Basics of Thermodynamics and Phase Transitions in Complex Intermetallics (Book Series on Complex Metallic Alloys)

Book Series on Complex Metallic Alloys – Vol. 1

BASICS OF THERMODYNAMICS AND PHASE TRANSITIONS IN COMPLEX INTERMETALLICS

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Book Series on Complex Metallic Alloys – Vol. 1

BASICS OF THERMODYNAMICS AND PHASE TRANSITIONS IN COMPLEX INTERMETALLICS

edited by

Esther Belin-Ferré Laboratoire de Chimie Physique-Matiere et Rayonnement Centre National de la Recherche Scientifique, Université Pierre et Marie Curie, France

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BASICS OF THERMODYNAMICS AND PHASE TRANSITIONS IN COMPLEX INTERMETALLICS Series on Complex Metallic Alloys — Vol. 1 Copyright © 2008 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN-13 978-981-279-058-3 ISBN-10 981-279-058-6

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FOREWORD

This volume assembles the texts of the lectures delivered at the first Euro-School on Materials Science, organised in Ljubljana, Slovenia, from 22 to 27 of May 2006 by the European Network of Excellence Complex Metallic Alloys (CMA) under contract NMP3-CT-2005500145. The central objective of the CMA Euro-School is to provide a lecture-style background to students graduating in the field of materials science, in particular in the physics of metals and metallurgy. Four annual sessions of the CMA-Euro-School are foreseen, from which a series of four books will be issued. It is my great pleasure to introduce here the first volume in the series. During the first session of the CMA Euro-School, emphasis was on the basics of thermodynamics, phase transitions, crystallography and electronic properties of complex metallic alloys. These subjects were accounted for owing to two types of lectures. On the one hand, plenary lectures referred to basic topics and on the other hand, shorter lectures reported on tutorial topics. The book begins with a general introduction to the field of CMAs. Each of the following chapters refers to a distinct lecture: chapters 2 to 6 to basic lectures and chapters 7 to 13 to tutorial lectures. The European Commission is warmly acknowledged for financial support. Special thanks go to all authors. They made editing this volume possible. Paris, April 2007

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CONTENTS

Foreword

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Chapter 1: An introduction to complex metallic alloys and to the CMA network of excellence Jean-Marie Dubois

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Chapter 2: Thermodynamics and phase diagrams Livio Battezzati

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Chapter 3: Permanent magnets and microstructure Paul McGuiness

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Chapter 4: Solidification Peter Gille

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Chapter 5: Diffusive phase transformations Yves Bréchet

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Chapter 6: Diffusionless transformations C. Duhamel, S. Venkataraman, S. Scudino J. Eckert

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Chapter 7: Intermetallics: characteristics, problems and prospects Gerhard Sauthoff

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Chapter 8: An introduction to electronic structure methods D. A. Papaconstantopoulos

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Chapter 9: Crystallography of complex metallic alloys Walter Steurer and Thomas Weber

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Contents

Chapter 10: Electronic properties of alloys Östen Rapp

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Chapter 11: Electron transport properties of complex metallic alloys Uichiro Mizutani

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Chapter 12: Chemical bonding and crystallographic features Yuri Grin

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Chapter 13: Plasticity of complex metallic alloys M. Feuerbacher

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CHAPTER 1

AN INTRODUCTION TO COMPLEX METALLIC ALLOYS AND TO THE CMA NETWORK OF EXCELLENCE Jean-Marie Dubois Institut Jean Lamour (FR 2797 CNRS-INPL-UHP), Nancy Université, Ecole des Mines, Parc de Saurupt, Nancy, F-5404 E-mail: [email protected] Now that new tools are available to solve the crystallographic structure of complex compounds in metallic alloy systems, a vivid interest manifests itself to discover new compounds in multi-constituent alloys. Several are yet known to contain hundreds or more atoms per unit cell. In the meantime, real efforts are made for better understanding of the properties of these compounds and the mechanisms that underpin the progressive loss of metallic character when the size of the unit cell increases. This introductory chapter focuses at a few examples of this atypical behavior of complex metallic alloys, including quasicrystals as the ultimate state of structural complexity in a crystal made of metals. Examples are transport properties, surface electronic structure, surface energy, wetting and friction. All examples show the same trend, namely apparent localization of electronic states, loss of conductivity, opening of gaps, softening with no work hardening, etc. All phenomena are reminiscent of what is observed in nanostructured metals, but together with the increase of the size of the unit cell. This effect is therefore coined “inverse nanostructuration” by the author who argues that complex metallic alloys help us revisit ancient problems in metal physics, while in parallel potential applications may be sorted out.

1. Introduction This introductory chapter aims at a short overview of the crystallographic peculiarities and physical properties of Complex Metallic Alloys (CMAs hereafter). Most of the experimental data presented here was obtained quite some time ago and has provided the basis that was used by K. Urban, L. Schlapbach and the author to file an application within the 6th 1

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Framework Program of the European Communities for granting a socalled Network of Excellence (NoE). This successful application came into force in July 2005 and allowed financing the first session of the European School in Materials Science of the CMA NoE1 to which the present book is associated. The focus of the 2006 school was on general aspects of phase transitions in materials, and more specifically on our current knowledge of the transformations taking place in CMAs, from their synthesis to their possible applications. Other sessions will be organized in the coming years, with emphasis on more specific aspects of CMAs. The chapters that follow in this book will give an impressive account of the great progress that was achieved in recent years on CMAs, and also of the many questions that remain only partially solved, or fully open. CMAs are crystalline compounds of the family of intermetallics that are characterized by a) large unit cells, containing up to thousands atoms, b) the occurrence of well-defined clusters, most often of icosahedral symmetry and c) some disorder, essentially due to the fact that icosahedra do not fill Euclidian 3-dimensional space. Therefore, quasicrystals belong to the family of CMAs, but clathrates also do so. The properties of CMAs are surprising, although they cannot be claimed unique1. In Al-based CMAs, electron transport properties (conductivity, Seebeck coefficient, etc.) are governed by the formation of a pseudo-gap (when not a real gap) in the Al 3p partial density of states at the Fermi energy, which results of a combination of Hume-Rothery scattering of electron waves and sp-d hybridization effects2, 3. It turns out that surface properties like surface energy, solid-solid friction or wetting also reflect the depth of the pseudo-gap balanced by the presence of d states at the Fermi energy. As a result, it turns out that the surface energy of highly complex intermetallics is so much smaller than that of the metallic constituents of the alloy4 that reduced friction or wetting against polar liquids was for long taken as the best examples of potential applications of CMAs1, 5.

a

For simplicity in this chapter, italics represent the CMA network, whereas normal capital letters (CMA) are for the complex metallic alloys themselves.

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Quite a few more properties show the same trend, namely apparent localization of electronic states, loss of conductivity, opening of gaps, mechanical softening with no work hardening, etc. All phenomena are reminiscent of what is observed in nanostructured metals, but take place with increasing the size of the unit cell. This effect is therefore coined “inverse nanostructuration” by the author6. In the end, it was admitted by a sufficiently large number of European scientists that the unexpected properties displayed by the few CMAs known so far are fascinating enough, and of strong enough potential interest for technology, to deserve the creation of a new field dedicated to complexity in metallic alloys7. The existence of the field was recognized by the foundation of the NoE labelled according to the same acronym CMA and funded by the Commission of the European Communities as briefly described at the end of this chapter. 2. Complexity in reciprocal and real space 2.1. A definition of CMAs An essential question to address at the beginning of this book is to know what we call a CMA. First, it is a compound, or a phase, or an alloy, essentially made of metals. This does not mean that the alloy is a metal itself, or an alloy characterized by metallic properties, because most often the metallic character of the alloy species has become poor or much weaker than in the pure metal constituents. In scarce cases like Al2Ru, it has turned to semi-conducting. It simply means that the major part of the constituents belongs to sp or d metals (Al, Ga, Sn, Fe, Ni, Pd, W, Rh, Re, etc.), possibly alloyed with semi-conductors (Ge, Si), chalcogenides (Se) and/or rare earths. In few cases, the situation is reversed: the major constituent is the semi-conductor like in clathrates. Oxides, although some may be structurally very complex, are excluded from the CMA family because they present no metallic behaviour whatsoever (except in few cases at very high temperature). The broad variety of chemical combinations that may be synthesized out of about 80 metals in the periodic table participates to the complexity of the compounds

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considered in this book. This also means that the potential for discovering new ternary, quaternary, etc. CMAs is huge. Second, it is often anticipated that complexity is expressed by a difficulty to describe the crystal lattice arrangement due to the large number of independent atomic positions in the unit cell. In many compounds, such a (weak) definition would be acceptable. However, many CMAs do not require lots of independent positions to be accounted for. Very frequently instead, a distribution of occupancy factors must be considered in order to match the chemical disorder inherent to the compound. This is the case for instance of the superstructures of the βCsCl –type cubic phase that forms in Al-Cu alloys8. The basic unit cell is the 2-atom, body-centred cubic unit cell of parameter 0.29 nm. Depending on the Al/Cu composition ratio, substitution vacancies order in the lattice and increase dramatically the size of the unit cell. The largest superstructure known so far forms at composition Al36Cu48V12 (V = vacancy), with a unit cell volume 47.7 larger than that of the conventional β-phase. The example of the superstructures of the β-cubic phase points out the difficulty to describe accurately complexity in real space due to the need to introduce a function that adequately fits the chemical disorder in the lattice, although atomic positions may be easily accounted for by a simple Bravais lattice like body-centred cubic. It is more relevant to call 'complex' an alloy, or a compound, essentially made of metals as presented above, whose reciprocal space exhibits complexity within the Jones zone. The Jones zone is the Brillouin zone constructed with the most important Fourier components in reciprocal space. For a simple crystal, it fits with the first Brillouin zone. When complexity arises, it corresponds to that zone which is built by taking into account the most intense diffraction peaks. A measure of complexity of the crystal is then supplied by the number of peaks that fall inside the Jones zone or equivalently, by the inverse of the reciprocal distance between the first diffraction peak and the origin of reciprocal space: the most complex CMA in a series presents a diffraction peak that falls at the shortest distance to the origin of reciprocal space. In many Al-based compounds, this is a quasicrystal (with in principle 0 distance of the first diffraction peak from the origin of reciprocal space). As an immediate consequence,

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metallic glasses are not considered to belong to the group of CMAs, although it is often observed that metallic glasses of specific composition crystallise in a CMA, because they show no sharp Fourier component is their diffraction pattern. The Jones zone is ill defined by the broad main halo in the diffraction pattern and the pre-peak, when it exists, does not fall that close to the origin of reciprocal space. Their physics however is often very much reminiscent of that of CMAs9. As an example of a typical CMA, Fig. 1 illustrates the case of triclinic Al11Mn4 (unit cell parameters a = 0.5087 nm; b = 0.8848 nm; c = 0.5052 nm; α =89.72°; β =100.54°; χ =105.37°; unit cell volume: 0.215 nm3). The major diffraction peaks fall in the vicinity of the wavevector q=30 nm-1 (q=4π sin θ/λ with θ the Bragg angle and λ the wavelength) whereas many other peaks of variable intensity are observed inside the range [0, q]. The average distance between opposite centres of the facets of the Jones zone is often labelled KP, so that here we have KP ≈ 30 nm-1.

Fig. 1. X-ray diffraction pattern (λ=KαMo) of triclinic Al11Mn4. The x-axis is labelled according to the wave vector q as explained in text. The position of 2kF is shown by a vertical bar, assuming valences of +3 for Al and -3 for Mn. (Courtesy of Dr M. Feuerbacher, CMA).

The number of peaks in the vicinity of q=1/2 KP reflects the degree of symmetry of the Jones zone. The larger the symmetry of the Jones zone,

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the closer the shape will be from a sphere. Such a resemblance to a sphere is actually achieved in many CMAs, for instance, quasicrystals but also γ-brass phases, etc. Furthermore, electronic concentration is very often naturally selected so that a close matching between Jones zone and Fermi surface is observed, which fulfils the Bragg condition KP = 2 kF (where kF is the Fermi vector). Such a selection is responsible for the opening of Hume-Rothery gaps and therefore for the enhanced stability of the compound. In Al11Mn4, the Fermi vector amounts to kF = 14.25 nm-1 if the contribution to the valence band by Al is taken equal to +3 electrons and that of Mn is assumed negative like in many other transition metals such as Fe, Ru, Re (but not Cu or Pd, see ref. 1 for more information on this point) and equal to –3 electrons according to its position in the periodic table along the 3d-metal series. Hence, it is observed that 2kF ≈ KP, a result which is indeed traditionally associated with the formation of a CMA in a given system. 2.2. The example of Al-Cr(-Cu)-Fe alloys After the pioneer contributions of Pauling10 and his successors, the Shoemakers, Samson and others, very little was published on Al-based CMAs until the discovery of quasicrystals re-launched the interest for such crystals. Driven by the need to find a quasicrystalline or approximant material offering high corrosion resistance against acids, we considered addition of Cr to Al-Cu-Fe icosahedral crystals5, 11. Above a small concentration in Cr species, the icosahedral phase is no longer stable and is replaced, depending on the Cu/(Cr+Fe) ratio, by orthorhombic or monoclinic compounds of large to very large unit cell. The same trend is observed in Al-Cr-Fe alloys, in which the icosahedral phase is metastable and can be formed only by rapid cooling of the melt12. Space is too limited here to give a brief description of all compounds discovered in Al-Cu-Fe-Cr alloys (for more details, see ref.1). In the following, I shall concentrate on one single compound, namely the O1-orthorhombic compound of lattice parameters a =3.25 nm, b =1.22 nm, c =2.37 nm, which contains 600 atoms per unit cell (117 independent atomic positions13) and has a hierarchical structural

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relationship with the more simple rhombohedral γ-AlCrFe brass-type phase12. A sketch of the structure is shown in Fig. 2. It consists of a stacking of planar and puckered layers (top left side of the figure). A projection of the planar and puckered layers, respectively, is shown in the bottom part of the figure. Using pentagonal and flattened hexagonal units, a tiling that although periodic is closely related to a Penrose tiling, is superimposed on the drawing. Close examination of the atom positions reveals the constitutive icosahedral units (Fig. 3) and a large amount of close-packed atomic planes that are interspaced by a distance of 0.43 nm. Such planes were shown by Mizutani3 to play the most important role in transport properties of CMAs because they establish a resonance by propagating electron waves with a Fermi wavelength of λF = 0.43 nm whereas the major Fourier lattice components are located at KP ≈ 28 - 30 nm-1 in reciprocal space (i.e. KP/2 ≈ 2π/λF). The very same situation is observed in the previous example of orthorhombic O1-AlCuFeCr (Fig. 3).

Fig. 2. Sketch of the orthorhombic O1-AlCuFeCr atom structure. The stacking structure of flat (bottom left) and puckered (bottom right) atom layers is shown in the top left part of the structure. Flat layers are very close to perfect pentagonal tiling. (Courtesy V. Demange, CMA).

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+

Fig. 3. Formation of icosahedral clusters in O1-AlCuFeCr and their arrangement that shows spacing by 0.43 nm. (Courtesy V. Demange, CMA).

3. The essential property of CMAs 3.1. The pseudo-gap at the Fermi energy This section is for Al-based CMAs and more specifically for the partial density of states (DOS) of aluminum that may be investigated by a large number of techniques, but preferably in the context of the present chapter by emission (XES) and absorption (XAS) X-ray spectroscopies (see 1 and 2 and references therein). As just mentioned above, so-called Hume-Rothery (HR) compounds (i.e. compounds for which it is observed that 2kF ≈ KP) are electronically stabilized. We shall restrict ourselves to such compounds in this section and the followings. The stabilization mechanism induces a depletion at the Fermi energy (EF) in the DOS, the so-called Hume-Rothery pseudogap14. Using the XES and XAS techniques, a series of Al-Cu(-Fe) HR compounds of different atomic structures and accordingly different Jones zones were studied in order to investigate the HR pseudo-gap. The valence band (VB) of all these compounds was analyzed together with the Al p and Cu d (-Fe d) conduction bands. In γ-Al35Cu65 as well as in

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all Al-Cu(-Fe) samples, the Al 3s-d and Al 3p sub-bands split in two distinct parts located on each side of the maximum of the Cu 3d distributions, around 4 eV15. Accordingly, the shapes of the Al 3s-d and Al 3p distribution curves depart dramatically from that in pure Al, which is a typical parabolic curve distorted by experimental artifacts and experimentally induced many-body effects (not shown here, see reference2). This result emphasizes that the free-electron model is no longer valid for HR alloys. The interaction of Cu with Al can be interpreted within the framework of a Fano-like interaction between highly localized states and extended states16. The fact that here, the Al partial spectral curves both display a marked depletion exactly at the energy of the maximum of the localized states points out that Al states still retain an extended-like character in these compounds. Using the same methodology as for the HR alloys, we have analyzed several CMAs of much larger unit cells than HR compounds, including icosahedral (i-) and approximants crystals (especially from Al-Cu-Fe system). Again, like in βAl55Cu33Fe12, it was observed that the valence band of Al-Cu-Fe samples shows that Fe 3d states overlap the Al sub-bands edges nearby EF, Cu 3d states are found about 4 eV below EF, whereas the Al sub-bands overlap each other over the whole extent of the VB, namely over about 12 eV. In all these samples, it was observed that the intensity of the Al subbands at EF departs from its value in fcc Al. It is lower compared to the pure metal, so that the corresponding valence edges have no longer their half-maximum intensity set at EF. This more or less pronounced depletion that appears in the Al DOS at EF, points out the formation of a pseudo-gap. We refer now mainly to Al 3p states because (a) in pure fcc aluminum, these states are originally of extended-like character and therefore are more sensitive to changes of the electronic interactions than d states, (b) they are obtained alone by XES whereas this technique gives always d and s states together. To quantify the pseudo-gap of the Al 3p partial DOS, we shall restrict ourselves to using only the intensity at EF, labelled n(EF) hereafter. It is expressed in arbitrary units, with a value n(EF) = 0.5 in the pure metal, since the inflexion point of the DOS is by

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nEF (un. arb.)

definition located at EF and at half-maximum in a free-electron system. A summary of a large number of numerical data for n(EF) is given in Fig. 4.

e/a Fig. 4. Variation of the partial Al3p DOS at the Fermi energy n(EF) as a function of the electron-to-atom (e/a) ratio in Al-Cu-Fe intermetallics1. The DOS is expressed in arbitrary units: 0.5 stands for pure, fcc Al (cross at the right upper corner of the figure). The diamonds in the middle of the figure are for Al-Cu compounds, open squares for B2, CsCl-type β-cubic phases, the open circle for tetragonal ω-Al70Cu20Fe10. The black symbols at the bottom of the curve (which serves only to guide the eye) are for the icosahedral Al62Cu25Fe13 compound and its approximants. Observe that there is no real difference of n(EF) for a quasicrystal and for its closely related crystalline approximants of large unit cell and nearly identical electron concentration. In contrast, the difference is much more marked with respect to smaller crystal unit cells, yet also at identical electron concentration.

Figure 4 demonstrates that the minimum of n(EF) is obtained for quasicrystals with i) a lattice of very high perfection and b) containing a transition metal (TM) of the mid-series, preferably a 5d TM alloyed to a TM of the right hand side of the series like Cu or Pd. However, clearly enough, the minimum of n(EF) cannot be taken as a unique property because CMAs of very large unit cell and nearly identical chemical composition exhibit almost identical values of n(EF). This conclusion holds true for all CMAs known so far: physical properties vary in inverse proportion to the size of the unit cell and undergo an extreme at the ultimate size of the unit cell, but no gap is observed when the size of the

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unit cell approaches its maximum, for instance infinite size like in an icosahedral quasicrystal. Actually, a pseudo-gap is expected in most CMAs due to the HumeRothery effect, i.e. the interaction of the Fermi Surface with the Jones zone, an effect which in turn stabilizes the crystal structure of the compound. Therefore, the more complex the compound, in other words the closer to a sphere the Jones zone, the more efficient the HR effect and henceforth the deeper the pseudo-gap. The formation of a pseudogap was very carefully studied by Mizutani in a large number of γ-brass compounds having 52 atoms in the unit cell3. This work clearly established the origin of the HR mechanism and assigned it to a resonance between Fermi electrons and certain Bragg planes spaced by about.4 nm. The number of such planes may vary, depending on the details of each crystal lattice. However, hybridization effects between sp and d states deepen and broaden the pseudo-gap more efficiently than the HR effect. This argument was demonstrated both by computations17 and experimental studies of compounds like Al2Ru18, which as a matter of facts shows the opening of a true, tiny gap of 0.17eV at Fermi energy. Much broader gaps are expected in CMAs containing 5d TM elements, but so far none was synthesized19. 3.2. Transport properties Space is too limited in this introductory chapter to comment all properties measured so far in CMAs. The reader should refer to reference 1 and to the present book for a state of the art regarding transport properties of CMAs. Experimental determinations of optical conductivity, thermal conductivity, thermo-electric parameters like the Seebek coefficient were achieved in a rather systematic way for most CMAs known so far and contrasted to theoretical analysis, especially by Macia et al.20. For the sake of illustration, a blend of several resistivity measurements between liquid helium and room temperature is presented in Fig. 5 for several Al-based CMAs21. Surprisingly enough, the low temperature resistivity of those samples varies inversely to the unit cell

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size and presents a change of the sign of the temperature coefficient of resistivity (TCR) when the unit cell becomes large, with remarkably a zero-TCR at a specific composition (and crystal structure).

Fig. 5. Electronic resistivity of a variety of CMAs with different unit cell sizes21. Icosahedral CMAs are located in the upper part of the figure, O1-orthorhombic compounds are labelled O-AlCrFe and O-AlCuCrFe whereas a large unit cell AlPdMn CMA and its superstructures22, noted Ψ and ξ’, respectively, fall in the middle of the figure with a zero TCR. The notation ω-AlCuFe is for the tetragonal ω-Al7Cu2Fe compound and AlCuB is for a superstructure of the AlCu β-phase8 doped with 1 at% of boron. Observe the TCR, which marks the transition from normal metallic behaviour (small unit cell size) to an unexpected behaviour for a system made of metals (large unit cells).

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This observation has triggered a large research effort to understand better the transition from usual metallic behavior (small unit cell size) to a behavior much more resembling that of a semi-conductor. In the first regime, the diffusion of charge carriers follows a ballistic law, i.e. is proportional to time t and Einstein’s conductivity applies in proportion to N(EF), the total DOS at EF. In the second regime, conductivity is instead proportional to [N(EF)]2 and diffusion follows a tβ regime, with 0≤ β ≤123. Essentially, it was concluded that the breakdown of Bloch's theorem at large to infinite unit cell size is accompanied by the formation of socalled critical states, neither extended, nor fully localized like in a totally disordered medium24. Further analysis by Mizutani3 and others25, 26 has concluded to a transport mechanism by hopping in highly complex CMAs such as quasicrystals and their approximants, in total contrast to the usual behavior of alloys. 3.3. Mechanical properties The study of large unit cell CMAs and of quasicrystals under compression stress at high temperature was for some time hesitating, concluding first that plasticity was carried by glide of dislocations and later by climb. Finally, climb assisted by phason jumps appeared to be the most important mechanism for icosahedral quasicrystals27, but a general view at all possible deformation mechanisms of CMAs remains still to be worked out. A mechanism specific to large unit cell CMAs, called metadislocation, was discovered to be able to generate plasticity on the basis of very local rearrangements of structural units28. The situation is quite similar as far as contact mechanical properties (at much lower temperatures) are concerned, although it was recognized soon after the discovery of quasicrystals that friction is significantly reduced against metallic antagonists like steel29. A fairly illustrative example of the contrast to be expected regarding friction between a normal metallic crystal and an aperiodic CMA riding against a steel antagonist is provided by Fig. 6. It shows the data recorded during a pinon-disk experiment performed at a low residual pressure (typically 10-5 mbar) and a velocity of the disk relative to the pin of 5.10-4 m s-1. These parameters were chosen in such a way that a full layer of oxide has no

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Fig. 6. Pin-on-disk experiment in vacuum (1.9 10-6 mbar), using a hard Cr-steel ball of 6 mm diameter riding under a normal load of 1N on a mono-domain Al-Ni-Co decagonal single crystal at a linear velocity of 5 10-4 ms-1. The upper curse (open dots) is for the friction coefficient μ and the lower curve for the vertical position of the pin (see text). Two successive maxima of this curve mark the length of a full circular trace of the indenter in contact with the surface of the single crystal (diameter of the trace 3 mm). At the beginning of the experiment, the decagonal sample is covered by a thin layer of its native oxide, namely amorphous alumina. Due to a slight misalignment of the specimen surface with respect to a perfectly horizontal position, the maxima of friction coincide with the middle of the ‘up-hill’ part of the trace (solid vertical bars). Therefore, right after the test has started, friction on the amorphous oxide is isotropic as expected in the sample surface plane. The oxide layer is removed after two turns of contact with the indenter and friction switches to another regime. New peaks appear, superimposed on the ones due to horizontal misalignment. More specifically, two maxima (respectively, minima) of friction show up during one single rotation (dashed vertical bars), which clearly fits with half the previous period. Careful examination of the angular position of the single crystal on the disk holder proves that the maxima of friction coincide with the pin riding along the periodic stacking direction in the crystal whereas the minima are observed when the rider moves perpendicular to this direction.

time to grow in the time interval left between two successive passages of the indenter on the disk. Immediately after the beginning of the test, the native surface oxide inherited from exposure to ambient atmosphere of the sample is broken through by the pin. In Fig. 6, this operation takes two turns of the disk (about 0.012 m). After this point, the two naked pin and disk surfaces come into contact. The data shown in Fig. 6 was obtained using a single grain of the AlNi-Co decagonal phase, a material characterized by periodicity along one

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direction, and loss of periodicity perpendicularly to this direction. The stacking period is about 0.4 nm. Therefore, it is possible to probe the respective effects of periodicity and aperiodicity using the very same sample. The key point is that a large difference of the friction coefficient μ = FT/FN is observed depending on whether friction occurs along the periodic stacking direction of the single crystal or perpendicular to it (FT represents the tangential force that works against the movement of the pin, and FN is the normal load applied to the pin, here 1N). In other words, the measurement of the tangential force FT (Fig. 6) leads to a friction coefficient μ =FT/FN that returns to the same value with a period equal to the length of one turn (or a time periodicity equal to the duration of one rotation) when the test starts whereas it shows half that periodicity when pin and decagonal sample come in contact (beyond 0.012 m riding distance). This means that friction on the amorphous native oxide is isotropic (the slight variation of μ when one rotation proceeds is due to a misalignment of the sample surface against a perfectly horizontal plane, which causes friction forces when the pin goes ‘down-hill’ and ‘up-hill’ to be different (the vertical position of the pin is also shown in figure 6, which in turn supplies us with a reference for the position of the pin along the circular trace). After two turns, the oxide layers are broken and the pin comes into contact with the naked surface of the specimen. Then, it appears that the period of μ is no longer that of one full rotation, but only half a turn. Within one rotation, two peaks of μ are now visible, one of rather large amplitude, going from μ = 0.25-0.30 at its minimum up to μ = 0.5 at its maximum whereas the other peak exhibits intermediate values of μ. This situation remains unchanged for quite a significant number of pin-on-disk turns, until wear debris and severe plastic deformation of the surfaces in contact disturb the quality of the experiment. The simple example above illustrates the variety of friction conditions experienced on CMAs, but so far, the example of the decagonal phase is unique as was first pointed out by Park et al.30. Many more data was collected by the author, especially on Al-based binary CMAs 1, 31. In the coming section, the main contribution to friction in vacuum is assigned to the (reduced) adhesion properties of Al-based CMAs and therefore to the (lower) surface energy of these materials.

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These results have direct relevance to technological applications, especially hindrance of cold welding in aerospace mechanisms or cutting tools. 3.4. Chemical properties At least for Al-based CMAs, chemical properties are disappointingly determined by the constituents introduced in the complex alloys: the presence of elements like Cr or Mo enhances corrosion resistance, especially when pitting corrosion occurs1, 32, oxidation resistance is pretty good, including up to temperatures far above room temperature (e.g. 700-900 K for alloys that melt in the range 1200-1500 K) thanks to the passivating role of aluminum dioxide that is well documented in conventional alloys of this element1, etc. Two different chemical properties that benefit from complexity of the CMA lattice have been pointed however. The first is the catalytic efficiency of nano-domains of Cu or Pd prepared by etching a quasicrystal in an alkaline solution33. The beneficial effect results from a combination of properties. One the one hand, pure metal nanograins of the late transition metal (Cu or Pd) may be grown out the surface of the powdered CMA by etching and releasing it in a mixture of amorphous oxides of the other elements (i.e. Al and Fe or Mn, respectively). On the other hand, coarsening of the grains is prevented by the presence of the second TM, with which the enthalpy of mixing is negative. As a consequence, the catalytic activity of this type of material, used for instance for methanol reforming, is at least equal, if not superior to that of the pure, ultra-divided metal whereas savings are achieved on the quantity of catalyst and its preparation processing. Another challenging chemical property is hydrogen storage, which raises great interest in our community, see the program of the second CMA EuroSchool that is going to take place in May 200734. The hydrogen capacity of Ti-Zr-Ni icosahedral compounds, to a lesser extent their crystalline approximant, reaches a H/M atom ratio of 2 (H: number of stored hydrogen atoms; M: total number of Ti, Zr and Ni atoms)35. This represents a considerable amount of hydrogen that is comparable or even superior to the performance of more conventional metal hydrides.

Introduction to CMAs and to the CMA Network of Excellence

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The key point is that a very large number of tetrahedral sites form in the lattice, in relation to the complexity of the icosahedral structure, which are favorable sites to insert a H atom. Furthermore, the chemical composition yields TM atoms (e.g. Zr) that preferably bong to H atoms and are sitting on one or two vertices of these tetrahedral sites. However, technological difficulties are still there, such as an insufficiently long lifetime, or equivalently a too small number of load/unload cycles that are required for a commercial battery, which so far did not open the way to the usage of this type of CMAs in commercial devices. Furthermore, lighter materials such as Mg look more promising in respect of the H/M weight ratio. This is also a good reason to pursue a research on Mg-based CMAs, again trying to combine atom compositions with a large affinity for H and the presence of the largest possible number of tetrahedral traps for H atoms. 4. Surface energy 4.1. Surface energy in general As far as Al-based CMAs are concerned, surface energy (noted γS in the following) combines chemical physics with contact mechanical properties as we try to demonstrate in this section. It is a very important property which determines the cleavage energy of a solid, the equilibrium shape of a crystal, the wetting properties of both solids and liquids, the nucleation rate of a second phase (via the energy born at the interface), etc. It is however very difficult to assess experimentally, either by contact angle measurements or flow stress measurements at the approach of the melting point because of the need to employ single crystals with no defects emerging at the surface. Indirect assessment of the surface energy may be based on contact angle measurements of liquids wetting the surface of interest, but this also is a very difficult task for experimental reasons (contamination of the surfaces, high temperatures, etc.) and because the energy of the interface between solid and liquid is most often ignored36. In the case of CMAs, the number of different samples and the difficulty to grow them all as mono-domain samples simply rules out this route. Computer physics is more efficient

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Jean-Marie Dubois

to calculate the surface energy of metals and simple alloys and was successfully applied to a large number of metals and pure elements37. Due to the limited power of computers, it is at present days limited to a small number of atoms per unit cell and cannot be applied to CMAs and even more so to quasicrystals. The importance of friction and wetting in potential applications of CMAs, and furthermore the need for a better understanding of the fundamentals that govern those properties, forced us to find a simpler method to assess approximately γS for a large number of CMAs of various compositions. This method, although far less accurate than the ones evocated above, is described in the following. 4.2. Experimental Placed in a vacuum chamber evacuated down to 10-6 mbar or less, a pinon-disk instrument allows us to measure the friction coefficient between a hard steel ball and the solid of interest without intervening artifacts like moisture or external contamination (refer to previous section). Tribooxidation however may play an important role as was noticed elsewhere for tests in ambient atmosphere38. This artifact is dramatically reduced if the relative velocity of the indenter to the disk is large enough to forbid the growth of a full oxide layer in the time interval elapsed between two successive passages of the indenter, whereas the native oxide always present at the surface of our samples is broken and disappears from the trace within very few passages at the beginning of the experiment. This experimental procedure was applied in a systematic way to a large number of samples prepared by sintering according to a standard procedure depicted in ref.1. After polishing these samples (diameter 20 mm, thickness 4-6 mm) down to mirror polish, the specimens were placed in a pin-on-disk set up housed in a vacuum chamber as explained already in the previous section. The friction coefficient μ was recorded continuously during each test, using the same parameters as given in the previous section. Careful examination of the contact trace was performed after each test, for both the steel ball indenter and the surface of interest, thus allowing an evaluation of the wear produced during the test. In most cases, but not all, wear appeared negligible.

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Theoretical models were developed to understand better the origin of Admontons law FT = μFN39. Some of them take adhesion forces explicitly into account, but unfortunately require some information that is missing in our experiments (e.g. the actual contact area). As a much easier to handle test model, we shall assume that: μ= α/HV + β ΩSP

(1)

where the material (Vickers) hardness is noted HV, the work of adhesion (under these specific experimental conditions) of the steel pin P on the surface S is ΩSP and α and β represent calibration parameters which may be determined easily by producing the same experiment, but for a series a materials of known hardness and surface energy as explained in reference 40. Since wear is most often negligible, we furthermore assume that ΩSP is actually the reversible adhesion energy of P onto S: ΩSP = γS + γP - γSP with γP the surface energy of the pin and γSP the interfacial energy at equilibrium between S and P materials. This is obviously a very drastic assumption that can only be marginally valid for a pin-on-disk test, but it is strengthened by the very slow motion of the pin relative to the CMA solid and the (near) absence of wear. On top of this, we take γP-γSP =0, which means that we overestimate the value of γS that can be obtained after inverting Eq. 1, or: γS ≤ (μ−α/HV)/β

(2)

Calibration of Eq. 1 using known materials (metals, alloys, oxides) leads to a linear fit characterized by a regression coefficient very close to 140, which makes us very confident in the validity of the method. Yet, it must be insisted on the fact that instead of measuring γS, we just supply an estimate of its upper limit for a large number of the CMAs of interest here. 4.3. Data analysis and upper limit of γS Table 1 and Fig. 7 summarize our findings regarding the estimation of the surface energy of Al-TM CMAs (in this sub-section, TM represents one or two 3d metals).

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Jean-Marie Dubois

Table 1. Pin on disk tests in vacuum on Al-based CMAs of varying composition and crystal structure: experimental results and estimated values of γS. Compound (at.% or atoms)

Crystal structure

Vickers Hardness (load 0.5 N) ± 8%

Friction coefficient ± 15%

Estimated γS (Jm-2) ± 25%

Al3Ti

tetragonal

604

0.6

1.72

AlTi

B2-cubic

277

0.7

1.86

Al3V

tetragonal

427

0.7

1.97

Al9Cr4

γ-brass

0.48

1.37

Al8Cr5 Al11Mn4

γ-brass triclinic

695

β-AlCo l-Al13Co4

720

0.49

1.40

B2-cubic

476 620

0.41 0.5

1.09 1.41

monoclinic

700-800

0.75-0.8

2.2 – 2.4

β-AlFe

B2-cubic

λ-Al13Fe4 Al3Ni

monoclinic

417 815

0.64 0.69

1.78 2.04

orthorhombic

662

0.45

1.27

Al2Cu γ-Al9Cu4

tetragonal

550 480

0.44 0.37

1.21 1.0

AlCu oF-Al3Cu4

hexagonal orthorhombic

806 710

0.3 0.24

0.82 0.62

φ-Al10Cu10Fe

cubic

650

0.32

0.9

cubic

β-Al55Cu30Fe15

B2-cubic

680

0.31

0.84

ω-Al70Cu20Fe10 Al62Cu25.5Fe12.5

tetragonal

640-640

0.6-0.75

1.7-2.0

icosahedral

780

0.26

0.55

Al59B3Cu25.5Fe12.5

icosahedral

790

0.21

0.54

Al70Pd20Mn10

icosahedral

750

0.3

0.82

First, one should notice that some CMAs, especially quasicrystals (large diamond symbols in Fig. 7) exhibit a particularly low surface energy compared to that of the constituent 3d metals (typically 2.2 J/m2 for Fe) and that of Al (1.15 J/m2) as well as to conventional binary intermetallics (for example, triangles in Fig. 7). Worth mentioning, AlCu binary compounds (squares in Fig. 7) are also characterised by a rather low surface energy. Second, the surface energy follows a smooth decrease with the filling-in of the valence band.

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Fig. 7. Variation of γS with the total number of s, p and 3d electrons in the molecular composition of Al-3d TM CMA samples. Large open and grey diamonds stand for iAlCuFe and i-AlPdMn compounds, respectively. Grey squares represent Al-Cu, open triangles Al-Ni, grey triangles Al-V (large symbol) and Al-Ti (small symbol), open squares Al-Cr, grey dots Al-Mn, small solid diamond Al-Fe CMAs, respectively. Black solid triangles and black solid dots are for Al13Co4 and ω-Al7Cu2Fe compounds (two separate measurements on two distinct samples each), respectively. The top grey triangle is for Al8V5, but this estimate is rather uncertain due to the contribution of wear to friction.

Scrutinizing Fig. 7 shows that γS is smaller for TM alloying elements that belong to the right hand side of the 3d series whereas elements like V, Ti, Mn, Co or Cr are associated with much larger values of γS (grey and black symbols in Fig. 7). Nevertheless, quasicrystals are located significantly below average at a given electron concentration, an effect most presumably related to the formation of critical states and their unusual contribution to the valence band at the Fermi energy. Similarly, large differences in γS are observed for compounds of comparable chemical composition, but different TM content. This is the case for instance for β- and γ- compounds of the Al-Co and Al-Fe systems (Table 1). The two types of compounds are characterized by

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Jean-Marie Dubois

values of γS that differ by a factor of nearly 2, a difference that must be related to the HR pseudo-gap existing at EF and to the respective abundance of s, p and d states in the surface electronic structure (see also Fig. 4). More work is in progress to understand better the possible role of the pseudo-gap in determining γS. 5. Inverse Nano-Structuration 5.1. Comparison to Conventional Nano-Structuration To sum up at this stage of the chapter, I shall remind the Reader that both the Hume-Rothery scattering of Fermi electrons and sp-d hybridization contribute to the formation of a deep pseudo-gap at Fermi energy in large unit cell CMAs. In few compounds, a true gap is open. The width and depth of the pseudo-gap are in proportion to the size of the unit cell (although no analytical exact expression that would account for the coupling has been worked out till now). A change of behavior of the transport properties is observed from normal metallic transport to a hopping mechanism when the size of the unit cell becomes comparable to the Fermi wavelength, i.e. when the lattice parameters reach approximately 1 nm or more. Associated with this fundamental characteristic of CMAs are numerous other effects that could not be introduced in the present chapter for the sake of brevity, but are illustrated elsewhere in this book and in review documents quoted in1. For the same range of lattice parameters, plasticity becomes one more essential property of CMAs and is carried by defects intrinsic to the lattice complexity of CMAs27. Specific chemical reactivity is not associated with the size of the unit cell of CMAs, although after etching it becomes quite clear that catalytic performance is only observed with Al-Cu-Fe CMAs of large unit cell, and not with more simple materials like the ω-Al7Cu2Fe phase. Altogether, such effects are considered a specific behavior of nanostructured materials, in association with the reduction of the size of constitutive (isolated) objects, or with the relatively higher importance of interfaces and surfaces compared to the volume of a bulk specimen synthesized from nanograined material. The comparison is made for

Introduction to CMAs and to the CMA Network of Excellence

23

illustration in Fig. 8 between plasticity under compressive stress of nanograined ultra-pure copper41 and an Al-Cu-Fe-Cr orthorhombic of approximant of the decagonal quasicrystal42. 400

True stress (MPa)

b 300

c

a

200

100

0

0

2

4

6

8

10

12

True strain (%) Fig. 8. True stress-true strain curves recorded during compression testing at room temperature of microcrystalline copper (a) and nanocrystalline copper (b) of average grain size 50 nm41 compared to orthorhombic Al-Cu-Fe-Cr CMA (c) at 650°C42. Observe the absence of grain coarsening on curves b) and c).

In the former sample, the average size of the Cu grains is about 50 nm whereas in the latter, the grain size is a fraction of a millimeter. The stress-strain curves are however nearly identical because plastic deformation is due to the collective movement under stress of localized atomic defects, which go with the disorder installed in the respective materials. The occurrence of localized-like (critical) states, the change. transport regime of conductivity, the plasticity associated with atom jumps, the surface chemical reactivity that raises with increasing the size of the unit cell beyond a significant fraction of a nanometer (typically 1 nm) is coined Inverse Nanostructuration (INS) by the author. It is directly associated with the potential of CMAs for technology, see next section An oversimplified picture of the main effects encountered in conventional nanostructuration, when the size of the individual objects or the grain size in a composite material becomes small (typically below few nanometers), and INS is given in Fig. 9 and its caption.

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Jean-Marie Dubois

Fig. 9. Simplified comparison between conventional nanostruturation (CNS, top part of the figure) and inverse nanostructuration (INS, bottom part of the figure). In CNS, the parameters that matter are the size A of the objects (large circles) and their respective interspacing distance a compared to the wavelength λ of the excitation. The presence or not of disorder (featured by small dots) between the separate objects, for instance contamination, dust, ill-grown particles, etc. also determines the response of the system to the excitation. The size of the system (e.g. a terrace on a single-grain wafer, light grey area) is supposed infinite, or much larger than the individual size of the objects. Effects of relevance to the small size of the system components manifest themselves when their size becomes small, i.e. when A ≈ λ whereas the separation distance a cannot be very large in order to observe a coupling between the objects, typically a ≥ λ. In INS, the situation is reversed. The individual objects are the atom clusters (large circles of diameter A) embedded in a periodic crystal having a unit cell of lattice parameter a, which is approximately the separation distance between individual clusters. Here also, there is some disorder between the clusters, often called ‘glue atoms’ (small dots). Effects due to INS manifest themselves when A ≈ λ, like in CNS, and when a ≥ λ, preferably when a >> λ. This supposes that the more complex the compound, equivalently the larger the unit cell, the more enhanced the effects of INS.

5.2. A great potential for future research Accessible due to very recent progress in materials science, CMAs offer great potential for innovation. Examples of this potential are heat insulation at low temperature (using e.g. Al-Cu-Fe compounds), hydrogen storage, thermoelectricity, enhanced catalytic efficiency at

Introduction to CMAs and to the CMA Network of Excellence

25

lower cost, reduced friction, optimised composites, nanostructuration of metallic aggregates or thin films, development of innovative coating processes adapted to complex surface shapes, etc. Thermoelectricity is of special relevance nowadays that green energies are foreseen. Most presumably, clathrates and skutterudites offer the best CMA candidates for this purpose with quite respectable figures of merit achieved so far. Already mentioned, there are some doubts about the actual usefulness of CMAs regarding hydrogen storage in competition with light materials and especially nanotubes, but the challenge is worth a serious research effort. To end with, composite materials containing CMAs are the subject of various attempts for application in view of enhanced mechanical performance, whether they are prepared by blending with polymers or light metals or by in-situ reinforcement produced by nanoprecipitation of particles. Thanks to their low surface energy, grain coarsening is prevented, but direct usage of the specific surface energy is also foreseen in order to prevent cold-welding of mechanical parts kept in contact under severe load, for instance in satellites or vacuum technology applications. 6. Goals and organisation of the NoE CMA On this basis, a European network uniting 20 high-level core institutions in Europe, with a staff of more than 300 scientists and 60 PhD students, was designed to strengthen the competitiveness of European industries wherever materials need to offer hybrid properties, being both structural and functional, or embody an extraordinary combination of properties that are mutually excluding in conventional materials. Innovative management procedures for knowledge handling and networking, grant administration, organisation of conferences, exhibitions, industrial open days and specific measures for personnel exchange, access to platforms and durable integration of women in science are being taken, together with an ambitious program of summer schools and personnel training. The main purpose of a European Network of Excellence is to counterbalance the fragmentation of research in the continent. The essential tool used by CMA to achieve integration is the creation and functioning of so-called VIls, or Virtual Integrated Laboratories, in

26

Jean-Marie Dubois

conjunction with VIUs, or Virtual Integrated Units. These latter VIUs are service units that assist the VILs along their scientific strategy and needs (executive board and network office, gender mainstreaming, publications, mission service, transfer of knowledge and innovation, legal expertise). A very important VIU in this respect is the EuroSchool that has produced the present book. Six VILs were created, which assemble the expertise found in various European countries regarding metallurgy (VIL A), crystallography (VIL B), physical properties (VIL C), surface physics, chemistry and nanosciences (VIL D), surface technologies (VIL E) and finally applied physics of CMAs (VIL E). The degree of integration through the creation and functioning of the VILs can be appreciated from Fig. 10, which for the sake of simplicity presents only partial information on the exchange of data and deliverables between VILs and the external scientific community.

Fig. 10. Virtual Integrated Units of the CMA network of excellence, their relationships to each other and to the external world (simplified).

Introduction to CMAs and to the CMA Network of Excellence

27

As an example, VIL A will supply the other VILs with wellcharacterized samples, for structure determination, property measurements or assessment of potential applications (shown by single-line arrows). On the other hand, output of VIL A relevant for the external world such as the discovery of new compounds, phase-diagram data and measured or computed thermodynamic properties, will be distributed via scientific publication (double-line arrows). There is therefore a clear will of CMA to establish integration both within a VIL, binding together various topically related laboratories, and between different VILs in order to forward co-operation between different communities. Acknowledgments Thanks are due to the Commission of the European Communities for partial support of my work over the years (Grants BRE 2 CT 92 0171, G5RD-CT-2001-00584 & NMP3-2005-CT-500145). The author also gratefully acknowledges the long-lasting support of his research by the local authorities in Nancy (Communauté Urbaine du Grand Nancy, Conseil Régional de Lorraine and Préfecture de la Région Lorraine). Thanks also go to Dr M. Feuerbacher, Forschungszentrum Juelich, Germany, for the provision of the X-ray data shown in Figure 1 and Prof. P. A. Thiel and her project team at Iowa State University, USA for the provision of the single grain decagonal sample used in part of this work. I am also deeply grateful to M. Sales, J. Brenner and A. Merstallinger, Austrian Research Centres, Seibersdorf, for providing access to the pinon-disk facility and experimental help. References 1. J.M. Dubois, Useful Quasicrystals (World Scientific, Singapore, 2005). 2. E. Belin-Ferré, J. Phys. Cond. Matter 14, R789 (2002). 3. U. Mizutani, The Theory of Electrons in Metals, (Cambridge University Press, Cambridge, 2001); U. Mizutani, in The Science of Complex Alloy Phases. Ed. T.B. Massalski and E.A. Turchi (The Minerals, Metals and Materials Society, Warrendale, 2005). 4. J.M. Dubois et al., Phil. Mag. 86-6-8, 797 (2006).

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Jean-Marie Dubois

5. J.M. Dubois and A. Pianelli, French Patent n° 2671808 (17-06-1994); US Patent n° 5432011 (11-07-1995). 6. J.M. Dubois, Robert F. Mehl Lecture 2007, (TMS Conf. Proceedings, Warrendale), to be published (2007). 7. For more details, see the website of CMA at: www.cma-ecnoe.org. 8. C. Dong, Q.H. Zhang, D.H. Wang and Y.M. Wang, Euro Phys. J. B 6, 25 (1998). 9. P. Häussler, J. Barzola-Quiquia, D. Hauschild, J. Rauchhaupt, M. Stiehler and M. Hackert, in The Science of Complex Alloy Phases, Ed.. T.B. Massalski and P.E.A. Turchi (The Minerals, Metals and Materials Society, Warrendale), pp.43-86 (2005). 10. L. Pauling, The Nature of the Chemical Bond, 3rd Edition, Chap. 11, (Cornell University Press, Cornell, 1960). 11. C. Dong and J.M. Dubois, J. Mat. Science, 26, 1647-1654 (1991). 12. V. Demange, J.S. Wu, V. Brien, F. Machizaud and J.M. Dubois, Mat. Sc. Eng. 294296, 79-81 (2000). 13. X. Z. Li, C. Dong and J.M. Dubois, J. Appl. Cryst. 28, 96-104 (1995). 14. A.P. Blandin, in Phase stability in metals and alloys, Ed. P.S. Rudman, J. Stringer, R.I. Jaffee (McGraw Hill, New York), 115 (1965). 15. V. Fournée, E. Belin-Ferré and J.M. Dubois, J. Phys.: Cond. Matter 10, 4231 (1998). 16. K. Terakura, J. Phys F: Met. Phys. 3, 1773 (1977). 17. G. Trambly de Laissardière, D. Nguyen Manh, L. Magaud, J.P. Julien, F. CyrotLackmann and D. Mayon, Phys. Rev. B 52, 7920 (1995). 18. D. Nguyen Manh, G. Trambly de Laissardière, J.P. Julien, D. Mayou and F. CyrotLackmann, Solid State Comm. 82, 329 (1992). 19. M. Krajci and J. Hafner, Mat. Res. Symp. Proc. 805, 121 (2004). 20. E. Macia, Phys. Rev. B 66, 174203 (2002); C.V. Landauro, E. Macia and H. Solbrig, Phys. Rev. B 67, 184206 (2003). 21. E. Belin-Ferré, M. Klansek, Z. Jaclic, J. Dolinsek, J. M. Dubois. J. Phys.: Condens. Matter 17, 6911 (2005). 22. L. Behara, M. Duneau, H. Klein and M. Audier, Philos. Mag. A 76, 587 (1997). 23. D. Mayou, in Quasicrystals, Current Topics, Ed. E. Belin-Ferré et al. (World Scientific, Singapore, 2000). 24. C. Sire, in Lectures on Quasicrystals,. Ed. F. Hippert and D. Gratias (Les Editions de Physique, Les Ulis, 1994). 25. C. Janot, Quasicrystals, a Primer, 2nd Edition (Clarendon Press, Oxford, 1994). 26. V. Demange, A. Milandri, M.C. de Weerd, F. Machizaud, G. Jeandel, J.M. Dubois, Phys. Rev. B 65, 144205 (2002). 27. F. Monpiou, D. Caillard and M. Feuerbacher, Philos. Mag., 84, 2777 (2004). 28. H. Klein, M. Feuerbacher, P. Shall and K. Urban, Phys. Rev. Lett. 82, 3468 (1999). 29. J.M. Dubois, S.S. Kang, J. von Stebut, J. Mat. Sc. Lett., 10, 537 (1991). 30. Jeong Young Park, D.F. Ogletree, M. Salmeron, R.A. Ribeiro, P.C. Canfield, C.J. Jenks and P.A. Thiel, Science 309, 1354 (2005).

Introduction to CMAs and to the CMA Network of Excellence

29

31. E. Belin-Ferré and J.M. Dubois, Int. J. Mat. Res. 97, 7 (2006). 32. D. Veys, C. Rapin, X. Li, L. Aranda, V. Fournée, J.M. Dubois, J. Non Cryst. Sol. 347/1-3, 1 (2004). 33. A.P. Tsai and M. Yoshimura, Mat. Res. Soc. Symp. Proc. 643, K16.4.1 (2001). 34. See the website of the Euroschool at: http://euroschool-cma.ijs.si 35. A.M. Viano, R.M. Stroud, P.C. Gibbons, A.F. McDowell, M.S. Conradi and K.F. Kelton, Phys. Rev. B 51-17, 12026 (1995). 36. N. Eustatopoulos, M.-G. Nicholas and D. Drevet, Wettability at High Temperatures, Elsevier, Amsterdam (1999). 37. L. Vitos, A.V. Ruban, H.L. Skriver and J. Kollar, Surf. Science 411, 186 (1998). 38. I.L. Singer, J.M. Dubois, J.M. Soro, D. Rouxel and J. Von Stebut, in Quasicrystals, Ed.S. Takeuchi and T. Fujiwara, World Scientific Singapore 769 (1998). 39. Jianping Gao, D.W. Luedtke, D. Gourdon, M. Ruth, J.N. Israelachvili and Uzi Landman, J. Phys. Chem. B, 108, 3410 (2004) and references therein.D. R. Bates, Phys. Rev. , 492 (1950). 40. J.M. Dubois, M.C. de Weerd and J. Brenner, Ferroelectrics 305, 159 (2004). 41. Y. Champion, C. Langlois, S. Guérin-Mailly, P. Langlois, J.L. Bonnentien and M.J. Hÿtch, Science, 300, 310 (2003). 42. S.S. Kang and J.M. Dubois, Philos. Mag. A 66-1, 151 (1992).

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CHAPTER 2

THERMODYNAMICS AND PHASE DIAGRAMS Livio Battezzati Dipartimento di Chimica IFMe Centro di Eccellenza NIS, Università di Torino, Via Pietro Giuria 7, 10125 Torino, Italy E-mail: [email protected] Phase diagrams are an extremely important tool for assessing alloy constitution, phase stability and for materials processing. In this tutorial chapter the strategy and practice of the construction of phase diagram are provided. The use of modern computer software for phase diagram calculation is outlined. Starting from basic thermodynamics, the free energy functions are derived for phases appearing in a given system: elements, solutions, compounds. The equilibrium conditions are defined for both metastable and stable states as a function of composition and temperature. The phase diagram is derived as an ensemble of equilibrium states with examples taken from binary systems. Examples of the use of thermodynamics for phase transformations are finally given.

1. Introduction The basic thermodynamics needed for the understanding and appropriate use of phase diagrams is given concisely in this chapter. The number of formulae has been kept as low as possible since extended treatises on the topic are readily available1-4. The equilibrium is defined for unary, binary and multicomponent systems composed of a single or multiple phases as function of temperature, pressure and composition. Throughout the chapter use is made of current computational tools for thermodynamic quantities and for assessment of phase diagrams5. Also the use of thermodynamics and phase diagrams is underlined for metastable equilibria often encountered in materials processing and phase transformations. Illustrations and examples are provided with numerical details so that they can be reproduced. 31

32

Livio Battezzati

2. Basic relationships and unary phase diagrams For a given system a reversible infinitesimal change of the Gibbs free energy, G, as a function of temperature, T, pressure, P, mole fraction, ni, is given by

dG = − SdT + VdP +

∑ μ dn + ..... i

i

(1)

i

with S, the entropy, V, the volume and μi the chemical potential of the i component, being the summation extended to all components of the system. The equilibrium condition is expressed by dG = 0........

(2)

d 2G ≥ 0...........

(3)

G being at a minimum or

and can be verified at constant T and/or P and/or ni,. The integral quantity, G = H - TS, can be made explicit when a reference state is defined. A standard element reference state (SER) is usually employed by posing H298 = 0 and S = S298 for the phase stable at 298 K. Therefore the enthalpy and entropy as a function of temperature are given by

H = H 298 +

S = S298 +

∫

∫

T

C p dT.....

(4)

C p d ln T .....

(5)

298

T 298

The specific heat is expressed through a series of empirical coefficients, ci multiplying various powers of temperature. Further possible contributions, such as magnetic, are accounted for by additional terms: C p = c1 + c2T + c3T 2 + c4 / T 2 + ... + C p ,mag + .......

(6)

Thermodynamics and Phase Diagrams

33

An example is now provided for a pure element: Fe, i.e. a unary system. The specific heat for the condensed phases having different structure, i.e. face centred cubic, fcc, body centred cubic, bcc, hexagonal closed packed, hcp, and liquid is computed using assessed coefficients. The specific heat curves are shown in Fig. 1. Note the cusp in the specific heat of the bcc structure due to the ferromagnetic to paramagnetic transformation. All curves have a small cusp at the melting point of the element. This is not an experimental fact but a feature introduced by the assessment of thermodynamic properties. Since the specific heat of a liquid (solid) element cannot be measured at temperatures below (above) the melting point, the quantity is estimated outside the field of existence of the phase from the ensemble of properties and from its behaviour in alloys. The specific heat of the liquid is brought towards that of the solid in the undercooling regime.

-1

Cp /J mol K

-1

60 bcc

50

liq

40 fcc, hcp 30 20

500

1000

1500

2000

Temperature /K Fig. 1. The specific heat of phases of Fe according to the compilation of the Scientific Group Thermodata Europe (SGTE)6.

The corresponding free energies of fcc, bcc, hcp and liquid phases of Fe at p = 1 bar are given in Figs. 1 and 2 using the free energy of the bcc phase as reference state for all temperatures and, therefore, set to zero. The equilibrium condition imposes that the stable phase at every temperature has the minimum free energy. This corresponds to the bcc phase from 298 K to 1184 K, the fcc phase from 1184 K to 1668 K (see insert in Fig. 2), the bcc phase again from 1668 K to 1809 K and the

34

Livio Battezzati

15000

Liquid

10000

Gibbs energy /J mol

Gibbs energy /J mol

-1

-1

liquid above 1809 K. At the temperatures just mentioned, two phases have the same free energy, i.e. the difference in their free energy, ΔG is nil. Therefore, the two phases coexist at equilibrium. 100 50 bcc

0 -50 -100 1000

fcc 1200

1400

1600

Temperature /K

5000

hcp fcc

0

bcc 0

500

1000

1500

2000

Temperature /K Fig. 2. The free energy of phases of Fe. The reference state at all temperatures is the free energy of the bcc phase stable in a wide temperature range. The insert is an enlargement of the plot in the temperature range of stability of the fcc phase.

Performing the same calculation at different pressures, the loci of equilibrium points in the p-T phase diagram are obtained as plotted in Fig. 3.

Temperature /K

3000 liq

2500 2000

fcc bcc

1500 1000

hcp bcc

0.0

10

10

2.0x10 4.0x10 Pressure /Pa

Fig. 3. The p-T of Fe. The phase stable in each field is marked in the figure.

Thermodynamics and Phase Diagrams

35

It is useful to consider areas, lines, and triple points where lines join according to the Gibbs phase rule: v = c – f + 2, where v is the variance, the number of degrees of freedom of the system, c the number of components, f the number of phases in which components are distributed and the number 2 counts the physical variables defining such state, here pressure and temperature. Areas contain bivariant states, lines monovariant states and triple points invariant states. When a system is invariant, three phases are at equilibrium and no physical variable can be modified if equilibrium has to be maintained. When a system is monovariant, only one of the physical variables can be modified. The other one must follow its change to keep equilibrium. In bivariant states the physical variables can be modified independently within certain limits without changing the state of the system. Other variables, of magnetic, electrical, etc. origine, can be used to define a given system. In the following the pressure will be kept constant at p = 1 bar to consider systems containing two components and the Gibbs phase rule will be used as v = c – f + 1. 3. Simple binary phase diagrams

The free energy for each homogeneous phase, termed solution, is written as Gϕ =

ref

Gϕ + id Gϕ + exGϕ

(7)

where refGϕ is the free energy contribution of the pure elements, Gi, in their respective mole fractions, idGϕ is the ideal contribution to the free energy when a solution is formed. This is an entropic term stemming from the random occupancy of the sites in the structure of the system. ex ϕ G is called excess free energy and contains all non ideal contributions to the free energy of the phase. The excess term represents specific properties of the phase and, therefore, is model dependent. It is now common practice to express it via polynomial functions of composition containing empirical interaction parameters, vLi,jϕ ex

G ϕ = x A xB

ν L ∑ ν

A, B

ϕ

( x A − xB )ν

(8)

36

Livio Battezzati

Simple approaches correspond to all nLA,B j = 0 implying exGϕ = 0 (ideal solution); 0LA,Bj = const and nLA,B j = 0 for n > 0 (regular solution).The sum of idGϕ and exGϕ is termed free energy of mixing, ΔGmix, in that it expresses the variation of free energy when the solution is formed from pure components in their respective mole fractions. If the solution is ideal there is no enthalpy contribution to the free energy but only the ideal entropic term. If the solution is regular, the 0LA,B j interaction parameter can be either negative or positive expressing the favourable or unfavourable attitude of components to mix in a homogeneous phase. Considering a binary system A-B, the free energy of mixing is written as

ΔGmix = xA μA + xB μB

(9)

where μA and μB are the partial molar free energies of component A and B relative to the appropriate reference state or chemical potentials of A and B in the solution phase. The chemical potential is defined as the derivative of ΔGmix with respect to mole fraction of a given component taken at constant composition: ⎛ ∂ΔGmix ⎞ ⎟ ⎝ ∂x A ⎠T , P , xB

μA = ⎜

(10)

It expresses the behaviour of the component in the solution. When the chemical potential of A and B are equal in two phases, ϕ and ζ,

μ Aϕ = μ Aζ

(11)

μ Bϕ = μ B ζ

(12)

the condition for phase equilibrium is reached and the two phases coexist in the system. As an example, a set of free energy curves is shown in Fig. 4 for a regular solution having a positive interaction parameter at various

Thermodynamics and Phase Diagrams

37

temperatures. At the highest temperature the free energy of mixing is negative and has positive curvature for all compositions: the solution is always stable in its homogeneous state. For all other temperatures, the free energy of mixing displays both positive and negative curvature. Taking its first derivative it can easily be shown that the chemical potential of both components is equal at the compositions of the two minima of the curve implying that two phases having such compositions must coexist in equilibrium.

Free energy /Jmol

-1

4000 2000 0 -2000 -4000 0.0

0.2

0.4

0.6

0.8

1.0

Mole fraction B Fig. 4. Free energy curves for a solution phase at various temperatures from 500 K to 1700 K at 200 K interval using the interaction parameter for a regular solution 0LA,Bj = 27000 Jmol-1.

Geometrically, the equality of chemical potential is represented by the construction of a common tangent to the two minima. The phases and the components have the same structure since their free energy is expressed with the same curve. Therefore, a single phase stable at high temperature, de-mixes on cooling forming two phases. The composition corresponding to the tangent points at every temperature are collected in a temperature-composition diagram containing a miscibility gap between components. The procedure followed up to now allows for the synthesis of a phase diagram from known free energy curves and reference states. For most practical cases, the free energy curves are only partially known from

38

Livio Battezzati

Free energy /Jmol-1

experiments. In addition points of the phase diagram are known where phase equilibria are established. The complete phase diagram is then optimised by fitting the set of relevant expression of free energy to the available data to obtain a reasonable number of interaction parameters. In the case of components having different structure, two free curves are needed. An example is shown in Fig. 5 for solution phases having negative interaction parameters. 0,β

GA

0,α

GB

5000

0 α

β

μA = μA

β

α

0.0

β

xB

α

xB

-5000 0.2

0.4

0.6

0.8

α

β

μB = μB

1.0

Mole fraction B Fig. 5. Free energy curves for two solution phases having different structure. GA0, β is the free energy of A in the β structure. GB0, α is the free energy of B in the α structure. The melting point of A, TmA, is 1200 K and its heat of fusion is ΔHm = 14 kJmol-1; The melting point of B, TmB, is 1000 K and its heat of fusion is ΔHm = 12 kJmol-1. Curves were computed at 600 K for two regular solutions with 0LA,Ba = -10000 Jmol-1 and 0LA,B b = -12000 Jmol-1.

The reference states are the pure A and B components in their stable state at the given temperature. Each curve extends from the stable state of one component (the zero of free energy) to the value of the free energy of the other component in the hypothetical state where the component has the structure of the other one. Such states are called lattice stabilities and need to be evaluated to draw a free energy curve for each phase. The lattice stabilities for elements have been compiled by the SGTE6. The equilibrium condition that the chemical potential be equal in the two phases is represented by the common tangent to the curves and, as above,

Thermodynamics and Phase Diagrams

39

the corresponding compositions are collected in the binary phase diagram. When two phases of a binary system are in equilibrium at constant pressure the Gibbs phase rule implies v = 1, i.e. a single degree of freedom in order the system remains in the same state. If the temperature is modified, the composition of phases will follow accordingly. 4. Three phase equilibria: invariant reactions

Let us consider a system made of components A and B having the same structure and admitting two solution phases, having positive enthalpy of mixing, α, and ideal behaviour, β, respectively. A possible free energy scheme is shown in Fig. 6 at a temperature where the stability is given by the common tangent to the two minima of the first curve.

Free energy /Jmol

-1

8 0 00

T = 6 00 K

6 0 00 liqu id 4 0 00 2 0 00 so lid so lu tio n 0 0 .0

0 .2

0 .4

0 .6

0 .8

1.0

M o l fra ctio n B Fig. 6. Free energy curves computed for a hypothetical A-B system with: TmA = 1000 K and ΔHm = 12 kJmol-1; TmB = 1200 K and ΔHm = 14 kJmol-1. Ideal solution for liquid; regular solution for crystal (0LA,Bj = 30 kJmol-1).

The second curve lies above the first one or above the common tangent for all compositions. When the temperature is increased, the two curves are displaced relative to one another and the second one crosses the common tangent to the first curve branches (Fig. 7). The chemical potential of components is now equal in three phases which coexist at equilibrium and the system becomes invariant. No change in physical or

40

Livio Battezzati

chemical variables is possible until the three-phases equilibrium is maintained. On increasing the temperature further, the β phase acquires a stability range and coexists with α in composition ranges defined by the common tangents to the free energy curves of the two phases. The phase diagram obtained by collecting all composition points at equilibrium at every temperature contains monovariant lines and a horizontal line joining the three compositions coexisting at equilibrium. If the β phase is liquid the phase diagram is called eutectic, if the β phase is solid it is called eutectoid. The temperatures marked by the horizontal lines take the same names.

Free energy /Jmol

-1

6000 Teut = 735.5 K

liquid

4000

solid solution

2000 0

0.0

x2

xeut

x1 0.2

0.4

0.6

0.8

1.0

Mol fraction B Fig. 7. Free energy curves computed for a hypothetical A-B system at the eutectic temperature. Parameters as in Fig. 6.

An analogous phase diagram is obtained when three phases of different structure compose the system, all of them displaying upward curvature in their free energy curves as those of Fig. 7. In all these cases a high temperature phase will decompose isothermally into two phases at the temperature where the equality of chemical potentials occur. For the eutectic l ⇔ α1 + α 2 , if the solid phases have the same structure, and otherwise l ⇔ α + β . For the eutectoid the l phase is replaced by a γ solid phase.

Thermodynamics and Phase Diagrams

41

T = 603.5

6000

liquid

4000

Free energy /Jmol

Free energy /Jmol

-1

-1

Should the relative position of the free energy curves differ from those employed up to now due to the values of the interaction parameters and/or of the lattice stabilities, other invariant reactions will appear. Fig. 8 shows a free energy scheme in which the liquid phase is placed at the extreme of a common tangent to three phases, two solid and a liquid one. 100 T = 603.5

0 -100 x2

xperi -200

0.96

0.98

Mol fraction B

2000 0

solid solution

xperi x2

x1

0.0

0.2

0.4

0.6

0.8

1.0

Mol fraction B Fig. 8. Free energy curves computed for a hypothetical A-B system with: TmA = 1000 K and ΔHm = 12 kJmol-1; TmB = 600 K and ΔHm = 6 kJmol-1. Regular solution assumed for both crystal and liquid (0LA,Bj = 20 kJmol-1 and 15 kJmol-1 respectively).

The resulting phase diagram has a reaction, named peritectic, in which on cooling a liquid reacts at a given temperature with a solid phase to form a single solid phase l + α1 ⇔ α 2 . As above, the structure of solid phases is often different and the reaction is l + α ⇔ β . If the l phase is replaced by a γ solid phase, the reaction is called peritectoid as is the corresponding temperature. In a binary system the occurrence of a miscibility gap in the liquid phase or of allotropic transformations of solid phases will cause modifications in the shape and relative position of free energy curves with the consequence of producing more invariant reactions. The local shape of the phase diagram, i.e. around the temperature where the reaction occurs, will be similar to those of the previous reactions,

42

Livio Battezzati

however the type and position of phases will be different and the reactions will deserve a new name as the corresponding temperature. A collection of all types of reaction is reported in Figs. 9a and b. liq 1

α

Temperature

liq 1 α

peritectic

liq

β

liq

α

β

eutectic

0.4 0.6 mol fraction B

α

α

α

0.2

0.8

liq

katatectic

α

Temperature

liq 2 monotectic

α

0.2

liq 2

syntectic

β

β1

monotectoid peritectoid β γ

eutectoid

0.4 0.6 mol fraction B

β2 γ

β

0.8

Fig. 9a and b. Upper panel, a: the position of monovariant lines at and close to eutectic, peritectic, monotectic, syntectic temperatures in hypothetical A-B systems. Lower panel, b: the position of monovariant lines at and close to eutectoid, peritectoid, monotectoid, katatectic temperatures in hypothetical A-B systems.

The monotectic implies a miscibility gap with two liquid phases in equilibrium with a solid phase. An analogous phase diagram can exist, made completely of solid phases, the monotectoid. The syntectic occurs

Thermodynamics and Phase Diagrams

43

when two liquid phases solidify in a single solid phase. The analogous reaction in the solid state would be indistinguishable from a peritectoid. The katatectic is a case of inverse melting, although only of a portion of the system. In fact, here a solid phase decompose on cooling into a different solid phase and a liquid one. The analogous reaction in the solid state would be indistinguishable from a eutectoid. 5. Compounds

The free energy of formation of an ordered compound from the components is ΔGfor = ΔHfor -TΔSfor. The enthalpy of formation, ΔHfor, is usually negative and the entropy of formation, ΔSfor, is small since the compound is ordered at all temperatures. The free energy is, therefore, negative, centred at the stoichiometry of the compound, AnBm, and V-shaped since any compositional deviation from the exact stoichiometry would cause a sharp increase in free energy. In some cases the resulting field in the phase diagram is so narrow that it is drawn as a vertical line (line compound). The deviation from stoichiometry in ordered phases is accounted for by the sublattice model also called compound energy formalism5. From the knowledge of the structure of the phase, a minimum number of sublattices is defined which is needed to represent the order within the compound. Each sublattice is assigned a number of sites, Ns, which will host one of the components in the case of full order; otherwise, the other components or vacancies can mix on the same sites. The site occupation by component i is given by yi = ni/Ns with ni the number of atoms of the i component. The free energy of the compound is then written according to that of section 2. The ideal term stems from the entropy of mixing in the various sublattices is: ideal S mix = − kT

∑N ∑y s

s

s i

ln yis

(13)

i

The excess free energy is given as a function of the mole fraction of species in the sublattice, yi, via parameters, Li,j, expressing the interaction of components i and j inside a sublattice and between different sublattices, Li,j:k via component k. Taking a compound containing four

44

Livio Battezzati

components in two sublattices with n = m =1 and assuming a sublattice contains the A and B components and the other contains the C and D components, the simpler form for the excess terms is written as: G ex = y1A y1B L0A, B + yC2 y D2 L0C , D

(14)

This is equivalent to having a regular solution on both sublattices. A term for a regular solution between sublattices will be written as yA1yB1yC2 LA,B:C. More extended interactions will be given the form of RedlichKister polynomia. The reference state is the average of the free energy of binary compounds, weighted on the mole fraction of components in the respective sublattice, having the same structure as AnBm, the so called end members. For the case taken above (A, B)(C, D) it is given by ref

0 0 0 0 G 0 = y A yC G AC + y B yC GBC + y A y D G AD + y B y D GBD ..

