Introduction to Surface and Superlattice Excitations

Introduction to surface and superlattice excitations

Introduction to surface and superlattice excitations

Michael G. Cottam Department of Physics, University of Essex, UK and Department of Physics, University of Western Ontario, Canada

David R. Tilley Department of Physics, University of Essex, UK

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Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 32 East 57th Street, New York, NY 10022, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia © Cambridge University Press 1989 First published 1989 British Library cataloguing in publication data Cottam, Michael G. Introduction to surface and superlattice excitation. 1. Materials. Surfaces. Structure & physical properties I. Title II. Tilley, David R. (David Reginald), 1937530.4 Library of Congress cataloguing in publication data Cottam, Michael G. Introduction to surface and superlattice excitations / Michael G. Cottam, David R. Tilley. p. cm. Bibliography: p. Includes index. ISBN 0 521 32154 9 1. Surfaces (Physics) 2. Exciton theory. 3. Lattice dynamics. 4. Superlattices as materials. I. Tilley, David R. II. Title. QC173.4.S94C68 1989 530.4'1-dc 19 87-36710 CIP ISBN 0 521 32154 9 Transferred to digital printing 2002

MC

Contents

Preface

ix

1 1.1 1.2 1.3 1.4 1.5

Introduction 1 Excitations in crystals 2 Surface reconstruction 12 Theoretical methods 15 Experimental methods 21 Problems 30

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

Surface waves on elastic media and liquids 33 One-dimensional lattice dynamics 34 Bulk elasticity theory 40 Surface elastic waves 45 Acoustic Green functions 50 Normal-incidence Brillouin scattering 56 Oblique-incidence Brillouin scattering 64 Inelastic particle scattering 68 Surface waves on liquids 72 Problems 83

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

Surface magnons 85 The magnetic Hamiltonian 85 Magnons in bulk Heisenberg ferromagnets Semi-infinite Heisenberg ferromagnets 90 Heisenberg ferromagnetic films 104 Heisenberg antiferromagnets 109 Experimental studies 117 Magnons in Ising ferromagnets 119 Problems 125

8£

Contents

VI

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

Surface magnetostatic modes 127 Dipole-dipole interactions in bulk ferromagnets 128 Magnetostatic modes for a ferromagnetic slab 130 Magnetostatic modes for a double-layer ferromagnet 136 Dipole-exchange modes for ferromagnets 137 Magnetostatic modes for antiferromagnets 142 Brillouin scattering 146 Other experimental studies 153 Problems 154

5 5.1 5.2 5.3 5.4 5.5

Electronic surface states and dielectric functions Single-electron surface states 158 General properties of dielectric functions 163 Collective properties of the electron gas 165 Dielectric function of ionic crystals 177 Problems 180

6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8

Surface polaritons 183 Bulk polaritons 184 Surface polaritons: single-interface modes 194 Surface polaritons: two-interface modes 208 Surface polaritons: nonlinear effects 214 Attenuated total reflection 218 Other experimental methods 228 Surface magnon-polaritons 235 Problems 244

7 7.1 7.2 7.3 7.4 7.5 7.6 7.7

Layered structures and superlattices 247 Continuum acoustics: folded acoustic modes Lattice dynamics: folded optic modes 256 Optical properties 263 Single-electron states 271 Plasmons 276 Magnetic properties 287 Problems 299

8 8.1 8.2 8.3

Concluding remarks 302 Mixed excitations 302 Nonlinear effects and interactions between excitations Excitations in wedges and at edges 305

157

249

303

Contents 8.4 Excitations in spheroidal and cylindrical samples 8.5 Surface roughness 309 Appendix Green functions and linear response theory A.I Basic properties of Green functions 311 A.2 Linear-response theory 312 A.3 The fluctuation-dissipation theorem 314 A.4 Problems 316 References Index

307

311

318

329

vn

Preface

The past twenty years have seen a great expansion in the study of surface properties. One part of this activity has been concerned with the various acoustic, magnetic and optic modes that propagate at the surface of a solid or liquid. These modes have a great deal in common, for example, they are often characterised by an amplitude that decays as an exponential (or sometimes a sum of exponentials) with distance from the surface. The generality of the concepts is well known to research groups working on surface modes, and most of them have made contributions across the board. However, although a number of excellent advanced monographs and review articles have appeared, there is no introduction to the field. The present work is designed to fill this gap. Our intention is to provide an introductory text for someone starting research on surface modes or extending their range from one type of surface mode to another. It is hoped in addition that much of the material will be useful for advanced undergraduate teaching. In keeping with this pedagogical character, we have provided problems at the end of each chapter, and the lists of references are extensive, although we do not claim that they are comprehensive. The experimental techniques employed for the study of surface modes are described, some in Chapter 1, and the more specialised techniques at the appropriate points in later chapters. The experimental data that are shown have been selected to clarify the discussion and not with any intention, for example, of always showing the latest available results. The theoretical description of surface modes can be given at two levels. First, homogeneous equations of motion can be solved for a dispersion equation (frequency versus wavevector) and related properties such as variation of amplitude with distance from the surface; this level is adequate for understanding most of this book. Second, inhomogeneous equations ix

