energy (Gehlen etal., 1972; Crocker etal., 1980; Vitek and Srolovitz, 1989). Hexagonal metals have generally presented a problem, because apart from Mg they are not nearly "free electron like" and the non-ideal axial ratios are not reproduced by pairwise potentials. Hence, generic pair potentials have been used, by which one hopes at least to understand general features which depend on the h.c.p. lattice, without modeling any particular metal (e.g. Serra and Bacon, 1986). A more realistic class of empirical or semi-empirical potentials for transition and noble metals goes beyond the purely pairwise description of the interaction. These are called second-moment models (Cyrot-Lackmann, 1967; Allan, 1970; Ducastelle, 1970; Allan and Lannoo, 1976; Rosato et al., 1989), effective medium mod-
els (Norskov and Lang, 1980; Stott and Zaremba, 1980; Jacobsen et al., 1987), embedded atom models (Daw and Baskes, 1984; Foiles etal., 1986), Finnis-Sinclair potentials (Finnis and Sinclair, 1984; Ackland et al., 1987), or the glue model (Ercolessi etal., 1986; Heine etal., 1990). We refer to this class of models as isotropic N-body potentials. They all have the following functional form for the energy at zero temperature: l
(9-4) (9-5)
where the summations are over lattice sites. The function F, called the embedding function or the glue function, is a square root in the second moment model and in Finnis-Sinclair potentials. This square root behavior is derived from a simplified tight-binding description of electronic densities of states. In the other approaches the embedding function is derived from the energy of an atom embedded in a homogeneous free electron gas of the local density Qt at the given atomic site. In either case it is a negative concave function of the density Q. The function 3> is the density function which is sometimes interpreted as the electronic charge density of a free atom, otherwise as a measure of the squared interatomic matrix elements of a tight-binding Hamiltonian, or even simply as an empirical quantity with which to define the local density of atoms or electrons. V(R) is a pairwise interatomic potential, containing parameters to be fitted to data. A detailed discussion of models of this kind is given by Finnis et al. (1988) and Carlsson (1990). Vitek and Srolovitz (1989) also has useful discussions and references. The first such potentials for hexagonal metals were derived by Oh and Johnson (1988) (Mg,
9.4 Simulation of Homophase Boundary Structures
Co, Ti, Zr) and by Igarachi et al. (1991) (Co, Zr, Ti, Ru, Hf, Zn, Mg, Be), and, in contrast to pair potentials, non-ideal c/a ratios are successfully included in the procedure of parameter fitting. For practical purposes the isotropic TVbody potentials are very similiar in behavior to pairwise potentials and almost as straightforward to compute. For f.c.c. pure metals they have been most thoroughly tested and seem to work well. Their limitation lies in the fact that they take no account of the bond-angle dependence of the total energy. Unfortunately the importance of the neglected angular forces is sometimes very obvious, as when the potentials fail to reproduce the negative Cauchy pressure of certain cubic crystals (Pettifor and Aoki, 1991; see also Sec. 2.3.3 and Table 2-4 of Chap. 2), or in the physical origin of the structural stability of b.c.c. metals (Pettifor, 1987), so circumspection is always called for in their application. The next step up the ladder of realism in metallic potentials takes us to 3- and 4body potentials and to tight-binding models beyond the second-moment approximation. These approaches are only just beginning to be used for the simulation of solidsolid boundaries, although results are in the literature for surfaces and dislocations in b.c.c. metals. In principle, the most accurate predictions would be made by using ab initio methods. They have not yet been applied to predict boundary structures in metals, due to their computational cost, but this situation is changing rapidly. For insulators, simulations have used the Born model (Born and Huang, 1954), in which the cohesive part of the total energy is the pairwise summation of the coulomb interaction of the ions. This is counterbalanced by a short ranged repulsive energy which is either fitted to the lattice parame-
551
ter and bulk modulus of the perfect crystal or determined by the electron gas method (Gordon and Kim, 1972; Kim and Gordon, 1974) or from a Hartree-Fock calculation (Pyper, 1986). The Born model has also been extended to include polarizable shells around the ions. Harding (1990) has reviewed further improvements and some applications of the ionic model to atomistic simulation. Details and references are given in the papers which have applied it to grain boundaries (Duffy and Tasker, 1983, 1986a, b; Duffy, 1986; Mackrodt, 1987; Bingham et al., 1989) and interfaces (Tasker and Stoneham, 1987; Cotter et al., 1988; Allan et al., 1989; Davies et al., 1989; Kenway et al., 1989); homophase and heterophase boundaries are treated no differently in principle, although the latter are likely to be more computationally expensive. In semiconductors, a number of empirical angular dependent interatomic potentials have been developed, motivated by the stability of the fourfold coordination which results from sp3-hybrid bonds. The best known are due to Stillinger and Weber (1985) and Tersoff (1988 a, b), although several others have been proposed. A Stillinger- Weber potential for Ge was parameterized by Ding and Anderson (1986). None can be regarded as completely reliable for structural predictions since they are fitted to bulk properties and the energy associated with dangling bonds or rehybridization remains poorly represented. The energy of stacking faults is also incorrectly described. The real-space tight-binding method has been applied to grain boundaries in Si to calculate both electronic and atomic structures (Paxton and Sutton, 1989). This method is more reliable than purely empirical methods, but two or three orders of magnitude more costly to compute.
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9 Structures of Interfaces in Crystalline Solids
In semiconductors ab-initio pseudopotential methods without the approximation of linear screening and the consequent pair potential representation have made rapid progress since 1985, when Car and Parrinello (1985) showed how the Schrodinger equation with a basis of thousands of plane waves could be solved for the wave functions with a simultaneous adjustment of the positions of the ions; either to solve their equation of motion or to minimize the total energy. There are several fully self-consistent methods of solving the Schrodinger equation and calculating total energies all of which are based on expanding the wave functions in a Fourier series of plane waves (Payne et al., 1986). The state of the art at the end of 1991 (Stich, 1991) is that the total energy has been minimized for a system of over 400 silicon atoms in a periodically repeated cell. The classical concept of explicit interatomic potentials or interatomic forces no longer applies to these calculations, which obtain the total force on each ion from all the electrons and other ions in the system. Other self-consistent ab initio methods such as the full potential linear combination of muffin-tin orbitals method (FPLMTO) or the linear combination of augmented plane waves method (FLAPW) have been applied to the calculation of interface energies, but can only handle far fewer atoms in the repeated cell. These latter methods are also computationally very costly and full relaxation of all atomic positions to find the minimum energy configuration is not yet carried out. 9.4.2 Simulation Techniques
The techniques of atomistic simulation of interfaces are simple in principle but complicated and diverse in their detailed implementation. In this section we outline
the principles without going into details of the algorithms, which are documented in the literature. The interface is usually constructed within a periodically repeated cell, called the simulation cell or just "the box". Sometimes the box is repeated in three dimensions, in which case the simulation actually represents an infinite stack of interfaces. More often the direction normal to the interface (z) is treated as special. One scheme is to introduce free surfaces at a reasonable distance from the interface such that the interaction between the surfaces and the interfaces is negligible. This usually needs several tens of atomic planes on either side of the interface. For computational convenience, so that the same periodic boundary conditions, apply in x, y and z directions, the resulting sandwiches are still sometimes periodically repeated in the z-direction with sufficient intervening vacuum that they do not interact. Another scheme is to assume that the layers are joined smoothly atom by atom at a suitable distance from the interface onto semiinfinite rigid crystals. These rigid crystals have three degrees of translational freedom, so they "float" on the sandwich of atoms containing the boundary. The latter scheme has the advantage of eliminating surface effects while keeping the number of atomic degrees of freedom as small as possible to minimize the computing cost. Given a starting structure, a choice of boundary conditions and an algorithm for calculating the total energy and forces on the atoms, there are various kinds of calculation which can be done to seek the equilibrium structure. First and foremost is the search for the static zero temperature equilibrium structure. The most direct method of doing this is an iterative adjustment of the atomic positions until all the forces vanish. We shall refer to such an approach as molecu-
9.4 Simulation of Homophase Boundary Structures
lar statics. For this purpose a number of algorithms have become established, e.g. the methods of conjugate gradients (Fletcher and Reeves, 1964) or generalizations of Newton-Raphson called variable metric methods, described for example in the book by Lootsma (1972). The choice of which method to implement in which systems has always involved a compromise between the three considerations of computer time, coding convenience and required memory. The Newton-Raphson and related methods are generally more efficient than conjugate gradients but require more memory as a matrix of second derivatives of the total energy with respect to atomic positions has to be held and updated at each iteration step. The second kind of calculation is the simulation of the system at a finite temperature. A critical review is given by Pontikis (1988). The most straightforward approach is a dynamical simulation, in which the classical equations of motion are solved numerically. Such dynamical simulations are referred to as molecular dynamics (MD). The positions of all the atoms are updated at discrete time steps by a method such as the Gear predictor-corrector algorithm. The simplest version of MD simulates a microcanonical ensemble (constant internal energy). More sophisticated versions can simulate constant temperature as well as constant stress conditions. The penalty to be paid for imposing these constraints of constant temperature or constant stress is that the atoms depart slightly from the trajectories of Newtonian mechanics. This effect is normally insignificant. By quenching the motions of the atoms, a MD program can also be used to seek the zero temperature equilibrium structure, and indeed it has the advantage over straightforward molecular statics or
553
energy minimization that local minima in the energy are less likely to trap the system. A slow quenching process based on MD is sometimes called simulated annealing. These techniques have been used in conjunction with the plane wave methods of total energy calculation as well as with the interatomic force models. A different type of finite temperature calculation is based on the Metropolis Monte Carlo (MC) algorithm (Metropolis etal., 1953; Binder, 1984). This approach cannot follow dynamical processes in the system under study. Its aim is rather to generate members of a microcanonical or canonical ensemble. The method has been most fruitful in the study of heterophase boundaries, where it has the advantage over MD of exploring within a reasonable computation time states in which an exchange of atomic species has taken place. With MD much less of the phase space which might be relevant is actually accessible to the system within the small time scale covered by a simulation, which is typically only a few picoseconds. MD also requires conservation of the number of particles of each species. Applications of MC are described in the literature (Foiles, 1985; Rogers et al., 1990; Bacher and Wynblatt, 1991; Bacher etal., 1991; Binder, 1992). In the case of first-principles calculations, there are in addition a number of algorithms which deal specifically with the problem of minimizing the total energy with respect to the electronic coordinates, sometimes simultaneously with respect to the positions of the ions (Payne et al. 1986; Remler and Madden, 1990). In first-principles MD calculations, of the type pioneered by Car and Parrinello (1985), the electronic coordinates are not usually completely relaxed between updates of the atomic coordinates.
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9 Structures of Interfaces in Crystalline Solids
Recently a very simple way of doing finite temperature calculations has been developed (LeSar et al., 1989; Sutton, 1989a, b, c; Najafabadi et al., 1991), in which a static energy minimization is done, but with an effective potential that embodies the free energy of the crystal. The idea is based on a simplification of the full quasi-harmonic phonon density of states. Given the atomic positions, one can in principle evaluate the dynamical matrix and then the phonon density of states, from which the free energy can be obtained by a quadrature. This is computationally expensive and offers no practical way of minimizing the free energy with respect to the atomic positions. If however only the trace of the dynamical matrix is retained, that is the second moment of the phonon density of states, an approximation to the thermodynamic functions can be derived which appears as an additional pairwise interatomic term. From the resulting effective potential, effective temperature-dependent forces are derived with which to seek the free energy minimum. The latter approach has not been applied to ionic crystals. These present a general problem for simulation because the Coulomb potential is long-ranged. Special techniques are applied to sum it over the periodically repeated cells in two dimensions parallel to the boundary (Heyes etal., 1977). A computer program called MIDAS (Tasker, 1978), which is commercially available, has been designed for the static relaxation of boundaries in ionic crystals, using the floating block approach to treat the boundary conditions far from the interface. 9.4.3 Results The results of atomistic simulation of the structure of interfaces can be summa-
rized according to the structures and bonding types: (a) Simple s-p bonded metals, using pseudopotential based pair-potentials, (b) f.c.c. metals using empirical and isotropic TV-body potentials, (c) b.c.c. metals using empirical and isotropic Nbody potentials, (d) b.c.c. metals using simplified tight-binding methods, (e) semiconductors using empirical non-pairwise potentials, (f) semiconductors using firstprinciples methods, (g) ionic materials. One can categorize the types of boundary studied as (i) symmetric tilt and twin, (ii) asymmetric tilt, (iii) simple twist and (iv) mixed tilt and twist. It would be ideal if all types of boundary in all types of material could be studied by the most accurate, that is ab initio, methods, but these are only just becoming computationally practical for the simplest systems. Fortunately in most cases useful information is obtained even from very simple models, as we summarize here. B.C.C. metals In b.c.c. metals, symmetrical twist (Wolf, 1989) and tilt boundaries (Wolf, 1990 a, b, c) and random boundaries (Wolf, 1991a, b) have been studied using an empirical pair potential for oc-Fe (Johnson, 1964) and the Finnis-Sinclair potential for Mo as modified by Ackland and Thetford (1986). Some properties of random boundaries were also discussed by Brokman and Balluffi (1981) and Sutton (1991 a, b). Wolf and co-workers and Sutton have found a particular correlation between structure and energy of boundaries which applies to what Wolf and Phillpot (1989) called typical high angle grain boundaries, which are far from any special (low I) boundary. Their conclusion, supported by a large set of atomistic simulations, is that the energy of a boundary in-
555
9.4 Simulation of Homophase Boundary Structures
creases as the spacing of the atomic planes on either side parallel to the boundary decreases. A larger grain boundary expansion is also correlated with a smaller interplanar spacing. In general, less simulations have been carried out on the b.c.c. metals because the potentials are less satisfactory than for f.c.c. Paidar (1990) used a polynomial potential in which the h.c.p. structure was stable to stimulate the (110) tilt boundaries in the metastable b.c.c. structure. He verified the structural unit description (see next section) for the b.c.c. case. In addition he observed particularly extended relaxation of the atomic structure at the (332) boundary, which could be identified as the nucleation of the h.c.p. structure. H.C.P. metals For hexagonal metals, the field is less developed than for cubic because it has been difficult to derive potentials which are reliable for specific metals. A series of simulations of twin boundaries on {10Tl}, {1012}, {1121} and {1122} in generic h.c.p. metals using empirical pair potentials has been done by Bacon and co-workers (Serra and Bacon, 1986, 1991; Serra et al., 1988, 1991; De Diego and Bacon, 1991; Pond et al., 1991). The latter workers have also made comparative studies with an isotropic N-body potential for Mg (Igarachi et al., 1991). The configurations and trends found in both models were qualitatively the same. It was concluded that in metals with a near ideal c/a ratio, twinning dislocations in {1012} and {1121} twins have a planar form widely distributed along the boundary and are very glissile. The twinning dislocations in {1011} and {1122} twins are highly localized and are almost sessile. The experimental results on twinning discussed by Serra et al. (1991) are
consistent with these conclusions from the atomistic simulations, because twinning on {1011} and {1122} is more difficult. It was also concluded that differences in these dislocation core properties are more significant than differences in the twin boundary energies. F.C.C. metals A number of general results can be stated, which have emerged from atomistic simulations mainly on the f.c.c. cubic metals Al and Cu, an example of which is shown in Fig. 9-9. One of the first to be established is that there is frequently a rigid body translation of the grains, which removes the CSL. It was a major feature of tilt boundaries in Al studied by Pond and co-workers (Pond, 1977; Pond and Vitek,
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Figure 9-9. Calculated structure of the incoherent 13 twin boundary in Cu viewed along [110]. Its mean orientation is 8° from (1T2), i.e. the [111] direction is exactly parallel to the boundary. The picture shows two layers of atoms (circles and crosses). The solid arrow and the dashed arrow show the [111] and the [1T2] direction, respectively. The corresponding HREM image is shown in Fig. 9-29.