(15)

As above, these free energies, as well as the interaction parameters, must be optimised by fitting to experimental data on thermodynamic quantities or phase diagram points. An example of such calculation is shown in Fig. 10. 1400 1400

T/K

T/K

1200 1200

liquid

1000 1000 800 800 600 600 400 400 0 0.0

Cu

Cu2Mg 0.2

0.4

CuMg2 0.6

x(Mg) x(Mg)

0.8

1 1.0

Mg

Fig. 10. The assessed Cu-Mg phase diagram. Parameters taken from7.

Thermodynamics and Phase Diagrams

45

Free energy /J mol

-1

The Cu-Mg system is made of three eutectics involving the terminal solid solutions and two intermetallic compounds. CuMg2 is stoichiometric and described as a line compound. Cu2Mg can deviate from stoichiometry at high temperature and is described with two sublattices. The resulting free energy curve extends from the free energy of the end members to the minimum (Fig. 11). Common tangents to the neighbouring phases will touch the curve close to but not on the minimum. -20.0k -25.0k -30.0k

Mg hcp fcc

liquid

-35.0k -40.0k 0.0

Cu2Mg

0.2

0.4 0.6 XMg

CuMg2

0.8

1.0

Fig. 11. Free energy curves and values for phases in the Cu-Mg system at 700 K. Parameters taken from7.

6. Metastable phases and phase transformations

Thermodynamics deals with system at equilibrium. However, in various instances metastable phases and microstructures can be discussed in thermodynamic terms as long as they remain in the same state long enough to perform measurements of specific heat capacity or they transform to another well defined state with a measurable enthalpy change. In same cases thermodynamics is applied locally to a part of the system where a change in free energy occurs. A few examples of such applications will be provided in the following. The extension of solid solubility, frequent in processes involving quenching, can be explained

46

Livio Battezzati

by making use of the T0 concept8. The T0 line which can be superimposed to a phase diagram, is the locus of temperaturecomposition points at which the free energy of two phases is equal. With reference to the free energy scheme of Fig. 5, the relevant composition is recognized at the crossing of the two curves. This is not an equilibrium state since the minimum of free energy for this composition lies on the common tangent. Examples of T0 curves referring to a liquid and two crystalline phases are reported in Fig. 12. For every composition they represent the temperatures at which the liquid phase could transform to a solid without partition of composition. The T0 temperature can be reached if the liquid is undercooled bypassing the equilibrium solidification.

Fig. 12. A eutectic phase diagram (full lines) with superposition of T0 curves (dashed lines), i.e. the loci of composition-temperature points at which either the α or the β free energy equal that of the liquid.

Quenching can be performed not only from the liquid but, perhaps more commonly, from the solid state. In such processes, a high temperature phase is frozen at low temperature where it should not exist. The free energy scheme of Fig. 13 illustrates this event. At high temperature a homogeneous solution phase is stable at all compositions. After quenching it remains homogeneous and its free energy can lie in between the minima marking the condition for phase separation and also

Thermodynamics and Phase Diagrams

47

within the spinodal points where the instability condition for the solution d2G < 0 is met. Any fluctuation in composition inside the solution would produce a decrease in free energy with the consequence that a spatially continuous decomposition of the solution can occur. The dashed line in the figure shows that the formation of a mixture of two arbitrary solutions having composition on both sides of the homogeneous one implies lowering of the free energy of the system. This is the starting point for describing the precipitation mechanism known as spinodal decomposition. 2000 Free energy /Jmol

-1

low T

0

spinodal points

-2000

-4000 0.0

high T 0.2

0.4

0.6

0.8

1.0

Mole fraction B Fig. 13. Free energy curves for a solid solution at high temperature (full miscibility) and low temperature (inside the miscibility gap). The curves are taken from Fig. 4. The dashed line indicates the free energy of the solution which has decomposed to some extent with a composition fluctuation.

Further, an example of more complex system will be provided with the aim of showing the procedure for drawing metastable phase diagrams. Fig. 14a reports the latest version of the Al-rich corner of the Al-Mn phase diagram9: the phases are solid and liquid solutions, 4 intermetallic compounds. All of the intermetallics, represented as line compounds, decompose peritectically on heating. In processes such as rapid solidification the compounds may not form and the system would be described by a metastable phase diagram. Ion terms of the calculation calculation of phase diagrams this means suspending a phase from the calculation and derive the equilibria as detailed above. Suspending the λ

48

Livio Battezzati

phase, the phase diagram of Fig. 14b results. The range of existence of Most phase transformations start with a nucleation event. The classical theory of nucleation evaluates the stability of a system containing a cluster of the new phase via the change in free energy, ΔGn, as a function of cluster volume, Vn and interfacial area, An: 1300 1300

T/K

T/K

1200 1200

liquid

μ

1100 1100

Al6Mn

1000 1000 900 900 800 800 700 700 0 0.00

Al

Al12Mn 0.06 0.06

0.12 0.12

0.18 0.18

0.24 0.24

x(Mn) x(Mn)

0.3 0.30

1300 1300 1200 1200

liquid λ

TT // K K

1100 1100

Al6Mn

1000 1000 900 900 800 800 700 700 0 0.00

Al

μ

Al12Mn 0.06

0.12

0.18

x(Mn) x(Mn)

0.24

0.3 0.30

Fig. 14a, upper panel: the assessed Al-Mn phase diagram. Parameters taken from9. Fig. 14b, lower panel: the metastable Al-Mn phase diagram where the λ phase has been suspended from calculation.

Al6Mn is extended although it still melts with a peritectic reaction.

ΔGn = Vn ΔGv + Anσ

(16)

Thermodynamics and Phase Diagrams

49

The driving force for the formation of a nucleus of critical size which will eventually grow, is the difference in free energy between the nucleus and matrix phases, ΔGv whereas the process is adversely affected by the interfacial energy term, σ. In a transformation involving compositional partition, such as precipitation or primary solidification, the composition of the most probable nucleus will correspond to the maximum free energy gain in the process. This is determined by means of the parallel tangent construction as shown in Fig. 15. for solidification.

Free energy

crystal at liquidus temperature

crystal at nucleation temperature liquid at liquidus temperature

Alloy composition

mol fraction B Fig. 15. Free energy scheme for solidification of a crystal phase derived from curves in Fig. 6, illustrating the parallel tangent construction and the composition of the more likely nucleus differing from that of the equilibrium crystal (arrows).

At the equilibrium liquidus temperature the chemical potential of elements is provided by the common tangent construction. However, at this temperature there is no free energy available for the system to be spent as work of formation of the interface between new and old phases. Such free energy becomes available on undercooling the matrix. At the temperature were nucleation occurs, the relative position of the liquid and crystal phases will be modified as drawn in Fig. 15 with the liquid being metastable. The maximum change in chemical potential of elements is obtained by drawing the tangent to the free energy curve of the crystal phase parallel to the tangent to that of the liquid phase. The composition of the nucleus will differ from that expected by merely

50

Livio Battezzati

considering the liquid-crystal equilibrium. When the small nucleus of radius r forms, the overall composition of the matrix will not change appreciably; they will coexist through an unstable equilibrium established locally at the crystal liquid interface and defined by dGn =0 dr

(17)

The quantity ΔGv can be derived from the optimisation of the phase diagram for all phases allowing the prediction of phase selection by nucleation10 whereas the common tangent construction will show the driving force available for growth of the crystal after nucleation. References 1. D. A. Porter and K. E. Easterling, chap. 1 in Phase Transformations in Metals and Alloys, Chapman & Hall, London (1992). 2. D. R. Gaskell, Introduction to Metallurgical Thermodynamics, McGraw & Hill, New York (1973). 3. M. C. H. P. Lupis, Chemical Thermodynamics of Materials, PTR Prentice Hall, Englewood Cliffs, New Jersey (1983). 4. M. Hillert, Phase Equilibria, Phase Diagrams and Phase Transformations, their Thermodynamic Basis, Cambridge University Press, Cambridge (1998). 5. N. Saunders and A. P. Miodownik, CALPHAD Calculation of Phase Diagrams, A comprehensive Guide, Pergamon, Oxford (1998). 6. A. T. Dinsdale, in Calphad.Computer Coupling of Phase Diagrams and Thermochemistry, 15, 317 (1991). 7. P. Jiang et al., in Calphad.Computer Coupling of Phase Diagrams and Thermochemistry, 22, 527 (1998). 8. W. J. Boettinger, J. H. Perepezko, chap. 2 in Rapidly Solidified Alloys, Eds. H. H. Liebermann. and M. Dekker, New York (1993). 9. COST 507, Definition of thermochemical and thermophysical properties to provide a database for the development of new light alloys, Thermochemical database for light metal alloys, I Ansara., A. T. Dinsdale, and M. H. Rand Eds., European Communities, vol. 2, Bruxelles (1998). 10. L. Battezzati and A. Castellero, in Materials Science Foundation, Eds M Magini. and F. H. Wöhlbier., Trans. Tech. Publications Inc., Zurich, vol. 15 (2002).

CHAPTER 3

PERMANENT MAGNETS AND MICROSTRUCTURE Paul McGuiness Jožef Stefan Institute, Ljubljana, Slovenia E-mail: [email protected] Permanent magnets are vital components in many types of technology, from PCs and iPods to electric motors and generators. The strength of a permanent magnet depends to a large extent on the elements that comprise the magnetic material, and on the microstructure – the grain size, the grain shape, the distribution of phases, the occurrence of precipitates, the intergranular phases, etc. – that results from the way the material is processed. The most powerful magnets available today are based on rare earths and transition metals. These materials are not only interesting because of their excellent magnetic properties; they are also very interesting from the processing point of view, because a wide range of different techniques can be used to produce them. In this chapter we will look mainly at Nd–Fe–B-type magnets and how processing them with various techniques, like powdering and sintering, meltspinning, and hydrogen disproportionation, can be used to produce microstructures leading to the optimum magnetic properties for a particular application.

1. Historical introduction to permanent magnets Permanent magnets have been known since the discovery of the lodestone by the ancient Greeks. The first application, the compass, was invented in China; the earliest recorded use of lodestone as a direction finder was in a 4th-century Chinese book: Book of the Devil Valley Master. However, a systematic study was not carried out until William Gilbert (1544-1603) wrote his famous book De Magnete, Magneticisque Corporibus, et de Magno Magnete Tellure (On the Magnet and Magnetic body, and on That Great Magnet the Earth), which was published in 1600. From his experiments, he concluded that the Earth was itself 51

52

Paul McGuiness

magnetic and that this was the reason compasses pointed north (previously, some believed that it was the pole star (Polaris) or a large magnetic island on the north pole that attracted the compass). Before the invention of the electromagnet by Sturgeon in 1925 the only permanent-magnet materials were the naturally occurring lodestone, a form of magnetitie Fe3O4, and various forms of iron-carbon alloys. Many of these early magnets were built up from wires or strips, since these were easily magnetised by stroking them with another magnet. The development of modern permanent magnets can be said to have begun around the end of the 19th century, with the introduction of the steel magnet. This was improved upon in about 1900 by using tungsten steel as a starting material. The first substantial improvement came with the appearance of Honda steel, in which about 35% of the iron in the FeW-C steel was replaced by cobalt. However, owing to the high price of cobalt compared to iron there were very few applications for this material. The introduction of MK steel (an alloy of Fe, Ni, Co and Al) by Mishima was of considerable significance: not only did it have much better magnetic properties than Honda steel, it was also considerably cheaper. MK steel can be considered as the forerunner of Ticonal II, which was developed in 1936. This led to the development of an anisotropic form of the same material, called Triconal III, produced by annealing in a magnetic field. The next step forward in the story of permanent magnets was made when it was realised that larger values of magnetic anisotropy were needed to produce higher coercivities. The coercivity of a permanent magnet is its ability to withstand the effects of an opposing magnetic field. High anisotropies where found in materials that had highly anisotropic crystal structures and hexagonal or tetragonal symmetries. A good example of a material with this type of structure is the common, household ferrite magnet. This is the type of magnet you will often find stuck to the door of a fridge, but is also the workhorse magnet for thousands of industrial, automotive and domestic applications. Ferrites are oxide materials with the general formula M(Fe2O3)6, where M is one or more of the divalent metals barium, strontium or lead. Ferrites have a relatively low magnetisation, but their high coercivity and low

Permanent Magnets and Microstructure

53

price mean that they dominate the market for permanent magnets, at least in terms of tonnage, even today. There were few developments in permanent magnets until the 1960s. The second world war had seen advances in the separation and purification of rare earths, and with these metals now available researchers began to look at combinations of rare earths and transition metals. 1967 saw the first reports of RCo5 (R = rare earth) materials with CaCu5-type structures, which soon led to the commercial availablilty of SmCo5 magnets with properties that literally dwarfed those of ferrites, AlNiCos and the steel magnets that had come before – albeit at a price. Within a short time it was realised that the magnetic properties of these SmCo5 magnets were being limited by the magnetisation of the cobalt sub-lattice, and so a new type of magnet, based on R2Co17, quickly followed. These Sm–Co magnets made possible a wide range of new applications and presented tremendous possibilities for miniaturisation because of their enormous energy densities, but the high – and perhaps more importantly, the variable – price of cobalt was a problem. This situation became even worse with the cobalt crisis of 1979–81. The crisis was due to a rebellion in Zaire, source of about half of the world’s cobalt, when many of the mines were flooded. The price of cobalt increased six fold as a result, and this intensified the search for high-energy cobalt-free permanent magnet materials. The first announcements of the successful production of magnets based on neodymium, iron and boron were made at a meeting in Pittsburgh, PA, in 1983. At the same meeting there were reports from Sumitomo Special Metals of Japan and General Motors of the US of a new generation of permanent magnets based on a material with the chemical formula Nd2Fe14B. This Nd–Fe–B-type was an improvement in many ways over the existing Sm–Co materials, and could be produced using a number of different techniques. The Japanese had produced their Nd–Fe–B magnets via a relatively conventional powder-metallurgy sintering route, whereas the Americans had used a novel method called melt spinning. Within a few years there would more reports of good magnetic properties from groups working with techniques such as mechanical alloying, screen printing, sputtering, ablation and techniques based on hydriding.

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One more major discovery has been made since the arrival of Nd–Fe– B, and that occurred in 1990, when a group in Ireland reported hard magnetic properties in nitrided Sm–Fe-based materials. The Sm2Fe17N3 magnet had excellent properties, comparable in many ways to the market-leading Nd–Fe–B-based magnets, but difficulties associated with nitriding bulk samples has kept them from anything other than a niche market. A summary of the improvements made in permanent-magnet materials over the past 100 years can be seen in Fig. 1.

Fig. 1. Progress in permanent magnets, in terms of energy product.

In this chapter we will look only at the example of Nd–Fe–B permanent magnets, showing how it is possible to prepare magnets using a number of different techniques, so producing a wide range of microstructures that result in magnets with a variety of magnetic properties. 2. Permanent-magnet properties In order to describe a permanent magnet quantitatively we need to measure its magnetic properties. To do this we subject the magnet to a large positive magnetic field, to saturate the magnet, then we apply a large negative field in order to assess it ability to withstand a reverse magnetic field. This form of measurement is shown schematically in Fig. 2 and described in more detail below.

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Fig. 2. Hysteresis loop of a permanent magnet.

The measurement of the permanent magnet’s properties begins with a completely demagnetised magnet in a zero magnetic field at the crossing point of the x axis (the applied field) and the y axis (the magnetisation of the sample). The state of the magnet is illustrated by the empty rectangle, i.e., the magnet is unmagnetised. The first part of the measurement involves applying a large positive magnet field (+H). At this point the magnet becomes fully saturated (the red arrow) while it exists in a large positive field. The next stage is to remove the applied field and look at the magnetisation state of the magnet while there is no external field. With a good-quality magnet the internal magnetisation (the red arrow) will remain, even in the absence of the applied field. This point on the y axis, Br, is referred to as the remanence.. This is followed by a demagnetisation stage, where a negative field (-H) is applied to the sample. With a sufficiently high field the magnetisation of the sample will be reduced to nothing (no red arrow), and this field is referred to as the intrinsic coercivity, Hci, of the magnet. For most permanent-magnet applications we are looking for magnets with a high remanence and a high coercivity, although there are some applications when a very high coercivity would be disadvantageous. There is one other point that is important from the applications point of view, the normal coercivity, Hcb. This is the point where the external demagnetising field is equal

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and opposite to the internal magnetisation; the system, in effect, is equal to zero. This measurement is of more interest to electrical engineers than to material scientists, who are more interested in the material’s performance than the performance of the system. In order that we can have a single quantity to describe the quality of a magnet, the term energy product has been introduced. To calculate the energy product of a magnet we draw a straight line from the remanence point to the normal coercivity and then measure the area of the largest rectangle that we can fit under this line. The best Nd–Fe–B magnets have energy products in excess of 50 MGOe (mega gauss oersted), typical Sm–Co magnets are in the range 25–30 MGOe, and ferrites are about 4 MGOe. 3. Some applications of permanent magnets Permanent-magnet applications can be divided into four distinct groups: • Applications that make use of the attractive force that a magnet can have on a soft magnetic material, like magnetic particles in a slurry, or the repulsion between to permanent magnets, like in the case of magnetic bearings. • Applications that use the magnet’s magnetic field to convert mechanical energy into electrical energy. Examples include generators and alternators. • Applications that use the magnet’s magnetic field to convert electrical energy into mechanical energy. This is by far the largest category of applications, and includes all kinds of motors, meters, actuators and loudspeakers. • Applications that use the magnet’s magnetic field to control electron beams. The most obvious application here is the cathode-ray tube, but there are also a lot of magnets used in wave tubes, wigglers and cyclotrons. A particularly good example of a product that uses a lot of magnets is the car. Apart from the obvious applications like the starter motor, the alternator, the windscreen wipers and the loudspeakers for the radio, there is an enormous number of magnets in applications like door locks, ABS systems, fuel and water pumps, electric windows, seat-position actuators (as many as 50 magnets per seat) aerial motors, ignition

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systems, crankshaft-positioning, air-conditioning, cruise control, CD player, speedometer/tachometer, and tens if not hundreds of sensors. There are many cars on the road today that contain more than 1000 permanent magnets. 4. The crystal structure of Nd2Fe14B The first crystal-structure determination of Nd2Fe14B was reported in 1984. The structure is relatively complex, and there are 68 atoms in the unit cell. The tetragonal structure belongs to the space group P42/mnm; it comprises six crystallographically inequivalent Fe sites and two crystallographically inequivalnet Nd sites. The homogeneity range of Nd2Fe14B is very small, or even absent; it is effectively a line compound. A schematic representation of the crystal structure of Nd2Fe14B is shown in Fig. 3.

Fig. 3. The crystal structure of Nd2Fe14B.

5. Phase relationships in the Nd–Fe–B system The Nd–Fe–B system is characterised by three ternary compounds: Nd2Fe14B (sometimes called the φ phase), Nd1+εFe4B4 (sometimes called the η phase or the boride pahse), and Nd2FeB3; however the last of these

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does not figure in the compositions from which permanent magnets are fabricated. Fig. 4 shows a pseudobinary-type vertical section of the Nd– Fe–B phase diagram, with 100% Fe on the left-hand side and Nd/B = 2 on the other.

Fig. 4. Pseudobinary-type vertical section through the Nd–Fe–B phase diagram.

This section shows clearly the peritectic reaction L + Fe --> φ at 1180ºC. From the metallurgical point of view this is a critical reaction. In effect it is saying that you cannot melt an alloy with the composition Nd2Fe14B and hope to get a single-phase solid. Unless you go to extraordinary lengths, such a sample will always consist of islands of peritectically formed Fe surrounded by Nd2Fe14B and a liquid relatively rich in Nd. In order to avoid the formation of any peritectic iron it is necessary to be cooling down on the right-hand side of the diagram so that the reaction sequence is L --> L + φ. For this reason the permanent magnets based on the Nd–Fe–B system usually have compositions richer in Nd than Nd2Fe14B, close to Nd2.6Fe13B1.4, or Nd15Fe77B8 as it is more commonly expressed, and the microstructures of solidified samples are normally composed of three phases: Nd2Fe14B (typically around 85%), Nd1+εFe4B4 (typically around 2–3%) and a phase that is very rich in neodymium, called the Nd-rich phase, which normally constitutes about 12–13% of the material.

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6. Coercivity and microstructure in Nd–Fe–B permanent magnets The origin of the coercivity in all rare-earth–transition-metal permanent magnets is their high easy-axis magnetocrystalline anisotropy. However, since the coercivity remains well below the value of the anisotropy field, by a factor of about four, the coercivity is clearly also very dependent on microstructure, with magnetisation reversal being the result of the nucleation and growth of reverse magnetic domains. Sintered Nd–Fe–B magnets show two main coercivity characteristics that suggest the coercivity mechanism is one of nucleation and growth rather than a pinning type mechanism: unmagnetised magnets have many domains per grain, and the coercivity increases with the size of the field used to magnetise the magnet. In order to generate high coercivities under such conditions it is important to produce materials with a small grain size, thereby limiting the surface areas of individual grains. This is in stark contrast to permanent magnet materials like the Sm2Co17-type magnets, where large coercivities can be generated in cast-and-annealed ingots with very large grains, and where the mechanism of the coercivity is related to the pinning of domains within the volume of the grains by precipitates. 7. Processing Nd–Fe–B permanent magnets Nd–Fe–B permanent magnets are multiphase metallic structures. Irrespective of the processing route employed to produce them, the starting point is nearly always an as-cast alloy with a composition in the region of Nd15Fe77B8. Depending on the casting conditions, in the as-cast state this material will exhibit the three phases mentioned above as well as, possibly, a small amount of dendritic iron. Although this ferromagnetically soft form of iron is potentially very detrimental to the permanent magnetic properties of the material, in particular the coercivity, the processing of the alloy into a permanent magnet has the effect of removing this iron, providing the dendrites are not too coarse. The aim when processing Nd–Fe–B magnets is twofold: • First, we need to reduce the grain size down to micron or submicron sizes, thereby maximizing the potential coercivity of the sample.

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• Second, we need to orient the grains as much as possible so that the c axes (which are the hard-magnetic axes) of the grains are pointing in the same direction, thereby maximizing the potential remanence of the sample. This is shown in Fig. 5, where Fig. 5a shows schematically a typical as-cast microstructure for a Nd–Fe–B alloy with a composition of Nd15Fe77B8, cast into a mould containing about 10–15 kg of material, and Fig. 5b shows schematically an idealised magnet microstructure with grains of less than 10 micrometers.

Fig. 5a (left panel): Schematic microstructure of Nd15Fe77B8 cast alloy. Fig. 5b, (right panel): Schematic microstructure of idealized permanent-magnet microstructure.

7.1. Processing Nd–Fe–B magnets via the sintering route The powder-metallurgy sintering route has for a long time been the main processing route for metals and alloys when shape and uniformity of properties are important. Sintering has also played a major role in permanent magnets, dating back to the earliest sintered AlNiCo and ferrite magnets of the mid-20th century. Sintering was also the processing route of choice for Sm–Co-type magnets too, because it made it possible to produce the extremely brittle Sm–Co magnets at close to net shape, and, more importantly, it made it possible to align the powder particles with the crystal c axes all in the same direction prior to sintering in order that the material can exhibit a high degree of remanence. The differences

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in the magnetic properties of an aligned sample (the anisotropic case) and an unaligned sample (the isotropic case) are shown in Figs. 6a and 6b.

Fig. 6a. Schematic diagram of an unaligned (isotropic) permanent magnet and the associated magnetic properties.

Fig. 6b. Schematic diagram of an aligned (anisotropic) permanent magnet and the associated magnetic properties.

Under a scanning electron microscope or an optical microscope the microstructures of these two types of magnets would look to a large extent the same, because it is not easy to tell the orientation of a grain of Nd2Fe14B simply from looking at a polished surface. However, a measurement of the magnetic properties would quickly reveal which sample was well aligned, because a well-oriented sample would exhibit a

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remanence not far short of the saturation magnetisation (Ms), whereas the non-aligned (isotropic) sample would have a remanence close to about half of the value of Ms. But before we look at aligning Nd–Fe–B powders, let us have a look at the first stage of the powder-metallurgy sintering process, casting. 7.1.1. Casting Nd–Fe–B-type sintered magnets usually begin as cast ingots produced from appropriate amounts of neodymium, iron and ferroboron. These materials are then induction melted at temperatures in the range 1400– 1500ºC in an inert atmosphere before being poured into thin, bookshaped moulds to provide fast cooling rates. Industrial castings range in size from about 10 to 100 kgs. 7.1.2. Powdering The next, and arguably the most critical, stage is producing the powder from the ingot. The target size for the powder is 2–5 microns, but this is very difficult to achieve in a single step. The usual procedure is to crush the ingots to centimetre-sized lumps, then hammer-mill these lumps to pieces a few millimetres in size, and then finally attritor-mill or jet-mill these pieces to the final size of a few microns. Of course due to the high reactivity of rare-earth-based powders all of these procedures must be carried out in a protective atmosphere. In the cases of crushing and hammering this atmosphere is usually nitrogen, but for jet-milling the gas used is normally argon. 7.1.3. Pressing and aligning The next stage after powdering is to press the powders into compacted pellets, usually referred to as green compacts, so that we can achieve the right shape of magnet, while at the same time aligning the individual powder particles so that the magnetic c axes are all in the same direction. Since the initial grain size of the cast alloy is in the range of tens or hundreds of microns, we can be reasonably sure that most of the powder

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particles consist of a single grain with a single c axis. The fields required depend on the size and shape of the magnet, but they are usually in the range of about 1 Tesla. Like with the powder production, the pressing and aligning procedures must be carried out in such a way as to avoid oxidation as much as possible. 7.1.4. Sintering Once enough green compacts have been produced they can be loaded into the sintering furnace. The individual green compacts are normally arranged on stainless-steel trays. In order to maximise the density of the resulting sintered magnets the green compacts are sintered in a vacuum. The sintering-temperature varies considerably depending on composition, particle size, required properties, but generally involves slow heating to the sintering temperature in the range 1050–1100ºC, followed by a hold at this temperature for 1–2 hours, with subsequent holds during cooling at 900–950ºC and 600–650ºC. The complete sintering and heat-treatment cycle could last 12 or more hours. 7.1.5. Machining, coating and magnetising The sintered samples usually need machining to meet dimensional requirements. Because of the very brittle nature and high reactivity of Nd–Fe–B magnets this normally involves centreless grinding using nonreactive fluids. Nd–Fe–B magnets also require coating, because of their sensitivity to the atmosphere, and the coatings are usually based on nickel. The final magnetisation sometimes takes place as part of the production process of the magnets; however, increasingly magnets are being magnetised after they are fixed to the assembly in the final product. Fig. 7 summarises the key steps in the production of sintered Nd–Fe– B permanent magnets.

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Fig. 7. The main stages in the processing of sintered Nd–Fe–B permanent magnets.

Microstructures of the as-cast Nd15Fe77B8 alloy and a sintered magnet are shown in Fig. 8. Fig. 8a is an optical micrograph; Fig. 8b was produced from a scanning electron microscope. Note the dramatic reduction in the grain size and the even distribution of phases, both of which are very important for the development of a high intrinsic coercivity. The sintered magnet is also highly oriented, although this is not visible in this type of micrograph.

Fig. 8a (left panel): Microstructure of as-cast Nd15Fe77B8 alloy. Fig. 8b, (right panel): Microstructure of sintered Nd15Fe77B8 magnet.

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7.2. Processing Nd–Fe–B magnets via the melt-spinning route The melt-spinning production route, unlike the powder-metallurgy sintering route, was not used for previous generations of permanent magnets. The combination of melt-spinning and Nd–Fe–B-type magnets was pioneered by General Motors in the US, a development that ran in parallel with Sumitomo Special Metals’ research on the sintered route for Nd–Fe–B. In the melt-spinning process a jet of molten alloy comes from material in an induction-melting crucible and hits a rapidly rotating water-cooled copper wheel. Under such conditions cooling rates can be as high as 106Ks-1. The Nd–Fe–B alloy tends to form in the shape of ribbons, a few centimetres long and about 30 microns thick, which are then thrown from the copper wheel and collected a metre or so away in a hopper. Like with the sintering process, everything is carried out in a protective, inert atmosphere. A schematic diagram of the melt-spinner and some crushed Nd–Fe–B ribbons are shown in Figs. 9a and 9b.

Fig. 9a, left panel: Schematic diagram of melt-spinner. Fig. 9b, right panel: Crushed melt-spun ribbons of Nd–Fe–B.

The usual procedure involves over-quenching the ribbons to produce a largely amorphous structure, and then heat treating them at 600–700ºC to produce Nd2Fe14B grains that are 0.5–1.0 microns in size. The grain size of melt-spun magnets tends to be smaller than with sintered magnets, but this is a consequence of many factors, not least of which is the radically different processing route. The most important feature of

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Nd–Fe–B melt-spun ribbons is that they are completely isotropic, i.e., there is no preferred orientation of the c axes in the material leading to relatively low values for the remanence. Nevertheless, the simplicity of the ribbon-production process, the intrinsic stability of the ribbons in the atmosphere and the ease with which these materials can be mixed with polymers and other binders and moulded into intricate shapes makes them a very attractive material. 7.2.1. Hot pressing melt-spun ribbons These Nd–Fe–B melt-spun ribbons are also suitable for hot pressing. In such a process the ribbon pieces are placed in a die and compacted under high loads and temperatures in the range 700–800ºC to form 100%-dense solids. Of course, since there is no possibility of aligning the grains in a magnetic field these materials are still isotropic, but their high density gives them an advantage over the polymer-bonded variants. 7.2.2. Die-upset forging of melt-spun ribbons A third possibility is to die-upset forge the compacted melt-spin ribbons. This rather expensive process involves first of all producing a straightforward 100%-dense solid from the ribbons as described above, and then repressing the dense compact in an over-sized die, so causing the material to flow in a direction perpendicular to the direction of pressing. The two hot pressing techniques are illustrated in Figs. 10a and 10b.

Fig. 10a (left panel). Hot pressing of melt-spun powder to produce a fully dense isotropic magnet and Fig. 10b (right panel): Die-upset forging of a fully dense isotropic magnet to produce an anisotropic magnet.

This flow under pressure at high temperatures causes the material to become highly oriented along the pressing direction. This reorientation is

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the result of the growth of favourably oriented grains in combination with grain-boundary sliding, boundary diffusion and diffusion slip. The process can be enhanced by small additions of elements such as gallium, although die-upsetting ratios of about four, i.e., the compact must be reduced to a quarter of its original height, are required. Microstructures of the hot-pressed melt-spun ribbon and the subsequently die-upset variant are shown in Figs. 11a and 11b.

Fig. 11a (left panel): Microstructure of hot-pressed Nd–Fe–B melt-spun ribbon. Fig. 11b (right panel): Microstructure of subsequently die-upset forged Nd–Fe–B material.

7.2.3. Processing Nd–Fe–B magnets via the hydrogenationdisproportionation-desorption-recombination route A third method for producing permanent magnets from a starting material of as-cast Nd15Fe77B8 alloy is called the hydrogenationdisproportionation-desorption-recombination process, or HDDR process, for short. The process involves heating the as-cast Nd15Fe77B8 alloy in an atmosphere of hydrogen to about 700–750ºC, holding for a period of minutes to hours, and then cooling the material to room temperature in a vacuum. During the first stage of the process the material reacts with the hydrogen to form interstial hydrides and a lot of cracks form in the material due to the expansion of the crystal lattice with the formation of the hydrides. However, as the temperature increases these hydrides become unstable and the material disproportionates to form iron,

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ferroboron and neodymium hydride. The disproportionation reaction can be represented as: Nd2Fe14BHx <=> 2HdHx/2 + 12Fe + Fe2B + ΔH

(1)

where ΔH is the heat of reaction. The value of x is dependent on the hydrogen pressure used and the exact composition of the starting alloy. The disproportionated mixture consists of a very finely divided mixture of iron, neodymium hydride and ferroboron. During the second stage of the process, when the material is subjected to vacuum conditions at high temperature the neodymium hydride desorbs to form neodymium metal, which then leads to the neodymium, iron and ferroboron recombining to form large amounts of Nd2Fe14B phase, together with some Nd–Fe intergranular material, only now the grain size of the Nd2Fe14B phase is in the 0.1–1.0-micron range. In simple terms the HDDR process converts a coarse-grained as-cast material into a very fine-grained powdered material via a reversible chemical reaction involving hydrogen. The process is shown schematically in Fig. 12.

Fig. 12. Schematic representation of the hydrogenation-disproportionation-desorptionrecombination (HDDR) process.

Like with melt-spinning the process is intrinsically isotropic, and the resulting HDDR powders tend to have remanences close to about half of

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the saturation magnetisation; however, in the case of the HDDR process it is possible to produce anisotropic material with the use of additives like zirconium, hafnium and gallium, and closely controlled processing conditions. HDDR-processed powder is also very suitable for hot pressing fully dense magnets in a similar to the procedure used for the melt-spun ribbons. 7.2.4. Other processing techniques Nd–Fe–B materials are remarkable in many ways, but perhaps their most remarkable characteristic is the number of different methods that can be used to produce high-coercivity permanent magnets from basically the same starting material. From the commercial point of view the three techniques already discussed – sintering, melt-spinning and HDDR – are the most important, but from the research perspective techniques like rapid casting, hot working, mechanical alloying, laser ablation, pulsedlaser deposition, rotary forging, gas atomisation and explosive compaction have provided valuable insights into the capabilities and limitations of the Nd–Fe–B system. 8. Magnetic properties The magnetic properties obtainable with Nd–Fe–B materials are in general higher than those available with other magnetic materials. In some situations Sm–Co materials might be a better choice because of a need to operate at high temperatures, or AlNiCo magnets, when temperature stability is required, but for most modern high-technology applications where cost is not the only factor, Nd–Fe–B magnets are the magnets of choice. Fig. 13 shows typical properties obtainable for Nd– Fe–B magnets produced by the three techniques described here. The highest remanences and energy products are the result of processing the material with a powder-metallurgy sintering route. Over the past 20 years improvements in laboratory and industrial processing have resulted in remanence increases from about 1.2 Tesla to more than 1.3 Tesla; this means that magnets with energy products in excess of 50 MGOe are available. Another magnet with a high remanence and energy

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product is the anisotropic die-upset-forged melt-spun magnet. Although the remanences are not quite as high as for sintered material, the magnets can be made net shape and can be considered as more than niche products. Anisotropic HDDR bonded magnets are still increasing their market share and are the only option when a 1 Tesla+ bonded magnet is required for an application. The processing of HDDR material is relatively high, comparable to the various processes associated with the melt-spinning option, and so HDDR magnets will never be in a position to displace sintered Nd–Fe–B from most applications.

Fig. 13. The magnetic properties of Nd–Fe–B magnets processed using a variety of techniques.

Lower down the energy-product scale we have isotropic hot-pressed melt-spun materials, which can be produced with very high coercivities, and isotropic HDDR and melt-spun bonded magnets which can be easily produced in large numbers through processes like injection moulding and extrusion.

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9. Summary This has been a brief overview of the processing of Nd–Fe–B magnets and the relationships between processing, microstructure and magnetic properties. We have looked at just one composition – the Nd15Fe77B8 ternary compound – and three types of processing – sintering, meltspinning and the HDDR process – and found that each of these techniques is able to produce high-quality magnets with useful properties for many applications, providing the right kind of microstructure is obtained. There are, of course, many other magnetic materials, some of which were mentioned in the text, that have other interesting processing– microstructure–properties relationships; however, that would require space than a single chapter. Further reading There is an enormous body of literature on permanent magnets, properties, and microstructures. Below are listed some widely available books that will provide you with plenty of information on all aspects of magnetism and magnetic materials. Permanent-magnet Materials and Their Applications. K.H.J. Buschow. Trans Tech Publications (ISBN: 087849796X) Permanent Magnet Materials and Their Application. Peter Campbell. Cambridge University Press. (SBN: 0521566886) Introduction to Magnetism and Magnetic Materials. David C. Jiles. CRC Press Inc. (ISBN: 0412798603) Rare-Earth Iron Permanent Magnets. Ed. J.M.D. Coey. Clarendon Press. (ISBN 0198517920) Hidden Attraction. Gerrit L. Verschuur. Oxford University Press. (ISBN 0195106555) Driving Force. James D. Livingston. Harvard University Press. (ISBN 0674216458) Modern Magnetic Materials: Principles and Applications. Robert C. O’Handley. John Wiley & Sons. (ISBN 0471155667)

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CHAPTER 4

SOLIDIFICATION Peter Gille Crystallography Section, Department of Earth and Environmental Sciences, Ludwig-Maximilians-Universität München, Theresienstrasse 41, D-80333 München, Germany E-mail: [email protected] Solidification of a metallic melt is basic to various technological processes like ingot casting, directional freezing of composite alloys, single-crystal growth and rapid solidification of metallic glasses. Apart from varying scientific or industrial goals and significant technical differences in these areas of application, most of the fundamental problems are common to these fields. Starting from slight deviations from equilibrium thermodynamics, various aspects of the transformation process of a melt to the solid state are treated in this tutorial chapter: homogeneous and heterogeneous nucleation, kinetic aspects of crystal growth, segregation phenomena, and interface instability caused by constitutional supercooling. An understanding of the mechanisms of solidification and how they influence practical processes and alloy properties are the main objectives rather than a complete treatment of all solidification techniques. A short overview of the most important methods of bulk crystal growth from the melt is given.

1. Introduction Solidification means any process transferring a fluid phase into the solid state. In a narrower sense it is understood as crystallizing a liquid caused by lowering its temperature below the melting point or its liquidus temperature. In this chapter, basic principles of solidification are treated that should be considered when crystallizing a binary or highercomponent melt. Solidification of well-defined samples or even single crystals may be regarded as the goal of these processes. The same

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principles, problems and equations are also fundamental to several other technical disciplines of solidification, like casting or welding. Often it is only the quantity of some parameter that makes the main difference between these applications: the extent of the deviation from equilibrium, the solidification rate or simply the amount of the melt to be frozen. 2. Thermodynamics and nucleation Thermodynamics gives the expressions how Gibbs free energy G of all phases that may occur changes with temperature. At each temperature T, the phase with the lowest content of Gibbs free energy is expected to exist. In Fig. 1 the principle course of Gibbs free energy curves of a solid and its melt in a one-component system is pictured.

Fig. 1. Volume free energy for a pure component as a function of temperature for solid and liquid phases. Dashed lines represent branches of the curves that are unfavourable with respect to energy minimization.

From equilibrium thermodynamics, lowering the temperature of a melt below its melting point should immediately start the process of solidification with a driving force ΔG that is proportional to the undercooling Tm − T :

Solidification

ΔG =

ΔH m (Tm − T ) Tm

75

(1)

with ΔHm being the latent heat of crystallization. But, this is only true for a specific volume of the melt that can reach a lower content of Gibbs free energy by becoming solidified. If the influence of the interface that separates the solid phase to be formed from the mother liquid is regarded as well, the total amount of Gibbs free energy might actually be even higher. This would prevent the occurrence of the new phase even at temperatures lower than the melting point. Since the influence of the surface strongly depends on the size of such a particle, these effects are only restricted to very small scales, typically in the nm-region. But, it is just this tiny size a new phase has to start with before it may form a large-scale crystal.

Fig. 2. Free energy change associated with homogeneous nucleation of a sphere of radius r.

In the nucleation theory usually spherical particles of the new phase, i.e. the crystalline phase of a radius r, are assumed to form in the fluid phase with a specific solid/liquid interface energy γ. The change in the total Gibbs free energy of the system is then composed of the volume-

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depending part being proportional to r3 and the surface-depending term with a r2-function:

4 ΔH m ΔG = − π r 3 ΔT + 4π r 2γ 3 Tm

(2)

that is pictured in Fig. 2. Once a particle as large as r* that is called nucleus has formed by some fluctuation, crystallization will spontaneously proceed accompanied by a lowering of the total Gibbs free energy of the system. The radius of such a nucleus can be derived from Eq. 2 to be: r* =

2Tmγ ΔH m ΔT

(3)

with an activation energy for nucleation that amounts to: ΔG* = ΔG (r = r*) =

16 Tm 2γ 3 π 3 ΔH m 2 Δ T 2

(4)

What is most important is the reverse proportionality to the degree of undercooling ΔT in these two equations. Thus, with a very low deviation from equilibrium, nucleation has almost no chance to occur and the nucleation rate is practically zero. If nucleation does not happen in the above mentioned homogeneous way but in contact with a third phase (ampoule wall, foreign particle etc.) that may act as a substrate, the process is called heterogeneous nucleation and the leading equations stay formally almost unchanged. It is a geometrical factor that includes the interface energies between the three different phases under consideration and will change the onset of nucleation to smaller nuclei and lower nucleation energies at the same amount of undercooling. Therefore, whenever foreign phases exist that are wetted by the melt, heterogeneous instead of homogeneous nucleation will occur because of its exponentially higher nucleation rate. This is schematically pictured in Fig. 3 assuming some detection limit for the nucleation rate, expressed by the number of nuclei N that may have formed within some time and in a given volume. The negligible chance of nucleus formation with small deviations from equilibrium is

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sometimes a problem during first solidification of a new phase. But on the other hand, it is of great help in single crystal preparation when the growth of the solid phase without the risk of a parasitic grain formation is the intention.

Fig. 3. Variation of the free energy of nucleation ΔG* with undercooling ΔT for homogeneous and heterogeneous nucleation (solid lines) together with the corresponding nucleation rates (dashed lines).

3. Growth kinetics

Once the solid phase exists, either formed by spontaneous nucleation or given by seeding, solidification may be regarded simply as the growth of the solid phase by the movement of the solid/liquid interface. Kinetically, this can be described as a 3-dimensional periodic attachment of specific building units. A very simple but powerful model has been suggested by Kossel1 describing the crystal as being constituted of cubic units and crystal growth as a process of the periodic arrangement of these cubes (see Fig. 4). For the sake of this chapter I will stay with these simple building units, but do emphasize that nothing is said about the nature of

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these cubes, and even their shape is not really important for this basic introduction. The elementary units may be single atoms or ions, molecules or even large clusters of atoms that have already formed in the melt prior to the interface attachment. According to Kossel’s model, the attachment energy of a new cube that enters the crystal will be different for the various sites pictured in Fig. 4. While for a single building unit attached not to a step but to the extended surface the gain in the free energy is the lowest, the kink position with three of six cubic faces to be attached to the already existing crystal is by far the best position. Apart from the highest gain in free energy that is obtained from the attachment at a kink position, it is the repeatability of this step that makes the kink position deciding for crystal growth kinetics. All other alternative sites pictured in Fig. 4 would qualitatively change the surface. But, after having occupied the kink position, a new kink has been formed for further growth.

Fig. 4. Surface of a simple cubic crystal (“Kossel’s crystal”) where the surface atoms or building units have various numbers of nearest neighbours in the crystal depending on the site occupied: (1) vacancy, (2) jag, (3) kink, (4) step site, (5) adsorbed unit.

Since the number of kink sites at a crystalline surface is so important for growth kinetics, the question arises which parameters do determine the atomistic state of a crystal surface. This problem has been first

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treated by Jackson2 for a simple one-layer boundary between the solid and the fluid phase similar to Kossel’s model. An atomically smooth interface with no empty site or additionally attached building blocks is the optimum with respect to the total enthalpy of the surface. But, at temperatures T > 0 K the entropic contribution to the free energy gain has to be regarded as well and the question arises whether or not an atomically rough interface is preferred that consists of many empty positions. The various states of surface roughness can be described by the occupation factor c of surface sites being the ratio of the number of occupied positions to the total number N of sites at the surface. The calculation of the free energy ΔF of the contribution of a specific surface (hkl) can be simply obtained by counting the number of dangling bonds within the boundary layer and using the statistic formula for the surface entropy

ϕ ⎡ ⎤ ΔF = NkT ⎢c (1 − c ) ξ hkl + c ln c + (1 − c ) ln (1 − c ) ⎥ kT ⎣ ⎦

(5)

with k being Boltzmann’s constant, ϕ being the latent heat of crystallization per building unit and ξhkl the anisotropic factor giving the ratio of bonds within the surface (hkl) compared to all bonds of a unit. The normalized plot of Eq. 5 is given in Fig. 5. The parameter α of the various curves stands for:

α=

ϕ kT

ξ hkl

(6)

From Fig. 5 it is clearly seen that the parameter α well divides the free energy curves into those with a local minimum at c = 0.5, i.e. with a preference to atomically rough surfaces (α < 2), and those (α > 2) having the local minimum of the symmetric curves next to c = 0 or c = 1 which means almost atomically smooth surfaces. Therefore, Jackson’s factor α is well suited to describe the tendency of a crystalline surface to become atomically smooth or rough. A high latent heat of crystallization, a low phase transition temperature and/or a dense-packed crystalline interface increase the tendency to form an atomically smooth surface. On the other hand, with

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a low Jackson factor the crystalline interface may be assumed as consisting of a huge number of kink sites that allow the easy attachment of the crystallizing building units. In these cases, the rate of crystallization will be proportional to the deviation from equilibrium, i.e. to the undercooling ΔT. Therefore, a linear dependence of the growth rate v will be observed.

Fig. 5. Plot of the relative free energy ΔF NkT as a function of surface occupation for various values of Jackson’s factor α.

With an atomically smooth surface, the question arises how crystal growth can kinetically take place. It is obvious that a new layer of elementary building blocks has to start with a very first block, that will

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create quite a large new interface but result only in a little gain in binding energy. This first step is similar to the above mentioned nucleation problem but reduced to a 2-dimensional nucleus of only elementary step height. Consequently, the conditions to form a stable 2-dimensional nucleus at the top layer of the crystal can be expressed by similar formulae like Eqs. 2 – 4. Once the nucleus of the new layer has been formed, it can be assumed that lateral growth of the new layer will immediately occur and soon be completed since the undercooling that has been accumulated for nucleation is high enough for step growth. Therefore, 2-dimensional nucleation is the limiting step and the growth rate of the crystal can be calculated from Boltzmann statistics using the activation energy that is required to start a new layer. It results in an exponential growth law: ⎛ A ⎞ v ∝ exp ⎜ − ⎟ ⎝ ΔT ⎠

(7)

with A being a constant factor coming from the specific surface. Burton et al.3 have suggested that a screw dislocation intersecting a growth interface can provide a continuous step on a surface for growth. Such a surface step winds up into an Archimedean spiral that leads to never ending steps. High dislocation densities are very common defects of crystals in almost all solidification processes, and spiral growth is therefore very likely to occur at smooth interfaces. The step edges overcome the problem of the 2-dimensional nucleation, can easily laterally proceed and result in a square law for the growth rate, v ∝ ΔT 2 . The three different kinds of how the growth rate of a crystal may depend on the degree of undercooling are summarized in Fig. 6. Of course, even with a parabolic or an exponential dependence in the case of spiral growth or 2-dimensional nucleation, respectively, the growth rate can never exceed that one of the continuous growth. With a very high supercooling, the density of steps and kinks at a formerly smooth interface will become as high as in the case of an atomically rough one, and growth will proceed in a continuous way.

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Fig. 6. Influence of interface undercooling ΔT on the growth rate for atomically rough and smooth interfaces.

4. Phase diagrams

Phase diagrams are very fundamental to the understanding of many effects and problems that occur in solidification. Whenever a melt is to be solidified that consists of more than one component some phase diagram knowledge is required. Regarding the preparation of multicomponent alloys, the growth of doped crystals or simply the casting of melts containing some impurity, there is nearly no technically interesting crystallization process in a true single-component system. Since basic knowledge on the thermodynamic background of phase diagrams is given in Chapter 2, here it is sufficient to state, that in each type of a binary phase diagram the liquidus and solidus lines may be approximated by linear slopes, at least in the narrow region under consideration. For not too large concentration changes, a constant segregation coefficient :

k0 = CS/CL

(8)

can be derived from the phase diagram with CS and CL being the equilibrium concentrations of the treated component in the solid (S) or in the liquid phase (L). For the scope of this chapter it is only necessary to know whether the equilibrium segregation coefficient k0 is lower or larger than unity.

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With k0 < 1, only a part of the regarded component of the melt can be incorporated into the solid phase. As a consequence, there should be expected an internal accumulation in the melt next to the phase boundary that depends on the rate of the interface movement. Assuming no other stirring mechanism in the vicinity of the phase boundary than diffusion, a steady-state concentration profile parallel to the growth direction will be formed that can be calculated from the 1-dimensional diffusion equation. This characteristic profile is pictured in Fig. 7 for a phase diagram region with k0 < 1.

Fig. 7. Solute concentration near an advancing solid/liquid interface.

From this, it seems to be reasonable to define an effective segregation coefficient keff that is again the ratio of the concentration in the solid to the one in the liquid. Contrary to the equilibrium segregation coefficient, now the melt composition at some distance to the phase boundary is regarded that is different from that one at the interface where equilibrium conditions are assumed. It is the problem of matter transport that makes the difference between keff and k0. Therefore the growth rate of the interface v, the diffusion coefficient D, and the diffusion boundary layer thickness δ enter the formula derived by Burton et al.4 : (9)

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The diffusion boundary layer that has first been introduced by Nernst5 should not be understood as a really stagnant layer, instead, there is a smooth transition between the dominant diffusive or convective transport and δ gives an implicit measure of the more or less strong convective mixing of the melt. 5. Interface stability

A stability analysis of the solid/liquid interface during growth of a solid phase has to answer the question of what happens if a disturbance suddenly occurs. Since temperature and concentration fluctuations are always present, stability or instability will depend only on the reaction of the specific system, whether it will decrease or increase a sudden fluctuation. This problem could have already been addressed to singlecomponent materials where temperature gradients have to ensure heat transport away from the growing solid phase. With a temperature maximum at the phase boundary position, i.e. with a decreasing temperature towards the melt, a sudden fluctuation of the interface position would be amplified. The resulting, locally enhanced, growth rate would produce even more latent heat that increases the original problem again. The result of such a thermally driven interface instability is a dendritic structure of the solid. In multi-component systems, the same phenomenon may occur even if the solid/liquid interface is not affected by a reverse temperature field but by a constitutional supercooling. This was first mentioned by Rutter and Chalmers6 and mathematically treated by Tiller et al.7. Since diffusive matter transport is orders of magnitude slower than heat dissipation, the stability limit that results from constitutional supercooling is much narrower than stability criteria originating from thermal problems. Starting with a concentration profile in a melt in front of a growth interface like that one in Figure 7, each position-dependent concentration has its own liquidus temperature. With the liquidus temperature being the lower limit of the single-phase melt stability, the question arises whether or not the position-depending specific liquidus temperature may have fallen below the actual temperature at some place. Therefore, the actual

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temperature gradient in the melt next to the interface gradTexp has to be compared to the fictive profile of the according liquidus temperatures TL(z). Assuming diffusion to be the only transport mechanism, the 1dimensional concentration profile CL(z) in the melt adjacent to an interface moving with the rate v amounts to: (10)

with C0 being the melt composition in some distance z to the interface, D the diffusion coefficient of the component under consideration and k0 the equilibrium segregation coefficient. According to the liquidus curve of the phase diagram with a linear slope, m = -dTL/dCL, the position-depending liquidus temperature of the melt can be calculated. The graphic construction is shown in Fig. 8.

Fig. 8. Schematic liquidus curve T(CL) of a binary phase diagram that projects a given exp solute concentration in the melt, CL(z), on the resulting temperature profile T(z). T1 exp and T2 represent different experimental temperature gradients that obey or violate the constitutional supercooling criterion.

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Since the highest gradient of the liquidus temperature occurs at the phase boundary ( z = 0 ), the toughest stability criterion may be expressed as:

(11) where Texp is the actual temperature profile in front of the growth interface. With Eq. 9 and the slope of the liquidus line, Eq. 11 turns to: (12) This expression derived by Tiller et al.7 has been known as constitutional supercooling criterion and may serve as a rough limit for the highest possible growth rate of a solid to be solidified from a multi-component melt of composition C0. Although, this expression has been derived for a diffusion-controlled regime, the problem of constitutional supercooling persists even with more or less intensive mixing by convection. One should always keep in mind that in the vicinity of a solid phase, convective motion becomes zero and diffusion is the only remaining process of matter transport. In the case of a too high solidification rate, the conditions in front of the phase boundary are unstable as mentioned above. In the melt, in some distance to the interface, there is a higher deviation from the equilibrium than at the interface itself. It is not just a real supercooling since the actual temperature may be well above that one at the phase boundary, but with respect to the actual melt composition at the specific position, it is constitutionally supercooled. Therefore, some interface fluctuation may reach a position where solidification can easier be achieved than at the solid/liquid interface. As a result of constitutional supercooling, typical dendritic structures may be formed that are morphologically not to be distinguished from those originated from thermally unstable conditions. Because of the relatively narrow limit of the constitutional supercooling criterion with respect to the applied temperature profile and the solidification rate used, these types of instability frequently occur in almost every casting process. On the other

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hand, they have strictly to be avoided in single crystal growth, independent of the technique that is used. Experimental conditions how to create steep temperature profiles next to the phase boundary as well as good mixing conditions in front of the growing crystal will be important arguments to judge about specific crystal growth methods. And, narrow growth rate limits have to be accepted not only with respect to an average rate, but also in the time intervals relevant for the ongoing solidification processes, i.e. within parts of seconds. 6. Segregation

From the phase diagram discussion it is obvious, excluding only a few very special cases, that a solid phase growing from a melt of composition CL will not solidify congruently but with a composition CS = kCL , with k being the segregation coefficient. Depending on the growth rate and the matter transport conditions, k may be taken from the effective segregation coefficient keff -formula (Eq. 8) or simply from the equilibrium phase diagram (k0). In any case, the incongruent solidification changes the composition of the remaining melt and the next infinitesimal layer to be crystallized starts from a changed situation. For a quantitative description of the resulting local distribution of some component in the solid phase (solid solution component, doping element or impurity), the mass balance of the liquid has to be calculated. This can easily be done for a 1-dimensional problem by assuming a complete mixing of the melt and no diffusion in the solid phase. For an ingot to be solidified from a finite amount of material that was completely molten at the very beginning, the problem of the component distribution has been solved by Scheil8. From an initial melt composition C0 one obtains: (13) with z being the axial position and L the total length of the ingot. The axial distribution obtained from the normal freezing process is pictured in Fig. 9 for various values of the segregation coefficient k.

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It is obvious from the mass balance that in systems with a segregation coefficient k < 1 , the distribution function is homogeneously increasing, while with k > 1 it results in a decreasing concentration curve. In the derivation of this normal freezing function (Eq. 13), the volumes of the melt and the solid have been considered instead of the axial position. Therefore, solidification problems in geometries that deviate from a constant cross section, the relative axial position z/L can be substituted for the volume portion of the already crystallized melt V/V0, and this formula may well be taken to explain component segregation in various geometrical configurations of casting processes and in crystal growth.

Fig. 9. Distribution curves for normal freezing, showing solute concentration in the solid versus distance in crystallized fraction, for various values of the distribution coefficient, k.

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Fig. 10. Schematic drawing of the geometry of zone melting.

There has been a second type of macroscopic segregation function that is fundamentally different from the normal freezing case. It has been derived by Pfann9 when he invented the principle of zone melting. Having an ingot with a constant cross section and an initial composition C0, a narrow zone is molten and made to pass along the ingot (see Fig. 10). If we assume the same rates of solidification and melting at the two solid/liquid interfaces, the molten zone has a fixed volume that can be expressed by a constant zone length l. The resulting distribution function amounts to: ⎡ ⎛ CS ( z ) = C0 ⎢1 − (1 − k ) exp ⎜ − k ⎝ ⎣

z ⎞⎤ ⎟ l ⎠ ⎥⎦

(14)

Again, k is the segregation coefficient and z the axial position within the ingot. From the plotted curves in Fig. 11, it can be seen that the zone melting distribution function CS(z) asymptotically reaches the initial composition C0 with the zone length being the characteristic length and depending on the segregation coefficient. With k-values next to unity, after a few zone lengths one gets the starting composition again, i.e. the molten zone has reached steady state conditions. On the other hand, with a very low segregation coefficient, the component under consideration is permanently accumulated in the liquid zone without any practical chance of the crystallized ingot’s composition to equal the initial one. This may be regarded as a powerful advantage if zone melting is carried out as a procedure to refine a material by passing a molten zone along the ingot. But, if a more or less macroscopically homogeneous solid is the aim, distribution coefficients much less than unity may be a severe problem.

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Fig. 11. Distribution curves for zone melting, showing solute concentration in the solid versus distance in zone lengths from beginning of charge, for various values of the distribution coefficient, k.

7. Techniques of single crystal growth

Some of the most important solidification methods for single crystal growth shall be briefly presented. According to the component distribution problems discussed in the previous paragraph, the techniques will be treated in two groups depending on their segregation characteristics. There are methods with totally molten material at the very beginning which leads to the normal freezing type of the component distribution (Bridgman, Czochralski, and Kyropoulos method) and those with zone melting characteristics (floating zone and Verneuil method).

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7.1. Bridgman method

The Bridgman method10 is regarded as the easiest to do, but nevertheless a very powerful crystal growth method from the melt. In this technique, an ampoule containing a melt is slowly lowered through a temperature field of a vertical tube furnace (Fig. 12).

Fig. 12. Bridgman crystal growth method.

Often a crucible with a pointed bottom is used to support nucleation, but seeded growth may be applied as well. Since a not too low temperature gradient at the solid/liquid phase boundary has been found necessary, e.g. to avoid constitutional supercooling, usually a twosegment furnace is used to produce a steep axial temperature profile at the interface with quite low gradients in the upper and lower parts of the tube furnace. This modification was suggested by Stockbarger11 and is nowadays included in almost all Bridgman-type experiments. For the process of solidification, it is not the decisive factor whether the crucible is mechanically passed through the temperature field of the furnace or the change of temperature is done by an electronically based temperature versus time program. Therefore, recent modifications of the Bridgman

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method may use multi-segment tube furnaces that are dynamically controlled to move a temperature field along the ampoule without any mechanical motion. 7.2. Czochralski method

Czochralski’s original idea12 was to measure the crystallization rate of metals by pulling a metallic wire from its native melt. Exceeding some upper limit of the pulling rate would separate the crystalline material from the liquid. This has been developed to the most important growth method for single crystals from the melt (Fig. 13) and is used for the large-scale production of electronic materials. Starting from a totally molten source material, a single-crystalline seed is brought in contact with the melt surface and wetted by the melt. After having reached equilibrium conditions, the seed is slowly pulled upwards which transfers the original interface to a lower-temperature position. Thus, the driving force for solidification is simultaneously created by the interface shift and the crystal growths downwards with approximately the pulling rate.

Fig. 13. Czochralski crystal growth method.

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It is really difficult to perfectly control the diameter of the growing crystal because it is influenced by the contact angle of the melt at the three-phase line (vapour/liquid/solid) where the meniscus of the melt touches the crystal interface. An increase of the crystal’s diameter, that is e.g. necessary at the end of the seeding procedure, is obtained either by lowering the pulling rate or by a slight reduction of the heating power. Conversely, a decreasing diameter is the result of an opposite change of one of these parameters. Counter rotation of the crystal and the melt is regarded as a decisive feature of the Czochralski method because it not only gives a better rotational symmetry but also ensures a good mixing of the melt in front of the growing interface. Thus, a more or less good homogeneity of the melt as assumed in the segregation analysis (Eq. 12) can much better be achieved than with the Bridgman method. The missing contact between the crystal and the crucible is one of the main advantages of this technique. 7.3. Kyropoulos method

The Kyropoulos method that is pictured in Fig. 14 looks technically very similar like the latter one. Now, the crystal grows into the melt instead of being pulled upwards. Therefore wetting does not play a dominant role, but solidification is exclusively driven by the temperature field. A slowly decreasing temperature, as well as the intensive cooling via the seed holder, make the crystal growing into the bulk melt. Crystal growth has to be completed by pulling the crystal rapidly out of the melt before the crystal might touch the crucible wall. Since the crystal is totally embedded during the solidification process, there are low thermal gradients in the solid. This is usually an advantage with respect to the mechanical stress that may be caused by inhomogeneous temperature fields. On the other hand, a low temperature gradient gives a rather narrow limit for the possible growth rate obtained from the constitutional supercooling criterion. With metallic melts, there is nearly no chance to in-situ observe the proceeding crystallization. Therefore, the Kyropoulos method is mainly used for transparent melts, i.e. for the growth of insulators and crystals for optical applications where large-diameter crystals are needed.

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Fig. 14. Kyropoulos crystal growth method.

7.4. Floating zone method

The floating zone method (Fig. 15) is the crucible-free modification of the vertical zone melting technique. Since it does not require any crucible, it can be used for crystals where a suitable container can hardly be found, either for reasons of its chemical reaction with the melt or because of the risk of impurity contamination at very high temperatures. The liquid is held by the surface tension that restricts the possible zone length to a height that depends on the liquid/vapour interface energy and the density of the melt. Growth proceeds by moving the heater that passes the liquid zone along the feed rod. Most frequently, a RF coil or an optical heating facility is used to create a steep axial temperature profile with a maximum in the floating zone. The feed material and the growing crystal are separately fixed at their ends and counter-rotated to enhance convective mixing as well as radial heat transfer. There are modifications with a much smaller diameter of the feed rod compared to the crystal to be solidified. In these cases, the feed rod must be moved relative to the growing crystal at a different rate.

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The separate preparation of the feed rod prior to the growth process makes the floating zone technique more complicated compared to other methods and therefore restricts it to the growth of ultra-pure materials and really high melting point substances for which suitable crucibles are missing.

Fig. 15. Floating zone crystal growth method.

7.5. Verneuil method

Verneuil was the first to develop a growth method for single crystals from the melt for commercial purposes15. His flame fusion method is especially dedicated to high melting point oxides like sapphire and spinel. Because of the heating by an oxyhydrogen burner, it can not be

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used for metals and other oxidizing substances, at least in the original version. In Fig. 16, the experimental set-up is shown that has almost not been changed during the last hundred years. The growing crystal is mounted on a ceramic rod that is rotated and slowly moved in vertical direction. Fine-grained powder is used as source material that trickles through a sieve driven by some vibration mechanism. Falling through the oxyhydrogen flame the powder grains become molten and enter the thin melt layer on top of the growing crystal. It is this thin layer that corresponds to the liquid zone in zone melting techniques. Since no other heating facility than the burner is used, the temperature field is simply the result of the flame and the ceramic insulating chamber and very steep temperature gradients occur in the crystal that produce considerable mechanical stress. Nevertheless, Verneuil’s method, which is a cruciblefree technique as well, has demonstrated its feasibility and high output, e.g. for the single crystal growth of gemstones. Before growing single crystals from the melt, one should thoroughly analyse all available properties of the substance to be solidified, including the melting behaviour, the vapour pressure of its components and the needs the crystal has to fulfil for application purposes. And the full range of crystal growth methods developed over the decades should be considered for growing this special crystal. There is no method that is suited for the whole spectrum of single crystals of interest. Only a short overview on the most prominent techniques of crystal growth from the melt has been given in this chapter. Detailed information on very special methods and a profound treatment of the phenomena that will influence the more or less successful growth experiments can be obtained from modern textbooks16, 17. References 1. W. Kossel., Zur Theorie des Kristallwachstums, Nachr. Akad. Wiss. Göttingen, Math.-phys. Kl., 135 (1927) (in German). 2. K. A. Jackson, Amer. Soc. Metals Cleveland, 174 (1959). 3. W. K. Burton, N. Cabrera and F. C. Frank, Philos. Trans. Roy. Soc., A 243, 299 (1951). 4. J. A. Burton, R. C. Prim and W. P. Slichter, J. Chem. Phys., 21, 1987 (1953). 5. W. Nernst, Z. phys. Chem., 47, 52 (1904) (in German).

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6. J. W. Rutter, and B. Chalmers, Can. J. Phys., 31, 15 (1953). 7. W. A. Tiller., K. A. Jackson, R. W. Rutter and B. Chalmers, Acta Met., 1, 428 (1953). 8. E. Scheil, Z. Metallkd., 34, 70 (1942) (in German). 9. W. G. Pfann, Trans. AIME, 194, 747 (1952). 10. P. W. Bridgman, Proc. Amer. Acad., 60, 303 (1925). 11. D. C. Stockbarger, Rev. Sci. Instr., 7, 133 (1936). 12. J. Czochralski, Z. phys. Chem., 92, 219 (1918) (in German). 13. S. Kyropoulos, Z. anorg. allg. Chem., 154, 308 (1926) (in German). 14. P. H. Keck, Phys. Rev., 89, 1297 (1953). 15. V. Verneuil, C. R. Acad. Sci. Paris, C 135, 791 (1902) (in French). 16. K. T. Wilke, and J. Bohm, Kristallzüchtung, Deutscher Verlag der Wissenschaften, Berlin, 1988 (in German). 17. D. T. J. Hurle, Ed., Handbook of Crystal Growth, 1-3, Elsevier, Amsterdam, (1993).

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CHAPTER 5

DIFFUSIVE PHASE TRANSFORMATIONS Yves Bréchet SIMAP, Institut National Polytechnique de Grenoble, Domaine Universitaire, PB75, 38402 Saint Martin d’Heres cedex, France E-mail: [email protected] This chapter outlines the main analytical tools available to treat diffusion controlled phase transformations. These tools will be applied on one side to precipitation reactions, and on the other side to interface migrations. Special emphasis will be laid upon the necessary improvement to go from binary to multicomponent systems, and to integrate the models of phase transformation into a global alloy and process optimisation. Examples will be taken from aluminium alloys and steels.

1. Introduction and motivation In recent years, in addition to the pure knowledge driven research, the need to optimize alloy and process design, together with the enhanced possibilities of computers has motivated the development of a physically based modeling of phase transformations. The aim being to obtain the best compromise between properties, and the tools available for that being either the composition, or the the parameters of thermomechanical treatments, the understanding of the relations between process parameters and generated microstructures, and between microstructures and the resulting properties has become a necessary step for metallic alloys improvements. The microstructural features under consideration are the granular microstructures (grains size, texture), and the phase microstructures (nature, size and morphology). These microstructures are strongly influenced by the treatments materials undergo in the solid state. We focus in this course on the phases transformations leading to a multiphase structure. These multiphase structures control mechanical 99

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properties such as yield stress, work hardening and ductility. They may also govern functional properties such as critical currents in super conductors or coercivity in magnets. They certainly influence damage properties such as toughness or fatigue or corrosion, though the relation microstructure / properties is in these situations far less understood. Depending on the property under consideration, the microstructural features of interest are different, but in most situations, the nature, the volume fraction, the shape and the scale of the phases present have to be considered. The yield stress of an alloy with spherical precipitates depends both on the volume fraction f of precipitates, and on their size R. If the precipitates are bypassed, the yield stress scales as f1/2/R, if they are sheared, as (fR)1/2. The way dislocation interacts with them depends on their chemical nature, and on their size. Most phenomenological models would provide , at best, a description of the volume transformed after a given heat treatment. This illustrates the need to develop a more sophisticated approach for microstructure development which would incorporate the transformation mechanisms. The study of solid state phase transformations in metals and alloys is traditionnally divided in two main streams: the displacive transformations in which atoms move in a cooperative manner at a velocity approaching the one of sound waves, the diffusive transformations in which atoms move in a non cooperative manner, by diffusion processes. The emphasis laid upon various aspects of phase transition is different in the two cases. The distinction is certainly not as clear cut as it sounds (see for instance years of controversy concerning Bainite) but it remains useful at least to set limiting cases.For the martensitic transformation, treated in another chapter of this book, crystallography and back stresses generated during the transformation are a central issue, the thermodynamics of the problem enters mainly into the conditions for nucleation of the new phase, and since the propagation of the transformation is very rapid, the kinetics of invasion during continuous cooling is controlled by the possibility of repeated nucleation. The patterns emerging from these transformations reflect both the crystallographic constraints and the elastic interactions between different variants.