Preface

can be solved to find Green functions by means of linear response theory. This method is harder, but it gives more complete information, including, for example, thermal fluctuation spectra and scattering cross sections. For those who are interested, we have given the basic formalism in the Appendix, and details of a Green function calculation are given just once, in Chapter 2. Somewhere in the middle of the gestation period of this book the authors got involved in the worldwide effort on superlattices. A system with a large number of interfaces is an obvious generalisation of a system with one or two interfaces, so naturally many ideas about surface modes carry over for superlattices. We have therefore included a chapter on this topic. The book is primarily concerned with acoustic, magnetic and optic properties, mainly because they are closely related, but also because that is where the bulk of our own experience lies. This means that some topics, although substantial, are dealt with only briefly. In particular, electronic properties do not feature largely, since to do more than we have would have required a lengthy digression into electron-band theory. Likewise, the discussion of liquid surfaces is restricted in scope. Our indebtedness to a large number of friends and collaborators will be clear from the book. It has been our privilege for the last ten years or more to be part of the theoretical surfaces group at Essex, and to develop many ideas with the help of Professor Rodney Loudon, Dr Mohamed Babiker and Dr Stephen Smith. Our graduate students, now established in their own careers, include John Nkoma, Enaldo Sarmento, Karsono bin Dasuki, Latiff bin Awang, Eudenilson Albuquerque, Fernando Oliveira, Bob Moul, Demosthenes Kontos, Marcilio Oliveros, Arnobio dos Santos, Aurino Ribeiro Filho, Nilesh Raj, Nic Constantinou, Roger Philp, Monkami Masale, Hossein Heidarpour and Heidar Khosravi. We are grateful to all of them. People elsewhere who have contributed to our understanding include the surfaces groups in Irvine, California; Natal, Brazil; Exeter, England, and Royal Holloway and Bedford New College, England. Their work features in many parts of the book. Finally, we should like to express our thanks to two people. Simon Capelin, the commissioning editor, has given us unfailing encouragement and allowed us a free hand over content. Carol Snape has typed the book with an accuracy which is no less amazing because we have become accustomed to it. Mike Cottam David Tilley June 1988

1 Introduction

In this book we are concerned with the ways in which surfaces or interfaces modify the properties of solids and liquids. Various different effects may be identified. First, there may be a modification to the equilibrium configuration in a medium close to a surface; this is known as surface reconstruction. For example, the atoms near to a surface may have a different crystallographic arrangement compared with those in the bulk, or they may be disordered. Another example is a ferromagnetic solid, in which the interactions between the magnetic moments at the surface may differ from those in the bulk, leading to a different value of the magnetisation. Clearly this type of effect may be temperature dependent, and it is particularly relevant when there is a phase transition (e.g. close to the Curie temperature in a ferromagnet). Second, the excitations within the system (such as the phonons in the lattice dynamics of a crystal or the magnons of a ferromagnet) are modified by a surface. In an infinite medium the bulk (or volume) excitations are characterised by an amplitude that varies in a wave-like fashion in three dimensions. When surfaces are present the bulk excitations are required to satisfy appropriate boundary conditions. Consequently there will, in general, be changes to the density of states of the bulk excitations and, in some cases to the bulk excitation frequencies. However, a more interesting effect of a surface on the excitation spectrum is that it can give rise to localised surface excitations (e.g. surface phonons or surface magnons). By contrast with bulk excitations, the surface excitations are wave-like for propagation parallel to the surface (or interface) and have an amplitude that decays with distance away from the surface. As indicated by the title of this book, we are primarily concerned with the dynamic properties of finite media, i.e. with the excitations. For the most part, we include discussion of the static properties of surfaces 1

Introduction (the surface reconstruction) only to the extent that these may influence the phase behaviour and the nature of the excitation spectrum. The general characteristics of bulk and surface excitations, together with their symmetry aspects, are described in §1.1, and §1.2 deals with surface reconstruction. This is followed in §1.3 by a brief account of some of the theoretical methods applied to bulk and surface excitations, while §1.4 contains a preliminary survey of the experimental methods. Specific types of excitations and/or specific surface structures are dealt with in the later chapters. 1.1 Excitations in crystals