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9 Structures of Interfaces in Crystalline Solids
1977; Smith et al., 1977; Pond et al., 1978, 1979; Vitek et al., 1980) using a pair potential for Al based on a pseudopotential (Dagens etal., 1975). Another general result is the existence of a multiplicity of metastable structures for the same boundary orientation (Weins etal., 1971; Vitek etal., 1980, 1983; Oh and Vitek, 1986). Sometimes the structures are related by a symmetry element of the bicrystal, in which case they are degenerative in energy. Distinct structures may also occur which differ in the relative translation of the grains. One is the equilibrium structure in the model and the others are metastable. There is in general a normal component of the rigid body displacement which is different for different metastable structures, so that they differ in their excess volume. This offers an interesting possibility for the existence of delocalized vacancies at the boundary in the form of regions of metastable structure of larger excess boundary volume. Examples and further discussion of the multiplicity of boundary structures are in the review by Sutton (1984). The predictions of computer simulation (Vitek et al., 1983) of the 15 boundary showed structural multiplicity due to different arrangements of structural units (capped trigonal prisms or tetrahedron units). These predictions have recently been verified by HREM on Au bicrystals (Krakow, 1991 a, b). The structures of [001] twist boundaries in Au have been a very fertile area of research and a source of some controversy. For Z113, Z25, £13, £17 and Z5 boundaries (twist angles 7.6°, 16.3°, 22.6°, 28.1° and 36.9°, respectively), the structures have been investigated by X-ray diffraction and computer simulation in a recent study by Majid et al. (1989). The most elusive structure has been that of the E 5 boundary, for which earlier work by Oh
and Vitek (1986) suggested a very long period or an aperiodic structure, composed of substructures they labeled A± and A 2 . The substructures, taken individually, were only metastable. This conclusion is contradicted by Majid et al. (1989), whose minimum energy structure is a CSL unit with relatively small lattice displacements (see Fig. 9-10), which also gives the best prediction of the structure factors of the X-ray diffraction pattern (see Sec. 9.6.1.6). Their simulations used the embedded atom model for Au. Recent ab initio calculations have been made by Needels etal. (1992) which confirm the lower energy of the Ma-
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(b) Figure 9-10. Two models for the Z5 (36.9°) [001] twist boundary in gold, (a) Model of Majid et al. (1989), which on this scale is indistinguishable from the perfect unrelaxed geometry, (b) Model of Fitzsimmons and Sass (1988 a, b), in which large atomic relaxation is visible. From Needels etal. (1992).
9.4 Simulation of Homophase Boundary Structures
jid, Bristowe and Balluffi structure. Finite temperature calculations (Srolovitz, 1992) suggest that in fact at room temperature a mixture of short period structural units is to be expected. Other atomistic simulations showed that the predicted equilibrium structures are dependent on the potentials. An example is the comparison of structures for [001] symmetric tilt boundaries and 13 twin boundaries in Cu and Al (Smith et al., 1977; Sutton and Vitek, 1983 a, b,c). The pair potential representing Al is positive and repulsive at nearest neighbors, whereas the potentials used for Cu have a minimum near the first neighbor distance, so the appearence of different structures is perhaps not surprising. More recent work (Wolf et al., 1992) on Cu suggests that other effects, in particular those responsible for the much lower stacking fault energy in Cu compared to Al, may be important in determining the structure of the incoherent Z3 boundary (Fig. 9-9). For Cu the EAM potential as well as an empirical pair potential (Crocker and Faridi, 1980) predicts the boundary at orientations near (211) to be exceptionally wide, extending over 1-2 nm, which is confirmed by high-resolution electron microscopy (see Fig. 9-34). This has been explained by Wolf et al. (1992) in terms of a thin boundary layer of the rhombohedral 9R phase which is formed in order to improve the atomic matching across the interface. This is the rhombohedral phase, previously known as a martensite structure in CuZn and CuAl alloys, which has also recently been identified in pure Cu precipitates within a b.c.c. Fe matrix (Othen et al., 1991). A most important result of atomistic simulations is how the crystallographic description in terms of secondary grain boundary dislocations manifests itself in the atomic structure. A useful concept in
557
this connection, which was based on the results of simulating f.c.c. metals, is the structural unit model, which we describe in Sec. 9.4.4. Semiconductors We mention here some grain boundary simulations on the semiconductors. Exploratory ab-initio calculations were made by Payne et al. (1986, 1987) on the Z5 (001) twist boundary in Ge. These authors relaxed the atomic positions by an iterative method related to the Car-Parrinello method (Car and Parrinello, 1985). Paxton and Sutton (1989) studied the electronic and atomic structures of symmetric (112), (310) and (111) tilt boundaries and stacking faults using a semi-empirical realspace tight-binding model. Kohyama et al. (1988 a, b) used a very similar method to model the (112) and (310) twins and obtained comparable results. Besides identifying a number of metastable structures, these authors found that the lowest energy structures contained only fourfold coordinated atoms. The (112) and (310) boundaries in Si were also simulated by Sutton (1989 a, b, c) using the semi-empirical Stillinger-Weber potential (Stillinger and Weber, 1985). The lowest energy structures were the same as those found by tightbinding. In the case of the (112) twin, five metastable structures were found both with tight-binding and with the Stillinger Weber potential. Although the two methods gave values for the boundary energies which differed for a given structure by up to a factor of two, the boundary structures were ranked identically in energy. The conclusion is that good qualitative information is obtainable from the semi-empirical potentials when the tetrahedral coordination is preserved, that is if the potentials are not applied to configurations too far
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9 Structures of Interfaces in Crystalline Solids
from those around which they were fitted. The Z9 [Oil] tilt boundary has been the subject of a number of simulations. Ab initio and semi-empirical tight-binding methods were used by DiVincenzo et al. (1986), which showed that the favored structure contains alternating five- and sevenfold rings of atoms. These structures were found in Z9, and also in the Z19 and £33 boundaries in diamond, Si and Ge studied by Narayan and Nandedkar (1991) using Stillinger-Weber and Tersoff potentials (Stillinger and Weber, 1985; Tersoff 1988 a, b), and are similar to the core structure of a (a/2)<110>{001} dislocation. Narayan and Nandedkar (1991) also studied in 13 twin boundary, which was found to contain bulk-like sixfold rings. They stated that the structures of 13 and 19 boundaries are consistent with their HRTEM observations on diamond. The structures obtained in diamond, Si and Ge were all very similar.
served by HREM (Merkle, 1990), that is the simulation indicates a larger interfacial excess volume than the HREM (see Sec. 9.8.3.1). A possible explanation for this discrepancy is extensive reconstruction of the grain boundary in reality, producing a configuration that could not be reached by the energy minimization procedure. Bingham et al. (1989) simulated grain boundary structures and ionic conductivity in tetragonal zirconia. They use two different empirical shell model potentials, PI and PII. The first of these took account of properties of both cubic and tetragonal crystals, while the second was fitted to cubic properties only. It is typical that the most relaxed boundary, namely the (112)/ [1T0] symmetric tilt, showed the most sensitivity of the structure to the potential used. The structures obtained with PI and PII were quite distinct. 9.4.4 Structural Unit Models
Ionic materials The state of the art in computer simulation of ceramics prior to 1981 was not satisfactory, as reviewed by Balluffi et al. (1981). Since then considerable progress has been made, in the simulation of bulk properties, then of surfaces and most recently of interfaces. Wolf (1987) calculated the energies for two symmetric tilt boundaries in NiO, (210) and (310), and for the asymmetric boundary (439) (100) at which the (439) plane on one side of the boundary is parallel to the (100) plane on the other. Duffy and Tasker (1983) investigated tilt grain boundaries in NiO and went on to study cation boundary diffusion (Duffy and Tasker, 1986 a, b). Their predicted rigid body translation normal to the interface of an Z5 boundary is much larger than ob-
The idea behind a structural unit model of a boundary is that any long period boundary can be decomposed into strained units of shorter period boundaries, as noted by Bishop and Chalmers (1968,1971). This concept was first formulated in a rigorous and testable way by Sutton and Vitek, in a series of papers in which they investigated symmetrical and asymmetrical tilt boundaries in Al and Cu by atomistic simulation (Sutton and Vitek, 1983 a, b, c). Sutton and Vitek concluded that all boundaries in a certain misorientation range are composed of mixtures of two structural units. One example is shown in Fig. 9-11. The misorientation range is defined by two delimiting boundaries of short period. The structural units are defined with reference to the structures of these boundaries so that each delimiting
559
9.4 Simulation of Homophase Boundary Structures
...ABBABB...
£"=37 ...AABABAABAB...
r=i ...AAAAAA...
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(a)
I
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Figure 9-11. Core structures of three symmetrical [001] tilt boundaries in a f.c.c. lattice, illustrating the structural unit model, (a) Geometrical construction of the interpenetrating lattices for 15: (210) planes parallel (dashed lines). The CSL unit cell is marked with solid circles and the two inequivalent planes of atoms are shown as triangles and open circles, (b) The relaxed configuration of the Z5 boundary. Four structural units are shown, denoted type B. (c) The relaxed configuration for 117: (530) planes parallel. Two types of structural units appear, denoted A and B, in the repeating sequence ABB . . . (d) As above for £37: (750) planes parallel. The repeating sequence of structural units is now AABAB . . . (e) The perfect crystal [IT. (110) planes parallel] is composed entirely of the units denoted A.
boundary is entirely composed of one of the types of structural units. If one of the delimiting boundaries is taken as a reference structure, as the boundary is tilted towards the other delimiting boundary, structural units of the other boundary start to appear. These minority units can be identified with the cores of secondary dislocations. Sutton and Vitek (1983 a, b,c) also found that the hydrostatic stress fields are consistent with this description of the boundary in terms of secondary dislocations centered on the minority structural units. The structural unit model was also applied (Sutton, 1982) to interpreting the results of atomistic simulation of twist boundaries. Three metastable structures of each of the Z5, £3, XI and £25 (001) twist boundaries in Cu and Ni were found by
Bristowe and Crocker (1978). The three higher Z boundaries were found to contain units of the 15 boundary and units of perfect crystal. The strain fields of localized secondary screw dislocations were predicted and observed (Schwartz et al, 1985). The limitations of the structural unit model have been emphasized in recent studies by (Sutton, 1989 a, b, c; Sutton and Balluffi, 1990). It was concluded that it only has predictive power for pure tilt and pure twist boundaries with low index rotation axes, namely <100>, <110>, <111> and possibly <112>. Although the model can be applied in principle to boundaries with higher-index rotation axes, in practice it was concluded that it is of no use because the number of delimiting boundaries rapidly increases and they span a correspondingly smaller range of orientations.
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9 Structures of Interfaces in Crystalline Solids
Nevertheless, a [100], 45° twist plus 17.5° tilt boundary in Al was recently studied (Shamsuzzoha et al., 1991) by high-resolution transmission electron microscopy and these authors claimed that the structural unit model was helpful in the description of its atomic structure. A hierarchical classification of boundaries in cubic crystals has been introduced by Paidar (1985,1987,1990). He systematically constructs a tree of orientations based on interplanar spacings, starting at the lowest level with boundaries having the largest interplanar spacings normal to the boundary. This enables the prediction of which boundaries should be of the special delimiting type, namely those low down on the tree. For asymmetric boundaries (Paidar, 1992) he finds that large interplanar spacings, which may be associated with interesting properties, are not necessarily of short period. A generalization of the structural unit model was made by Sutton to irrational tilt boundaries (Sutton, 1988). He showed, following some ideas of Rivier (1986) that these boundaries can be regarded as onedimensional quasicrystals perpendicular to the tilt axis, comprising a quasiperiodic sequence (Levine and Steinhardt, 1986) of the structural units of the delimiting boundaries. This theory enables the sequence of structural units expected in the boundaries to be constructed. As in long period rational boundaries, the tendency is to keep the spacing of minority units as wide as possible.
9.5 Simulation of Heterophase Boundary Structures The atomistic simulation of heterophase boundary structures has a shorter history
than the simulation of grain boundaries in single phase materials. A major reason for this in the case of metals has been the lack of any reliable interatomic potentials for the interaction between different types of atom. Another problem is that the block of atoms needed for simulating a heterophase boundary is necessarily large even in the simplest cases if the misfit is taken into account. The box must contain at the very least a two-dimensional unit cell of a lattice of equivalent O-points, that is an interfacial unit cell of the combined crystals. Normally the lattices of the two crystals do not exactly match, so that to create a periodic unit cell which is manageably small, one or both of the crystals must be strained. To avoid any spurious effects of this strain, which is introduced only for computational convenience and has no physical significance, it must be less than one or two percent. For example in Ag and Cu, at room temperature the lattice parameters are approximately in the ratio 9:8. So the technique as used by Dregia et al. (1989) was to construct a slab of Cu measuring 9 lattice translations on the side in contact with a slab of Ag measuring 8 lattice translations on the side. Perfect matching at the edges and periodic boundary conditions were achieved by distributing a strain of only 0.5 % between the slabs in such a way as to minimize the resulting strain energy. Alternatively, the strain can be entirely accommodated in one halfcrystal, which would be appropriate for a simulation of epitaxial behavior in which the substrate should not be strained. Commonly, several thousand atoms are required to meet this condition. Computers sufficiently powerful to handle problems of this size have only recently become generally available. At present the field is developing rapidly; the models of the interatomic forces and the algorithms are im-
9.5 Simulation of Heterophase Boundary Structures
proving and faster computers with larger memory are being exploited. 9.5.1 Interatomic Potentials The simulation studies on metals have used a generalization of the isotropic Nbody potentials to allow a different embedding function F and density function $ for each atomic species (Foiles etal., 1986; Voter and Chen, 1986; Ackland and Vitek, 1990): (9-6) (9-7) where vf and Vj denote the types of atom occupying sites / and j respectively. The reliability of this class of potentials cannot be regarded as quantitative. Some of the parameters are determined by fitting to alloy properties such as heats of mixing. Nevertheless, a wide range of interesting physical results has been obtained by their use. As confirmation of their usefulness we cite the calculations of the segregation energy of a Ni atom at an Ag (100) surface (Foiles etal., 1986). These predicted the favored segregation site to be in the second layer, in agreement with experiment. For an Ag atom at a Ni surface on the other hand calculations and experiment agree that the Ag resides on the outside. For semiconductors, Tersoff (1989) proposed a generalization of his earlier model for Si to describe Si-Ge and Si-C bonds which describes their elastic properties. Ashu and Matthai (1989) proposed a generalization of the Stillinger-Weber potential to describe Si-Ge. It gave the atomic relaxations in the (Si)n(Ge)n (001) superlattice structures in good agreement with ab initio total energy calculations (Froyen etal., 1988).