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For the diffusive transformation, the central role in modelling is given to mass transport via diffusion processes. The scale of the microstructure results from the competition between available free energy (driving force) and interfacial energy. The kinetics results from diffusive transport, and limited mobility of the interfaces. Although crystallography and elastic stresses may play a role they are not central to the main issue: transformation kinetics and morphology of the reaction products. The present contribution is limited to diffusive phase transformations. Table 1. Overview of the key differences between displacive and diffusive phase transformations. Displacive transformations Atoms move on interatomic distances a, in a cooperative manner Transformations occur below a critical temperature, at a rate independent of temperature The volume transformed depends on temperature only The chemical composition of the parent and daughter phases are identical There are strict crystallographic relations between the two phases

Diffusive transformations Atoms can move on distances 106a, in stochastic manner Transformation rate is highly dependant on temperature The volume transformed depends on time and temperature The chemical composition of the parent and daughter phases may differ There may be crystallographic relations between the two phases

2. Variety of situations in diffusive phase transformations Within the class of diffusive phase transformations, three other classifications can be proposed, in relation with the possible role of structural defects (homogeneous vs. heterogeneous), with the type of chemical evolution of the mother phase (continuous vs. discontinuous), and with the kinetics of transformation (linear or parabolic). In some situations, the transformation is homogeneous: it takes place in the grain interior, and as long as there is a thermodynamic driving force, it progressively transforms the mother phase. This process is typified by the example of fine precipitation in aluminium alloys obtained after quench and annealing below the solubility limit. But the

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transformation may also be heterogeneous, in the sense that it requires structural defects such as dislocations and grain boundaries to operate. Heterogeneous precipitation very often leads to coarser structures, with less interesting properties. Heterogeneous precipitation (similar to heterogeneous nucleation from the liquid, treated elsewhere in this book) occurs generally when the driving force is not sufficient, and the help of a structural defect decreases the nucleation barrier, but the scale remains large. In some situations, the transformation leads to a continuous evolution of the chemical composition of the mother phase. This is the case for the traditional precipitation reaction, where the precipitates progressively deplete the matrix supersaturated in solute. By contrast, in other situations, the decomposition process takes place by the migration of an interface through the mother phase. Across this moving interface, the composition, and even the crystallography may change in a discontinuous manner. These types of reactions can be found very often in steels (transformation from austenite to ferrite, or eutectoid reaction known as pearlite) but also in other systems. For instance, the same phase leading in sole circumstances to continuous precipitation, can lead to discontinous precipitation where a moving grain boundary sweeps a supersaturated solid solution to leave as a daughter phase a solid solution with less supersaturation, and regularly spaced lamellar or rod structures as precipitates. The third classification is according to the reaction rates. Depending on the morphology, the linear dimensions of the product phase may have different time dependence. In the case of non conserved shapes such as spherical precipitates growing from a matix, mass transport by diffusion, in absence of any interface limiting reaction, would lead to a parabolic growth rate. In the case of conserved shapes, such as needle growth or planar fronts, the situation is more complex. For planar fronts, the only way to obtain a constant velocity is when the daughter phase and the mother phase have the same average composition: this is the case for pearlite or for discontinuous precipitation. If the daughter phase has a composition different from the mother phase (such as for austenite to ferrite allotriomorphic transformation), the need for mass transport by diffusion imposes a parabolic growth. In the case of needles (such as

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Widmanstatten ferrite), the “volume of the mother phase explored” by the growing needle increasing while transformation proceeds, a constant velocity is possible even in the case of a non conserved composition. This description of the kinetics is highly idealized, one should rather speak of “constant velocity” vs. “decreasing velocity”. However, it sets the scene for the basic models of diffusive growth that are currently used in microstructural modeling. 3. Diffusion and diffusion equations 3.1. Basics of diffusion Diffusion processes in crystalline solids are closely related to point defects (vacancies and intersticials) and the activation energy for this thermally activated process is the sum of the defect formation energy, the defect migration energy and the binding energy between the defects and the solute atoms diffusing. This activation energy is lowered when diffusion takes places at dislocations (by a factor 0.6), at grain boundaries (0.4) and at surfaces (0.2). The details of the diffusion processes are beyond the scope of this paper, but it is worth keeping in mind that diffusion via intersticials is far more rapid than the one taking place by vacancies. This fact is the reason for some rich features in ternary Fe-X-C alloys. The phenomenological description of diffusion relies on the thermodynamics of irreversible processes. A simple way to derive it for perfect solid solutions is the following: the driving force F for diffusion is the opposite of gradient in chemical potential. The velocities of atoms under this driving force is v=MF (assumption of linearity), and the mobility M is related to diffusion via Einstein relation M=D/kT. The flux is J=cV. For perfect solutions, the chemical potential is kTLn(c). The resulting flux is therefore J=-D grad (c) , known as the first Ficks law. The second Ficks law is simply the mass balance, and, in three dimensions, the governing partial differential equation for diffusion is:

∂c = D.Δc ∂t

(1)

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Modeling diffusive phase transformations relies entirely on the solution of Fick’s diffusion equations with the appropriate boundary conditions which reflect the thermodynamics of the system. 3.2. Classical exact solutions

In diffusion controlled phase transformation, an essential tool for modelling is the classical solutions of Fick's equation in various geometries. We will successively investigate planar and spherical geometries leading to the so called "Parabolic solutions", and the needle like geometries leading to shapes propagating at constant velocities. 3.2.1. Parabolic solutions The family of parabolic solutions relies on the classical solution of Fick's equation:

C = A + Berf ( erf (u ) =

2

π

x ) 4 Dt

(2)

u

∫

. exp( −λ )d λ 2

0

In such solutions it is possible to impose a constant concentration at a position defined by:

ξ = K 4 Dt

(3)

K = 0 corresponds to a static interface, whereas a non zero K corresponds to an interface propagating with a decreasing velocity. Depending on the boundary conditions, A, B and K can be determined, leading to a prediction of the diffusion field, and of the propagation rate. For instance, the case of precipitation of a β phase of composition Cβ from a solid solution of composition Cα larger than the equilibrium concentration Cαβ, leads to the following results for the diffusion field and the parabolic constant K:

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⎡1 − erf ( x / 4 Dt ⎤ C α ( x) = C α∞ + (C αβ − C α∞ ) ⎢ ⎥ ⎣ 1 − erf ( K ) ⎦

(4)

( −1/ π ) (C K≈

αβ

− C α∞ )

(C β − C αβ )

(5)

This can be readily extended with the same method to situations where the composition of the precipitate is allowed to vary, where diffusion is allowed in the product phase, where an interface separates a one phase region from a two phase region, etc. For a better grasp of the physical meaning of the method, it is worth going into detail through the derivation of the growth rate of a spherical precipitate from a supersaturated solid solution. In spherical coordinates, Fick’s equation can be rewritten under the assumption of an invariant diffusion field: d 2 dc (r . ) = 0 dr dr

(6)

With appropriate boundary conditions at the interface and at infinite distance (i.e. assuming a diluted precipitation), one gets the expression for the diffusion field: C (r ) − C α∞ = ( C αβ − C α∞ ) .

R r

(7)

The mass balance at the interface writes then: 4π R 2 .dR.(C β − C αβ ) = 4π R 2 .Jdt

(8)

The expression of J can be then readily obtained from the expression of the diffusion field, from which the rate of growth of precipitate radius can be derived and by direct integration one finds the expression of the precipitate size as function of its initial size, of the diffusion coefficient, and of the supersaturation: R 2 (t ) − R02 = 2 D.

C α∞ − C αβ .t C β − C αβ

(9)

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3.2.2. Self preserving shapes These solutions correspond to C(x-Vt). As can be readily seen, a planar solution is possible (with an exponential variation of the concentration field) only if the product phase has the same overall composition as the parent phase (as is the case with massive transformation, pearlitic reaction or discontinuous precipitation). But there is another family of solution propagating at constant velocity and preserving its shape: the cases where the surface of the reaction front is a quartic, and the most relevant situation in phase transformations, when it is a paraboloïd. In order to prove this, the corresponding total differential equation for the function of x-Vt is rewritten in a system of confocal parabolic coordinates. In such a system, the parabolae can be lines of constant concentration. Solving the equation in this system of coordinates leads to an expression of the diffusion field, and the mass balance produces a relation between the dimensionless supersaturation S and the Peclet number P defined by:

S =

C αβ − C α ∞ C αβ − C β

P=

Vρ 2D

(10)

This relation can be derived for all paraboloïds. The most frequently used are the cylindrical paraboloïd and the circular paraboloïd. For each of these geometries the relations between S and P are respectively: 2

∞

exp(−u π∫

S = π P .exp( P ).

2

)du

(11)

P

∞

exp(−u ) du u P

∫

S = P.exp( P ).

(12)

These exact solutions do not survive to the introduction of capillary effects, which introduces higer order terms due to curvature, leading to similar mathematical difficulties to the ones found in the theory of dendritic solidification.

Diffusive Phase Transformations

R 2 (t ) − R02 = 2 D.

C α∞ − C αβ .t C β − C αβ

107

(13)

3.3. Classical approximate solutions

The examples given above clearly indicate both the mathematical complexity, and the limitations of exact analytical solutions of diffusion equations. One can bypass these difficulties using numerical solutions, but it is however often enlightening, for the sake of a physical understanding of the observed transformations, and even for a relatively accurate quantitative description within the limits of experimental accuracy and of the quantitative knowledge of thermodynamic and diffusion data, to obtain approximate solutions. These “classical approximate solutions” are still of great interest. 3.3.1. Growth of a sphere: the constant field approximation The constant field approximation amounts to solve Fick’s equation assuming the diffusion field is stationary in the frame of the growing precipitate. The concentration field in spherical coordinates decays as 1/r (r being the distance to the origin), and the growth rate for a precipitate of β growing from a supersaturated solution α is given by: R 2 (t ) − R02 = 2 D.

C α∞ − C αβ .t C β − C αβ

(14)

3.3.2. Diffusional growth of a planar front: Zener approximation Zener approximation is a very simple approach which considers that the diffusion field facing a moving planar front is linear and extends on a distance ΔX. Mass conservation provides a relation between the layer thickness X, the diffusion extension ΔX, and the various concentrations indicated on Fig. 1. ΔC.Δx = 2 (-Cα+C°).X

(15)

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Cαγ

ΔC

C° ΔX

Cα

X Fig. 1. Diffusion profile in Zener’s approximation.

The next step is the solution of Ficks first equation, which, via a mass balance, gives an expression of the velocity: (Cα-Cαγ).V=D.(dC/dx)

(16)

The final integration provides an expression for the layer thickness at a function of time:

X2 =

(Cαγ

D(ΔC ) 2 .t − Cα )(C ° − Cα )

(17)

3.3.3. Diffusionnal growth of a needle: Hillert approximation For the growth of a needle of tip radius r, with a similar approximation of a linear diffusion gradient, Hillert proposed an expression for the constant growth rate V: V=

2 D Cαγ − C ° . r Cαγ − Cα

(18)

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4. Precipitation

We will first outline the most frequently used experimental methods, then summarize the classical understanding of the phenomena, and give indications on recent developments.

Fig. 2. Precipitation of Copper into Iron observed by 3D Atom probe (D.Blavette).

4.1. Experimental methods

Informations on precipitation can be obtained by a variety of manners, both by direct and indirect methods. Their relevance to get reliable informations on the size, shape, volume fraction and chemistry of precipitation are summarized in the following table, italics indicate some information which may be attaignable, with difficulty, in certain cases. Table 2. Experimental methods for the study of precipitation. Direct Methods

Transmission Electron Microscopy

Size, shape, Volume fraction, chem

Neutron and X Rays Scattering

Size, volume fraction, shape

3D atom probe

Size, Chemistry

Resistivity, Thermoelectric power

Evolution of the solid solution

Indirect

Calorimetry

Nature and volume fraction

Methods

Hardness

Combination of size and volume fraction

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4.2. Classical picture

Precipitation from a supersaturated solid solution starts usually by a nucleation step where fluctuations of compositions providing a sufficient gain in bulk energy to afford the cost in surface energy can grow. This defines a critical radius which is essentially the ratio of the surface energy to the bulk energy, and a nucleation barrier which scales as the inverse cube of the available driving force: the larger the available free energy , the finer will be the structure, the smaller the energy barrier and the more abundant the nucleation rate. The classical nucleation theory presented in the chapter on solidification can be translated, with some minor modifications, for the solid state. In recent years, numerical simulations, namely Kinetic Monte Carlo and cluster dynamics, have shed a new light on this nucleation process. When temperature is changing, both the transport efficiency and the driving force for precipitation are modified. At temperature close to the solubility limit, the driving force for precipitation is low, and, in spite of a fast transport, the transformation kinetic is slow. At low temperatures, the driving force is large but the transport by diffusion is limited: again the transformation is slow. The transformation rate will be maximum at an intermediate temperature where both the driving force is important, and the diffusion coefficient is large enough. This is the origin of the shape so called C curves plotting the time necessary at a given temperature to perform a prescribed percentage of the transformation. While precipitation proceeds, the solid solution is progressively depleted, the available driving force for precipitation decreases and the nucleation of new germs of the precipitate phase becomes less favourable. In the first stages new germs appear: it is the nucleation stage, then only germs already present can grow: it is the growth stage. Finally only the largest germs which have a lower surface energy cost can grow: it is the coarsening stage. In these three stages, the scales emerging are given, in the nucleation stage by the critical radius, in the growth stage by the parabolic law: R 2 (t ) − R02 = 2 D.

C α∞ − C αβ .t C β − C αβ

(19)

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and in the coarsening stage, the growth follows the so called “Lifschitz Slyozov Wagner equation”, where the surface energy γ enters directly: 8 DγΩC αβ 2 R 3 − R(0)3 = . t kT 9

(20)

4.3. Recent developments

This classical distinction between the different stages in the precipitation sequence is somewhat artificial. In recent years, the so called “class models” which are describing a population of precipitate, and not simple an average, have allowed to account for progressive transitions between the different regimes. They have also allowed to deal with non isothermal treatments, and with phenomena such as reversion of precipitation when precipitates become unstable due to an increase in temperature resulting in a rapid increase in critical radius. This possibility to deal with non isothermal treatments has opened the path for the modelling of microstructure gradients in heat affected zones in the proximity of welds, an example of “integrated modelling”. The classical models in the literature are derived for binary alloys. Most of the situations encountered in real systems are multicomponents. The adaptation of the models to these situations is not necessarily straightforward. A simple hypothesis is to deal with a “quasi binary system”, where the alloying elements are not distinguished. This is by no way satisfactory, even if the inaccuracy of available data may end to make this drastic simplification admissible. To deal rigourously with multicomponent systems is nowadays understood only when the precipitates are stoechiometric. In ternary alloys for instance the equilibrium conditions at the interface can be given by any of the equilibrium conodes of the phase diagram. For each of these possibilities, one can compute the flux of the two elements, and only one among these possibilities will fulfill also the ratio on the fluxes imposed by the stoechiometry conditions. This selects uniquely the operating conode, and sets the evolution of the boundary conditions when the solid solution gets depleted.

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5. Interface migration

Phase transformations associated with interface migration are very important in systems presenting several allotropic phases such as steels (as shown in Fig. 3). But they can also occur without cristallographic changes as is the case in discontinuous precipitation. For the sake of simplicity in this section, we will focus on the austenite to ferrite transformation, where partitioning of C is thermodynamically favoured. • traditional

• decarburization

200 μm

Fe-0.1C-0.1Mo; 800C 1min

~ 4 mm

Fe-0.54C-0.51Mo; 825C 128min

Fig. 3. Examples of interface migration in the austenite to ferrite transformation (H.Zurob, C.Hutchinson) a) during an isothermal treatment, b) during a decarburization treatment leading.

The main features of these types of transformations is to start at the boundaries of the mother phase (austenite) and to generate microstructure whose length scale if of the order of microns. The modeling of these reaction rates remains at the continuous level, and the key issue is to understand the interplay between thermodynamics and kinetics to define the interfacial conditions governing the transformation velocity. 5.1. Experimental methods

While precipitation requires often characterization methods at very small scales, interface mediated transformations are more often observed at the micron scale. As a consequence, optical metallography and Scanning Electron Microscopy remain the major tools for their investigation (Table 3).

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Table 3. Experimental methods for the study of precipitation.

Direct Methods

Indirect Methods

Scanning Electron Microscopy Microprobe

Size, shape, Volume fraction, chem

Optical microscopy Neutron and X Rays Scattering

Volume fraction

3D atom probe

Interfacial Chemistry

Dilatometry

Evolution of the volume fraction

Calorimetry

Nature

Hardness

Combination of size and volume fraction

5.2. Classical picture

For binary alloys, a Zener type of description, assuming equilibrium conditions at the interface for the local carbon composition, describes accurately the kinetics. When one considers the ternary alloys (such as Fe-Ni-C), an extra difficulty appears, to decide whether or not the substitutional element partitions between the two phases according to the phase diagram. If this is the case, due to the slow diffusion of Ni compared to C, the transformation would be always very sluggish. As it is observed not to be so in many cases, the possibility of transformation without long range partitioning of Ni had to be considered. Carbon diffusion is controlled by the interfacial concentration which depends on the concentration in alloying element X. The best solution from a thermodynamics viewpoint, and always possible, is “Full local equilibrium with partitioning” (Fig. 4.a) but since both X and C have to be partitioned, the predicted kinetics is very slow (being controlled by the diffusion of X). Less favourable from the thermodynamics view point, the “local equilibrium with negligible partitioning, or LENP (Fig. 4.b) is still a local thermodynamic condition at the interface. There is no long range partitionning of X, but a short range spike. The kinetics is controlled by C diffusion, the presence of the X spike modifies the carbon concentration at the the interface. This gives a fast reaction. This condition is possible only for compositions below dashed line in Fig. 4.b

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Fig. 4. The classical interfacial conditions : a) Local equilibrium, b) local equilibrium with negligible partitioning, c) paraequilibrium.

(no partition limit): above, the C concentration would be such that C would go toward α, instead of being rejected from α. In the “Paraequilibrium hypothesis” (Fig. 4.c), the interfacial conditions are given by a constrained equilibrium: the composition in X is forced to be the same in both a and b. The dashed lines are obtained by a double tangent construction between the curves G(C, X=cte) for a and g. There is no long range partition of X, no X spike. This is a fast

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115

reaction, only possible between the dashed lines on Fig. 4c indicating the «paraequilibrium limit». As is shown in Fig. 5, the choice between the different interfacial conditions has significant consequences on the predicted kinetics. 450

Fe-0.51C 775°C

450

Layer thickness (microns)

400 350

LE

Fe-0.50C-0.97Ni 775°C

400

Layer thickness (microns)

500

350

PE LENP

300

300

250

250

200

200

150

150

100

100

50

50

0

0 0

50

100

150

200

250

0

300

50

100

150

200

250

300

time(min)

Time(min) 350

400

PE

Layer thickness (microns)

200 150

LENP

100 50

PE

Fe-0.38C-1.03Ni 800°C

350

Layer Thickness (m icron

Fe-0.50C-1.66Ni 775°C 250 300

300 250

LENP

200 150 100 50

0

0

0

50

100

150

200

250

300

0

time(min)

Fig. 5. Modelling of decarburization kinetics temperature (C.Hutchinson, H.Zurob).

50

100

150

Time (min)

200

250

for various alloy compositions and

5.3. Recent developments

The selection criteria leading to a given interfacial condition is still a matter of active research. In recent years, some evidence has been given of the fact that the interfacial conditions are not constant when transformation proceeds. It seems that the interfacial condition at nucleation is close to paraequilibrium, whereas it progressively evolves building up a spike at the interface, toward LENP in the last stages of the

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reaction. It seems also that this transition depends itself on the interface velocity. The relative importance of solute drag (diffusive dissipation in the interface) and of cross interface diffusion is still under investigation. 6. Conclusions

This brief overview of diffusive phase transformations is far from being exhaustive. We have not dealt with lamellar structure such as discontinuous precipitation, or eutectoid transformations, neither with the puzzling aspects of the massive transformation, nor of the controversial nature of Bainite. The reader is referred to extensive treaties for these issues. We have presented an overview of the commonly admitted concepts and methods used in the understanding of diffusive phase transformations. More details on their application can be found in the lecture notes which were presented at the school and are on the website. To conclude, in recent years substantial progress has been made in the field due to better thermodynamic databases, better understanding of the coupling between kinetics and thermodynamics. Reliable systematic experimental investigations of model systems, and comparison between analytical modeling and atomistic simulations have clarified a lot the issue of nucleation in precipitation. For the case of interfacial migration, the evidence of the key role of the interfacial conditions is now well established, but the issue of nucleation is basically untouched, and the question of morphological instabilities remains open. For both precipitation and interface mediated transformation, the question of the interplay and the relative importance of thermodynamics, kinetics and crystallography is totally open. Acknowledgments

The author wants to thank his colleagues Prof A.Deschamps (Grenoble), Prof G.Purdy and H.Zurob (McMaster) and Dr C.Hutchinson (Monash) for invaluable discussions and collaborations over the years in the field of diffusive phase transformations. Constant support of Arcelor research is gratefully acknowledged.

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References In a chapter of a course, a list of specialized references would give an excessive weight to the topics selected which somewhat reflect the interest of the author. The field of phase transformations in metals and alloys has a long history, which has also led to a jargon which somewhat makes it less accessible to specialists of other fields. As a result, the general knowledge on precipitation for instance, is much more developped than the one on phase transformations controlled by interface migration. The general references listed here are, from the view point of the author, good overviews for a non specialist to enter the field. G1. H.Aaronson "Lectures in the theory of phase transformations",TMS (1999), recently reedited and updated is a must. The articles by M.Hillert on "thermodynamics" (Chapter 1) and by R.Sekerka on "Moving boundary problems" (Chapter 5) are masterpieces. G2. G.Kostorz "Phase transformations in Materials", Wiley VCH, (2002), is an updated version of the Treatise on Materials Science and Technology , volume 5 edited by R.Cahn, P.Haasen, E.Kramer. Of special relevance to the present topics are the review papers by R.Wagner et al. on "Homogeneous second phase precipitation" (Chapter 5) and G.Purdy et al. on "Transformations involving interfacial diffusion" (Chapter 7). G3. J.Philibert, "Atom movements, Diffusion and Mass transport in solids", Editions de Physique (1991) provides an excellent textbook on diffusion and its applications. G4. M.Hillert "Phase diagrams and Phase Transformations", Cambridge University Press (1999) is the key reference on the application of thermodynamics to phase transformations. G5. J.W.Martin, R.D.Doherty, B.Cantor "Stability of micostructures in metallic systems", Cambridge University Press (1997) is an encyclopedic visit of the world of microstructural evolution, and an invaluable source of references to original papers.

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CHAPTER 6

DIFFUSIONLESS TRANSFORMATIONS C. Duhamel1,2, S. Venkataraman1,2, S. Scudino1,2 J. Eckert2 1

FG Physikalische Metallkunde, FB 11 Material- und Geowissenschaften, Technische Universität Darmstadt, Petersenstraße 23, D-64287 Darmstadt 2 IFW Dresden, Institut fürKomplexe Materialien, Postfach 270116, D-01171 Dresden E-mail: [email protected] Diffusionless transformations, also called displacive transformations, are solid state transformations that do not require diffusion, i.e. long range movements of atoms, for a change in the crystal structure to occur. They result from correlated atomic displacements where the atoms move less than one interatomic spacing and retain their relationship with their neighbours. The classical example is the martensitic transformation occurring in steels. The word “martensite” was originally used to name the hard and fine constituent formed in quenched steels. Later, other materials, such as non-ferrous alloys1 or ceramics2, were found to exhibit diffusionless transformations. The name “martensite” now refers more generally to the resulting product of such a transformation. The purpose of this chapter is to describe the basic features regarding the processes occurring during diffusionless transformations. First, the crystallographic theory of martensitic transformation will be presented. The crystallographic theory provides a description of the overall displacements involved in the transformation, consistent with the observed geometrical features. Then, the nucleation and growth of the martensitic phase will be described. In the last part, the mechanical effects in martensitic transformations will be discussed.

1. General characteristics of diffusionless transformations Although many of them occur at high temperature, diffusionless transformations are athermal. The transformation occurs at a very high velocity, independent of the temperature, which can be reached because

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of the absence of long-range atomic movements. The overall kinetics depends on the nucleation and growth steps and is limited by the slower of these two stages. The interface between the martensite and the parent phase is glissile and thermal activation is not required for its movement. Depending on the material, this interface can be fully coherent or semicoherent. For example, in ferrous alloys, the martensite / austenite boundary is semi-coherent. Only localized regions of the interface are coherently accommodated. On the contrary, the fcc → hcp transformation in pure Co leads to a fully coherent interface. The martensitic transformation implies coordinated structural changes with lattice correspondence and a planar parent / martensite interface, which gives an invariant-plane strain shape deformation. At a finer scale, inhomogeneities such as slip, twinning or faulting can be observed in the martensite, suggesting that a second deformation process occurs. This process is part of the overall diffusionless transformation. It gives the invariant plane conditions at a macroscopic scale and provides the semicoherent interface between martensite and the parent phase. 2. Martensite crystallography 2.1. General features Fig. 1 shows optical micrographs of typical martensite morphologies in iron-base alloys. The martensitic phase can have a lath, a lenticular or a

Fig. 1. Example of martensite morphologies in iron-base alloys: (a) lath in a Fe-9% Ni – 0.15% C alloy, (b) lenticular in a Fe – 29% Ni – 0.26% C alloy, (c) thin plates in a Fe – 31% Ni – 0.23% C alloy (From www.msm.cam.ac.uk/phase-trans/2005/Maki.ppt).

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thin plate shape. In steels, the orientation variants and the morphology of the plates depend on the alloy composition. The plates extend over the whole diameter of the grains and their growth occurs in a limited number of orientations. The contrast observed in the micrographs results from different lattice orientations with the initial parent grain. Different regions of martensite have undergone distortion in different ways with respect to the initial surface. This macroscopic distortion is known as shape deformation. The shape deformation exhibits features that are similar to simple shear, but is, in fact, associated with an invariant-plane strain where the plane of reference is the undistorted and unrotated habit plane. The invariant-plane strain is a homogeneous distortion in such a way that the displacement of any point is in a common direction and proportional to the distance from a plane of reference, the invariant plane, which is not influenced by the strain. It consists of an expansion (or contraction) normal to the invariant plane and a shear in a direction lying in the invariant plane. However, this invariant-plane strain is not sufficient to describe the martensitic transformation. 2.2. The Bain model This model was proposed by Bain3 in 1924 to explain how martensite in steel can be obtained from austenite with a minimum of atomic movement and a minimum of strain. 2.2.1. The solid solution of carbon in iron In this case, the parent phase is the austenite phase with fcc structure. In a fcc (or hcp) structure, two different types of sites can accommodate interstitial atoms: the tetrahedral sites with 4 near-neighbour atoms and the octahedral sites with 6 near-neighbour atoms. Assuming that the atoms are close-packed spheres, the maximum size of an interstitial atom that can be accommodated into these sites is given by: dt = 0.225×D = 0.568 Å do = 0.414×D = 1.044 Å

tetrahedral sites octahedral sites

with the diameter D of the lattice atoms. The numerical values are given for an iron structure.

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The diameter of a carbon atom is 1.54 Å. In the case of austenite, the two types of sites are too small to accommodate the carbon atoms. A distortion of the lattice is thus necessary and the carbon atoms will preferentially fill the octahedral sites. In a bcc structure, the maximum sizes of interstitial atoms are given by: dt = 0.291×D = 0.733 Å tetrahedral sites octahedral sites do = 0.155×D = 0.391 Å There is less space available for the interstitials in this structure. Although, the octahedral sites are smaller, the carbon atoms will still occupy these positions. The distortion induced is then huge and asymmetric. The lattice expands along the z direction leading to a bodycentered tetragonal (bct) structure. 2.2.2. The Bain model Bain3 revealed a particular correspondence between austenite and martensite. He has shown that a bcc cell can be drawn within two fcc cells (Fig. 2).

(a)

(b)

Fig. 2. The Bain’s model: (a) conventional unit cell, (b) relationship between the fcc and the bct cells. (From www.msm.cam.ac.uk/phase-trans/2002/encyclop.martensite.html).

There are other ways to generate a bcc cell from a fcc structure but the one shown in Fig. 2 involves the smallest strain. This strain is called the Bain strain. It involves a contraction of about 20% along the z-axis and an expansion of about 12% along the x- and y-axis. However, such

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distortion will leave no plane invariant (undistorted and unrotated) and is inconsistent with the experimental proof of invariant-plane strain shape deformation. A lattice-invariant distortion should be added in the form of slip or stacks of twins. This additional deformation has to be latticeinvariant insofar as the structure change was already done with the Bain distortion. The purpose of the lattice-invariant distortion is to shear the cell resulting from the Bain distortion in order to obtain an undistorted interface between the martensite and the parent phase. This additional deformation is known as inhomogeneous shear or complementary shear. It leads to the formation of a substructure inside the martensite phase. However, even after this second deformation, the plane of contact between the martensite and the parent phase is still rotated and an third deformation, a rigid body rotation, should be added to maintain the habit plane unrotated. Microscopically, the habit plane of the martensite plates results from a succession of slipped planes or thin twins (Fig. 3). By averaging the distortion over many individual slipped planes or twins, the “net” distortion is found equal to zero. By adjusting the width and the angle of the individual features, the habit plane can adopt a large variety of orientations.

Fig. 3. Simplified mechanism for the martensitic transformation. From left to right: original austenite; effect of Bain strain; additional deformation by slip, additional deformation by twinning.

2.2.3. Comparison with experimental results A wide scatter in experimental measurements exists for the habit planes of a given type of steel. Additional elements also modify the habit planes. The reason why the martensitic transformation can lead to different habit planes is still unclear. Most probably, it results from

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different operative modes of the inhomogeneous shear (i.e. different planes or directions) or activation of multiple modes. However, a trend can be extracted. When the carbon contents increases, the transition between the habit planes obeys the following scheme:

{111}γ → {225}γ → {259}γ For low-carbon steels (up to 0.4% C), the habit plane is {111}γ and the martensite phase has a lath morphology or consists of bundles of needles lying on the {111}γ planes. The typical dimensions of a lath are 0.3 × 4 × 200 µm. The laths are untwinned but contain a high density of dislocations suggesting that the inhomogeneous shear is slip. The movement of interface dislocations affects the lattice-invariant deformation. When the habit plane is {225}γ and {259}γ, the martensite consists mainly of twinned plates or has a lens morphology. The lenticular plates are promoted because of their low-energy shape. They usually contain a high density of twins revealing that the inhomogeneous shear occurs via twinning. However, an accurate description of the morphology of martensite is difficult as, after growth, the martensitic phase has generally an irregular shape. 2.2.4. Summary

The martensitic transformation involves three different steps: - the Bain strain, which converts the crystal structure of the parent phase to that of the martensite phase - an inhomogeneous shear, which gives the lattice-invariant distortion - a rigid body rotation, which leaves the habit plane unrotated. 3. Martensite nucleation

Like numerous other phase transformations, the formation of martensite also occurs by a nucleation and growth mechanism. However, the whole transformation occurs at a velocity close to the speed of sound. A martensite plate, once formed, grows to its full size in 10-5 to 10-7 s. Due to the extreme rapidity of this process, it is difficult to study the transformation experimentally. One way of measuring the growth

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Fig. 4. Schematic representation of the resistivity change during the martensitic transformation of a Fe-Ni alloy. (adapted from reference 4).

velocity of a nucleated martensite is to measure the resistivity change caused upon transformation from the parent austenite (γ phase) to the martensite (α phase). After a small initial increase in resistivity due to the initial strain in the austenite lattice caused by formation of the martensite nucleus, a steep decrease is observed upon transformation from γ to α (Fig. 4). The initial increase suggests that the austenite and the newly formed martensite nucleus are coherent. 3.1. Free energy change of martensitic transformation

Fig. 5(a) shows the free energy – temperature diagram for a diffusionless transformation.

(a)

(b)

Fig. 5. (a) Free energy – temperature diagram for the diffusionless transformation from the γ-phase to the α-phase; (b) variation of the martensite fractions with temperature upon cooling and heating.

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T0 is the temperature at which the parent phase and the martensite are in thermodynamic equilibrium. The reaction begins at the martensite start temperature (Ms) and is completed at the martensite finish temperature (Mf) below which further cooling does not increase the amount of martensite anymore (Fig. 5(b)). Theoretically, all austenite should have transformed into martensite at Mf. However, in practice, a small amount of austenite remains. The reverse transformation from martensite to austenite begins at the austenite start temperature (As) and is completed at the austenite finish temperature (Af). The chemical driving force to initiate diffusionless transformations is larger than for diffusional processes and is given by:

⎛ T − Ms ⎞ ΔG γ→α = ΔH γ→α ⎜ 0 ⎟ ⎝ T0 ⎠

(1)

where ΔH γ−α is the enthalpy for the transformation from the parent phase (γ-phase) to the martensitic phase (α-phase). 3.2. Formation of martensite nuclei

The increase in Gibbs free energy due to the formation of a coherent inclusion of martensite nuclei in the parent phase is expressed as:

ΔG = Aγ + VΔG s − VΔG v

(2)

where ΔG is the free energy change, A and V are the surface area and the volume of the nucleus, respectively, ΔGs is the strain energy, ΔGv is the volume free energy and γ is the interfacial free energy. Eq. (2) does not consider additional energies that might be active during martensite nucleation coming from thermal stresses generated upon cooling, addition of external stresses and stresses produced ahead of growing martensitic plates. Similar to other nucleation events, the surface and elastic terms tend to increase ΔG, while the volume component tends to decrease it. It has to be noted that the strain energy of the coherent nucleus contributes significantly towards the total free energy while the contribution from the interfacial (surface) component is relatively small.

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Let us consider the nucleation of a thin ellipsoid shaped nucleus, having a radius a, a semi-thickness c and volume V. The free energy increase can be given as: 4 2 2 ⎛ 2(2 − ν ) ⎞ c ΔG = 2πa 2 γ + 2μV ( s / 2 ) ⎜ ⎟ π − πa c.ΔG v ⎝ 8(1 − ν ) ⎠ a 3

(3)

where γ is the interfacial energy, ν is the Poisson’s ratio of the austenite, μ is the shear modulus and ΔGV is the free energy difference at the Ms temperature. For ν = 1/3, Eq. (3) can be simplified to: ΔG = 2πa 2 γ (surface) +

16 4 2 ( s / 2 ) μac 2 (elastic) − πa 2c.ΔG v (volume) (4) 3 3

The assumptions made are that nucleation occurs homogeneously without the aid of lattice defects like grain boundaries and that shear is responsible for nucleus formation. Additionally, the interface is coherent. The critical free energy minimum is obtained by differentiation of Eq. (4) with respect to a and c and can be expressed as: 4

512 γ3 ⎛s⎞ 2 ΔG = μ π joules/nucleus . . 2 ⎜ ⎟ 3 ( ΔG v ) ⎝ 2 ⎠ *

(5)

The critical nucleus size (c* and a*) are given by:

2γ c = ΔG v *

and

16γμ( s ) 2 2 a = 2 Δ G ( v) *

(6)

Metallographic studies on Fe-Ni single crystal particles have shown that upon cooling from the Ms temperature, martensite nuclei form heterogeneously. Several reasons have been given to account for this observation: 1. Cooling from below Ms down to 4 K does not result in complete transformation. 2. The number of nuclei formed is of the order of 104 per mm3, which is less than what is expected for homogeneous nucleation. 3. The number of nuclei increases with increased supercooling. 4. Surfaces and grain boundaries do not seem to be a preferred site for nucleation.

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This suggests also that the transformation is initiated at other defects: dislocations. 3.3. Role of dislocations in martensite nucleation

Based on nucleus density, it has been predicted that dislocations are preferred sites for martensite nucleation. It was first demonstrated by Zener5 how the movement of dislocations during twinning can generate a thin bcc lattice region from an fcc one. Fig. 6 shows layers of a fcc close packed plane, numbered 1,2,3.

Fig. 6. Zener’s model for nucleation of martensite by half-twinning shear (based on6).

The normal twinning vector b1 is formed by the dissociation of a 110 dislocations into two partials. 2

i.e.:

b = b1 + b2 a a a [110] = ⎡⎣ 211⎤⎦ + ⎡⎣ 12 1 ⎤⎦ 2 6 6

(7) (8)

For the generation of a bcc structure, all the “full circle” atoms 1 a (Level 3) jump by: b1 = ⎡⎣ 211⎤⎦ . 2 12

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This results in a thin nucleus. A thicker nucleus can be formed at dislocation pile-ups due to reduced slip vectors. An alternative approach of martensite nucleation was proposed by Venables7, with respect to martensite nucleation in stainless steels. According to Venables, the martensite forms via an intermediate phase having an hcp structure, denoted as epsilon martensite (ε):

γ→ε→α

(9)

Thickening of this ε structure occurs by non-homogeneous halftwinning on every {111}γ plane. However, there has been no direct evidence for this and electron microscopy studies on martensitic stainless steels have shown that the γ → ε or γ → α transformations occur independently of each other8. In case of pure cobalt, it has been demonstrated that half-twinning shear can induce a martensitic phase transition. This fcc → cph transformation occurs at 390°C. Transmission electron microscopy studies indicate that this transformation is a result of the formation of large number of dislocations with stacking faults appearing at the grain boundaries. Other models describing the role of dislocations in martensite nucleation have been reported9. They won’t be detailed here. 3.4. Dislocation strain energy assisted transformation

Let us now consider that the nucleation barrier necessary for the formation of coherent nuclei is reduced by the help of the elastic strain field of a dislocation. In this case, the dilatation associated with the extra half plan of the dislocation contributes to the Bain strain. Hence, the total Gibbs free energy can be represented as:

ΔG =Aγ + V ΔGs − V ΔGv − ΔGd

(10)

where ΔGd is the dislocation interaction energy which reduces the nucleation energy barrier. ΔGd, in turn, can be represented as:

ΔGd = 2μ sπ .ac.b where s is the shear strain of the nucleus and b is the Burgers vector.

(11)

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Fig. 7. Schematic diagram showing the need for twin nucleation when the martensitic plate reaches a given size.

Thus, using Eq. (10) and recalling Eq. (3), the total energy can be written as:

ΔG = 2π a 2γ +

16π 4 2 ( s / 2 ) μ ac 2 − π a 2c.ΔGv − 2μ sπ ac.b 3 3

(12)

It is clear from the above expression that the energy depends on the diameter (a), the thickness (c), the degree of assistance from strain field and whether it is twinned or not, as it is shown in Fig. 6. A fully coherent nucleus can reach a size of about 20 nm diameter and two to three atoms in thickness. However, further growth or thickening occurs only upon formation of twins or slip, which tends to reduce the strain energy. This theory offers the advantage to combine the crystallographic features of the Bain strain and inhomogeneous shear. The Ms temperature depends on the orientation or configuration of the dislocations. This is because large undercooling below Ms is necessary for nucleation. It suggests that a large chemical driving force is needed for the transformation and the presence of ideally oriented dislocations remains a statistical probability. 4. Martensite growth

Martensite growth occurs once the nucleation barrier has been overcome. It involves the edgewise propagation of the plate across the parent crystal

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and the thickening of the plate. Growth occurs rapidly until obstacles such as another plate or high angle grain boundaries are hit. Initially, very thin plates (with large a/c ratio) are formed and subsequently they thicken. A typical feature in high carbon martensites is the “midrib” of fine twins. Low carbon martensites are usually lath-shaped with a high dislocation density. Since the growth is a very high speed process, the interface between the parent phase and the martensite is most often a glissile semicoherent one. It consists of coherent regions separated by matching dislocations or twin boundaries. The movement of the glissile interface induces homogeneous transformation of the coherent areas as well as gliding of the matching dislocations in their slip planes or extension of twin boundaries. When the interface is moving, all the atoms in the parent phase are integrated in the martensite structure. This process involves no diffusion, no thermal activation and atomic movements of less than one interatomic spacing. It is thought that the Ms temperature dictates the mode of the inhomogeneous shear. At lower temperatures, slip-twinning transition is associated with increasing difficulty for nucleation of the dislocations necessary for slip. However, the chemical energy for the transformation is independent of the Ms temperature. Another factor affecting the growth is the mode of formation of the nucleus, i.e. by homogeneous Bain deformation or inhomogeneous shear. In the following, some aspects of the martensite growth in steels will be described. 4.1. Growth of lath martensite

The morphology of a lath of dimensions a > b >>c growing on a {111}γ plane involves nucleation and glide of the transforming dislocations, which move on distinct ledges behind the growing front. Dislocations nucleate due to the large misfit between the bct and fcc lattices. The required condition is that the stress at the interface should exceed the theoretical strength of the material. Using Eshelby’s approach10, the maximum shear stress at the martensite/austenite interface is given by:

σ ≅ 2 μ sc / a

(13)

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Fig. 8. Eq. (13) plotted versus the a/c ratio (modified from reference 6).

where μ is the shear modulus of the austenite. According to Kelly11, a theoretical shear strength of 0.025 μ can be used for fcc materials as a minimum or threshold stress for nucleation of dislocations. Assuming s = 0.2 (typical value for bulk and lath martensite), Eq. (13) is plotted in terms of a/c ratio in Fig. 8. The Kelly threshold stress can be exceeded in the case of lath martensite, but it is unlikely in the case of thinner plates. When nucleation of dislocations occurs at a highly strained lath interface, the misfit energy gets reduced and the lath is able to grow. Using internal friction measurements, it has been shown that, in lath martensite, the carbon density is higher at cell walls than inside the cells. This suggests that a limited amount of carbon might diffuse during transformation. Conventionally, the lath morphology is associated with higher Ms temperatures. This also allows dislocation climb and cell formation after transformation. The amount of retained austenite between laths is small in lath martensite, which is important for the mechanical properties of low-carbon steels. 4.2. Plate martensite

Plate martensite is a more common feature in medium and high-carbon steels. It is associated with a low Ms temperature and more retained

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austenite. The martensite is usually completely twinned. The morphology of the plates is much thinner compared with lath martensite and bainite. Once nucleated, twinned martensite grows very rapidly. Hence, the mechanism is not yet clear. The transition from twinning to dislocations can result from changes in the growth rate. Martensite formed at higher temperatures or slower rates grows by a slip mechanism while martensite formed at lower temperature or higher rate grows by twinning. Frank12 proposed a model for the formation of a dislocation-generated martensite. He considered that by inserting a set of screw dislocations at the interface, the lattice misfit could be reduced to a minimum. This is illustrated in Fig. 9.

Fig. 9. The Frank model (from reference 6).

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The close-packed planes of the fcc and bcc structures, (111)γ and (101)α respectively, meet along the habit plane (Fig. 9(a)). In order to reduce the misfit and to equalize the atomic spacings of the (111)γ and (101)α planes at the interface, a rotation ψ is introduced but is not sufficient. Inserting an array of screw dislocations with a spacing of six atom planes removes the misfit (Fig. 9(b)). 4.3. Stabilization

Stabilization can be defined as a process that occurs when a sample is cooled to a temperature between Ms and Mf and held there for some period of time prior to cooling again. The transformation to martensite does not immediately continue and so the total amount of martensite is less than that obtained by continuous cooling. The existing martensite plates do not grow. Instead, new plates nucleate. The amount of stabilization is a function of the time held at particular temperature between Ms and Mf. This phenomenon is not fully understood, yet. It seems that carbon atoms have enough time to diffuse to the interface, since the plate growth is under high stress. Additionally, local atomic relaxation at the interface increases the nucleation barrier necessary to generate dislocations 4.4. Effect of external stresses

External stresses aid martensite nucleation if the elastic strain components of the stress contribute to the Bain strain and raise the Ms temperature. Upon plastic deformation, there is an upper limit of Ms temperature defined as Md. The Ms temperature can be suppressed upon hydrostatic compression, since the increasing pressure stabilizes the close-packed austenite and lowers the driving force for the transformation to martensite. However, in the presence of a magnetic field, the Ms temperature is raised, since it favours the formation of a ferromagnetic phase. Too much plastic deformation can also hinder martensitic transformation. Although it increases the dislocation density and hence the number of nucleation sites, it also introduces restraints to the growth of nuclei. This increases the number of nucleation sites, and

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hence, refines the plate size. This phenomenon is called the “ausforming” process. The high strength of ausformed steel is due to a combination of fine plate size, solution hardening and dislocation hardening. The mechanical effects on martensitic transformations will be further discussed in section 6. 4.5. Role of grain size

Even though the grain size does not affect the number of martensite nuclei in a given volume, the final plate size is a function of grain size. In a large grain size material, the effect of large residual stresses can cause quench cracking and leads to an increase of the dislocation density of martensite. In general, fine grain-sized alloys along with a smaller martensite plate size exhibit superior mechanical properties13. 5. Tempering of ferrous martensites

The formed martensite always requires some heat treatment in order to improve the toughness and in some cases the strength of the steel. This is achieved by tempering. It is an ageing process where the martensite transforms to a mixture of ferrite and ε-carbide or ferrite and cementite.

α ' → α + ε − carbide or α ' → α + Fe3C

(14)

In the presence of strong carbide forming elements like Mo, Ti, Nb or V, the stable precipitates can be an alloy carbide instead of cementite. The phases that form depend on the heat treatment practice adopted. It is also possible that carbon segregation occurs during tempering. Especially in low-carbon steels, martensite starts to form at relatively high temperatures and can have sufficient time during quenching to segregate or precipitate as ε-carbide or cementite. ε-carbide has a hexagonal crystal structure and precipitates in the form of laths. Cementite (Fe3C) forms in most steels upon tempering between 250°C and 700°C. It has an orthorhombic structure. In case of alloyed steels having sufficient carbide-forming elements, the cementite composition can be represented

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as (FeM)3C where M is the carbide forming element. The precipitates form in situ whereby the alloy carbide can nucleate at several points at the cementite/ferrite interfaces and grows until the cementite disappears. Alternatively, they can form by a separate nucleation and growth process whereby the alloy carbides form heterogeneously within the ferrite on dislocations, lath boundaries and prior austenite boundaries. The actual mechanism depends on the alloy composition. Alloy carbides induce strength to the material and are advantageously used in high-speed tool steels. Tempering of molybdenum steels results in the formation of alloy carbides by a process commonly termed as secondary hardening. The effectiveness of carbides as strengtheners depends on their fineness and volume fraction. Finest precipitates are obtained from VC, NbC, TiC, HfC. These compounds are all close-packed intermetallics. The fineness depends on the free energy of formation. The volume fraction depends on the solubility of the alloy carbide in the austenite prior to quenching, relative to the stability in ferrite at a given tempering temperature. In most steels, containing more than 0.4 % C, austenite is retained upon quenching. It subsequently decomposes to bainite in the temperature range of 250-300 °C. As-quenched lath martensite contains high-angle grain boundaries, low-angle cell boundaries and dislocation tangles. Recovery occurs above 400 °C and leads to elimination of tangles and cell walls. However, the lath like structure is retained. Though the aim of tempering is to improve the ductility, in some steels, tempering in the range 350-575 °C can lead to embrittlement and thus loss of ductility. This is due to the segregation of impurity atoms like P, Sb or Sn to the prior austenite boundaries. 6. Non-thermoelastic and thermoelastic martensitic transformations 6.1. Non-thermoelastic transformations

6.1.1. General features Let us consider the cooling of a non-thermoelastic alloy. During cooling, nucleation and growth of martensite occur. If the cooling is stopped, the

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growth ceases and subsequent cooling does not lead to further growth of the martensite phase. The interface between the martensite and the parent phase apparently becomes immobilized. When the martensite is heated again, the interface does not move backward. The reverse martensite → parent phase transformation takes place by nucleation of small platelets of the parent phase inside the martensite. In steels, the usual stages of martensite tempering occur. For the non-thermoelastic transformations, a large difference between MS and Mf is observed.

6.1.2. Mechanical effects The martensitic transformation can be seen as a “shear” process that implies a cooperative shear movement of atoms. This process can be aided by an applied elastic stress. The non-thermoelastic martensitic transformation is associated with an apparent immobilized interface. If martensite formation is strain- or stress-induced, no retransformation of martensite into austenite is supposed to happen after unloading. The martensite phase is stable. It has been shown that ferrous martensite formed during deformation leads to a considerable increase of the work-hardening rate and elongation. This phenomenon is known as transformation-induced plasticity (TRIP)14-17.

6.1.3. Mechanical driving force When a stress is applied to the austenite at T1 (Fig. 10), a mechanical driving force U is added to the chemical driving force, ΔG Tγ→α . 1 Stress-induced martensite formation occurs when the stress reaches a critical stress σc. For this stress, the contribution of the mechanical and γ→α the chemical driving force is equal to the total driving force ΔG M s necessary to induce the martensitic transformation: γ→α ΔG M =ΔG Tγ→α + U' s 1

(15)

The mechanical driving force depends on the stress and the orientation of the forming martensite. It can be expressed by:

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Fig. 10. Schematic diagram showing the contribution of the mechanical driving force γ→α . (U’) and the chemical driving force ΔG Tγ→α to the total driving force ΔG M 1

s

U = τγ 0 + σε n

(16)

where τ is the shear stress along the transformation shear direction in the martensite habit plate, γ0 is the transformation shear strain along the shear direction in the habit plane, σ is the normal stress perpendicular to the habit plane and εn is the dilational component of the transformation strain. The applied stress σa can be decomposed into a shear component τ and a normal component σ, which are given by:

τ = (1/ 2) σa (sin 2θ) cos α and

σ = ±(1/ 2) σa (1 + cos 2θ)

(17)

where θ is the angle between the applied stress and the normal to the habit plane and α is the angle between the transformation shear direction and the maximum shear direction of the applied stress in the habit plane (Fig. 11). In Eq. (17) the plus (+) and the minus (-) correspond to a loading in tension or in compression, respectively. From Eqs. (15-17) follows: U = (1/ 2) σa ⎡γ ⎣ 0 (sin 2θ) cos α ± ε n (1 + cos 2θ ) ⎤⎦

(18)

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Fig. 11. Schmid-factor diagram. S is the direction of the shape strain for the martensite, Sm is the maximum shape strain elongation parallel to the habit plane HP, N is the normal to the habit plane.

When the martensitic transformation begins, due to the applied stress, the martensite plates, which are oriented in such a way that the mechanical driving force is maximum, form first. The maximum mechanical driving force is obtained for α = 0 and dU/dθ = 0 and Eq. (18) becomes: U ' = (1/ 2) σ c ⎡γ ⎣ 0 (sin 2θ ') cos α ± ε n (1 + cos 2θ ') ⎤⎦

(19)

where σc is the critical applied stress for the martensitic transformation to begin. The chemical driving force decreases linearly when the temperature increases for T > Ms. The critical applied stress σc is thus expected to increase linearly with the temperature in the same temperature range. This is true up to a certain temperature M sσ (Fig. 12). For T> M sσ , another trend is observed. Let us consider the deformation of the austenite for the temperature T = T2. Under the applied stress σa, the austenite will first deform elastically. Then, when σa = σ1, the plastic

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deformation of the austenite begins. Strain-hardening occurs and the stress increases up to σ2 where the martensitic transformation begins. As it can be seen in Fig. 12, σ2 is lower than the critical applied stress σc predicted by the linear trend. This decrease of the onset stress for martensitic transformation is due to plastic deformation of the austenite. However, the origin of this phenomenon is still unclear. One hypothesis is that the martensite nucleation is strain-induced. Another one is that the stress is locally concentrated at obstacles, such as grain boundaries, because of plastic deformation of the austenite. The local stress reaches the critical stress level σc, promoting the formation of martensite.

Fig. 12. Critical stress to induce martensite formation as a function of the temperature.

6.2. Thermoelastic transformations

6.2.1. General features Let us consider now the cooling of a thermoelastic alloy. During cooling, nucleation and growth of martensite occur. If the cooling is stopped, the growth ceases. However, in this case, if cooling starts again, the growth of the existing martensite continues. When martensite is heated, the

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interface moves backwards and shrinkage of the martensite plates is observed. The parent phase elastically accommodates the martensite plates. No dislocations are observed at the interface and the interface remains glissile. The stored elastic energy constitutes the driving force for the reverse transformation.

6.2.2. Mechanical effects The thermoelastic transformations are associated with remarkable mechanical effects in both the martensite and the parent phases. The most famous one is the shape memory effect (SME). However, this general name covers a large variety of interesting and unusual mechanical behaviours18,19. Some of them will be briefly described in the following section. Several alloys exhibit the SME. Among them, the classical examples are Ni-Ti alloys and their ternary variations20. The SME has also been revealed in Cu-based alloys21, 22 and Fe-base alloys23. In these alloys, the martensite is internally twinned or faulted, which is a sign of the inhomogeneous shear of the crystallographic theory.

6.2.3. The shape memory effect (SME) The shape memory effect is the ability of an alloy to be severely deformed and to return to its original shape after heating. The process to regain the initial shape is closely associated with the reverse transformation of the deformed martensite. The processes involved in this mechanical behaviour can be roughly described as follows (Fig. 13). Upon cooling, the parent phase transforms into martensite plates with different orientations. Usually, a single crystal of parent phase gives 24 orientations of martensite. During deformation, the different orientations of martensite reorganise to give one single orientation, which can be achieved by twinning and movements of certain interfaces. The remaining orientation is the one that will lead to the maximal elongation of the specimen as a whole in the tensile direction. When the material is heated again, the single orientation of martensite gives a single orientation of the parent phase and the material recovers its initial shape.

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The reverse transformation from martensite to the parent phase can be compared to an “unshearing” process.

Fig. 13. Schematic description of the shape memory http://www.cs.ualberta.ca/~database/MEMS/sma_mems/sma.html).

effect.

(From

6.2.4. The two-way shape memory effect The two-way shape memory effect (TWSME) is characterized by a spontaneous deformation of the specimen upon cooling from Ms to Mf and an “undeformation” of the specimen upon heating from As to Af. In order to obtain such an effect, a “training” of the material is necessary. This training can be done following two ways: (i) SME cycling: the specimen is cooled below Mf, deformed and then heated again above Af. The processes described previously occur. This procedure is repeated several times. (ii) SIM cycling: the specimen is deformed above Ms to produce stress-induced martensite (SIM). Then, the specimen is unloaded and the reversal of the SIM occurs. This procedure is repeated several times. After sufficient training, the first step of the TWSME occurs upon cooling where a large proportion of the martensite adopts a preferred

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orientation, leading to a spontaneous strain. The SME cycling seems to be less efficient than the SIM cycling to obtain the TWSME.

6.2.5. Pseudo-elastic effects The pseudo-elastic effects refer to the ability of a material, upon unloading at a constant temperature, to completely recover large strains, well above the elastic limit. Two categories of pseudo-elastic behaviours can be distinguished, depending on the nature of the driving forces and the mechanisms involved: Superelasticity: under the effect of an applied stress above Ms, mechanically elastic martensite is stress-induced and will disappear if the stress is released. Schematic stress-strain curves characteristic of a superelastic behavior are shown in Fig. 14. When the applied stress reaches the critical stress σc, stress-induced martensite is formed (upper plateau). Upon unloading, reversal of the stress-induced martensite occurs. Plates of the parent phase nucleate and grow (lower plateau). The plateau stresses depend on the temperature. The stress-strain curves in Fig. 14 are known as superelastic loops.

Fig. 14. Schematic illustration of the superelastic behavior.

Rubber-like behavior: although the superelasticity leads to a rubberlike behavior, this terminology refers to a mechanical effect that does not involve a phase transformation and is related to the reversible movement of twin boundaries or martensite boundaries.

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7. Summary

One of the purposes of this chapter was to give an overview on diffusionless transformations. A diffusionless transformation (or martensitic transformation) is a solid state transformation which is athermal, diffusionless and involves cooperative movements of the atoms less than one interatomic spacing. The phenomenological theory describing the crystallography of diffusionless transformations is based on the assumption that the macroscopic strain (or shape deformation strain) is associated with an invariant-plane strain. The invariant plane is the undistorted and unrotated habit plane. This macroscopic strain can be divided into three different components: – a Bain strain, which gives the lattice correspondence between the parent and the product phases – a inhomogeneous shear strain which leaves the invariant (or habit) plane undistorted – a rigid body rotation which leaves the invariant plane unrotated The nucleation and growth of martensite occur at a high velocity. The nucleation is heterogeneous and the transformation is mainly initiated at dislocations. Depending on the alloy composition, the temperature and the strain rate, the martensitic phase grows by dislocation slip or twinning. From a technological point of view, diffusionless transformations are of great interest because they are associated with improved mechanical properties such as transformation-induced plasticity or the shape memory effect. References 1. 2. 3. 4. 5. 6.

K. Otsuka, and X Ren,, Mater. Sci. Eng., 275, 89 (1999). P.M. Kelly, and L.R., Francis Rose, Prog. Mater. Sci., 47, 463 (2002). E.C., Bain, Trans AIME, 70, 25 (1924). R. Bunshah, and R.F., Mehl, Transactions AIME, 197 1251 (1953). C.Zener, Elasticity and Anelasticity of Metals, Univ. of Chicago Press, 1948. D.A. Porter and K.E. Easterling, Phase transformations in Metals and Alloys, 2nd Ed., Chapman & Hall, 1992. 7. J.A., Venables, Phil. Mag., 7, 35 (1942).

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8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

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J.W., Brooks, M. H. Loretto, and R.E., Smallman, Acta Metall., 27, 1839 (1979). G.B. Olson, and M. Cohen, Metall. Trans., 7A, 1897 and 1905 and 1915, (1976). J.W Eshelby, Proc. Roy. Soc. A, 241, 376 (1957). A. Kelly, Strong Solids, Clarendon, Oxford, 1966. F. C. Frank, Acta Metall., 1, 15 (1953). D. H. Shin, W. G. Kim, J. Y. Ahn, et al., Mater. Sci. Forum, 503-504, 447 (2006). J. R. Patel, and M. Cohen., Acta Metall., 1, 531 (1953). T. Angel., J. Iron Steel Inst., 177, 165 (1954). P. J. Jacques, Curr. Opin. Sol. State Mater. Sci., 8, 285 (2004). S. Turteltaub, and A. S. J. Suiker, J. Mech. Phys. Sol., 53, 1747 (2005). C. M. Wayman, Prog. Mater. Sci., 36, 203 (1992). R. James, and K. F. Hane, Acta Mater., 48, 197 (2000). K. Otsuka, and X. Ren., Prog. Mater. Sci., 50, 511 (2005). C. Y. Chung, and C. W. H. Lam, Mater. Sci. Eng., 275, 622 (1999). F. C. Lovey, and V. Torra, Prog. Mater. Sci., 44, 189 (1999). N. Stanford, and D. P. Dunne, Mater. Sci. Eng., 422, 352 (2006).

Further reading:

C.M. Wayman, Phase transformation, non diffusive, Chap. 15, in Physical Metallurgy, 3rd Ed., ed. R.W. Cahn and P. Haasen, (North-Holland, Amsterdam), 1983. C.M. Wayman, Introduction to the Crystallography of Martensitic Transformations, Macmillan, New York, 1964. J.W. Christian, Theory of Transformation in Metals and Alloys, Pergamon, Oxford, 1965. J. Perkins, The shape memory effect in Alloys, Plenum, New York, 1975.

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CHAPTER 7

INTERMETALLICS: CHARACTERISTICS, PROBLEMS AND PROSPECTS Gerhard Sauthoff Max-Planck-Institut für Eisenforschung GmbH, 40074 Düsseldorf, Germany E-mail: [email protected] Intermetallics is the short and summarizing designation for intermetallic compounds (IMCs) and phases, which result from the combination of various metals and which form a tremendously numerous and manifold class of materials. Intermetallics with outstanding physical properties led to the development of functional materials in the past. During the last 20 years intermetallics have aroused enormous and still-increasing interest in materials science and technology with respect to applications at high temperatures. Various new structural materials are being developed around the world, in particular in the United States, Japan, and Germany. The present report overviews thevarious intermetallics which have already been selected for materials developments or which have been and still are regarded as promising for materials developments with emphasis on structural materials.

1. Introduction An overview is given on the vast and manifold field of intermetallics which results from a tutorial for students of the 1st European School in Materials Science of the EU Network of Excellence "Complex Metallic Alloys" CMA. The aim is to point out the characteristics, problems and prospects of intermetallic phases and compounds and of related more or less complex alloys by using appropriate examples. Intermetallics have been subject of a long series of excellent reviews which is not to be continued by the present overview. This overview is based on a recent compilation of the knowledge on intermetallics1 and shorter reviews2, 3. Data of intermetallic materials were collected in reference 4. 147

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Applications of intermetallics are handicapped by their brittleness which is the reverse of the high strength and which is not sufficiently understood. The present still continuing enhanced interest in intermetallics results from the need for stronger metallic materials for applications at higher service temperatures e.g. in energy conversion for increasing the thermal efficiency. Thus the following discussion is done from the position of a physicist who tries to understand what mechanisms control the behaviour of the intermetallics in order to explore the possibilities for developing novel structural materials based on intermetallics for high-temperature applications. 1.1. Definition of intermetallics Intermetallics is the short and summarizing designation for the intermetallic phases and compounds which result from the combination of various metals and which form a tremendously numerous and manifold class of materials as will become visible in the following sections. An example is given by the binary Ni-Al phase diagram (Fig. 1) which comprises the phase NiAl with compositions around 50 at.% among other Ni-rich and Al-rich phases. This NiAl phase has a significantly higher melting temperature than the constituent metals Ni and Al indicating a much stronger bonding between the unlike Ni and Al atoms than between the alike Al atoms and Ni atoms. Its crystal structure is known as B2 structure (Strukturbericht designation) which is a bcc structure with atomic ordering and which clearly differs from the fcc structure of the constituent metals. Accordingly there is a simple general definition5, 6: intermetallic phases and compounds are chemical compounds of metals the crystal structures of which are different from those of the constituent metals. Examples are given by the phase diagrams of metal systems. The composition of an intermetallic may vary within a restricted composition range known as homogeneity range. This homogeneity range may be narrow or vanishing as is the case for a line compound and such phases are usually addressed as intermetallic compounds. Phases with a wide homogeneity range are usually addressed as intermetallic phases. Phases may exist only in a restricted temperature range. There are phases which show an order-disorder transition when heated above a critical

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temperature as is exemplified by the Cu3Au phase which is fcc ordered (L12 structure) at lower temperatures and disorders above the critical disordering temperature to form the familiar fcc solid solution. The latter phases are known as Kurnakov phases7. The Russian mineralogist Nikolai Semyonovich Kurnakov (1860 - 1941) in Sankt-Peterburg/Russia and the German chemist Gustav Tammann (1861 - 1938) in Göttingen/Germany initiated intensive and broad-scale scientific work on intermetallics at the beginning of the 20th century from which a large number of directive papers resulted8-13.

Fig. 1. Ni-Al phase diagram2

1.2. Historical Intermetallics have been made use of since the beginning of metallurgy as is visible in Table 1.

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Gerhard Sauthoff Table 1. Some applications of intermetallics4.

Only the applications in the 20th century were based on an understanding of the underlying metallurgical principles. The early applications as coatings made use of the high corrosion resistance of the respective intermetallics. The use of amalgams as dental restoratives is an early example of applications as structural materials. The more recent applications as functional materials rely on advantageous physical properties of particular intermetallics. The relatively high strengths of intermetallics are attractive for applications as structural materials at high temperatures. However, these high strengths are usually accompanied by brittleness, and the respective materials developments in the second half of last century have not yet led to widespread structural applications. Topical intermetallics of present interest are listed in Table 2.

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Table 2. Topical intermetallics of present interest4. Alloy

Majority phase

Application

Functional materials (physical) A15 compounds Cu-Al-Ni, Cu-Zn-Al Fe-Al-Si rare earth magnet materials GaAs

Nb3Sn β-Cu-Al-Ni,β-Cu-Zn-Al Fe2Si SmCo5, Sm2Co17 Sm2Fe17(C,N), Nd2Fe14B

Superconductor shape-memory alloy thermoelectric generator permanent magnet

GaAs

transistor, laser diode, optoelectronic device, solar cell, acoustoelectric device electric heating elements

Kanthal Super, Mosilit Ni-Ti Permalloy Permendur Sendust silicide thin film

MoSi2 NiTi Ni3Fe FeCo Fe3(Si,Al) transition metal silicides,

alloys Terfenol-D Thermoelectrics Functional materials (chemical) M-Cr-Al-Y Pt aluminide rare-earth hydrides

precious metal silicides (Tb,Dy)Fe2 Bi2Te3

silicides Structural materials (under development) Advanced nickel

transition metal silicides

aluminide alloys alpha-2 titanium aluminide alloys B2 nickel aluminide alloys gamma titanium aluminide alloys iron aluminide alloys molybdenum disilicide alloys

NiAl Pt3Al LaNi5

Ni3Al

shape-memory alloy high permeability magnetic alloy high permeability magnetic alloy magnetic head material microelectronics, silicon integrated circuits giant magnetostrictive actuator thermoelectric generator

coating coating hydrogen storage, rechargeable battery coating

high-temperature-wear-resistant components

Ti3Al

jet engine components etc.

NiAl

gas turbine & automotive components

TiAl

aircraft, jet engine, automotive components etc.

Fe3Al, FeAl MoSi2

chemical engineering aircraft, automotive components etc.

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2. Principles There are a huge number of intermetallic phases which differ by stability, crystal structure and by their physical, chemical and mechanical properties. Stability, crystal structure and properties are a function of the particular atomic bonding. However, the principles controlling this function are still unclear as is discussed in the following. If the relation between bonding, phase stability and properties were understood, then it would be possible to predict the properties of given intermetallic phases and alloys which is a prerequisite for specific materials developments. 2.1. Bonding – stability – structure The examples in the various regarded metal systems -e.g. Ni-Al, Cu-Zn, Cu, Au - show that the relation between crystal structure type and atomic properties of the constituent atoms is not a simple one. Thus various criteria have been deduced in the past for correlating structure type and phase type, i.e. for predicting the crystal structure for a given phase or phase group. Compounds of metals from the left-hand side of the periodic table of elements with metals from the right-hand side are known as Zintl phases (named after the German chemist Eduard Zintl (1898 - 1941)). They are characterized by completely filled electron orbitals, in particular by a full octet shell in the normal case14-16. Thus they may be regarded as valence compounds which satisfy the familiar chemical valency rules6, 17. The Zintl phases have crystal structures which are characteristic for typical salts - e.g. NaTl with cubic B32 structure or Mg2Si with cubic C1 structure18 - and therefore ionic bonding is expected. However, all types of bonding - ionic, metallic and covalent - and mixtures thereof have been observed. Accordingly the Zintl phases may be regarded as electron compounds, the crystal structures of which are related to particular valence electron concentrations (average number of valence electrons per atom)17. Electron compounds are best exemplified by the Hume-Rothery phases (named after the British metallurgist William Hume Rothery (1899 - 1968)), the crystal structures of which are related to the valence electron

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Table 3. Some electron compounds (with data from6). VEC

Structure

Phase

3/2

cubic

B2

complex cubic

A13

Zn3Co , Cu5Si

7/4

close-packed hexagonal close-packed hexagonal

A3 A3

Cu3Ga

21/13

complex cubic

D82

FeAl,, CoAl , NiAl Ag3Al

CuZn3 Ag5Al3 Cu5Zn8 Fe5Zn2

concentrations (VEC) of the phases - see Table 3. However, it has been found also in these cases that the bonding is not purely metallic in spite of the metal-like band structure. The bonding in NiAl is primarily covalent with some metallic character and no ionic component19 which can be understood in view of the band structures of Ni, Al and NiAl20. Ab-initio calculations have shown that - strictly speaking - the B2 aluminides are not Hume-Rothery electron compounds21. Obviously the Hume-Rothery rules, which relate crystal structure to valence electron concentration, are very simplifying rules which describe reality surprisingly well. There are crystal structures which have been derived from the size ratios and packing schemes of the constituent atoms. The respective intermetallic phases are known as size-factor compounds or topologically close-packed intermetallics or Frank-Kaspar phases6,17,22,23. The best-known size-factor compounds are the Laves phases with composition AB2 (named after the German mineralogist and crystallographer Fritz Laves (1906 - 1978)). They represent the most numerous group of intermetallics, which crystallize in the closely related hexagonal C14, cubic C15 or hexagonal C36 structures22. These structures result from an ideal packing of spherical atoms with a size ratio of √(3/2) = 1.225. However, nature is more complex as is visualized by Fig. 2. Obviously the ideal size ratio is more the exception than the rule. In addition it is noted that the valence electron concentration VEC seems to control the crystal structure at least in the case of the binary compounds AB2 of transition metals as is indicated by the border lines in Fig. 2a. However, there are ternary Laves phases A(B1-xCx)2 of transition metals which do not adhere to these border lines (Fig. 2b) and in the case of ternary Laves phases A(B1-xCx)2 with Al these border lines are no longer valid (Fig. 2c).

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Radius ratio

Radius ratio

C15 C14

C15 C14 Zr(Co1-xCr)2

Valence electron concentration Valence electron concentration

a

b Radius ratio

C15 C14 a

c

a

a c c b

c b

b

b

Valence electron concentration Fig. 2. Atomic radius ratio as a function of valence electron concentration for various binary Laves phases of transition metals (data from27) with C14 (o) and C15 ( ∆) structure and the phase TaNi2 with C11 structure (the filled symbols indicate a temperature-dependent phase transition whereas indicates conflicting information on the crystal structure) (a), ternary Laves phases of transition metals (the interconnecting continuous / interrupted lines indicate the reported / supposed miscibility ranges, respectively) (b), and ternary Laves phases of transition metals and Al (a: Ti(Fe1-xAlx)2 2 , b: Nb(Co1-x,Alx2 , c: Ta(Fe1-xAlx)2 ) – see text28.

Obviously atomic size ratio and valence concentration are not sufficient for characterizing the complex bonding behaviour of the Laves phases. However, it is noted that the Laves phases show primarily metallic bonding according to the scarce information available5, 17. The complex

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behaviour of the Laves phases is already exemplified by the binary phase diagrams of the systems of Cr-Ti, Co-Nb and Fe-Zr, which all contain all three crystal variants C14, C15 and C36 of the respective Laves phases24. The case of the ternary Laves phase Zr(Fe1-xAlx)2 is particularly intriguing since x can be varied between 0 and 1 which produces a sequence of crystal structure transformations with C15 → C14 → C15 → C1425. This cannot be rationalized by any argument based on atomic size ratios and valence concentrations since these parameters vary monotonously with x. Cooperative work is in progress for improving the understanding of the Laves phase behaviour26. A more specific characterization of the type of bonding should take account of the particular electron distribution of the respective atoms which is reflected in a simple way by the positions of the atoms in the periodic table. Accordingly David Pettifor numbered the atoms in the periodic table following the increase in the number of electrons in the electron shells to receive the so-called Mendeleeff numbers of the atoms29. Using the Mendeleev numbers as coordinates of binary intermetallic phases, Pettifor constructed structure maps, in which the phases with common crystal structure were located in restricted areas. However, the areas were not always cohesive, the borders were not well defined, areas overlapped, and there were phases in “wrong” areas. Thus the characterization of electronic structure by Mendeleev numbers is not sufficient and the predictive power of such structure maps is restricted. It has to be concluded that the bonding character and crystal structure of an intermetallic phase can be predicted only on the basis of a quantum-mechanical ab-initio calculation17, 30, 31. Much progress has been made in this respect32, 34 and for various important phases with not too complicated structures the crystal energy has been calculated as a function of lattice structure in order to determine the phase stability35-46. However, these calculations are very time-consuming even in simple cases - i.e. for small unit cells with few atoms - and more progress is necessary in the future with respect to computing power and understanding in order to give guidance to practical materials developments on the basis of multinary phases with less simple crystal structures.