We now introduce some of the general properties of surface excitations, stressing the symmetry aspects. It is instructive to do this by comparing and contrasting them with bulk excitations in infinite media, for which the concepts are more familiar. The results for bulk excitations will be summarised first; details are to be found in any of the standard textbooks on solid-state physics (e.g. Ziman 1972; Elliott and Gibson 1974; Ashcroft and Mermin 1976; Kittel 1986). 1.1.1 Bulk excitations An ideal crystal can be described as a basis of one or more atoms or ions located at each point of a lattice. A lattice is an infinitely-extended regular periodic array of points in space. All lattice points are equivalent, and the system possesses translational symmetry. The lattice may be defined in terms of three fundamental translation vectors a l5 a2 and a3, which are non-coplanar. If a translation is made through any vector that is a combination of integral multiples of these basic vectors the crystal appears unchanged. The end points of such vectors R, defined by R = nx2Lx + n2a2 + n3a3 (1.1) with nl9 n2 and n3 integers, form the space lattice. In addition to the translation operations, there may also in general be other symmetry operations, such as certain rotations and reflections, which leave the crystal apparently unchanged. All such symmetry operations must leave both the space lattice and basis unchanged. An important part is played by the unit cell, defined as the smallest volume based on one lattice point such that the whole of space is filled by repetitions of the unit cell at each lattice point. The specification of the unit cell is not unique: one possible choice would be the parallelepiped subtended by the basic vectors a1? a2 and a3, with the cell arbitrarily centred on one of the atomic positions. Some choices of unit cell are

Excitations in crystals illustrated in Fig. 1.1, which depicts for ease of representation a two-dimensional lattice with basic vectors ax and a2. Two examples of parallelograms as the unit cell are shown, both with sides defined by ax and a 2 but centred at different positions on the space lattice. A hexagonal unit cell also is indicated, obtained by drawing the perpendicular bisectors of the lattice vectors from a central point to the nearby equivalent sites (a construction known generally as the Wigner-Seitz cell). The translational symmetry of the crystal structure implies that various position-dependent physical quantities, such as the electron density or the electrostatic potential, are the same within each unit cell. These quantities must be multiply-periodic functions satisfying /(r + R)=/(r)

(1.2)

for all points r in space and for all vectors R defined by (1.1). In one dimension it is a well-known mathematical result that a periodic function can be expanded as a Fourier series of complex exponentials. The analogous result in three dimensions enables us to write )

(1.3)

Q

where the vectors Q must satisfy exp(iQ.R)=l

(1.4)

for all lattice vectors R. The end points of all vectors Q satisfying (1.4) form a lattice known as the reciprocal lattice (since Q has the dimension of a reciprocal length). The reciprocal lattice can be generated from the basic vectors bl9 b2 and b 3 satisfying a£-b, = 27cay

(ij=l,2,3)

(1.5)

Fig. 1.1 General oblique lattice in two dimensions showing the basic vectors a, and a2. Three possible forms of the unit cell are indicated - two are parallelograms with different centres, and the other (the Wigner-Seitz cell) a hexagon obtained by drawing the perpendicular bisectors of the lines from a central point to nearby lattice points.

Introduction where dtj is the Kronecker delta (defined by Stj = 1 if i = 7, dtj = 0 if i Explicitly, the definitions are x a3

b

b b

27ta3 x ^

fc

fc

27cat x a2

2 3 a 1 .(a 2 xa 3 ) a 1 -(a 2 xa 3 ) a 1 -(a 2 xa 3 ) A general reciprocal lattice vector Q then takes the form Q = Vibi + v2b2 4- v3b3 (1.7) where vl5 v2 and v3 are integers. The proof that (1.6) and (1.7) lead to the property (1.4) is left to the reader (see Problem 1.1 at the end of this chapter). The unit cell of the reciprocal lattice is conventionally obtained using the Wigner-Seitz construction described earlier; it is known as the Brillouin zone. Its volume is the same as the parallelepiped formed by the basic vectors b l5 b2 and b 3 , which can be shown from (1.6) to be 8TT3/^O where Vo is the volume of the unit cell in real space. The easiest example is a simple cubic space lattice, for which the vectors a l9 a2 and a3 may be taken as a2 = fl(0, 1,0) a3 = a(0,0,1) (1.8) ax = fl(l, 0,0) where a is the lattice parameter (the nearest-neighbour separation). The basic vectors of the reciprocal lattice are easily found to be 1

b 2 = — (0, 1,0) b 3 = — (0, 0, 1) (1.9) b, = — (1,0,0) a a a Hence the reciprocal lattice is also simple cubic in this case. It is fairly straightforward to show that the reciprocal lattice of a body-centred cubic (bcc) space lattice is face-centred cubic (fee) and the reciprocal lattice of a fee space lattice is bcc. The details are given in solid-state textbooks. We are now in a position to discuss how the elementary excitations of a crystal are influenced by the symmetry. The important property is embodied in a result that is generally known to solid-state physicists as Bloch's theorem (Bloch 1928); it is also related to Floquet's theorem in mathematics (e.g. see Whittaker and Watson 1963). Here we outline a simple proof applicable to any type of excitation in a crystal. Examples of excitations that we shall consider in later chapters include lattice vibrations (phonons), spin waves (magnons), electronic modes (such as plasmons), and so forth. All of them are excitations of the whole region, rather than localised excitations of particular atoms, and this leads to common symmetry features. A consequence of the periodicity of the crystal lattice is that a position-dependent quantity, such as the Hamiltonian operator //(r), is unaltered by a translation through the vector R in (1.1). For example,