561
For insulators, the Born model and shell model can in principle be used both at heterophase and at homophase interfaces; the limiting factor in their application is the computational cost of working with a sufficiently large box. Useful references are Tasker and Stoneham (1987), Cotter et al. (1988), Allan etal. (1989), Davies etal. (1989), Kenwayetal. (1989). Surprisingly, the first atomistic simulations for a metal-ceramic interface have used a first-principles method, namely the FP-LMTO method (Methfessel, 1988; Methfessel et al., 1989). Mixed semiconductor interfaces have also been calculated with this method (Methfessel and Scheffler, 1991). The supercells used in such methods are small, and the atomic positions are not all fully relaxed. Only simple boundaries and a restricted number of degrees of freedom can be studied at the present time. Simple models for the atomistic simulation of metal-semiconductor and metalinsulator interfaces, analogous to those used for metal-metal interfaces, are least of all developed. Matthai and Ashu (1990) have devised a model for the NiSi 2 -Si interface using an isotropic TV-body potential for the Ni-Ni and Ni-Si bonds and a Tersoff potential for the Si-Si bonds. A new approach for metal-ceramic interface simulation based on the image interaction has been proposed by Finnis (1991) but not yet tested for a realistic system. 9.5.2 Results A number of authors have studied the structures of interfaces between the noble metals and nickel, using the embedded atom model. The results are briefly reviewed here. Epitaxial Cu/Ag interfaces in parallel cube-on-cube orientation were simulated
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9 Structures of Interfaces in Crystalline Solids
by Dregia et al. (1989). Coherent and semicoherent interfaces were studied with inclinations parallel to (001), (Oil) and (111). The expected misfit dislocations were observed after relaxing the system. The relaxation fields of the misfit dislocations forming a square array on (001) were seen to be qualitatively in accord with the van der Merwe model (van der Merwe and Ball, 1975) for a unidirectional distribution of misfit dislocations. The misfit dislocations in the simulations were twice as wide as predicted by the van der Merwe model. Further work cited below has also shown that the continuum theory underestimates the degree of delocalization of the dislocation cores. An important result which appears to be general concerns the relationship between structure and energy: in spite of atomic relaxation, the interfacial energy was dominated by the structural component, that is the energy associated with the dislocations. The interfacial energy of the corresponding coherent interfaces was an order of magnitude less, and sometimes negative in value. Twist boundaries on (001) Ag/Ni were simulated by Gao et al. (1989 a, b), who studied the variation of interfacial energy with azimuthal orientation. The minimum energy was found at an angle of 26.56° corresponding to a 15 against 14 construction, in which a 3.3% strain was required to produce the exact coincidences. Exact coincidences in two dimensions are needed to satisfy the periodic boundary conditions of the simulation cell. A second higher minimum was found in the energy at 4.4°. Both these favored orientations were confirmed experimentally by the rotating crystallite method. Further similar calculations were made on (lll)Ag/(001)Ni twist boundaries and interfaces between low index planes of Ag and Ni (Gao et al., 1989 a, b). It was shown
that among low index interfaces, those containing the (111) plane have the lowest energy while those containing (110) have the highest. On a (111)/(111) interface the net of misfit dislocations, which would be hexagonal according to the O-lattice construction, is dissociated into a triangular net, as discussed in detail by Gumbsch and Fischmeister (1991). Still within the embedded atom description of the interatomic potentials, the segregation of Au was studied at Ag/Ni (Dregia et al., 1986) and Ag/Cu (Dregia etal., 1987) boundaries. Further extensive studies of boundaries in the f.c.c. metals are described in Gumbsch et al. (1989, 1990, 1991), Gumbsch (1991), Gumbsch and Fischmeister (1991), Wolf etal. (1991), which have confirmed the above conclusions. Furthermore the thermal stability of various Ag/Ni boundaries was investigated (Gumbsch, 1991) by the Monte Carlo method, with the result that (110) and (100) interfaces tended to form facets of (111) orientation. Matthai and Ashu (1990) have studied the energies of strained and unstrained Ge layers on Si(100) substrates as a function of the number of layers. The critical thickness for dislocation nucleation was found to be less than 12 layers, in agreement with elastic continuum theory (Matthai and Ashu, 1990). The same authors simulated (111) NiSi2/Si interfaces, which occur in two types, labelled A and B. They found that the separation of the boundary in A is hardly altered by relaxation from its ideal value, whereas in type B it is reduced by about 0.01 nm. This, they note, is consistent with the experimental value obtained indirectly via the reduced Schottky barrier height of B compared to A. Phase boundaries in ionic materials have received little attention because the
9.6 Experimental Studies of Interface Structures by X-Ray Techniques
problems of mismatch and uncertainty in the potentials are particularly acute. Tasker and Stoneham (1987) investigated a boundary between the different dielectrics NiO and BaO with regard to the importance of image charge effects, which tend to attract charged defects to the interface. In their ab initio calculations for Ag/ MgO and Ti/MgO (001) cube-on-cube interfaces, Schonberger et al. (1992) relaxed the distance between the terminating layers of each phase, the first interlayer spacing of the metal phase and the relative translation parallel to the boundary. They found that the lowest energy structure is with the metal sites above the oxygen sites. The Ti-O bond appeared to be much more covalent than the Ag-O bond and weaker than in bulk TiO 2 . The length and force constant of the AgO bond on the other hand resembled that in Ag 2 O. This suggests that the Ag at the interface is in the charge state + 1 , as it is in Ag2O (see also Sec. 9.8.4.2).
9.6 Experimental Studies of Interface Structures by X-Ray Techniques 9.6.1 X-Ray Diffraction 9.6.1.1 Introduction
In recent years X-ray diffraction has emerged as an experimental technique with the potential to reveal the full three-dimensional structures of grain boundaries (Sass, 1980; Sass and Bristowe, 1980; Budai and Sass, 1982; Budai et al., 1983; Fitzsimmons and Sass, 1988a, b; Taylor etal., 1988; Chap. 8, Vol 2 A of this Series). The atoms in the thin boundary region are generally displaced with respect to the positions they would normally occupy in the crystals adjoining the boundary.
563
These displacements produce an extra scattering, and the measurement and analysis of this scattering over sufficient regions of reciprocal space can then, in principle, reveal the full 3-D atomistic structure of the boundary region. In practice, as much of this scattering is measured as is practicable. It is then compared to the corresponding scattering predicted for various atomistic models of the boundary structure in a search for consistency. Unfortunately, the boundary scattering is relatively weak and has a complicated distribution in reciprocal space. It is therefore difficult to make sufficiently adequate measurements to ensure reliable structure determinations, particularly for long period boundaries with large unit cells containing many atoms. In this section, we first discuss briefly the general form of boundary diffraction and then go on to list the special experimental difficulties which arise in measuring it. The various strategies for deducing the boundary structure from the experimental diffraction measurements are then described including those based on the use of the measurements alone and those based on a combination of measurements and computer modeling. Some recent results are then cited for purposes of illustration. Finally, an agenda for future diffraction studies is briefly described. 9.6.1.2 General Form of the Boundary Diffraction
All of the X-ray work has been concentrated on boundaries which have been assumed to be ideally periodic. Boundaries as close to ideally periodic as possible have been prepared (see below) and studied experimentally. The results have then been compared with corresponding calculated results for ideally periodic models. All ex-
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9 Structures of Interfaces in Crystalline Solids
perimental specimens have had the form shown in Fig. 9-12 where the boundary is assumed to possess a 2-D structure corresponding to the ideal periodicity of the coincidence site lattice (CSL) in the plane of the boundary. The thicknesses of crystals 1 and 2 are larger than the thicknesses of the boundary region in each crystal defined as the regions over which significant atom displacements are present. Such a specimen can be regarded as a 2-D "crystal" composed of the unit cells shown in Fig. 9-12 where ax and a2 define the 2-D periodicity in the boundary plane, and t is a vector normal to the surface and of magnitude equal to the total specimen thickness. The diffracted intensity is then expected to fall on a lattice in reciprocal space, i.e., the boundary diffraction lattice (BDL) (Taylor etal., 1988; Majid etal., 1989; Brokman and Balluffi, 1983), composed of an array of line elements which, as shown in Fig. 9-13, run perpendicular to the boundary and project on the boundary plane in a pattern corresponding to the reciprocal lattice of the 2-D boundary lattice defined by a1 and a2- The unit vectors of the BDL are taken to be a* and if, and k. The vector k is parallel to t, and \ convenient choice for its magnitude is the smaller of the spacings of the planes in the reciprocal lattices of crystals 1 and 2 which lie parallel to the boundary. A general vector in this space is then represented by All reflections from the perfect crystals 1 and 2 fall on a subset of the elements of the BDL. The additional scattered intensity due to the displaced atoms in the boundary region falls more generally on elements of the BDL in the form of patches (Fig. 9-13) corresponding to relrods (i.e. reciprocal lattice rods) which are elongated along L because of the relatively small thickness of the boundary region. This behavior may
Crystal 1 Unit cell
Crystal 2
Figure 9-12. Layered bicrystal containing a grain boundary in its midplane. Unit cell of the bicrystal is shown. Area sh corresponds to the area of 2-D cell in boundary plane.
be understood simply on the basis of the model suggested by Brokman and Balluffi (1983) who pointed out that the displacement field of the boundary can be represented by a Fourier summation of static sinusoidal displacement waves. Each wave produces a set of satellite reflections around each lattice reflection on elements of the BDL. The lengths of the relrods on
Relrod patch element Figure 9-13. Boundary diffraction lattice (BDL).
9.6 Experimental Studies of Interface Structures by X-Ray Techniques
each BDL element are roughly equal to the reciprocal of the thickness of the boundary displacement field in the L direction. 9.6.1.3 Experimental Difficulties with Measuring the Boundary Diffraction
Weak Scattering and Poor Signal-toNoise Ratio. A major experimental problem is the fact that the boundary scattering is relatively weak and that this can lead to a problem owing to a poor signal-to-noise ratio. For example, for a Z5[001] twist boundary in Au, the intensity of the weaker diffracted beams corresponds to about that expected for only 1/25 of a monolayer of Au, or less (Majid et al., 1989)! The background, as usual, comes from a wide variety of sources including: Compton scattering, scattering from air and other obstacles, thermal diffuse scattering, etc. The thermal diffuse scattering is particularly bothersome since it tends to peak at the positions of the lattice peaks (Warren, 1969) which are near regions where many of the important boundary peaks occur. The situation can be aided by obtaining better counting statistics through the use of high intensity X-ray sources such as synchrotrons (Fitzsimmons and Sass, 1988 a, b) and rotating anode units, and reducing the background through the use of the thinnest possible specimens, and other techniques. The thin bicrystal specimens, with the geometry shown in Fig. 9-12, have generally been prepared by welding face to face epitaxially evaporated single crystal thin films of predetermined geometry (Sass, 1980; Sass and Bristowe, 1980; Budai and Sass, 1982; Fitzsimmons and Sass, 1988 a, b; Taylor etal., 1988; Majid etal., 1989). The specimens (of total thicknesses 60150 nm) have then been mounted on thin polymer films, also of minimum possible
565
thickness. This geometry has the advantage that a relatively large ratio of boundary volume to specimen volume is achieved because of the exceptionally small thickness of the evaporated films. Evacuated beam lines have been used to reduce air scattering. The thermal diffuse scattering from each crystal could be further reduced by cooling with a specimen cryostat, but this technique has not yet been used. Preparation of High Quality Bicrystal Specimens. Great difficulties have been encountered in producing bicrystal specimens which are sufficiently thin and at the same time sufficiently perfect and flat to produce nearly ideal diffraction effects. Details of specimen preparation are described by Fitzsimmons and Sass (1988 a, b), Taylor et al. (1988) and Majid etal. (1989). Presence of Allowed and Forbidden Lattice Reflections on the BDL. As already noted, all lattice reflections from crystals 1 and 2 fall on the BDL. Since it generally is not possible to separate cleanly a boundary reflection from a lattice reflection on an element of the BDL (Fitzsimmons and Sass, 1988 a), boundary reflections which fall on these elements cannot be measured. Problems also arise with respect to normally forbidden lattice reflections on the BDL. If crystals 1 or 2 in Fig. 9-12, contain a non-integer number of unit cells, extra diffracted intensity may appear at locations on the BDL not associated with normal bulk diffraction (Cherns, 1974). This extra scattering is relatively weak, but it generally falls on the BDL and is comparable to that expected from boundaries in many cases. It must therefore be taken into account as an undesirable complication in
566
9 Structures of Interfaces in Crystalline Solids
grain boundary scattering measurements. As an example, consider the <001> symmetric tilt boundary along {310} illustrated in Fig. 9-14 a and the corresponding L = 0 plane of the BDL for this boundary shown in Fig. 9-14 b. The supercell ABCD shown in crystal 1 of the bicrystal consists of a stack of 10 {310} planes and contains 5 complete unit cells. The structure factor for this cell, or any integer multiple of this cell, is zero everywhere in reciprocal space except at the normal f.c.c. lattice reflection positions, such as those shown in Fig. 914b. However, if extra {310} planar layers are present which do not fit into integral numbers of ABCD-type supercells, it is easily shown that a residual intensity appears on the elements of the BDL in Fig. 9-
t.
1*
1
K
t
(020)
(622) (311) f
{
oi
b)
(602)
Figure 9-14. (a) Z5[001] symmetric tilt boundary parallel to {310} in f.c.c. structure. Edge-on view is along tilt axis: different symbols indicate ABAB . . . stacking sequence, (b) Portion of L = 0 plane of BDL of tilt boundary in (a): x's indicate f.c.c. lattice reflections.
14 b. For example, if a single extra {310} plane is present, the residual intensity which appears is that due to a single {310} plane. Since the periodicity of the boundary is identical to the periodicity of the {310} plane, the intensity falls on all of the elements of the BDL. In the example illustrated in Fig. 9-14, the crystal periodicity in the boundary plane is the same as the boundary periodicity. Fortunately, this will not generally be the case. It is inevitable that extra layers leading to "forbidden" reflections of the type just described will be present in each of the crystals making up actual specimens. For example, measurements in our laboratory from specimen forbidden lattice positions due to the presence of extra (002) layers on the (001) films used to fabricate these specimens. In view of these results, measurements of boundary diffraction on the BDL at the positions of normally forbidden lattice reflections must be avoided. This phenomenon, therefore acts as a restriction on the number of boundary reflections which can be measured. Double Diffraction on the BDL from Crystals Adjoining the Boundary. Since crystals 1 and 2 are in a CSL misorientation, a special danger exists that double diffraction from crystals 1 and 2 may occur which will fall on the BDL at a location where a boundary diffraction measurement is in progress. Since the intensities of such doubly diffracted beams are generally at least comparable to the intensities of the boundary reflections, a measurement error can then result. The situation is illustrated in Fig. 9-15. The incident primary beam, I1? first falls on crystal 1 and is diffracted by the vector k±. This diffracted beam is then incident on crystal 2 as I 2 where it is diffracted by the vector k2 into the same direction, D 2 , as another beam, D b , dif-
9.6 Experimental Studies of Interface Structures by X-Ray Techniques Ewald spheres
Incident on crystal 1 I.,
Diffracted by boundary Db Diffracted by crystal 2 D2 Diffracted by crystal 1 D1 Incident on crystal 2 I 2
Figure 9-15. Diagram indicating condition for double diffraction on BDL from crystals 1 and 2 adjoining the boundary. Lx and L 2 are reciprocal lattice points of crystals 1 and 2, respectively. kY and k2 are the corresponding reciprocal lattice vectors. kh is a reciprocal lattice vector of the boundary. Further details are given in the text.
fracted by the boundary. The necessary condition is then kh = k1 + k2, where kv and k2 are vectors of the reciprocal lattices of crystals 1 and 2, and kh is a vector which falls on the BDL. Unfortunately, this condition is automatically satisfied whenever a diffracted beam in crystal 1 excites a beam in crystal 2. It is well known that the totality of all the possible vector sums (±kl±k2) defines a 3-D point lattice which may be described as the DSC-Lattice (Smith and Pond, 1976) formed by the reciprocal lattices of crystals 1 and 2. This DSC-Lattice, which may be regarded as the double-diffraction lattice (DDL) for the bicrystal, has been shown by Grimmer (1974) to be just the reciprocal lattice of the CSL formed by crystals 1 and 2 in real space. We therefore have the important result that whenever the bicrystal is oriented for double diffraction, the doubly diffracted beam is automatically in the direction of a possible grain boundary reflection. The special double diffraction situation shown in Fig. 9-15 is not usually satisfied.