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The challenge of this problem is best exemplified by the case of the three simple binary phases FeAl, CoAl and NiAl which all crystallize in the simple bcc-ordered B2 structure and show similar melting temperatures in similar phase diagrams. The behaviour of these phases is well known, and in particular it is known that the most pronounced behaviour difference is observed for the transition from FeAl to CoAl. However, corresponding ab initio calculations showed that this FeAl-CoAl transition does not much affect the character and directionality of bonding as indicated by the valence electron density difference plot, whereas a distinct directional bonding increase is produced by the transition from CoAl to the most similar NiAl47. A common crystal structure is obviously not a sufficient criterion for similar behaviour. The problem is still more aggravated by the fact that the stability of a crystal structure may sensitively be affected by the presence of interstitial impurities as is demonstrated by Table 4. The expected normal case of solution of impurities without affecting the crystal structure is shown by the fcc-ordered Ni3Al phase which dissolves appreciable amounts of carbon without change of crystal structure. In contrast to this, the dissolution of less than 1 C atom per unit cell in the bcc-ordered Fe3Al phase produces a change of crystal structure to the fcc-ordered L12 structure with the C atom in the unit cell centre which is designated as L1’2 structure, i.e. this phase is also regarded as complex carbide with the perovskite-type structure48, 49. The most intriguing case is the case of the M5Si3 silicides with M = V, N, Ta, the crystal structure of which is affected even by much less than 1 C or N atom per unit cell50. As to the relation between crystal structure and properties, it is well known that there is a correlation of crystal symmetry and unit cell size on the one side and plastic deformability - i.e. brittleness - on the other side: the larger the unit cell and the less symmetric the lattice is, the less deformable the phase is. However, the correlation is a very loose one, and the plot of brittle-to-ductile transition temperatures as a function of the number of atoms per unit cell shows a huge scatter51. From the present discussion follows that the intermetallics do not form a homogeneous group of materials at all. Instead the term intermetallics comprises a huge variety of phases which differ drastically with respect to bonding, crystal structure and properties. Thus intermetallics cannot be

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discussed generally with respect to all intermetallics, but can be discussed only by referring to specific groups of intermetallics. In view of the complexity of the classification of intermetallics, intermetallic phases are often grouped according to more practical criteria which refer to similarities in behaviour. Table 4. Crystal structures of phases without and with interstitial impurities (data from18). Phase Pearson Symbol cP4 (filled) cP4

Structure StructureBericht Ll2 Ll2

Fe3Al Fe3AlC<1 (complex carbide)

cF16 filled cP4

DO3 Ll’2 = E21

BiF3 CaTiO3 (Perowskite)

(V,Nb,Ta)5(Si,Ge)3 (V,Nb,Ta)5(Si,Ge)3(C,N)<<1

tI32 filled hP16

D8m

W5S3 Mn5S3 (C/N-stuffed)

Ni3Al Ni3AlC<1

Structure type AuCu3 AuCu3

2.2. Energy – properties It is well known that basic properties of materials scale with the melting temperature which is made use of by Ashby’s deformation mechanism maps52. Fig. 3 shows that the melting temperature is proportional to the formation enthalpy of some intermetallic phases with B2 structure whereas the deviating behaviour of the phase TiAl with L10 structure indicates another relation. The observed proportionality is not trivial since both properties are controlled by energy differences: the melting temperature at which the Gibbs free energy of the phase in the liquid state is equal to that in the solid state, and the formation enthalpy is the difference between the phase enthalpy and the sum of the enthalpies of the constituent metals in the solid state, i.e. it is not the total enthalpy of phase formation from the gaseous metal atoms.

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Fig. 3. Melting temperature as a function of formation enthalpy for various intermetallic phases2.

Figure 4 shows data for the activation energy of diffusion as a function of the total phase enthalpy of formation from the gaseous state corresponding to the sublimation enthalpy for the fcc metals Al and Ni and for the nickel aluminides Ni3Al and NiAl with L12 and B2 crystal structures respectively. Obviously the data indeed indicate a close correlation in spite of the varying crystal structures. Ni3Al NiAl Ni

Al

Fig. 4. Activation energy of diffusion as a function of total phase enthalpy for various metals and phases2.

This correlation is valid even in a more general context as is visible in Fig. 5 which shows data for a larger variety of metals and phases not only

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for activation energies of diffusion, but also for activation energies of creep deformation which is controlled by diffusion. Clearly the correlation of the activation energies and the sublimation enthalpies of the materials holds for the metals with the various crystal structures and the Laves phases with hexagonal C14 and cubic C15 structure. The deviations of some creep activation energy data are supposed to be due to temperaturedependent changes in the respective complex creep mechanisms.

+

+++ + + + + +

Fig. 5. Apparent activation energies of creep (filled symbols) for Laves phases with C15 structure (♦: own work, ■: other work) and Laves phases with C14 structure (▲) in comparison to activation energies of diffusion for a Laves phase with C15 structure (), for a Laves phase with C14 structure (∇) and for pure metals with fcc structure (), bcc structure (+) and hcp structure (∆) as a function of the respective sublimation enthalpies53.

Already in the past cohesive energies of crystals have been calculated theoretically and thermodynamics allows for deriving the elastic moduli from the crystal energy. A striking example for this is given by Grüneisen's first rule (named after the German physicist Eduard Grüneisen (1877 - 1949)), according to which the elastic bulk modulus K is proportional to the lattice energy U and which relies on a simple

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Fig. 6. Elastic bulk modulus K (multiplied by molar volume Ω) as a function off sublimation enthalpy for various C15 Laves phases (©) in comparison to alkali metals (+). alkali halides (O). transition metals (□) and alkaline earth metals (Δ)53.

modelling of the attractive and repulsive interatomic forces: K = (m n/6Ω) U, where the parameters m and n characterise the energy potential of the atoms and Ω is the molar volume54-56. Fig. 6 shows data for the bulk modulus, which is not affected by the particular crystal structure, of various metals and phases as a function of the sublimation enthalpy which is supposed to substitute for the cohesive energy of the respective material. Clearly Grüneisen's first rule is fulfilled by the alkali and alkaline earth metals and alkali halides with data on straight lines through the origin. The data for the C15 Laves phases again indicate a linear relationship with, however, an abscissa section, i.e. the interpolating straight line is not directed at the origin which does not correspond to the simple Grüneisen's rule. More data are available for Young’s modulus and in particular for the intermetallic phases of more practical interest as are shown in Fig. 7.

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Fig. 7. Young's modulus as a function of total phase enthalpy at room temperature for the fcc metals Al and Ni, the fcc-ordered Ni3Al with L12 structure, the bcc-ordered FeAl, NiAl and CoAl with B2 structure, and the cubic Laves phases CaAl2, YAl2, LaAl2, NbCr2, ZrCo2 and HfCo2 with C15 structure2.

Clearly the data for the cubic C15 Laves phases show the same behaviour as the bulk modulus data, i.e. they again indicate a linear relationship with abscissa section. The data for the fcc metals Al and Ni are near the corresponding straight line whereas the data for the B2 phases deviate with a large scatter. It is noted that part of the scatter may be due to experimental problems in view of the sensitive dependence of Young’s modulus on the sample quality. In any case it is concluded that the cubic C15 Laves phases show a behaviour which is near that of pure metals in contrast to the cubic B2 phases with a more complex behaviour. In view of structural applications, strength in general and yield stress in particular are of interest, which usually scale with the elastic constants, i.e. with Young’s modulus or the shear modulus. Fig. 8 shows yield stress data at 1100 °C again as a function of the sublimation enthalpy for some Laves phases which are formed as compounds of the transition metals Zr, Nb, Ta from groups IVb and Vb of the periodic table and Fe and Co from group VIII. Obviously the data can indeed be interpolated by parallel straight lines, one for the Laves phases with Co, which all show the cubic C15 structure, and the other for the Laves phases with Fe, of which ZrFe2 again shows the C15 structure whereas the other two phases NbFe2

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Fig. 8. Compressive 0.2% yield stress at 1100 °C as a function of sublimation enthalpy Hsubl for various AB2 Laves phases with B-element Fe or Co with hexagonal C14 structure (Ο) or cubic C15 Structure (∆)53.

and TaFe2 show the hexagonal C14 structure. This demonstrates that common crystal is only a secondary symptom of similarity. A rather simple, but complex parameter for characterising the strength of a material is the hardness at room temperature. Figure 9 shows data for various Laves phases with C14 and C15 structure, respectively, as a function of the sublimation enthalpy. The data for the six Laves phases of Fig. 8 may again be interpolated by straight lines, which are no longer parallel. However, the additional data for other Laves phases clearly indicate that there is no simple relationship, i.e. there is only a loose correlation of the hardness data with the sublimation enthalpies with a large scatter of data. It is concluded from this discussion that there is a strong correlation between total phase enthalpy and activation energy of diffusion (and creep) as well as elastic properties, yield stress and less for hardness only as long as the phases are sufficiently similar. This requested similarity is controlled by the bonding of the constituent metal atoms and cannot by

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Fig. 9. Room-temperature hardness data (full symbols: own data, open symbols: other data) for various transition-metal Laves phases with hexagonal C14 structure (circles) or cubic C15 structure (triangles) as a function of sublimation enthalpy53.

characterised by simple easily accessible parameters. In particular, a common crystal structure is no reliable indication of similarity. Thus any discussion of the characteristics, problems and prospects of intermetallics has to be referred to specific groups of intermetallics. Whether a phase belongs to a particular group, depends on criteria which are related to the behaviour of the group members and are specific to the particular group. Accordingly the following discussion refers only to some intermetallics in metal systems which are of interest for possible hightemperature applications. 3. Practice Subjects of the following discussion are intermetallic phases with Al which have been considered since long as candidate phases for developing novel high-temperature materials. The reason for this is that such phases

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Gerhard Sauthoff

with sufficient contents of Al can form protective adherent oxide scales of α-Al2O3 at temperatures above 1000 °C which provide a sufficient oxidation resistance and is a prerequisite for hightemperature application. Phases with Cr are not useful for applications above 1000 °C since they form scales of Cr oxides which evaporate above 1000 °C. 3.1. Titanium aluminides & related phases 3.1.1. Ti3 Al The Ti-rich aluminide Ti3Al with hexagonal D019 structure, which is known as α2 phase, was selected already in the 1950s as candidate phase for materials development because of its low density. Figure 10 shows the temperature dependence of the strength and ductility of this phase. Obviously this phase is brittle up to a brittle-to-ductile transition temperature (BDTT) of about 600 °C. Only above this temperature a yield stress can be determined, which usually is the 0.2 % proof stress, i.e. the stress for 0.2 % plastic strain. Below the BDTT only a fracture stress was determined without prior plastic deformation. The observed deformability

Fig. 10. Tensile strength and plastic deformability as a function of temperature for single-phase polycrystalline stoichiometric Ti3Al: _____ fracture stress (below 600 °C) or ultimate tensile strength, _ . _ yield stress, __ __ __ apparent deformability including microcracking, ----- estimated deformability without microcracking2, 57.

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was accompanied by microcracking which illustrates the difficulties of determining the plastic ductility in a reliable way. A necessary condition for plastic ductility is given by the von Mises criterion (named after the Austrian mathematician, engineer and physicist Richard Edler von Mises (1893 - 1953)), according to which 5 independent slip systems are necessary for homogeneous plastic deformation58. In the D019 structure of Ti3Al 5 independent slip systems can be activated for the movement of dislocations59 which would satisfy the Von Mises criterion. However, single-crystal studies on Ti3Al have shown that the yield stresses for the different slip systems are widely different60, 61, and thus not all 5 slip systems are activated during the deformation of polycrystalline Ti3Al. Only little tensile ductility has been found for basal slip, whereas prism slip results in very large tensile elongations62. Since there is no stress-relieving twinning as in hexagonal metals, the insufficient number of slip systems - together with the observed planarity of slip - leads to strain incompatibilities and stress concentrations at grain boundaries from which cleavage fracture results59, 63, 64. Various materials developments on the basis of Ti3Al were carried out in order to increase both the strength and plastic deformability64, 65. The most advanced Ti3Al alloys were obtained through alloying with large amounts of Nb as is exemplified by Fig. 11. Ti3Al -base alloys with engineering significance are known as α2 alloys and super- α2 alloys with 10-30 at.% Nb64, 65. Such large amounts of Nb lead to multiphase alloys which additionally contain β-Ti in the disordered state with fcc structure or in the ordered state with B2 structure and/or the orthorhombic O phase. The deformation behaviour of these Ti3Al -Nb-base alloys and in particular the balance of strength and ductility is controlled by the nature and distribution of the various phases in the alloys, i.e. by the multiphase microstructure which is produced by the thermal and mechanical processing of the alloys66. Ti3Al-Nb-base alloys were used to produce and test gas turbine components. In particular, a combustor swirler was precision cast for a demonstrator engine, and a high pressure compressor casing was tested under service conditions67, 68. Problems are - apart from brittleness - the environmental effects. Oxidation occurs at high temperatures in oxidative

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atmospheres and the growing oxide scales are not protective since the Al activity in these alloys is not sufficiently high for the formation of a protective αAl2O3 scale69. In addition, oxygen diffuses into Ti3Al as solute due to a comparatively high solubility for oxygen, which leads to further embrittlement and crack formation at the surface59, 65, 70, 71. Consequently, the interests have shifted to the other titanium aluminide TiAl with higher Al activity and still lower density, which is discussed in the following. Ti-24Al-11Nb

UTS Ti3Al

elongation Ti-24Al-11Nb Ti3Al

Fig. 11. Temperature dependence of the ultimate tensile strength UTS and the tensile and respectively), In elongation of Ti-24%Al-11at%Nb ( comparison to that of Ti3Al and , respectively)2, 65.

3.1.2. TiAl The titanium aluminide TiAl - known as γ phase - crystallizes with the tetragonal L10 structure which is basically an fcc lattice with atomic ordering and tetragonal distortion. Ist properties differ positively from those of the Ti3Al and compare favourably with those of the Ni-base superalloys as is illustrated by the data in Table 5. The titanium aluminide TiAl - known as γ phase - crystallizes with the tetragonal L10 structure which is basically an fcc lattice with atomic

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Table 5. Properties of alloys based on the titanium aluminides Ti3Al and TiAl as compared to conventional titanium alloys and nickel-base superalloys59, 64, 67, 72, 73. Property structure density [g/cm3] thermal conductivity [Wm-1K-1] Young’s modulus [GN/m2] at room temperature yield strength [MN/m2] at room temperature tensile strength [MN/m2] at room temperature temperature limit [°C] due to creep temperature limit [°C] due to oxidation tensile strain to fracture [%] at room temperature tensile strain to fracture [%] at high temperature fracture toughness Kk [MN/m3/2] at room temperature

Ti-base Ti3Al-base Hexagonal/ D019/bcc/B2 bcc 4.5-4.6 4.1-4.7 21 7

TiAl-base Ll0/D019

Superalloys fcc/Ll2

3.3-3.9 22

7.9-9.1 11

95-115

100-145

160-180

195-220

380-1150

700-990

400-650

250-1310

480-1200

800-1140

450-800

620-1620

600

760

1000

1090

600

650

900

1090

10-25

2-26

1-4

3-50

12-50

10-20

10-60

8-125

high

13-42

10-20

25

ordering and tetragonal distortion. Its properties differ positively from those of the Ti3Al of the Ni-base superalloys as is illustrated by the data in Table 5. The decisive features for the application of TiAl alloys are the density, temperature limit due to oxidation and the fracture toughness. Figure 12 shows the temperature dependence of the strength and ductility of the TiAl phase. As in the case of the Ti3Al alloys, the phase is brittle up to a BDTT of about 700 °C, above which plastic deformation through thermally activated dislocation motion occurs. The materials developments on the base of TiAl for improving strength and ductility rely primarily on a reduction of the Al content which produces two-phase TiAl alloys with Ti3Al as second phase.

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This effect is clearly visible in Fig. 13 which shows the variation of hardness and ductility with the Al content with 50 at.% Al being the lower limit for the Al content of single-phase TiAl. fracture stress 0.2% yield stres s

UTS

elongation

Ti3Al + TiAl

TiAl

as c

ast

annealed

elongation in arbitrary units

Vickers hardness in kg/mm2

Fig. 12. Tensile strength i.e. fracture stress, ultimate tensile strength (UTS), and 0.2% yield stress- and plastic deformability-i.e. tensile elongation- as a function of temperature for single-phase-polycristalline TiAl with 54at.%Al2, 74 .

Al content in at.%

Fig. 13. Vickers hardness and tensile elongation at room temperature as a function of Al content for single-phase and two-phase TiAl alloys2, 73 .

d i ne gr a

elongation in %

fine duplex

am e lla r

YS

coarse lamellar

fine

duplex

169

UTS

rse l

TMP p rocess ed

co a

strength in MN/m2

Intermetallics: Characteristics, Problems and Prospects

El coarse grained

Temperature in °C

Fig. 14. Temperature dependence of ultimate strength (UTS), tensile yield strength (YS), brittle-to-ductile transition temperature (BDTT) and tensile elongation (EL) for two-phase TiAl alloys with various microstructures produced under various processing conditions, in particular thermomechanical processing (TMP)2, 76.

The balance of strength and ductility is then controlled by the amounts and distribution of phases which is optimized by appropriate thermal and mechanical pre-treatments. Representative data are shown in Fig. 14 which illustrates the variation of properties as a function of microstructure variation. The balance of strength and ductility is then controlled by the amounts and distribution of phases which is optimized by appropriate thermal and mechanical pre-treatments. Representative data are shown in Fig. 14 which illustrates the variation of properties as a function of microstructure variation. The reduced Al content - typically 48 at.% or less - affects the oxidation resistance since at least 50 at.% Al are necessary for a sufficiently high Al activity for α-Al2O3 scale formation69, 75. In addition, oxidation at the surface reduces the Al content of TiAl with formation of a layer of brittle Ti3Al - usually with cracks. Below these scales and surface layers internal oxidation is observed with oxide dispersions in the bulk69. The worldwide developments of TiAl-base alloys are much advanced and components have been produced for demonstrating the feasibility of

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using such alloys for various applications. A sheet technology was developed for producing TiAl alloy sheet material through isothermal forging and subsequent modified pack rolling, which can be formed superplastically for various kinds of application77. However, the sheet production has been ceased because of presently insufficient market opportunities. Gas turbine airfoils have been produced and successfully tested under service conditions78, 79 as well as automotive engine valves79-82. Passenger cars have been equipped with turbocharger wheels, the performance of which excels that of conventional superalloy wheels because of the low density of the used TiAl alloy83, 84. Present ongoing developments, which make use of the effects of large alloying additions of Nb85 and/or special processing schemes86-88, are aimed at further improvements of strength, oxidation resistance and ductility. 3.1.3. TiAl3 The trialuminide TiAl3 crystallizes with the tetragonal D022 structure which is common to various other trialuminides. The high Al content is advantageous not only with respect to lowered density, but also with respect to enhanced oxidation resistance. However, TiAl3 is a line compound, i.e. oxidation with Al consumption leads to decomposition. Ternary alloying with V, Nb, or Ta widens the homogeneity range of TiAl3 which opens possibilities for alloy development. Other ternary trialuminides with compositions corresponding approximately to Al66Ti25M9 with M = Cr, Mn, Fe, Co, Ag, Cu show the cubic L12 structure89. Since a reduced brittleness is expected for cubic structures with sufficient numbers of slip systems for plastic deformation in accordance with the von Mises criterion58, these trialumindes have been regarded as promising for materials developments and have been subject of extensive studies90-92. Problems are the restricted areas of homogeneity leading to multiphase alloys in most cases and the still prevailing brittleness.

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3.2. Nickel aluminides & related phases 3.2.1. Ni3Al The nickel aluminide Ni3Al with cubic L12 structure is the familiar strengthening γ’ phase in the Ni-base superalloys93 and thus is the best known and most studied intermetallic. Its enormous strengthening effect in the Ni-base superalloys results from its anomalous temperature dependence of the yield stress, i.e. the yield stress of Ni3Al increases with increasing temperature up to a critical temperature, and only above this temperature the familiar normal softening with rising temperature occurs. The reason for this anomaly is the Kear-Wilsdorf mechanism94: instable dislocations, which are the only mobile ones at lower temperatures, cross-glide to become stable and immobile. Monocrystalline Ni3Al is ductile, but polycrystalline Ni3Al is brittle, and the reasons for this embrittlement are not yet clear. However, the microalloying of substoichiometric Ni3Al with B results in a ductile polycrystalline Ni3Al alloy95. This “ductilization” of Ni3Al through the boron effect provoked worldwide research and development activities. Nevertheless the reasons for this boron effect are still in discussion. The centre of activities was in the Oak Ridge National Laboratory (ORNL) where it was found that the optimum balance of yield stress and ductility is obtained with a B content of the order of 500 wt.ppm for Ni3Al with 24 at.% Al96. The subsequent successful ORNL development lead to novel Ni3Al alloys for high-temperature applications which are known as advanced aluminides and which compare favourably with the familiar nickel-base superalloys97. Remaining problems are the insufficient oxidation resistance because of the too low Al activity98 the hot embrittlement at temperatures of the order of 800°C99, which is prohibitive for hot forming, and the environmental embrittlement which is a hydrogen embrittlement resulting from the presence of moisture in the atmosphere100. In spite of the problems, Ni3Al alloy wire has been cast and roller hearth plate hardening furnace rolls have been produced101. Apart from “ductilizing” Ni3Al through the boron effect, stoichiometric Ni3Al foil could be cold-rolled using a pure directionally solidified Ni3Al alloy which was strongly textured and nearly monocrystalline102.

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3.2.2. NiAl NiAl, which was already mentioned in the Introduction because of its congruent melting temperature being higher than those of the constituent metals, has a wide homogeneity range, which offers possibilities for studying the effects of off-stoichiometry on the mechanical behaviour, and there are manifold possibilities for ternary alloying, which are attractive for alloy development103. The effects of off-stoichiometry and of ternary alloying are illustrated by Fig. 15. Deviations from stoichiometry produce constitutional defects, which are primarily anti-site atoms for Al contents below 50 at.% and vacancies for Al contents above 50 at.%104-106. Such point defects are immobile at low temperatures thus being obstacles for moving dislocations, i.e. they contribute to hardening and increase the yield stress at low temperatures as is visible in Fig. 15. At high temperatures the point defects become mobile through thermal activation thus enhancing the diffusibility107, 108, which contributes to softening109. The dissolved ternary Fe atoms in the stoichiometric NiAl are dislocation obstacles and increase the yield stress at all temperatures.

Fig. 15. Yield stress (0.2% proof stress in compression at 10-4 s-1strain rate) as a function of temperature for binary and ternary NiAl phase, i.e. stoichiometric NiAl (circles), stoichiometric (Ni0.8Fe0.2)Al (squares) and off - stoichiometric (Ni1.0Fe0.2)Al0.8 (triangles)2, 110.

Plastic deformation at high temperatures under constant stress occurs through continuous creep with thermally activated non-conservative

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dislocation motion which is controlled by diffusion. Indeed the creep resistance, which is the stress for a given secondary creep rate, of NiAl and similar B2 phases was found to be inversely proportional to the effective diffusion coefficient as demonstrated by the data in Fig. 16. The remarkable fact is that a partial substitution of Fe for Ni in NiAl can increase the creep resistance through reduction of the diffusion coefficient111 though the FeAl phase is much less creep resistant than NiAl. NiAl is most attractive for high-temperature applications because of its excellent oxidation resistance through formation of a protective α-Al2O3 scale due to the high Al activity118, 119. Besides the sufficiently high Al activity, a sufficiently high Al diffusibility is a prerequisite for a high oxidation resistance which is the case for NiAl because of its open B2 structure. Thus NiAl compares favourably in particular with Ni3Al which has only a too low Al activity and a much lower Al diffusibility due its fccordered, i.e. close-packed L12 structure.

Fig. 16. Creep resistance of binary and ternary stoichiometric B2 aluminides at 900°C (in compression with 10-7 s-1strain rate) as a function of dfiffusion coefficient found in the literature or estimated from available data 2, 112-117.

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Fig. 17. Transmission electron microscopical micrograph of a two-phase Ni-44at.%Fe28at.%Al with substoichiometric B2 (Ni.Fe)Al matrix and about 15vol.% bcc Fe precipitate particles after compressive creep at 900°C120.

Fig. 18. Secondary creep rate at 900°C as a function of compressive stress for various NiAl alloys: two-phase Ni-44at.%Fe-28at.%Al with substoichiometric B2(NiFe)Al matrix and about 15% vol. bcc Fe precipitate particles (alloy 4), single phase Ni-10at.%Fe-40at.%Al i.e. substoichiometric B2(NiFe)Al (alloy 5) corresponding to the matrix in alloy 4, and single phase Ni-10at.%Fe-50at.%Al i.e. substoichiometric B2(NiFe)Al (alloy 6)120, 121.

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The open B2 structure, which is advantageous for oxidation resistance, is disadvantageous for structural applications at high temperatures since it enables faster diffusion and thereby creep, i.e. the creep resistance of NiAl is comparatively low. As in conventional alloys, the creep resistance can be increased through precipitation hardening. Even precipitate particles, which are softer than the matrix, are obstacles for moving dislocations see Fig. 17 - and improve the creep resistance as is illustrated by Fig. 18. Clearly the precipitated bcc Fe particles shift the creep rate curve to higher stresses corresponding to a higher creep resistance. However, the matrix phase is off-stoichiometric due to the thermodynamic local equilibrium between precipitate particles and matrix phases, and the decrease of creep resistance due to off-stoichiometry is larger than the increase due to precipitate hardening. A NiAl-based materials development aiming at applications in flying gas turbines has used the precipitation of the Heusler phase Ni2AlHf (the Heusler phases are named after the German mining engineer and chemist Friedrich Heusler (1866 - 1947)) and the G phase Ni16Hf6Si7 for hardening to provide a sufficient high-temperature strength122. Turbine blades have been produced successfully and tested under service conditions. A remaining problem is the brittleness with a BDTT, which increases with increasing deformation rate. Thus particles, which impinge a NiAl alloy blade at very high rates as is characteristic for erosion tests, produce impact damage with brittle fracture even at the very high service temperatures, for which the conventional laboratory tensile tests with deformation rates of about 10-4 s-1 predict ductile behaviour. This insufficient impact damage resistance precludes any applications as turbine blade materials, i.e. the respective developments are now focussed at less demanding applications, e.g. combustor liner panels. Another NiAl-base alloy development has relied on hardening through precipitation of the ternary Laves phase TaNiAl with hexagonal C14 structure123-128. The obtained NiAl-base Ni-Al-Ta-Cr alloy with 45 at.% Al, 2.5 at.% Ta and 7.5 at.% Cr, which is known as alloy IP75, contains a comparatively coarse distribution of Laves phase particles and a fine distribution of Cr-rich particles in a NiAl matrix. The high temperature strength and creep resistance as well as the oxidation resistance allows for applications up to about 1200 °C. The alloy is brittle, but it is

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shock-resistant in spite of the brittleness. Both ingot metallurgy and powder metallurgy processing has been established. Combustor liner model panels have been precision cast and tested successfully. 3.3. Iron aluminides and related phases Fe-Al alloys including the iron aluminides are most attractive for high-temperature applications because of their excellent hot-gas corrosion resistance129, 130. Fe-Cr-Al electric heating elements are well known131 and the application of FeCrAl alloys in automotive catalytic converters is in development 132. The iron alumindes Fe3Al with bcc-ordered D03 structure and FeAl with bcc-ordered B2 structure represent a special case since in general they are not separated by a two-phase field in the Fe-Al phase diagram133-135 and this is also true with respect to the Al-poorer bcc disordered Fe-Al solid solution, i.e. the disordered can transform with rising Al content into Fe3Al and then into FeAl through 2nd order phase transitions. The ductility of the disordered Fe-Al solid solution decreases with increasing Al content136 and the aluminides Fe3Al and FeAl are known as being brittle. However, the yield strength and the ductility of the latter ordered phases varies in a complex way with degree of ordering, i.e. there is a relative ductility maximum with about 8 % elongation at about 28 at.% Al137. The yield strength depends very sensitively on the prior heat treatment138 which is in contrast to the usually observed behaviour of intermetallics. The reason for this special behaviour is the presence of excess vacancies which are easily created at high temperatures by thermal activation - the more the higher the temperature is - because of the very low enthalpy of formation and which anneal out at lower temperatures only very slowly because of their high enthalpy of migration138, 139. Hardness and yield strength increase with increasing vacancy concentration, and the annihilation of the excess vacancies needs long annealing times at low temperature, i.e. several days at 400 °C138. Apart from the vacancy effect, the deformation behaviour is affected by the presence of even small amounts of impurities. As an example, only 50 wt.ppm carbon are sufficient for precipitating the complex carbide Fe3AlC

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with perovskite-type structure140. In spite of the various problems FeAl sheet of 0.2 mm thickness could be manufactured powder-metallurgically by roll compaction, cold rolling and annealing with final coiling141. As in the case of the NiAl alloys, there is a demand for increased high-temperatue strength and creep resistance in view of structural applications at high temperatures. This can be accomplished by ternary alloying in order to make use of solid-solution hardening, hardening through enhanced ordering at higher temperatures, precipitation hardening and/or combinations of the various possibilities142. Of particular interest is the case of Fe3Al with carbon as ternary alloying element143. Fe3Al has only a very low solubility for C at low temperatures140, i.e. excess C is precipitated as fine Fe3Al-C particles first on grain boundaries and second within the grains which produces precipitation hardening. Fe3Al and Fe3AlC are separated by an extended two-phase field in the isothermal section of the Fe-Al-C phase diagram144, i.e. the Al content of the Fe3Al can be varied to vary the degree of order and the amount of Fe3AlC can be varied between 0 and 100 %. The yield stress at temperatures below 500 °C increases with increasing C content, i.e. increasing volume fraction of Fe3AlC, for alloys with about 25 at.% Al corresponding to the stoichiometric composition and about 28 at.% Al, which is expected, whereas the yield stress for the alloys with about 23 at.% Al is not much affected by the variation of the C content. However, the ductility also increases with increasing C content according to the data in Fig. 19, which is in contrast to expectation. The reasons for these surprising effects are not yet understood and need further investigation. Another case of particular interest is the ternary Fe-Al-Ta system. There is an equilibrium between the Fe-Al solid solution including the disordered bcc phase, the D03-ordered Fe3Al phase and the B2-ordered FeAl phase on the one hand and the ternary Laves phase Ta(Fe,Al)2 with hexagonal C14 structure on the other hand, and in both the Fe-Al solid solution and the Laves phase the Al content can vary between 0 and 50 at.% according to the available phase diagram information28, 145-147 (it is noted that the additional equilibrium between FeAl and a µ’ phase, which was reported in148, could not be validated) - see Fig. 20. Accordingly such Fe-rich Fe-Al-Ta alloys with precipitated Laves phase in a Fe-Al matrix

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Fig. 19. Tensile ductility at room temperature of various Fe3Al-C alloys with precipitated Fe3AlC as a function of the integral C content of the alloys (full symbols: as cast, open symbols: after homogenisation at 1200 °C for 24 h)143.

Fig. 20. Tentative isothermal section at 800 °C of the Fe-rich corner of the Fe-Al-Ta system146, 147.

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are promising for alloy developments for high-temperature applications since strength, ductility, corrosion resistance and the balance of these can be controlled - and optimized with respect to the envisaged application by controlling the Al content, which controls the atomic ordering in the matrix, and the Ta content, which controls the amount of Laves phase. Yield stress and ductility have been studied with systematic variation of the Al and Ta contents145, 148. The microstructure studies of the Laves phase-strengthened Fe-Al-Ta alloys have revealed that the expected exclusive Laves phase precipitation occurs in the Fe-Al-Ta alloys with about 25 at.% Al only at temperatures of 800 °C and above first at grain boundaries and second in the grains whereas at lower temperatures an unexpected Heusler phase Ta2FeAl with L21 structure is precipitated homogeneously as fine particles with cubic orientation and only later Laves phase precipitate particles are observed see Fig. 21146, 147. The still open question is whether this Heusler phase is a stable phase or a metastable phase which is kinetically favoured. In any case the finely

Fig. 21. Tentative temperature-time-transformation diagram for Fe-25at.%Al-2at.%Ta alloys (if not stated otherwise) with data at 700 °C and 800 °C referring to annealing and data at 650 °C and 750 °C referring to creep146, 147.

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distributed Heusler phase particles contribute significantly to improving the creep resistance of these Fe-Al-Ta alloys and this effect is larger than that of the Laves phase precipitate as is visible in Fig. 22. The comparison of the data for the present Fe-Al-Ta alloys with those for the heat resistant martensitic/ferritic 9-12%Cr steels in Fig. 22 clearly shows that the Fe-Al-Ta alloys allow for surpassing the creep resistance of the established martensitic/ferritic 9-12%Cr steels, which are not yet fit for application at 650 °C in conventional power plants149.

Fig. 22. Minimum compressive creep rate (with stepwise loading corresponding to secondary creep) at 650 °C as a function of applied stress for various Fe-Al-Ta alloys in comparison to data for some martensitic/ferritic 12wt.%Cr model steels (DT4-3, DT4-4, DT4-1149, 150) and the martensitic/ferritic 9wt.%Cr heat-resistant steel P92149 (left) and micrograph (transmission electron microscopy) of the microstructure of the Fe-25at.%Al-2at.%Ta alloy with precipitated Heusler phase (L21 structure) in the bcc Fe-Al matrix (A2 structure) after creep for 600 h at 650 °C146, 147, 151.

4. Conclusions The intermetallics - i.e. the intermetallic phases and compounds constitute a multifaceted group of materials, the physical, mechanical and chemical properties of which vary within very wide limits. The atomic bonding is complex, i.e. in general there is a mix of metallic bonding,

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covalent bonding and ionic bonding. The principles, which control the crystal structure of a particular phase, its stability, its physical and mechanical properties and its behaviour as a function of stress and temperature, are not yet sufficiently understood for predicting the properties of a given phase as a function of its constitution. This problem is most clearly expressed by the brittleness which is little understood even for conventional metallic alloys. Thus intermetallics are a huge challenge to solid state physicists and chemists. It is hoped that first principles calculations will be possible for phases with crystal structures with larger numbers of atoms per unit cell including point defects and impurities as a function of temperature which allow for calculating theoretically the thermodynamic and elastic properties. It is further hoped that the modelling of the microscopic fracture mechanics allows for rationalizing the conditions of brittle fracture for a phase that is characterized by crystal structure, constitution and microstructure apart from stress and temperature. Much more theoretical and experimental work is necessary for fulfilling the hopes. As to practice, there are various more or less advanced successful developments of novel intermetallic alloys for structural applications at high temperatures. The usual brittleness makes processing difficult and handicaps any general application of intermetallic alloys. Instead the task is to define a specific application, for which an appropriate promising intermetallic alloy is available and for which an appropriate conventional alloy is not available, to optimize the properties of this intermetallic alloy with respect to the specific application and to develop the appropriate processing with minimum costs in close cooperation with the production engineers after having convinced the respective controllers that the development is worth the necessary money. References 1. Intermetallic Compounds, 4 Volume Set, EDs. J.H.Westbrook and R.L.Fleischer, John Wiley & Sons, Chichester (2000). 2. G. Sauthoff, in Intermetallics, Wiley-VCH, Weinheim (1995). 3. G. Sauthoff, Intermetallics, Ullmann's Encyclopedia of Industrial Chemistry, Seventh Edition, 2006 Electronic Release, Wiley-VCH, Weinheim (2006). D. R. Bates, Phys. Rev. , 492 (1950).

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H. A. Lipsitt, D. Shechtman, and R. E. Schafrik, Metall.Trans., 6A 1991 (1975). S. Taniguchi, T. Shibata and S. Itoh, Mater.Trans. JIM, 32 151 (1991). Y. W. Kim and D. M. Dimiduk, JOM-J.Min.Met.Mat., 43 40 (1991). G. Das, P. A. Bartolotta, H. Kestler, and H .Clemens, in: Structural Intermetallics 2001 (Proc. ISSI-3), Eds. K. J. Hemker, D. M. Dimiduk, H. Clemens, R. Darolia, H. Inui, J. M. Larsen, V. K. Sikka, M. Thomas, and J. D. Whittenberger, TMS, Warrendale 121 (2001). A. Gilchrist and T. M. Pollock, in: Structural Intermetallics 2001 (Proc. ISSI-3), Eds. K. J. Hemker, D. M. Dimiduk, H. Clemens, R. Darolia, H. Inui, J. M. Larsen, V. K. Sikka, M. Thomas, and J. D. Whittenberger, TMS, Warrendale 3 (2001). H. A. Lipsitt, M. J. Blackburn, and D. M. Dimiduk, Structural Intermetallics 2001 (Proc. ISSI-3), Eds. K. J. Hemker, D. M. Dimiduk, H. Clemens, R. Darolia, H. Inui, J. M. Larsen, V. K. Sikka, M. Thomas, and J. D. Whittenberger, TMS, Warrendale 73 (2001). H. Clemens and H. Kestler, Adv. Eng Mater., 2 551 (2000). M. Blum, G. Jarczyk, H. Scholz, S. Pleier, P. Busse, H. J. Laudenberg, K. Segtrop, and R. Simon, Mater. Sci. Eng. A, 329 616 (2002). K. Gebauer, Intermetallics, 14 355 (2006). T. Abe, H. Hashimoto, H. Ishikawa, H. Kawaura, K. Murakami, T. Noda, S. Sumi, T. Tetsui, and M. Yamaguchi, in: Structural Intermetallics 2001 (Proc. ISSI-3), Eds. K. J. Hemker, D. M. Dimiduk, H. Clemens, R. Darolia, H. Inui, J. M. Larsen, V. K. Sikka, M. Thomas, and J. D. Whittenberger, TMS, Warrendale 35 (2001). Y. Nishiyama, T. Miyashita, T. Nakamura, H. Hino, S. Isobe, and T. Noda, in: Proc. 1987 Tokyo International Gas Turbine Congress, Gas Turbine Society of Japan, Tokyo III-263 (1987). F. Appel, in: 4th International Conference on Processing and Manufacturing of Advanced Materials, (THERMEC'2003), PTS 1-5, 91 (2003). J. Beddoes, W. R. Chen, and L. Zhao, J .Mater. Sci., 37 621 (2002). M. Takeyama and S. Kobayashi, Intermetallics, 13 993 (2005). F. Appel, M. Oehring, and J. D. H. Paul, Adv. Eng. Mater., 8 371 (2006). K.S.Kumar, Int. Mater. Rev., 35 293 (1990). W. S. Chang and B. C. Muddle, Metals and Materials International, 3 1 (1997). K. S. Kumar, in: Structural Intermetallics, Eds. R. Darolia, J. J. Lewandowski, C. T. Liu, P. L. Martin, D. B. Miracle and M. V. Nathal, TMS, Warrendale 87 (1993). E. P. George, D. P. Pope, C. L.Fu and J. H. Schneibel, ISIJ INT., 31 1063 (1991). M. McLean, in: High-Temperature Structural Materials, Eds.R. W. Cahn, A. G. Evans, and M. McLean, Chapman & Hall, London 1 (1996). P. Veyssiere, Mater. Sci. Eng. A- 309 44 (2001). K. Aoki and O. Izumi, J. Jpn .Inst. Metals, 43 1190 (1979). C. T. Liu, C. L. White, and J .A .Horton, Acta Metall., 33 213 (1985). S. C. Deevi, V. K. Sikka, and C. T. Liu, Prog. Mater. Sci., 42 177 (1997).

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98. G. H. Meier, D. Appalonia, R. A. Perkins, and K. T. Chiang, in: Oxidation of High-Temperature Intermetallics, Eds. T. Grobstein and J. Doychak, TMS, Warrendale 185 (1989). 99. C. T. Liu and V. K. Sikka, Journal of Metals, 38 19 (1986). 100. E. P. George, C. T. Liu, H. Lin, and D. P. Pope, Mater. Sci. Eng .A- 193 277 (1995). 101. V. K.Sikka, M. L. Santella, P. Angelini, J. Mengel, R. Petrusha, A. P. Martocci and R. I. Pankiw, Intermetallics, 12 837 (2004). 102. M. Demura, Y. Suga, O. Umezawa, K. Kishida, E. P. George and T. Hirano, Intermetallics, 9 157 (2001). 103. G. Sauthoff, Intermetallics, 8 1101 (2000). 104. D. B. Miracle, Acta Metall. Mater., 41 649 (1993). 105. R. D. Noebe, R. R. Bowman, and M. V .Nathal, Int. Mater .Rev., 38 193 (1993). 106. Y. L. Hao, R. Yang, Y. Song, Y. Y. Cui, D. Li, and M. Niinomi, Materials Science and Engineering A, 365 85 (2004). 107. C. Herzig and S. Divinski, , Intermetallics, 12 993 (2004). 108. G .F. Hancock and B. R. McDonnell, Phys. Status Solidi, 4 143 (1971). 109. R.R.Vandervoort, A.K.Mukherjee, and J.E.Dorn, Trans.ASM, 59 930 (1966). 110. M. Rudy and G. Sauthoff, in: High-Temperature Ordered Intermetallic Alloys, Eds. C .C. Koch, C. T. Liu and N. S. Stoloff, MRS, Pittsburgh 327 (1985). 111. S. Divinski, F. Hisker, W. Loser, U. Sodervall and C.Herzig, Intermetallics, 14 308 (2006). 112. I. Jung, M. Rudy and G.Sauthoff, in: High-Temperature Ordered Intermetallic Alloys II, Eds. N. S. Stoloff, C. C. Koch, C. T. Liu, and O. Izumi, MRS, Pittsburgh 263 (1987). 113. G. Sauthoff, in: Proceedings of the International Symposium on Intermetallic Compounds - Structure and Mechanical Properties - (JIMIS-6), Ed. O. Izumi, The Japan Institute of Metals, Sendai, 371 (1991). 114. S. Shankar and L L. Seigle, Metall. Trans., 9A 1467 (1978). 115. H. C. Akuezue and D. P. Whittle, Met. Science, 17 27 (1983). 116. G. H. Cheng and M. A. Dayananda, Metall. Trans., 10A 1415 (1979). 117. T. D. Moyer and M. A. Dayananda, Metall. Trans., 7A 1035 (1976). 118. G. H. Meier, N. Birks, F. S. Pettit, R.A. Perkins, and H. J. Grabke, in: Structural Intermetallics, Eds. R. Darolia, J. J. Lewandowski, C. T. Liu, P. L. Martin, D. B. Miracle and M. V. Nathal, TMS, Warrendale 861 (1993). 119. H. J. Grabke and G. H. Meier, Oxidat. Metal., 44 147 (1995). 120. I. Jung and G. Sauthoff, Z. Metallk., 80 484 (1989). 121. I. Jung and G. Sauthoff, Z. Metallk., 80 893 (1989). 122. R. Darolia, Intermetallics, 8 1321 (2000). 123. W. Kleinekathöfer, A. Donner, H. Meinhardt, M. Hengerer, G. Sauthoff, B. Zeumer, G. Frommeyer, and H. J. Schäfer, in: Symposium Materialforschung - Neue Werkstoffe des Bundesministeriums für Forschung und Technologie (BMFT), Eds. U. Dahmen, I. Gilbert, D. Lillack, and S. Runte, KFA-PLR, Jülich 1014 (1994).

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124. B. Zeumer and G. Sauthoff, Intermetallics, 5 563 (1997). 125. B. Zeumer, W. Sanders, and G.Sauthoff, Intermetallics, 7 889 (1999). 126. M. Palm and G. Sauthoff in: Structural Intermetallics 2001 (Proc. ISSI-3), Eds. K. J.Hemker, D. M. Dimiduk, H. Clemens, R. Darolia, H. Inui, J. M. Larsen, V. K. Sikka, M. Thomas, and J. D .Whittenberger, TMS, Warrendale 149 (2001). 127. M. Palm, J. Preuhs, and G. Sauthoff, J .Mater. Process. Technol., 136 105 (2003). 128. M. Palm, J. Preuhs, and G. Sauthoff, J. Mater. Process. Technol., 136 114 (2003). 129. J.Klöwer, High temperature corrosion behaviour of iron aluminides and iron-aluminiumchromium alloys, Mater.Corros., 47 (1996) 685-694. 130. P. F. Tortorelli and K. Natesan, Mater. Sci. Eng. A- 258 115 (1998). 131. H. Thomas, in: Steel - A Handbook for Materials Research and Engineering Vol. 2: Applications, Ed. Verein Deutscher Eisenhüttenleute, Springer-Verlag, Berlin 445 (1993) . 132. J. Klöwer, A. Kolb-Telieps, R. Brück, L. Wieres, J. Lange, M. Brede, and H. Bode, in: Werkstoffwoche 1998, Bd. II: Werkstoffe für die Verkehrstechnik, Eds. R. Stauber, C. Liesner, R. Bütje, and M. Bannasch, 279 (1999). 133. Binary Alloy Phase Diagrams, Eds. T. B. Massalski, H. Okamoto, P. R. Subramanian and L. Kacprzak, 2 edn.,ASM, Materials Park (1990). 134. W. Köster and T. Gödecke, Z Metallk., 71 765 (1980). 135. P. G. Gonzales-Ormeno, H. M. Petrilli and C. G. Schon, Scripta Mater., 54 1271 (2006). 136. J. Herrmann, G. Inden, and G. Sauthoff, Acta Mater., 51 2847 (2003). 137. C. G. McKamey, in: Physical Metallurgy and Processing of Intermetallic Compounds, Eds. N. S. Stoloff and V. K. Sikka, Chapman & Hall, London 351 (1996). 138. J. L. Jordan and S.C. Deevi, Intermetallics, 11 507 (2003). 139. J. Wolff, M. Franz, A. Broska, R. Kerl, M. Weinhagen, B. Köhler, M. Brauer, F. Faupel, and T. Hehenkamp, Intermetallics, 7 289 (1999). 140. J. Herrmann, G. Inden, and G. Sauthoff, Steel Research International, 75 343 (2004). 141. D. G. Morris and S. C. Deevi, Mater. Sci. Eng. A- 329 573 (2002). 142. M. Palm, A. Schneider, F. Stein, and G. Sauthoff, in: Integrative and Interdisciplinary Aspects of Intermetallics, Eds. M. J. Mills, H. Inui, H. Clemens, and C. L. Fu, MRS, Warrendale 3 (2005). 143. A. Schneider, L. Falat, G. Sauthoff, and G. Frommeyer, Intermetallics, 13 1322 (2005). 144. M. Palm and G. Inden, Intermetallics, 3 443 (1995). 145. D. D. Risanti and G. Sauthoff, Mater. Sci. Forum, 475-479 865 (2005). R. Bates and H. S. W. Massey, Proc. R. Soc. London Ser. A192, 1 (1947). 146. D. Risanti and G. Sauthoff, Entwicklung ferritischer Eisen–Aluminium-Tantal -Legierungen mit verstärkender Laves-Phase mit höchster Zeitstandfestigkeit in korrosiven Atmosphären (DFG-Abschlussbericht), (2006).

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147. D. D. Risanti and G. Sauthoff, to be published, (2007). 148. D. D. Risanti and G. Sauthoff, Intermetallics, 13 1313 (2005). 149. V. Knezevic, G. Sauthoff, J. Vilk, G. Inden, A. Schneider, R. Agamennone, W. Blum, Y. Wang, A. Scholz, C. Berger, J. Ehlers, and L. Singheiser, ISIJ INT., 42 1505 (2002). 150. V. Knezevic and G. Sauthoff, in: Advances in Materials Technologgy for Fossil Power Plants (Proc. Fourth International Conference Oct. 25-28 2004 Hilton Head Island, South Carolina), Eds. R. Viswanathan, D. Gandy, and K. Coleman, EPRI, Palo Alto/CA 1256 (2005). 151. M. Palm, A. Schneider, F. Stein, and G. Sauthoff European Congress on Advanced Materials and Processes - EUROMAT 2005, Prague 6-9-2005.

CHAPTER 8

AN INTRODUCTION TO ELECTRONIC STRUCTURE METHODS D. A. Papaconstantopoulos Department of Computational and Data Sciences George Mason University, Fairfax, VA USA E-mail: [email protected] This chapter presents an introduction to electronic structure methods with emphasis on the APW and tight-binding techniques. The purpose and the output of such methods are described and examples of the results are given with appropriate comparisons to experimental quantities. These methods are practical implementations of the density functional theory and can be used to solve the Schrodinger equation and obtain from first principles various electronic and mechanical properties of solids including metals, insulators and semiconductors. By calculating the total energy of a material the stable crystal structure, the equilibrium volume and the elastic constants are determined in very good agreement with experiment. Calculating the energy bands and the density of states leads to the evaluation of the Fermi surface, the electron-phonon coupling for superconductivity and criteria for the occurence of magnetism. The tight-binding approach that I will describe is based on fitting to the APW results and extends the capabilities of the APW method to larger systems and leads to performing molecular dynamics simulations that are outside the computational limits of the first-principles methods.

1. Introduction In this article I present an introduction to electronic structure methods with emphasis on the APW method. The purpose and the output of such methods will be described and examples of the results will be given with appropriate comparisons to experimental quantities. These methods are practical implementations of the density functional theory and can be used

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to solve the Schrödinger equation and obtain from first-principles various electronic and mechanical properties of solids including metals, insulators and semiconductors. By calculating the total energy of a material the stable crystal structure, the equilibrium volume and the elastic constants are determined in very good agreement with experiment. Calculating the energy bands and the density of states enables the evaluation of the Fermi surface, the electron-phonon coupling for superconductivity, and criteria for the occurrence of magnetism. 2. Band Theory and Density Functional Theory The band theory of solids was formulated by Slater1 and others in the thirties following the development of quantum mechanics. The main task is to solve the Schrödinger equation: Hψ = Єiψ in the reciprocal space as an one-electron problem. The introduction of the density functional theory (DFT) by Kohn and coworkers2, 3 in the sixties provided a firm foundation to include the evaluation of the total energy of a system. DFT is based on the following two theorems: (a) The total energy, E, of a system of interacting electrons such as an atom, a molecule or a solid, is given by a functional of the ground state electronic density ρ. (b) The electronic ground state density is the density that minimizes E(ρ). The DFT basically reduced a formidable many-body problem into an one-body problem that is tractable for accurate numerical solution. Walter Kohn received the 1998 Nobel Prize in Chemistry for his pioneering work on the DFT. In addition to original articles by Hohenberg and Kohn2 and by Kohn and Sham3 there are several comprehensive reviews of DFT (see for example Callaway and March4). According to DFT the total energy of a system is written as: E(ρ)= Eh(ρ)+Exc(ρ)

(1)

where the Hartree energy can be written as, Eh(ρ) = T (ρ) + Ee-e(ρ) + Ee-n(ρ) + En-n(ρ)

(2)

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where T(ρ) is the single particle kinetic energy found from the sum of one electron eigenvalues in the Scrödinger equation, Ee-e(ρ) represents the Coulomb interaction between electrons, Ee-n(ρ), is the interaction energy between electrons and the nuclei, and En-n(ρ) describes the interaction energy between nuclei. The term Exc(ρ) is the exchange and correlation energy that is not treated exactly by DFT. The evaluation of Exc(ρ) requires an approximation known as the local density approximation (LDA). The exchange and correlation term Exc(ρ) is given by the following integral: (3) where Єxc(ρ)is approximated by a fit to the energy of a uniform electron gas. The minimization of E(ρ)leads to a set of single particle Schr¨odinger equations as follows: (4) The charge density ρ(r) is the usual summation over occupied states, i.e. 2 ρ = Σ ψ . The solution of the Schrödinger equation is coupled with Poisson’s equation that relates ρ(r) to the Coulomb potential Vc(r) i.e. 2

∇ (Vc(r)) = 8πρ(r)

(5)

The potential in Eq.(4) is given by V (r) = Vc(r)+Vxc(r)

(6)

where Vxc(r) is the exchange potential and correlation potential. We will discuss below how Vxc(r) is calculated within the LDA. 3. Local Density Approximation There are several different prescriptions of the LDA, i.e. HedinLundqvist5, Perdew-Wang6, Wigner and others7. The results obtained by the different forms of the LDA have very small differences in the resulting electronic spectrum. A substantial difference in obtaining the equilibrium

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lattice parameter is found when the so-called generalized gradient approximation (GGA) is used as discussed in the next section. The first practical applications of the LDA were made by Slater in 19518, independently of the formal DFT theory of Kohn, Hohenberg and Sham, and revised by Schwarz in19729. This form of LDA is known as the Xα method. In the Xα method an exchange potential is constructed that has the form: (7) The coefficient α was varied between the values of 2/3 and 1 to match the Hartree-Fock total energy for the atom. To obtain the results presented in this article we used the form of LDA proposed by Hedin and Lundqvist that includes correlation5. The exchange and correlation potential Vxc(r) is given by the expression: Vxc(r) = αβ(rs)Uex(r)

(8)

where, α = 2/3 is the Kohn-Sham parameter and β is the so called correlation enhancement factor defined by (9) Where rs=(3/4πρ(r))1/3, x = rs/21 and B =0.7734. An alternative expression for β(rs) was given by Wigner7 i.e. (10) which in some cases as in alkali metals gives better agreement with measured lattice constants. The LDA is surprisingly accurate and for most systems gives very good results. It has been established that the LDA often overbinds molecules. In weakly bonded systems these errors are exaggerated and bond lengths are too short. In metallic solids such as transition metals the LDA works very well. It also works for sp bonded systems such as semiconductors with the exception that it seriously underestimates the energy gap.

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4. Generalized Gradient Approximation There have been several attempts to improve the LDA by using a dependence of exchange correlation energy on the derivatives of electron density. A so-called “Generalized Gradient Approximation” (GGA) was introduced by Perdew in recent years10. In the GGA approximation, the exchange and correlation energy is not only a functional of ρ(r), but it depends on the gradient ∇ρ as well, and on high spatial derivatives of the total charge density. The generic functional form is written as: (11) where, is the exchange-correlation energy per particle, and n(r) is the electron density. The GGA approximation significantly improves the total energy of atoms, binding energies and vibrational frequencies for the simple metals and 3d transition metals. It also correctly predicts the ground state of Fe, therefore correcting a notable failure of the LDA. However, for the remaining transition metals, the GGA does not give any significant improvement over the LDA for equilibrium lattice constants and bulk moduli. In fact, very often the LDA results for transition metals and semiconductors are in better agreement with experiment. 5. Born-Oppenheimer Approximation The so-called Born-Oppenheimer approximation is made when solving the above equations of band theory. This assumes that the nuclei are at rest at the positions they would occupy in the crystal at temperature T = 0K. This means that the Schrodinger equation is solved for the motion of the electrons around the fixed nuclei. Thus the electronic motion is separated from the nuclei motion. This is justified because of the large difference in mass between electrons and nuclei. 6. Units in Band Theory The usual choice for units in band theory is: Atomic unit of length (a.u) = Radius of the first Bohr orbit = 2 / me 2 = 0.529 A Unit of energy =

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Ionization energy of the hydrogen atom = me4/2ħ2 = 1 Rydberg. It is useful to note that 1 Ry = 13.6058 eV. Another unit of energy is the Hartree, in Eq. which is equal to 2 Ry. As a result of using these units the factor (4) becomes unity and therefore the Schrodinger equation is written as: (12) It should also be noted that the potential energy of an electron in the Coulomb field of the nucleus, (-Ze2/r) becomes (-2Z/r). Therefore we have ħ = 1, mass of electron = 1/2 and charge of electron e2 = 2. 7. Electronic Structure Methods

The Schrodinger equation is solved by an expansion of the wave function (13) This expansion converts the Schrödinger equation to a system of linear algebraic equations (eigenvalue problem). The form of ϕ distinguishes different electronic structure methods from each other. For example, in the augmented plane wave (APW) method, ϕ has a spherical harmonics form inside a sphere surrounding the atomic sites and a plane wave form outside these spheres (muffin-tin spheres). In the muffin-tin orbitals method the wave functions are expressed in terms of Bessel functions inside the spheres and Neumann functions outside. Another method is the KKR method, which is based on multiple scattering theory. These methods are known as all-electron methods because they solve for all the electrons in the system. Another class of methods freezes the core electrons, separating them from the valence electrons. These methods, known as pseudopotential methods, use plane waves to describe the wave function. Many groups that are practicing the numerical implentation of these methods have written their own computer codes. The best known commercially available electronic structure codes are the Wien code based on the LAPW method and the VASP based on the pseudopotential method. The all electron codes have become more efficient using the linearization procedure proposed by Andersen11.

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8. Augmented Plane Wave Method (APW)

In this section we present, the APW method in some detail as an example. This method was originally proposed by Slater in 193712. In this method, the muffin-tin approximation (MTA) is often made especially for close-packed cubic structures. Although this method has been developed in general form without the MTA, for pedagogical reasons we will discuss it within the MTA. In the MTA each atomic site is surrounded by a sphere. Inside the sphere, the potential V(r) is a spherical symmetric function , and in the interstitial region, the potential is assumed to be constant Vc. Correspondingly, the wave function inside the muffin-tin spheres is expanded in spherical harmonics Ylm, (14) where, Alm is determined by the boundary conditions at the muffin-tin sphere to assure the continuity of the wave function outside the muffin-tin spheres, where it has the form of a plane wave,

ϕ(r) = exp(ik r)

(15)

the ul is computed by solving the radial equation at each k point in the Brillouin zone: (16) The potential V(r) is computed by solving Poisson’s equation through a self-consistent procedure. Since the potential V(r) is a periodic function, the wave function satisfies the Bloch condition13: Ψ(r + Gn,k) = e

ikGn

Ψ(r, k)

(17)

The secular equation is found by substituting into the Schrödinger equation eq. 17. This leads to a system of N algebraic equations: (18)

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The APW matrix elements include Legendre polynomials, spherical Bessel functions, and the logarithmic derivatives u′l/ul from eq. (16) evaluated at the MT radius Rs. Having evaluated the matrix elements one then solves an eigenvalue problem by diagonalizing the matrix. 9. Scalar Relativistic Approach

For elements heavier than the 3d transition metals, generally, the relativistic effects can not be ignored. The relativistic Hamiltonian is (19) where, the first term on the right side of the above equation denotes the non-relativistic Hamiltonian. The second term is called the mass-velocity term and represents the relativistic correction to the kinetic energy p2/2m The third term is called the Darwin term and represents the correction of the centrifugal potential. The fourth term represents the spin-orbit coupling. The mass-velocity term is negative and is larger for s states. The Darwin term affects only the s-like wave function. The mass-velocity and Darwin terms together consist of the semi-relativistic effects, known as the scalar relativistic correction14. The main effect of this correction is lowering of the s states. The relativistic effect becomes important for atomic numbers Z> 50. The spin-orbit term becomes important for 5d elements and higher, and in semiconductors such as Ge. The spin-orbit doubles the size of the secular equation and splits certain degenerate bands. In the results presented here we have used the scalar relativistic approach and omitted the spin-orbit interaction. 10. Self-consistency Cycle

Inside the muffin-tin sphere, the starting charge density is given by superposition of the atomic charge density ρ0(r) obtained from relativistic atomic structure calculations. Outside the muffin-tin sphere, the potential is a constant. The self-consistent solution process consists of three major steps: (1) making an initial guess of the charge density from superposition of atomic charge densities; (2) solving the scalar relativistic Schrödinger

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equation to compute the new charge density of the valence electrons and separately the Schrödinger equation for the atomic-like core states to obtain the new core charge density;(3) solving Poisson’s equation to get a new potential; (4) mixing the old and new charge densities puting them back to the Schrödinger equation as input to repeat the process. The procedure must be repeated until the electron density has converged to within certain tolerance, or the total energy has reached a convergence criterion of about 0.1 mRy. 11. Technical Details

In Fig. 1 we show schematically on top the input and at the bottom the output of an electronic structure calculation. It is clear that this is a first-principles procedure requiring to specify the atomic number, the number of valence electrons and the crystal structure as the only input. Of course, the crystal structure can also be determined since it is possible to perform these calculations for several crystal structures and select the one with the lowest energy.

Fig. 1: Electronic Structure calculation

In Fig. 2, Fig. 3, Fig. 4 and Fig. 5, we show the first Brillouin zone for the bcc, fcc, sc and hcp lattices. We note in these figures the standard notation used to label the different directions ink-space which is also used in the energy band plots that depict the results of band structure calculations.

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Fig. 2: First Brillouin zone of bcc structure

Fig. 3: First Brillouin zone of fcc structure

Fig. 4: First Brillouin zone of sc structure

Fig. 5: First Brillouin zone of hcp structure

11.1. Band Structure Nomenclature and Group Theory

Symmetry operations transform a crystal into itself. All symmetry operations form a Space Group. A Space Group consists of, 1. Translation Group. 2. Point Group. The Point Group consists of, a) Rotation symmetry. b) Mirror symmetry. c) Inversion symmetry.

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In the figures that show the Brillouin zone for various crystal structures a notation introduced by Bouckaert, Smolukowski, and Wigner15 is used. The different symbol sindicate points in reciprocal space that have high symmetry including symmetry lines and planes. In the case of cubic crystals, point group operations allow for only 1/48th of the Brillouin zone volume (irreducible BZ) to be used for sampling a k-point mesh necessary to solve for the electronic states of a given system. At certain k-points the secular equation can be block-diagonalized. Hence, this leads to matrix representations that are called ”irreducible”. This reduction of the size of the secular determinant results in considerable savings in computational costs which, however, are often ignored in many electronic structure codes because they are not significant due to the high speed of modern computers. But block-diagonalization is still useful because it is important in identifying the angular momentum character of the electronic states that is needed in many applications. As an example of point group symmetry we consider in Table I the group Oh at the center of the cubic BZ at k=0. This point is denoted by the Greek letter Γ. The point H(100) in the bcc lattice and the point R(111) in the simple cubic lattice possess the same symmetry. 11.2. Further Symmetry Considerations

The results obtained from band structure calculations determine the symmetry and angular momentum character of the states along various directions in the Brillouin zone. As an example Table 1 gives this information for the high symmetry points in the fcc, bcc, and simple cubic Table 1: Symmetry at the Point Γ Label

Degeneracy

Γ1 Γ12 Γ25’ Γ15 Γ2’ Γ25

1 2 3 3 1 3

Angular Momentum Character s d(eg) d(t2g) p f f

Basis Function 1 (x2-y2), (3z2-t2) xy, yz, zx x, y, z xyz z(x2-y2)

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lattices. In the energy band figures shown below, the connectivity of the bands is determined by group theory. In Table 2 we show the so-called compatibility relations that are obeyed in plotting energy band diagrams. These relations have been implemented by Boyer16 to interpolate APW results on a fine k-point mesh which subsequently is used to calculate the DOS. In order to calculate the DOS, the eigen values are calculated on a uniform grid in k-space in the 1/48 th of the cubic Brillouin zone. These are shown in Table 2. Table 2: Compatibility Relations Γ1

Γ2

Γ12

Γ’15

Γ’25

Γ’1

Γ’2

Γ’12

Γ15

Γ25

Δ1 Λ1 Σ1

Δ2 Λ2 Σ4

Δ1Δ2 Λ3 Σ1Σ4

Δ’1Δ5 Λ2Λ3 Σ2 Σ3Σ4

Δ’2Δ5 Λ1Λ3 Σ1Σ2 Σ3

Δ’1 Λ2 Σ2

Δ’2 Λ1 Σ3

Δ’1 Δ’2 Λ3 Σ2Σ3

Δ1 Δ5 Λ1Λ3 Σ1Σ3Σ4

Δ2 Δ5 Λ2Λ3 Σ1Σ2 Σ4

X1

X2

X3

X4

X5

X’1

X’2

X’3

X’4

X’5

Δ1 Z1 S1

Δ2 Z1 S4

Δ’2 Z4 S1

Δ’1 Z4 S4

Δ5 Z2Z3 S2S3

Δ’1 Z2 S2

Δ’2 Z2 S3

Δ2 Z3 S2

Δ1 Z3 S3

Δ5 Z1Z4 S1S4

M1

M2

M3

M4

M5

M’1

M’2

M’3

M’4

M’5

Σ1 Z1 T1

Σ4 Z1 T2

Σ1 Z3 T’2

Σ4 Z3 T’1

Σ2 Σ3 Z2Z4 T5

Σ2 Z2 T’1

Σ3 Z2 T’2

Σ2 Z4 T2

Σ2 Z4 T1

Σ1 Σ4 Z1Z3 T5

The k-point meshes of 285, 505 and 165 k-points for bcc, fcc and sc respectively serve as an input in well converged calculation of the DOS. An efficient way to calculate the DOS is the tetrahedron method which is briefly outlined in Section 12. Table 3: k-point grids for the cubic lattices Div. Along (100) Num. k-points

4 14

bcc 8 55

16 285

4 20

fcc 8 89

16 505

2 10

sc 4 35

8 165

11.3. The Fermi Surface

One or more bands may be partially filled, i.e. they are intersected by the

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Fermi level EF. For each partially filled band there is a surface in k-space separating the occupied from the unoccupied energy levels. The set of all such surfaces is called the Fermi surface. Clearly only metals have a Fermi surface; semiconductors and insulators do not. The Fermi surface of the n-th band is defined by En(k) = EF

(20)

So the Fermi surface is a constant energy surface in k-space at energy equal to the Fermi level. 12. Electronic Structure Results

We now present the results of the energy bands and DOS for typical elements. Many more results for the elements may be found in Papaconstantopoulos’s book17. Fig. 6 shows the energy bands along high symmetry directions in the Brillouin zone (BZ) for bcc Nb. Since EF crosses the bands, the material is predicted correctly as a metal. This figure also shows the symmetry of the states at the center of the BZ, i.e. Γ1 (pure s), Γ25′ (pure dt2g ) and Γ12(pure deg ). The p-states (Γ15) lie very high and are not shown. In Table I we list the angular momentum character of states at high symmetry points of the BZ for the bcc, fcc and sc structures. In Fig. 7, we show the energy bands along high symmetry directions in the BZ for fcc Pd. Again, EF crosses the bands confirming that Pd is a metal. The d-bands of Pd, characterized by the energy difference E(Γ12)-E(Γ25′ ), are significantly narrower than those of Nb. In Fig. 8 we present the energy bands of Si, a semiconductor as shown by the formation of a gap at EF. Si similarly to Ge and C has the diamond structure. In this structure the notation for symmetry points in an fcc BZ corresponds to a different assignment of l-character of states. For example, while the Γ1 state has s-character, the Γ25′ has p-character like Γ15 does. On the other hand, the Γ2′ state has s-character in the diamond structure unlike the situation in the other cubic structures where Γ2′ is of f-character.

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Fig. 6: Band structure of bcc Nb

Fig. 7: Band structure of fcc Pd

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Fig. 8: Band structure of Si in Diamond structure

We have shown energy bands that involve s, p and d electrons. The heavy elements have energy bands that involve f electrons as well. These bands are extremely narrow resembling atomic states. A very important quantity resulting from electronic structure calculations is the density of states (DOS), which is defined as the number of states per unit energy. In practice an accurate determination of the DOS requires an interpolation of the first-principles results in k-space. An efficient way to calculate the DOS is the tetrahedron method18in which the eigen values ε(k) are interpolated linearly between four k-points placed on the vertices of a tetrahedron according to a formula: (21) where, ε0 and b are determined by the energies of the corners of the tetrahedron. In the next few figures (Figs. 9-11) we show the DOS for the materials we presented energy bands. We note the decompositions of the total DOS into its angular momentum components. The calculated DOS denoted as N(E) can be used to compare with a variety of spectroscopic measurements. It is important to note that for Nb and Pd the DOS is dominated by the d-component with s and p states having much smaller contribution. On the other hand in the Si DOS we should note that the

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Fig. 9: Density of states of Nb in BCC structure

main contributors are s and p states. The value of the DOS at the Fermi level, N(EF), is used to calculate several quantities that are measured by experiment. First we refer to the specific heat given by the expression, C = γT + αT

2

(22)

the coefficient γ of the linear term is known as the electronic specific heat coefficient and is proportional to the N(EF ) i.e. γ = 0.1734(1+ λ)N(EF)

(23)

where the numerical constant is chosen in such a way that N(EF) is expressed in states/Ry per atom for both spins and γ is given in mJmol−1deg−2 . The term (1 + λ) is known as the mass enhancement factor. Usually, accurate DOS calculations give a calculated γ that is smaller than experimental value, thus providing an estimate of λ that enters the BCS theory of superconductivity as the electron-phonon coupling constant. A list of the bare electronic specific heat coefficient compared with measured values is given in Table 4 for selected metals. One can deduce from this the mass enhancement factor.