Excitations in crystals the Hamiltonian might take the form — (#2/2m)V2 + V(r) appropriate to an electron (mass m) in a periodic potential V(r). The translational invariance property of the Hamiltonian can be stated as if (r + R) = H(r). It follows that the corresponding Schrodinger's equation H(T)II/(T) = EII/(T)

(1.10)

must be invariant under the translation r - ^ r + R. Hence, if ^(r) denotes the wavefunction of a stationary state (with energy eigenvalue E), then i^(r + R) is also a solution describing the same state of the system. This implies that the two functions must be related by a multiplicative factor, and we write iA(r + R) = ciA(r) (1.11) It is evident that c must have unit modulus, otherwise the wavefunction would tend to infinity if the translation through R (or — R) were repeated indefinitely. Hence c must be expressible as c = exp(iq.R)

(1.12)

where q is an arbitrary (real) constant vector: it has dimensions of reciprocal length. The general form of the wavefunction having the above property is
(1.13)

where Uq(r) is a periodic function: Uq(r + R) = Uq(r). The overall phase factor exp(iq«r) gives a plane-wave variation to the wavefunction in accordance with the properties (1.11) and (1.12) deduced from the translational symmetry of the lattice. An alternative (and more rigorous) proof of Bloch's theorem (1.13) can be formulated using group theory (e.g. see Wherrett 1986). For convenience we have expressed the result in terms of the wavefunction {//(T), as would be the case for electronic states. More generally, Bloch's theorem would be written in terms of whatever variable is used to describe the amplitude of the excitation (e.g. an atomic displacement in the case of phonons). The wavevector q is clearly not unique: values differing by any reciprocal lattice vector Q as defined in (1.7) would equally satisfy (1.12) because exp[i(q + Q)«R] =exp(iq«R). In general, the energy eigenvalue in (1.10) will depend on q, and it is periodic in the reciprocal lattice. Denoting E = /jco(q), where co(q) is the excitation frequency, we must therefore have co(q + Q) = co(q)

(1.14)

For an infinite crystal q can take any real value but, because of the above property, it is unnecessary to define it outside the first Brillouin zone, defined to be the Brillouin zone centred at q = 0. For example, earlier in

Introduction this section we showed that the Brillouin zone of a simple cubic lattice is a cube with sides of dimension 2n/a. Hence it is sufficient in this case to take q values corresponding to — n/a
(1.15)

— n/a
(1.16)

where z is a position coordinate, follows from the requirement that the wavefunction is single-valued. However, the ID case of (1.11) is t(z + a) = aKz)

(1.17) N

for a symmetry translation through a. Thus it follows that c = 1, which has the roots c = exp(iqza) where

and mz is an integer. Within the first Brillouin zone, mz will take the N consecutive integer values in the range —jN<mz^jN (if N is even). The generalisation of the above argument to 3D is obvious: each component of the vector q will be given by an expression analogous to (1.18). Thus we obtain for the allowed wavevectors a set of uniformly distributed points in q-space. The 'volume' per allowed wavevector is a constant, namely (2n/Na)3. Since N is supposed to be very large this distribution is effectively continuous, and it is permissible to convert summations over discrete q values to integrations over continuous variables by the prescription dqxdqydqz JJJ

where V = (Na)3 is the volume of the crj^tal. 6

(1.19)

Excitations in crystals 1.1.2 Surface excitations In the presence of one or more surfaces, many of the symmetry arguments presented in the preceding subsection need to be modified. To illustrate this we restrict attention initially to the effect of a single plane surface. Specifically we consider the situation represented in Fig. 1.2, where the surface is taken to lie in the plane z = 0 of a Cartesian coordinate system. The thickness of the crystal and its dimensions in the x and y directions will be taken as sufficiently large (effectively infinite) that the crystal may be regarded as filling the half-space z ^ 0. For the present we ignore any rearrangement or distortion of the surface atoms (i.e. surface reconstruction) and deal with an 'idealised' surface. If the orientation of the surface relative to the crystal axes is such that two of the basic translation vectors (ax and a2, say) are parallel to the surface, while a 3 is not, then the set of translations R y defined by the vectors R

H = ni*i + n2a2

(1-20)

with nx and n2 integers, are symmetry operations of the crystal. However, translation operations involving a3, such as those given by (1.1) with n3 ^ 0, are not symmetry operations: they do not leave the crystal apparently unchanged, because they connect points that have different positions relative to the surface plane. The end points of the set of vectors R|l form the 2D space lattice. There are in fact only five space lattices (called Bravais lattices) in 2D, compared with fourteen in 3D. These are depicted in Fig. 1.3. For the corresponding 2D reciprocal lattice there are just two basic vectors bx and b 2 . They are defined as the vectors in the xy plane that satisfy urbJ = 2nSiJ