567
When it does occur, it can be readily destroyed by rotating the bicrystal around kh until kx and k2 are no longer excited, and the boundary reflection corresponding to kh can then be measured in the absence of double diffraction. 9.6.1.4 Strategies for Determining the Boundary Structure
The general procedure for the determination of boundary structure consists of the following three main steps: (i) measurement of appropriate integrated intensities by diffractometry; (ii) determination of boundary structure factors, Fh(H,K,L), from the integrated intensities; and (iii) comparison of the measured values of Fh (//, K, L) with corresponding calculated values for various atomistic models of the boundary in a search for consistency. Measurement of Relative and Absolute Structure Factors. A procedure for obtaining relative values of Fh(H,K,L) from measurements of integrated intensities has been described by Fitzsimmons and Sass (1989 a, b) while a method for determining absolute values has been described by Taylor et al. (1988). In the former method, appropriate scans are made through the boundary reflections using 4-circle diffractometry in order to obtain integrated intensities. The resolution function of the experimental arrangement is measured (or estimated, when necessary), and values of |Fb (H,K,L)\2 are then extracted by deconvolution. The values of |Fb (//, K, L) \2 measured are relative values usually normalized to the value for a relatively strong boundary reflection. In the method described by Taylor et al. (1988), scans are also made through boundary reflections in order to obtain integrated intensities. However, the scans are
568
9 Structures of Interfaces in Crystalline Solids
carried out in a lower resolution mode in a manner which eliminates the need for a determination of the resolution function and the use of deconvolutions. In addition, integrated intensities of lattice reflections which are near the boundary reflections in reciprocal space are measured. Upon taking the ratio between the integrated intensity for the boundary and that of the nearby lattice reflection, common unknown factors cancel out, and an expression for \Fh(H,K,L)\2 is obtained in which all quantities are either known, can be measured, or can be calculated with acceptable accuracy. This method has the advantage that it yields absolute structure factors. On the other hand, the relatively low resolution of the technique, at least in its present form, is disadvantageous in cases where fine details of the intensity distribution over reciprocal space are desired. However, this could be improved (at the expense of some of the above advantages) by introducing higher resolution procedures involving the use of the resolution function and deconvolution. Determination of Structures by Means of Structure Factor Measurements Alone. The first step in deducing an atomistic boundary structure solely from a set of \Fb(H,K,L)\2 measurements is to search for boundary diffraction symmetry elements and any selection rules which may be present. If any can be established, corresponding symmetry elements in the atomistic structure can be identified. The second step involves a search for the structure possessing a set of |Fh (H, K,L)\2 values which best matches the measured set. In principle, it should be possible to find the correct structure by this procedure providing that a sufficient number of measured | F b (//, K,L)\2 values is available and a sufficient number of models is tested.
However, in practice, problems may be encountered with such a procedure because of the experimental difficulties in obtaining large numbers of reliable measurements (which were discussed previously) and the large number of models which must be tested. This was demonstrated explicitely by Balluffi et al. (1989) for a X 5 (001) twist boundary in an f.c.c. structure. In this structure, all of the significant atomic displacements occur in the first two planes of crystals 1 and 2 facing the boundary. If the boundary is in the CSL translational state, the boundary structure can then be completely characterized by four independent parameters corresponding to the x and y displacements of single atoms in the first and second planes (Fitzsimmons et al., 1988 a). Balluffi et al. (1989) suggested the following procedure: (1) calculate a set of N exact structure factors for the known boundary structure; (2) assign random errors to these structure factors up to a given limit in order to simulate a set TV "measured" structure factors; (3) run through all possible combinations of the four displacement parameters to find the boundary structure which yielded the closest fit to the set of N "measured" structure factors as determined by obtaining a minimum reliability factor R (Budai et al., 1983) whereby R is defined exactly in the same way as for crystal structure determination (Schwartz and Cohen, 1977) as
R=
(9-8)
where FJh and FJal are observed and calculated structure factors of the jth reflection, and Wj represents the weighting factor (0< Wj< 1); and (4) calculate the degree of fit (again in terms of an 7^-factor) between the known actual structure and the TV
9.6 Experimental Studies of Interface Structures by X-Ray Techniques
"measured" structure factors. In addition, this procedure was carried out using both relative structure factors (normalized to one strong reflection) and absolute structure factors. In the search for the lowest 7^-factor structures, the atoms were moved around on a fine 2-D grid possessing a spacing which was about 1/11 the size of the maximum atomic displacement in the boundary. One example of a set of results is shown in Fig. 9-16 where the 7?-factors of the "best-fit" structures and the actual structure are plotted versus TV under the conditions indicated. It may be seen immediately that, for small N9 structures other than the true structure can be found which have smaller i^-factors than the true structure. Hence, these structures would unfortunately be identified as the most likely structures in these situations. However, as N increases, the structure with the lowest /^-factor also tends to be the true structure. The true structure therefore always tends to emerge as the structure with the lowest i^-factor when N becomes sufficiently large and, hence, it can then be correctly identified. Of considerable importance in the present context is the result that smaller numbers of measured structure factors are required to resolve clearly the true structure when absolute structure factors are available rather than relative val-
I 0.3
0.2
I
569
ues. It should be noted that the statistical aspects of finding the most likely structure have been described in many places in the voluminous crystallography literature, e.g. Ibers and Hamilton (1974). 9.6.1.5 Some Current Results
Essentially all quantitative work to date has concentrated on the study of the structures of [001] twist boundaries in gold. For the 15 (twist angle 0 = 36.9°) boundary (Fitzsimmons and Sass, 1988 a, b) measured 19 relative structure factors in the L = 0 plane of the BDL and verified that the boundary possessed the symmetry elements characteristic of the "CSL translational state" (Bristowe and Crocker, 1978). They then assumed the 4-plane/ 4-parameter model described (Budai and Sass, 1982) and, by systematically varying the 4 parameters, found a 2-D projected structure which produced a best fit with the 19 measured structure factors. Fitzsimmons and Sass (1988 b) measured for the Z13 ( 0 = 22.6°) boundary relative structure factors for 50 reflections in the L = 0 plane and also carried out 390 "relrod profile" measurements along nine different relrods on the BDL. Fitzsimmons and Sass (1988 b) again assumed a 4-layer boundary model which, in this case, possessed 24 independent x, y9 and z coordinates which
I
o True structure * Lowest-/? structure 40% max. error o •o o o o i
0.1
(a) 80 120 N (Relative)
160
40
80 120 N (Absolute)
Figure 9-16. ifc-factor versus number TV of "measured" structure factors for calculated boundary structure and structure with the lowest i^-factor. Maximum error in "measured" structure factors is 40%; (a) results using relative structure factors, (b) results using absolute structure factors (Balluffi et al., 1989). Black dots mean coincidence.
570
9 Structures of Interfaces in Crystalline Solids
were necessary to describe the positions of the 8 atoms which were not related by symmetry. A best-fit 3-D structure of the boundary was then obtained by systematically varying these 24 parameters. More recently, Majid et al. (1989) carried out a systematic series of absolute structure factor measurements for a series of [001] twist boundaries in Au which included the 1113 ( 0 - 7 . 6 ° ) 125 {© = 16.3°), £13 ( 0 = 22.6°), £17 ( 0 = 28.1°), and 15 ( 0 = 36.9°) boundaries. The measurements consisted mainly of absolute structure factors measured in the L = 0 plane along with a few measured in the L = 1 and 4 planes and along relrods. The number of (//, K, 0) absolute structure factors which was measured for each boundary was as follows: 33 (I'll3); 8 (125); 4 (113); 4 (117) and 14 (15). In order to measure absolute structures, it is necessary to have a precise knowledge of both the grain boundary area and the quality of the boundary (with respect to defects) which produces the diffraction. A careful procedure was applied for the identification of specimens which possess the required quality (Majid et al., 1989). Majid et al. (1989) calculated the structures of this series of boundaries by computer simulation using molecular statics and the embedded atom model (Daw and Baskes, 1984) as described in Sec. 9.4.1 and 9.4.2. The structures were tested against the measured diffraction data for consistency. The results obtained by these studies may be summarized as follows: (i) Good agreement was found (as determined by 7^-factor analysis) between the absolute structure factors of the calculated structures and the measured absolute structure factors for the entire series of boundaries. (ii) It was found that a 4-plane/4-parameter model of the projected structure of the
15 boundary which produced a best-fit (as determined by /J-factor analysis) with the 14 absolute structure factors measured for this boundary was in good agreement with the corresponding structure calculated by computer simulation using the embedded atom model. (iii) Good agreement was found by Majid et al. (1989) between the structure of the 113 boundary obtained by Fitzsimmons and Sass (1988 b), the structure calculated by Majid et al. (1989) using the embedded atom model and the absolute structure factors measured by the latter authors, see Sec. 9.4.3 for f.c.c. metals. (iv) The same authors found that the projected structure obtained by Fitzsimmons and Sass (1988 a) for the Z5 boundary possessed absolute structure factors (and atomic displacements) which were much too large compared with the measured absolute structure factors of Majid etal. (1989). Some of their results are shown in Figs. 9-17 and 9-18 (see captions for details). Hence, they concluded that the structure obtained by Fitzsimmons and Sass (1988 a) is incorrect. (v) The structures of the boundaries in the series were found by Majid et al. (1989) to vary systematically as 0 increased. The displacements consisted mainly of rotations around O-elements (Brokman and Balluffi, 1983) in the boundary, and their magnitudes decreased monotonically with increasing 0 . Majid et al.'s observations represent reliable quantitative X-ray scattering experiments where good agreement has been obtained between boundary structures determined by X-ray diffraction and by calculation using a physical model. The reason for the failure of the structure model by Fitzsimmons and Sass (1988 a, b) in spite of the low i^-factor (0.16) is the use of relative structure factors, i.e. the model is
9.6 Experimental Studies of Interface Structures by X-Ray Techniques
o Measured -^Calculated
9 8 -
•FS
7
2 1 0 -
20 £(deg)
30
Figure 9-17. Measured and calculated (embedded atom model) values of the quantity | F° | 2 /I2 (a measure of the scattering power per unit area of boundary) plotted as a function of the twist angle 0 for the series [001] twist boundaries in Au studied by Majid et al. (1989). Also shown are values (FS) predicted by the models of Fitzsimmons and Sass (1988 a) for the Z13 (0 = 22.6°) and 15 (0 = 36.9°) boundaries. All values are for the grain boundary reflection which lies on the H axis equidistant from the [200]! and [200] 2 lattice reflections in reciprocal space. It may be seen that the Fitzsimmons and Sass (1988 a, b) result for the 15 (0 = 36.9°) boundary is anomalously large.
571
based on wrong absolute intensities. The use of even smaller numbers of absolute structure factors by Majid et al. (1989) led to a totally different and correct model. As it was pointed out, the measurement of larger numbers of structure factors will be a difficult task experimentally. Progress can be expected by producing more perfect, flatter bicrystal specimens resulting in sharper reflections and by reduced background noise. At present an advanced UHV bicrystal bonding machine is in existence at the Max-Planck-Institut fur Metallforschung at Stuttgart (Fischmeister et al., 1988). With such a machine, high purity thin films with well-characterized ultraclean surfaces could be bonded under controlled conditions. The use of synchrotron radiation of wavelength ^0.07 nm would roughly double the radius of the Ewald sphere used to date, and would allow the measurement of additional reflections. Those studies are still in progress. 9.6.2 Structure Determination by Grazing Incidence X-Ray Scattering
T5
9.6.2.1 The Technique Calculated: present work Z"25
Fitzsimmons> Sass model i
i
i
i
i
i
i
i
J
-0,1nm
Figure 9-18. Magnitudes of the largest displacement vectors projected onto the (001) boundary plane for [001] twist boundaries in Au calculated statically with the embedded atom model by Majid et al. (1989), and for the Fitzsimmons and Sass (1988 a, b) models. Again the Fitzsimmons and Sass result for 15 is anomalously large.
A new promising technique for studying interfaces by X-rays became recently important. This scattering of X-rays under grazing angles, grazing incidence X-ray scattering, GIXS (Marra etal., 1979; Eisenberger and Marra, 1981). In this technique, a grazing incidence X-ray beam of high intensity is reflected not only by the bulk material, but also by the reflectivity of interfaces buried slightly underneath the surfaces, see Fig. 8-32 in Chap. 8, Vol. 2 A of this Series. Following the recent publication of the first GlXS-experiments on structural relaxations at the Al-GaAs interfaces in 1979 (Marra et al., 1979) subsequent demonstrations of the monolayer
572
9 Structures of Interfaces in Crystalline Solids
surface sensitivity were performed in 1981 (Eisenberger and Marra, 1981). The potential of this technique has now been demonstrated in several studies and details are described e.g. in Chap. 8, Vol. 2 A of this Series. Recently, it also was applied to the studies of atomistic structure of the Nb-Al 2 O 3 interface, actually similar films as studied by HREM (Lee etal., 1991; Liang et al., 1992). In the following GIXS will be described for a heterointerface between an epitaxial overlayer of Nb onto A12O3. 9.6.2.2 Experimental Studies Studies of buried interfaces between Nb and A12O3 (sapphire) were carried out in a Z-axis surface scattering spectrometer (Fuoss et al., 1990). The X-rays from the electron storage ring were focused by a bent cylindrical mirror and monochromatized by a double crystal monochromator of Ge(lll). Pairs of slits are employed as an analyzer in the diffracted beam with in-plane resolution in the reciprocal lattice of about 0.0005 nm" 1 . The Nb thin films were grown on the sapphire substrates by molecular beam epitaxy (MBE) (Du and Flynn, 1990). X-ray measurements were carried out in both surface-parallel (i.e. in-plane) and surfacenormal directions at different grazing incidence angles. The critical angle of X-rays of Nb is 0.294° for the 0.1129 nm X-rays employed in this study. The thickness of the Nb films is about 12 nm as determined from specular X-ray reflectivity measurements (Liang, 1992). The crystal mosaic of the sapphire substrates is 0.006°. Nb films were found to plane despite a miscut of ^1.4°. The in-depth structure of the films was probed with GIXS by varying the incidence angle of X-rays.
9.6.2.3 Experimental Results For Nb(lll) films of thickness 12 nm grown on the (0001) surface of the sapphire substrate, the in-plane orientation was found to be [110]Nb || [2110]s with a sixfold symmetry. The miscut angles of the substrate and lattice misfits of the films studied are given in Table 9-1. From the measured lattice constants, it is observed that the Nb lattice is expanded in the inplane direction by ^0.4% and contracted in the surface-normal direction by ^1.6%. Figure 9-19 shows the in-plane radial scans near the Nb(220) peak at incidence angles varying from 0.15° to 0.6°. In this figure three features are quite noticeable: (i) a very sharp sapphire (4220)-peak A which is instrument resolution limited, (ii) a broad Nb(220) peak for which the peak position is shifted from that for bulk Nb (labelled C) and the lineshape changes with incidence angle of X-rays, and (iii) a clear satellite peak D which appears at lower Q as the incidence angle is increased (Lee etal., 1991). It is noted in Fig. 9-19 that the diffraction profile of Nb(220) observed at 0.15° incidence angle has a simple Gaussian lineshape (coherence length « 20 nm) with its peak position shifted from the bulk Nb lattice position (labelled C) toward a coherent lattice match position. At this inciTable 9-1. A summary of crystallographic measurements on Nb(lll) film on sapphire (0001) substrate. Experimentally determined e. |( and s + denote the strain in in-plane and surface-normal directions, respectively. In-plane orientation
Miscut
Misfit Cal.