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Table 4: Electronic Specific Heat Coefficients in mJ mol−1deg−2 Na Al V Nb Pd Ta Au

γth

γ expa

1.17 0.95 4.31 3.44 5.58 2.96 0.69

1.38 1.35 9.26 7.79 9.42 5.90 0.729

Fig. 10: Density of states of Pd in FCC structure

13. McMillan theory of Superconductivity

Using the BCS theory, McMillan developed an approach to determine the electron - phonon coupling, constant λ and the critical transition temperature, Tc, for superconductivity19. McMillan’s strong coupling theory defines an electron-phonon coupling constant by: (24)

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Fig. 11: Density of states of Si in Diamond structure

where (25) where M is the atomic mass; η is the Hopfield parameter20 which equals the product of the total density of states, N(EF), at the Fermi level, EF, 2 times which is the square of the electron-ion coupling matrix element 2 at EF averaged over the Brillouin zone; <ω > is the renormalized phonon frequency. Eq. (26) is due to Gaspari and Gyorffy and is known as the rigid muffin-tin approximation21. In this expression δl are scattering phase shifts at EF, Nl are the angular momentum components at EF, and Nl (1) are the free-scatterer DOS defined from the following equation: (26)

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The above theoretical works have provided us a way to calculate the transition temperature for superconductivity based on energy band calculations. McMillan proposed the following equation for Tc: (27) The average phonon frequency <ω> is estimated from the Debye temperature ΘD following the approximation:

<ω > =

ΘD 2

(28)

Results using this method can be found in reference 22. An alternative procedure is to perform calculations based on linear response theory. These calculations are potentially more accurate but often suffer from inadequate convergence in the Brillouin zone integrations. A more serious issue in these approaches is the estimation of the Coulomb pseudopotential, µ*, which is usually given values between 0.1 and 0.2 which seriously affect the final value of Tc. Recently ,this approach has been used by Shi et al.23 to predict the variation of the superconducting temperature under high pressure. Theoretical and experimental works have shown that several metals outside the transition metal series become superconductors under pressure. Notable examples are Li, Y, and Ca which approach or exceed temperatures of 20K. 14. Calculation of X-ray Spectra

The results of band structure calculations and in particular the angular momentum components of the density of states can be used to evaluate the K, L, and M x-ray emission and absorption spectra. The K and L spectra are in generally good agreement with experiment in the overall shapes and widths. Variations between the calculations and measurements are similar in magnitude to variations between measured spectra from different laboratories. The calculated M spectra are usually too narrow, and in poor agreement with experiment. The one-electron approach appears inadequate for spectra in which significant interaction occurs between a weakly bound core-level vacancy and the partially filled d band.

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Here we follow the formalism of Goodings and Harris24, who derived the formulae given below for the x-ray intensity in terms of densities of electronic states and matrix elements formed by the core and band radial wave functions. The x-ray intensity of the K spectrun is given by (29) and that of the L2,3 or M2,3 spectra by (30) In Eqs. (28) and (29) the quantities Ns(E), Np(E), and Nd(E) are the l-components of the DOS for a band energy E. Ec is the core energy corresponding to the 1s, 2p, and 3p levels for K, L, and M spectra respectively. The matrix elements M for the appropriate transition are given by the following integral: (31) where unl(r) is the core wave function, ul+1(r) is the valence band wave function and RMT is the muffin-tin radius. The corresponding absorption spectra are given3 by similar expressions as (28) and (29) after replacing the factor (E-Ec) with 1/(E-Ec). In order to compare the x-ray intensities obtained from the above equations with the measured spectra, a Lorentzian broadening function is applied. 15. Total Energy

In addition to the energy bands and DOS, the LDA calculations provide an accurate determination of the total energy and hence an evaluation of the equilibrium lattice parameter and the bulk modulus. The total energy computed for a given crystal structure and several volumes is expanded by the Birch-fit formula25 according to the expression:

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(32) where ai are expansion coefficients and N is the order of the fit. A third order fit is usually providing accurate description of the total energy. The second derivative of E with respect to volume gives the bulk modulus: (33) Examples of total energy calculations for Nb and Pd are shown in Fig. 12 where the minimum shows the equilibrium volume. It is evident from these plots that these calculations predict correctly that the bcc structure is the equilibrium structure in Nb and the fcc structure is favored in Pd.

Pd

fcc

bcc

Total energy (Ry)

Total energy (Ry)

Nb

bcc

fcc

Volume (a.u.3)

Volume (a.u.3)

Fig. 12: Total energies of states Nb and Pd in the fcc and bcc structures

Calculations for these metals in other structures such as the simple cubic, diamond and hcp always place the energy higher than the ground state. In the following article by Sigalas et al. tables of LDA results for the lattice constants are given26 which show 1-2% discrepancy from experiment for most elements except the alkali metals where the discrepancy is larger. For the bulk modulus the LDA error with respect to experiment is much larger (10%) due to the fact that the calculation of B involves a second

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derivative. It is interesting to note here that the alkali metals have an energetic difference between fcc and bcc that is extremely small (less than 0.5 mRy). For typical transition metals, these differences are an order of magnitude larger providing an unambiguous determination of the ground state. On the other hand, for the alkali metals it becomes problematic to show that bcc is the ground state within the constraint that the DFT calculations are done at T = 0K. It should be mentioned here that with the GGA for the 3d series, one can obtain a closer agreement to experimental values of the lattice constants. 16. Elastic Constants

The power of LDA/GGA calculations can be further demonstrated by computing elastic constants Cij. The theory and computational procedure for performing calculations of Cij is discussed by Mehl et al.27. To summarize we confine ourselves here to cubic systems where symmetry reduces Cij to only three independent elastic constants C11,C12 and C44. Applying an orthorhombic strain e, it can be shown that the strained energy E, follows a linear dependence on e2as follows: E = E0 + V (C11 - C12)e

2

(34)

where E0 is undistorted energy and V is the fixed volume of the unit cell. So from the slope of the E v. e2 straight line we determine the quantity C11 - C12 known as the tetragonal shear modulus. Similarly, a volume conserved orthorhombic strain results in the expression: (35) which determines the elastic constant C44. To evaluate C11 and C12 separately we use their relationship to the bulk modulus: (36) Figs. 13 and 14 depict the total energy E as a function of e2 for the intermetallic compound YCu. The elastic constants of stable or metastable

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structures of materials must satisfy certain conditions. The following rules determine the mechanical stability for cubic materials: B>0

(37)

C11 - C12 > 0

(38)

C44 > 0

(39)

Fig. 13: Elastic constants C11- C12 of YCu

Fig. 14: Elastic constant C44 of YCu

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17. The STONER Criterion

Many years ago Stoner28 proposed that the following inequality: N(EF)I > 1

(40)

can be used as a criterion for the occurence of ferromagnetism. As was shown by Vosko and Perdew29, the parameter I can be accurately calculated using the results of band theory as follows. (41) where ul(EF) is the Fermi level value of the radial wave function and K(r) is a kernel giving the exchange and correlation enhancement of the external field due to magnetization. We note again the key role of the angular momentum decomposed DOS, Nl(EF). The first implementation of this theory was made by Janak30 who demonstrated that the Stoner criterion is satisfied for Fe and Ni with values clearly exceeding one, while for all other elements the values obtained by Janak were well below 1.0 (Fig. 15).

Fig. 15: Stoner criterion in Fe-Ni

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For Co in the fcc structure, Janak reports a value of 0.97. Sigalas and Papaconstantopoulos31, doing the calculations in the bcc lattice, found a value of 1.35. It is safe to assume that in the hcp strucure, which is the ground state for Co, a value over 1.0 is a reasonable expectation. This method has been extended to binary systems by Papaconstantopoulos32 who performed calculations for all the3d, 4d, and 5d monohydrides in the NaCl structure. These calculations predicted that CoH exceeds the Stoner criterion, which makes it a ferromagnet in agreement with experiment. 18. Spin Polarized Calculations

The above discussion on the Stoner criterion is based on paramagnetic calculations. Although this approach is satisfactory for the prediction of magnetic instabilities, it is necessary to perform spin-polarized calculations to get more information on magnetic properties such as the magnetic moment of a given material. Spin-polarized calculations are also possible using the codes described in this article. The basis of these calculations is that we now have two distinct potentials, one for the up spin and another for the down spin electrons. The difference between these two potentials is that they have different exchange and correlation parts. So these calculations result in two distinct sets of bands and DOS corresponding to spin up and down electrons. The difference between spin-up and spin-down energy bands of the same symmetry in these bands defines the exchange splitting that characterizes the magnetic anistropy, a quantity that can be measured in ferromagnetic systems such as Fe. These calculations also give, by integrating the DOS up to EF, the number of electrons for each spin and thus the value of the magnetic moment. Another important part of these calculations is that when comparing with the total energy of the paramagnetic state one can predict which state is the stable one. It is interesting to note here that in the case of Fe only the GGA correctly predicts the bcc ground state. 19. Virtual Crystal Approximation

The standard LDA/GGA codes can be used to calculate the electronic structure of substitutionally disordered alloys. For this purpose supercells

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canbe constructed that include two or more elements. Such calculations significantly increase the computational cost and need to be checked carefully for convergence with respect to size of the supercell. A simpler approach is the virtual crystal approximation (VCA).In the VCA the computation of the electronic structure of an alloy AxB1-x assumes an ideal new element C whose atomic number as well as the number of valence electrons, denoted by Z, are both calculated by the following equation: Z = xZA +(1-x)ZB

(42)

where ZA and ZB are the atomic numbers(or the number of valence electrons) of the component elements A and B. This approximation is reasonably accurate when the elements A and B are in the same row of the periodic table. It should be mentioned here that certain features of the band structure can be captured by the rigid band approximation (RBA). For example, in Fig. 6 of the band structure of Nb, the RBA amounts to shifting EF to the right to obtain the band structure and DOS of Mo. In Fig. 15 we give the results for the Stoner criterion in the Fe-Ni alloy using the VCA. We note that for the bcc lattice the system is predicted to be ferromagnetic in the Fe-rich region while it is ferromagnetic in the Ni-rich region of the fcc lattice. These results are consistent with the experimental situation in Fe-Ni steels. 20. Calculations for Multi-atom Systems

Although in the previous sections I alluded to results for binary materials, for pedagogical reasons I have confined myself to mainly discussing examples of electronic structure in monatomic materials. The electronic structure codes are capable of treating much larger systems including the complicated multi-atom structures of the high-temperature superconductors. Discussion of these results is beyond the scope of the present article. Instead I present here the band structure of some typical binary materials of interest. Figure 16 shows the band structure of PdH in the NaCl structure (also known as the B1 structure) which places hydrogen in the octahedral sites.

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Fig. 16: Band structure of PdH

These calculations establish the following characteristics of the band structure of PdH: (a) hydrogen has a small effect on the Pd d-states, which are centered around the Γ25′ and Γ12 k-points, (b) hydrogen makes the Fermi level to move above the Pd d-bands so that N(EF) now has a small value in contrast to pure Pd where as it can be seen from Figs. 7 and 10, N(EF) is very large, (c) hydrogen pulls down the bonding levels at the Γ1 point by approximately 2eV, and (d) hydrogen forms an antibonding band at about 3eV above the Fermi level at the high Γ1 k-point. Figure 17 shows the energy bands of NbC in the NaCl structure. Examining this figure we find the lowest split-off occupied valence band to be primarily a carbon s band. Following is a gap and then a set of bonding bands consisting of carbon p states mixed with Nb d states. The dotted line, indicating the Fermi level, crosses the bands at a low density of states which results in a low N(EF) as can be seen from33. Clearly at EF there is hybridization between d-Nb and p-C states. An example of the accuracy with which lattice parameters can be predicted in binary materials is shown in Fig. 18. This shows the results of total energy

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Fig. 17: Band structure of NbC

Fig. 18: Lattice constant andbulk modulus as afunction of the number of valence electrons in 3d-transition-metal aluminides

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calculations for the 3d series of transition-metal aluminides in the CsCl (orB2) structure. These calculations were performed using the GGA and the LAPW method. For VAl, FeAl, CoAl, and NiAl that are known to crystalize in the CsCl structure the agreement with the measured lattice constants is excellent to be clearly lower in energy than the CsCl. This figure also shows the variation of the calculated bulk modulus as a function of the number of valence electrons. We are not aware of any experimental values to compare. Also shown in Fig. 18 are results for the 50-50 alloys generated with the VCA approach. Finally, we point out that several of these materials form in the L10 structure. Indeed, for TiAl we found this structure. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

J. C. Slater, Rev. Mod. Phys. 6, 209 (1934). P. Hohenberg, and W. Kohn, Phys. Rev. 136, B 864 (1964). W. Kohn, and L. J. Sham, Phys .Rev. 140,A 1133 (1965). J. Callaway and N. H. March, ”SolidStatePhysics”, H. Ehrenreich and D. Turnbull, Eds., 38, 135 (1984). L. Hedin, and B .I .Lundqvist, J .Phys. C4, 2064 (1971). J. P. Perdew and Y. Wang, Phys .Rev. B 33, 8800 (1986). E. P. Wigner, Phys .Rev .46, 1002 (1934). J. C. Slater, Phys .Rev. 81, 385 (1951). K. Schwarz, Phys. Rev. B 5, 2466 (1972). J. P. Perdew, Phys. Rev .Lett. 55, 1665 (1985). O. K. Andersen, Phys.Rev.B 12, 3060 (1975). J. C. Slater, Phys.Rev. 51, 151 (1937). F. Bloch, Z.Physik 52, 55 (1928). D. D. Koelling, and B.N.Harmon, J. Phys. C 10, 3107 (1977). L. P. Bouckaert, R. Smoluchowski, and E. P. Wigner, Phys. Rev. 50, 58 (1936). L. L. Boyer, Phys. Rev.B 19, 2824 (1979). D. A. Papaconstantopoulos, ”Handbook of the Band Structure of Elemental Solids”, Plenum(1986). G. Lehmann, and M. Taut, Phys .Status Solidi 54, 469 (1972). W. L. McMillan, Phys .Rev. 167, 331 (1968). J. J. Hopfield, Phys.Rev. 186, 443 (1969). G. D. Gaspari, and B. L. Gyorffy, Phys. Rev. Lett. 28, 801 (1972). D. A. Papaconstantopoulos, L. L. Boyer, B.M. Klein, A.R. Williams, V.L. Mozuzzi and J.F. Janak, Phys. Rev. B 15, 4221 (1977).

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23. L. Shiand, D. A. Papaconstantopoulos, and M. J. Mehl, Solid State. Commun. 127, 13 (2003); L.Shiand, D. A. Papaconstantopoulos, Phys. Rev. B 73, 184516 (2006). 24. D. A. Goodings and R. Harris, J. Phys. C 2,1808 (1969). 25. F. Birch, Phys. Rev. 71, 809(1947). 26. M. Sigalas, D. A. Papaconstantopoulos, and N. C. Bacalis, Phys .Rev. B 45, 5777 (1992). 27. M. J. Mehl, B. M. Klein, and D. A. Papaconstantopoulos, in Intermetallics Compounds, Vol. 1, (Wiley, London 1994). 28. E. C. Stoner, Proc. Roy. Soc. London Ser.A 154,656(1936). 29. S. H. Vosko, and J. P. Perdew, Can.J.Phys.53, 1385 (1975). 30. J. F. Janak, Phys.Rev.B 16,255 (1977). 31. M. Sigalas and D. A. Papaconstantopoulos, Phys. Rev. B 50, 7255 (1994). 32. D. A. Papaconstantopoulos, Europhys. Lett. 15 ,621 (1991). 33. B. M. Klein, D. A. Papaconstantpoulos, and L. L. Boyer, Phys. Rev. B 22, 1946 (1980).

CHAPTER 9

CRYSTALLOGRAPHY OF COMPLEX METALLIC ALLOYS Walter Steurer and Thomas Weber Laboratory of Crystallography, Department of Materials, ETH Zurich, Wolfgang-Pauli-Strasse 10, CH-8093 Zurich, Switzerland E-mail: [email protected] The physical properties of a material are based on its chemical composition, crystal structure and microstructure. The inherently anisotropic distribution of chemical bonds leads to the inherent anisotropy of physical properties. After an introduction into the crystallographic description of crystal structures we will focus on structure/property relationships. A second focus will be on the power of X-ray diffraction techniques for structure analysis and the study of order/disorder phenomena.

1. Introduction What is crystallography? This seems to be a simple question, however, even crystallographers have strongly differing opinions on this point1. Some see crystallography just as a toolbox of structure determination methods. Others regard it as a scientific discipline with its own mode of argument and reasoning, i.e. the structural way of thinking. Crystallography overlaps with parts of physics, chemistry, molecular biology and materials science (Fig. 1). It involves experimental and theoretical structural studies on all types of materials, from amorphous to crystalline, from quasicrystals to nucleosomes, on meta-materials such as photonic or phononic crystals, on a pico-second timescale, in excited states, at pressures and temperatures of the Earth’s core or close to zero K, on bulk materials, fibers, nanocrystals or surfaces and interfaces; it also includes the study of the self-assembly of matter, of crystal growth at ambient conditions, under high pressure or micro-gravity, and the 219

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Fig. 1. Crystallography and its relationship to related scientific disciplines.

investigation of the relationship between the crystal structure of a material and its physical properties. What is the crystallographic information we are interested in? This is usually the idealized, averaged crystal structure. An organic synthetic chemist, for instance, routinely checks the structure of a new compound. He/she is usually less interested in the full crystal structure then in the structure of its constituents, the molecules and the chemical bonding therein. This is also true for a molecular biologist who is interested in the way a biological macromolecule functions and not how it packs in its crystallized form. Physicists or materials scientists who are interested in structure/property relationships of materials need the full picture, structure and bonding. Since physical properties can be greatly influenced by small structural variations, the real structure is of interest. The real structure takes into account all deviations from the ideal structure, i.e. disorder as well as defects. By disorder, which can be displacive and/or substitutional, we mean all structural deviations from an ideal structure that exist in thermodynamic equilibrium. The only exceptions are thermal vacancies, which belong to the defects as well as dislocations, stacking faults, grain boundaries etc., which are not in thermodynamic equilibrium. Disorder can have a strong influence on particular physical properties. The defect structure (dislocations, stacking faults) or the microstructure in case of polycrystalline and/or multiphase

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Fig. 2. Relationship between the chemical composition of a compound, its crystal structure, physical properties and potential applications. The crystal structure may also depend on applied fields or, in the case of nanocrystals, on the particle size. Physical properties are influenced by the defect structure and microstructure as well.

materials (shape and distribution of single-crystalline grains), which is of great importance for materials properties in applications, is usually beyond the crystallographic focus. One of the goals of crystallography is to fully understand why a material with given chemical composition has exactly the particular structure it has at a given temperature and pressure. This understanding is a prerequisite for predicting crystal structures based on chemical information only. To clarify how the physical properties are related to a crystal structure is the next step (Fig. 2). The ultimate goal is to be able predicting what kind of chemical composition and crystal structure a material with desired physical properties should have, and if such a material is physically possible at all. Is there anything special with the crystallography of complex metallic alloys (CMAs)? Basically, no, the structure analysis may just be a little bit more demanding, perhaps. CMAs are characterized by large to giant unit cells, containing up to several thousand atoms. However, crystals of biological macromolecules, which may contain millions of atoms, easily surpass this. The difference is in the factors controlling the ordering of the atoms. In case of polypeptides, the sequence of amino acids can be

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determined by automated methods. Structure analysis then means the determination of the folding of the chains, driven mainly by hydrogen bonds, i.e. local interactions. In CMAs, there are local and non-local interactions. Covalent bonding contributions are responsible for preferred symmetries of coordination polyhedra (AET ... atomic environment types), details of the electronic band structure may be responsible for particular distortions or ordering phenomena (superstructure, pseudosymmetry, modulated structures). The structure is the result of competing factors minimizing the Gibbs energy. The stoichiometry influences the electron concentration, energetically favorable interactions can be controlled by the AET and packing density, crystal structure and electronic structure are mutually conditional. The structural ordering principles can comprise several length scales. They may range from small AETs to hierarchical clusters or from sub-unit-cells to multiples of basic unit cells such as in superstructures (Fig. 3). How to determine the crystal structure of a CMA? The first step should always be a single crystal X-ray diffraction (XRD) structure analysis, if possible. If there is not sufficient contrast between the scattering powers (scattering factor, atomic form factor) of the constituent atoms, neutron scattering (NS) can overcome this problem

Fig. 3. Schematic representation of CMAs as (a) superstructures and (b) cluster-based structures, respectively. In (a), there is the length scale of the sub-unit cell and that of the modulation wave, in (b) that of the pentagonal unit cluster and the one of the large unit cell.

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usually. If no single crystals of the necessary size are available, powder diffraction methods can be applied. The structure determination delivers space group symmetry, lattice parameters, atomic coordinates, occupancy factors and atomic displacement parameters (ADPs). If highquality diffraction data are available with sufficient resolution, electron density maps can yield valuable information on chemical bonding. If the structure analyses are performed as a function of temperature or pressure additional information can be derived. The structural variation as function of temperature, T, gives information on the potentials the atoms are exposed to. The application of high-pressure, p, allows to study structural changes related to those in the electronic band structure. The variation of composition, x, over the whole stability region of a CMA usually leads to local relaxations and may influence the long-range order via changing the electron concentration. In case a phase transition takes place, a microscopic model of its mechanism may be derived from T, p and/or x dependent structural information. The second step is the exploration of the ordering phenomena reflected in disorder diffuse scattering. If necessary, this type of diffuse scattering has to be separated from thermal diffuse scattering or the usually very weak and negligible contributions from Compton scattering or defect scattering. To determine the full structure of a disordered material one, therefore, needs full reciprocal space information, Bragg reflections as well as disorder diffuse scattering. The 6 information derived from Bragg intensities gives the average structure, i.e. the real disordered structure modulo one unit cell. Any deviation from the average positions and site occupancies leads to diffuse scattering. It shows a characteristic structure (streaks, diffuse intensity rings, stars, sheets, etc.) depending on the correlations between the disordered structural constituents. Understanding a crystal structure does not only mean to understand its architecture. Atomic distances and the electron density distribution function contain valuable information on the chemical bonding in the structure. For a deeper insight in the bonding system, first-principles quantum-mechanical structure modelling and calculation of the electronic density of states is crucial. This also allows a prediction of physical properties to some extent.

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2. Description of crystal structures No matter how complex a CMA might be, the symmetry of its idealized (average) structure can always be described by one of the 230 threedimensional (3D) space groups. The only exceptions are aperiodic crystals such as incommensurately modulated structures, composite crystals and quasicrystals. These are characterized by a Fourier module of rank n, with n > d, the dimensionality of the structure (usually three). Their symmetries can be properly classified using higher-dimensional space groups. In the following, we focus on the symmetry of regular, 3D periodic crystal structures and that of their properties. For illustration, the structure of the prototype of A15 superconductors, cP8-Cr3Si, will be used (Fig. 4). It has the cubic space group Pm 3 n (No. 223) and the lattice parameter a = 4.5599(3) Å2. Si occupies Wyckoff position 2 a ( m 3. ), 0 0 0, and Cr is in site 6 c ( m 3. ), 1/4 0 1/2 (see Figs. 5, 6, 7).

(a)

(b)

Fig. 4. The structure of cP8-Cr3Si in two different representations. (a) The icosahedral coordination polyhedron around Si (grey) is emphasised, b) Cr atoms (black), linked by bonds along the shortest distances, form chains in all three directions.

Figure 4 nicely shows the ambiguity in the description of a (not very) complex crystal structure. It seems to be arbitrary whether one describes the structure as packing of Si-centered Cr-icosahedra or by a network of Cr-chains embedded in a body-centered cubic (bcc) Si-substructure.

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Only a detailed analysis of atomic distances shows that the second description is more physical. All distances from the Si atom in the center to the Cr-atoms at the corners of the icosahdral coordination polyhedron have the same length, 2.549 Å. The icosahedron, however, is far from regularity. The shortest distances between the chain-forming Cr-atoms are 2.280 Å; those between Cr-atoms of different chains are 2.792 Å. For comparison, the shortest atomic distance, i.e. the atomic diameter, in bcc chromium is 2.498 Å. Indeed, the atomic chains are the crucial structure elements responsible for superconductivity in the respective representatives of the A15 type. To summarize, coordination polyhedra (AETs) can be very helpful for visualizing crystal structures, however, they are not necessarily atomic clusters of physical relevance. 2.1. Crystallographic symmetry groups

As mentioned above, the characteristic property of any geometric model of a crystal structure is its translational periodicity. This means, it is possible to describe the structure mathematically by a lattice, Λ (Eq.1), decorated with atoms ⎧ Λ = ⎨r = ⎩

⎫

3

∑ n a | n ∈ Z ⎭⎬ i i

i

(1)

i =1

with basis vectors ai and Z the set of integers. The set of all vectors x1a1 + x1a2 + x1a3 with 0 ≤ xi ≤ 1 is called a unit cell of the vector lattice. Since the absolute orientation of the lattice in space is usually not of interest for a crystal structure description, instead of basis vectors the lattice (or cell) parameters ai, i = 1…3, and αi or a, b, c and α, β, γ are given. If each unit cell is decorated in the same way by a structure motif (set of atoms, ions, molecules), an ideal infinite crystal structure model results. There exist 14 different 3D lattices. They are called Bravais lattices and described by the symbols aP, mP, mC (mS), oP, oC, oI, oF, tP, tI, hR, hP, cP, cI, cF. The lowercase letter gives the crystal system, anorthic, monoclinic, orthorhombic, tetragonal, hexagonal and cubic, respectively. The uppercase letters indicate the type of centering, P primitive (i.e. noncentered C or SC-face or side-centered (two-fold primitive), respectively,

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I body centered, bcc (two-fold primitive), R rhombohedral centered (three-fold primitive), F all-face centered, fcc (four-fold primitive). Crystal structures with the trigonal (rhombohedral) crystal system are usually described using a hexagonal coordinate system. Then, the primitive trigonal lattice gets threefold primitive (rhombohedral centering). n-fold primitive means that the unit cell contains n vertices. The reciprocal lattice, Λ*, is spanned by the basis (Eq. 2), a*i =

a j × ak , j = i + 1mod3, k = i + 2mod3, i = 1K 3; ai ⋅ a*j = δ ij (2) V

with δij the Kronecker symbol. The reciprocal lattice shows the same symmetry as the direct lattice with one exception. The bcc direct lattice has a fcc reciprocal lattice and vice versa. Lattices are not only invariant under translational symmetry but also under point symmetry group operations. A point group operation always leaves a point, line or plane invariant, the symmetry element: center of inversion, 1 (read "one bar"), mirror plane, m, n-fold rotation axes (proper rotation or symmetry element of the first kind), N, n-fold rotation-inversion axes (improper rotation or symmetry element of the second kind), N . The 32 crystallographic point groups (crystal classes) are the subset of point groups that leave at least one type of 3D lattice invariant, i.e. n is restricted to 1, 2, 3, 4, 6. The full symmetry of the 14 Bravais lattices is given by the following symmorphic space groups (Bravais groups): P 1 , P2/m (read "two over m"), C2/m, Pmmm, Cmmm, Immm, Fmmm, R 3 m, P4/mmm, I4/mmm, P6/mmm, Pm 3 m, Im 3 m, Fm 3 m. They result from the direct product of the translation group of the lattice and the holohedral point group of the respective crystal system. For the description of the symmetry of crystal structures, the space groups are needed. The 73 symmorphic space groups (24 of the first and 49 of the second kind) result from the direct product of the 14 translation groups with the respective, symmetrically compatible, point groups. The remaining 157 hemisymmorphic and asymmorphic space groups are their subgroups. They contain a new type of symmetry operations, consisting of a point group symmetry operation and a translation operation: screw axes Nm (proper operations), consisting of a rotation around an angle of 2π/N combined with a translation t = ai/N along the axis ai, with N =

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2,3,4,6 and m = 1...(N-1) and glide planes (improper operations). Glide planes combine a mirror plane with a translation parallel to it, either along an edge of the unit cell or along particular diagonal directions. Two times the same glide plane operation applied results in a translation by one period in lattice direction. The site symmetry of a point in a unit cell is the subgroup of the space group without translations. The Wyckoff position consists of all points of a particular site symmetry. Different Wyckoff positions may have the same site symmetry but are positioned on different symmetry elements. The set of all points symmetrically equivalent to one particular point is called the crystallographic orbit (point configuration, equipoint set). Every crystallographic orbit belongs to a particular Wyckoff position with a given multiplicity (important for the stoichiometry of a compound), which reflects the order of the point group symmetry at that site. A lattice complex corresponds to the set of all crystallographic orbits that can be generated within one type of Wyckoff position in all the space groups where it occurs. The 1731 Wyckoff positions of the 230 space groups can be uniquely assigned to 402 different lattice complexes. This concept may be useful for comparing crystal structures with different space group symmetry, having subsets of atoms forming the same type of polyhedra, for instance. The asymmetric unit (fundamental domain) of a space group is that part of the unit cell which fully fills space by application of all symmetry operations of the space group. Its boundary planes and edges are mirror planes and rotation axes, respectively, if there are any. In our example cP8-Cr3Si, there are two atoms in the asymmetric unit, which is defined in Fig. 5 (under the drawings). The asymmetric unit the in structurefactor-weighted reciprocal space is that part of it which fully fills space by application of all symmetry operations of the point group the space group belongs to. In the usual, experimentally easily accessible case of intensity weighted reciprocal space, the size of the asymmetric unit, i.e. the set of unique reflections, is determined by the Laue group. Laue groups are the 11 crystallographic point groups containing the centre of inversion. They describe the symmetry of diffraction patterns (intensity distribution), which do not contain phase information.

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Fig. 5. First page of information on space group No. 223, Pm 3 n , given in International Tables for Crystallography Vol. A (2002)3. In the headline, from left to right are listed: short international (or Hermann-Mauguin) symbol, Schoenflies symbol, point group (crystal class), which the space group belongs to, crystal system. Second line: space group number, full Hermann-Mauguin symbol, Patterson symmetry. Below the second line, the space group diagram is drawn. It shows the projection along [100] of the framework of symmetry elements of one unit cell. The symmetry elements inclined to the projection plane are given in their stereographic projections (diamond glide planes d and threefold axes along <111>) or in a perspective view (twofold axes and screw axes along <110> as well as threefold screw axes along <111>) . Below the space group diagram projections are shown of one unit cell (inclined to the projection plane by 0º, 6º and 12º, respectively) with one set of equivalent points in general position. Below these drawings, the location of the origin of the unit cell with respect to the symmetry elements is given. In the last two lines the asymmetric unit is defined , i.e. the smallest connected part of the unit cell from which, by application of all symmetry operations of the space group, the whole space is filled.

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Fig. 6. Second page of information on space group N°223, Pm 3 n , given in International Tables for Crystallography Vol A (2002). In the second line, the set of generators used for space generation is listed. The numbers in brackets refer to the point group symmetry operations listed on the next page. Below the information on atomic position is given. First column: multiplicity, i.e. number of equivalent atomic sites per unit cell related by symmetry; second column: Wyckoff letter for numbering positions; third column: point group symmetry of the site; fourth column: coordinates of symmetrically equivalent positions in a unit cell; last column: reflection conditions, i.e. reflections which are systematically extinct due to translational components of symmetry operations.

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Fig. 7. Third page of information on space group N°223, Pm 3 n , given in International Tables for Crystallography Vol A (2002). On top, the symmetry (plane groups) of special projections is given, together with the axes and origins of the projected cells. Under the heading maximal non-isomorphic subgroups are given: I translationengleiche subgroups (all translations retained), IIa klassengleiche subgroups (all point symmetry operations retained)obtained by decentering the conventional cell, IIb klassengleiche subgroups obtained by enlarging the conventional cell. For each entry, first is listed the index of the subgroup in brackets followed by the space group symbol with short symbol and space group number in brackets. On the right, the coordinate triplets retained in the subgroup a given by their numbers used for the general position in Fig. 6. Subsequently, the maximal isomorphic subgroups of lowest index are given, it corresponds to a 3 x 3 x 3 supercell in the shown case. This is followed by the minimal nonisomorphic supergroups, with I and II in the same meaning as before. Finally, the symmetry operations are given, which transform a point in x, y, z into the position listed under the same number in the list of general positions in Fig. 6.

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2.2. Symmetry of physical properties

How are physical properties of a material related to its crystal structure? The density of a material, for instance, not only depends on its chemical composition but also on its crystal structure. For example, the density of the different modifications of SiO2 (quartz) ranges between 2.20 Mgm-3 for the cubic high-temperature (HT) phase β-cristobalite and 4.35 Mgm3 for the tetragonal high-pressure (HP) phase Stishovite. The density is related to the mean atomic volume, which itself depends on the atomic environment type (AET), in particular the coordination number. The mass density is a scalar property, this means that it is isotropic on macroscopic scale. Generally, physical properties are described by tensors taking the intrinsic anisotropy of crystals into account. This anisotropy results from the anisotropic distribution of the electron density and atomic interactions (bonds) in a crystal structure and has the same symmetry as the crystal structure. For the macroscopic physical properties we are usually interested in, the translational part of the symmetry group is irrelevant. Consequently, the transformation properties of the materials tensors are determined by the point symmetry group of the crystal only. The point group symmetry of the property, Gproperty, cannot be lower than that of the crystal, Gcrystal , (Neumann's principle, Eq. 3) G property ⊇ Gcrystal

(3)

If we bring a crystal in a field it will respond to it depending on the way of interaction. Thus, we define an influence tensor Aij.. (temperature T, electrical field Ei, stress σij, etc.) of rank p and an effect tensor Bkl.. (electrical polarization Pi, strain εij, etc.) of rank q. Then the physical property is described by the materials tensor fijkl… (thermal expansion αij, electrical susceptibility χij, piezoelectricity dijk, elasticity cijkl, etc.), which has to be of rank (p + q), relating these two physical quantities to each other (Eq. 4) (4) or, in short form, in the Einstein notation (Eq. 5)

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(5) The above mentioned tensors are polar tensors. The magnetic field, for instance, is described by the axial vector Hi (pseudo tensor of rank 1). Polar and pseudo tensors differ by their transformation properties. While polar tensors of even rank are invariant under the inversion operation and those of odd rank not, for pseudo tensors exactly the opposite is true. Consequently, quartz will show the piezoelectric effect (polar tensor of rank 3) only in the noncentrosymmetric modifications such as trigonal αquartz, point group 32, and not as cubicβ−cristobalite, point group m 3 m . For the full symmetry description of physical properties, additionally to the 32 crystallographic point groups also the 7 Curie groups are needed (Table 1). These are the continuous limiting groups of the crystallographic point groups, i.e. ∞ , ∞m , ∞/m , ∞2 , ∞/m2/m , 2∞ , 2/m∞. They describe the symmetries of a cone (e.g. electric field Ei), a cone with an angular momentum around its axis, of a double cone with the same (e.g. magnetic field Hi), the opposite and no angular momentum (e.g. tensile stress σij), a sphere with angular momentum on every point of its surface and a sphere without angular momentum (e.g. temperature), respectively. The Curie groups also describe the symmetry of some textured materials. Textured materials are polycrystalline, with a crystallite distribution other than statistical. A texture mostly originates from processing, for instance, by sheet-metal rolling or wire drawing. Important for the existence of some effects are that some groups contain polar axes, have no inversion center or no mirror plane (Table 1). If we study physical properties of a crystal which is already in an external field, for instance under uniaxial pressure (see Fig. 7), then we have to consider for the crystal symmetry those symmetry elements that are common to the crystal without the field and the field without the crystal (Curie's principle) field field G property ⊇ Gcrystal = Gcrystal ∩ G field

(6)

For the example shown in Fig. 7, we have the following relationship (Eq. 7)

Crystallography of Complex Metallic Alloys field field G property ⊇ Gcrystal =

4 2 2 4 2 4 2 ∞ 2 = 1 = 3 ∩ mmm m m m m mm

Table 1. Characteristics of the 32 crystallographic and 7 continuous point groups (Curie groups).

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(7)

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(a)

(b)

Fig. 7. Cubic crystal with components of the stress tensor drawn in. (a) without external field, (b) with uniaxial stress applied.

3. Fundamentals of X-ray diffraction The kinematic theory describes a diffracted X-ray beam as superposition of elementary waves arising in a crystal independently from each other. Based on this approach, crystal structure and diffraction pattern are mathematically related by the Fourier transform. The crystal structure is defined in direct space, the diffraction pattern in reciprocal space (Fourier space). The integrated intensity, I(H), of the diffracted beam results to ⎛ e2 ⎞ 2 λ 3V I ( H ) = I 0 ⎜ 2 ⎟ pL • F ( H ) TEG ⎝ mc ⎠ ω Vuc2

H=

3

∑h a

* i i

or

(8)

H = ha* + kb* + lc*

i =1

with H the diffraction vector, I0 the intensity of the primary beam, p the polarization factor, L the Lorentz factor, ω the angular velocity, V, Vuc the volumes of the crystal and its unit cell, respectively, T the transmission factor, E the extinction factor and G the correction for anomalous scattering. The structure factor F(H) corresponds to the Fourier transform of the electron density distribution function, ρ(r), and contains the full structural information

Crystallography of Complex Metallic Alloys

F (H) =

∫

ρ ( r ) exp ( 2π iHr ) dr and ρ ( r ) =

unitcell

1 V

235

∑ F ( H ) exp ( 2π iHr ) H

(9) Due to the translational periodicity of an ideal crystal it is sufficient to integrate over just one unit cell and F(H) ≠ 0 only at the reciprocal lattice points

⎧ Λ* = ⎨H = ⎩

3

∑h a

* i i

i =1

⎫ | hi ∈ Z ⎬ ⎭

(10)

In case of disorder breaking the translational symmetry, the integration has to go over the whole crystal and diffracted intensities are also found in between the reciprocal lattice nodes. The intensity of Bragg reflections is given by

( )

I (H) = I H = F (H) F* (H) = F (H)

2

(11)

We see from Eq. 11 that the intensities of reflections H and H are equal. This relationship is called Friedel's law. It has the consequence, that the symmetry of the intensity-weighted reciprocal space is inherently centrosymmetric. It can be described by the 11 Laue groups, i.e. the centrosymmetric crystallographic point groups. One also sees from Eq. 11 that from experimentally accessible intensities only the amplitudes of the complex structure factors can be determined. The phase information is lost and structure solution mainly means recovering phases either computationally or by complimentary experiments. If we rewrite Eq. 11 in the form

I (H) =

N

N

∑∑ f ( H ) f ( H )T ( H )T ( H ) cos ⎡⎣2π H ( r k

l

k

l

k =1 l =1

k

− rl ) ⎦⎤ (12)

then we get the Patterson function (autocorrelation or pair-correlation function) P(r) by inverse Fourier transform of the experimentally directly accessible intensities P (r ) =

∑ I ( H ) cos ( 2π H ⋅ r ) H

(13)

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The Patterson function is the vector map of the structure, the convolution of the structure with its inverse. Therefore, the Fourier transform of the intensity-weighted reciprocal space is sometimes called vector space. Its possible symmetries, the Patterson symmetry, can be described by one of the 24 symmorphic centrosymmetric space groups. Their respective point group symmetry correspond to the 11 Laue groups. The continuous electron density (Eq. 9) can in good approximation be described as superposition of isolated spherical atoms located at rj

ρunitcell ( r ) = ∑ ρ atom j ( r )

(14)

j

and its Fourier transform by the spherical atomic scattering factor or form factor fj(H) f (H ) =

∫

ρ atom ( r ) exp ( 2π iHr ) dr

(15)

atom

When scattering is treated in the second-order Born approximation, two correction terms, f j' ( H , λ ) and f j" ( H , λ ) , are needed for X-ray energies close to the absorption edges of atoms to account for resonance scattering. f j ( H ) = f j0 ( H ) + f j' ( H , λ ) + if j" ( H , λ )

(16)

Using the atomic form factor we can rewrite Eq. 9 F (H) =

N

∑f j =1

j

( H )T j ( H ) exp ( 2π iHr j )

(17)

with the temperature factor (Debye-Waller factor), T j ( H ) , considering the 'smearing' of electron density by thermal vibrations. It can be written in the form 2 ⎡ 2 ⎛ uj ⎞ T j ( H ) = exp ⎢ −2π ⎜ ⎟ ⎢⎣ ⎝ d hkl ⎠

⎤ ⎥ ⎥⎦

(18)

in the exponent is the expectation value (ensemble average) of the mean square displacement of the j-th atom from its equilibrium position, in a

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direction perpendicular to the reflecting lattice plane (hkl) with distance dhk. The ensemble average over all atoms of a crystal is equivalent to the time average of a single atom. Thus, the temperature factor results from the combined displacements of N atoms from the 3nN independent vibrational modes. With 1 d = H we can rewrite Eq. 18 2 T j ( H ) = exp ⎡ −2π 2 ( u j ⋅ H ) ⎤ ⎥⎦ ⎣⎢

(19)

and expanded to T j ( H ) = exp ⎡⎣ −2π 2 h 2 a*2u x2 + k 2b*2u 2y + l 2c*2u z2 + 2hka*b*u x u y + 2hla*c*u xu z + 2klb*c*u y u z ⎤⎦

(20) or, simpler, in matrix form T j ( H ) = exp ⎡⎣ −2π 2 HT ⋅ U j ⋅ H ⎤⎦

(21)

where Uj is a second rank tensor with U klj = uk ul . The expression exp ⎡⎣ −xT ⋅ U j ⋅ x ⎤⎦ describes the probability of finding the atom j in an infinitesimal box displaced by the vector x. The equation xT ⋅ U j ⋅ x = C describes an ellipsoidal surface on which the probability density is constant (‘thermal ellipsoids’). Instead of Uj, sometimes Bj is used, defined as Bklj = 8π 2U klj and the temperature factor becomes ⎡ 1 ⎤ T j ( H ) = exp ⎢ − HT ⋅ B j ⋅ H ⎥ 4 ⎣ ⎦

(22)

In case of the 'isotropic temperature factor’, the probability ellipsoid is just constrained to a sphere. Sometimes it is useful to calculate the 'equivalent isotropic temperature factor’ from the anisotropic displacement factor. It is defined by Bj =

(B

j 11

+ B22j + B33j 3

)

(23)

The atomic displacements are approximately inversely proportional to the strength of chemical bonds and to the atomic mass, and directly proportional to the temperature. Typical values at 300K are: BAl = 0.88 Å2, BNi = 0.37 Å2, UAl = 0.0112 Å2, UNi = 0.0047 Å2 (atomic masses Al:

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26.982, Ni: 58.693). This corresponds to amplitudes (root mean square displacements) of u ≈ 0.11 Å and u ≈ 0.07 Å, respectively. The intensity of thermal diffuse scattering (TDS) in a distance q from the reciprocal lattice point H arises from the vibrational modes with wave vector q. While the contribution from optic modes remains roughly constant across the Bragg peak, that of the acoustic modes varies with 1/q2 and, therefore, is peaked just beneath the Bragg peak, that of the acoustic modes varies with 1/q2 and, therefore, is peaked just beneath the Bragg peak (Fig. 8). In a first approximation the TDS intensity is proportional to I ( H ) exp ( c sin 2 θ λ 2 ) , with c a constant.

Fig. 8. Bragg reflection (Gaussian profile) with underlying Lorentzian-shaped thermaldiffuse scattering (TDS) background. If the TDS is not separated from the Bragg intensities, too small thermal parameters would result from the structure refinements.

In structure refinements, the temperature factor is a fit parameter that does not only account for atomic displacements of dynamic origin. It includes also static displacements due to disorder or crystal defects (relaxation of atoms close to defects, impurities, etc.). This part can be singled out by temperature-dependent measurements. Since this is usually not done, the factor Tj(H) is nowadays called atomic displacement factor and the former thermal parameters are now denoted atomic displacement parameters (ADP). It should be kept in mind that the ADPs can be strongly biased if the data reduction is not properly done (missing or incorrect absorption and extinction correction, missing or incorrect separation of Bragg intensity and thermal diffuse scattering,

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incomplete data set,...), symmetry is wrongly chosen or atomic positions are not properly occupied (wrong type of atom, incorrect occupancy factor,...). 3.1. Extinction

The kinematic theory neglects that the primary beam loses energy due to absorption and that the secondary, diffracted waves themselves interfere with the primary beam as well as with each other (multiple diffraction, 'Umweganregung'). The theory taking into account all these phenomena is called dynamic theory. However, since dynamic effects develop gradually as the primary beam penetrates the crystal, the kinematic approach yields fairly accurate results for sufficiently small thicknesses (Ak ≈ 1 µm), as long as the total scattered intensity is small compared to that of the incident beam and extinction effects can be neglected:

λ

Fabs k A ≤1 V uc

(24)

In practice, the kinematic theory can be applied to crystals exceeding Ak by several orders of magnitude due to their real structure. This behavior has been described in terms of a mosaic crystal. According to this approach, a crystal consists of mosaic blocks of ≈ 0.1 µm diameter (i.e. smaller than Ak) slightly misoriented with respect to each other. The mosaic spread is of the order of 0.001º in perfect crystals like silicon and may reach values of 0.1º and more in crystals of poor quality. The coherent interaction of waves occurs only within the individual mosaic blocks. The total diffracted intensity then is the incoherent superposition of these contributions. In case of perfect crystals with non-mosaic structure and sizes exceeding Ak, the dynamic theory has to be applied. In most cases dynamical effects weaken the strongest reflection intensities only. This can be corrected by the extinction factor. It corrects for primary extinction, which accounts for the slight drop in intensity due to dynamic effects in each mosaic block as well as for secondary extinction considering the continuous weakening of the primary beam by reflection in each mosaic block (Fig. 9).

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(a)

(b)

Fig. 9. Schematic drawing visualizing extinction effects. (a) Primary extinction takes multiple diffraction into account; (b) secondary extinction considers the weakening of the primary beam due to energy transfer into the reflected (diffracted) beam.

There are mainly two approaches for describing dynamic scattering, Darwin's treatment and Laue-Ewald's treatment. The term dynamic theory comes from the dynamic interaction (energy flow) between incident wave and reflected (diffracted) waves. Darwin considered a crystal of thickness A = Nd as a stack of N thin crystals of thickness d < Ak for which the kinematic theory is valid. Taking the continuous energy loss of the primary beam in each thin crystal plate into account, one obtains for the integrated reflectivity

I ( H ) = I0

8 ⎛ e 2 ⎞ 1 + cos 2θ Nλ 2 F (H) ⎜ ⎟ 3π ⎝ mc 2 ⎠ 2sin 2θ

(25)

The main difference compared to the integrated intensity according to the kinematic theory is that the intensity is now proportional to |F(H)| instead to |F(H)|2. According to the Laue-Ewald approach, the interaction of electromagnetic waves (X-rays) with the crystal, i.e. a periodic electron density ρ(r), dielectric constant ε and polarization χ = ε -1, is analyzed by a solution of Maxwell's equation for the induction vector D ∂ 2D D = −c 2 curlcurl 2 ε ∂t

(26)

Resulting from the interference of the initial and all excited waves in the crystal, an electromagnetic wave field is set up in the crystal. Its amplitude varies with the lattice periodicity. If the diffraction condition

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is visualized with the Ewald sphere, the reflection condition is fulfilled with the lattice point not exactly on the surface of the Ewald sphere. This is caused by refraction of the X-ray beam at the vacuum/crystal interface, which changes the direction of the X-ray beam in the crystal slightly. 3.2. Multiple diffraction in the Ewald construction

The Ewald construction is a very powerful visualization of Bragg's law (Eq. 27)

k − k0 = H , k = k0 =

1

λ

, H =

1 or 2d hkl sin θ = nλ d hkl

(27)

with the wave vector of the primary beam, k0, of the diffracted (reflected) beam, k, the diffraction vector H = ha* + hb* + hc* , reflection indices h, k, l, reciprocal space basis a* , b* , c* , wave length λ, distance between net planes of type (hkl), dhkl, diffraction angle θ, and order of the diffracted wave n. An example for the Ewald construction is shown in Fig. 10. Bragg's law is met for any reciprocal lattice point that sits exactly on the Ewald sphere. This can be achieved by either rotating the reciprocal lattice around its origin (monochromatic radiation) or by varying the size of the Ewald sphere as it is the case for the Laue method, where white radiation is used. If more than one reciprocal lattice point, besides the origin, is on the surface of the Ewald sphere, then multiple diffraction takes place. In the example shown in Fig. 10, the primary beam is diffracted into the directions k and k'. All reciprocal lattice points on the Ewald sphere, 3 10 and 6 10 , can act as new origin for the diffraction vector H" = 900 or -H" = 900, connecting them. This means, if the indirectly excited reflection 900 is strong (large structure amplitude), intensity from the 3 10 reflection would flow into the 6 10 reflection. Thus, a weak or even systematically extinct 6 10 reflection could appear strong due to multiple scattering ('Umweganregung'). The larger the unit cell and the higher the perfection of a crystal, the larger is the problem of multiple diffraction. The worst case applies for high-symmetry icosahedral quasicrystals because their reciprocal space is densely filled with Bragg reflections. Therefore, multiple diffraction is

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pervasive. However, since most reflections are too weak to be observed by in-house measurements (X-ray tube, rotating anode) they also do not contribute significantly to multiple diffraction. This is different for synchrotron radiation. On the one hand, the high brilliance of synchrotron radiation makes it possible to measure a wealth of very weak reflections, which contain important structural information. On the other hand, the weak reflections are those which are most biased by multiple scattering. How to overcome this problem? If the data are collected on a four-(or more)-circle diffractometer, the method of choice would be to perform Ψ-scans (i.e. scans around the diffraction vector) or to measure the intensities at positions where multiple diffraction has the least influence. In case of area detectors, the crystal could either be measured several times mounted differently in each case, or, with a single mount, several times at slightly different wavelengths.

Fig. 10. Ewald construction showing the geometry of multiple Bragg scattering. The origin of the reciprocal lattice and the primary beam have both to coincide on the same point on the Ewald sphere. Bragg's law is met if a reciprocal lattice node sits exactly on the surface of the Ewald sphere. The relative sizes of the Ewald sphere and the reciprocal lattice are related to CuKα-radiation (λ = 1.54051 Å) and the Samson phase, β-Mg2Al3 (a = 28.239 Å). If more than one reflection is on the Ewald sphere, in our example 3 10 and 6 10 , multiple diffraction takes place.

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3.3. Twinning and intergrowth on the example of CoAl3

A nicely faceted sample looking like a single crystal not necessarily is always a single crystal. By careful inspection one may find facets with reentrant angles, which do not exist in case of single crystals. This indicates twinning, i.e. an intergrowth of at least two single-crystal individuals in a definite orientation relationship. The mutual orientation is defined by the twinlaw. The twinning symmetry operation can be a rotation (axial twins), a reflection on a plane (reflection twins) or on a point (inversion twins), translation for a part of the lattice (translation twins) or combinations of these. Twins can result from phase transformations where the lost symmetry elements of the HT phase appear as twinning element of the LT phase. Twinning can already occur during growth due to pseudosymmetry, for instance, if a tetragonal crystal shows a pseudocubic metric. It can also occur, in special cases, under the influence of external fields. For instance, in case of ferroelastic crystals just under uniaxial stress. The twin individuals can have a common sublattice (coincidence lattice). Then the twin index n is defined by the ratio of the number of lattice nodes per unit cell of the twin lattice to that of the crystal lattice of the twin individuals. In case n = 1, the twin individuals have a common reciprocal lattice, all reflections of the twin individuals perfectly overlap. In this case, called merohedral twinning, the twin and its individuals have the same Laue symmetry and the twin operation belongs to the Laue class of the crystal. In case n > 1 (n = 2 hemiedry, n = 4 tetartoedry etc.), the Laue symmetry of the crystal is lower than that of his lattice. Then the twin operation transforming the n twin individuals into each other belongs to the Laue group of the crystal lattice but not of the crystal itself. In all these cases, twinning would not be detected just by inspecting the diffraction patterns except the twin ratio clearly deviates from one. Only a careful examination of an statistical analysis of the diffraction intensities and particularly the ADPs resulting from the structure refinements may indicate presence of twinning. An example, where twinning can already be seen on the diffraction pattern is depicted in Fig. 11. This is a growth twin of CoAl3 (τ2Co4Al13) , a = 39.71(8) Å, b = 8.11(8) Å, c = 32.09(8) Å, β = 108.01(8),

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P21/m. It just has a 2D common sublattice, i.e. the (011) plane that is also the twin plane. In reciprocal space, this corresponds to the hk0 section. The reflections of type hk0 overlap completely, all others to a varying amount. This makes quantitative data collection and integration of Bragg reflection intensities very difficult.

Fig. 11. h0l reciprocal space sections of twinned monoclinic CoAl3 taken from a sample with nominal composition Al70Co15Ta15. The reciprocal lattices belonging to the two twin orientations are marked (black and dark grey) as well as reflections of oriented intergrown tetragonal TaAl3 (white) (ctetr||bmono). The picture at bottom left is an enlarged part of the upper figure. The relationships between the reciprocal lattices of the twin individuals are shown at bottom right.

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4. X-ray diffraction data based structure analysis Basically, highly accurate quantitative structural data can be obtained from • diffraction methods about the globally averaged crystal structure if based on Bragg reflections (XRD, SAED, CBED, NS,...); • diffraction methods about the real crystal structure if based on full reciprocal space information (XRD, NS, coherent XRD); • spectroscopic methods about globally averaged local structures such as AETs or clusters (NMR, EXAFS,...). Qualitative or at best semi-quantitative structural data can be obtained from • electron microscopic methods (HRTEM, HAAD-STEM,...), • surface imaging methods (STM, AFM,...). In the following, we will focus on the method of choice in most cases, single-crystal X-ray diffraction. In case of polycrystalline samples, the 3D reciprocal space information is projected onto one dimension. This makes ab initio structure analysis basically much more difficult and for CMA with giant unit cells almost impossible. A structure analysis consists of three parts: data collection and reduction, structure solution, structure refinement and modelling. If only Bragg data are to be collected, usually in-house experiments are sufficient. If diffuse intensities are to be taken into account, synchrotron radiation will be very advantageous. Employing an area detector for the measurements, automatically yields the full reciprocal space information. There is no danger of overlooking weak superstructure reflections, reflection splitting due to twinning etc. Furthermore, Bragg and diffuse scattering can be measured at the same time on a reasonable time scale (hours to days). With a point detector a higher spatial and intensity resolution can be achieved together with a lower background and, if needed, a good energy resolution. For giant unit cells, however, even the Bragg (not to speak about diffuse intensities) data-collection time becomes unfeasibly large, i.e. of the order of weeks to months. The data reduction entails the correction of raw data for the background, absorption, Lorentz-polarization, extinction etc. as well as the integration of the Bragg peaks. The solution of the crystal structure means the determination of the phases of the

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experimentally accessible structure amplitudes. This can be done by a toolbox of methods: statistical direct methods based on relationships between reflections (triplets, quartets, etc.), Patterson techniques in combination with anomalous dispersion effects and many more. Once the phases for at least a subset of reflections are restored, the resulting structure model is refined. This is usually done by the least-squares method, minimizing a residual factor against the observed intensities. The refined model has not only to show a good fit but it has to be chemically and physically reasonable. A good check of the model would be a quantum-mechanical simulation based on first principles. What do we want to know about a crystal structure, what are the parameters defining the model to be refined? We want to know its space group symmetry, its metrics (lattice parameters), atomic coordinates, atomic displacement parameters (ADPs), site occupancies (probability of finding an atom at this site), kind of disorder (distribution functions). Furthermore, it may be of interest to know all these parameters as function of external fields such as temperature and pressure. The symmetry of the crystal structure gives information on the principal existence or non-existence of physical properties. The coordinates not only allow the geometrical description of a crystal structure (structure model), they are also needed as input for quantum-mechanical calculations. The electron density distribution contains information on chemical bonding, the thermal parameters (i.e. the dynamic part of the ADPs) allow the derivation of interaction potentials and force constants. In case of disorder, the entropic contribution to the stabilization of a compound can be calculated. 4.1. Crystal quality

A prerequisite of an accurate structure analysis is a perfect single crystal. However, depending on the growth method, as-grown crystals are usually chemically not fully homogenous (e.g. radial gradient in composition). Thermal annealing can lead close to an equilibrium state at a given temperature, T. This is possible on a laboratory time scale (days to months) if T is not too far from melting temperature (T< 1/2 Tm). Otherwise even geological timescales (centuries to millions of years)

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may no be sufficient for equilibration. A few words on terminology: we can never get an ideal crystal, this is just a mathematical object (infinite, strictly ordered, at rest). The goal is to obtain a perfect crystal, the best possible real crystal in its thermodynamic equilibrium state. It still contains thermal vacancies in statistical distribution, impurities, phonons. One has to keep in mind that also a disordered state can be an equilibrium state at finite temperature! This means, a strongly disordered crystal can be a perfect crystal. If defects are introduced such as dislocations, the crystal quality deteriorates and the crystal is no more perfect. With other words, the average-structural correlation, i.e. the average long-range order, is maintained in the case of a disordered structure but not in case of a highly defective structure. In Fig. 12, an example of a not well-equilibrated crystal is shown. By cooling down from the growth temperature, the HT-structure with all its inherent disorder (thermal vacancies, Al/Mg disorder, etc.) is frozen in over a large temperature range (depending on the cooling rate). On the way to RT, the structure partially relaxes until it stops somewhere in an illdefined metastable structural state. The larger the crystals are (e.g. Czochralski grown.), the larger is the problem due to a smaller cooling rate and a larger radial temperature gradient. After annealing and quenching, the quality of the sample has greatly improved.

Fig. 12. hk0 reciprocal space sections of Mg38.5Al61.5 reconstructed from imaging-plate data4. The upper image was taken at ambient temperature, the lower one at 150 ºC. A comparison of the Bragg reflections in the white-outlined rectangles shows that reflections are sharper and more intense in the annealed sample. The diffuse scattering also gets much weaker (see, e.g., encircled area).

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4.2. Structure refinement

The refinement of the crude structure model resulting from a structure solution procedure gives the best possible model of the average structure. Its quality depends on the quality and size of the data set as well as on the model parameters refined. The quality of the refinement is difficult to determine but very important for comparing the quality of different models. For CMAs this is particularly difficult, one has to use R-factor plots as function of reflection classes and statistical tests. One has to keep in mind that Bragg reflections contain only information of the average structure. In a refinement based on Bragg data, an averagestructure model has to be refined. It may contain split positions, chemically disordered and partially occupied sites. This makes modelling and the parametrization of the model much more complicated as well as quantum mechanical simulations. There are mainly two different ways used for refining a structure model against diffraction data. Traditionally, the weighted differences between observed and calculated structure amplitudes have been minimized. However, it makes more sense from a physical and statistical point of view to minimize the weighted differences between the observed and calculated intensities. The weight used for a reflection intensity is usually taken inversely proportional to its standard deviation. It is impossible to estimate ⌠(|F|) from ⌠(|F2|) when F2 is zero or (as a result of experimental error) negative. The diffraction experiment measures intensities and their standard deviations, which after the various corrections give Fo2 and ⌠(Fo2). It is very important, to include all measured reflection intensities into a structure refinement and not, as is often the case, only reflections with intensities larger than some threshold value (say, I > 3 ⌠(|F2|)). Particularly in case of pseudosymmetric or twinned structures weak intensities are crucial to distinguish between different models. The quality of a structure refinement is checked by reliability factors (R-values, R-factors, R-indices). Usually are given (see Eq. 28) an unweighted R-factor, R1, an weighted one, wR2 as well as the goodnessof-fit, S, which in the ideal case equals one,

Crystallography of Complex Metallic Alloys 1

n

∑F

i

R1 =

− Fi

obs

i =1

N

∑F

i

i =1

obs

clc

249 1

2 2 2 2 ⎧ n ⎧ n ⎡ obs 2 − F clc 2 ⎤ ⎫ ⎡ obs 2 − F clc 2 ⎤ ⎫ i i ⎪ wi ⎣⎢ Fi ⎪ ⎪ wi ⎣⎢ Fi ⎪ ⎥ ⎥ ⎦ ⎦ ⎪ ⎪ ⎪ i =1 ⎪ = , wR 2 = ⎨ i =1 N , S ⎬ ⎨ ⎬ 2 n− p obs 2 ⎤ ⎡ ⎪ ⎪ ⎪ ⎪ wi Fi ⎢⎣ ⎥⎦ ⎪⎩ ⎪⎭ ⎪⎩ ⎪⎭ i =1

∑

∑

∑

(28) where n is the number of reflections and p is the total number of parameters refined. The common weighting scheme is wi = 1/[ σ2(Fo2) ]. One cosmetic disadvantage of refinement against F2 is that R-indices based on F2 are larger than (more than double) those based on F. The R-index for Maximum-Entropy (MEM) calculations is not a reliability factor. Its value is meaningless, it is just a convergence indicator. Besides these R-values, further measures of the accuracy and reliability of results are • counting statistics; • data-set merging R-values (Rint, internal R-factor); • completeness and redundancy of the data set; • good absorption correction (min/max transmission); • good extinction correction • low residual electron density; • small standard deviations of refined parameters; • physical and chemical plausibility of the structure. Factors affecting the quality of diffraction data are • crystal quality (mosaic spread, high defect density, inclusions of a second phase, chemical inhomogeneity); • winning (domain structures); • strong absorption; • strong extinction; • strong fluorescence; • multiple scattering; • exture in case of powder XRD; • .....

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5. Structural disorder and diffuse scattering The investigation of average crystal structures based on Bragg diffraction data is nowadays a matter of routine. Highly optimized instruments for measuring Bragg reflections and sophisticated computer programs for solving, refining and validating crystal structures are available and thus several hundred thousand accurately described crystal structures are now deposited in databases. In contrast thereto detailed information about disorder in crystals is known for only a few examples. A reason for this discrepancy is that quantitative measurement and modelling of disorder based on diffuse scattering is less straightforward and requires more expert knowledge than the investigation of average structures. A qualitative or semi-quantitative interpretation of diffuse scattering, however, can be relatively easily given. Often this is sufficient for a basic understanding of the underlying disorder phenomena. Consequently, we will provide a toolbox that enables beginners to 'read' diffuse intensities. Of course, the discussion is restricted to very simple examples, which illustrate basic principles. For a more comprehensive overview the reader is referred to review articles and textbooks5-8 An on-line tutorial about diffraction physics, which also covers disorder and diffuse scattering, may be found under reference9. 5.1. What can we learn from diffuse intensities?

It is well known that a crystal structure obtained from Bragg scattering alone represents a superposition of all unit cells in a crystal. As a consequence only single site properties like coordinates, occupancy and ADPs are obtainable and information about pair correlations is lost. Figs. 13a-c show three simple, chemically disordered structures, which illustrate this effect. In the first case, atomic sites are randomly occupied by black or white atoms, in the second case the atoms prefer same atoms as next neighbors, while a tendency for an alternating sequence of black and white atoms is found in the third example. The structures clearly represent different chemical information, but the Bragg intensities and therefore the average structures cannot be distinguished: the average unit cells consist of one atomic position, which is half-and-half occupied by a

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black and by a white atom. From Bragg scattering alone nothing can be said about neighborhood relationships in the real structure. The structures, however, can clearly be distinguished by means of diffuse scattering, because information about pair correlations is preserved.

a

b

c

d

e Fig. 13. Crystal structures and diffraction patterns from some 1D disorder models. Chemical disorder is shown in a) - d), displacive disorder is present in e).

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5.2. Reading diffuse scattering

5.2.1. Angular dependence Angular dependence of diffuse intensities relative to Bragg intensities allows distinguishing displacive/orientational disorder from chemical disorder. A typical sign for presence of displacive disorder is that diffuse scattering is weak close to the origin of reciprocal space and tends to become stronger with increasing scattering angles (Fig. 13e). Diffuse intensities from chemical disorder show an angular dependence similar to Bragg scattering: close to the origin of reciprocal space diffuse scattering is strong and becomes weaker with increasing scattering angles (Fig. 13a-d). Initial information about the nature of disorder and about atoms involved in disorder may also be obtained from the average structure. Partially occupied sites are a hint for the presence of chemical disorder. Unusual large and/or strongly anisotropic ADPs indicate strong displacive or orientational disorder. These interpretations are to be understood as geometrical descriptions, which may differ from chemical interpretations. For example, a well resolved split position is described by two or more partly occupied sites, but chemically it would be understood as displacive or orientational disorder. 5.2.2. Width The width of diffuse intensities gives information about the length scale over which information about local order is transported or about the size of well ordered domains. It can therefore be understood as a measure for the degree of order within disorder. Broad features indicate that order is restricted to small regions, narrow features represent large correlation lengths. An example is shown in Figs. 13c and 13d. The inverse of the half-width of a diffuse feature gives an estimate about the typical propagation length of order. 5.2.3. Location The location of diffuse maxima provides information about the local translation symmetry of the disordered objects. In Fig. 13b diffuse

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intensities are found beneath the Bragg positions, what means that the local translation symmetry is one average unit cell and thus a homogenous sequence of atoms is present. In Fig. 13c the diffuse maxima are located in-between the Bragg positions. Consequently, the local translation is twice as long as it is in the real structure, i.e. an alternation of black and white atoms is preferred. 5.2.4. Dimension The dimension of diffuse features is a measure for the dimension of disorder. In general, nD diffuse features result from objects that are disordered in n dimensions and long-range ordered in 3 – n dimensions. In the case of one dimensional, streak-like diffuse scattering the disordered units are long-range ordered in directions perpendicular to the streaks, while disorder propagates along the streak direction. Typical representatives are stacking faults in layered structures, e.g. disorder in the sequence of hcp and fcc domains. Other examples for streak-like scattering are lamellae-like structures or scattering from interfaces between well-ordered domains (domain walls). The intensity modulation along the streaks gives information about correlations along disorder propagation. In an hcp/fcc disordered structure this could be the probability that an AB sequence is followed by an A or by a C layer. Diffuse scattering condensed in sharp layers is coming from structural units that are long-range ordered perpendicular to the layers and disordered in the other two directions. Examples are chains or 1D stacks, which are perfectly ordered along the chain/stack direction but disordered relative to neighboring chains/stacks (e.g. position, orientation or chemical composition). It is frequently found in host-guest systems where the host builds 1D tunnels, which accommodate a guest structure that is well ordered within a tunnel but weakly correlated with neighboring channels. Finally, in the case of 3D diffuse scattering disorder is effective along all directions. Diffuse scattering coming from thermal displacements (‘thermal diffuse scattering’, TDS) is a common 3D diffuse feature. Another example for 3D disorder, which is frequently found in metallic alloys, is the presence of structure motifs (‘clusters’)

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that are well ordered within a finite range, but weakly correlated with neighboring clusters. 5.2.5. Extinctions The interpretation of systematic extinctions in diffuse scattering is analogous to Bragg scattering. Extinction rules are not always strictly fulfilled. In such cases one can conclude that corresponding symmetry (glide plane, screw axis, Bravais centering) is preferred but subject of disorder. Additionally, diffuse scattering may show systematic extinctions, which are not observable in Bragg patterns. Here we restrict the discussion to the most important case, i.e. 1D or 2D sections through the origin of reciprocal space that are free of diffuse intensities. Since sections in reciprocal space correspond to projections in real space, it can be concluded that the structure appears to be perfectly long-range ordered if projected onto the corresponding line or plane.

a)

b)

Fig. 14. a) shows an up-down disordered structure (top) and the corresponding diffraction pattern (bottom). The structure is perfectly ordered when projected along the vertical direction and therefore no diffuse scattering can be observed in the zero layer. In b) the structure remains disordered, when projected vertically and therefore diffuse scattering is also present in the zero layer. The origin of reciprocal space is in the center of the diffraction patterns. Note that the average structure is the same in both cases.

The structure in Fig. 14a shows up-down disorder that is not visible in a vertical projection and thus no diffuse intensity is visible in the zero

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layer of the diffraction pattern. A similar diffraction pattern would be observed if the chains would be well ordered orientationally, but randomly displaced along vertical directions. In Fig. 14b also the projected structure is disordered and consequently diffuse scattering is present in the zero layer. 5.3. Diffuse scattering in β '-Mg3 Al 2

The rhombohedral β'-Mg2Al3 phase is the low temperature phase of Mg2Al3, which was comprehensively described in reference4. It forms from the structurally closely related cubic high temperature phase, βMg2Al3, and is twinned, showing cubic diffraction symmetry. The average structure of β'-Mg2Al3 shows no indication of disorder, i.e. there are no partly occupied atomic sites and no unusually large ADPs. Nevertheless, a synchrotron diffraction experiment shows a weak, but rich pattern of 1D and 3D diffuse scattering phenomena (Fig. 15). Since there is no significant disorder in the bulk structure of β'-Mg2Al3 it may be presumed that a major amount of observed diffuse intensities is coming from domain walls and strain between twin domains or from residuals from the high temperature phase. This assumption is supported by the observation that diffuse intensities clearly decrease after annealing the melt-grown sample, i.e. during annealing twin domains grow and the total volume covered by domain walls is reduced. The diffuse pattern can be divided in three main classes. First, diffuse streaks coming from domain wall scattering connect Bragg reflections along cubic <100>, <110> and <111> directions. Streaks along <111> are very weak what means that cubic {111} domain walls are rare. The other two streak systems have approximately the same integral intensities, but different profiles and reciprocal space dependencies. In both cases the streaks show maxima at Bragg positions, but streaks along <110> have a smaller half-width than <100> streaks, indicating thicker {110} domain walls. Further, <110> streaks are extinct in the hh0 sections, i.e. {110} domain walls are coming from displacements perpendicular to face diagonals of the cubic cell. The second system of diffuse intensities is a pattern of strong 3D diffuse maxima beneath Bragg reflections, which intensities generally become stronger with increasing scattering angles. The

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appearance resembles patterns from thermal diffuse scattering, but it was found that diffuse intensities decrease after annealing the sample. Therefore it is more likely that the underlying disorder is not coming from lattice dynamics but from displacive distortions in interfaces between twin domains, e.g. long-range relaxations around voids.

Fig. 15. Reciprocal space section of twinned β'-Mg2Al3 at room temperature after annealing the sample. The indices refer to a pseudo-cubic setting. The data set was measured at Swiss-Norwegian beamline at the synchrotron source ESRF.

Finally, the third dominant kind of diffuse intensities is a hollow diffuse sphere around the origin. The exact radius of the sphere cannot be determined exactly because the diffuse intensities are highly structured, but it corresponds roughly to interatomic distances. The correlation length of disorder estimated from the reciprocal of the radial half width is about 10Å, what is smaller than a unit cell. The very broad diffuse background at high scattering angles could be a smeared-out second order maximum of the diffuse sphere, similar to diffraction patterns from

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liquids. Since the diffuse intensities from this pattern slightly decrease with decreasing temperature the origin of the diffuse feature might be a dynamic effect e.g. atomic diffusion, but the available experimental results do not allow a definite conclusion. References 1. 2. 3. 4. 5. 6. 7.