(U=l,2)

(1.21)

A general 2D reciprocal lattice vector Q(| can then be expressed as Q\\=v1b1+v2b2

(1.22)

Fig. 1.2 Choice of coordinate axes for a semi-infinite medium.

z

•

I I I I I y '

Semi-infinite medium I I

Introduction with vx and v2 integers. The Wigner-Seitz unit cell of the 2D reciprocal lattice defines the 2D Brillouin zone: its area is equal to 4n2/A0 where Ao is the area of the unit cell for the lattice planes parallel to the surface. The periodicity property (1.2) satisfied by any position-dependent physical quantity /(r) in an infinite 3D system no longer holds in the present case because there is translational symmetry only in the xy plane. The more limited property applicable to the semi-infinite crystal is that /(rl|+R||,z)=/(r||,z) (1.23) where we denote r = (r||,z) for the 3D position vector and r(| = (x,y). From the properties of the 2D reciprocal lattice vectors and from the lattice periodicity in the xy plane we may define a 2D Fourier series expansion by )=

Z,F(Qn,z)ap(iQVTn)

(1.24)

This is the analogue of (1.3) for the infinite 3D case. The proof of Bloch's theorem also requires modification in the presence of a surface. We may follow the same type of argument as in the preceding section: equation (1.11) must be replaced by ^(r(| -\-R^9z) = cil/(r^9z) because R|t is now the general operation of translation symmetry. This leads to the conclusion that c = exp(iq,rR,|)

Fig. 13 The five Bravais lattices in two dimensions, defined in terms of the basic vectors a t and a2. »2,

Square •il = la2l 0 = 90°

Hexagonal laj = |a2| = 60° (or 120°)

.0 Primitive rectangular l»ih 0 = 90°

•

Centred rectangular

1 # Ia2| = 90°

Excitations in crystals where q|( = (qx, qy) is a real 2D vector, and the 2D analogue of Bloch's theorem becomes ^(r,,, z) = exp(iq,| -r^U^r^z)

(1.25)

Here t/q||(r|,z) has the 2D periodicity property that l/q (r| + R|,z) = It is not possible to use symmetry arguments to make a rigorous statement about the dependence of Uq(r^,z) on the coordinate z perpendicular to the surface. The procedure for deducing the z dependence in any specific problem is frequently to obtain a differential equation (or, in some cases, a finite-difference equation) for Uq (ry,z); this would be Schrodinger's equation in the case of electronic states or the appropriate equation of motion in the case of other excitations. The differential equation must be solved subject to the relevant boundary conditions; in the present example, these are at the surface z = 0 and at z -> — oo. It is usually found that this differential equation has more admissible solutions than is the case in the corresponding infinite crystal problem. Suppose that the differential equation for the semi-infinite medium admits solutions of the form expiiq^z) where q(zj) may be complex and take a series of values (labelled by j) determined from the differential equation. The solution for Uq (i*||, z) would be formed from a linear combination of such terms: U (r,,, z) = X ^ ( r , , ) exp(i^z)

(1.26)

j

where ^ ( r ||) is an amplitude factor. Because the right-hand side of (1.26) must remain finite at large distances from the surface (z -• — oo), we either have q{p is real or qzj) is complex with lm(q{zj)) < 0. The former possibility corresponds to a bulk excitation, since cxp(iq(zj)z) has a constant modulus (equal to unity) for all z. The quantity q{zj) is just the third component of the 3D wavevector q = ( q ^ ^ ) describing the propagation of the excitation. The bulk excitation is affected by the surface in that it must satisfy a boundary condition at z = 0, and this will enter into the calculation of the corresponding ^-coefficient in (1.26). The second of the above possibilities, i.e. I m ^ ^ ) < 0, corresponds to a surface excitation localised near the z = 0 surface, because exp^g^z) -• 0 as z -• — oo within the crystal. If q(zj) is pure imaginary we may denote gO)_ _1K (wjth K real and positive), and the surface excitation has a simple exponential decay proportion to exp(/cz) as z -• — oo. The attenuation length (or decay length) is l//c. More generally, q{j] is complex and this corresponds to an excitation that oscillates within an exponentially decaying envelope function as z - • — oo. Hence, in such cases, the surface excitation can be characterised by a 2D wavevector q{!