[110]/[1120] [1T2]/[1IOO]
1.5° 0.0°
Strain
Exp.
-1.8% -0.6% 0.4% -1.6% -1.8% -0.7%
9.6 Experimental Studies of Interface Structures by X-Ray Techniques
A:AI203U220)
incidence angle increases (i.e. the X-rays probe deeper toward the interface), the Nb peak becomes asymmetric. The observed lineshape suggests that a new peak (labelled B) is emerging at a position close to that for bulk sapphire (labelled A). This new structural feature is believed to be associated with the interface. As the incidence angle is increased, it is noted that the satellite peak (D) at the lowQ side can only be seen at the grazing incidence angle higher than 0.3°, indicating that the satellite originates from the interface. We suggest that the satellite peak comes from more-or-less regularly spaced misfit dislocations at the interface. From the observed separation between the satellite D and the principal Bragg peak B, the distance between dislocations is estimated to be about 6 nm. Similar results are seen on the (112)Nb || (01l0) s sample with scans taken along the inplane [110]Nb || [2110]s direction. In addition, the intensity distribution of the two peaks B and D in Fig. 9-19 were measured normal to the specimen surface, see Fig. 9-20. The scans were taken at a fixed incidence angle of a = 0.4° (c.f. Fig. 9-19). The rod profile of the satellite D is shown to be relatively flat as compared with that of the main Nb peak B. This indicates that the thickness of the interface
1
B:Nb(220) C:Bulk Nb(220)
5.5 Q [A"1]
Figure 9-19. In-plane radial scans around Nb (220) peak with incidence angles of X-rays varying from 0.15° to 0.6°. Q corresponds to 2sin0/A. For the explanation of the peak positions B, C and D, see text.
dence angle, the X-rays can only penetrate a few nanometers below the surface. This observation indicates that the surface of the 12 nm Nb film is still strained by the substrate. This result is supported by HREM studies (Mayer et al., 1992). As the 1E+6
1
d •
i
i
i
i
a
|
i
i
i
i
I
1
incident angle 0.4°
I
I
!
diffraction
-
—
a
— -
crystal
• • •
— * *
>
D
X
Nb(220) (B)
/
•
n a
X X
e x t r a satellite (D) 1
0.1
_
n
X
s*
-
573
0.2
.
.
.
.
1 0.3
x . . . .
|
•
0.1*
•
•
i
Figure 9-20. Comparison of the rod profiles of the principal Nb (220) peak and its satellite; the incidence angle of X-rays was fixed at 0.4°. The same specimen was used as for Fig. 9-19.
574
9 Structures of Interfaces in Crystalline Solids
layer associated with the satellite is smaller than that associated with the Nb peak. Noting that an ideal two-dimensional structure should have a constant rod profile, the observed drop in intensity of the satellite with increased AQ suggests a finite thickness of this structure, presumably located at the interface. It is interesting to see that X-ray reflectivity measurements (Fig. 9-21) reveal an amplitude-modulated fringe pattern. A reasonable fit of the data is obtained by assuming a three-layer model. The model includes an oxide layer at the top surface, a normal Nb layer, and an interface layer between Nb and sapphire (insert of Fig. 9-21). The reflectivity data is fitted by following the procedure of Tidswell et al. (1990). The fitting is relatively insensitive to the oxide layer within a reasonable thickness range ( < 1 nm). The most striking result to come from this analysis is evidence for the existence of the interface layer with a thickness about lA that of the Nb film thickness and an electron density (1.66 10 24 e/cm 3 ) between those of Nb (2.16 10 24 e/cm 3 ) and sapphire (1.17 10 24 e/cm3). This interface layer could be a Nb layer with a high density of dislocations. The GIXS studies give detailed results on the interface structures, however, very
large sample areas are required (typically 0.2 cm x 1 cm) under which an interface is buried. The GIXS study averages over this area. Films of extreme homogeneity are required. Those films can only be obtained for epitaxial films of semiconductors or metals grown on rigid substrates such as semiconductors or ceramics.
9.7 Experimental Studies of Interface Structure by High-Resolution Electron Microscopy 9.7.1 Introduction
The point-to-point resolution of highresolution electron microscopes (HREM) (for a definition see e.g. Chap. 1, Vol. 2 A of this Series) is now better than 0.17 nm for instruments with an acceleration voltage of 400 kV. The next generation of instruments pushes the resolution limit to ^0.1 nm. High quality experimental images can be obtained. Nevertheless, the interpretation of the HREM micrographs is not possible on a naive basis owing to the aberration of the rotation symmetrical magnetic lenses. Therefore, the micrographs have to be analyzed by comparing
Figure 9-21. The experimental and calculated X-ray reflectivity curves (see text). For the measurements the incidence angle of the X-rays is 0.1° which is smaller than the angle of total reflectivity.
9.7 Experimental Studies of Interface Structure by High-Resolution Electron Microscopy geometrical optics
object
_j|L
J
back focal plane
(a)
image
wave optics transmission function qfx.y)
575
(b) pourier into
diffraction pattern
transfer function exp[ix(u.v)\
Fourjer
into
complex crystal of large periodicity
- f(u,V;Lf,Cs)
them to simulated images. Considerable advances have been made in this analysis (Spence, 1988; Busek et al., 1988; O'Keefe, 1985). Methods and programs have been developed (O'Keefe, 1985; Stadelmann, 1987), which allow the simulation of HREM images of any given atomic arrangement. The recent developments enable us to use HREM as an important method for solving problems in materials science. Atomic structures of different lattice defects, such as phase boundaries, grain boundaries and dislocations, can be determined by HREM (see e.g. Chap. 1, Vol. 2 A of this Series). Over the last decade very important information has been obtained on the structure of lattice defects using HREM, particularly for grain boundaries in semiconductors (Chap. 7, Vol. 4 of this Series). The new generation of instruments allows also characterization of defects in ceramics and metals. This paper summarizes these possibilities. HREM observations for interfaces
image amplitude
Figure 9-22. Image formation by the objective lens of a transmission electron microscope, a) Geometrical optical path diagram b) wave optical description (see text for explanation)
will be reported in Sec. 9.8. The more general application of the various techniques of transmission electron microscopy (TEM) is described in Chap. 1, Vol. 2 A of this Series. 9.7.2 Comments on Direct Lattice Imaging of Distorted Materials The geometric beam path through the objective lens of a TEM is shown in Fig. 9-22 a. Beams from the lower side of the object travel both in the direction of the incoming and diffracted beam. All beams are focused by the objective lens in the back focal plane to form the diffraction pattern. In the image plane, the image of the object is produced by interference of the transmitted and diffracted beams. Fig. 9-22 b uses wave optics to describe physical processes which contribute to the image formation. From the lower side of the foil, a wave field emerges that can be described by a transmission function
576
9 Structures of Interfaces in Crystalline Solids
q(x, y) where x and y are the object coordinates. For an undistorted lattice, q{x,y) represents a simple periodic amplitude and intensity distribution. The transmission function of a complex lattice with a large periodicity (e.g., a periodic grain boundary) is very complicated and q(x,y) is a non-periodic function for the distorted region of a crystal. The intensity distribution in the diffraction pattern is given by the Fourier transformation Q (u, v) of the transmission function q(x,y) where w, v are the coordinates in the diffraction plane (reciprocal space). Since spherical aberration cannot be avoided with rotationally symmetrical electromagnetic lenses (Spence, 1988; Busek etal., 1988), the beams emerging from an object at a certain angle (Fig. 9-22 a) undergo a phase shift relative to the direct beam. Imaging with a small defocusing distance Af leads also to a phase shift. This depends on the sign and magnitude of the defocusing distance A/, which is defined as the distance between lower foil surface and imaging plane. The influence of the lens errors and the defocus on the amplitude Q(u,v) of the diffraction patterns is described by the contrast transfer function (CTF) (see Chap. 1, Vol. 2 A of this Series; Spence, 1988). The image (Fig. 9-22) is formed by a second Fourier transformation of the amplitude distribution in the diffraction pattern multiplied by the contrast transfer function. The amplitude in the image plane, W(x,y) is not identical to the wave field in the object plane (transmission function q(x,y)). The image is severely modified if scattering to large angle occurs, since the influence of the spherical aberration increases strongly with increasing scattering angle. The modification is most severe if the components in the diffraction pattern coincide with the oscillating part of the
CTF (large values of w, v). If, however, the wave vectors lie within the first wide maximum of the CTF, it can be assumed that characteristic features and properties of the object can be directly recognized in the image (Spence, 1988; Busek etal., 1988). Good HREM imaging requires that the first zero value of the CTF under the optimum defocus distance (so-called Scherzer defocus, i.e., that defocus distance where the first zero value of the CTF is maximum, its reciprocal value d1 {Scherzer resolution) is minimum, see Chap. 1, Vol. 2 A of this Series) must be at sufficiently large reciprocal lattice spacings. Good imaging conditions are fulfilled for lattices with large lattice parameters (Busek, 1985; Mitchell and Davies, 1988; Barbier etal., 1985; Merkle, 1987). If, however, deviations in the periodicity exist, components of the diffraction pattern also appear at large diffraction vectors. It is then most likely that certain Fourier components possess reciprocal lattice distances larger than the first zero value of the CTF. Lens aberrations and defocusing cannot be neglected. 9.7.2.1 Resolution The image contrast obtained in HREM is produced by interference of the diffracted beams with the transmitted beam. The interference can also be represented as an impulse response in the form of a delta function, the location and shape of which is highly dependent on the focusing distance (Spence, 1988; Bourret, 1985). From this description, it can easily be concluded that the resolution is not uniquely defined and that various distances can be adopted to define the resolution: d1 is defined as the Scherzer resolution, which is determined from the focusing distance A/s (Scherzer defocus) at which the influence of negative
9.7 Experimental Studies of Interface Structure by High-Resolution Electron Microscopy
defocusing compensates as far as possible for the influence of the spherical aberration (Spence, 1988; Busek et al., 1988; Chap. 1, Vol. 2 A of this Series). The information limit, d2, represents the smallest resolvable distance on an image at an optimum defocus distance. For an HREM analysis of crystalline materials, the first step is to establish the number and location of the positions of atom rows. It is assumed that the lattice to be investigated is periodic in the direction of the incoming beam so that the projected potential on the lower surface of the object is formed by the superposition of identical atoms that lie upon each other in the direction of the incoming beam (Fig. 9-23). The evaluation of the number of atomic rows and their exact positions are discussed in the next section. Different zone axes of crystalline materials can be used for the evaluation of crystal structures and of defects if the lattice plane distances are larger than the Scherzer resolution d1 of the instrument. If lattice defects must be analyzed by HREM, one must first establish the number of atomic rows that exist in the surroundings of such a lattice defect. Usually,
~ 5-10nm
Figure 9-23. Direct lattice imaging by HREM. The crystalline specimen must be adjusted so that the direction of the incoming electron beam coincides directly with the orientation of atomic rows. The schematic drawing includes a grain boundary. HREM can be successfully performed for pure tilt boundaries with tilt axis parallel to the direction of the incoming beam.
577
no information can be obtained in any HREM investigation on defects which are not parallel to the direction of the incoming electron beam. HREM images always represent a two-dimensional projection of electron distribution on the exit surface of the transmitted foil. For the image evaluation, it is implied that the atomic rows are exactly parallel to the incoming electron beam. In order to discuss the possibilities of determining the atomic structure, it is necessary to introduce a parameter d0 representing the minimum distance of two neighboring atom columns in the region of the lattice defect. Comparing d0 with d1 and d2 enables us to decide whether the distorted structure (or structural unit) in the lattice defects can be resolved in all details or not. Three cases can be distinguished: a) do>d1. In this case, all atom columns can be observed; artefacts are not introduced into the micrograph taken at a defocusing distance A/s (Scherzer image). For thin foils (typical thicknesses 5 to 10 nm), channels in the lattice structure appear as white spots on the image. If certain information on the atom configuration is available, the observed HREM micrographs can be directly compared with the atomic structure (Spence, 1988; Busek et al., 1988). b) d1>d0>d2. The determination of the number of atom columns of a distorted structure requires micrographs taken under a series of defocusing distances A/j which differs typically by small defocusing values, %5nm (Spence, 1988; Chap. 1, Vol. 2 A of this Series). A quantitative evaluation requires the comparison of micrographs taken under such different but specific focusing distances with computer simulated images (simulated for the same A/j) in order to differentiate interferences
578
9 Structures of Interfaces in Crystalline Solids
produced by artefacts and real lattice distortions such as additional atom columns. c) do
9.7.2.2 Determination of the Atom Column Locations
After determining the number of atom columns in the vicinity of the lattice defect, the coordinates of the individual atom columns must be determined with the best possible accuracy. The evaluation demands a comparison between experimentally obtained micrographs and computersimulated images. A "trial and error" method is usually applied. The comparison is made by superimposing the two images and visually comparing the location of the individual contrast features. It is important that the location of all contrast maxima and minima coincide. Very reliable data of atomic coalescence coordinates are obtained for do>dx: the accuracy is ± 0 . 0 1 ^ (Busek, 1985; Bourret, 1985). The accuracy of the atom coordinate determination is thus substantially better than the actual resolution. Since very small changes of the phases at different waves may result in an observable change in the intensity distribution and in the location of the intensity maxima during formation of a phase contrast image (Spence, 1988; Busek etal., 1988). The relative displacement of perfect lattices against each other (e.g., in the neighborhood of phase boundaries or even at grain boundaries) can be analyzed from HREM micrographs. The values obtained by HREM are often sufficiently accurate for a differentiation between different grain boundary models.
9.8 Experimental Results and Comparison to Results of Simulation Many studies were performed for the determination of the structure of homophase boundaries and heterophase boundaries.
9.8 Experimental Results and Comparison to Results of Simulation
The results are summarized in proceedings on special conferences on structure and properties of interfaces (Murr, 1975; Chadwick and Smith, 1976; Hagege and Nouet, 1982; Riihle et al., 1984 a, b; Ishida, 1986, 1987; Raj and Sass, 1987; Aucouturier 1984, 1989; Iwamoto and Suga, 1984; Pugh, 1991; Moller etaL, 1989; Werner and Strunk, 1991; Romig etaL, 1991; Clark etaL, 1992; Forwood and Clarebrough, 1991). This section will be subdivided into parts in which results on experimental studies on grain boundaries in semiconductors, metals, ceramics are reported separately. Studies on heterophase boundaries between metals and ceramics conclude the chapter. 9.8.1 Structures of Grain Boundaries in Semiconductors
Experimental observations on the structure of grain boundaries in semiconductors were recently reviewed in Chap. 7, Vol. 4 of this Series and Sutton (1991 a, b).