W. Steurer, , Z. Kristallogr., 217 267 ( 2002). J.-E. -Jørgensen., S., E., Rasmussen, Acta Crystallogr. B 38, 346 (1982). International Tables for Crystallography Vol. A. Kluwer (2002). M. Feuerbacher et al., Z. Kristallogr., submitted to, 2006. T. R., Welberry, B. D Butler, Chem. Rev. 95 2369 (1995). F Frey, Z. Kristallogr., 212 257 (1997). J. T. R Welberry, in: Diffuse x-ray scattering and models of disorder, Oxford University Press: Oxford (2004). 8. J. M. Cowley, Diffraction Physics, Elsevier: Amsterdam (1995). 9. http://www.uni-wuerzburg.de/mineralogie/crystal/teaching/teaching.html. Further reading: C. Giacovazzo: Fundamentals of Crystallography. Oxford Science Publications (1992). B.K. Vainshtein: Fundamental of Crystals. Springer (1994). V.K. Pecharsky & P.Y. Zavalij: Fundamentals of Powder Diffraction and Structural Characterization of Materials, Springer (2005). B.T.M. Willis & A.W. Pryor: Thermal Vibrations in Crystallography. Cambridge University Press (1975). R.E. Newnham: Properties of Materials. Oxford University Press (2005).

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CHAPTER 10

ELECTRONIC PROPERTIES OF ALLOYS Östen Rapp Solid State Physics, IMIT, KTH-Electrum 229, 164 40 Stockholm-Kista, Sweden E-mail: [email protected] Electronic properties of alloys consisting of metals are reviewed starting from the level of an undergraduate curriculum in solid state physics. These alloys occur in all three different forms of solid matter, viz. crystalline alloys, amorphous alloys, and quasicrystals. A selection of electronic properties of these materials will be discussed, including examples from conduction in good metals to electron transport in insulators. The presentation is intended as a background to complex metallic alloys, the main theme of the School.

1. Introduction 1.1. Useful concepts This subsection is a brief summary of some useful concepts for descriptions of electrons. It is intended as a reminder for students who have only briefly encountered the subject before. In metals the Free Electron Model provides a simple and useful tool for describing electrons in metals. It must be warned from the beginning however, that in almost any real material, the electronic structure is too complex for this model to be quantitatively valid. The usefulness of the model is instead related to the fact that the simplicity of its concepts has a remarkably general validity, albeit not for calculations the more so for an illustrative physical description of electrons in metals. In free electron-like models the number of electrons means the number of electrons in the conduction band, those electrons which no longer belong to the electronic shell of any particular atom, but to the 259

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whole crystal volume. Assume that each ion in a metal crystal contributes Z electrons to this conduction electron gas. Z is an integer, which for simple elemental metals,(those with only s and p electrons at the Fermi surface) can be directly read off from the column number in the Periodic Table, i.e. Z = 1 for Na and Cu, 2 for Mg and Zn, 3 for Al and In, etc. Noninteger values of Z are found in alloys. E.g. for brass (Cu0.5Zn0.5), the average Z = 1.5. In transition metals the concept of an easily accessible value of Z is less clear, as discussed below. With N atoms on crystal volume V, each contributing Z electrons to the conduction electron gas, the electron density n in the crystal is (1) NA is Avogadro’s number, ρd the mass density, and M the molecular weight. n is of order ~1028-1029m-3. A convenient parameter for labelling all these electrons is their wave vector k in the free electron wave function Ψ ~ ei kr. The vectors k = (kx, ky, kz) form wave vector space or k-space, which for crystals is also the reciprocal space. For free electrons k scales with velocity v (k = mv/ħ), and k-space is then a scaled velocity space. The free electron model1 is based on two assumptions: (i) electrons obey the Pauli principle, and there is only one particle in each quantum state. In the state k there can then be at most two electrons, each in one quantum state, one is (k, spin↑), the other (k, spin ↓). (ii) The only contribution to electron energy is the kinetic energy. Hence there is no spatially dependent potential, and the electron energy ε is obtained from : (2) Allowed k points cannot be arbitrarily densely packed. This can be realized from the wave function by considering e.g. periodic boundary conditions over a crystal, volume V, of lengths Lx, Ly, Lz so that for an orthogonal structure V = Lx .Ly .Lz.. The boundary condition Ψ(r) = Ψ(r + Li), i = x, y, z then leads to the following set of allowed k-points;

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(3) The whole electron gas requires half as many k points as the number for n given above (2 spins/k-point). For free electrons the quadratic form of the energy, Eq. (2), makes it clear that filling of electrons (increasing n) proceeds in spherical shells in k-space to minimize electron energy. When all electrons are accommodated, they fill the Fermi sphere, of radius kF, the Fermi wave vector. kF is readily calculated; (4) This equation says that the total number of electrons equals (two electrons/k-point).(number of k-points per unit volume of k-space in a crystal of volume V). (volume in reciprocal space of a sphere with radius kF). From Eq. (3) the distance between two adjacent k-points is 2π/Li, in three dimensions the volume of one k-point is (2π)3/V, and the density of k-points in reciprocal space is the inverse of this number. Taking n = 5.1028m-3 in Eq. (4), one finds kF ≈ 1.1.1010 m-1, and from Eq. (2) the Fermi energy εF = ε(kF) of order ~ eV. Thus kF is huge on the scale of the distance between k-points for all macroscopic samples (L~µm or larger). The drift velocity of electrons acquired in an electric field applied over the metal is of order some cm/s, and the corresponding change in the electron |k| is of order a few 100 m-1, a small fraction of kF. Similarly the thermal energy at temperature T, i.e. kBT, is, in the solid state generally at most some percent of εF. Therefore, under most experimental conditions, the state of an electron in the interior of the Fermi sphere cannot change: all states within the additional energy provided by the experiment are already occupied by other electrons. Only electrons with k states within some kBT :s from εF can change states. This is the region where the Fermi factor, (the probability that an available electron state is occupied) is <1 and >0. These electrons can contribute to e.g. electrical or thermal conduction. It is this minority of the conduction electrons which give metals their characteristic properties. An important concept is the electron density of states, N(ε), the number of available electron states per unit energy range. The electrons

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discussed above, which change their state in (normal) solid state experiments, is then of order N(εF) kBT. To present N(εF) as a material property independent of sample size, it is useful to give it in units of (number of states)/(eV. atom), or (number of states)/(J. m-3) or (number of states)/(J.kg). N(εF) can be readily calculated, in the free electron model from the formalism above by counting the number of electrons in a spherical shell at energy ε, of a thickness δk corresponding to δε, or e.g. by taking the derivative of Eq. (4) with respect to ε; (5)

1.2. Alloying in simple metals When an element is dissolved into another element, the concentration limit for a single phase alloy depends markedly on the elements considered. A first condition for extended mutual solubility is comparable ionic radii of the elements. E.g., for Cu and Ni the radii differ by ~2, and there is a full range of solubility; CuxNi1-x can be produced in the fcc phase for 0≤ x ≤1. However, strong chemical inclination to form intermetallic compounds can result in quite restricted solubility, also when the ratio of ionic radii is favorable. For Cu and Zn, the radii differ by ~9%. Extended solid solubility is not possible, and this phase diagram is instead characterized by a number of intermediate phases. The Hume-Rothery rules describe empirical correlations between Z, the number of electrons/atom, and the solubility limits of simple alloys. These special Z values were obtained from the distance between the Fermi surface and the closest point on the Brillouin zone boundary. With an fcc alloy as example; the first diffraction peak in fcc is (111). The shortest reciprocal lattice vector G is then |G(111)|. The distance from center to zone boundary, |G(111)/2 = π√{3}/a for cubic lattice parameter a. Z at Fermi surface contact with zone boundary is then obtained from Eq. (4) with kF = π√{3}/a and V/N = a3/4 for fcc. One finds Z π√{3}/4 ≈ 1.36, close to the observed solubility limit in several fcc phases.

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Table 1. Hume-Rothery rules for Cu-Zn alloys Structure of phase fcc bcc γ hcp

Z obs. 1.38 1.48 - 1.51 1.58 – 1.66 1 78 – 17 87

Z calc. 1.36 1.48 1.62 1.69 for ideal hcp

Table 1 illustrates Hume-Rothery rules for Cu-Zn alloys. For the various intermediate phases, the Table shows $Z^{obs}$ as calulated from the observed concentration at the fcc solubility limit, or the observed phase boundary ranges for the other phases, and $Z^{calc}$, the value calculated from the free electron model for Brillouin zoneFermi surface contact. The γ structure is somewhat more complicated than the others with a cubic cell of 52 atoms and ideally 84 valence electrons in the unit cell. The ideal Z value for this structure is thus Z ≈1.62. For the hcp structure the c/a value can vary substantially from the ideal value √ 8/3 and the trend from Table 1 is therefore qualitatively correct. The conclusion from this and further comparisons is that the HumeRothery rules are often well obeyed for simple alloys with not too different ionic radii, in fact remarkably well obeyed, considering the simplification of the free electron model. Electron band theory can give some physical understanding for this result. Everywhere on the zone boundary in Fig. 1 there is an energy gap. When the Fermi surface reaches the zone boundary, it thus costs the additional energy of this gap to continue growing in the same direction. The electrons may therefore minimize their energy by a change of the crystal structure to one which can accommodate a relatively larger free electron sphere without touching the zone boundary. This is the case in the sequence fcc, bcc, γ, hcp. 1.3. Electrical resistivity The forces on an electron in a metal in electric field E are –eE from the field, and a breaking force - mvd/τ , oppositely directed to the drift velocity vd, and thus || E. m is the electron mass, and τ the relaxation

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Fig. 1. A section of the 1:st Brillouin zone of an fcc metal is shown. It is built up by (111) and (200) surfaces. These surfaces are normal to one of the two shortesst reciprocal lattice vectors and cut them in half. Two free electron Fermi surfaces are shown. For Z=1 it is entirely within the zone. First zone contact, the larger circle, occurs at Z ≈ 1.36. Experimentally phase transformations are observed e.g. for 38 at% of 2-valent Zn in Cu, or 12 % of 4-valent Ge in Cu, in rather close agreement with the calculated Z=1.36.

time, the mean time between collisions. In a stationary condition their sum =0; (6) Then j = -ne vd = (ne2τ/ m)E. The proportionality factor between j and E is the conductivity, σ The resistivity ρ=1/σ is (7) Here D is the diffusion constant for electrons = in three dimensions. vF is the Fermi velocity. The last expression of Eq. (7) can be derived from the middle one by free electron formalism. It nevertheless represents a great improvement in physical insight, since it emphasizes that it is not the total n which is relevant but, besides τ other Fermi surface properties such as v and D(εF). Therefore one often uses this generalized Eq. (7) qualitatively in situations where the free electron model is not strictly valid.

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Scattering gives resistance to electric conduction. In simple crystalline alloys the two most important scattering mechanisms are phonons and impurity scattering. Phonon scattering is inelastic. Impurity scattering is the sum of contributions from all types of defects, foreign atoms, impurities, etc., and is elastic For alloys without magnetic scattering one thus considers the T dependent electron-phonon scattering ρph(T), and ρi from the constant impurity scattering. In the formalism of the first members of Eq (7): (8)

τie is the elastic scattering time, τ the elastic scattering time. The rates of these scattering mechanisms are added in the parenthesis. Since the number of phonons increases with increasing temperature, phonon scattering becomes more frequent at elevated temperatures and τιε decreases with increasing T. Fig. 2 illustrates ρ(T) for good metals. The simplest application of Eq. (8) is to dilute alloys in the approximation that ρph does not depend on impurity concentration, c, and ρi increases linearly with c. Curves for the observed ρ(T) vs T for varying c are then parallel, and displaced for increasing c at distances proportional to c.

Fig. 2. ρ(T) for three alloys with the same host element are schematically shown. To first approximation the curves are parallel. Characteristics of good metallic behavior are (i) the flattening of ρ(T) at low temperatures, where impurity scattering dominates, and (ii) the linear ρ(T) at high temperatures, with dρ/dT>0.

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This relation is named Mathiesen's rule. Today it can be considered as a zero'th approximation, useful for rough estimates of ρ(c,T) in dilute alloys. Deviations from Mathiesen's rule, DMR, Δ(c,T) are ubiquitous. This field of study has been challenging enough to be a research area on its own. One can define Δ(c,T) by: (9) Δ(c,T) is a complicated function of c and T. It is not understood except for certain limits. An example will be discussed in Sec. 1.7. The linear ρ(T) at high T is expected from phonon scattering. At T above about the Debye temperature θ, phonons have attained their maximum energy kBθ , and further increase of the energy of the crystal is achieved by excitation of more phonons. The number of phonons is α T in this region, and the rate of scattering of electrons by phonons is α (the number of phonons). At low temperatures the number of phonons increases much faster than linearly with increasing T, (the specific heat α T3 at low enough T’s). In addition, the small phonon energy at low T only allows electron-phonon scattering with reduced scattering angles. In this temperature region the resistivity increases much faster than linearly with T since electron-phonon processes both become more frequent and more effective. 1.4. Magnetoresistance and Hall effect The magnetoresistance and the Hall effect are the two most prominent effects in electron transport in magnetic field. The magnetoresistance is the change of ρ when an external magnetic field B is applied. It depends on B and T. One usually defines it as (10) However, for giant and colossal magnetoresistance materials, where ρ decreases by orders of magnitude for small B’s, one reverses the sign in Eq. (10) and takes ρ (B,T) in the denominator instead. Numbers in excess of 106 can be obtained in such cases. Here the definition of Eq. (10) is used.

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The Hall effect refers to the appearence of a transverse electric field in a magnetic field. With B ┴ to j this Hall field, Ey, is ┴ to both j and B. These effects can be described as follows: in the relaxation time approximation, the probability for scattering at a time between t and (t+dt) is dt/τ. The electrons that do not scatter, [i.e. a fraction (1- dt/τ) conserve their momentum ħk(t)]. In addition, the external force F contributes with momentum Fdt during dt. The momentum thus develops in time as

Here F = -e(E + vxB) has been inserted. In a stationary state the current does not depend on time. With B║z, the x and y components of the last equation are therefore both =0. Using j =-neħk/m, one then has: (11)

(12) For B = 0, Eq. (7) is retrieved from both equations. In a magnetoresistance experiment there is no transverse current and e.g. in Eq. (11) jy = 0. Hence σ retains its zero field value, ne2τ /m. There is no magnetoresistance in the free electron model. For the Hall effect, with current in the x-direction, and after application of a magnetic field in the z-direction, the initial internal current in the y-direction due to the Lorentz force ceases when an internal electric field Ey has been built up in the -y direction which exactly cancels the Lorenz force. In the steady state, and with jy = 0 one finds from Eq. (12): (13)

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RH, the Hall coefficient, is the proportionality constant between the Hall field Ey and jxBz. In the free electron model RH = -1/ne. It is convenient to introduce the cyclotron frequency ωc=eB/m into the description of magneto-transport phenomena. The mean angle an electron turns between collisions is ωcτ.ωc τ 1 defines the low field 1 the high field region. For simple crystalline alloys one region, ωc τ is almost always in a low field region, since increased scattering from foreign atoms makes τ small enough that the high field region is out of reach. The magnetic field is then a weak perturbation. In such materials the magnetoresistance, Δρ / ρ, is small and >0. Typically Δρ / ρ ~ B2 up to at most Δρ / ρ ≈10-6-10-5 in fields of some Teslas. This phenomenon is small and featureless. A two band model with charge carriers of different signs in different bands (electrons and holes) can account for this small magnetoresistance. In simple metals it can occur by deviations from a free electron model and spherical Fermi surfaces. The free electron model for noble metals is an oversimplification. In reality, bands flatten out towards zone boundaries (ZB). Instead of dε / dk > 0 at the ZB, as in Eq. (2), dε / dk = 0 there. More electrons are then accommodated in this direction without much increasing energy. The Fermi surface swells and makes contact with the ZB in that direction. Due to the proximity of the ZB in the [200] directions it also grows slightly in that direction. Since the number of electrons is constant the Fermi surface shrinks in other directions. This is illustrated in Fig. 32. On the belly the charge carriers are electrons, and on the necks they are hole-like. This describes a two band model. Elemental pure metals can display a rich variety of magnetoresistance phenomena including huge values of the magnetoresistance. One is then usually in the high field limit; electrons make many turns in between scattering events. Features of the Fermi surface can then be reflected, also in very pure polycrystalline samples. This field is outside the present subject. Other cases with substantial magnetoresistance in simple crystalline alloys are ferromagnets close to the Curie temperature, and superconductors close to their Tc. A magnetic field aligns spins, reducing spin scattering, and decreasing the resistivity. A magnetic field breaks superconducting pairs, which exist close to and above Tc. Their

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[111]

Fig.3. Fermi surface for noble metals, (Z = 1). In the 8 [111] directions the free electron Fermi sphere has swollen and makes contact with the zone boundary. The volume is conserved from Fig. 1. (Figure adopted from 2).

contribution to the remaining supercurrent is then destroyed. This magnetoresistance is thus positive. The Hall effect gives by Eq. (13) a convenient way to roughly estimate n from experiments. However, such estimates must be taken with caution. An example of deviations from Eq. (13) already in simple binary alloys, Cu1-xAux, is shown in Fig. 43.

Fig. 4. –RH for fcc Cu1-xAux (redrawn from 3) Horizontal lines were calculated from Eq. (13) with n(Au) and n(Cu) from Eq. (1). RH varies over a larger range than expected from Eq. (13), and shows significant and varying temperature dependence.

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Here there is a complete fcc miscibility, Z is constant, and the lattice parameter increases by 13% when going from Cu to Au. It can be seen that the magnitude of -RH,, its concentration dependence, and its temperature dependence all deviate from Eq. (13). Again a two band model, based on electrons at the belly and holes in the necks can describe the observations. When two channels contribute to RH, each component is weighted with a function of σ in that channel. Therefore relaxation times, and effective masses of each channel enter, and RH is no more ~1/n. For Cu1-xAux it was found4 that for increasing x the relaxation time on the necks decreased relative to that on the belly and hence also that σ (neck) decreased with increasing x. In this way a good description of data was obtained. This result is semi-empirical. The Hall effect can be quite complicated also in simple crystalline alloys. RH may depend on both T and B, in addition to unexpected alloy concentration dependence, and can also display sign changes as a function of these parameters. 1.5. The thermoelectric power When a metal bar, say along x, (or a semiconductor) is subjected to a temperature gradient, electrons at the hot end diffuse to the cold end where they find states of lower energy. Excess negative charge on the cold end sets up an electric field, counteracting thermal diffusion. For an open circuit in steady state these two mechanisms balance exactly. The thermoelectric power S is defined from this electric field and the temperature gradient by (14) In practise one must connect wires of a different material to the ends of the sample. If not, the circuit consists of two identical parts in opposite orientation and E = 0. Experiments thus employ junctions between different materials. S is evaluated by application of Eq. (14) to all parts of the circuit. One must now distinguish between three electron velocities; (i) The Fermi velocity, vF (= ħkF/m), here v for short, (ii) the drift velocity vd, Eq. (6), and (iii) vth from the thermal gradient.

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|v| (=|(vx, vy,vz|) is of order 105-107m/s, |vd| and |vth| several cm/s. In a stationary state (ii) + (iii) cancel:

vd + vth = 0

(15)

One can derive an expression for S in a free electron model. To find vth consider first a (1-dim) bar along x, with dT/dx >0. Half of the electrons arrive to x from the right, and had their last collision at x + vxτ, where the velocity is vx(x + vxτ). The other electrons travel towards larger x with velocity vx(x - vxτ). The resulting electron mean velocity vth at x is then

To make contact with 3-D thermal properties, assume an isotropic velocity distribution, so that = = = /3. Then

Cv/n is the specific heat per electron. Only electrons within some kBT’s of εF can contribute to the thermal properties. They are a fraction ~kBT /εF of n and they each contribute 3kBT/2 to the energy, U. Thus U~3(kBT)2n/2εF, and Cv/n = (dU/dT)/n ~ 3kB2T/εF. A more precise evaluation of Cv gives1 Cv/n=(π2kB2T)/( 2εF). Hence (16) With Eqs. (6) and (16) in Eq. (15) one obtains a relation between Ex and dT/dx as in Eq (14), giving S: (17) This S is called the diffusion thermopower. S< 0 for electron charge carriers, and is expected to decrease linearly with T in the free electron model.

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Fig. 5. Thermoelectric power for fcc Ag1-xAux alloys [redrawn from Ref 4]. The dashed line is Eq. (17) with the free electron value for Ag of εF=5.48 eV.

Results for Ag1-xAux alloys4 are shown in Fig. 5. Ag and Au are fairly free electron-like. There is a complete mutual solubility in the fcc phase, a constant Z = 1, and the difference in lattice parameters of the elements is < 1%. This is thus one of the simplest binary alloy systems one can find. Eq. (17) is roughly approached at high T's and not too low Au concetrations. There are marked deviations for small x at all temperatures. At low T there is a peak in S which is attenuated for increasing x. In a general treatment one must take into account the energy dependence of the relaxation time: τ = τ(ε). The thermopower is more complicated than the conductivity, since it also depends on energy derivatives at the Fermi surface. An expression for the diffusion thermopower is then5 (18) The positive sign of S at small dopings and T above about 100K in noble metals is believed to be due to a negative sign of dτ/dε over at least part of the Fermi surface. Phonon-drag has been neglected in Eqs. (17-18). It is due to phonon influence on electron diffusion. At high T, phonon-phonon scattering is more intensive than phonon-electron scattering, and the dominating

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mechanism for dissipation of heat. The influence on electron drift is then negligible; there is no phonon drag. As T is reduced, phonon mean free path increases, and scattering from other phonons becomes insufficient for phonon thermal relaxation. Phonon-electron scattering then provides this relaxation. Electrons take up momentum from the phonons, affecting S. At still lower T the number of phonons decreases fast and these processes vanish. In the simplest case electrons take up momentum in the direction of the heat flow. It then appears that phonons drag electrons with them to the low temperature end, thus the name phonon-drag. In this case S has a negative peak. However, when electron-electron Umklapp processes occur, the situation can be reversed, as shown in Fig. 6.

Fig. 6. Electron-phonon Umklapp scattering. Two adjacent cubic Brillouin zones are shown with free electron Fermi spheres. An electron in state kF in zone 1 is scattered by a phonon K into kF’ on the Fermi surface in zone 2. kF’ is equivalent to kF”. The conservation rule kF + K-G} = kF” is obeyed. kF”has a significant component opposite to phonon heat flow (║K), giving a positive phonon-drag contribution to S.

Some of these features can be seen around 25 K in Fig. 5. With Umklapp processes the phonon drag peaks are positive. In Fig. 5 they are strongly attenuated with increasing Au concentration, since alloying shortens phonon life times. 1.6. Superconductivity Superconductivity is one of the most striking electronic phenomena of

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matter and a large research field of its own. Due to the remarkable properties, the questions of the magnitude of the critical temperature Tc and how to enhance it have always accompanied this research. More specifically; what alloys are superconducting and is there a guiding principle for selecting elements and their concentration in order to make high-Tc alloys? A first question is: how is Tc affected by alloying? Excepting magnetic impurities one might guess that since superconductivity is an electronic phenomenon, more electrons, i.e. a second element with larger Z, would be favourable for Tc .This conjecture is untenable. Nb (Z = 5) is a bcc metal in the 4-d series. But Tc of the bcc phase increases with Zr (Z = 4) in Nb while it decreases with Mo (Z = 6) added to Nb. In fcc Al (Z = 3) addition of Mg (Z = 2) increases Tc (except for a small initial decrease due to smearing of the energy gap anisotropy). Also, Tc can change substantially with isoelectronic alloying, e.g. in bcc V with susbtitution of Ta (Z = 5), where a valence rule does not work. The Bardeen, Cooper, Schrieffer (BCS) theory from 1955 has proved to be highly successful for conventional superconductors, i.e. possibly excluding high Tc's. In a form later found by McMillan6 the main ingredients in a Tc formula are (19) Θ is the Debye temperature. µ* is a measure of the Coulomb repulsion between electrons. It is screened in metals, and falls off exponentially instead of the Coulomb 1/r term. λ is the electron-phonon interaction parameter, an attractive interaction mediated by phonons. It can be factorized as (20) Here is a squared electron-phonon interaction matrix element averaged over the Fermi surface, M the ionic mass and <ω2> the average of the square of the phonon frequencies. Tc in Eq (19) is linear in Θ and depends exponentially on λ-µ*. µ* does not vary much, and is often taken to be 0.1 for sp-metals and 0.13 for transition metals. The single

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Tc(K)

4-d bcc

N(εF) (states /eV/atom)

3-d bcc

N(εF) (states /eV/atom)

Tc(K)

most important parameter determining Tc is therefore λ. In situations where Θ and do not vary substantially, as in alloying in a single phase between neighbouring elements, Tc is thus mainly governed by N(εF). Fig. 7 shows Tc from experiments and N(εF) from band structure calculations for bcc alloys in the 3-d and 4-d series from reference 6. Z is again simply the column number. It can be seen that Tc in both series varies similarly to N(εF) (up to Z ≈ 6.2), justifying the approximation that the variation of the other parameters in Eq. (20) is negligible. This correlation gives a useful overview of Tc in transition metal alloys. For superconducting compounds related empirical rules exist as found already in the 50's by Berndt Matthias. If one simply identifies Z with the column number, one runs into problems when continuing after a complete transition metal series. E.g. in the 3-d series one has Z = 1, 2, 3, 4, 5,…..9, 10 in the sequence K, Ca, Sc, Ti, V, ……Co, Ni. What valence should Cu, the next element, have? Matthias took Z = 11, 12, for Cu, Zn, alloyed with transition metals, and their usual values (Z = 1, 2) in sp-metal alloys. Thus the full d-shell was assumed to participate in such alloys. This is clearly an oversimplification. One can say quite

Fig. 7. Relation between Tc [open symbols] and N(EF) [filled symbols] for 3-d and 4-d bcc alloys. For 3-d the bcc range is limited to TiV and VCr alloys. 4-d metals have a wider bcc range. A few results for MoRe (triangles) have been included. Re is a 5-d metal below Tc in the Periodic Table, a metal which is difficult to handle [redrawn from reference 6].

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generally that the convenience of a simple parameter Z is an attractive feature, and therefore it has been used also in situations where it is fitted or determined from conditions of that particular situation. The differences between such various approaches illustrate that the physical significance of Z then is weakened. However, Matthias' assumption works quite well for a great number of superconducting alloys. By compilation of available experimental data he found that Tc has maxima around Z = 4.7 or 6.5, with minima or small values in between or beyond these numbers, and that superconducting compounds with different Tc's tend to cluster around those values. Tcmax and the corresponding value of Z are shown for several crystal structures in the Table. It is remarkable that a similar rigid band-like N(εF) as in Fig. 7 survives in a number of different crystal structures. Table 2. Tcmax and the corresponding value of Z for several crystal structures

This relation is illustrated in some detail in Fig. 8 for the A15 structure with data from Roberts7. A15 is cubic of the form AB3 with Aatoms in bcc positions and B atoms in chains on each of the cube sides. Superconducting compounds cluster in two groups with Tcmax around Z = 4.7 and 6.5. When these rules were introduced, the largest Tc known, (also an A15 alloy), was about 17 K. Matthias’s rule served as a guideline where to search for new high Tc 's for 30 years up to the discovery of high Tc's. There are also exceptions to this rule. In the C14 hexagonal Laves phase, Tcmax occurs at 6.8K for ZrRe2 where the value of Z is 6. The most notable failure occurs for the high Tc superconductors. In YBa2Cu3O7-δ, Tcmax is 92K. Oxygen is not included in a rule for metallic valencies, but has an essential role for superconductivity in the CuO2 planes of the structure, where Cu valence is close to or somewhat larger than 2.

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Attempts to scale Tc in other ways with simple parameters have nevertheless continued. Examples are Tc vs superfluid density, ns, that is density of superconducting charge carriers, or Tc vs σ above Tc, or most successfully, ns vs Tc .σ.

Tc(K)

A 15

Z

Fig. 8. Matthias` rules for the A15 structure. Superconducting alloys cluster around Z = 4.7 and 6.5.

1.7. The electron-phonon interaction The electron-phonon interaction is the dominating source of the electrical resistivity at elevated temperatures as well as the main parameter determining Tc.. One can therefore expect a relation between ρ and Tc. Qualitatively this is already apparent from a look at ρ(273K) and Tc of elements.Cu, Ag, and Au have small ρ~ 1.5-2 μΩcm at 273K, indicating a small electron-phonon interaction. None of them are superconductors. The largest Tc’s for elements (at normal pressures) are 9.2 and 7.2K for Nb and Pb respectively, indicating strong electron-phonon interaction. In qualitative agreement ρ(273K) is 13.5 and 19.3 μΩcm, respectively. When studying the relation between ρ and λ two points must be considered in order to make a direct comparison:

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(i) The electron-phonon interaction in transport, λtr, is not exactly the same as λ in superconductivity. It differs by a factor (1-cosθ), an average over the Fermi surface of the scattering angle θ, taking into account that large angle scattering is more effective in resistivity. When there is no strong variation of Fermi surface shape with doping, this factor is ≈ constant. (ii) The strong temperature dependence of τ in ρ of Eq. (7) has no counterpart in superconductivity, which is not a transport phenomenon. However, a simplificaton is possible at temperatures above about the Debye temperature Θ, where ρ(T) is linear in T, and τie, the only (significantly) temperature dependent parameter in ρ(T), takes the simple form ~ħ/kBT. By studying instead dρ/dT, this factor thus becomes a constant. Combining these considerations one can write (21) The absence of observable superconductivity in the noble metals has been a challenge. It was found that superconductivity in fcc noble metal alloys could be achieved by dissolving into them small amounts of elements to the right in the Periodic Table, such as Al, Ga, In8. Some results for Ga-doping are shown in Fig. 9.

Tc(mK)

Au1-cGac

Ag1-cGac

Cu1-cGac

c Fig. 9. Tc vs Ga concentration c in noble metals (redrawn from reference 8).

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Data indicate an exponential growth of Tc with Ga doping, c, suggesting by Eq. (19) an increased electron-phonon interaction with c. By inserting Eq. (21) into Eq (19) it can be rewritten as (22) For dilute alloys the c-dependent variables are Tc and dρ/dT. Eq. (22) is illustrated in Fig. 10 for fcc Au-Ga and Ag-Ga samples.

Fig. 10. Eq. (22) for Au-Ga and Ag-Ga alloys (Ref 9). Straight lines are fitted separately to each alloy system. The lines are almost parallel reflecting small changes with doping of similarly shaped Fermi surfaces.

This result shows that an increasing electron-phonon interaction is responsible for the appearance of superconductivity. Furthermore, the intercept of the straight lines of Eq. (22) is -µ*, giving the estimates of µ*≈ 0.11 and 0.10 for Au- and Ag-doped alloys respectively. A full analysis, including some neglected factors in Eq. (19) gives µ*≈ 0.12 for both alloy systems9.

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Eq. (21) can also be used to study deviations from Mathiesen’s rule10. A c-dependent dρ/dT reflects a finite dΔ/dT, [from Eq. (9)], and a finite dλ/dc [by Eq. (21)].Formally from Eqs. (9) and (21);

Eliminating k by Eq (21) this is equivalent to (23) This gives a quite general result: deviations from Matthiesen’s rule which depend on temperature at T>Θ are due to a concentration dependent electron-phonon interaction. 2. Amorphous Metals 2.1. Consequences of the amorphous state Amorphous materials are characterized by absence of long range atomic order. Long range is important in this definition. For short distances, say 3-4 nearest neighbour distances, short range order exists, and the environment of an atom resembles that of the crystalline counterpart. At longer distances correlation between atomic positions vanishes. Absence of long range order implies loss of translational order and unit cell, and loss of the most powerful tool to describe crystals; the lattice and the reciprocal lattice. For electrons, Eqs. (1) and (2) are still useful. k is no more a good quantum number, but electron momentum is defined from ħk = mv as for free electrons. k-points in reciprocal space (metric m-3) form a discrete set due to the Pauli principle. There is no net of unit cells over which Eq. (3) can be applied. However, an uncertainty relation (Δx)(Δp)>h gives V.ħ3Δ3k ~ h3 in 3-dimensions, [Δx3 = V], and the volume Δ3k of a k-point is thus of the same order as for crystals, 8π3/V.. Eq. (4) is qualitatively correct. With no lattice, electron properties are isotropic, and the Fermi surface is spherical. It is not sharp as for high quality crystals. In the

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amorphous state there are many defects, and a short life time Δt for an electron wave ~eikr. By another uncertainty relation, Δt ΔεF ~ h. Hence ΔεF is large and the Fermi sphere is blurred. Due to the short range order in amorphous metals, X-ray diffraction is still a useful structural tool. Although nearest neighbour distances are not as well defined as in crystals, variations around mean values are small enough that an amorphous sample shows several somewhat smeared peaks. With standard methods in X-ray diffraction 3-4 such peaks are usually observed. The Brillouin zone is not defined. However, there is a shortest scattering vector, Kp = 4πsinθmin/λ. For crystals this expression for Kp = ⎪G|min, is the shortest reciprocal lattice vector. This is used to construct a pseudo zone in reciprocal space according to the same recipe as for Brillouin zones of crystals, with planes ┴ to each equivalent Kp through the points Kp/2. These concepts are of importance also for electron properties of amorphous metals. Examples will be given below on phenomena related to an electron filling kF ≈ Kp /2. One also finds simplifications in amorphous metals. A foremost one is an extended mutual solubility. In crystals the solubility limit often hampers further studies. E.g., in the fcc alloys in Section 1.7 one cannot much increase Tc beyond the values of Fig. 9, due to Hume-Rothery type arguments (Sec. 1.2). In amorphous alloys this restriction does not occur. One often encounters a much extended amorphous state. However, in A1-xBx, amorphization becomes increasingly difficult or unfeasible for x > 0.9 (x < 0.1). 2.2. Overview of electrical resistivity The study of physical properties of amorphous alloys accelerated in the 1960's, when the liquid quenching technique was developed. The frequent observation in amorphous metals of a negative dρ /dT attracted attention from the beginning. A compilation by Mooij of data for crystalline and amorphous alloys11 showed that the sign of dρ /dT was correlated with ρ(273 K). Fig. 11 suggests that for increasing ρ, there is a change of sign of (1/ρ) dρ /dT from positive to negative values when ρ(273 K) exceeds ~100-150 μΩcm.

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Fig. 11. The Mooij band (Ref. 11). Available data (1/ρ) dρ /dT(273 K) vs ρ were restricted to the area between two straight lines. dρ /dT = 0 for ρ ~100-150 μΩcm.

Typical results for ρ(T) of disordered alloys are shown in Fig. 12. Panel a) shows data12 for amorphous Ni1-xPx. dρ /dT decreases for increasing x and changes sign at x ≈ 0.24. ρ(293K) increases for increasing x (not shown) and is ≈160 μΩcm at x =0.24 in agreement with the Mooij rule11. Another way to introduce disorder is to damage a crystalline lattice by irradiation. Panel b) shows a LuRh4B4 thin film irradiated with α−particles13. Irradiation destroys the lattice, and leads eventually to an amorphous state. ρ saturates when, roughly, each atom has been knocked out of position by an α−particle. Saturation is apparent at the higher doses. In this case, the sign change of dρ /dT occurs at ρ ≈ 250 μΩcm, somewhat above the Mooij limit. However, a similar trend within one alloy system is maintained. In Fig. 12 the experimental control parameter is P-doping or irradiation dose. ρ and dρ /dT develop similarly in both cases. This can be qualitatively discussed in terms of electronic disorder quantified by the parameter F = ħ/(τ εF). τ is usually the elastic scattering time. For an ideal metal τ = ∞ and F = 0. From F and Eq. (7) it follows that increasing ρ corresponds to increasing disorder, by e.g. a decreasing τ from increased scattering, and/or a decrease in number density and density of states.

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Fig. 12. Panel a): Normalized resistivity of amorphous Ni1-xPx (Redrawn from Ref. 12). Panel b): Resistivity of a pristine and α−particle irradiated LuRh4B4 thin film. Irradiation doses in units of 1016 cm-2 are given on the curves. (Redrawn from 13).

The amorphous structure itself is thus neither necessary, nor sufficient for observing dρ /dT <0. One should also warn against the practice sometimes used, to associate a sign change of dρ /dT with a metal-insulator transition. Moderately large negative values of dρ /dT are compatible with a metallic state. The common trend in Fig 12 is that each alloy series develops from a (rather) good metal to metals which are weakly disordered. 2.3. Examples of explanations for dρ /dT < 0 in metallic alloys Several mechanisms have been suggested to explain a negative dρ /dT. This was hotly debated in the 70's and 80's. Agreement was never reached, and the problem instead faded away. A few theories are exemplified. (i)-(iii) below are not limited to low temperatures, the subject in focus here. (i)-Mott s-d model. The chemical potential η of a system is the energy required to change the number of particles by 1. For electrons in a metal at T = 0, this energy is the Fermi energi, εF. At T > 0, η decreases with T due to the broadening of the Fermi factor;

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(24) This effect is sometimes negligible. However, consider a metal with a narrow d-band overlapping a broad s-band (Fig. 13). Assume that conduction is primarily by s electrons. The scattering rate 1/τ is proportional to the number of available states, i.e. to N(ε) in the final band. Nd(η)>>Ns(η) and s-d scattering dominates. Hence 1/τ ∝ Nd(η).For increasing T, Nd(η) can decrease (Fig. 13), and hence s-d scattering and ρ also decrease. The leading T-dependence is as in Eq. (24), ρ(T) ~ T2, and d ρ /dT < 0.

d-band

s-band

Fig. 13. Mott s-d scattering. s electrons at η are scattered elastically into the d-band at η at a rate (1/τ) ∝ the d-band density of states. When η decreases slightly for increasing T, this scattering rate decreases and ρ can decrease with increasing T.

(ii) The tunnelling model. In an amorphous solid the atoms are in a landscape of a non-periodic potential V(r). Compared to crystals, V(r) has larger values at atomic positions, and varies somewhat between different sites (Fig. 14). In such a two-level system (TLS) tunnelling can occur between neighbouring sites. Considering a distribution of states in a real sample, thermal properties of amorphous insulators have been described. An example is the linear low T term in their specific heat, traditionally associated with an electronic contribution. Cochrane and co-workers generalized this idea to amorphous metals. They pointed out that an observed -lnT behaviour of ρ(T) indicated the existence of a low energy degree of freedom to which electrons can

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couple, and argued that the lack of periodicity could provide this effect. A contribution to the resistivity was found of the form (25) c is a constant >0 and Δ<1K. ρTLS is constant at very low T. For increasing T the first deviations are of the form ρ ~ -T2.

Fig. 14. Potential landscape in an amorphous metal. Coupling of electrons to an ensemble of two level systems like this one, can contribute to transport by tunnelling.

(iii) Ziman model and diffraction models. This topic will be treated by Prof Mizutani at this School, and is described in a textbook by him14 One can n ote that this theory gives a model for the temperature dependence of ρ over a wide range of T. The low temperature limit is of the form

ρ ∼ αΤ 2

(26)

The sign of α depends on the balance between a positive contribution from inelastic phonon scattering and a negative contribution from the Debye-Waller factor, describing the change of the static structure factor with temperature. A negative dρ(T)/dT is found in this model when kF ≈ Kp/2, where Kp is the first maximum of the structure factor. (iv) Quantum corrections. In crystalline alloys electron-phonon scattering usually dominates and τ >τie. However, in amorphous metals the strong impurity scattering can lead to τ << τie at low T (already below about 20-30 K). Then corrections to the Boltzmann theory occur. They belong to one of two major groups: (1) Weak localization (WL), and (2) Electron-electron interactions (EEI).

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(1) WL is a one-electron effect. During τie many elastic scattering events interfere. Partial one-electron waves can travel by elastic scattering in different closed loops back to the same point in space with preserved phase coherence. This interference gives enhanced probability for the electron to remain where it is, hence the name weak localization. ρ thus increases. However, spin-orbit interaction, characterised by a scattering time τso, can destroy this contribution, since it does not preserve the spin part of an electron. When τso <<\ τie, (strong spin-orbit scattering), the effect on WL is reversed. (2) EEI. In the Boltzmann picture all scattering events are independent. With intense elastic scattering this is no longer the case. EEI is a two-electron effect arising from interference between two scattered electrons.The corrections, Δρ /ρ, to the Boltzmann theory are: (27) The first term is from WL and -C√T is the EEI term. C contains a factor (e2/ ħ) . (D/kB)1/2, and a function of the screened Coulomb interaction Fσ . For superconductors there are additional terms. Usually, C > 0. For strong screening it can be < 0. An illustration of the two parts of Eq. 27 and low-T data15 for amorphous Cu65Ti35 are shown in Fig. 15.

Fig. 15. Illustration of WL and EEI in the resistivity of an Cu65Ti35 amorphous alloy15. The full curve is a fit of Eq. (27) to data, the dashed curve is a fit to the WL part only, the first term in Eq. (27).

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The temperature dependence of the WL part is due entirely to τie(T). One often takes τie = τo T^{-p (p > 0). τso is temperature independent. The importance of τso in Eq. 27 can be monitored by changing temperature. At low (enough) T one has τie >> τso and the second √ in [ ]is ~ constant. Then (Δρ(T)/ρ)WL ~ 1/√4 τie = Tp/2/2 τo1/2, which increases with increasing T. At high T, 1/ τie >>1/ τso, the terms within [ ] become -2/√4 τie, and (Δρ(T)/ρ)WL ~-Tp/2/τo1/2, decreasing with increasing T. These two WL-regimes are visualized by the dashed curve in Fig. 15. One can conclude that it is difficult to decide on a mechanism for a negative dρ/dT at low T from observations of ρ(T) alone. ρ(T) is a smooth function of T. Almost any theory with a small number of parameters can be fitted to data. Also, more than one conduction mechanism may contribute. 2.4. The magnetoresistance and quantum corrections The magnetoresistance of amorphous alloys is much larger than in simple crystalline alloys. Fig. 16 shows typical results.

Cu65Ti35

Fig. 16. Magnetoresistance of amorphous Cu65Ti35 in the range 45 mK < T < 9.2K, 0 < B < 4 T15.

It can be seen that Δρ /ρ reaches about 0.1% compared to ~ 10 ppm or less in simple alloys. This phenomenon is explained by quantum

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corrections from weak localization and enhanced electron-electron interactions. The following forms show the parameters of these contributions: (28)

(29)

Δρ (T,B) / ρ(T,D)

g* is the Landé factor, Fσ the screened Coulomb interaction. To first order Δρ / ρ = ρ . Δσ. Within quantum correction theories the large Δρ (Β) / ρ of amorphous metals is thus qualitatively explained by the large ρ. Eqs. (28, 29) are more complex than Eq. (27). Their usefulness for the experimentalist is based on two important facts: (i) The whole (B,T) plane is available for studies, giving more stringent conditions for fitting stability. (ii) Several parameters, τie(T), τso, D and Fσ, appear in both Eq. (27) and Eqs. (28, 29). Co-analyses of ρ(T) and Δρ (Β) / ρ can thus improve analyses. An overview of the form of the WL terms is given in Fig. 17. At t1 [low T] in Fig. 17, Δρ / ρ ~ B1/2, as for Δρ / ρ in Fig. 16 at most temperatures shown. At t2, Δρ / ρ ~ + B2, and at still larger t's one has Δρ / ρ ~- B2 with an amplitude which decreases with increasing t.

t = τSO / τie(T)

t1 t2

t4 t3 B (arb.units)

Fig. 17. The weak localization in magnetoresistance as a function of B and t = τso/ τie (T). t1 < t2
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The ~ +B2 region is reached at 9K in Fig. 16 at small B, with a B1/2-behavior at larger B. At still larger B a maximum in Δρ / ρ is found from Eq. (28), followed by a sign change to negative values. The curves in Fig. 16 illustrate the best description obtained with Eqs (28, 29). The magnitude of Δρ (B) / ρ , and its B and T dependences are qualitatively well described. The bulk of the literature in this field shows agreement between data and quantum interference theories at this level. Descriptions are seldom quantitatively accurate. This is seen in Fig. 16 at higher T. It has been found that in some cases a constant factor, somewhat larger than 1 in front of Eq. (28) may remedy such discrepancies. In the absence of a clear explanation, this remains empirical. For a moderately strong spin-orbit alloy, such as in Fig. 16, it is difficult to reach the region of a negative magnetoresistance by increasing the temperature. This is due to the increasing complexity to regulate temperature in magnetic field. A given accuracy of regulation, δT, gives resistivity fluctuations δρ [=δ T. dρ / dT ] which must be << Δρ (B) / ρ. However, Δρ (B) / ρ rapidly decreases for increasing T, while Δρ /dT is roughly constant. A sign change will also occur for increasing B. A parameter in Eq. (28) of type Bτso has a similar effect as τso / τie in Eq. (27): spin-orbit interaction is quenched when B (or 1 / τie) is large, and Δρ (B) / ρ (or dΔρ(T)/dT) becomes < 0. Few studies have been made since often large fields are required. An example is amorphous Cu57Zr43 in pulsed fields exceeding 30T16. This result is shown in Fig. 18. It provides evidence without any detailed calculations that Δρ (B) / ρ is controlled by weak localization. The evidence that quantum corrections describe the low temperature magnetoresistance suggests that also the low temperature ρ (T) can be analyzed within the same theory (such as e.g. in Figs. 15-16). 2.5. Hall effect and Thermoelectric power From a spherical Fermi surface and isotropic scattering one expects that the Hall effect and the thermopower of amorphous metals would be

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Fig. 18. The magnetoresistance as a function of √B for amorphous Cu57Zr43 up to fields in excess of 30T (redrawn from16).

simple and nearly free electron like. In general this is not the case, however. 2.5.1. Hall effect RH for amorphous metals is often positive, not negative as in Eq. (13). In addition it shows a small temperature dependence in contrast to Eq. (13). The positive sign of RH has been much debated. One possible explanation is hybridization between s and d electrons, which could lead to a negative δε /δk in the region where the d band is roughly half filled. At low temperatures quantum corrections are expected. For the Hall effect the weak localization vanishes, and only EEI contribute. The magnitude is twice the EEI term in Δρ (T) / ρ. In the notation of Eq. (27) (30) An example is shown in Fig. 19 with accurate data for RH of amorphous Cu43Ti5717. The temperature dependence of RH is often weak. In Cu43Ti57, RH decreases by 2% from 10 to 77K. Data from ~ 4-30K follow a √T dependence with a slope fairly close to that expected from Eq. (30). The discrepancy of 10% [ΔRH / RH) / Δρ ./ρ) ≈ 2.2 instead of

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2]17, could be due to a WL term which has not saturated, but still depends on T, and gives a contribution as in the left end of Fig. 15. The slope of Δρ ./ρ vs √T then underestimates C of Eq. (27).

Fig. 19. The Hall coefficient vs √T for amorphous Cu43Ti57. The full line is a fit of Eq. (30). EEI corrections are observed up about 30 K [redrawn from17].

2.5.2. Thermoelectric power The thermopower S of amorphous metals often displays an approximately linear T-dependence, with a positive slope instead of the negative one expected from Eq. (17). This can be explained within the diffraction model. When kF ≈ Kp/2 a positive contribution to S may dominate, similarly to the negative sign in dρ (T)/dT. In fact, a correlation between a negative sign of dρ (T)/dT and a positive S(T) is often observed in amorphous metals, consistent with this model. Fig. 20 shows an example of S(T) for amorphous alloys with data from18 . Quantum corrections to S are expected in the form of electronelectron interactions with a √T term in S(T). As mentioned, the thermopower is more complicated than other transport properties since it depends on energy derivatives at the Fermi surface. Quantum corrections to S of order a few percent are too small to be be clearly verified in the somewhat complex set of contributions to S(T). This field has been little studied.

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Fig. 20. The thermopower vs temperature for two amorphous alloys. S(T) is positive, and approximately linear for both alloys. [calculated from data of Reference 18].

In Fig. 20 there is no phonon drag peak in contrast to simple alloys, Sec. 1.5. This is due to the short phonon mean free path in amorphous metals, giving adequate thermal relaxation. In this case the amorphous structure thus leads to a simpler description of physical properties. A result based on this property will be discussed in Sec. 2.7. 2.6. Superconductivity A main consequence of amorphization on the electronic band structure is a smearing of the density of states. Peak values of N(εF) will decrease. In a valley, at Z ≈ 5.5 in Fig. 7, N(εF) increases. The overall N(εF) in a d-band is single peaked and a considerably smoother function of electron/atom number than in Fig. 7. Consequently, from the exponential relation between Tc and N(εF) from Eqs. (19, 20), smoothing of Tc is also expected. A relation between Tc and Z for amorphous metals was first found by Collver and Hammond in 1973 on Mo-Ru and W-Re films, Fig. 21. The fairly sharp peak and its position cannot be explained solely by band smearing. Further work has shown that this curve is rather an envelope for Tc data of amorphous metals, similar to Fig. 8 for the A15 structure. The valence difference ΔZ between the elements in Mo-Ru and

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25

20

Tc (K)

15

10

5 0 3

4

5

6

7

8

Z

Fig. 21. Superconducting Tc vs Z for amorphous superconductors shown by the dashed line in comparison with the full curve from Fig. 8.

W-Re is 2 or 1, and the peak value of Tc is ≈9K. The maximum Tc decreases with increasing ΔZ, and e.g. is 2K in Zr-Cu, where ΔZ = 6. The simple picture of band smearing in amorphous alloys is useful as a first approximation. Generally Tc for amorphous alloys goes up compared to the crystalline state for low density of states alloying elements, and down in the opposite case. It is useful to recall also what can be learned from mistakes: N(εF) is large in Nb, small in Al. Before the relation in Fig. 21 was known, Al in disordered thin films was found to have Tc ≈ 5.7K, compared to 1.2K in bulk. The search for high Tc superconductors then took up the idea to quench condense Nb [Tc = 9.2K, Fig. 7] with the plan to enhance Tc by a factor of 5 to 45 K. Tc was instead reduced by ≈ 50%. This was a first step towards establishing a relation as in Fig. 21. Disorder effects in superconductors When the effect of disorder is investigated in 2-D superconductors, the experimentalist can monitor an increase of disorder by decreasing the thickness of thin films. In 3-D alloys one can irradiate the sample to disturb or destroy the lattice. In this case however, the effect of Tc is a combination of a change of the density of states and the introduction of lattice disorder, which complicates the interpretation and may mask

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disorder effects. The smeared density of states of an amorphous superconductor can be utilized. Since N(εF) is already smeared, further smearing by irradiation is small, and the effect of electronic disorder is more straightforwardly studied. Quantum correction theories give the reduction of Tc as a function of the disorder parameter ħ/(εFτ). To first order this relation is (31) Tco is the starting value of Tc. [ ] contains superconducting parameters and logarithms of ħ/(εFτ). This expression shows that Tc.vanishes almost exponentially with increasing electronic disorder ħ/(εFτ). To compare with experiments one can approximate ħ/(εFτ). with the normal state ρ. From Eq. (7) one has ħ/(εFτ) = ρ (2e2n1/3) / (3π2/3ħ).n is the electron number density, and the slowly varying factor n1/3 is neglected. Fig. 22 shows Tc for neutron irradiated amorphous Zr-Cu, Zr-Fe, and Hf-Co samples, for a range of neutron doses19. Disorder induced depression of Tc of amorphous superconductors can thus be controlled by neutron irradiation. An initial value, [at Δρ = 0], of εFτ / ħ ≈ 1.7 is suggested by the results. With εF of order ≈ 1 eV, this corresponds to τ ≈ 5.10-16 s, illustrating the intense electronic scattering in amorphous metals.

Fig. 22. The reduction of Tc /Tco vs = the increase of Δρ / ρ for neutron irradiated amorphous superconductors. The curves were calculated from quantum interference theories for a few values of the initial εFτ / ħ (from19).

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2.7. Electron-phonon interaction 2.7.1. Estimates from S(T) It was shown by Opsal and coworkers (1976) that the electron-phonon interaction, which gives an electron mass enhancement by a factor 1 + λ(T), should be observable in S(T). λ(T) vanishes at high temperatures, and increases for decreasing T. At low T, λ(T), = λ in the BCS theory [Eq.(19)]. The absence of phonon drag in S(T) makes it possible to observe this effect in amorphous metals. In this case one has18 (32) S(T)low T is a zero temperature value estimated at low temperatures, and S(T)high T a value from the region where S ∝ T, i.e. S/T is constant. This analysis is facilitated by plotting data as S(T)/T vs} T, (Fig. 23). Data clearly show the mass enhancement factor 1+λ(T) which →1 at about 50 K in amorphous Nb-Ni and 100K in the Cu-Ti alloy. Estimates of λ from Eq. (32) are often somewhat larger than expected. Further work has shown that there are additional terms in Eq. (32) due to corrections to τ and electron velocity, which are ∝λ(T). These terms are difficult to calculate, but imply corrections in the direction to decrease λ.

Fig. 23. The data of Fig. 20 on the form S(T)/T vs T (redrawn from18).

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2.7.2. Estimates from dρ/dT As mentioned (Sec. 2.3), the negative sign of dρ /dT arises in the diffraction model from a balance between a positive term from the phonon scattering and a negativ term from the T-dependence of the Debye-Waller factor. It was shown20a that both these terms are ∝dρ /dT evaluated at a high temperature. The sign of dρ/dT is then determined by the sign of [(1-S(2kF)]. S is the structure factor. Amorphous Zr-based superconductors were studied in this model20b. λ was evaluated from measurements of Tc and (full) Eq. (19), and dρ /dT from measurements around 280K. The results are shown in Fig. 24. Eq. (21) can thus be extended to amorphous Zr-based alloys with dρ /dT <0. The slope of the line in Fig. 24 is k = -0.025K (nΩcm)-1, roughly of the magnitude estimated from the diffraction model. In Zr-based alloys electron spectroscopy has shown that Fermi surface properties are dominated by Zr d-electrons, while electrons from other ions have low lying states, and do not participate. In this sense, a second element in amorphous Zr-based alloys represents a way of changing only Zr properties. This fact is likely one reason that the relation in Fig. 24 works well with a single k. In amorphous Nb-Ni one finds disagreements in similar analyses.

Fig. 24. λ vs dρ /dT at 280K for amorphous Zr-based superconductors. In order of decreasing Tc and λ, the second element is Co, Ni, Pd, and three alloys of increasing Cu concentration20.

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3. Quasicrystals 3.1. Overview of the quasicrystalline state Quasicrystals are characterized by long range atomic order, and the absence of translational symmetry. Long range order is verified by sharp, crystalline-like peaks in X-ray diffraction. Absence of translational order is shown in electron microscopy, where symmetries such as five fold rotation axes have been observed. Such cells cannot fill space without overlap or empty space in between. Hence a unit cell does not exist. This is a remarkable state of matter, and a great surprise at the time of its discovery (1984). The basic question of solid state physics "where are the atoms?'' does not yet have a unique answer. Quasicrystals, QC’s, of several different forms exist. The 3-dimensional QC’s are generally icosahedral, and can be thought of as projections to 3-D from 6-D space, where the quasicrystals are periodic and form a simple 6-D hypercube lattice (P-type) or an fcc lattice (F-type). In 2-D there are several groups of QC’s which can be described as quasiperiodic planes of different symmetries (8-fold, 10-fold, or 12-fold), which are stacked periodically along the perpendicular direction. In 1-D the distinction between an incommensurately modulated phase and a quasicrystal has become diffuse. Quasicrystals thus have long range order in common with crystals, and absence of a unit cell in common with amorphous metals. This suggested that they would form an intermediate state between crystals and amorphous metals. The first QC’s discovered seemed to support this conjecture, since physical properties were in between those of their crystalline and amorphous counterparts. A striking example is Mg8Zn3Al2 in crystalline, amorphous, and quasicrystalline states. The results for the QC are in between those for the crystalline and amorphous alloys for ρ, dρ /dT, N(εF), the Debye Θ, sound velocity, superconducting Tc, and λ. If this situation had persisted, quasicrystals would have remained a subject in crystallography. However, with the discovery of stable icosahedral (i-) quasicrystals around 1990, the situation changed. These QC’s were found to have properties not previously observed. Clearly the QC structure can have a

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profound influence on the physical properties. Focus here will be on the electronic transport properties of these QC’s. Similarly to amorphous metals, the loss of reciprocal space complicates the study of quasicrystals. One can use the two strongest scattering vectors in X-ray diffraction to construct a quasi-Brillouin zone. This concept has sometimes been useful and has also served as a guideline in search for new QC’s, much along the lines of Hume-Rothery phase stabilization, described in Sec. 1.2. However, values of Z for the elements of a QC have been assigned on the basis of empirical rules and an assumed charge transfer between s,p bands and the d band, resulting in non-integer values, sometimes negative (e.g. for Fe, Z = -1.71). The usefulness of this method has been demonstrated by its success to predict new quasicrystals. Nevertheless it lacks the rigour of the rule based exclusively on column number in the Period Table. Another useful concept is the approximant. An approximant is a crystal with a structure related to the QC it approximates. It is designated by one of the ratios of two successive numbers in the Fibonacci chain; 2/1, 3/2, 5/3, 8/5 etc. This ratio → (√5+1)/2, the golden mean, at ∞. In the projection to a lower dimension, the golden mean corresponds to an irrational slope of the cut in hyperspace, defining how a real QC is obtained by projection. If the slope of this cut-line is a rational number, a crystal is obtained, not a QC. Since 3/2 < 8/5 < (√5+1)/2 < 13/8 <5/3 <2, it is clear that the sequence of Fibonacci ratios gives successively improved approximations to the QC. Theoretically this is a powerful tool, since calculations can be made in a series of approximants from which firmer statements can be made, by convergence or by extrapolation, on the properties of the quasicrystal. Experimentally it is also interesting, since in several cases the lower approximants exist and can be studied to infer how properties develop when the QC is approached. 3.2. Remarkable electronic transport anomalies Several anomalies in electron transport properties are briefly described. (i) ρ is large. frequently observed. One can compare ρ with simple crystalline alloys

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and amorphous metals. The sequence of Fig. 2 (simple alloys), Fig. 12 (amorphous metals), and Fig. 25 [i-QC's], suggests that ρ increases roughly by two orders of magnitude in each step.

Fig. 25. ρ (T) for an icosahedral AlPdRe sample. The ratio ρ (4.2K) / ρ (295K) ≈ 4.

(ii) Large|d ρ (T)/dT|. It is useful to define an average temperature dependence by the resistance ratio R: (33) In Fig. 25, R ≈ 4. Fig 26 illustrates the development of Rmax in the 90-ties. New record values have not been found after that. The discovery of stable QC’s brought about a dramatic change of scene. This is illustrated by the fact that neither simple crystalline alloys, R ≈ 0.05-0.2, Fig. 2, nor amorphous metals, with Rmax ≈ 1.01 or 1.08 in Fig. 12 a) and b), can be distinguished from the R = 0 line in Fig. 26. For simple simple crystalline alloys R is typically << 1, e.g. in the range 0.05-0.2 in Fig. 2. Amorphous metals, as mentioned, often have dρ / dT < 0, i.e. R > 1. In Fig. 12 the maximum values in each alloys sytem are 1.01 and 1.09.

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Fig. 26. The maximum R-value of quasicrystals vs time of discovery. The alloy systems are from left to right ∇: i-AlCuLi, Δ: i-AlCuFe, square: i-AlCuRu, o: i-AlPdRe.

annealed

cubic AlFe type

i - 4/8 0/0 0 /0

i-2/4 4/6 0 /0

arbitrary units

(iii) Removal of impurities leads to increased resistivity. The two strongest diffraction peaks and ρ (T), for an i-AlCuFe sample in two different states21 are shown in Fig. 27.

as-quenched

ρ (μΩcm)

θ (degrees)

T (K)

Fig. 27. X-ray diffraction and ρ (T) for i-Al63Cu25Fe12. As quenched: dotted curve in upper panel and lower curve in bottom panel. After annealing: full curve in upper panel and upper curve in lower panel (from21).

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An as quenched sample (dotted line in top panel) contains impurities, shows broads diffraction peaks, and has a small ρ and a weakly negative d ρ /dT (lower curve in bottom panel). After annealing, the impurity peak is not observable, diffraction peaks are narrow, and ρ and |d ρ / dT| have increased. This property is completely counterintuitive to conventional thinking of impurities as scattering centers. (iv) Large Hall effect and unusually strong concentration dependence. In some cases quasicrystal properties display exceptional sensitivity to small concentration differences. Data at 4.2 K for the conductivity σ and RH of i-AlCuFe are shown in Fig. 2822. Fe content was varied in the narrow range 12-13 at %. σ (4.2 K) changes by a factor of 2 in this range, and RH (4.2 K) shows a sign change close to 12.5 % Fe. The values of | R H | in Fig. 28 are much larger than in simple crystalline alloys (Fig. 4) and amorphous metals (Fig. 19). The difference is of order a factor 100, indicating a much reduced charge carrier concentration n in icosahedral quasicrystals. For the more resistive quasicrystals a reduction of electron density of states is also inferred from specific heat measurements.

Fig. 28. The Fe-concentration in i-AlCuFe is varied in a narrow range. The conductivity and RH are shown. The vertical dashed line in the right panel shows the sharp boundary between negative and positive RH (redrawn from22).

(v) Thermopower. For completeness the thermopower of icosahedral quasicrystals is mentioned. For the more resistive QC’s the simplifications which occur

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for amorphous metals do not occur. Instead S is at least as complicated as it can be in crystalline alloys, e.g. with strong temperature dependence and change of sign of S(T) or dS(T)/dT. Although not understood in detail, S of quasicrystals is not as outstandingly anomalous as the other properties discussed here. (vi) Magnetoresistance. The magnetoresistance in some icosahedral quasicrystals is unusually large. Fig. 29 shows an example for the same sample as in Fig. 2523. The magnetoresistance in this case is of order 10 %, i.e. it is again about a factor 100 larger than for typical amorphous metals (Fig. 16). The curves through data points were calculated from quantum interference effects, discussed in Sec 3.3. The consistent description of data by these theories indicates that this sample is still in the metallic state.

Fig. 29. The low temperature magnetoresistance for an i-AlPdRe sample with R = 4. The curves are fits to Eqs. (28,29) for disordered metals (from23).

The overall maximum magnetoresistance in a particular alloy often occurs at much larger B's than available in the laboratory. For comparison one can then use the maximum value reported for each alloy. Laboratory conditions vary strongly, but the trend of data are clearly seen on logarithmic scales. Fig. 30 shows such a compilation24 for fcc CuGe, amorphous metals, decagonal and icosahedral QC’s and crystalline approximants.

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X

X

Fig. 30. Maximum observed magnetoresistance |Δρ / ρ |max vs ρ (4 K). The straight line, ⎪Δρ / ρ (B,T)/ρ(0,T)|max ≈ ρ1.3 summarizes data up to ~105 μΩcm (from24).

The correlation is obeyed over 4 orders of magnitude of the magnetoresistance, which suggests a common mechanism for this phenomenon. In Sec. 3.3 it will be further discussed that this mechanism is quantum corrections. It should be noted that there are no data above ρ (4.2 K) ≈ 105 μΩcm in Fig. 30. This limit corresponds to a metalinsulator transition which will be discussed in Sec. 3.5. Thus, it is only the metallic magnetoresistance which is illustrated in Fig. 30. 3.3. Quantum corrections In alloys with not too large R-values, say R < 2, temperature can be regulated with sufficient precisions to measure the magnetoresistance in a wide range up to room temperature. Results for i-AlCuFe are shown in Fig. 3124. Comparing these data with Fig. 17, one can see that Δρ /ρ in i-AlCuFe develops in the same way, from ~ +√B at low T, to ~ +B2 at intermediate temperatures, and to negative values of form ~ -B2 which decrease in amplitude for further increasing T. This detailed qualitative evidence supports the presence of quantum corrections in the magnetoresistance, already before any quantitative analyses in terms of Eqs. (28, 29) has been made.

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Fig. 31. Magnetoresistance of i-Al65Cu25Fe12.5 from 80 mK to 280 K (from24).

The curves in Fig. 30 show fits of Eqs. (28, 29) to data. The result contrasts with corresponding efforts for amorphous metals (e.g. Fig. 16). Not only is the task of fitting data to quantum correction theories more demanding for the QC's, with Δρ /ρ varying over more than four orders of magnitude, including a sign change, but the result is also quantitatively accurate. This precision gives strong evidence for the detailed validity of the theories for quantum corrections in 3 dimensions,

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a point in question when data for only amorphous metals, or otherwise disordered metals, were available. Quantum corrections are also present in other electron transport properties of quasicrystals. Fig. 32 shows examples for i-AlCuFe from the Hall effect21 and from the temperature dependence of ρ (T)24.

Fig. 32. Temperature dependence of Hall effect and resistivity of two i-AlCuFe samples. Panel a) RH vs √T (data from21). Panel b) Δρ (Τ) /ρ (T) vs √T (from 24). The straight lines in both panels have slopes calculated from magnetoresistance results for Fσ.

Electron-electron interaction theory (Eqs. 27, 30) accounts for this behavior; WL does not enter in RH, and in Δρ (Τ) at low T, τie has saturated and does not contribute to the T-dependence. The slope of the straight lines were taken from magnetoresistance results for Fσ. Panel a) is a low resistivity sample, Fσ ≈ 0.72 (< 8/9), and the slope is negative. Panel b) a high-ρ sample, Fσ = 1.16 and the slope changes sign. It should be noted that there are zero adjustable parameters in both panels. Similar results are obtained for other icosahedral samples, but quantitative fits do not have a similar extended validity. E.g. in i-AlPdMn, contributions from the magnetic state of Mn affect both direct spin scattering and dephasing in weak localization, which much complicates analyses. In i-AlPdRe, as indicated, |dρ (T)/dT| is so large that measurements are limited to a (low) temperature region where the magnetoresistance is still substantial.

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3.4. Electronic conduction in insulators In this subsection an important general theory for conduction in insulators is discussed. Electrons in insulators are localized on ionic sites, with some extension of the wave function over the localization length ξ (Fig. 33). In a metal ξ → ∞ corresponding to itinerant electrons. N(εF) can still be >0. It is the number of electron states per unit energy, and localized as well as itinerant electrons can contribute. Jumps from one atomic site to a nearby site gives a current. In e.g. ionic conduction, the ions jump. Such nearest neighbour hopping (NNH) is however not the only possibility for electrons.

Fig. 33. Electron wave functions on two sites in an insulator ξ measures the extension of the wave functions, r is the hopping length. The wave functions shown are not centered on nearest neigbour ions, but can be hundreds of interatomic distances apart.

Mott introduced variable range hopping, VRH, and derived an expression describing how σ → 0 when T → 0. The probability for a long electron hop decreases exponentially with hopping length, r. An advantage is also gained, because the electron then probes a larger phase space, with increased probability of finding a state close to its energy, reducing energy cost. When does the maximum probability for hopping occur? In a sphere of radius r, the volume into which the electron hops, there are (4πr3/3)N(εF) electrons per unit energy. The spacing between energy levels in this volume is then W(r) = 3 / (4πr3N(εF)). With a larger r, energy levels are more closely spaced and smaller energy is expended.

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The jump frequency p is (34)

ωph is a phonon frequency, taken to be the attempt frequency. The maximum, pmax, of p is easily found by derivation;

(35) This hopping corresponds to a current. The Mott VRH conductivity is (36) To' is defined by factors in the exponent of Eq. (35). Its physical significance is (roughly) the temperature below which hopping can occur, which corresponds roughly to the region where r > ξ. When Coulomb repulsion between charge carriers is considered, Efros-Shklovskii found that the temperature dependence of the conductivity should instead follow (37) with a different, generally smaller characteristic temperature To and a square root behavior in the exponent. In the VRH magnetoresistance there is one term at small B, which is negative and linear in B and is due to interference of the contributions to the amplitude of the hopping probability. Shrinking of electronic wave functions in magnetic field gives a positive contribution to the magnetoresistance, which increases as B2. At still larger fields the B dependence is weaker. Hence in moderate fields (38)

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Bo is the field for which one flux quantum e/h passes through the interference area (r3ξ)1/2. r has a different temperature dependence for Mott and Efros-Shklovskii hopping; rM ~ ξ (T'o/T)1/4 and rES ~ ξ (T'o/T)1/2, with the characteristic exponents of 1/4 and 1/2 respectively. This gives different temperature dependencies in the B2 region of Eq. (38). (39)

(40) cM and cES are characteristic constants for each hopping mode. The behavior of Eqs. (39, 40) is limited to a moderately low magnetic field region where the magnetic length ℓB = √ħ/eB < ξ. This limit marks the cross-over from the B2 region to a B2/3 behavior at larger fields. 3.5. A metal-insulator transition (MIT) in i-AlPdRe The strong increase of ρ (T) with decreasing T for QC's with large R has led to the conjecture that such samples are insulators. However, there is no energy gap as in semiconductors, and no factor ~ exp(-ε/kBT) in σ (T). On the other hand, theories for the metallic magnetoresistance break down at large R-values. Attempts to describe σ (T) from Eqs. (36, 37) have led to seemingly acceptable descriptions in various regions of T, but often with so different parameters that firm conclusions have not been obtained. A main difficulty is the existence, apparently in all QC's, of a zero temperature conductivity σ (0) >0, experimentally revealed by a flattening out of σ vs T at low T, sometimes only at T<20 mK. This could signal a metallic ground state. On the other hand, if the sample is insulating a finite σ (0) must be included in Eqs. (36, 37); (41)

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ρ (Ωcm-1)

ν is 1/4 or1/2. With 4 adjustable parameters and an exponential function Eq. (41) is too flexible for smooth σ (T) data. Due to these difficulties, the most common way to verify an insulating ground state, i.e. by VRH in σ (T), has not led to conclusive results. Evidence for a finite σ (0), must include consideration of some experimental concerns; (i) possible impurity effects, or (ii) an artefact due to poor thermal contact between sample and thermometer at ultralow temperatures.