Introduction describing its propagation parallel to the surface and by a decaying amplitude in the direction perpendicular to the surface. The concepts of bulk and surface excitations may be illustrated by considering some aspects of bulk and surface waves in a semi-infinite isotropic elastic medium. This problem was first examined by Rayleigh (1887) and we return to it in detail in Chapter 2. The general equation of motion is the wave equation —j — v \ u = 0

(1-27)

where u denotes any component of the vectors uL and uT for longitudinal and transverse displacements in the elastic medium, and v = vL or vT is the corresponding velocity. Assuming coordinate axes as in Fig. 1.2 we seek a plane-wave solution to (1.27) propagating parallel to the surface with wavevector qy and frequency co: u(r, t) = exp(iq,, -r,|)/(z) exp(-iatf)

(1.28)

On substituting (1.28) into (1.27) the equation for /(z) is =0

(1.29)

It can now be seen that there are two types of solution for /(z). If q\ < a>2/v2 we have f(z) = B, exp(iqzz) + B2 exp(-iqzz)

(1.30)

where (1.31) This describes a bulk wave. The two terms in (1.30) describe a wave propagating towards the surface and a reflected wave. The other type of solution of (1.29) occurs when q\ > w2/v2 and it corresponds to /(z) = £ 3 exp(/cz)

(1.32)

K=( q2

(1.33)

where °>

\ v

This describes a surface wave decaying with distance from the surface (recall that z < 0 within the solid). To proceed further with this calculation one would need to form the total displacement (uL + uT) and then bring in the boundary conditions at z = 0. The details are given in §2.3.1. In the discussion so far, we have restricted attention to the simplest case of a semi-infinite medium, so that only one surface is involved. The extension to a parallel-sided slab (or film) of finite thickness involves 10

Excitations in crystals boundary conditions at two surfaces. However, the symmetry considerations in terms of the 2D Brillouin zones and the modified Bloch's theorem (1.25) are still applicable. Numerous applications to thin films and slabs are given in the following chapters. A general point, which we may usefully emphasise here, is that (1.18) for the discrete qz values of a bulk excitation is an artefact of employing cyclic boundary conditions. When the proper boundary conditions appropriate to the surface problem are applied ,(1.18) is modified and the discrete qz values may no longer be evenly separated in reciprocal space. A specific example of this occurs in Chapter 3, where we derive the discrete qz values for a ferromagnetic film (see (3.67)). We shall also be describing excitations in multilayer systems, where there are many surfaces or interfaces between different media. Of particular interest are superlattices: these are structures in which the composition and thicknesses of the layers are such as to give a quasi-periodicity. For example, Fig. 1.4 shows a two-component superlattice consisting of alternate layers of media labelled A and B with thicknesses dx and d2 respectively. The periodicity length of the superlattice is (1.34) D = d! + d2 If the superlattice has very many layers (so that it may be assumed to extend indefinitely) then translations through D, or multiples thereof, in Fig. 1.4 Schematic illustration of a superlattice with alternate layers (labelled A and B) of thicknesses dt and d2. 2 I

/ i

A B Periodicity length D = dx

A B A B A B j

11

Introduction the z direction are well-defined symmetry operations of the structure. This gives rise to useful and distinctive properties, and superlattices are currently of widespread interest for device applications. They form the topic of Chapter 7. 1.2 Surface reconstruction

We have already mentioned at the beginning of this chapter that under certain circumstances the atoms at a surface may have a different equilibrium configuration from those in the bulk, a phenomenon known as surface reconstruction. As a result, the 2D unit cell of the surface layer (and possibly adjacent layers) may be different from that of a parallel layer well inside the crystal. The surface layer may even be disordered. If the reconstructed surface region is sufficiently thin (e.g. corresponding to a single layer of adsorbed atoms) it could be argued that the effect on the excitations will be through a modification to the boundary conditions at the surface. In this case a necessary condition on the thickness of the reconstructed region is that it should be small compared with the attenuation length l//c for a surface excitation or compared with the wavelength 2n/qz for a bulk excitation. Such requirements can often be satisfied for long-wavelength excitations. In other cases the reconstructed surface region would have to be treated explicitly as a finite-thickness layer (sometimes called the selvedge). Some circumstances where surface reconstruction occurs include the adsorption of overlayer foreign atoms onto the surface of a crystal, or an impure material where the concentration of impurities may be different at the surface. However, surface reconstruction can also occur in a chemically clean material, as we shall discuss. A good review account of surface reconstruction has been given, for example, by Somorjai (1975). Here we mention a few situations in which surface reconstruction is known to occur; the experimental evidence for the surface crystallography comes from a variety of techniques including low-energy electron diffraction (LEED) and Auger electron spectroscopy (AES). Some of the methods are discussed in §1.4.1; full accounts are to be found in the books by Prutton (1983) and Woodruff and Delchar (1986). The clean crystal surfaces of metals have been thoroughly studied using LEED. The surfaces corresponding to low Miller indices (e.g. see Problem 1.3 or Kittel 1986) in Al, Cu and Ni each have a 2D unit cell that appears to be very similar to a layer in the bulk. However, in some cases there is evidence for an expansion or contraction of the outermost atomic layer in the direction normal to the surface. For example, the outer layer of the (110) face of Al is apparently moved inwards by 10-15% relative to 12