579
All HREM work on grain boundaries in semiconductors is included in Chap. 7, Vol. 4 of this Series. Table 9-2 summarizes grain boundaries in Si and/or Ge that have been characterized. Low angle boundaries have been excluded in Table 9-2 because their structures may be described by (complex) networks of (dissociated or undissociated) crystal lattice dislocations. All the [001] tilt boundaries appeared in bicrystals grown from specially oriented seeds by the Czochralski method. An important aspect of research is to correlate the structures of grain boundaries to properties. Only few experiments have been done where identical grain boundary structure and properties are determined (Chap. 7, Vol. 4 of this Series). Recently, TEM and deep-level-transientspectroscopy were applied for an understanding of the electrical activity of grain boundaries in germanium (Wang and Haasen, 1991, 1992).
Table 9-2. Grain boundaries in Si and/or Ge that have been analyzed by HREM. s.t.: symmetrical tilt; a.t.: asymmetrical tilt; tw: twist; R: rotation axis; 0: misorientation in degrees. Plane (441) (552) (221) (114) (111)/(115) (995) (332) (443) (111) (112) (910) (710) (510) (810)/(740) (11,3,0) (310) (115)
I
0
Type
33 27 9 9 9 187 11 41 3 3 41 25 13 13 65 5 9
20.05 31.59 38.94 38.94 38.94 42.89 50.48 55.88 70.53 70.53 12.68 16.26 22.62 22.62 30.51 36.87 120.00
s.t. s.t. s.t. s.t. a.t. s.t. s.t. s.t. s.t. s.t. s.t. s.t. s.t. a.t. s.t. s.t. tw.
R [110] [110] [110] [110] [110] [110] [110] [110] [110] [110] [001] [001] [001] [001] [001] [001] [115]
Reference Bourret and Bacmann (1986) Vaudin et al. (1983) Bourret and Bacmann (1986) Garg et al. (1989) Bourret and Bacmann (1986) Poutaux and Thibault (1990) Poutaux and Thibault (1990) Poutaux and Thibault (1990) Bourret and Bacmann (1986) Bourret and Bacmann (1986) Rouviere and Bourret (1990) Rouviere and Bourret (1990) Rouviere and Bourret (1990) Bourret and Rouviere (1989) Rouviere and Bourret (1990) Bacmann et al. (1985) Cheikhetal. (1991)
580
9 Structures of Interfaces in Crystalline Solids
9.8.2 Structures of Grain Boundaries in Metals
The structure of grain boundaries in metals was, so far, studied predominantly by X-ray diffraction experiments as discussed in Sec. 9.6. Recently, some quantitative HREM studies were performed on highly symmetrical tilt grain boundaries in niobium (Campbell et al., 1992a, b). Those grain boundaries were fabricated by diffusion bonding of two oriented single crystals of Nb (Fischmeister, 1988). Some representative results will be described here in detail. The examples emphasize the importance of quantitative HREM for the elaboration of grain boundary structures in metals and are based on recent results. The smaller lattice constants of metals (compared to semiconductors) require a high-resolution instrument. In the specimens of Nb bicrystals (Campbell et al., 1992a, b) prepared for HREM, regions could be identified where the interface was nearly symmetric and
atomic resolution imaging could be carried out. One image that was analyzed is shown in Fig. 9-24. A periodic contrast can only be identified over a short segment of the boundary. This may be caused either by the deformation introduced during bonding or owing to the fact that the grain boundary structure is not stable. Nevertheless, an analysis of the image has been carried out and the image has been compared with images simulated based on atomic structures predicted using potentials of the isotropic A/-body type. An experimental micrograph of a region of this interface (outlined in Fig. 9-24) was found to be nearly symmetric and close to the Z5(210) orientation. The outlined region is shown enlarged in Fig. 9-25 a. The experimental image (Fig. 9-25 a) has been averaged along the direction of the boundary to reduce noise. Three periods of this averaged image have been assembled for comparison with the simulated images. The tilt-angle was measured to be 15.6 + 0.7° on the left side of the boundary
Figure 9-24. HREM image of nearly symmetric region of the diffusion bonded bicrystal of Nb (Z5(210)/[001]). A representative section of the interface is marked. The interface is not flat. Many jogs and steps can be identified. A short flat part of the interface is marked. This set is used for further evaluations.
9.8 Experimental Results and Comparison to Results of Simulation
(a) (b) Figure 9-25. Comparison of experimental micrograph (a) with simulated image (b) based on EAM potential (Fig. 9-26a) for a Z5(310)/[001] symmetrical tilt grain boundary in Nb.
and 16.4 + 0.6° on the right side. The 4.9° deviation from exactly £5 (210) is due to deformation that occurred during bonding and owing to the curvature of the single crystal surfaces created during polishing prior to bonding. The experimental image is not mirror symmetric with respect to the boundary plane. HREM images were simulated for calculated grain boundary structures resulting from different atomic potentials (Fig. 9-26) and for the experimental conditions
581
derived by comparing simulated micrographs. The best agreement between simulated and experimentally obtained micrographs was obtained for the EAM structure (Fig. 9-26 a). HREM image simulations for the structures depicted in Fig. 9-26 b and 9-26 c resulted in micrographs which were disctinctly different from the experimental image of Fig. 9-25 a. However, the analysis of this 15 boundary in Nb was not very satisfactory or convincing due to the lack of long sections of the boundaries possessing the same symmetry. Often only 3 or 4 identical "structural units" (Sutton and Vitek, 1983 a, b) were adjacent to each other and "defects" within the grain boundary were seen to shift to other structures. This "unstable" behavior may be caused by the experimental conditions during diffusion bonding resulting in the non-equilibrium state of the boundary. Detailed observation should result in a better understanding of the grain boundary structure. It is well established, however, that the energies of twin boundaries are much smaller than those of Z5 boundaries. This should result in a more stable grain boundary configuration. Therefore, (310) symmetrical twin boundaries were investigated. A typical HREM micrograph of a Z3(310)/[001] is shown in Fig. 9-27. Both crystals adjacent to the twin are viewed in
•VV/.V.V.' (a)
(b)
(c)
Figure 9-26. Simulated atomic structures of a symmetrical I"5(310)/[001] tilt grain boundary in Nb (projected along [001]). (a) Lowest energy structure predicted using EAM. (b) Next lowest (metastable) energy structure predicted using EAM. (c) Structure predicted using a modified potential MGPT. This structure is not stable for EAM potentials.
582
9 Structures of Interfaces in Crystalline Solids
001] • ? ™30 nro 2 lift!
Figure 9-27. HREM micrograph of a (310) twin in Nb. The electron beam is parallel to [001] in both crystals and crossed (110) planes are imaged (lattice spacing d=0.23 nm). Long atomistically flat regions of the grain boundary exist indicating a stable structure of boundary. (Defocusing distance A/= — 30 nm, foil thickness 5.0 + 0.3 nm).
the [001] direction. Crystallographically flat, straight sections of the boundary on (310) planes in both crystals are seen to be separated by small regions of severe mismatch. The experimental conditions for imaging can be derived by standard techniques (O'Keefe, 1985) and utilized for an evaluation of the HREM micrographs. Theoretically predicted models of grain boundary structure are compared to the experimental HREM micrograph through image simulation. The comparison of the simulated image formed using the relaxed CSL structure (Campbell, 1992 a, b) and the experimental image is shown in Fig. 9-28 a for the determined value of focus. The contrast in the image simulation is relatively insensitive to thickness when in the range of less than one extinction length for the main imageforming reflection (Spence, 1988). The match is good for the defocusing value A/n = — 30 nm in Fig. 9-28 a as well as for A / = +40 nm in Fig. 9-28 b. However, detailed examination of the contrast in the interface of the simulated image (left part
of Fig. 9-28 b) shows more intensity between the bright spots than seen in the experimental image. The twin has been found to be formed naturally in Nb after annealing a heavily deformed microstructure (Segall, 1961), but apparently the presence of a high concentration of interstitial oxygen also enables its formation (Hartley, 1966). The chemical analysis of the crystal does not show a significant amount of oxygen in the single crystalline Nb. Segregation to the interface should be considered. Under the imaging conditions used for Fig. 9-28 a, b, where areas of high projected potential appear dark, the presence of additional atoms at the interface would decrease the intensity. Proper image simulation, though, requires an appropriate model of the geometry of oxygen segregation. A complete structural determination of the interface would allow the choice of just one model structure as the best for that interface. Determining the projected structure of the interface in another direction, when combined with the above projected
9.8 Experimental Results and Comparison to Results of Simulation
foot]* A* -- 40 nm 5 A
583
Figure 9-28. Comparison of the experimental image (enlargement of Fig. 9-27) with the simulated image (left part) and overlay of the atom positions for Model 1. (a) Defocus value A/=-30nm. (b) A/=+40nm.
(b)
structure determination, would give a complete characterization. But structural imaging of interfaces by HREM is subject to strict geometrical requirements (Pirouz and Ernst, 1990). The plane of the interface, as well as zone axes in both crystals, must be aligned with the electron beam. Such conditions do exist in other directions contained in the interface, but they require the resolution of the (310) planes which are parallel to the interface and have a spacing of 1.0 A, a resolution that is currently not obtainable.
Recently, theoretical and experimental studies were performed for the evaluation of the structure of incoherent twins in Cu and Ag (Ernst etal., 1992). As shown in Sec. 9.4.3 (Fig. 9-9) equilibrium atomistic structures and grain boundary energies were calculated by static energy minimization using an embedded atom potential. Copper bicrystals of the same boundary orientations were fabricated by welding (Fischmeister etal., 1988) of Cu single crystals. The atomistic structure of the {211} twin boundary was investigated by
584
9 Structures of Interfaces in Crystalline Solids
HREM. Figure 9-29 depicts a HREM micrograph. It is evident that significant atomic rearrangements occur extending over several planes normal to the boundary. The rearrangements encompass large translations. A striking feature of the structure in the transition region is a bending of the {111} planes which run continuously through the boundary. Wolf et al. (1992) explained the structure in terms of a rhombohedral 9R phase of Cu and Ag forming a thin (1 to 2 nm) layer at the boundary. The presented results show the strong impact of HREM on the determination of grain boundary structures. More exciting results are expected owing to the better resolution of the high-resolution electron microscopes available in the very near future.
Figure 9-29. HREM micrograph of an incoherent twin in Cu. The interface is not represented by an abrupt transition but by a transition region. The transition regions consist of a different structure which corresponds to the 9R structure (Wolf et al., 1992; Ernst et al., 1992), see also Fig. 9-9. The arrow marks a dislocation.
9.8.3 Structures of Grain Boundaries in Ceramics 9.8.3.1 Grain Boundaries in Simple Oxides
Relatively few studies have been directed at determining the structure of grain boundaries in ceramics, especially compared to the extended literature devoted to the structure of grain boundaries in metals. Balluffi etal. (1981) and Riihle (1982, 1985, 1986) reviewed the different studies on the crystallographic structure of grain boundaries but the reviewed work was restricted to single-phase ceramics, mostly of relatively simple structure and, with a few exceptions, structures with a small unit cell. Many of the studies were performed on NiO. Tilt boundaries have been studied (Riihle, 1985, 1986; Riihle and Sass, 1984; Ruhle et al., 1984) by conventional TEM and by electron diffraction. Merkle and Smith (1987a, b), Merkle (1989,1991) performed quantitative HREM studies on the structures of large angle tilt boundaries in NiO, studying not only rigid body shifts but also grain boundary core structures, i.e. regions adjacent to the interface. Both the £5(210) and £5(310) grain boundaries show strong deviations from the calculated structures, while the observed structure for the £13(320) grain boundary is in qualitative agreement with the calculated structure (Merkle, 1989, 1991). Although the NiO grain boundaries generally appear to have a more open structure than the metallic grain boundaries, the £5 grain boundaries have a quite dense arrangement of atomic columns, essentially containing one additional atomic plane (or one extra atomic column per structural unit) compared to the calculations by Duffy and Tasker (1983). The rigid-body displacement normal to the grain boundary, the so-called volume
9.8 Experimental Results and Comparison to Results of Simulation
expansion for the (310) grain boundary is approximately 0.03 nm compared to 0.11 nm for the calculated structure (Merkle and Smith, 1987 a, b). The total excess volume of the boundary may, however, be greater than that given by the rigid-body shift when vacancy-type defects are present at the grain boundary. Some or all of the atomic columns at the interface would then not be fully occupied. The strong Fresnel contrast behavior which is invariably observed for these grain boundaries indicates that this is indeed the case in NiO. Consequently, HREM images should be expected to reflect the partial occupation of columns. A comparison of experimental micrographs with simulated figures shows that there is quantitative agreement only if point defects (preferably vacancies) are introduced in the interface. For a £5(310) grain boundary in NiO, agreement between the experimental and calculated micrograph is reached if the boundary core is surrounded by atomic columns containing 25 % vacancies. Although the differences between simulated and experimental HREM images are quite clear (Merkle, 1991), subtle contrast effects at the interface may also be caused by small variations in specimen thickness, since the latter is only approximately 10 lattice parameters. Therefore, HREM must be supplemented by other techniques, such as Fresnel contrast (Boothroy e t a l , 1986). Atomistic computer simulation studies of twist grain boundaries in NiO have suggested that a reconstruction, which is equivalent to the introduction of Schottky pairs, greatly stabilizes such boundaries (Tasker and Duffy, 1983; Wolf, 1984). It appears that incorporation of vacancytype defects into the grain boundary structure is important for tilt grain boundaries in ceramic oxides.
585
9.8.3.2 Grain Boundaries in Polyphase Ceramics
Most structural ceramics applied in different technologies are polyphase materials. They are either composites of different crystalline phases or contain an intergranular vitreous phase in addition to a single crystalline phase. The vitreous intergranular phases can be caused (i) as a result from a liquid phase sintering process, which is used to densify the ceramic (e.g. in SiAlON ceramics, alumina . . .), (ii) by an incomplete crystallization of a glass (glass ceramic) and (iii) by a condensation of impurities present in the single phase component at the grain boundaries (e.g., silicates in zirconia). Techniques exist, which allow the identification of the amorphous phase (Clarke, 1979). In the materials noted above, most grain boundaries are covered with a glassy film and, in addition, glass is present at grain junctions. The amorphous phases were revealed by TEM observations in alumina (Philips and Hansen, 1983), hotpressed silicon nitride (see e.g. Kleebe and Riihle, 1992) and zinc oxide varistors (Clarke, 1990), which indicate that most grain boundaries are indeed covered with an amorphous film, some special boundaries are not (Schmid and Riihle, 1984). Clarke (1987) explained this observation by plotting (see Fig. 9-30) the energy of the grain boundary as a function of misorientation for crystalline and wetted boundaries. The curve of a crystalline boundary has the form generally used to indicate the angular dependence of low angle grain boundaries and the existence of so-called special misorientations (cusps). In contrast, the energy curve of the wetted grain boundary should be independent of orientation due to the isotropic nature of the glass. On the basis of such descriptions,
586
9 Structures of Interfaces in Crystalline Solids
CRYSTAL-CRYSTAL BOUNDARIES
MISORIENTATION, B
Figure 9-30. On the wetting of grain boundaries by a grain boundary energy/orientation relationship. The grain boundary will only be free of a vitreous intergranular film if the energy is lower than that of a wetted boundary, i.e., for misorientations for which the crystal-crystal boundary energy (full line) is lower than that of the wetted boundary (dashed line). After Clarke (1987).
low angle grain boundaries (misorientations smaller than a critical angle) will be free of glass, whereas all large angle grain boundaries will contain an intergranular glass phase with the exception of deep "cusp" orientations (the "special" boundaries). Since both curves will show a substantially different dependency on temperature, different misorientation regions will
be wetted at different temperatures (Clarke, 1987). Recently, Kleebe and Riihle (1992) determined the thickness of the amorphous grain boundary film for different Si 3 N 4 ceramics, one example is shown in Fig. 9-31. The experimental studies showed that the thickness of the films is constant for a specific chemical composition. This result is expected from Clarke's theory (1987). Only small angle boundaries and some special boundaries (Schmid and Ruhle, 1984) did not contain the amorphous grain boundary film. 9.8.4 Structures of Metal/Ceramic Interfaces The application of modern engineering materials often requires that two different materials be bonded. The resultant interfaces must typically sustain mechanical and/or electrical forces without failure. Consequently, interfaces exert an important, and sometimes controlling, influence on performance in such applications as composites (Mehrabian, 1983; Dhingra and Fishman, 1986; Lemkey et al, 1988; Suresh and Needleman, 1988), electronic packaging systems used in information
Figure 9-31. HREM image of a polyphase boundary in Si 3 N 4 ceramics. Lattice planes can readily be observed in the crystals. The grain boundary is wetted with a thin amorphous glass film. The film possesses a constant thickness.