T (K) Fig. 34. σ (T) for an i-AlPdRe sample, R=110, at B=0, 2, and 6 T, from top to bottom.

Fig. 34 illustrates one piece of evidence that σ (0) is an intrinsic property25. The saturation of σ (T) observed in zero field below 20 mK is lifted by a magnetic field. At B = 6T, σ (0) is linear in T. This supports that thermal contact is adequate. Absence of impurity effects is discussed in Sec. 3.6 An alternative to analyzing σ (T) is to study the magnetoresistance. One can find empiricial evidence for an MIT using only the curve forms of the observed magnetoresistance. Fig. 35 shows examples for two iAlPdRe samples, one with R = 11, the other for R = 160. At R = 11, Δρ(B)/ρ(0) starts as B2 and gradually shifts to B1/2, reflecting a disordered metallic state. For R = 160, increasing B gives the sequence -B, B2 and B2/3, in agreement with VRH theories for insulators. An MIT occurs in between these R-values.

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Fig. 35. Magnetoresistance at 4.2 K for two i-AlPdRe samples. Curve segments of characteristic magnetic field dependencies are shown on various parts of the curves (from25).

3.6. The resistance ratio R i-AlPdRe can be prepared in states of widely varying R values by different annealing procedures. In samples of identical nominal composition, R can vary from 3 to 300. The reason for such a dramatic change of electronic properties is not understood. Nor can one decide what R value to produce by a certain annealing, only a limited range of R-values within which the resulting R will fall. The usefulness of R as a parameter to characterize i-AlPdRe has therefore been questioned. In particular, although samples with widely different R are phase pure icosahedral in standard X-ray diffraction, a possible role of impurities is a question of concern. One method is to vary the state of a single sample by some cleaner method than preparation of different samples and annealings. High energy neutron irradiation disturbs icosahedral order, but penetrates the sample, and leaves negligible irradiation products in the sample. With destroyed icosahedral order one expects that ρ and |dρ /dT| decrease (cf Fig. 27). Irradiation of a high-R sample may thus cause an insulator-

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Fig. 36. ρ (T) vsT for an i-AlRePd sample, with increasing neutron irradiation doses from 5.1017 n/cm2, R = 57, (top curve) to 7. 1019 n/cm2, R = 1.2 (bottom curve).

metal transition. Fig. 36 illustrates the decrease of ρ and R with neutron irradiation in an i-AlPdRe sample (originally R =67) for a series of increasing irradiation doses. There is no particular feature in these curves which would mark a transition from insulating to metallic behaviour with increasing irradiation dose and decreasing R. This, again, illustrates the difficulty to use ρ (T) alone in MIT studies of i-AlPdRe. At R = 1.2 the temperature dependence seems insignificant. This impression is due to the double logarithmic scales. In fact |dρ /dT| is still much larger than for amorphous metals, and X-ray diffraction shows a single icosahedral phase, albeit with weaker reflections. Δρ (B) ρ(0) for one i-AlPdRe sample in two states is shown in Fig. 37. These results are similar to those in Fig. 35. Δρ (B)/ρ(0) at moderately large R is characteristic for a weakly disordered metal. For large R variable range hopping is observed. Neutron irradiation induced damage thus provides a method to monitor R in i-AlPdRe. It also demonstrates that the variation of R is not due to impurities.

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Fig. 37. The field dependence of the magnetoresistance for an i-AlPdRe sample, originally with R = 67. After irradiation with about 4.1018 neutrons/cm2, R has decreased to 13 and the magnetoresistance is metallic (from25).

3.7. Results at the MIT of i-AlPdRe Some details are described here which show how quantitative information on ξ and To can be obtained from σ (T) and Δρ (B,T)/ρ (0,T). The different T dependences of the magnetoresistance in the B2 region, Eqs. (39, 40), allows to determine whether the variable range hopping is of Mott or Efros Shklovskii type. For i-AlPdRe one finds that this T dependence is ~T-3/2 (Fig. 40 below) corresponding to ES- VRH, and hence that ν =1/2 in Eq. (41). σ (0) in Eq. (41) can be estimated from measurements taken to below 20 mK. Two parameters in Eq. (41) have then been eliminated, and one can analyze σ (T) in the form (42) from which To can be determined from the slope of a straight line. This is shown in Fig. 38 for an R = 40 sample.

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Results for To for a series of i-AlPdRe samples are shown in Fig. 39. To → 0 at the MIT. The results suggest that the value Rc of R at the MIT is roughly of order 20.

Fig. 38. Determination of To from Eq. (42). For this sample To ≈5.2 K.

Fig. 39. To,determined as in Fig. 38 for a series of i-AlPdRe samples. To → 0 when approaching the MIT from the insulating side (from 25).

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From Eq. (40), with cES = 0.0015, one has in the B2 region that δ2[Δρ(B,T)/ρ(0,T)]/ δB2δT-3/2 = 0.0015(e/ħ)2ξ4To3/2. With results for To (Fig. 39), and the slope Bo-2 of the magnetoresistance in B2 region vs T-3/2, as in Fig. 40, one can thus calculate ξ. 8

R = 220

B0-2 (10-3T2)

6

4 2 0 0.0

0.2

T-3/2

0.4

0.6

(K-3/2)

Fig. 40. The magnetoresistance in the B2region for an i-AlPdRe sample with R =220. The coefficient Bo-2 of B2in Eq. (38) is shown to have the characteristic temperature dependence of Efros-Shklovskii VRH Eq. (40).

The localization length ξ diverges when approaching the MIT from the insulating side. This behaviour can be analyzed by a power law; (43) Rc is taken to be 20 in Fig. 41, and γ is the exponent describing this divergence. Data are fairly well described by a divergence of this form. The result for γ is close to 1/3. Hence R-Rc is roughly inversely proportional to the localization volume; (R-Rc) ~ ξ-3. This suggests the physical picture that R increases into the insulator roughly as the inverse of the decreasing volume of the localized electrons.

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Fig. 41. log ξ vs the logarithm of the distance to the MIT, measured by R-Rc with Rc =20 (from25).

4. Brief Overview and Conclusions Selected electronic transport properties of metallic alloys have been briefly described. Three groups of such materials exist; crystalline alloys, amorphous metals, and quasicrystals, hereafter referred to as CR-, A-, and QC-metals. The presentation has been limited mainly to simple crystalline alloys, binary metal-metal amorphous alloys, and (polygrain) icosahedral quasicrystals. ρ (4 K) increases roughly by two orders of magnitude between each of these groups, from several μΩcm for to several hundreds μΩcm, and to several tens of mΩcm respectively. A unifying concept in this development is disorder. However, an important distinction must be recalled; CR and QC's are atomically well ordered, whereas A-metals have short range atomic order over a few atomic distances, and no correlation between atomic sites at larger distances. It is instead the electronic disorder which is relevant, and which increases in the sequence of CR, A, and QC for these metals (Fig. 42). This electronic disorder is quantified by F = ħ/τεF. The close relation between F and resistivity gives a first illustration of this increasing electronic disorder.

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Fig. 42. Simple crystalline alloys, metal-metal amorphous alloys, and icosahedral QC's are shown in the order of increasing electronic disorder. Arrows illustrate the direction of changes brought about by irradiation.

The trend of more strongly anomalous transport properties for increased electronic disorder is also apparent in several other electronic properties of the materials discussed. In the sequence CR, A, and QC, R goes from values in the range 0.1, to 1, and to above 100, and the magnetoresistance Δρ (B) /ρ from values of order 10-5, to 10-3, to values in the range up to 1. Disorder can be monitored by irradiation. It is therefore a useful experimental tool to control electronic disorder. To some extent the method of irradiation can also bridge the gaps between CR-, A-, and QCmetals. Fig 42 is a reminder of the direction of the changes brought about by irradiation. In other aspects the picture is more complex. Both the thermopower and the Hall coefficient are more complicated properties, sensitive to minute changes of electronic structure. Some noteworthy features are the simplifications for S in amorphous metals, and large values of S and RH for QC's. The QC superconductors have properties in between CR and A-alloys, and electron-electron interaction and superconducticvity do not therefore show new type of results. One can also note that the topics of electron-phonon interaction and superconductivity seldom enter for quasicrystals. The QC superconductors are of the types of QC’s first discovered, with properties in between CR- and A- alloys. In a text where emphasis is on explaining some phenomena, the message how little is understood is easily overshadowed. Therefore it should be stressed that a large number of electronic properties are still open problems. The magnitude of ρ is poorly understood for QC's and cannot be quantitatively calculated even for A-metals. For dρ /dT in Ametals, a number of competing mechanisms exist, and a definite statement is difficult. For QC's quantum corrections can likely describe ρ(T) for some materials up to room temperature, but are not likely a valid description of the continued decrease of ρ (T) at still higher temperatures,

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and anyway fail completely over any extended temperature range in iAlPdRe with its huge |dρ / dT|. Quantitative descriptions of S and RH are also lacking as mentioned. The magnetoresistance in the metallic state is, however, a remarkable exception. Not only can quantum correction theories describe the observations. The descriptions for quasicrystals, where the effect is the largest, can be made over much larger ranges of temperature, and they are more quantitatively accurate, than in any other other 3-D material. It should be noted that this statement is limited to the metallic state. In i-AlPdRe there is a metal-insulator transition, in itself a remarkable phenomenon in an alloy consisting of three good metals. It was illustrated that Δρ (B)/ρ on the insulating side appears to be well understood within variable range hopping theories, and can be used to verify this MIT. However, this is valid for the temperature range 1-10 K. At lower temperatures the negative initial part of the insulating Δρ (B)/ρ(0) becomes unobservable, and similar analyses break down. This magnetoresistance and the complications of the very low temperature σ (T) are not understood and add to the numerous open questions for quasicrystals. Clearly a great number of challenging problems in electronic transport of alloys are open. Acknowledgments I am grateful to graduate student Rickard Fors for his many fruitful comments on the manuscript. References 1. Ch. Kittel, Introduction to Solid State Physics (John Wiley 7th ed. 1996, 8th ed. 2005). 2. H. P. Myers, Introductory Solid State Physics, (Taylor and Francis, London 1990). 3. R. D. Barnard, J. E. A. Alderson, T. Farrel, and C. Hurd,, Phys. Rev.,176, 761 (1968). 4. R. S. Crisp and J. Rungis, Philos Mag 22, 217 (1970). 5. N. W. Ashcroft and N. D. Mermin, Solid State Physics, Brooks/Cole, pp 256-259 (1976). 6. W. L. McMillan, Phys. Rev. 167, 331 (1967).

318 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

Östen Rapp B. W. Roberts, J. Physical and Chemical Reference Data 5, 581 (1976). R. F. Hoyt and A. C. Mota, Solid State Communications 18, 139 (1976). Ö. Rapp and M. Flodin, Phys. Rev. 26, 99 (1982). R. Fogelholm and Ö. Rapp, J. Phys. F 7, 667 (1977). H. J. Mooij, Phys. Stat. Solidi A 17, 521 (1973). P. J. Cote, Solid State Communications 18, 1311 (1976). R. C. Dynes, J. M. Rowell, and P. H. Scmidt, in Ternary Superconductors, Eds. G. K. Shenoy, B. D. Dunlap, and F. Y. Fradin, p.169, (North Holland, 1981). U. Mizutani, Introduction to Electron Theory of Metal, (Cambridge University Press, 2000). P. Lindqvist and Ö. Rapp, J. Phys. Condens. Matter 1, 4839 (1989). J. B. Bieri, A. Fert, and G. Creuzet, Solid State Communications 49, 849 (1984). A. Schulte, W. Haensch, G. Fritsch, and E. Lüscher Phys. Rev. B 40, 3581 (1989). B. L. Gallagher, J. Phys. F; Metal Phys 11, L207 (1981). A. Nordström, U. Dahlborg, and Ö. Rapp, Phys. Rev. B 48, 12866 (1993). Ö. Rapp, J. Jäckle, and K. Froböse, J. Phys. F: Metal Phys 11, 2359 (1981). T. Klein, A. Gozlan, C. Berger, F. Cyrot-Lackmann, Y. Calvayrac, and A. Quivy, Europhys. Lett.13, 129 (1990). P. Lindqvist, C. Berger, T. Klein, P. Lanco, F. Cyrot-Lackmann, and Y. Calvayrac, Phys. Rev. B 48, 630 (1993). M. Rodmar, M. Ahlgren, D. Oberschmidt, C. Gignoux, J. Delahaye, C. Berger, S. J. Poon, and Ö.Rapp, Phys. Rev. B 61, 3936 (2000). Ö. Rapp, in Physcial Properties of Quasicrystals, Ed. Z. Stadnik, (Springer Solid State Sciences Berlin 1999). V. Srinivas, M. Rodmar, R. König, S. J. Poon, and Ö. Rapp, Phys. Rev. B 65, 94206 (2002), A. E. Karkin, B. N. Goshchitskii, V. I. Voronin, S. J. Poon, V. Srinivas, and Ö. Rapp, Phys. Rev. B 66, 92203 (2002), Ö. Rapp, V. Srinivas, P. Nordblad, and S. J. Poon, J. Non Crystalline Solids 334-335, 356 (2004), Ö. Rapp, V. Srinivas, and S. J. Poon, Phys. Rev. B 71, 12202 (2005).

CHAPTER 11

ELECTRON TRANSPORT PROPERTIES OF COMPLEX METALLIC ALLOYS Uichiro Mizutani Toyota Physical & Chemical Research Institute, Nagakute, Aichi, 480-1192, Japan E-mail: [email protected] Several key theories on electron transport properties of complex metallic alloys are first reviewed, which include the Boltzmann transport equation, weak localization theory and Mott’s theories: one on expanded liquid mercury and the other on variable range hopping. Emphasis is laid on the effect of the pseudogap formed across the Fermi level on the electron transport mechanisms. By choosing two representative amorphous alloy systems VxSi100-x (0<x<74) and (Ag0.5Cu0.5)100-xGex (0<x<100) and a variety of quasicrystals and their approximants, we demonstrated that scattering mechanism successively changes with increasing the depth of the pseudogap at the Fermi level and finally crossing the metal-insulator transition. Systematic studies above will provide a comprehensive understanding of the electron transport mechanism in non-magnetic complex metallic alloys.

1. Introduction The electron transport properties refer to properties conduction electrons manifest in response to electric field, magnetic field, temperature gradient or a combination of these applied to a given specimen. First, we survey the key theories, on which interpretation of experimentally derived data rely1-5. Emphasis is laid on the fact that the Boltzmann transport equation is the most powerful tool to analyze the electron transport properties. Here, however, one must be well aware of limitations for use of theories based on the Boltzmann transport equation. Indeed, the Boltzmann transport mechanism fails, when the mean free path of conduction electron becomes comparable to an average atomic distance. Here weak localization effect 319

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emerges and eventually the variable-range hopping mechanism plays a key role upon crossing the metal-insulator transition. The complex metallic alloys (hereafter abbreviated as CMAs) are referred to as the metallic and marginally metallic alloys, the unit cell of which contains atoms far greater than those in ordinary crystals like body-centered cubic (bcc) and face-centered cubic (fcc). Typical examples are the gamma-brass, 1/1-1/1-1/1 approximants, quasicrystals, in which 52, more than 100 and an infinite number of atoms are accommodated in the unit cell, respectively. Amorphous alloys may be also treated as CMAs, since the unit cell cannot be defined because of the lack of lattice periodicity. The increasing structural complexity from bcc, fcc, gamma-brass, approximant, quasicrystal to amorphous alloy is reflected in an increase in the number of the Brillouin zone planes interacting with the Fermi surface. The greater the number of atoms in the unit cell, the more spherical a polyhedron bounded by the relevant Brillouin zone planes is. Moreover, the higher the number of zone planes is, the deeper the pseudogap across the Fermi level. This affects significantly the electron transport properties of CMAs. Following the brief description of the essence in existing key theories, we discuss the occurrence of different types of scattering mechanisms with the emphasis on the effect of the pseudogap on the electron transport mechanisms by choosing two representative amorphous alloy systems VxSi100-x (0<x<74) and (Ag0.5Cu0.5)100-xGex (0<x<100) and a variety of quasicrystals and their approximants. 2. Fundamentals in electron transport properties First, we assume a semi-classical model, in which electron is treated as a classical particle or more rigorously speaking, a wave packet constructed from wave functions. Now the free electron Fermi sphere is driven as a whole by the external electric field in the reciprocal space. The equation of motion is then given by

dp x dk x = = ( − e) E x dt dt

(1)

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where E x is the electric field applied along x-direction. Its integration immediately tells us that the Fermi sphere moves indefinitely toward a direction opposite to the electric field. However, this is not physically accepted, since an infinitely large current would flow. To establish the steady-state, we have to introduce a friction term into Eq. (1). Drude (1900)6 introduced the concept of the drift velocity given by ⎛ n ⎞ v D = ⎜ ∑ vi ⎟ n and added a friction term v D /τ proportional to the ⎜ ⎟ ⎝ i =1 ⎠ drift velocity to Eq.(1) to obtain the more general equation of motion: ⎛ dv D

m ⎜⎜

⎝ dt

v ⎞ + D ⎟⎟ = (−e)E

τ ⎠

(2)

where m is the mass of electron, τ is called the relaxation time and E is the applied electric field. The drift velocity is easily found to be proportional to the external field by setting dv D / dt = 0 when the steady-state is established. The well-known Ohm’s law is obtained by inserting the drift velocity thus derived into the expression for the current density J = n(−e) v D :

σ=

ne 2τ m

(3)

where n is the number of electrons per unit volume, (-e) is the electronic charge and the relaxation time τ characterizes a time to restore a steady-state upon switching the external field on or off. The relaxation time τ plays a key role in the rest of the discussion on the electron transport properties. It describes an average time for an electron to travel in between two successive scatterings in the presence of an external field. The mean free path Λ F representing a distance that an electron can travel in each passage is also an important quantity, which is given by the product of the relaxation time and the velocity of an electron. Table 1 lists the resistivity and conductivity at 300 K along with temperature coefficient of resistivity called TCR for elements in the periodic table5. Because of the presence of lattice periodicity, the

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resistivity of pure metals is low in the neighborhood of a few μΩcm. Instead, the resistivity of amorphous alloys without possession of the pseudogap at the Fermi level is much larger than that of pure metals because of the lack of lattice periodicity and ranges over 30 to 200 μΩcm. Table 1. Electron transport properties of pure metals at 273K element Li Na Cu Ag Au Mg Ca Zn Al Pb Bi Ti V Fe Zr W

conductivity σ 273K -1

(x106Ω m-1) 11.8 23.4 64.5 66 49 25.4 28 18.3 40 5.17 0.93 2.38 0.54 11.5 2.47 20.4

TCR resistivity α273K ρ273K -3 (μΩ cm) (x10 /K) 8.5 4.27 1.55 1.5 2.04 3.94 3.6 5.45 2.50 19.3 107 42 18.2 8.71 40.5 4.89

4.37 5.5 4.33 4.1 3.98 4.2 4 4.20 4.67 4.22 5.5 6.57 4.0 4.83

(From reference 5).

When a pseudogap grows across the Fermi level, the resistivity value at 300 K increases further, as will be discussed later. In the case of quasicrystals and their approximants, whose electronic structure is always characterized by the pseudogap at the Fermi level, the resistivity at 300 K ranges over 30 to 104 μΩcm. As discussed above, the resistivity of pure crystalline metal element is extremely low even at 300 K. This certainly owes to the Bloch theorem. At this stage, it is worthwhile noting the importance of the Bloch theorem in the electron transport properties. The Bloch wave function is described as the plane wave modulated by an arbitrary function possessing the lattice periodicity uk (r ) = uk (r + l ) , where l is the lattice vector. The most important to be kept in mind is that the wave vector appearing in the Bloch

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wave function is different from that employed in the free electron model. The wave function in the unit cell specified by the lattice vector l is given simply by the wave function in the cell at origin times a phase shift exp(ik ⋅ l ) specified by the lattice vector l 5. Hence, the wave vector k in the Bloch wave function serves as extending the wave function defined in a given cell to all other unit cells throughout a crystal. It means that the electron can propagate without any scattering throughout a crystal, leading to the absence of resistivity, as long as the ionic potential is perfectly periodic, i.e., the Bloch theorem holds. In other words, the resistivity emerges only when the periodicity of the lattice potential is disrupted by some reason. Further emphasis is laid on the difference between the Bloch wave vector and that appearing in the free electron model by using the empty periodic lattice model. The empty periodic lattice is defined as the periodic lattice possessing an infinitesimally small potential3, 5. We retain the periodic nature of the lattice but the wave function in this limit must be described by the plane wave exp(ik ⋅ r ) . The plane wave may be rewritten as

ψ k (r ) = exp(ik ⋅ r ) = exp [i (k ± g ) ⋅ r ] ⋅ exp(∓ig ⋅ r ) = exp [ik ′ ⋅ r ] ⋅ ug (r ) (4) where g is the reciprocal lattice vector satisfying the relation exp(±ig ⋅ l ) = 1 with the lattice vector l and ug (r ) = exp ( ∓ig ⋅ r ) . The function ug (r ) is found to possess the lattice periodicity by replacing r with r + l : ug (r + l ) = exp [ ∓ig ⋅ (r + l ) ] = exp [ ∓ ig ⋅ r ] = ug (r )

(5)

Hence, the free electron wave function in the empty periodic potential indeed satisfies the Bloch theorem. In contrast to a single-parabolic dispersion relation centered at origin in the free electron model, electronic state of wave vector k in the empty periodic lattice model must be the same as that of wave vector k+g. This means that electronic states on a series of parabola centered at any reciprocal lattice vector g are solutions for electrons propagating in the empty periodic lattice. This is illustrated in Fig. 1.

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(a) Electrons in the free electron model

(b) Electrons in empty periodic lattice model

Fig. 1. (a) energy dispersion relation in the free electron model and (b) that in the empty periodic lattice model. The electronic state marked as A is identical to the state A’ the reciprocal lattice vector g apart. The shaded area is called the reduced Brillouin zone. (b) is reproduced from reference 5.

When a periodic potential becomes finite, the energy gap opens at the wave vector, where two parabola intersect, as shown in Fig. 2. E

Δ E100

−

3π a

−

2π a

−

π a

0 k100

π a

2π a

3π a

Fig. 2. Energy dispersion relation in the nearly-free-electron model. ΔE100 indicates the energy gap on the (100) zone. The shaded area is the first zone bounded by -π/a
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Such an energy gap appears in a direction perpendicular to any set of lattice planes and forms a polyhedron in the reciprocal space. This is called the Brillouin zone. There is one-to-one correspondence between the crystal structure in the real space and the Brillouin zone in the reciprocal space. The Brillouin zone can be constructed by perpendicularly bisecting the shortest, the second shortest and further successive reciprocal lattice vectors in the reciprocal space and, hence, the Brillouin zone is not unique. The Brillouin zone, with which electrons near the Fermi surface are interacting, is important. Typical examples are depicted in Fig. 3.

Fig. 3. Unit cell and the Brillouin zone interacting with electrons near the Fermi level in bcc, fcc and different CMAs. The atom distribution of the Zr-Ni-Al amorphous alloy due to courtesy by Prof.T.Fukunaga, Kyoto University, Japan.

With increasing the number of atoms in a unit cell, the number of zone planes increases. In the bcc structure, the number of atoms involved in the unit cell is two and the Brillouin zone is a rhombic dodecahedron consisting of 12 zone planes. The fcc structure contains four atoms in its

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unit cell and its Brillouin zone is a truncated octahedron consisting of 14 zone planes. Alloys containing more than a few tens of atoms in unit cell may be grouped in CMAs. For example, gamma-brass contains 52 atoms in its unit cell and the Brillouin zone is bounded by 36 zone planes. In the case of the 1/1-1/1-1/1 approximant containing 160 atoms in a unit cell, the number of zone planes is increased to 84, approaching more to a spherical Brillouin zone. The diffraction spectrum consists of a large number of sharp peaks in quasicrystals. For example, the (222100) diffraction line yields a polyhedron having 60 identical zone planes in the reciprocal space. As noted in Introduction, the Fermi surface-Brillouin zone interaction leads to the formation of the pseudogap at the Fermi level and significantly affects the electron transport properties. The larger the number of atoms in a unit cell, the more the number of zone planes responsible for the Fermi surface-Brillouin zone interaction and generally the more significant its effect on the formation of the pseudogap near the Fermi level. In the case of amorphous alloys, the unit cell can be no longer defined but the diffuse Brillouin zone may be still constructed from the first main peak in the structure factor. The diffuse Brillouin zone must be isotropic and, hence, spherical, as shown in Fig. 3. Now we consider how an electron in a metal, whose electronic structure is characterized by its own energy dispersion relation, is accelerated upon application of external electric field. As discussed above, the energy gap exists across the Brillouin zone and the slope of the dispersion relation becomes zero at the boundary, as shown in Fig. 4. Let us consider the electron at k=0. The electron starts to travel along the direction opposite to the applied field, following Eq. (1) in the absence of a friction term. Since the wave vector increases in proportion to time, an electron changes its position and reaches the point B corresponding to the one side of the Brillouin zone. As emphasized earlier, the position B is equivalent to the position C the reciprocal lattice vector g apart. The electron at B is alternatively said to be that at position C. Remember that this has nothing to do with scattering. It goes further back to the origin k=0. This motion is indefinitely repeated, unless the friction is present. The

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slope of the dispersion relation is known to be proportional to the velocity 1 of an electron, v = ∇ k ε (k ) , and the second derivative yields the inverse

⎛ 1 ⎞ 1 ∂ 2 ε (k ) of the effective mass ⎜ = ⎟ ⎝ m* ⎠ij 2 ∂ ki∂ k j

1-5

. All information about

the band structure is contained in the energy dispersion relation ε (k ) and the band structure effect is brought into the transport properties through the velocity and effective mass of an electron.

Fig. 4. (a) ε-k relation of the electron in the first Brillouin zone, (b) the corresponding group velocity and (c) the effective mass. The points A and D correspond to the inflection point in the ε-k relation. The electric field E is applied as indicated by an arrow. g is the reciprocal lattice vector. Reproduced from reference 5.

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3. Theories based on the Boltzmann transport equation The Boltzmann transport equation describes the motion of electron in the steady-state in the presence of external fields like electric field, magnetic field, or a temperature gradient or a combination of these. Its derivation is described elsewhere3-5 and is written as

− v k ⋅ ∇ f (r , k ) −

( − e)

( E + vk × B ) ⋅

∂ fk ⎛∂ f ⎞ = −⎜ ⎟ ∂k ⎝ ∂ t ⎠ scatter

(5)

where f (r, k ) is the electron distribution in the steady-state in the presence of external fields at position (r , k ) in the phase space,

⎛∂ f ⎞ represents a change in electron distribution due to scattering ⎜ ⎟ ⎝ ∂ t ⎠ scatter event, v k is the velocity of electron and E + v k × B is the Lorentz force due to electric field E and magnetic field B . For its practical use, it is linearized by ignoring the second order correction to the Ohm law (see Appendix 1). In addition, the relaxation time approximation is introduced to simplify the calculation of the scattering term. The linearized Boltzmann transport equation with the relaxation time approximation is explicitly given by Eq. 6: ⎛ ∇ς ⎞ ⎪⎫ φ (r, k ) ∂φ (−e) ∂φ ⎪⎧ ⎛ ε (k ) − ς ⎞ ⎛ ∂ fo ⎞ + vk ⋅ + ( vk × B ) ⋅ ⎟⎬ = ⎜− ⎟ v k ⋅ ⎨− ⎜ ⎟ ∇ T + ( − e) ⎜ E − − ∂ε τ ∂ ∂ T ( e ) r k ⎝ ⎠ ⎠ ⎝ ⎠ ⎭⎪ ⎩⎪ ⎝

where φ (r, k ) = f (r, k ) − f o (ε k , T ) , the difference in the electron distribution between the steady-state f (r, k ) and the equilibrium state f o (ε k , T ) at temperature T, i.e., the Fermi-Dirac distribution function, is

φ (r , k ) ⎛∂ f ⎞ called the deviation function and − ⎜ = (ref. 3 and 5) ⎟ τ ⎝ ∂ t ⎠ scatter Various transport phenomena can be formulated by using the linearized Boltzmann transport equation. The electrical conductivity is derived under the conditions that ∇T = 0, ∇ς = 0, ∂φ / ∂ r = 0 and B = 0 in Eq. (6), i.e., the temperature gradient is zero, chemical potential is uniform throughout a sample, the deviation function is also uniform, magnetic field is absent. The linearized Boltzmann transport equation is reduced to:

Electron Transport Propoerties of Complex Metallic Alloys

329

f (r, k ) − f o (ε k , T ) ⎛ ∂ fo ⎞ ⎜− ⎟ v k ⋅ ( − e) E = τ ⎝ ∂ε ⎠

(7)

The derivation of the electrical conductivity will be described later. The electronic thermal conductivity and thermoelectric power are calculated under the conditions that ∂φ / ∂ r = 0 and B = 0 in Eq. (6), i.e., the deviation function is uniform throughout a sample and magnetic field is absent. The Boltzmann transport equation is then expressed as

⎛ ∂ f o ⎞ ⎡ ⎛ ε (k ) − ζ ⎜− ⎟ vk ⎢− ⎜ T ⎝ ∂ε ⎠ ⎣ ⎝

⎛ ∇ζ ⎞ ⎤ f (r, k ) − f o (ε k , T ) ⎞ , ⎟⎥ = ⎟ ∇T + ( − e) ⎜ E − τ ( − e) ⎠ ⎦ ⎠ ⎝ (8)

from which the Seebeck coefficient S and the electronic thermal conductivity κel are formulated as:

S=

π2

⎡ ∂ ln σ ⎤ ⋅ k B2T ⎢ ⎥ 3(−e) ⎣ ∂ε ⎦ ε = EF

(9)

and

κ el =

π2

k B2 σT 3 ( − e) 2 ⋅

(10)

respectively. The electronic thermal conductivity κel is found to be proportional to the electrical conductivity  σ in Eq. (10). This immediately leads to the demonstration of the well-known Wiedemann-Franz law:

κ el π 2 k B2 = ⋅ ≡ L0 σ T 3 ( −e) 2

(11)

where L0 is a constant called the Lorenz number. The Hall coefficient is also derived under the conditions that ∇T = 0, ∇ς = 0, ∂φ / ∂ r = 0, B = (0, 0, B) and E = ( E ,0, 0) in Eq. (6), i.e., the temperature gradient is zero, chemical potential is uniform, the deviation function is also uniform and magnetic and electric fields are

330

Uichiro Mizutani

applied along z- and x-directions, respectively. The Boltzmann transport equation is then deduced to be:

( − e) ⎛ ∂φ ∂φ ⎞ f (r, k ) − f o (ε k , T ) ⎛ ∂ fo ⎞ (12) B ⎜⎜ v x − vy ⎟= ⎜− ⎟ v x ⋅ ( −e) E + ∂ k x ⎟⎠ τ ⎝ ∂ε ⎠ ⎝ ∂ ky from which the Hall coefficient is derived as RH = 1/n(-e), where n is the number of electrons in a unit volume. As shown above, the Boltzmann transport equation is the most powerful tool to formulate almost all electron transport phenomena. Since the electrical conductivity is the main subject, we spend a bit more time on its formulation. The current density is given by J = n(−e) v , i.e., the product of the number of electrons per unit volume, electronic charge (-e) and the velocity of an electron. 1 Since n = 3 f (k )d k holds, the current density is rewritten in 4π the form:

∫∫∫

J = n ( − e) v =

( − e) 4π 3

∫∫∫ v

k

f (k ) d k =

( − e) 4π 3

∫∫∫ v ( f (k ) − f (k ) ) d k k

o

(13)

( − e) v k f 0 (k ) d k to Eq. (13) is harmless, 4π 3 since v k f 0 (k ) is an odd function with respect to the variable k and its integration over a whole reciprocal space is obviously zero. By inserting Eq. (7) derived from the Boltzmann transport equation into Eq. (13), we obtain where the addition of the term

J=

e2 4π 3

∫∫∫

e2 ⎛ ∂ f0 ⎞ k d = ⎟ 4π 3 ⎝ ∂ε ⎠

∫∫∫τ vk ( vk ⋅ E ) ⎜ −

∫∫∫τ v ( v k

k

⎛ ∂ f ⎞ dSd ε ⋅ E)⎜ − o ⎟ ⎝ ∂ε ⎠ ∇ k ε (14)

⎛ ∂f ⎞ where ⎜ − 0 ⎟ , the energy derivative of the Fermi-Dirac distribution ⎝ ∂ε ⎠ function, takes finite values only very near the Fermi level and is otherwise zero. This is the reason why only electrons at the Fermi level

Electron Transport Propoerties of Complex Metallic Alloys

331

can contribute to the current density. Since the conductivity σ is defined as J = σ E , it obviously becomes a quantity of tensor. In an isotropic system, into which many CMAs are grouped, the conductivity becomes a scalar quantity and is simplified as

σ=

e 2 τ v i2 dS F e 2τ v 2F e 2τ v F S F dS = ⋅ = F v k⊥ 4π 3 4π 3 v F 3 12π 3

∫

∫

(15)

or alternatively as

σ=

e 2τ v 2F 12π 3

∫

ε =ε F

dS e2 = Λ F v F N (ε F ) ∇ k⊥ ε 3

(16)

where S F is the area of the Fermi surface, v F is the Fermi velocity and N (ε F ) is the density of states (DOS) at the Fermi level. The conductivity formula in Eq. (15) was often employed by Mott, as will be discussed later. Eq. (16) is an alternative expression for conductivity. This is more transparent to recognize that the conductivity is proportional to the mean free path Λ F , the Fermi velocity v F and the number of electrons at the Fermi level N (ε F ) . The mean free path is a quantity determined by the degree of disruption of the periodicity, i.e., geometry-dependent parameter while the product of v F and N (ε F ) is determined solely by the electronic structure of a metal concerned. At this stage, it is timely to summarize conditions, under which the linearized Boltzmann transport equation with the relaxation time approximation can be validated. First of all, the wave vector k characterizing the electronic state of conduction electron must be well defined. To validate this, the mean free path of conduction electron Λ F must be longer than an average atomic distance a. In the second place, the higher-order terms like the deviation from the Ohm law are neglected. We limit ourselves to a linear response regime. As already stated, the Bloch theorem guarantees the propagation of conduction electron in a periodic potential in the form of ψ k (r ) = eik ⋅r uk (r ) and, hence, assures that the wave vector k remains unchanged throughout a crystal. This is equivalent to saying that electrons are not scattered, as long as a potential is perfectly periodic. In other words, any disturbance to periodic potential acts as a source of scattering: the Matthiessen rule5 states that the resistivity is

332

Uichiro Mizutani

given by the sum of a temperature dependent term ρ(T) caused by the inelastic electron-phonon interaction and a temperature independent residual resistivity ρ0. Different models have been put forward in the framework of the Boltzmann transport equation, depending on how the relaxation time τ is approximated under different conditions. First, we consider Ziman’s electrical resistivity theory7. It evaluates the relaxation time τ under the assumptions that (a) the linearized Boltzmann transport equation is valid (ΛF>a), (b) ionic potential is so weak that scattering is treated in the Born approximation, (c) scattering of electrons with ions is elastic. The assumption (b) is generally valid for metals and alloys consisting of only non-transition metal elements like noble metal alloys Cu-Zn, Cu-Al, Cu-Sn etc. They are hereafter referred to as “simple” metals and alloys. The scattering probability, which is inversely proportional to τ, is calculated under the conditions above: 2 ⎛ V ⎞⎛ 1 ⎞ ⎛ 3π 2 ( N / V ) ⎞ =⎜ ⎟ U p (K ) a (K ) (1 − cosθ ) sin θ dθ ⎟⎜ ⎟ ⎜ 2 τ ⎝ 2π N ⎠⎝ ⎠ ⎝ mv F ⎠

1

∫

(17)

where K is the scattering vector defined as the difference in wave vectors k and k ′ before and after scattering, respectively, θ is the scattering angle between k and k ′ , a( K ) is called the structure factor5 and | U p ( K ) |2 is the square of the pseudopotential. An insertion of Eq. (17) into Eq. (3) leads to the well-known Ziman’s resistivity formula: 3πΩ o ⎞ 2 2 4 ⎟ ⎝ 4e v F k F ⎠ ⎛

ρ =⎜

2kF

∫ a( K ) U

p (K )

2

K 3dK

(18)

0

where the upper limit of the Fermi diameter 2k F in the integral corresponds to the maximum scattering angle θ = π , i.e., backscattering. Since elastic scattering is assumed, k = k ′ = k F holds. Ziman was successful in applying his model to account for the occurrence of a negative TCR in divalent liquid metals and alloys5,7. It is of great importance to consider why the Ziman model is valid for liquid metals. This is because Ziman assumed the elastic scattering of electrons with ions and the assumption is valid either at very low temperatures,

Electron Transport Propoerties of Complex Metallic Alloys

333

where the interaction of electrons with vanishing phonons obviously diminishes or at temperatures higher than the Debye temperature ΘD like in liquid state. It is recalled that ions can no longer be treated as an independent particle but collective motions of ions become essential at intermediate temperatures lower than ΘD, where electron transport properties of CMAs are measured5]. Here inelastic electron-phonon interaction has to be rigorously treated. Figure 5 compares the 2kF/Kp dependence of resistivity between “simple” liquid alloys and “simple” amorphous alloys. ( a ) 貴金属を基としたグループ(V)液体合金 100

ρ (μ Ω − cm)

ρ and TCR in liquid Cu-Sn and Cu-In

- +++ ++ + -+ ++ ++Sn + ++++ In + ++

50

0

0.5

1. 5

1. 0 2 kF Kp

(b ) グループ(V)アモルファス 500

合金

simple amorphous alloys

-

--

ρ 300 K ( μΩ − cm)

400 300 200 100 0 0 .5

-

--------- - -+---+-+ +-+- +++++ 1.0 2 kF Kp

1. 5

Fig. 5. (a) resistivity as a function of 2kF/Kp in Cu-Sn and Cu-In liquid alloys and (b) resistivity at 300 K as a function of 2kF/Kp for simple amorphous alloys. Plus and minus signs indicate that of TCR. Reproduced from reference 5.

334

Uichiro Mizutani

Attached to the data points are signs in TCR. The value of 2kF is deduced from the measured Hall coefficient and the wave number Kp from the first peak of the structure factor. Note that the resistivity was measured in the liquid state, i.e., at temperatures much higher than the corresponding ΘD in Fig. 5(a) but at 300 K in Fig. 5(b). As shown in Fig. 5(a), the resistivity maximum occurs in line with a negative TCR at 2kF/Kp=1.0 in liquid alloys. This is well consistent with the Ziman theory. However, such tendency is missing in amorphous alloys. Instead, a negative TCR appears when the resistivity exceeds some critical value of 50~60 μΩcm, regardless of the ratio 2kF/Kp5. We realize that the Ziman theory fails at temperatures lower than ΘD, where the assumption on elastic scattering of electron with ions is no longer valid and inelastic electron-phonon interaction should be properly taken into account. The Baym resistivity formula can be derived under the assumptions that (a) the mean free path is greater than an average atomic distance, (b) the relaxation time approximation is valid, (c) scattering is so weak that the Born approximation is valid but (d) scattering of electrons with ions is now treated in the context of inelastic electron-phonon interaction. The scattering probability is derived in the following form5: ⎛ 2π ⎞ dω =⎜ τ ⎝ N ⎟⎠ 1

∫ ∑U

p (K )

k′

2

a(K , ω ) (1 − cosθ kk ′ ) δ ( ε k ′ − ε k + ω ) βω n(ω ) (19)

where β = 1/ k BT , a(K , ω ) is called the dynamical structure factor and n(ω ) is the Planck distribution function. An insertion of Eq. (19) into Eq. (3) yields

ρ=

3πΩ o 4e 2 v 2F k F4

2kF

∫ 0

2

K 3 U p ( K ) dK

∞

∫ a( K ,ω )βω n(ω )dω

(20)

−∞

Eq. (20) is called the Baym-resistivity formula. More details of its derivation will be found in reference 5. One can also easily prove that the Ziman resistivity formula (18) is derived as a high temperature limit of Eq. (20). The Baym theory is applicable to both simple crystalline and amorphous alloys at T<ΘD, as long as the ionic potential is well

Electron Transport Propoerties of Complex Metallic Alloys

335

approximated by weak pseudopotential. The famous Bloch-Grüneisen law is derived by applying the Baym resistivity formula to a crystalline metal5, while the Baym-Meisel-Cote theory is derived when it is applied to an amorphous alloy8. In the case of crystalline metals, the resistivity follows T5-law at low temperatures and T-linear dependence at higher temperatures in conformity with the Bloch-Grüneisen law. In the case of amorphous alloys, a successive change in temperature dependence from so called type (a) to (c) has been observed with increasing resistivity in low-resistivity simple amorphous alloys5. This is indeed consistent with the Baym-Meisel-Cote theory. As will be discussed later, a change in the ρ-T types from (a) to (c) is caused by a decrease in the mean free path down to an average atomic distance of about a few A. Note that a positive TCR is observed only in type (a) at room temperature. This explains why a positive TCR is observed only when resistivity at 300K is lower than about 50 μΩcm in simple amorphous alloys. Once the resistivity is increased to about 200 μΩcm, the temperature dependence of resistivity becomes more linear over a wide temperature range, which is designated as type (d) and eventually concave as shown in Fig. 6. This is called type (e). Both types (d) and (e) cannot be explained within the framework of the Boltzmann transport mechanism but need an incorporation of weak localization effect. The temperature dependences of resistivity for simple crystalline metals and amorphous alloys are shown in Figs. 6 (a) and (b), respectively. In summary, the resistivity formula including the Ziman theory for simple liquid metals and the Baym-Meisel-Cote theory for simple amorphous alloys and the Bloch-Grüneisen law for crystalline metals are constructed on the basis of the linearized Boltzmann transport equation with the choice of pseudopotential. All these theories are valid only when the mean free path of conduction electron is greater than an average atomic distance. What happens when the mean free path is shortened to an average atomic distance and/or when the pseudopotential approach breaks down? Theories taking into account these conditions have been developed. One is the Kubo-Greenwood theory, which has been applied to systems containing transition metal elements even in the vicinity of Λ F ≈ a . Its derivation is described elsewhere (See Section 11.13 in reference 5). In the next Section, we discuss the high-resistivity regime, where weak

336

Uichiro Mizutani

(b)

(a)

ρ ∝T 1− αT 2

Tmax

ρ ∝T 5

Tmax

+T 2

Fig. 6. (a) Temperature dependence of the electrical resistivity for various pure metals. All data fall on a master curve when both the resistivity and temperature are normalized with respect to that at the Debye temperature and the Debye temperature, respectively. T5-dependence below about T<0.2ΘD and T-linear relation above about T>0.3ΘD hold. (b) Temperature dependence of resistivity in simple amorphous alloys. The types (a) to (e) successively occur with increasing resistivity at 300 K. Type (a): T2-temperature dependence below about 20 K and T-linear with a positive TCR above about 30 K. Type (c): (1-αT2)-dependence below about 20 K and T-linear with a negative TCR above about 30 K. Type (b): type (a) remains at low temperatures but type (c ) appears at higher temperatures, resulting in resistivity maximum at Tmax at intermediate temperatures. Tmax decreases with increasing resistivity and type (b) eventually switches to type (c). Type (e): concave over a whole temperature range. Type (d): intermediate between type (c) and (e). Almost T-linear over a wide temperature range. Reproduced from reference 5.

localization and Mott’s variable-range hopping mechanism play a dominant role. 4. Weak localization and Mott variable-range hopping mechanism Anderson considered the motion of electron in both periodic and non-periodic lattices9. In the case of periodic lattice, electrons having an average energy E0 form the Bloch wave with band width W. In the case of non-periodic potential fields, electrons are localized, when V0/W exceeds

Electron Transport Propoerties of Complex Metallic Alloys

337

some critical value, where V0 represents the degree of disorder in the potential. The basic concept of weak localization phenomenon may be envisaged by using Bergman’s description as a guide10. He considered the situation, in which the electron of the wave vector k is scattered into the state –k by repeating elastic scatterings with impurities. This is illustrated in Fig. 7. Starting from the state k, electron is scattered into either k1’, k2’ and k3’ or k1”, k2”, k3” before reaching -k through the momentum transfers K1, K2 ,K3 and K4 or its opposite sequence of K4, K3, K2 and K1. Scattering amplitudes for the complementary passages may be expressed as A′ = A′ eiθ ′ and A′′ = A′′ eiθ ′′ . Here A′ = A′′ holds, since it is proportional to the product of the Fourier component of scattering potential U (K1 )U (K 2 )U (K 3 )U (K 4 ) . Furthermore, the elastic scattering assures the relation θ ′ = θ ′′ between the two passages. As a result, 2 2 A′ = A′′ = A , A′∗ A′′ = A and A′A′′∗ = A hold. Thus, we found that, thanks to the phase coherency, the probability density of the electron state of –k upon multiple elastic scatterings turns out to be 2 2 2 2 A′ + A′′ = A′ + A′′ + A′∗ A′′ + A′A′′∗ = 4 A . This is twice as large as 2 that of 2 A when the phase coherency θ ′ = θ ′′ is lost. This is nothing but

Fig. 7. Weak localization effect followed by Bergman10. It is alternatively called 2kF scattering. The conduction electron of the wave vector is scattered into the state by repeating elastic scatterings with impurities in reciprocal space. Reproduced from reference 5.

338

Uichiro Mizutani

the enhancement in electron localization tendency. This is possible only when elastic scattering is involved and the phase coherency is guaranteed. Elastic scattering is essential in both Anderson localization and weak localization. Remember that, at absolute zero, inelastic scattering due to the electron-phonon interaction vanishes but only elastic scattering survives. Hence, the localization effect is prone to occur in systems possessing a high residual resistivity. It should be also noted that, when the electron tends to be localized, the electron-electron interaction is enhanced and significantly affects low temperature electron transport, as proposed by Altshuler and Aronov11. Indeed, the enhanced electron-electron interaction coupled with the weak localization effect yields the following square-root temperature dependence of both conductivity and Hall coefficient at low temperatures:

( ) (0) (1 + β T )

σ (T ) = σ 0 1 + α T RH = RH

(T ≤ 20 K )

(T ≤ 20 K )

(21) (22)

and

β = 2α

(23)

A decisive conclusion about the development of weak localization effect can be drawn by proving experimentally the validity of Eqs. (21) to (23). So far we discussed the electron scattering mechanisms from the metallic side of the metal-insulator (hereafter abbreviated as MI) transition. The variable-range hopping model proposed by Mott12 describes the scattering mechanism from the insulating side of the metal-insulator transition. The electron conduction even in its insulating side becomes possible at finite temperatures through the phonon-assisted hopping of electrons. Mott assumed that the electronic states at EF are still finite but localized at 0 K. First, he considered that the localized wave function is no longer described by the Bloch wave but should decay exponentially at large distance. The two localized states centered at Ri and Rj can interact through the overlap of the localized wave functions in the form of ψ ∗ (r − R i )ψ (r − R j ) dr ≈ exp ( − R / a ) . The probability of a

∫

Electron Transport Propoerties of Complex Metallic Alloys

339

transition from Ri to Rj is therefore expressed as P ∝ exp(−2r / a) . Even when R is large enough to make the overlap integral to be very small, the hopping may still occur, if the energy difference Ei − E j is compensated by the absorption or emission of phonons. Mott gave the phonon-assisted transition probability as follows:

(

)

P = ν ph exp − Δ ij / k BT exp ( −2 R / a )

(24)

where Δ ij = Ei − E j and ν ph is the characteristic frequency of phonons. The transition probability is further rewritten by inserting the relation 4π 3 R N ( E F )Δ ij ≈ 1 into Eq. (24) and expressed as 3

⎡ 3 2R ⎤ P = ν ph exp ⎢ − − 3 a ⎥⎦ ⎣ 4π R N ( E F )k BT

(25)

The optimum value of R is determined by taking the condition dP / dR = 0 . By using the optimum R thus determined, Mott arrived at the well-known conductivity formula: ⎡ B ⎤ 1/ 4 ⎥ ⎣ T ⎦

σ (T ) ∝ ν ph exp ⎢ −

(26)

1/ 4

−1/ 4 −1/ 4 ⎛ 8 ⎞⎛ 9 ⎞ = 2.062 ⎡⎣ N ( EF )k B a 3 ⎤⎦ . where B = ⎜ ⎟⎜ ⎟ ⎡⎣ N ( EF ) k B a 3 ⎤⎦ ⎝ 3 ⎠⎝ 8π ⎠ The exponentially dependent T-1/4 behavior has been frequently observed on the insulating side of the MI transition, as will be demonstrated below.

5. Electron Transport properties of VxSi100-x amorphous alloys

As shown in Fig. 8 (a), we can form an amorphous single phase from pure Si up to at least 74 at%V by sputtering method in the V-Si alloy system13. The electrical conductivity at 10 K, shown in Fig. 8 (b) decreases rapidly with decreasing V concentration. Its behavior is very similar, irrespective of the choice of the transition metal elements like V, Ta and Ni. The shaded area represents an insulating regime, as will be discussed below.

340

Uichiro Mizutani WeightPercent per cent Silicon Weight Silicon

10

(a)

(c)

x=74

8

x=53

V Si 2

6

x=21

Atomic per cent Silicon

V

Si

metal-insulator transition

4

ln σ (Ω -1 cm -1 )

4 2

metallic regime

x=17

x=14

0

(b) 3

-2

logσ 10K (Ω -1 cm -1 )

2

-4

1 0 -1 -2 -4 0

x=7

-6 VxSi100-xV-Si TaxSi100-x Ta-Si NixSi100-x Ni-Si 10 20 30 40 50 Solute content (at.%)

insulating regime

1

2

3 logT

4

5

60

Fig. 8. (a) Amorphous alloys were prepared at compositions marked with arrows in the V-Si phase diagram. The composition at the MI-transition is also marked, (b) the solute concentration dependence of conductivity at 10 K on logarithmic scale for three alloy systems, and (c) temperature dependence of conductivity for a series of VxSi100-x amorphous alloys on logarithmic scale. Reproduced from reference 13.

Indeed, Fig. 8 (c) clearly shows that a dramatic change in the temperature dependence of conductivity is found in the vicinity of 15 at%V. The MI-transition was studied first from structural changes by using a combination of neutron and X-ray diffraction measurements13. We could determine the Si-Si partial radial distribution function (RDF) quite reliably by taking a full advantage that the neutron scattering amplitude for the V atom is zero. The results are shown in Fig. 9 (a). We could confirm from the area of the first peak in amorphous Si that the

Electron Transport Propoerties of Complex Metallic Alloys

341

50

(b) Metallic character reflecting Si2V is found at x=29 and 37 40

(a)

Si-Si partial distribution function

30

R D F (r) S i-S i

x=29 x=24 x=21 x=19

x=14 x=7

Si2V

x=37

V-V x=29

0

x=12

0

10 0

x=15

0

20

Si-V

10

5

a-Si 1

2

3

4

r (A)

5

6

7

8

0

0

1

2

3

4

5

6

7

8

S i 2 V c o o rd in a tio n n u m b e r

R D F (r) S i-V

x=37

0

r (A)

Fig. 9.(a) Si-Si partial radial distribution function (RDF) for a series of VxSi100-x amorphous alloys. An arrow indicates the position where Si-Si atomic pair existing in VSi2 compound grows, and Fig. 9(b) Si-V partial RDF for x=29 and 37 alloys. Reproduced from ref. 13).

coordination number is four, being consistent with tetrahedrally bonded network of Si atoms. The area in the first peak slightly decreases upon the addition of V atoms. However, a small peak emerges at about 3 A, as marked with an arrow. This corresponds to the Si-Si distance existing in hexagonal Si2V compound. The X-ray diffraction measurement can extract more information from V atom than from Si atom. A combination of these two diffraction spectra allowed us to determine the Si-V partial RDF, as shown in Fig. 9 (b). The local structure near 30 at%V is found to resemble that of Si2V metallic compound. Figs. 10 (a) and (b) show the V concentration dependence of both distance and coordination number in Si-Si and Si-V atomic pairs, respectively13. The total coordination number around Si atom is almost four up to x=15 but begins to increase sharply, when the V content exceeds x=15. This means that Si atoms in the tetrahedrally bonded network are randomly substituted by V atoms up to about x=15 but the metallic bonding characterized by a higher coordination number begins to

342

Uichiro Mizutani

Distance (Å)

2.7

2.8

(a)

Si-Si

Distance (Å)

2.8

2.6 2.5

Si - V

2.6

2.4

2.4 2.3

2.2

Si-Si

4

4

Coordination Number

(A)

3 2 1 0 0

Si - V

5

2.2 Coordination Number

(b)

10

20 30 at.% V

40

50

(B)

3 2 1 0

(A) 0

10

20 30 at.% V

40

50

Fig. 10. (a) Si-Si and (b) Si-V atomic pair distances and coordination numbers as a function of V concentration for a series of VxSi100-x amorphous alloy system. The line (A) is drawn through the data points in V-poor region. The sum of coordination numbers due to Si-Si and Si-V on line (A) is about 4 below x=15. Reproduced from ref. 13.

develop above x=15. From this structural studies, we conclude that the MI-transition takes place at about x=15 in this system. The composition dependence of the electronic specific heat coefficient in VxSi100-x amorphous alloys is shown in Fig. 11(a)13. One can clearly see that the value of the electronic specific heat coefficient decreases sharply with decreasing V content, indicating that the pseudogap grows across the Fermi level and its depth is deeper and deeper, as we approach towards a semiconducting amorphous Si. Another important to be noted is that the value of electronic specific heat coefficient remains finite even upon entering into the insulating regime. This strongly suggests that, in the insulating regime, the Fermi level falls on the mobility edge, where the wave function is localized. In such circumstances, the variable-range hopping mechanism proposed by Mott12 is expected to hold. Indeed, as shown in Fig. 11 (b), the data for x=7 and 14 exhibit its characteristic

Electron Transport Propoerties of Complex Metallic Alloys

343

temperature dependence of T-1/4 on the logarithmic scale. Instead, the temperature dependence of conductivity is negligibly small on this scale in the metallic regime. Upon its conversion into resistivity, we could confirm that it is typical of the type (e), indicating that weak localization is dominant. Indeed, the square-root temperature dependence of conductivity is observed at low temperatures for samples in the metallic regime, as shown in Fig. 11 (c). 3.0

x10 5

weak localization regime

VxSi100-x

2.0

1.0

10

30 20 V (at.%)

T (K) 300 100 50 30 20

10

40

50

5

x=21 2

1

5

0

x=74

x=53

0

1

2 T

5

3 1/2

[K 1/2 ]

300

x=21

ρ (μΩcm)

ln σ (Ω -1cm-1)

4

3

x=53

0

V74Si26 amorphous alloy 280

type (d) or (e)

260 240

x=14

220

-5

x=7 200 0

-10 0.2

25

4

0.0 0

10

15

1.5

0.5

(b)

T (K) 10 5

1

(c)

σ-σ 0 (Ω -1cm -1)

γexp (mJ/mol.K2)

2.5

-5

(a)

0.3

0.4 T

-1/4

0.5 (K

0.6

Temperature (K) 50

100

150

200

250

0.7

-1/4

)

Fig. 11. (a) V concentration dependence of electronic specific heat coefficient in VxSi100-x amorphous alloys, (b) temperature dependence of conductivity and resistivity for selected samples and (c) square-root temperature dependence of temperature dependent conductivity for x=21 and 51 samples. Reproduced from reference 13.

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Uichiro Mizutani

VxSi100-x

Intensity (arb. units)

xx=19 = 19 ▽ x=14 ∇ x = 14 x=12 + x= x=7 12 x=7 pureSi

12

& pure Si

3

10

8

6

2

4

1

2

EF

EF

-1

-2

Binding Energy (eV) Fig. 12. X-ray photoemission valence band spectra for a series of VxSi100-x amorphous alloys. The data near the Fermi level are magnified in its inset. Reproduced from [13].

The XPS photoemission valence band spectra for a series of V-Si amorphous alloys are shown in Fig. 12. In pure Si, one can clearly see the opening of the gap at the Fermi level. But the DOS apparently becomes finite, as soon as V is introduced into amorphous Si. This is consistent with the electronic specific heat data and supports our interpretation based on the variable-range hopping mechanism for dilute V samples. 6. Electron transport properties of (Ag0.5Cu0.5)100-xGex amorphous alloys

The Ge concentration dependence of the resistivity at 300 K along with its temperature dependence normalized with respect to that at 273 K is shown in Fig. 13(a)14. The resistivity at 300 K is only 20 μΩcm in binary Ag-Cu alloy. But the addition of Ge enhances the resistivity eight orders of magnitude over a whole composition range. The temperature dependence of resistivity changes from types (a) to (c) in accordance with the prediction from the Baym-Meisel-Cote model up to x=35 at%Ge. Further

Electron Transport Propoerties of Complex Metallic Alloys

109

(a)

1.2

108

(e)

1.1

3 0 0K

106

(Ag0.5Cu0.5)75Ge25

(d)

ρ ( Τ )/ ρ 27 3 Κ

107 ρ (μ Ω cm )

345

(d) (c) (c) (c) (b) (b) (a)

1.0

105

Cu-3d

(a)

Ag-4d

EF

0.9

104

0

100

200 T (K)

300

103 102 10

(a) (a)

0

(c) (c) (b) (b)

20

(c)

(d)

40

(d)

(e)

60

(b)

80

100

x (at.%Ge)

Fig. 13. (a) Ge concentration dependence of resistivity at 300 K and temperature dependence of resistivity normalized with respect to that at 273 K for (Ag0.5Cu0.5)100-xGex amorphous alloys and Fig. 13 (b) X-ray photoemission valance band spectrum for (Ag0.5Cu0.5)75Ge25 amorphous alloy. Reproduced from references 14, 15 and 21.

increase in Ge content beyond x=35 gives rise to types (d) and (e) before entering the insulating regime. From this, one can naively assume that the free electron-like electronic structure persists up to about x=35. Indeed, the photoemission valence band spectrum in Fig. 13(b) for the sample with x=25 clearly shows that the DOS near the Fermi level is dominated by free-electron-like sp-electrons15. The Ge concentration dependence of the Hall coefficient is shown in Fig. 14 along with that of the electronic specific heat coefficient in its inset14. The deviation from the free electron curve evidently occurs above x=35 in both sets of data. An upward deviation of the Hall coefficient indicates a decrease in carrier concentration relative to the free electron values. Indeed, the electronic specific heat coefficient reflecting the DOS at the Fermi level can be taken as a straightforward demonstration for the formation of the pseudogap above x=35 in good agreement with the onset

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Uichiro Mizutani

o ctr ele e e fr ve r cu

n

(a)

(a) (e)

(b) (c)

(c) (c)

(d) (c)

(c)

(d)

free electron curve

RH = −

1 ne

Fig. 14. Ge concentration dependence of the Hall coefficient and electronic specific heat coefficient in the inset for (Ag0.5Cu0.5)100-xGex amorphous alloys. The dashed curves represent the respective free electron values. Reproduced from references 14 and 22.

of the weak localization, as manifested by the ρ-T types (d) and (e) shown in the inset to Fig. 13 (a). The Ge concentration dependence of the Hall coefficient is shown in Fig. 14 along with that of the electronic specific heat coefficient in its inset14. The deviation from the free electron curve evidently occurs above x=35 in both sets of data. An upward deviation of the Hall coefficient indicates a decrease in carrier concentration relative to the free electron values. Indeed, the electronic specific heat coefficient reflecting the DOS at the Fermi level can be taken as a straightforward demonstration for the formation of the pseudogap above x=35 in good agreement with the onset of the weak localization, as manifested by the ρ-T types (d) and (e) shown in the inset to Fig. 13 (a). As shown in Fig. 15, the Ge-rich samples exhibit not only the square-root temperature dependence of both conductivity and the Hall coefficient at low temperatures but also the coefficient α of the Hall coefficient is twice as large as the coefficient β of the conductivity, taking

Electron Transport Propoerties of Complex Metallic Alloys

347

Fig. 15. Square-root temperature dependence of both conductivity (solid circles) and Hall coefficient (open triangles) for (Ag0.5Cu0.5)100-xGex amorphous alloys with x=50 and 70. σ(0) and RH(0) refer to the values extrapolated to absolute zero. Reproduced from refs. 14 and 22.

all as evidences for the development of the weak localization coupled with enhanced electron-electron interaction in the metallic regime, where the ρ-T types are characterized by either (d) or (e). The ratio of the measured electronic specific coefficient over the corresponding free electron value is plotted as a function of the Ge concentration in Fig. 16(a) along with the inverse of the ratio of the measured Hall coefficient over the corresponding free electron value14. The ratio of the electronic specific heat coefficient is obviously used as a measure to assess quantitatively the depth of the pseudogap at EF, since it is proportional to the DOS at EF. As can be seen in Fig. 16(a), the inverse ratio of the Hall coefficient may be alternatively used for this purpose. To study the effect of the pseudogap on the electron transport properties, we plotted in Fig. 16(b) the conductivity at 300 K as a function of the inverse ratio of the Hall coefficient on the logarithmic scale. Included are the data

348

Uichiro Mizutani 10

○● ○ ○

(a)

○

(b)

102 ○ ○ ●

γ R free g = exp or H γ free RH 0

20

types (d) or (e)

○

0.5

0

(b)

(a)

60 40 x (at.%Ge)

○ ○ ○

80

(c) (Ag0.5Cu0.5)100-xGex

300K

RHfree/RH and γexp/γfree

○

●●● ○○ ○

ρ (μΩcm)

○ ○

1.0

103

Expanded liquid Hg

100

2

104 0.1

0.2

0.3 0.4 0.5

0.8 1.0

g=RHfree/RH Fig. 16. (a) The Ge concentration dependence of the ratio of measured electronic specific heat coefficient over the corresponding free electron value and the inverse ratio of the Hall coefficient over the corresponding free electron value for (Ag0.5Cu0.5)100-xGex amorphous alloys and (b) the g-parameter dependence of the resistivity at 300 K for (Ag0.5Cu0.5)100-xGex amorphous alloys (solid circles) and expanded liquid Hg (open triangles). Reproduced from references 14 and 16.

for expanded liquid Hg measured by Even and Jortner in 197216. The ordinate referring to the degree of the depth of the pseudogap at the Fermi level may be hereafter called the g-parameter. Indeed, Even and Jortner used the inverse ratio of the Hall coefficient as the g-parameter and pointed out that the data fall on the straight line with a slope of +2 in good agreement with the Mott g2-theory. Surprisingly, the data of the Ag-Cu-Ge amorphous alloys characterized by the possession of types (a) to (c) fall on the vertical line at g=1 whereas those characterized by types (d) or (e) beyond x=35 fall on a straight line with the slope of +214. This encouraged us to rely on the Mott g2-theory for the Ag-Cu-Ge amorphous alloys a. a

As is clear from Fig. 16 (b), the data for Ag-Cu-Ge amorphous alloys and expanded liquid Hg start to fall on a straight line with a slope of +2 when the resistivity increases beyond 200 and 400 μΩcm, respectively. The difference in the resistivity may be explained by using the critical resistivity discussed in Section 8. See also Appendix 2.

Electron Transport Propoerties of Complex Metallic Alloys

349

7. Metal-Insulator transition in various CMAs

Mott in 196917 introduced the word “pseudogap”in his paper dealing with MI-transition of expanded liquid Hg under high temperatures and high pressures and modified the conductivity formula σ = e 2 S F Λ F /12π 3 derived from the Boltzmann transport equation by replacing the area of the Fermi surface SF by g2SFfree and the mean free path ΛF by an average atomic distance a in the following form:

σ = e 2g2 S Ffree a /12π 3 .

(27)

This is the conductivity formula Mott employed to predict the g2-dependent conductivity expected for expanded liquid Hg. He conjectured that the free electron behavior holds and the Boltzmann transport mechanism works when g=1 but weak localization effect sets in and g2-dependent conductivity will occur, as g is lowered below unity. Encouraged by our finding that the transport data for Ag-Cu-Ge amorphous alloys are scaled with g2 in the same way as expanded liquid Hg, we extended Mott’s theory to cover any kinds of CMAs including not only amorphous alloys but also quasicrystals and their approximants, as will be described below5. Before doing this, it may be important at this stage to confirm that CMAs like quasicrystals and their approximants also undergo successive changes in the ρ-T types from types (a) to (e) with increasing resistivity. Remember that many quasicrystals and approximants are characterized by the possession of the pseudogap at the Fermi level, though its magnitude is strongly dependent on the alloy system chosen. The temperature dependence of resistivity is shown in Figs. 17 (a) and (b). When the resistivity is low like in Al-Mg-Zn approximants, type (a) is observed in spite of the fact that the pseudogap exists at the Fermi level18. A systematic change in the ρ-T types from (a) to (e) with increasing resistivity is indeed observed in various quasicrystals, as shown in Fig. 17(b)19. As an additional evidence for the development of weak localization. Fig. 18 shows the square-root temperature dependence of conductivity in high-resistivity quasicrystals Al-Li-Cu (type (d)) and Al-Ru-Cu (type

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Uichiro Mizutani

(b) (a)

ρ300K (µΩcm)

7 (e)

1.08

(b)

(c)

1.04

Al 15Mg 44Zn41 QC

ρ(T)/ ρ(273K)

ρ300K=146 μΩcm

1.00

1 2 3 4 5 6 7

Mg-Al-Cu Mg-Al-Cu Mg-Al-Cu Mg-Al-Cu Mg-Al-Cu Mg-Al-Cu Mg-Al-Cu

70 86 90 96 139 800 1600

6 (d)

0.96 X=30.5 (a) x=30.5

0.92

x=25.5 X=25.5

0.88

(a) 5 (c)

(a) X=45.5 x=45.5

0.84 0

Alx Mg39.5 Zn60.5-x approx.

4 (c)

ρ300K=44~67 μΩcm

100

T (K)

200

3 (b) 2 (b)

300

1 (a)

Fig. 17. (a) Temperature dependence of resistivity normalized with respect to that at 273 K for Al-Mg-Zn quasicrystals and their approximants18 and (b) Temperature dependence of resistivity normalized with respect to that at 300 K for different quasicrystal19. The ρ-T types are indicated with resistivity values at 300 K.

Al-Ru-Cu quasicrystal

0.1

1

Al-Li-Cu quasicrystal

1/2

0.01

1 1/2 0.001 1

100

10 T (K)

Fig. 18. Temperature dependence of conductivity for Al55.0Li35.8Cu9.2 and Al68Ru15Cu17 quasicrystals on logarithmic scales. The σ(0) refers to the conductivity extrapolated to absolute zero. Reproduced from ref. 14.

Electron Transport Propoerties of Complex Metallic Alloys

351

(e))14. Judging from all data so far discussed, we now believe that the electron transport of CMAs may be universally described by extending Mott g2-theory developed originally for expanded liquid Hg. To discuss the effect of the pseudogap in CMAs on the electron transport properties, we have proposed to use the alternative conductivity formula σ = e 2 v F N ( EF )Λ F / 3 and replaced vF and N(EF) by gvFfree and gN(EF)free, respectively, while leaving the mean free path ΛF as a variable. Thus, the equation we have employed5 is given by

⎛ e2 ⎞ 2 free free ⎟ g N ( EF ) v F Λ F 3 ⎝ ⎠

σ =⎜

(28)

where both g and ΛF act as variables. We explain the occurrence of types (a) to (c) in CMAs by using this modified equation: A successive change in types (a) to (c) is understood as the process of shortening the mean free path down to an average atomic distance. Here g=1 holds in the case of amorphous alloys but g lower than unity is possible in the case of crystals like approximants because of the presence of lattice periodicity. This is consistent with the Baym-Meisel-Cote model based on the Boltzmann transport equation. When types (d) and (e) are observed, the mean free path must be constrained by an average atomic distance a and g2-electronic structure effect is expected to emerge for any CMAs including quasicrystals and approximants. The g2-law may be rewritten by inserting the definition of g = N ( EF ) / N ( EF ) free into the Mott formula: ⎛ e2 ⎞ 2 free free ⎟ g N ( EF ) v F a ⎝ 3⎠

σ =⎜

2

⎛ e 2 ⎞⎛ N ( E F ) ⎞ = ⎜ ⎟⎜ N ( EF ) free v Ffree a free ⎟ N E 3 ( ) F ⎠ ⎝ ⎠⎝ free 2 ⎛ e ⎞⎛ vF a ⎞ 2 = ⎜ ⎟⎜ ⋅ N ( E F )} free ⎟ { ⎝ 3 ⎠⎝ N ( E F ) ⎠

(29)

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Uichiro Mizutani

Now we find that the g2-law is equivalent to saying that the conductivity is proportional to the square of DOS at EF or that of the electronic specific heat coefficient. We plotted in Fig. 19 the room temperature resistivity value against the measured electronic specific heat coefficient for a large number of quasicrystals and their approximants on logarithmic scalesb,20.

Fig. 19. Resistivity at 300 K against the measured electronic specific heat coefficient γexp on a log-log scale for Al-Mg-Zn, Al-Mg-Pd, Al-Mg-Cu, Al-Mg-Ag, Mg-Ga-Zn, Al-Li-Cu, Al-Cu-Fe-Si, Al-Mg-Pd, Al-Cu-Ru, Al-Pd-Re quasicrystals and their approximants. The data fall on the line with a slope of -220.

The data are found to fall on a straight line with a slope of -2. This is taken as the confirmation of the Mott g2-law for quasicrystals and approximants. On this diagram, a line is drawn by assuming that a minimum conductivity would exist in the metallic regime very close to the MI transition, where the ordinary conductivity formula is still valid. The conductivity formula in this limit may be written as b

Here we intentionally used the resistivity at 300 K to minimize the weak localization effect, since the Mott g2-law has little to do with it. But the difference in the value between 300 K and low temperatures like 4.2 K is of minor importance on the logarithmic scale in the metallic regime.