Surface reconstruction

the bulk spacing, while for the (100) and (111) faces the effect is of order 5% or less (Martin and Somorjai 1973). Similar changes in the interlayer spacing at the surface occur in some of the alkali halides, for example, in LiF (Laramore and Switendick 1973). It is appropriate here to mention some notational conventions for classifying a surface layer. Cases such as those just described above, where the surface layer is identical to the 2D symmetry of a parallel plane in the bulk of the sample, are referred to as ( l x l ) . This indicates that the basic vectors ax and a2 in the surface layer are identical to those in the underlying bulk region (or substrate). In other cases the surface structure may have unit cells that are integral multiples of the substrate unit cell. For example, the notation W(211) - ( 2 x 2 ) applied to the (211) face of tungsten is used to describe the situation of a surface layer having basic vectors ax and a2 twice that of the substrate. This would, in fact, be the structure appropriate to an adsorbate layer of hydrogen atoms on a W(211) surface. A similar notation can also be employed for cases where the surface unit cell is rotated with respect to the substrate. This is illustrated in Fig. 1.5, where the c(2 x 2) structure on a square lattice substrate (with the letter c denoting a centred unit cell) can equivalently be designated as {Jl x Jl) - R45°.The latter notation means that the primitive unit cell of the surface structure has sides that are a J2 multiple of those for the substrate and that it is rotated through 45° with respect to the substrate. Other examples of surface reconstruction are provided by the (111), (100) and (110) faces of Si and Ge. In particular, detailed studies have been Fig. 1.5 Surface layer of adsorbed atoms (full circles) on a square lattice substrate (open circles). The basic vectors a } and a 2 of the substrate layers are marked. The unit cell of the adsorbed layer can alternatively be chosen as the square marked in broken lines, designated as c(2 x 2), or the square marked in full lines, designated as (J2 x J2) - R45° [after Somorjai 1975].

13

Introduction made of the Si(lll) surface (e.g., see Monch 1973; Joyce 1973). For a freshly cleaved sample prepared at room temperature the structure is Si(l 11) - (2 x 1). On heating to about 700-1000 K the structure eventually converts to Si(l 11) - (7 x 7). The temperature at which this transformation takes place seems to depend sensitively on the presence of trace impurities (such as Fe or Ni) and there may be other intermediate structures. The cause of surface reconstruction in Si(lll) is the covalent bonding between atoms. A Si atom in the bulk is bonded to its four nearest neighbours in a diamond cubic structure, and consequently an 'unreconstructed' (111) surface will have a layer of atoms with one bond 'dangling' into the vacuum in the direction normal to the surface (see Fig. 1.6). Such an arrangement of dangling bonds is unfavourable energetically, and surface reconstruction may take place in such a way as to reduce the overall energy of the bonds. A different type of surface reconstruction may occur in magnetically ordered solids. For example, even if there is no distortion or rearrangement of the atomic positions at the surface of a ferromagnet or antiferromagnet, there may be a different orientation of the spins (or magnetic moments) compared with the bulk material. This could occur due to thermal effects at the surface being different from those in the bulk or due to modified exchange interactions and anisotropy energies at the surface. Examples are given later in Chapter 3. Fig. 1.6 Dangling bonds from the 'unreconstructed' (111) surface of a covalently bonded diamond cubic structure [after Prutton 1983]. [Ill] 'Dangling' bonds

14

Theoretical methods 1.3 Theoretical methods Here we summarise some of the theoretical considerations relevant for studying bulk and surface excitations in media of restricted geometry. The details of the calculations will, of course, depend on whether one is concerned with (for example) phonons or magnons or some other type of excitation, and these cases are dealt with individually in the subsequent chapters. Nevertheless, it is helpful to mention some features that they have in common. 1.3.1 Dispersion relations In many cases one is interested only in calculating the dispersion relations (i.e. frequency versus wavevector relations) for the bulk and surface excitations. As discussed in §1.1.2, the frequency of a bulk excitation depends on a 3D wavevector q = (qy, qz), while the frequency of a surface excitation depends on the 2D wavevector q M. The surface mode is also characterised by an attenuation factor (usually dependent on q M >. The calculation of the excitation frequencies normally involves obtaining a set of homogeneous equations satisfied by the amplitude variable, which we denote by w(r, t), describing the excitation. For example, for a phonon w(r, t) would simply be an atomic displacement and for a magnon u(r, t) would be a spin vector. The next step is to utilise symmetry considerations in accordance with Bloch's theorem: if all the surfaces and interfaces are planar and perpendicular to the z axis, then u(r, t) can be written as exp[i(qn -ru — cot)~\f{z) as in (1.28). This leads to a set of homogeneous equations to be solved for the spatial dependence of f(z). To reach this stage we may have followed either a microscopic approach or a continuum (macroscopic) approach. In the former case, the discrete lattice structure is taken into account when forming the equations of motion for u(r, t), usually from a Hamiltonian. This leads to a set of finite-difference equations satisfied by /(z) in the different lattice planes parallel to the surface. In the latter case, the medium is treated as a continuum and the description of the excitation is in terms of macroscopic variables. The approach is appropriate only for small wave vectors q or q,l (such that the excitation wavelength, and attenuation length in the case of a surface excitation, are large compared with the interatomic distances). This is usually a simplification that leads to a differential equation for /(z), rather than finite difference equations. An example of a continuum calculation was given in §1.1.2 for vibrational waves in a semi-infinite elastic medium. We use both methods in the following chapters. The connection between the two methods is explored in 15