9.8 Experimental Results and Comparison to Results of Simulation
processing (Giess et al., 1985; Jackson et al., 1986), thin film technology (Gibson and Dawson, 1985; Yoo et al., 1988; Huang etal., 1991; Kasper and Parker, 1989), and joining (Ruhle et al., 1985; Ishida, 1987; Doyema etal. 1989; Nicholas, 1990; Shimida, 1990). Furthermore, interfaces play an important role in the internal and external oxidation or reduction of materials. The importance of systematic studies in this area was emphasized by different international conferences (Ruhle et al., 1985, 1990, 1992). In this section results will be reported with respect to structural studies at specific metal/ceramic interfaces. The structure (to the atomic level) can either be detected by high-resolution electron microscopy (see Sec. 9.7) or by X-ray scattering (see Sec. 9.6). The structure of different metal/ceramic interfaces has been studied by different authors (see conference proceedings, Ruhle et al., 1985, 1990, 1992). In this paper two systems will be described and discussed extensively. The two case studies demonstrate the complexity of the problem. 9.8.4.1 Structure of the Interface Between Sapphire (a-Al 2 O 3 ) and a Single Crystalline Nb Film Grown by Molecular Beam Expitaxy (MBE)
Several methods are capable of generating well defined metal/ceramic interfaces. At the most fundamental level, ultra clean, flat surfaces readily bond at moderate temperatures and pressures (Fischmeister et al., 1988). Interfaces can also be produced by internal oxidation of metallic alloys (Mader, 1989), where small oxide particles in different metals (Nb, Pd, Ag, . . .) are formed by oxidation of a less noble alloying component such as Al, Cd, etc. Interfaces produced by the internal oxida-
587
tion process usually show a well-defined low-energy crystallographic orientation relationship between the two components and were thus used as model systems. A third method for manufacturing metal/ ceramic interfaces is by evaporation of metals onto clean ceramic surfaces (molecular beam epitaxy: MBE). This method allows control over both substrate material/ orientation and overlayer composition and by this well defined interfaces can be obtained (Flynn, 1990). Nb/Al 2 O 3 serves as an excellent "model" system since Nb and A12O3 possess nearly the same thermal expansion coefficients and most thermodynamic quantities (solubility, diffusion data, etc.) are well established for both components. Nb/Al 2 O 3 composites are used in different applications such as Josephson junctions and as components for structural materials. To date, only a few detailed studies have been reported concerning the atomistic structure of Nb/Al 2 O 3 interfaces formed after diffusion bonding (Florjancic et al., 1985; Mader and Riihle, 1989; Mayer etal., 1990a), internal oxidation (Mader, 1989; Kuwabara etal., 1989), and after thin film deposition (Knowles et al., 1987; Mayer etal., 1989; Mayer etal., 1990b). The studies were all performed by high resolution electron microscopy (HREM). Orientation relationships (OR) were evaluated from diffraction studies, either by Xrays, or by selected area diffraction (SAD) patterns obtained in a transmission electron microscope (TEM). The OR between Nb and A12O3 is determined by the manufacturing route (Mayer et al., 1989). While the OR is preset for interfaces prepared by diffusion-bonding, topotaxial or epitaxial OR develops during internal oxidation and epitaxial growth, respectively. During internal oxidation a topotaxial relationship forms between Nb and A12O3
588
9 Structures of Interfaces in Crystalline Solids
(Mader, 1989; Kuwabara etal., 1989), so that close-packed planes of both systems are parallel to each other, i.e.: (0001)s||(110)Nb and [O1IO]S || [001]Nb (S = sapphire) 1
(9-9)
Epitaxial growth of very high quality single-crystalline overlayers of Nb on sapphire has been a subject of recent experiments. There is experimental evidence (Flynn, 1988, 1990) that for most sapphire surfaces a unique three-dimensional epitaxial relationship between Nb and A12O3 develops which is given by the following OR: (0001) s ||(lll) N b [1010]s||[121]Nb
and (9-10)
The Nb layers were fabricated in a MBE growth chamber equipped with electron beam sources for evaporating refractory metals. The sapphire substrates were parallel to (0001)s, (112O)S, and (1100)s, respectively. The substrates were preheated and cleaned by annealing (Arbab et al., 1989). A special technique was used to obtain transmission electron microscopy (TEM) cross-section samples of the Nb/Al 2 O 3 interface (Mayer et al., 1990b). The HREM studies were performed on the Atomic Resolution Microscope (ARM) at the National Center for Electron Microscopy (NCEM), Berkeley, CA, USA. The microscope was operated at 800 kV at a well defocused CTF (Hetherington et al., 1989). The OR between Nb and A12O3 was evaluated from selected area diffraction (SAD) patterns taken from different Nb films on sapphire substrates in three differ1
The orientation relationship (OR) between two crystals of different lattice structure is uniquely described by one coinciding plane (in both lattices) and one set of coinciding directions in that plane.
ent orientations. For all films a unique OR was evaluated which agrees with the observations of Flynn (1988, 1990), Eq. (9-10). The OR is represented in Fig. 9-32. The following low index directions of sapphire and Nb, are parallel to each other [2IIO]S || [H0] Nb
(direction A)
(9-11)
(direction B)
(9-12)
or [10l0]s || [121]Nb
From crystallographic symmetry arguments the angle between the directions A and B is found to be 30°. The three-fold axes of Nb and A12O3 (sapphire), are parallel. The OR described by Eqs. (9-10) to (9-12) is independently identified at epitaxially grown Nb/Al 2 O 3 interfaces for substrate surfaces parallel to: (000l) s , (lT00)s, and (1210)s. In this paper only results for (0001 ) s substrates are reported. Observations for the other substrate orientations are reported elsewhere (Mayer etal., 1990 b). [0001 ] s
Figure 9-32. Orientation relationship between sapphire substrates and niobium overlayers. (a) The threedimensional orientation relationship [0001]s || [lll] N b (S = sapphire) holds for all substrate orientations, (b) orientation relationship for different directions within the (0001 ) s || (lll) N b plane.
9.8 Experimental Results and Comparison to Results of Simulation
589
Figure 9-33. High-resolution images of a Nb/Al 2 O 3 interface. Direction of incoming electrons parallel to [110]Nb and defocusing distance A/= — 70 nm (Scherzer defocus — 55 nm). Lattice planes can clearly be identified in both sapphire and Nb. Foil thickness ^10 nm. At the interface, regions of good matching (M) and poor matching (D) alternate. S steps in the substrate.
Direct lattice imaging of near interface regions allows the determination of the atomistic structure of the interface as well as the analysis of defects associated with the interface, such as misfit dislocations, etc. To obtain interpretable HREM images the electron beam should be incident along high symmetry directions in both crystals and should be parallel to the plane of the interface. A three-dimensional analysis of the structure requires HREM images taken under different directions of the incident electron beam with respect to the interface orientation. These conditions are fulfilled if the electron beam is parallel to direction A and B, respectively, Eq. (9-11) and Eq. (9-12), see Fig. 9-32 b. High-resolution electron micrographs were taken from the same interface in both directions
by simply tilting the specimen inside the ARM. Figure 9-33 shows a large area of a nearinterface region. The atomic distance corresponding to the (200) planes with d— 0.165 nm is clearly visible. Other lattice planes can readily be identified in A12O3 and Nb. The foil thicknesses of Nb and A12O3 are identical. In Nb regions of good matching (mark M) and poor matching (mark D) alternate at the interface. Steps can also be identified (S). The region of good matching (Fig. 9-33, M) is imaged at a higher magnification in Fig. 9-34. Figure 9-34 a shows the interface with the electron beam parallel to direction A. Lattice planes transfer continuously from Nb to A12O3. Figure 9-34b is a micrograph of the same interface viewed along orienta-
590
9 Structures of Interfaces in Crystalline Solids
(a)
(b)
Figure 9-34. High-resolution image of Nb/Al 2 O 3 interface. Region of good matching (M), section of Fig. 9-33. (a) Direction A with [2lT0]s || [110]Nb. The indicated lattice spacing in Nb corresponds to a (112) plane, (b) Direction B with [1010]s || [121]Nb. The indicated lattice spacing in Nb corresponds to the distance of the (101) plane.
tion B. Only (101) lattice planes with a spacing of 0.233 nm are visible in the Nb crystal in regions of good matching M. In both orientations (Figs. 9-34a, b) a perfect match of the Nb and A12O3 lattice at the interface is visible. The mismatch of the (0ll0) s and (112)Nb planes which are perpendicular to the interface is only - 1 . 9 % and this misfit is accommodated by localized defects (misfit dislocations) in the regions D of poor matching (Fig. 9-33 and Fig. 9-35) (Mayer etal., 1992). This allows the Nb lattice in
between these defects to expand slightly along the interface (the Nb lattice possesses the smaller lattice plane spacing) resulting in extended regions of perfect matching. The expansion of the lattice plane spacings parallel to the interface is limited to regions close to the interface. The lattice planes of Nb, especially near the misfit dislocations, are bent resulting in a continuous transition to the undistorted Nb lattice further away from the interface (this can be seen by viewing Fig. 9-35 under grazing incidence). No dislocations or lattice distortions could be seen in the A12O3 lattice. The misfit dislocations in the Nb do not show any "stand-off" distance from the interface (Mader, 1989). In HREM images only projections of the foil on its exit surface can be analyzed. Dislocations can thus only be identified if viewed edge on, i.e. if the dislocation line lies parallel to the beam. The dislocation shown in Fig. 9-35 fulfils this requirement and by making use of the continuous transition of lattice planes between the niobi-
Figure 9-35. High-resolution image of a region of poor matching section of Fig. 9-31. A misfit dislocation forms with no stand-off distance from the interface. The core of the misfit dislocation can readily be identified, as outlined in the picture. The arrows indicate inserted planes in the Nb; black and white arrow: additional (110) plane, arrow: additional (002) plane in Nb.
9.8 Experimental Results and Comparison to Results of Simulation
um and the sapphire a Burgers circuit can be performed. A projected Burgers vector of b = 1/2 [llT] Nb was determined. Since only the projection can be seen in the HREM image this does not preclude a possible screw component parallel to the viewing direction. While in HREM individual dislocations can be viewed edge-on, imaging of a twodimensional array of dislocations lying parallel to an interface requires that this interface be inspected in plan-view. The dislocations can then be imaged with conventional TEM techniques. In the case of our system two difficulties arise: (i) All the low-indexed reflections of the Nb which can be used to image the dislocations are very close to corresponding reflections of the sapphire (the distances between the reflections are given by the 1.9% misfit). In bright-field or darkfield images this causes strong moire contrast with the same periodicity as the misfit dislocations, (ii) In order to obtain strong diffraction contrast of the dislocations (utilizing dark field imaging techniques) bent lattice planes have to be present in the vicinity of the dislocation core. In our system the core of the misfit dislocations is located directly at the interface. The bent lattice planes around the dislocation core terminate abruptly at the interface and additional relaxations have to be expected in the niobium at the interface to the rigid sapphire lattice. It is thus not clear whether the simple g • b criterion which is used to select possible imaging conditions for bulk dislocations is also valid in the case of misfit dislocations without "stand-off". A schematic drawing of the atomistic structure of a (1/2) [llT] Nb dislocation in bulk Nb is shown in Fig. 9-36. The model is purely geometrical and relaxations in the vicinity of the dislocation core have not been taken into account. The dislocation
591
Figure 9-36. Schematic drawing of a (1 /2) [111] dislocation in bulk Nb. Relaxations of the three inserted (222) planes have not been taken into account. One unit-cell including the Burgers-vector is outlined. The position of the terminating (111) plane at the Nb/sapphire interface is marked (* — *).
core consists of three inserted (222) planes. The position of the (111) plane forming the terminating plane at the interface is indicated in the figure. On the experimental micrograph (Fig. 9-35) it is virtually impossible to identify geometrically the three additional (222) planes forming the dislocation core (b = (1/2) [lll] Nb ). However, if viewed along the arrowed directions an additional (110) plane (corresponding to a Burgers vector of (1/2) [110]Nb) and an additional (002) plane (corresponding to a Burgers vector of (1/2) [001]Nb) can clearly be identified in both the experimental image and the schematical drawing. This description is equivalent to decomposing the Burgers vector into two components according to: (l/2)[llT] Nb = (l/2)[110]Nb + (1/2) [001]Nb • Especially for the (1/2) [110]Nb
592
9 Structures of Interfaces in Crystalline Solids
component strong bending of the lattice planes and a strong localization of the dislocation core can be recognized in the HREM image (Fig. 9-35). The diffraction contrast caused by the dislocations and the moire fringes can only be distinguished by weak-beam imaging (Pond, 1984). In weak-beam images diffraction contrast is caused by a specific strain component (Cockayne, 1972). The image obtained from a dislocation is narrower (1-2 nm) because the imaging conditions are such, that only areas close to the dislocation core, where lattice planes are strongly bent, contribute to the image. Mayer et al. (1992) concluded that the misfit dislocations are forming a triangular array at the interface between the Nb and the sapphire. We were not able to image the other two sets of parallel dislocations in one sample area because of the limited tilt angle capacity of the microscope. Quantitative HREM requires computer simulation. The atomistic configuration is obtained if experimental images are identical to images simulated for specific atomistic models. The analysis requires the knowledge of the exact focusing value and the foil thickness. These values have to be determined in a first step (Spence, 1988). The determination of the foil thickness is most accurate if observed images of the Nb and A12O3 are compared to tableaus of calculated lattice images for varying thicknesses and focus values (Stadelmann, 1987). Good agreement between the simulated images and the experimental image is obtained for a defocus of A/= — 40 nm and a foil thickness of / = 7 nm in Nb and A12O3. From the simulated images it is also possible to determine the positions of the atoms in both crystals with respect to the intensity distribution in the experimental image. The next step is to identify the translational state T of the two crystals
with respect to each other. Such a translational state only exists in the areas of good matching. In these areas the two crystals become commensurate at the interface by expanding the lattice plane spacing of the Nb (Fig. 9-37). The translational vector T(Tl9 r 2 , T3) can be constructed so that the components 7j and T2 lie within the interface plane and T3 is perpendicular to it. An inspection of the HREM images (Figs. 9-34 and 9-38 a) reveals that certain lattice fringes in Nb and A12O3 transfer
11 a2
7
ki f
Figure 9-37. The lattice plane spacings of two different materials are incommensurate. If two lattice planes of both materials are brought together in one point the same match will never occur again along the interface. Due to relaxations along the interface an epitaxial fit can be achieved at metal/ceramic interfaces. The mismatch is accommodated by localized misfit dislocations in the metal (in circles). In the regions of good matching between these defects a unique translational state is obtained between both lattices which can be described by a translational vector J.