Electron Transport Propoerties of Complex Metallic Alloys

σ min =

353

e2 ( v F Λ F )min N ( EF ) ≡ e 2 Dmin N ( EF ) 3

(30)

1 1 ( v F Λ F )min = gv Ffree a 3 3

(31)

where Dmin =

Here Dmin refers to a possible minimum electron diffusion coefficient, which is roughly evaluated as 0.25 cm2/s by inserting the minimum g value of 0.2 suggested by Mott17 and atomic distance of 4 A for a and the free electron Fermi velocity of 108 cm/s for v Ffree . The line with D=0.25 cm2/s may be called the MI-line as a rough measure to locate the boundary between the metallic and insulating regimes5. It is interesting to note that all resistivity data for quasicrystals fall just below the MI-line drawn in the diagram. To make the g2-law more comprehensive, we attempted to incorporate the data for amorphous alloys into this diagram. In the case of Ag-Cu-Ge amorphous alloys, the set of resistivity value at 300 K and the electronic specific heat coefficient can be read off from Figures 13 and 15, respectively14, 21, 22. Since the electronic specific heat coefficient near the MI transition has not yet been measured, we have roughly estimated its magnitude by extrapolating the Hall coefficient data shown in Fig. 16 (a) to the Ge concentration up to 90 at%, where the MI-transition is expected to occur. The data for the V-Si amorphous alloys were taken from Figs. 8 and 11(a)13. The datasets are easily read off from these two figures. On top of them, we also incorporated the data taken on TixSi100-x23 and CexSi100-x 24 amorphous alloys. The resulting diagram is shown in Fig. 205. As mentioned in relation to Fig. 19, the data for quasicrystals form a line with a slope of -2 on the log-log scale. The data for Ag-Cu-Ge amorphous alloys happen to be well superimposed onto the line on quasicrystals and approximants c1. This serves as a key point in the discussion in Section 8. The data in the c

Note that the composition for Ag-Cu-Ge amorphous alloys in Fig.20 is shown in the form of (Ag0.5Cu0.5)xGe100-x in accord with other amorphous alloys TMxSi100-x (TM=V, Ti and Ce).

354

Uichiro Mizutani 10000000 x=4

insulating regime x=7

1000000 x=12

ρ300K (μΩcm)

100000

x=14

x=6 x=10

10000

1000

x=9 x=9.5 x=20

x=21

x=13.3

x=15

x=29

x=17.5 x=51

x=40 x=60

100

10 0.1

x=80

x=41

metallic regime 1

x=63

x=74

x=83 D=0.25 cm2/sec

10

100

D=0.01 cm2/sec D=0.1 cm2/sec

1000

γexp (mJ/mol.K2 )

Fig. 20. Resistivity at 300 K against the measured electronic specific heat coefficient γexp on a log-log scale for different CMAs. (open crosses) quasicrystals and their approximants20, (circles) (Ag0.5Cu0.5)xGe100-x amorphous alloys, (triangles) TixSi100-x amorphous alloys23, (squares) VxSi100- amorphous alloys13, (diamonds) CexSi100-x amorphous alloys24. Solid and open symbols refer to data in metallic and insulating regimes, respectively. Note that the chemical formula for (Ag0.5Cu0.5)xGe100-x amorphous alloys is different from that in the text.

insulating regime is shown with an open circle. The data for V-Si amorphous alloys in the metallic regime are shown with solid squares and those in the insulating regime open squares. We see that the data are not only shifted to the right but also the metallic regime persists up to the region above the MI-line of D=0.25 cm2/s. Nevertheless, the data in the metallic regime apparently falls on a line with slope of -2 while those in the insulating regime do not fall on its extrapolation. The data for Ti-Si amorphous alloys23 falls in between the line of Ag-Cu-Ge and V-Si amorphous alloys. This seems to reflect the fact that Ti has one 3d electron less than the V atom. The data in the metallic regime again form a line with the slope close to -2. We can further add the data for Ce-Si amorphous alloys24. The data for Ce-Si amorphous alloys

Electron Transport Propoerties of Complex Metallic Alloys

355

are shifted further to the right, as shown in Fig. 20. The slope of the data in the metallic regime is closer to -1 rather than -2. The MI boundary seems to be located near an extremely low value of D=0.01 cm2/sec line. The Ce-bearing alloys often exhibit an extremely large electronic specific heat and are known as heavy fermion systems24. The present results indicate that amorphous alloys are not exception. By taking all data into account, we may draw the experimentally determined metallic regime as indicated by shades in Fig. 205. Its theoretical interpretation needs further studies in future. 8. Evaluation of the critical resistivity for the Boltzmann transport mechanism

As discussed in Section 7, the data for the Ag-Cu-Ge amorphous alloys fall on the same line as those for quasicrystals and approximants in Fig. 20. Here we emphasized that the pseudogap emerges across the Fermi level only after the concentration of metalloid element Ge increases beyond 35 at% and the covalent bonding develops. Instead, the pseudogap is always present in many quasicrystals and approximants. Nevertheless, the free electron-like ρ-T type (a) is observed in low-resistivity approximants, as if the pseudogap were absent. Moreover, the resistivity ranges over 40 to 104 μΩ-cm in quasicrystals and approximants, indicating that the pseudogap at EF differs from a system to system and affects the resistivity value significantly. We consider the lattice periodicity of approximants to play an important role. To shed more light on this, we evaluated the critical resistivity ρ0, across which the Boltzmann transport terminates and weak localization sets in, as evidenced from the switch of ρ-T types from (a)-(c) to (d) or (e)25. First, we inserted appropriate values of a=4 A, vFfree=108 cm/s and N(EF)free=0.8 mJ/molK2 into the Mott’s g2-relation ⎛ e2 ⎞ σ = ⎜ ⎟ g2 N ( EF ) free v Ffree a and roughly obtained a simple relation ⎝ 3⎠ ρ0=200/g2 (see Appendix 2). Now we evaluate the critical resistivity in different quasicrystals and approximants by using the data for amorphous alloys as a reference.

356

Uichiro Mizutani

We already knew that weak localization regime appears when Ge concentration exceeds about 35 at% in (Ag0.5Cu0.5)100-xGex amorphous alloys. This is also the composition, where the pseudogap at EF begins to grow. Hence, the critical resistivity of about 200 μΩcm is reached still at g=1 in Ag-Cu-Ge amorphous alloys. In other words, we can say that the lowest value of ρ0 is reached for amorphous alloys because of the lack of lattice periodicity. Note here that this argument cannot be applied for other amorphous alloys like VxSi100-x, which do not fall on the same line in Figure 20 as those for quasicrystals. First, we discuss Al-Mg-Zn approximants and quasicrystals possessing a rather shallow pseudogap across the Fermi level. According to the LMTO-ASA band calculations26, the depth of the pseudogap in the Al-Mg-Zn approximant is calculated to be about g=0.7, as shown in Fig. 21.

Fig. 21. The DOS calculated in the LMTO-ASA method for the Al30Mg40Zn30 1/1-1/1-1/1 approximant. The g-parameter is approximately 0.726.

The critical resistivity is estimated to be 400 μΩcm by inserting g=0.7 into the relation ρ0=200/g2. This is well consistent with the data in Fig. 17(a), where ρ-T type of only (a) with low resistivities around 50 μΩcm in approximants and that of only (c) with around 150 μΩcm in quasicrystals are observed18. The presence of lattice periodicity in approximants must be responsible for the appearance of type (a) in spite of the possession of the pseudogap at the Fermi level. Moreover, we can

Electron Transport Propoerties of Complex Metallic Alloys

357

understand why the weak localization does not develop even in the quasicrystalline phase. This is simply because an increase in resistivity up to only 150 μΩcm is small enough to overwhelm the critical resistivity of 400 μΩcm.

(a)

(d)

ρ(T)/ρ(273K)

integrated density of states (states/cell)

density of states (states/Ry.cell)

(b)

(a)

EF E F

Fig. 22. (a) DOS calculated for Al-Li-Cu 1/1-1/1-1/1 approximant27. The g-parameter is 0.4-0.5 and (b) temperature dependence of resistivity normalized with respect to that at 273 K for Al-Li-Cu quasicrystal and its approximant. The resistivity at 4.2 K is 870 and 200 Ωμcm for quasicrystal and its approximant, respectively29.

In the case of Al-Li-Cu approximant, the band calculations shown in Fig. 22(a) revealed the depth of the pseudogap at the Fermi level to be about 0.527, 28. Its insertion into relation ρ0=200/g2 yields the critical resistivity of 800 μΩcm. The possession of the type (a) for the approximant and the type (e) for the quasicrystal in Fig. 22(b) is quite reasonable. Note that the resistivity of 200 μΩ-cm for the Al-Li-Cu approximant29 is higher than that of the Al-Mg-Zn quasicrystal in Fig. 17 (a). This certainly reflects the difference in the depth of the pseudogap at the Fermi level between them. As a third example, the DOS calculated for the Al-Cu-Fe approximant is shown in Fig. 23(a)30. The depth of the pseudogap at the Fermi level is found to be about g=0.3. An insertion of g=0.3 into the relation ρ0=200/g2 leads to the critical resistivity of 2220 μΩcm. The temperature

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Uichiro Mizutani

dependence of resistivity normalized with respect to that at 300 K is shown in Fig. 23(b)25. The ρ-T type (a) is observed in the Al53Cu25.5Fe12.5Si9 approximant having the resistivity of about 600 μΩcm with subsequent successive changes from (a) to (c) below ρ0. However, the resistivity at 300 K for the Al62Cu25.5Fe12.5 quasicrystal reaches about 5000 μΩ-cm and well exceeds ρ0. A switch to the type (e) is indeed observed.

R e s is t iv it y ( x 1 0 3 μ Ω c m )

(b)

(e)

(a) (c)

6 5 4 3 2 1 0 0

2

4

6

8

10

Si content

(c) (b)

(a) (a)

Energy (Ry)

Temperature K

Fig. 23. (a) DOS calculated for Al-Cu-Fe 1/1-1/1-1/1 approximant 30. The g-parameter is about 0.3, and (b) Temperature dependence of resistivity normalized with respect to that at 300 K for two series of Al62-ySiyCu25.5Fe12.5 and Al55Si7Cu38-xFex alloys. Sample with y=0 refers to quasicrystal and otherwise approximants25.

The g-parameter dependence of the resistivity at 300 K is plotted in Fig. 24 for three representative quasicrystals and their approximants we discussed along with the critical resistivity line on the logarithmic scale25. We found that, the deeper the pseudogap at the Fermi level, the higher the critical resistivity, across which weak localization effect emerges. The confirmation of the ρo=200/g2 relation for quasicrystals and their approximants can be taken as demonstration for the validity of Eq. (28),

Electron Transport Propoerties of Complex Metallic Alloys

359

i.e., the extension of the Mott g2-theory to cover CMAs with the emphasis on the role of lattice periodicity in the pseudogap system. 100000 weak localization regime

ρ300K (μΩcm)

10000

QC

(e)

AC

(c) (d) (b) (a) QC

1000

100

AC (a)

QC (c) (a) AC

10 0.1

Boltzmann-transport regime

0.3 0.5 0.7 1 Al-Cu-Fe-Si Al-Mg-Zn Al-Li-Cu

g-parameter Fig. 24. Resistivity at 300 K as a function of the g-parameter for three different systems: Al-Mg-Zn (g=0.7), Al-Li-Cu (g=0.5) and Al-Cu-Fe-Si (g=0.3). The line refers to the critical resistivity given by ρ0=200/g2 described in the text. The region below g=0.2 would be in the insulating regime25.

Before ending this Section, we show the last example to remind the importance of the interrelation between the pseudogap and lattice periodicity. Fig. 25 is reproduced from the data from reference 31, who reported the temperature dependence of electrical resistivity for the Al-Pd-TM (TM=Fe, Ru and Os) 1/0-1/0-1/0 approximants over the range 12-300 K. The room temperature resistivity values informed by Tamura are added. Though one may naturally assume that the Boltzmann type transport mechanism would not last when the resistivity exceeds 1000 μΩcm, the Al70Pd19Os11 approximant having the resistivity of 1020 μΩcm is found to exhibit the ρ-T type of (a). We consider the critical resistivity in this system to be higher than 1000 μΩcm and the Boltzmann-type scattering mechanism to remain active owing to the coexistence of a deep pseudogap at EF and the lattice periodicity of 15.5 Å31.

360

Uichiro Mizutani

(c)1000 μΩcm (b)720 μΩcm

(a)1020 μΩcm

(a) 590 μΩcm

Fig. 25. Temperature dependence of electrical resistivity for Al-Pd-TM (TM=Fe, Ru and Os) 1/0-1/0-1/0 approximants. The room temperature resistivity and ρ-T types are also included31.

9. Conclusions

Electron transport properties and scattering mechanisms involved in CMAs have been surveyed by choosing the family of non-magnetic quasicrystals, their approximants and amorphous alloys. In the case of Ag-Cu-Ge amorphous alloys, the free electron-like DOS persists up to the concentration of 35 at%Ge and g=1 holds. Here an increase in metalloid concentration from zero up to 35 at%Ge causes a decrease in the mean free path of electrons down to an average atomic distance and accompanies the successive change in the ρ-T types from (a) to (c). All these changes can be interpreted within the framework of the Boltzmann transport mechanism. When the Ge concentration exceeds this threshold value, the pseudogap appears and the Mott g2-law is activated and weak localization effect dominates below 300 K. The critical resistivity of 200 μΩcm is reached in the Ag-Cu-Ge amorphous alloys by inserting g=1 into the ρo=200/g2 relation. In the case of quasicrystals and approximants having a rather shallow pseudogap, the critical resistivity is increased to about 400

Electron Transport Propoerties of Complex Metallic Alloys

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μΩcm. A critical resistivity is further increased above 2200 μΩcm, when the pseudogap is deepened and the g-parameter is decreased to about 0.3. Thus, we can say that the presence of lattice periodicity coupled with the decreasing g-parameter determines the magnitude of the critical resistivity in approximants and weak localization likely develops upon the formation of the quasicrystal counterpart as a result of the loss of lattice periodicity. The conclusions above are illustrated in Fig. 26.

Fig. 26. Summary for the electron transport mechanisms in non-magnetic CMAs.

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The metal-insulator (MI) transition was also studied on the ρ300K versus γexp diagram on the log-log scale. The data of both quasicrystals and Ag-Cu-Ge amorphous alloys fall on a common master curve with a slope of -2 in good agreement with the Mott g2-law. The MI-line on the diagram is apparently crossed, when the electron diffusion coefficient D reaches approximately 0.25 cm2/s. This is quite reasonable in accordance with the Mott g2-law. However, the data for Ti-Si, V-Si and Ce-Si amorphous alloys are shifted to the direction having a higher γexp value, as if the effective mass involved is enhanced in this sequence. We also found that the MI-transition is shifted to a lower value of D and apparently occurs at D=0.01 cm2/s in the Ce-Si amorphous alloys. Acknowledgments

The author wishes to express deep thanks to Prof. Dr. J. Dolinsek, J. Stephan Institute, Slovenia, and Dr. E. Belin-Ferré, France, and other members in the organizing committee for giving him an opportunity to present the lecture at the 1st Euroschool on Complex Metallic Alloys, held at Mons Hotel, Ljubljana, Slovenia. Appendix 1

By inserting the relation f (r, k ) = f o (ε k , T ) + φ (r, k ) into the Boltzmann transport Eq. (5), we can rewrite it as − vk ⋅

∂ f0 ( − e) ∂f ∂f ∂φ (−e) ∂φ ∇T − + vk ⋅ + ( E + v k × B ) ⋅ 0 = − ⎛⎜ ⎞⎟ ( E + vk × B ) ⋅ . ∂T ∂k ∂r ∂k ⎝ ∂ t ⎠ scatter

(A-1) Upon formulating the electrical conductivity, we derived Eq. (7):

φ (r, k ) ⎛ ∂ fo ⎞ ⎜− ⎟ v k ⋅ ( − e)E = τ ⎝ ∂ε ⎠ Hence, the term E ⋅

(A-2)

∂φ in the right-hand side of Eq. (A-1) is obviously ∂k

proportional to E 2 and is ignored as the second-order correction to the Ohm law.

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Appendix 2

The derivation of the relation ρ 0 = 200 / g 2 is made as follows: ⎛ e2 ⎞ 2 free free ⎟ g N ( EF ) v F a ⎝ 3 ⎠

σ0 = ⎜ =

(4.802 × 10 −10 ) 2 [ esu ]2 g 2 ⋅ 2.646 × 1011 γ f 3

⎡ states ⎤ −8 8 ⎡ cm ⎤ ⎢ erg .atom ⎥ ⋅ 10 ⎢ s ⎥ ⋅ a × 10 [ cm ] ⎣ ⎦ ⎣ ⎦

⎡ [esu ]2 [ s ] ⎤ = 2.034 × 10 −8 g2 ⋅ γ f a ⎢ ⎥ ⎣ [ g ][ atom] ⎦ ⎡ [cm]2 ⎤ 2.034 × 10 −8 2 g a γ = ⋅ f ⎢ ⎥ 9 × 1011 ⎣ [Ω].[ atom] ⎦ = 2.26 × 10 −20 g2 ⋅ γ f a ×

6.02 × 10 23 d A

⎡ 1 ⎤ ⎢ [Ω][cm] ⎥ ⎣ ⎦

⎛ g2 ⋅ γ f ad ⎞ ⎡ 1 ⎤ = 1.360 × 10 4 ⎜ ⎟⎟ ⎢ ⎥ ⎜ A ⎝ ⎠ ⎣ [Ω][cm] ⎦

(A-3)

where A is the atomic weight in g, d is the density in g/cm3, γf is the free electron electronic specific heat coefficient in mJ / mol.K 2 and a is an average atomic distance in Å. The Avogadro number, atomic weight and density are introduced to convert units from per atom to per cm3. The [esu ]2 [ s ] 1 [cm]2 relation is also inserted. The critical resistivity = [g] 9 × 1011 [Ω] ρ 0 is obtained by taking the inverse of the conductivity above in (A-3):

ρ0 =

⎛ ⎞ 1 A ⎟ [Ω][cm] 4 ⎜ 2 1.360 × 10 ⎜⎝ g ⋅ γ f ad ⎟⎠

⎛ ⎞ A = 7.353 × 10 ⎜⎜ 2 ⎟⎟ [ μΩ][cm] ⎝ g ⋅ γ f ad ⎠

(A-4)

In the case of (Ag0.5Cu0.5)100-xGex amorphous alloys, the ratio A / d is about 10 by taking a weighted mean of constituent elements. The electronic specific heat coefficient in the free electron model 5 is expressed as

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γ f = 0.136( A / d ) 2 / 3 (e / a)1/ 3 mJ / mol.K 2 ,

(A-5)

where e/a is an average number of electrons per atom and turns out to be 2.05 at 35 at%Ge, where the pseudogap starts to grow. An insertion of A / d =10 cm3 and e/a=2.05 yields γ f = 0.80 mJ / mol.K 2 . This is close to the measured value shown in Figure 14. The relation ρ 0 =

229 μΩcm is g2

obtained by inserting γf=0.80 mJ/mol.K2 and a=4 Å into (A-4). We 200 employed the relation ρ 0 = 2 μΩcm in the text as a rough guide. g In the expanded liquid Hg, the value of A / d , at which the pseudogap starts to grow and the data start to fall on a line with a slope of 2 in Figure 16(b), is roughly given as 18.2 cm3 from the measured density of d=11.0 g/cm3 (16). An insertion of A / d =18.2 and e/a=2.0 into Eq. (A-5) results in γ f = 1.18 mJ / mol.K 2 . Since an average atomic distance at d=11 g/cm3 is estimated to be about 3 A, the critical resistivity ρ 0 =

378 μΩcm is g2

obtained. This explains why the critical resistivity for expanded liquid Hg is read off to be about 400 μΩcm from Figure 16(b) and much higher than that for (Ag0.5Cu0.5)100-xGex amorphous alloys. References 1. C.Kittel, Introduction to Solid State Physics, (Sixth Edition, 1986, John Wiley & Sons, Inc.). 2. N.F.Mott and H.Jones The Theory of the Properties of Metals and Alloys (Dover, 1936). 3. J.M.Ziman, Principles of the Theory of Solids (Cambridge University Press, 1964). 4. N.W.Ashcroft and N.D.Mermin, Solid State Physics, Saunders College, West Washington Square, Philadelphia, PA 19105 (1976). 5. U.Mizutani, Introduction to the Electron Theory of Metals, (Cambridge University Press, 2001) Chapters 10, 11 and 15. 6. P.Drude, Ann.Physik, 1 566 (1900). 7. J.M.Ziman, Phil.Mag. 6 1013 (1961) . 8. L.V.Meisel and P.J.Cote, Phys.Rev.B16 2978 (1977); ibid B17 4652 (1978). 9. P.W.Anderson, Phys.Rev. 109 1492 (1958).

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10. G.Bergman, Phys.Rev. B28 2914 (1983). 11. B.L.Altshuler and A.G.Aronov, Electron-Electron Interactions in Disordered Systems, A.J.Efros and M.Pollak Eds., (Elsevier Science Pub., 1985) pp.1-153. 12. N.F.Mott, J.Non-Cryst.Solids 1 1 (1968); Phil.Mag.19 835 (1969). 13. U.Mizutani, T.Ishizuka and T.Fukunaga, J.Phys.:Condens. Matter 9 5333 (1997). 14. U.Mizutani, Phys.Stat.Sol. (b) 176 9 (1993). 15. U.Mizutani, R.Zehringer, P.Oelhafen, V.L.Moruzzi and H.-J.Güntherodt, J.Phys.:Condens.Matter 1 1365 (1989). 16. U.Even and J.Jortner, Phys.Rev.Lett. 28 31 (1972). 17. N.F.Mott, Phil.Mag. 19 (1969) 835; ibid 26 1015 (1972). 18. T.Takeuchi and U.Mizutani, Phys.Rev. B52 9300 (1995). 19. U.Mizutani, Electronic Properties of Liquid, Amorphous and Quasicrystalline Alloys, Materials Science and Technology-A Comprehensive Treatments- edited by R.W.Cahn, P.Haasen and E.J.Kramer (VCH, 1993), vol.3B, Chapters 9, pp.97-157. 20. U.Mizutani, J.Phys.:Condensed Matter 10 4609 (1998) . 21. U.Mizutani, K.Sato, I.Sakamoto and K.Yonemitsu, J.Phys.F:Metal Phys. 18 1995 (1988). 22. I.Sakamoto, K.Yonemitsu, K.Sato and U.Mizutani, J.Phys.F:Metal Phys. 18 2009 (1988). 23. A.Y.Rogatchev, T.Takeuchi and U.Mizutani, Phys.Rev. B61 10010 (2000). 24. T.Biwa, M.Yui, T.Takeuchi and U.Mizutani, Materials Transactions, 42 939 (2001). 25. U.Mizutani, 7th Int.Conf.on Quasicrystals, (Stuttgart, Germany, 1999), “Electron transport mechanism in the pseudogap system: quasicrystals, approximants and amorphous alloys”, Mat.Sci.Eng. 294-296 464 (2000). 26. H.Sato, T.Takeuchi and U.Mizutani, Phys.Rev.B64 094207 (2001). 27. J T.Fujiwara and T.Yokokawa, Phys.Rev.Lett. 66 333 (1991). 28. H.Sato, T.Takeuchi and U.Mizutani, Phys.Rev.B70 024210 (2004). 29. K.Kimura, H.Iwahashi, T.Hashimoto, S.Takeuchi, U.Mizutani, S.Ohashi and G.Itoh, J.Phys.Soc.Jpn 58 2472 (1989). 30. G.T.de Laissardiére and T.Fujiwara, Phys.Rev.B 50 5999 (1994). 31. R.Tamura, T.Asano, T.Tamura, S.Takeuchi, T.Shibuya, Mat.Res.Soc.Symp.Proc. Vol.553 373 (1999).

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CHAPTER 12

CHEMICAL BONDING AND CRYSTALLOGRAPHIC FEATURES Yuri Grin Max-Planck Institut für Chemische Physik fester Stoffe Nöthnitzer Str. 40, 01187 Dresden, Germany E-mail: [email protected] Local pentagonal or pseudo pentagonal symmetry is considered as one of the key fingerprints of complexity for the crystal structures of complex metallic alloys (CMA). The question, how the chemical bonding influences the formation of CMA, i.e. are CMA possible with local atomic arrangements without pentagonal or pseudo pentagonal symmetry, is still under discussion. The appearance of fingerprints of structural complexity in simple crystal structures with low coordination numbers of atoms (3-5) is analyzed on examples of Al0.9B2, Eu2-xGa3+2x, Eu3-xGa8+2x, Ir13Al45, Ba6Ge43 and Rb8Sn44. The formation of defects in these crystal structures is caused by requirements of chemical bonding. In case of ordering of different patterns (defects), the crystal structures with giant unit cells occur also for the smaller coordination number of atoms as it was observed for the traditional CMAs and without local (pseudo) pentagonal symmetry.

Local pentagonal or pseudo pentagonal symmetry is one of the basic characteristics of the crystal structures of approximants (cf. Mackay or Bergman clusters). Several binary and ternary transition-metal compounds with aluminum and gallium in the concentration region close to quasicrystalline phases and their approximants (called complex metallic alloys, CMA) reveal this feature resulting in different variations of structural complexity. Structurally complex alloy phases are based on crystal structures characterized by giant unit cells, cluster arrangements of atoms with a large number of different atomic environments, where icosahedral coordination plays a prominent part; inherent configurational,

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chemical or substitutional disorder, partial site occupancy and split positions1. The question, how the chemical bonding influences the formation of CMA, i.e. are CMA possible with local atomic arrangements without pentagonal or pseudo pentagonal symmetry, is still under discussion. The aim of this work is to analyze the appearance of fingerprints of structural complexity in simple crystal structures with low coordination numbers of atoms (3-5). Aluminium diboride is one of the oldest known intermetallic compounds and was considered a long time as one of basic crystal structures among this class of materials (Fig. 1). From the point of view of the Zintl-Klemm concept, the compound contains so-called excess electrons, i.e., if the bonding in this compound is following the octet rule, the whole amount of aluminium in the structure is not necessary. Recently, single crystals of aluminium diboride (space group P6/mmm, no. 191) a = 3.0050(1) Å, c = 3.2537 (8) Å) were prepared by the aluminium flux method. Crystal structure refinement shows defects at the aluminium site and resulted in the composition Al0.894(9)B2 ≈ Al0.9B2. The defect structure model is confirmed by the measured mass density of .exp = 2.9(1) g/cm3 in comparison with a calculated value of .calc = 3.17 g/cm3 for full occupancy of the aluminium position. In addition, the results of 11B NMR measurements supported the defect model and are in agreement with the structure as obtained by X-ray diffraction2.

Fig. 1. The crystal structure of aluminium diboride: (left) unit cell; (right) three-bonded boron network with the embedded aluminium atoms.

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Electrical resistivity measured on a single crystal parallel to its hexagonal basal plane with ρ(300K) - ρ(2K) = 2.35 µΩcm shows a temperature dependence similar a metal. Charge is dominantly carried by holes (Hall-coefficient R = +2×10-11 m/C). The respective p-type conductivity is confirmed by calculations of the electronic density of states (Fig. 2).

Fig. 2. Electronic density of states for aluminium diboride with partial contributions of different states.

Chemical bonding in aluminium diboride was discussed using the electron localization function (ELF).The electron localization function (ELF, η) was evaluated according to references 3 and 4, applying the program Basin5. The isosurface for η = 0.65 (Fig. 3) reveals attractors on the B-B contacts in agreement with a graphite-like net. Applying the theory of gradient vector fields (procedure as proposed for the electron density6), the whole 3D field of ELF values can be divided into the basins of core and bonding attractors. Integration of the electron density5 within these basins gives the number of electrons in the basins, the socalled electron count of a respective attractor. Integration of the density of valence electrons in AlB2 reveals 2.7 electrons per B-B attractor. This means that bonding is of the order 1.35. Thus, for the stabilization of the boron net only about 4 electrons per boron atom are necessary instead of

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4.5 electrons available in stoichiometric AlB2. Therefore, the composition should be (Al0.67)3+(B2)2- in full accordance with the Zintl count. Apparently, additional electrons are responsible for the metallic behavior of this material and for the (covalent) interaction between aluminium and boron atoms along [001]: (Al0.67+0.23)3+(B2)2-(0.7e-) = Al0.9B2. A similar interaction between metal and boron was recently described for magnesium diboride7.

Fig. 3. Electron localization function in AlB2: the isosurface with η = 0.65 illustrates the covalent character of the boron-boron interactions.

As was shown recently by full-potential total energy calculations, aluminium diboride is stable only with vacancies. The HRTEM investigation on a single crystal of Al0.9B2 did not show the formation of a strongly periodic CMA, but revealed the formation of defects in the aluminium substructure which are locally clustered exhibiting a pseudo periodical pattern with a coherence length of approx. 40 Å8. Europium digallide was previously found to have a hexagonal structure of the AlB2 type9,10. Later investigations showed a orthorhombic structure of the KHg2 type11, 12. The AlB2–type phase was

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explained either as high temperature modification of EuGa212 or as stabilized by impurities11. By single crystal X-ray investigation, the orthorhombic structure of stoichiometric EuGa2 was confirmed (space group Imma, a = 4.644 Å, b = 7.626 Å, c = 7.638 Å)13. Nevertheless, also single crystals with the hexagonal symmetry (a = 4.354 Å, c = 4.485 Å) were obtained from a sample with composition Eu30Ga70. No additional reflections changing the unit cell were found. The refinement of the pure AlB2 model (Fig. 4, left) led to the respectable Rgt(F) value of 0.041. Analysis of the electron density map at this stage revealed additional gallium position with an occupation of 4.7 % (Fig. 4, middle). The final refinement resulted in Rgt(F) = 0.015 showing also the defect occupation of the Eu position (93%). The obtained distribution of the electron density can be described as a partial replacement of europium atoms by a triple of gallium atoms (Fig. 4, right) leading to a composition Eu1-xGa2+3x (x = 0.08). This replacement appears mostly statistically. But in several regions, having a size of 2-5 nm, an ordered replacement (a‘ = a√3 and c‘ = 2c) was found by HREM14.

Fig. 4. Crystal structure of Eu1-xGa2+3x: (left) starting model of the AlB2 type with the section region; (middle) difference electron density map in the section region; (right) three-fold replacement of europium by gallium in the crystal structure.

The atomic arrangement in most digallides of rare earth (RE) metals with a crystal structure of the AlB2 type are characterized by covalently bonded graphite-like network of three-bonded gallium atoms. Quantum chemical analysis of the bonding in the orthorhombic EuGa2 shows the formation of a framework formed by four-bonded gallium atoms (Ga1-): [Eu2+][Ga1-]2. The structural motif of Eu1-xGa2+3x is stabilized by

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incorporation of a second kind of four-bonded gallium atoms (forming the triangles, Ga1+) between the graphite-like nets according to the equation: Eu1-xGa2+3x = [Eu2+]1-x [(3b)Ga1.5]2-6x [(4b)Ga1+]3x+6x ,

(1)

xcalc = 0.0625, xexp = 0.07-0.10 .

(2)

This mechanism can explain the formation of extended homogeneity ranges for REGa2 of light RE metals and AEGa2 compounds of the alkaline earth (AE) metals, as well as the compound YbGa2.6314, 15. The same mechanism is responsible for the formation and position of the homogeneity range of the phases Eu3-xGa8+3x = [Eu2+]3-x [(5b)Ga2-]2 [(3b)Ga1.5+]4-6x [(4b)Ga1+]2+3x+6x, xcalc = 0.125; xexp = 0.12-0.2016 (Fig. 5) and Sr3-xGa8+3x17.

Fig. 5. Partial ordering in the crystal structure of Eu3-xGa8+3x.

All afore-mentioned examples show the occurrence of crystallographic features of CMA on a local level, i.e., without periodicity. Replacement of the more electropositive components like alkali-, earth alkali or rare earth metals by transition metals changes the situation. Detailed investigations were performed in binary systems of 4d and 5d

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transition metals (i.e., rhodium and iridium) with aluminium and gallium. Several new compounds with large crystal structures and different levels of complexity occur in the aluminium- or gallium-rich regions. Here, the complexity appears without local pentagonal or pseudo pentagonal symmetry. E.g., the binary compound Ir13Al45 is the first representative of a new structure type (Pearson symbol oP232)18. The structure shows a partial local disorder in the vicinity of some aluminium positions. This feature resembles many approximant structures. But, contrary to the latter, covalent interactions between Ir and Al result in a very irregular coordination of the iridium atoms with relatively small coordination numbers. This causes low-symmetrical columnar packing of aluminum polyhedra centered by iridium atoms showing a pseudo pentagonal motif only in very crude approximation. Chemical bonding in different structures belonging to this group is tentatively interpreted applying the electron localization function, e.g., for the more simple case of IrGa219. This kind of bonding analysis allows a preliminary classification of interatomic contacts into different groups and reveals a transition from two-centre to multi-centre bonding also in the neighboring regions of the crystal structure. Recent detailed investigations on the intermetallic clathrates Ba8Ge43 (Pearson Symbol cI404)20 and Rb8Sn44 (Pearson Symbol cI408)21 showed that a similar level of structural complexity (in the sense of large unit cell parameters with hundreds of atoms per cell) can be achieved without a pentagonal or pseudo pentagonal arrangement of atoms (Fig. 6). The polyanionic part of these structures is formed by three- (3b) and four-bonded (4b) atoms. The defects in the framework polyanion (Fig. 6a) of the initial clathrate-I crystal structure (Fig. 6c) are caused by bonding reasons according to the following balances: Ba2+]8[(3b)Ge1-]12[(4b)Ge0]36.3 • 4e-

(3)

[Rb1+]8[(3b)Sn1-]8[(4b)Sn0]36.2 .

(4)

Ordering of the defects leads to a formation of a CMA-like superstructure (Figs. 6b, d). In particular, it was shown that especially kinetic factors play an important role in the formation of the complex ordered clathrate structures20.

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Fig. 6. Formation of CMA-like crystallographic features in Ba8Ga43: (a) ordering of the defects in the germanium framework along [100]; (b) superstructure reflections leading to the doubling of the initial unit cell; (c) packing of polyhedrons in the initial clathrate-I structure (Pearson Symbol cP54); (d) packing of the polyhedrons in the Ba8Ge43 structure (Pearson Symbol cI404).

Conclusion Covalent bonding is a substantial reason for disorder, a characteristic crystallographic feature of intermetallic compounds belonging to the

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group of CMA. The 'disorder' seems to be an expression for locally wellordered interatomic interactions within a rigid symmetry insufficient for. Thus, a disordered structure can be understood in the sense of different locally ordered atomic patterns with a defined bonding situation. Different patterns tend to order (cluster) within the crystal structure, but the full ordering is not always observed. In case of ordering of different patterns, the crystal structures with giant unit cells occur also for the smaller coordination number of atoms as it was observed for the traditional CMAs and without local (pseudo) pentagonal symmetry. References 1. K. Urban, M. Feuerbacher. J. Non-Cryst. Sol. 334&335 143 (2004). 2. U. Burkhardt, V. Gurin, F. Haarmann, H. Borrmann, W. Schnelle, A. Yaresko, Yu. Grin. J. Solid State Chem. 177 389 (2004). 3. A. D. Becke, K. E. Edgecombe, J. Chem. Phys., 92 5397 (1990); A. Savin, A. D. Becke, J. Flad, R. Nesper, H. Preuss, H. G. von Schnering, Angew. Chem. 103 421 (1991); Angew. Chem. Int. Ed. 30 409 (1991). 4. A. Savin, O. Jepsen, J. Flad, O. K. Andersen, H. Preuss, H. G. von Schnering, Angew. Chem. 104 186 (1992); Angew. Chem. Int. Ed. 31 187 (1992). 5. M. Kohout, Program BASIN 4.4. User’s Guide. Max-Plankck-Institut für Chemische Physik fester Stoffe, 2006. 6. R. F. W. Bader, Atoms in Molecules: A Quantum Theory, Oxford University Press (1990). 7. J. Schmidt, W. Schnelle, Yu. Grin, R. Kniep. Solid State Sci. 5 535 (2003). 8. K. Koch, H. Rosner, R. Ramlau, U. Burkhardt, V. Gurin, Yu. Grin. (2006) in preparation. 9. A. Iandelli. Z. anorg. allg. Chemie 330 221 (1964). 10. D. I. Dzyana, V. Ya. Markiv, E. I. Hladyshevskii. Dopovidi AN URSR 9 1177 (1964). 11. V. Ya. Markiv, N. N. Belyavina, T. I. Zhunkovskaya. Dopovidi AN USSR 2 84 (1982). 12. K. H. J. Buschow, D. B. de Mooij. J. Less-Common Met. 97 L5 (1984). 13. O. Sichevich, R. Cardoso Gil, Yu. Grin. Z. Kristallogr. NCS 221 261 (2006). 14. O. Sichevich, R. Ramlau, M. Schmidt, R. Niewa, W. Schnelle, Yu. Grin. (2006) in preparation. 15. S. Cirafici, M. L. Fornasini. J. Less-Common Met. 163 331 (1990). 16. O.Sichevich, Yu. Prots, Yu. Grin. Z. Kristallogr. NCS 221 265 (2006). 17. F. Haarmann, Yu. Prots, S. Göbel, H. G. von Schnering. Z. Kristallogr. NCS 221 257 (2006).

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18. M. Boström, R. Niewa, Yu. Prots, Yu. Grin. J. Solid State Chem. 178 339 (2005). 19. M. Boström, Yu. Prots, Yu. Grin. Solid State Sci. 6 499 (2004). 20. W. Carrillo-Cabrera, S. Budnyk, Y. Prots, Yu. Grin, Z. Anorg. Allg. Chem. 630 2267 (2004). 21. F. Dubois, T. F. Fässler. J. Am. Chem. Soc. 127 3264 (2005).

CHAPTER 13

PLASTICITY OF COMPLEX METALLIC ALLOYS M. Feuerbacher Forschungszentrum Jülich GmbH, 52425 Jülich, German E-mail: [email protected] The plasticity of complex metallic alloys (CMAs) involves novel deformation mechanisms. In this chapter, we will first give a brief review of the concepts of plastic deformation in ordinary metals and alloys. Then we will describe the salient features in the plasticity of CMAs, discussing macroscopic as well as microscopic aspects for different CMA phases.

1. Introduction Complex metallic alloys (CMAs) possess salient structural features distinguishing them from ordinary simple metals and alloys. The most prominent characteristics are the large lattice parameters, which are naturally connected with a high number of atoms in the unit cell, ranging from some ten to some thousand. Furthermore, most CMAs possess a particular type of local order – the atoms are arranged in icosahedralsymmetric shells or concentric sets of the latter. This particular local order is frequently referred to as cluster substructure1. These structural features determine many of the physical properties of CMAs. This particularly holds for the plastic properties – as a direct result of the large lattice parameters conventional dislocation-based deformation mechanisms become energetically unfavourable. In the present chapter, we will give a brief overview over the field of plasticity of CMAs. We will present macroscopic and microstructural features of a number of selected CMA phases. We will start, however, with a brief review of the concepts of plasticity in metals and alloys, in which the basic terms, required for the understanding of the subsequent chapters, are introduced. 377

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2. Concepts in plasticity of metals and alloys 2.1. Dislocations Plastic deformation is, for the huge majority of materials and experimental situations, mediated by dislocations. Dislocations are onedimensional structural defects, which, driven by an applied stress, move through the crystal lattice and thereby cause a relative displacement of lattice planes2. In its simplest form, a dislocation can be understood as an inserted extra half plane in an otherwise perfect crystal lattice. Figure 1a illustrates this in a simplified view. At the bottom side of a projected crystal, represented by light-grey circles, an extra half plane, represented by dark-grey circles, is introduced. The edge of the inserted half plane is a line defect extending along the viewing direction, around which the other atoms elastically relax and form a local displacement field.

Fig. 1: Definition of the Burgers vector of a dislocation by a Burgers circuit according to the FS/RH (finish-start/right hand) rule (see text).

The major quantity characterizing a dislocation is the Burgers vector, which describes the main component of the dislocation strain field. It can be constructed by a Burgers circuit in the dislocated crystal: A sequence of lattice vectors is taken to form a closed right-handed (RH) circuit around the dislocation core (Fig. 1a). The same sequence of vectors is then taken in the perfect lattice (Fig. 1b), where it is found that it fails to close. The closure vector FS (finish – start) is the Burgers vector of the dislocation. With this so-called FS/RH convention3 the magnitude as well as the direction of the Burgers vector is uniquely defined.

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The character of a dislocation can be further characterized by the angle between the Burgers vector and the line direction. If the latter is perpendicular to the Burgers vector, as it is the case in Figure 1a, the dislocation is called an edge dislocation. If, on the other hand, the Burgers vector and the line direction are parallel, the dislocation is called a screw dislocation. The line direction of the dislocation may change along its length, i.e. the dislocation may be curved, while the Burgers vector is constant for a given dislocation. Therefore, these terms may only apply for small local parts of the dislocation, which are then referred to as edge and screw segments, respectively. This is, for example, the case in dislocation loops, i.e. closed circuits of dislocation line in a crystal. Those parts of the loops, which are not locally perpendicular or parallel to the Burgers vector, are generally referred to as mixed segments. Dislocations mediate plastic deformation of a crystal by movement under a stress field in the lattice. Figure 2a shows an edge dislocation, represented by the symbol ⊥, in an initial stable position. Subjected to a shear stress τ (Fig. 2b), the dislocation opens atomic bonds in direction of motion (dotted line) and closes bonds in its back (solid line), thereby moving ahead by one Burgers vector length (Fig. 2c). This process can be repeated sequentially, leading to movement over macroscopic distances.

Fig. 2: (a) Schematic of an edge dislocation (⊥) in a crystal lattice, (b) movement of the dislocation under action of the shear stress τ, (c) new position of the dislocation.

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Dislocation movement leaves the structure of the crystal unaltered if and only if the Burgers vector corresponds to a translation vector of the crystal lattice. In that case, the dislocation is called a perfect dislocation. In contrast, a partial dislocation, possesses a Burgers vector which is not a translation vector of the lattice. As a direct consequence, a partial dislocation is necessarily connected to a planar fault, i.e. a stacking fault or antiphase boundary, in the lattice4. Finally, let us discuss different possibilities of dislocation motion in a crystal lattice. The most frequently observed situation is that dislocation motion takes place with the Burgers vector lying in the plane of movement. This type of motion is called dislocation glide (or sometimes dislocation slip). On the other hand, if the Burgers vector of the moving dislocation has a component pointing out of the plane of motion, it is referred to as dislocation climb. The special case of movement perpendicular to the Burgers vector is called pure climb. In contrast to dislocation glide, climb motion generally requires transport of matter in the crystal, from or to the dislocation core, i.e. additional diffusion is necessary. Therefore, this type of movement is referred to as nonconservative and it generally takes place at relatively high temperatures only. Revisiting Fig. 2 in these terms, we can specify that the sequence illustrates the special case of glide movement of a perfect edge dislocation. 2.2. Activation thermodynamics In the previous chapter a description of the microstructural processes taking place during the deformation of crystalline solids, the movement of dislocations, was given. In this chapter we will briefly recall a formalism for the description of the macroscopic phenomena of plastic deformation. For further reading please consult e.g. references 4, 5 and 6. Macroscopic deformation involves large numbers of dislocation processes on the microscopic scale – the dislocation densities, i.e. the dislocation line length per unit volume, involved in deformation of a real crystal are of the order of 107 to 1010 cm-1. The description of the macroscopic phenomena can therefore only be carried out considering

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averages over large numbers of contributing dislocations, in terms of a thermodynamic formalism. Assume a crystal containing dislocations at a density ρ possessing a Burgers vector length b. Movement of these dislocations over an average distance x leads to a plastic strain

ε plast = ρ bx

(1)

The time derivation of equation (1) yields the Orowan equation

ε plast = ρ bv

(2)

where v is the average dislocation velocity and ε plast is the plastic strain rate. The Orowan equation is of central importance, as it relates the microscopic parameters ρ, b and v to the macroscopic parameter ε plast . An external stress σ which is applied to the sample in compression direction causes a shear stress τ on the glide plane of

τ = msσ

(3)

where the Schmid factor ms = cos φ cos λ (φ and λ denote the angles between compression direction and slip plane and compression direction and slip direction, respectively) is a geometric factor assuming values between 0 and 0.5. A dislocation driven by the shear stress τ is simultaneously subjected to the oppositional internal stress τi, originating from long-range stress fields of other dislocations counteracting its movement. The effective stress acting on a moving dislocation τeff is then given by

τeff = τ - |τi |

(4)

The dislocation velocity v, in general terms, is limited by energetical obstacles to be overcome during dislocation movement. The friction stress on the dislocation originating from this process is τf. If τeff > τf, the dislocation can overcome the obstacles and continuously move through the crystal. If, on the other hand, τeff < τf at an obstacle, dislocation motion is halted. However, even in the case τeff < τf, an obstacle can be overcome by thermal activation. This means that dislocation motion is

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aided by thermal fluctuations leading to a non-zero probability to overcome the obstacle. The energy barrier overcome by thermal fluctuations is then given by x2

∫

ΔG = (τ f − τ eff )lbdx

(5)

x1

where ΔG is the Gibbs free energy and l the length of the dislocation line at the obstacle. The probability of thermal activation to overcome the obstacle is P = exp

−ΔG kT

(6)

where k is Boltzmann´s constant and T is the absolute temperature. The total free energy to overcome the obstacle is ΔF = ΔG + ΔW, where ΔW is the work-term, given by the energy contribution of the effective stress, i.e. x2

∫

ΔW = τ eff lbdx

(7)

x1

The corresponding energy contributions are shown in Fig. 3 in a schematic force-distance diagram. The respective forces are related to the friction stresses by lbτ. The dislocation, assisted by thermal fluctuations, follows the reaction path from the stable equilibrium position x1 to the unstable position x2 overcoming the activation distance Δx = x2 − x1 . If thermal activation is the rate-controlling process, the dislocation velocity is given by v = ν0ΔxP, where ν0 is the attempt frequency. Hence we obtain, using Eq. (6),

ε plast = ε0 exp with ε0 = ρbΔxν0.

−ΔG kT

(8)

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Fig. 3: Schematic force-distance diagram of a dislocation overcoming an energetical obstacle. ΔG is the Gibbs free activation energy and ΔW is the work term.

The Gibbs free energy is a thermodynamic variable of state which depends on the temperature and the stress. The differential is d(ΔG) = -ΔS dT – V dτ with the definitions

ΔS ≡ −

∂ ( ΔG ) ∂T

(9) τ eff

and

V ≡−

∂ ( ΔG ) ∂τ eff

(10) T

ΔS is the activation entropy. V is called the activation volume and can be written as V = lbΔx = bΔA

(11)

The activation area ΔA can be interpreted as the area that is passed by a dislocation line during the thermally activated overcoming of an obstacle. This is illustrated in Fig. 4. Before activation (solid line) the dislocation is in a stable position corresponding to x1 in Fig. 3. Upon activation the dislocation moves by Δx to the position given by the dashed line. The area between both lines is the activation area ΔA.

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Fig. 4: Schematic of a dislocation segment of length l overcoming an obstacle. Upon activation the dislocation moves by Δx and sweeps over the activation area ΔA.

2.3. Connection to experiment

The experimental determination of thermodynamic activation parameters is performed by incremental tests during plastic deformation. Most important are stress-relaxation and temperature-change experiments, which are employed in the investigation of CMAs as described below.

2.3.1. Stress-relaxation experiments The experimental determination of the activation volume is possible according to

V =−

∂ ( ΔG ) kT ∂ (ln ε plast ) = ∂τ eff T mS ∂σ T

(12)

where Eq. (8) was used. This relation contains the experimentally accessible parameters ε plast , the plastic strain rate, and σ , the applied stress. In a stress-relaxation experiment the deformation machine is suddenly halted during plastic deformation and the stress is measured as a function of time. As the total strain of the sample remains constant, the total strain rate is given by ε = ε plast + εelast = 0 , i.e. ε plast = −εelast . For the elastic strain ε elast ∝ σ holds according to Hooke´s law. Hence, during a stress relaxation experiment we have ε plast ∝ −σ . With Eq. (12) this yields a relation for the experimental determination of the activation volume V=

kT ∂ (ln − σ ) mS ∂σ T

(13)

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The comparison of the experimental activation volume V with the atomic volume Va , i.e. the average volume per atom in the unit cell, of a given crystal structure permits inference on the mode of plastic deformation. For the case of a diffusion-controlled mode of plastic deformation the experimental activation volume is expected to be of the same order of magnitude as Va. If dislocation motion, on the other hand, is controlled by the thermally activated overcoming of obstacles the experimental activation volume is rather of the order of the obstacle volume. 2.3.2. Temperature changes The Gibbs free energy of activation is not directly accessible in deformation experiments. However, the activation enthalpy ΔH can be determined via ΔH = ΔG + T ΔS = − kT 2

∂ ln ε plast ∂σ

T

∂σ ∂T

(14) ε plast

During a constant strain-rate test, a change of deformation temperature is used to determine Δσ ΔT ≈ ∂σ ∂T . In practice, the temperature is changed by an amount ΔT after unloading the sample. After thermal equilibrium is attained in the deformation machine the sample is reloaded. The temperature dependence of the stress Δσ/ΔT together with the result of the stress relaxation experiments (Eq. 13) yields the activation enthalpy ΔH. 4. Macroscopic deformation of CMAs 4.1. General comments

The plasticity of CMAs is a novel field of materials science. To date, only a very limited number of different CMA phases have been experimentally investigated. It is a common property of all these materials, that they are brittle at room temperature. Ductility sets in at temperatures of the order of 70 % of the melting point of the individual

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materials and higher. This is a feature discriminating CMAs from essentially all other simple crystalline metals – the latter typically show ductile behaviour at room temperature and even below. In this respect, the plastic behaviour of CMAs rather resembles that of covalently bond crystals such as e.g. silicon. In the following, we will discuss experimental results on three CMA phases, ξ’-Al-Pd-Mn, β-Al-Mg, and Al13Co4. The orthorhombic phase ξ’-Al-Pd-Mn has space group is Pnma and the unit cell contains 316 atoms. The lattice parameters are a = 23.54 Å, b = 16.56 Å, and c = 12.34 Å7. The main structural building blocks are 52-atom Mackay-type clusters, consisting of concentric shells of local icosahedral symmetry. ξ'-Al-Pd-Mn is the basic phase of a family of related structures with equal a- and b- but different c-lattice parameters. The most prominent of these, Ψ-Al-Pd-Mn, possesses a c-parameter of 58 Å8. ß-Al-Mg is a cubic phase, space group Fd 3 m. The lattice parameter is a = 2.82 nm and the unit cell contains about 1168 atoms9. The coordination polyhedra in the structure comprise 672 icosahedra (ligancy 12), 252 Friauf polyhedra (ligancy 16), 24 polyhedra of ligancy 15, 48 polyhedra of ligancy 14 and 172 more or less irregular coordination shells of ligancy 10-16. Because of incompatibilities in the packing of the Friauf polyhedra this structure features a high amount of inherent disorder which is apparent as displacement disorder, substitutional disorder and fractional site occupation. Al13Co4 is an orthorhombic phase (space group Pmn21) with lattice parameters a = 8.2 Å, b = 12.3 Å, and c = 14.5 Å10, 11, 12. The unit cell contains 102 atoms. The main structural features are pair-connected pentagonal prismatic channels which extend along the [1 0 0]-direction. The experimental results shown in the following were obtained in uniaxial compression using a ZWICK Z050 testing machine. All deformations were carried out at constant strain rate. Additionally, incremental tests, as described in the previous section, were carried out in order to determine the thermodynamic activation parameters of the deformation process. The tests were carried out, if not stated otherwise, on single crystalline samples.

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4.2. Stress-strain curves

Figure 5 shows the stress-strain curve of a ξ’-Al-Pd-Mn singlecrystalline sample, deformed at 700 °C with a strain rate of 10-5 s-1. The compression axis was chosen parallel to the [0 1 0] lattice direction. Let us, for a discussion of the main features of the stress-strain curve, ignore the three sharp dips – they are the result of stress-relaxation tests, which will be discussed below. Consider the course of the curve alone, which is at the positions of the relaxation dips interpolated by dotted lines.

Fig. 5: Stress-strain curve of a ξ’-Al-Pd-Mn single-crystal at 700 °C with a strain rate of 10-5 s-1 along a [0 1 0] compression axis.

At very small strains ε, the curve shows an almost linear behaviour. This is the elastic regime, where the deformation is reversible and, according to Hooke’s law, the stress is proportional to the strain. Plastic deformation sets in at about 0.70 % strain, where first deviations from a linear course occur. At 1 % strain the curve reaches a maximum at about 350 MPa. The corresponding point in the curve is called the upper yield point. Subsequently, the stress decreases down to a value of about 280 MPa, where it reaches a small plateau at about 2.5 %. The corresponding point in the curve is called lower yield point, and the difference between the upper and lower yield stress is referred to as the yield drop. After the lower yield point, the stress-strain curve goes through two further stages.

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First, from about 3 to 5 % strain, the stress decreases with strain, i.e. the material gets softer during straining, and we accordingly term it a work-softening stage. A corresponding stage with opposite behaviour, i.e. increasing stress at increasing strain, would be referred to as a workhardening stage. Second, from about 5 % to the termination of the experiment at 8 %, the stress-strain curve is essentially horizontal. In this stage, the material is in a dynamic steady state, where hardening and softening processes in the microstructure are balanced. First, from about 3 to 5 % strain, the stress decreases with strain, i.e. the material gets softer during straining, and we accordingly term it a work-softening stage. A corresponding stage with opposite behaviour, i.e. increasing stress at increasing strain, would be referred to as a workhardening stage. Second, from about 5 % to the termination of the experiment at 8 %, the stress-strain curve is essentially horizontal. In this stage, the material is in a dynamic steady state, where hardening and softening processes in the microstructure are balanced.

Fig. 6: Stress-strain curves of Al13Co4 single crystals at temperatures between 650 and 800 °C at a strain of 10-5 s-1 along a ⎡⎣ 6 4 5⎤⎦ compression axis.

Figure 6 displays a set of stress-strain curves of Al13Co4 samples, deformed along the ⎡⎣ 6 4 5⎤⎦ direction13. The deformations were carried out at a strain of 10-5 s-1 and at temperatures between 650 and

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800 °C. Each curve shows signatures of additional temperature cycling tests and stress-relaxation test, marked “TC” and “R”, respectively, in the uppermost curve. The corresponding results will be considered below. Let us first discuss some general features of the stress-strain curves. At all temperatures the curves have common qualitative features. After the elastic regime, a strong yield-point effect is observed in the strain range between 0.25 and 0.55 %. At 700 °C, for instance, a stress difference as large as 45 % between the lower and upper yield stress was measured. Additional yield point effects are seen after the temperature changes and after stress-relaxation. At high strains, above about 2 %, the curves show an almost constant flow stress or, at some temperatures, a very weak work hardening stage. The deformation behaviour is strongly temperature dependent: the stress strongly decreases with increasing temperature, leading to high-strain flow stresses between about 320 MPa at 650 °C to 120 MPa at 800 °C.

Fig. 7: Stress-strain curves of ß-Al-Mg samples at 10-4 s-1. The black curves: single crystalline samples along [1 0 0]; grey curves: polycrystalline samples.

Figure 7 displays stress-strain curves of ß-Al-Mg samples at 10-4 s-1. The black curves represent deformations of single crystalline samples along the [1 0 0] direction, the grey curves represent deformations of polycrystalline samples (grain size about 20 µm). Temperature cycles

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and relaxations were carried out during most of the experiments. The single crystal deformations show similar features as those of Al13Co4, with a generally smaller yield-point effect. The polycrystals deformations, on the other hand, behave considerably different. The yield points are much broader, the curves show work softening, and the high-strain flow stresses are considerably smaller than for the single-crystalline case. Also, the single crystalline samples can be deformed at temperatures down to about 225 °C, while the polycrystals are ductile only above 300 °C. 4.3. Activation parameters

The activation parameters are the quantities entering the equations which describe the macroscopic deformation behaviour of a material in terms of a thermodynamic formalism as described in section 2.2. These quantities can be determined by dedicated experiments, which are carried out during the deformation test as described in section 2.3.

Fig. 8: Stress relaxation: (a) stress as a function of time of a ξ’-Al-Pd-Mn single crystal at 750 °C. (b) Representation of the time derivative of the stress as a function of the stress.

Figure 8a shows the stress as a function of time during a stressrelaxation experiment of a ξ’-Al-Pd-Mn single crystal at 750 °C. The stress decreases monotonically, following a logarithmic law. According to Eq. (13) the time derivative of the stress is taken and plotted as a

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function of the stress (Fig. 8b). From the slope of this curve the activation volume can be calculated according to Eq. (13). Figure 9 displays the activation volume of ξ’-Al-Pd-Mn, determined according to this procedure as a function of stress.

Fig. 9: The activation volume of ξ’-Al-Pd-Mn single crystals as a function of stress.

The activation volume is strongly stress dependent. It decreases with increasing stress, following a hyperbolic curve. The absolute values vary within the range of about 0.5 to 2 nm3. This stress dependence and the absolute values of V are typical for CMAs. Let us compare different CMA phases for a given stress value of 300 MPa: Values of V = 0.45 nm3 (ξ’-Al-Pd-Mn), V = 0.8 nm3 (Al13Co4), and V = 0.6 nm3 (β-Al-Mg) are found. Scaled by the respective atomic volumes, we find V/Va = 30 for ξ’-Al-Pd-Mn. For Al13Co4 and ß-Al-Mg we find V/Va = 53 and V/Va = 32, respectively. The values found for the activation volume of different CMAs obviously exceed the atomic volumes by more than an order of magnitude. As indicated in section 2.3.1, we can therefore conclude that large obstacles containing some tens of atoms control dislocation motion. Recall that we have accounted the presence of a cluster substructure as a distinct structural feature of CMAs. Accordingly, it was concluded for several CMA phases, that the cluster substructure provides the rate controlling obstacles for dislocation motion14, 15.

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Fig. 10: The activation enthalpy (squares) and the work term (circles) of β-Al-Mg single crystals as a function of deformation temperature.

Figure 10 shows the activation enthalpy and the work term of β-AlMg single crystals as a function of deformation temperature. The experimental data was obtained from combined temperature-change and stress-relaxation tests as described in 2.3.2. The values for the activation enthalpy ΔH were calculated according to Eq. (14) and are shown as solid squares. Values increasing with temperature from about 1.8 to 2.6 eV are found. A linear fit under the boundary condition ΔH(T = 0 K) = 0 eV is shown as dashed line. The work term (equation (7)), corresponding to the part of the energy which is supplied by the applied stress, is calculated as ΔW = τ V. In Fig. 10 the calculated values are shown as circles. The work term is roughly constant in the observed temperature range and amounts to about 0.4 eV. The activation enthalpy ΔH is by about a factor of six larger than the work term. It can hence be concluded that the deformation is a thermally activated process. Similar behaviour of the energetic activation parameters is also found for other CMAs. For Al13Co4 ΔH = 2.2 eV13 and for ξ’-Al-Pd-Mn ΔH = 5 eV is found14. The activation enthalpy is always much larger than the work term, and the enthalpies are considerably larger than the corresponding self-diffusion energies. As the latter finding indicates that the deformation-rate-controlling mechanism is not

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given by a single-atom diffusion mechanism, this is consistent with the conclusions drawn from the results for the activation volumes. 5. Microstructure and defects 5.1. General comments

One of the basic structural characteristics of CMAs is the presence of large lattice parameters. As a direct consequence, conventional dislocation-based deformation mechanisms are prone to failure. The elastic line energy of a dislocation is proportional to b2, where b is the length of its Burgers vector3. For most materials, Burgers vector lengths larger than about 5 Å are energetically unfavourable. Accordingly, perfect dislocations in CMAs, which would have Burgers vectors exceeding 10 Å in length, are highly unfavourable and not likely to form. In common materials, dislocations with large Burgers vectors split into partials. However, for the case of very large lattice constants, as it is the case for CMAs, splitting into a large number of partials, each possessing its individual energy cost would be required. Moreover, the corresponding planar faults, which necessarily have to be introduced as soon as partials are involved (see section 2.1), cost additional energy. On the other hand, it has been shown that CMAs, at least at high temperatures, are ductile and that in all cases studied dislocations mediate plastic deformation14, 16, 15, 13. It is therefore a central question in CMA research, to explore the deformation mechanism operating in CMAs, and, in particular, what is the structure of the defects involved. 5.2. Experimental examples

Figure 11 displays transmission electron micrographs of dislocations in a deformed Ψ-Al-Pd-Mn single crystal, a superstructure of the previously discussed ξ‘-Al-Pd-Mn phase. Figure 11 a is taken under two-beam Bragg conditions17 using the (10 0 0) reflection close to the [0 1 0] axis. A high density of dislocations is seen in end-on orientation. In particular, no stacking fault contrast is seen in the image, i.e. the dislocations appear to be perfect dislocations.

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Fig. 11: Transmission electron micrographs of dislocations in a deformed Ψ-Al-Pd-Mn single crystal under two-beam conditions (a-left panel) and using a symmetric selection of reflections of the (0 1 0) zone axis (b-right panel).

Contrast-extinction analysis17 shows that the Burgers vectors of the dislocations are parallel to the [0 0 1] direction, i.e. they are pure edge dislocations. Fig. 11 b shows the same sample area imaged using a symmetric selection of reflections of the (0 1 0) zone axis. Under these conditions, it can be seen that each dislocation position is decorated by a small area of bright contrast. Figure 12a shows a single dislocation at a higher magnification. It can clearly be seen that it consists of a dislocation-like structure (c.f. Fig. 2c) with six inserted half planes. Note, however, that the dislocation-like structure resides on a length scale which is larger than the atomic scale by about one order of magnitude. Fig. 12b is a schematic representation of the defect structure of Fig. 12a in terms of a tiling description18. The dislocation core is represented by the dark-grey polygon in the image centre. The upper and lower edges as well as the right hand side of the figure are represented by a tiling composed of pentagons, banana-shaped nonagons, and flattened hexagons in two different orientations. This is the representation of the ideal Ψ-Al-Pd-Mn structure8. On the left hand side is a triangle-shaped area, the tiling of which consists of flattened hexagons in alternating orientations. This tiling represents the ideal

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Fig. 12: Metadislocation in Ψ-Al-Pd-Mn: (a) Transmission electron micrograph; (b) Schematic representation in terms of a tiling model.

ξ’-Al-Pd-Mn structure7. In the upper and lower vicinity of the ξ’-Al-PdMn triangle, the nonagon-pentagon arrangement, which in the undistorted Ψ-Al-Pd-Mn structure forms straight lines along the [1 0 0] direction, relaxes around the ξ’-Al-Pd-Mn triangle. G The Burgers vector of the dislocation can be determined as b = c τ 4 ( 0 0 1) by forming a Burgers-circuit around the dark-grey core. The Burgers-vector length amounts to 1.83 Å i.e. the dislocation is a small irrational partial. The complete defect structure is inseparably formed by the partial dislocation on the atomic scale and the dislocationlike structure on the larger length scale. The latter accommodates the partial dislocation to the lattice in such a way that the ideal Ψ-Al-Pd-Mn structure can be continued above and below the dislocation core. As a direct consequence, the defect structure as a whole can move through the lattice without introducing any additional planar defects. This novel type of defect, discovered 199919, was termed a metadislocation. Later on, other types of metadislocation with 4, 10, and 16 inserted superstructure half planes were discovered20. It was demonstrated that metadislocations mediate the plastic deformation process in ξ’- and Ψ-Al-Pd-Mn14. The mode of dislocation motion has not been directly identified, but strong evidence was found that the movement takes place by a pure climb process21.

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Fig. 13: Transmission electron micrograph of a deformed Al13Co4 single crystal under two-beam conditions. A dislocation is marked by a black arrow, a stacking fault is marked by a white arrow.

Figure 13 displays a transmission electron micrograph of a deformed Al13Co4 single crystal13. A high density of dislocations (black arrow) trailing planar defects (white arrow) can be seen. The dislocations have [0 1 0] Burgers vectors and [1 0 0] line direction, and their movement takes place in (0 0 1) planes. That is, the dislocations are of pure edge type and move by pure glide. Imaging the dislocation cores at higher magnification in the transmission electron microscope reveals more insight into the nature of the defects. Figure 14a is a lattice-fringe image of the dislocation core. A tiling representing the unit cell projection along the [1 0 0] direction is superposed. Rectangular tiles represent the orthorhombic Al13Co4 phase (c.f. section 4.1), rhombshaped tiles represent a closely related monoclinic modification11, 10. The dislocation core is localized in the open centre of the image, and the stacking fault stretches out to the right. It can clearly be seen that the planar fault is realized by a slab of monoclinic structure within the

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Fig. 14: Dislocation in Al13Co4: (a) Lattice-fringe image of a dislocation core. Unit cells of the orthorhombic and monoclinic unit cells are shown around the dislocation core. (b) Representation in terms of a pentagon-tiling.

otherwise orthorhombic lattice. Figure 14b is a schematic of the defect in terms of a pentagon-tiling22. The superposed unit cell projections correspond to those shown in the experimental image. The dislocation core is represented by the dark-grey tile. The planar defect corresponding to the slab of monoclinic phase stretches out to the right and is represented by a parallel arrangement of pentagon and rhomb tiles, while the surrounding orthorhombic phase is represented by an alternatively oriented arrangement of pentagon and rhomb tiles. Figure 15 displays two transmission electron micrographs showing defects in ß-Al-Mg. Both micrographs show the same specimen area. The left micrograph was recorded under two-beam Bragg-conditions using the (1 1 1 ) reflection for imaging. A set of planar defects with different habit planes, showing fringe-contrast, is seen. On the right image, which was taken using the ( 0 1 1) reflection, all defects are out of contrast. A full contrast-extinction analysis23 reveals that the majority of planar defects are parallel to {1 1 1} planes and were trailed by dislocations with [2 1 1] Burgers vectors. Less frequently, planar defects due to the movement of 1 1 1 dislocations on {2 1 1} planes are found. Dislocation motion in these cases takes place by pure glide. Further analysis reveals that the defects are stacking faults introduced by partial dislocations.

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Fig. 15: Transmission electron micrographs of defects in ß-Al-Mg. (a-left panel) Imaged using the 1 1 1 reflection. (b-right panel) Contrast extinction using the 0 1 1 reflection.

(

)

(

)

6. Conclusions

The plasticity of CMAs is a new field of materials science, bearing, as a direct consequence of the large lattice parameters, fundamental questions on deformation mechanisms and defect structures of these materials. Only a few CMAs have been experimentally characterized to date. Their macroscopic deformation behaviour shows some similarities: the stressstrain curves show no or only very weak hardening stages, sometimes even work softening. The thermodynamic activation parameters indicate that dislocation motion is controlled by the cluster substructure characteristic of CMA materials. Microstructural analysis of the defects mediating plastic deformation holds some surprises. For ξ’- and Ψ-AlPd-Mn a completely novel deformation mechanism, involving a new type of defect, the metadislocations, was found. It is one of the characteristics of this type of defect, that it inevitably involves local phase transformation to a closely related structure. This observation was also made for Al13Co4 and for some other CMA phases not mentioned in this chapter (e.g. Feuerbacher et al. 2004). On the other hand, for example in the case of ß-Al-Mg, mechanisms, which qualitatively do not differ from those in ordinary simple metals, were observed.

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Acknowledgments

The author thanks Dr. M. Heggen, M. Lipinska, and S. Roitsch for stimulating discussions and making their results available before publication. References 1. K. Urban and M. Feuerbacher, J. Non Cryst. Sol. 334 & 335, 143 (2004). 2. R. E. Smallman and R. J. Bishop, Modern Physical Metallurgy & Materials Engineering, (Oxford: Butterworth-Heinemann, 1999). 3. J. P. Hirth and J. Lothe, Theory of Dislocations (New York: Wiley,1982). 4. P. Haasen, Physical Metallurgy (Cambridge: Cambridge University Press, 1986). 5. A. G. Evans and R. D. Rawlings, Phys. Stat. Sol. 34 9 (1969). 6. U. F. Kocks, A. S. Argon and M. F. Ashby, Thermodynamics and kinetics of slip (Oxford: Pergamon Press, 1975). 7. M. Boudard, H. Klein, M. De Boissieu, M. Audier and H. Vincent, Phil. Mag A74 939 (1996). 8. H. Klein, PhD Thesis Inst. Nat. Polytech. de Grenoble (1997). 9. S. Samson, Acta Cryst. 19 401 (1965). 10. R. C. Hudd and W. H. Tailor, Acta Cryst. 15 441 (1962). 11. J.Grin, U. Burkhardt, M. Ellner and K. Peters, J. Alloys & Comp. 243 1994 (2006). 12. T. Gödecke and M. Ellner, Z. Metallkd. 87 11 (1996). 13. M. Heggen, D. Deng and M. Feuerbacher, Intermetallics, submitted (2006). 14. M. Feuerbacher, H. Klein and K. Urban, Phil. Mag. Lett. 81, 639 (2001). 15. S. Roitsch, M. Heggen, M. Lipinska and M. Feuerbacher, Intermetallics in press (2006). 16. M. Feuerbacher, M. Heggen M and K. Urban, Mat. Sci. Engng. A 375 – 37 7 84 (2004). 17. D. B. Williams and C. B. Carter, Transmission Electron Microscopy (New York: Plenum Press, 1996). 18. L. Beraha, M. Duneau, H. Klein and M. Audier, Phil. Mag. A76, 587 (1997). 19. H. Klein, M. Feuerbacher, P. Schall and K. Urban, Phys. Rev. Lett, 82 3468 (1999). 20. H. Klein and M. Feuerbacher, Phil. Mag. 83 4103 (2003). 21. M. Feuerbacher and M. Heggen, Phil. Mag. 86 985 (2006). 22. K. Saito , K. Sugiyama and K. Hiraga, Mat. Sci. & Eng. A294-296, 279 (2000). 23. M. Lipinska and M. Feuerbacher, in preparation.