9.8 Experimental Results and Comparison to Results of Simulation
593
Figure 9-38. Comparison of (a) the experimental image taken along direction A, (b) the same image after Fourier-filtering and (c) to (f) simulated images for various conditions: (c) simulated image showing the best fit to the experimental image, (d) as in (c) but with 0.04 nm shorter distance between the two lattices (component T3 of translation vector J ) , (e) as in (c) but with 0.04 nm longer distance and (f) after removing the terminating oxygen layer of the sapphire.
continuously into each other across the interface. Furthermore, the positions of the atoms with respect to these lattice fringes in the experimental images are known. However, due to the noise present in the experimental image the position of the atomic columns can not be located very accurately (Fig. 9-38 a). In order to reduce the noise of our experimental images, a Fourier filtering technique was applied (Mobus et al., 1992). The result of the filtering is shown in Fig. 9-38 b. From the known positions of the atoms in both the Nb and the A12O3 we have determined the translational state of both lattices across the interface. From this a model of the atomic structure at the interface can be derived. If we assume that the lattice continues undisturbed up to the interface, then
the atomistic structure represented in Fig. 9-39 a is obtained: the A12O3 is terminated by an oxygen layer and the Nb atoms of the first Nb layer fit accurately in the A sites of the oxygen layer under which no Al ions are positioned. An obvious choice for the origin of the translational vector T is given by one of the Al-sublattices as indicated in Figs. 9-39 a, b. The first Nb layer possesses exactly the same threefold symmetry and the same atomic distances as the individual layers of the Al-sublattice parallel to the interface. The third and final step is to simulate HREM images for the structure model shown in Fig. 9-39 a and to vary the individual parameters until a best fit is obtained. Four examples of the simulated images for different parameters are shown in
594
9 Structures of Interfaces in Crystalline Solids [110]
[2110]8
[10T0]
[1014]
[0112]
Figure 9-39. Schematic drawings of the atomic positions at the Nb/Al 2 O 3 interface, (a) Projected view parallel to the interface. O2 " ions: light large circles, Al3 + : black circles, Nb atoms: grey circles. Note the continuous transition of (110)Nb to (1014)s planes and (001)Nb to (0112)s planes, (b) View perpendicular to the interface: O2~ ions: light large circles, Al3 + : dark sections of small circles, Nb atoms: bold medium sized circles. It is assumed that the O2 ~ ions form the outermost layer. The letters A, B, C mark the "hole" site in this layer to their upper left. The Nb atoms of the first Nb layer are positioned above the empty sites of the first Al3 + layer.
Fig. 9-38 c to 9-38 f. Figure 9-38 c shows the image with the best fit from which we determined the translational vector T. From Fig. 9-38 c the components Tl9 T2 and Ty of the translational vector result in: Tt = a 0 ,73/6 = 0.137 nm T2 = a0/2 = 0.238 nm T3 =
where a0 and c0 are the lattice constants of sapphire. The experimental error in determining each individual component is ±0.02nm. The vector (Tl9T2,T3) given above is a lattice vector of the Al sublattice indicated in Fig. 9-39 a. Therefore, within the experimental error, the Nb atoms of the first layer are positioned exactly in the sites where the Al-ions of the next layer
would be placed if the sapphire lattice would be continued. If the distance T3 is changed by ±0.04 nm, the simulated contrast changes to that shown in Fig. 9-38 d, e. The relative shift of the lattice planes can be seen by viewing the images under grazing incidence. Figure 9-38 f shows an image which was obtained by removing the last oxygen layer of the sapphire lattice. This results in a contrast along the interface which is clearly different from the one in the experimental image and also leaves a gap at the interface where the atoms would not be in touch with each other. No twins could be identified in the (nearly) perfect Nb film. It was demonstrated that a unique atomistic relationship exists between the sapphire surface
9.8 Experimental Results and Comparison to Results of Simulation
and the monoatomic Nb layer. There exists only one set of Nb atom positions on the unreconstructed sapphire surface which leads to a twin free film (Fig. 9-39 b). Nb atoms must be located on the terminating O2~ layer of sapphire on top of "empty sites". Comparison of experimental HREM images with the corresponding simulated images verified this hypothesis. For the first time all three components of the translational vector between the two lattices joining at a metal/ceramic interface could be determined. The location of the Nb atoms on top of the terminating O 2 ~ layer is such that the Al sublattice which would form the next layer of the sapphire is continued. This implies that the distance between a Nb atom of the first layer and the three neighbouring O 2 ~ ions is shorter than the distance which would be expected for neutral Nb atoms. However, a good match is obtained for the radius of the Nb 3 + ion. A layer of Nb 3 + would also account for the charge balance across the interface assuming that a complete O2~ layer is terminating the sapphire at the interface. In conclusion the epitaxial growth of Nb on basal plane sapphire seems to be dictated by a continuation of the cation sublattice of the sapphire in both location and ionicity of the Nb atoms of the first layer. Nb grows well on sapphire because the resulting first layer has the same symmetry and atomic arrangement as the (111) plane of bulk Nb, with only a 1.9% mismatch. The model we present here was derived mainly from a geometrical evaluation of the HREM images. It therefore does not take into account the effects of possible interdiffusion leading to a partial substitution of e.g. Nb by Al near the interface. HREM as a method is fairly insensitive to such a change in chemistry which leaves the structure unaffected. The chemistry of
595
the interface therefore will have to be studied by high resolution chemical analysis. 9.8.4.2 Structure and Bonding at the Ag/MgO Interfaces
For an understanding of the fundamental physics of bonding between the metal (Ag) and the ceramic (MgO) it is important to compare the experimentally determined structure with models derived from physical principles. Both, experimental structure determination and numerical modeling of interface structures are presently limited to particular model-like situations, and therefore the variety of interfaces that can simultaneously be studied experimentally and theoretically is rather small. {100}Ag/MgO interfaces are promising systems in this respect. Sphere-on-plate experiments of Au and Cu on {100} MgO substrates suggest that the energy of an {100}Ag/MgO interface reaches its absolute minimum when the Ag and the MgO lattices are in parallel orientation (Fecht and Gleiter, 1985). The lattice mismatch S of Ag and MgO, defined as 3=
(9-13)
amounts to 3 %. aMgO and aAg are the lattice parameters of MgO and Ag respectively. The mismatch is sufficiently small for ab initio modelling of commensurate regions of the parallel epitactic Ag/MgO interface, as carried out by the group of Andersen (Blochl et al., 1990; Schonberger etal., 1992). In these calculations the lattice mismatch was balanced by a tetragonal distortion of the Ag lattice, while the high common symmetry of the Ag and MgO crystals in parallel orientation permitted a tractable small supercell.
596
9 Structures of Interfaces in Crystalline Solids
The calculated electronic structure of the Ag/MgO interface indicates that the bonding between Ag and MgO is weak and ionic in character. Another important result of the ab initio calculations is the energetically most favorable translation state of Ag and MgO. The atoms in the terminating {100} layer of the Ag crystal tend to sit on top of the O ions in the first {100}MgO layer. A value of 0.25 nm was calculated for the spacing between the atomic layers terminating the adjacent crystals. This value corresponds to an excess volume of about 20% in the interfacial planes (Schonberger et al., 1992). The results of the ab initio calculations should be compared to experimental observations of the Ag/MgO interface by HREM. As the {111} and {200} spacings of both Ag and MgO are larger than the point resolution of the HREM, the structure of the {100} interface between Ag and MgO in parallel orientation can be imaged in a common <100> as well as in a common <110> direction of Ag and MgO (Trampert etal., 1992). Besides the translation state our interest s also focused on the structure of mismatched regions in the Ag/MgO interface, vvhich were not included in the ab initio calculations. Considering the non-directional nature of electrostatic bonding it is an interesting question to what extent the contact of Ag and MgO introduces coherency strains in the elastically softer Ag crystal. In the closely related {100} interface between Au and MgO in parallel orientation (3 = 3%) Hoel et al. (1989) could not detect the strain fields of such dislocations; neither they could be detected by HREM in noble metal/oxide interfaces with larger lattice mismatch, such as Cu/ A12O3 (5 = 10%), Pd/Al 2 O 3 ((3 = 12%) or Ag/CdO ((5 = 14%) (Ernst etal., 1991;
Muschik and Riihle, 1991; Necker and Mader, 1988). In the following an HREM investigation of the {100} Ag/MgO interface will be reported. Ag layers were deposited onto {100} MgO substrates in an MBE growth chamber (Flynn, 1988) with a typical growth rate of one monolayer per second (Flynn, 1990). For details see Trampert etal. (1992). Defect structure of the interface. Figure 9-40 presents an overview of the Ag/MgO interface. The micrograph was recorded at optimum defocusing distance A/s (Scherzer defocus, Spence, 1988). At this focus setting the lattice image exhibits {200} planes in Ag and MgO with fringes parallel to the Ag/MgO interface. Along the Ag/MgO interface regions of good (G) and poor (P) lattice match alternate. In regions of poor lattice match the Ag lattice planes are bent in such a way that this bending restores the continuity of Ag and MgO {200} planes across the interface, a misfit dislocation is introduced. The average spacing of the misfit dislocation cores in the experimental image amounts to 32 + 5 {200} lattice planes, which corresponds to a distance D = (6.53 + 1.02) nm. In order to image the lattice plane bending more clearly, the experimental image of specimen A was digitally Fourier filtered (Mobus etal., 1992). In the filtered image (Fig. 9-41 a) the position of the misfit dislocation core can be determined unambiguously. Parallel to the Ag/ MgO interface the misfit dislocation has a long-range strain field; the bending of lattice planes is still observable up to ten {200} spacings away from the core position. The Burgers circuit around a misfit dislocation core in Fig. 9-41 b yields a Burgers vector component of b1 = (l/2)aAg <100>
9.8 Experimental Results and Comparison to Results of Simulation
597
Figure 9-40. HREM image of the {100} Ag/MgO interface. Direction of incoming electron beam parallel to <100>. Regions of good (G) and poor (P) lattice matching are marked. The micrograph was recorded with an instrument operated at 400 kV and with a point resolution of 0.175 nm. The objective aperture included four 200 and four 220 beams. {200} lattice planes are resolved in both crystals.
normal to the electron beam and parallel to the Ag/MgO interface. The spacing of the misfit dislocations agrees well with the geometrically expected spacing D = bjd. Translation state. In the commensurate regions between the misfit dislocations, where the coherency strains are minimal, two components of the relative translation between the Ag and the MgO crystal can be determined. In Fig. 9-42 the bright spots on every side of the interface represent the atomic columns. In unstrained regions the bright spots form straight lines without kinks across the interface. This means that the ions in the final Ag layer sit on top of either the Mg or the O ions of the first MgO layer. Which case is realized cannot be decided on the basis of the present HREM investigations. The translation component normal to the Ag/MgO interface can be sensitively determined from the course of {220} fringes crossing the interface at 45°. The
fringes show no kink in the interfacial region (Fig. 9-43). This is only possible if the spacing between the terminating Ag {200} layer and the first MgO {200} layer is (20 ± 5 ) % larger than the MgO {200} spacing. Thus, the excess volume predicted by the ab initio calculations is experimentally confirmed within the limits of error. Duffy et al. (1992) proposed that the adhesion between a noble metal and an oxide originates from Coulomb forces between the ions of the oxide and image charges in the metal. According to their model, the energetically most favorable translation state would be a "lock-in" configuration of the Ag and the MgO crystal because the interfacial energy decreases with decreasing distance between the ions and their image charges. Our experimental observations show, however, that the most favorable translation state of Ag and MgO corresponds to a "ball-on-ball" rather than a "lock-in" configuration.
598
9 Structures of Interfaces in Crystalline Solids
(a)
Ag
MgO
(b) Figure 9-41. HREM image of a mismatched region (P) of Fig. 9-40. (a) After digital filtering (Mobus et al., 1992); (b) Burgers circuit marked.
599
9.9 Concluding Remarks
Ag
MgO
Figure 9-42. Commensurate region of interface (region G of Fig. 9-40) identifying the atom positions on each side of the interface. The arrow marks a terminating atom plane at the interface.
Ag
MgO
9.9 Concluding Remarks It is quite remarkable how much progress has been made in the understanding on the structure of internal interfaces in crystalline solids. About six years ago (1986) many results were available only by computer simulations. Experimentally a few results on the structure of grain boundaries in semiconductors and metals were available from qualitative X-ray diffraction studies and transmission electron microscopy. The field has rapidly changed: Results of computer simulations can be checked by advanced experimental techniques, with emphasis on quantitative high-resolution electron microscopy. Also techniques utilized by surface scientists are now being tried by those working on internal interfaces. In addition, the applicability of local probes such as
Figure 9-43. HREM image showing a region of good matching (G) of Fig. 9-40. The course of {220} fringes across the interface depends sensitively on the translation component T3 normal to the interface.
Mossbauer spectroscopy, positron annihilation, EXAFS (extended X-ray absorption fine structure) or ELNES (energy loss near edge structure) are being tried and encouraging results are emerging. An important experimental problem seems to be the control of the purity of the grain boundaries investigated. In view of the sometimes drastic effects of segregants on grain boundary structure, improved control of the impurities at the grain boundaries seems to be essential. There are ongoing efforts to prepare highly controlled grain boundaries (Fischmeister
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9 Structures of Interfaces in Crystalline Solids
etal., 1988). Very encouraging are results emerging from a direct link of computer simulation and experimental studies of identical internal interfaces. From those studies exciting results can be expected in the near future.
9.10 Acknowledgements The authors acknowledge helpful and fruitful discussions with colleagues from the "interface community". Particularly, we would like to thank Drs. R. W. Balluffi, F. Ernst, H. F. Fischmeister, P. Gumbsch, W. Mader and A. Sutton. Ms. E. Pfeilmeier efficiently processed the many versions of the manuscript.
9.11 References Ackland, G. X, Thetford, R. (1986), Phil. Mag. A 56, 15-30. Ackland, G. X, Tichy, G., Vitek, V., Finnis, M. W. (1987), Phil. Mag. A 56, 735-756. Ackland, G. X, Vitek, V. (1990), Phys. Rev. B 41, 10324-10333. Allan, G. (1970), Ann. Phys. 5, 169-202. Allan, G., Lannoo, M. (1976), /. Phys. Chem. Solids 37, 699-709. Allan, N. L., Kenway, P. R., Mackrodt, W. C , Parker, S. C. (1989), /. Phys.: Condensed Matter 1, SB119-SB122. Arbab, M., Chottimer, G. G., Hofmann, R. W. (1989), Mater. Res. Soc. Symp. Proc. 153, 63-69. Ashu, P., Matthai, C. C. (1989), /. Phys.: Cond. Matter 1, SB17-SB20. Aucouturier, M. (1984), Les joints de grains dans les materiaux. Les Ulis: Les Editions de Physique. Aucouturier, M. (Ed.) (1989), Proc. Int. Congr. Intergranular and Interphase Boundaries in Materials, Coll. de Physique 51-CL Les Ulis: Les Editions de Physique. Bacher, P., Wynblatt, P. (1991), Mater. Res. Symp. Proc. 205, 375-380. Bacher, P., Wynblatt, P., Foiles, S. (1991), Acta Me tall. Mater. 39, 2681-2691. Bacmann, X X, Papon, A. M., Petit, M., Silvestre, G. (1985), Phil. Mag. A 51, 697-713. Balluffi, R. W. (1977), in: Interface Segregation: Johnson, W. C , Blakely, X M. (Eds.). Metals Park, OH: American Society for Metals.
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