Introduction particular detail for the case of magnons in semi-infinite Heisenberg ferromagnets (see §3.3) and Heisenberg ferromagnetic films (see §3.4). Solutions of the differential equation, or set of finite-difference equations, for /(z) may be found by various standard mathematical techniques. In many examples (such as the semi-infinite isotropic elastic medium in §1.1.2) it is convenient to study the bulk excitations by seeking wave-like solutions of the form (1.30). This introduces the real wavevector qz in the z direction. For a surface excitation in a semi-infinite medium (occupying the half-space z ^ 0), one would seek localised solutions for f(z) having the form of (1.32). In general, K is not necessarily real, but it must satisfy Re(jc) > 0 to ensure the decay of |exp(fcz)| as z -» — oo in the medium. More generally, for a medium of finite thickness (e.g. a slab of thickness L confined between the surface planes z = 0 and z — — L) it would be necessary to generalise (1.32) for the surface excitation to f(z) = B3 exp(fcz) + J54 exp( - KZ)

(1.35)

Also the boundary conditions would need to be applied at the two surfaces z = 0 and z= —L. 1.3.2 Linear-response theory For many applications one may need to know more about the excitations than just their dispersion relations: it may be important to determine the statistical weightings of the bulk and surface excitations and the dependences on the wavevector q(|. For example, the statistical weighting would enter into the calculation of the thermodynamic properties of the excitations or the intensities measured by inelastic scattering (of light or particles) from the excitations. Linear-response theory is a convenient method for investigating the spectrum of excitations. Basically it involves calculating the response of a system to a small applied stimulus. The result may be expressed in terms of a response function: this provides information about the dispersion relation of any excitation and, in addition, the power spectrum of the thermally excited fluctuations in the excitation amplitude may be deduced. The response functions are directly related to quantum-mechanical Green functions, which can be calculated by various other methods. However, linear response theory has a more direct physical appeal than many of the alternative methods. It is helpful to give a simple mathematical example to illustrate some of the basic concepts of linear-response theory. The more formal aspects, including the relation to Green functions, are treated in the Appendix. A general account of linear-response theory is to be found in the review 16

Theoretical methods article by Barker and Loudon (1972), and the application to surface problems has been comprehensively treated by Cottam and Maradudin (1984). Other references are given in the Appendix. Consider a ID damped harmonic oscillator, where u specifies the instantaneous displacement of a mass m subject to an elastic restoring force and a damping force (see Fig. 1.7). We let f(t) be a fictitious force that acts on the mass in such a way that the interaction Hamiltonian has the following simple form linear in both f(t) and the displacement: H^-ufit)

(1.36)

The driven equation of motion for the oscillator can then be written as d U 2

"™ " ' — (1.37) dt dt where co0 denotes the natural resonance frequency and the term proportional to F describes the effect of damping ( F > 0 ) . Fourier transforms of the time-dependent quantities u(t) and f(t) can be defined by m

1

(1.38)

I

2n I _ (1.39)

2»J-

where co is a frequency label. Suppose the Fourier-transformed displacement U(co) is averaged over an ensemble of oscillators (with randomly assigned phases) to give a value denoted by U(co). In the absence of a driving force we would have simply U(a>) = 0, but when F(co) is non-zero U(co) acquires a non-zero value linear in F(co). We may use this to define a linear-response function (or

Fig. 1.7 The mechanical damped harmonic oscillator used to illustrate linear response theory.

Mass m

AAAA/VWWyVM* Displacement u

17

Introduction generalised susceptibility function) by U(co) = x(co)F(co)

(1.40)

From (1.37)-(1.40) it is easy to show that *v~,

, 2 . ^ (1-41) 2 m(o>o — co — nco) This complex function of co displays a resonant behaviour at co = ±co0, as can be seen from Fig. 1.8 where the real and imaginary parts of x(co) are plotted. Note that the symmetry property X(-co) = x*(co) (1.42) is satisfied, implying that Re[/(co)] and Im[%(co)] are respectively symmetric and antisymmetric functions of co. The asterisk in (1.42) denotes complex conjugation. Equation (1.40) may be written in an alternative (but less convenient) form as u(t)

-f

X(t')f(t-t')dt'

(1.43)

where X(t') is the Jo time-dependent Fourier transform of x{co). The limits of integration in (1.43) are in accordance with causality: the value of u at time t can depend only on the force at preceding times, i.e. t' ^ 0. Fig. 1.8 The real and imaginary parts of the response function (1.41) plotted as a function of co, taking T/co0 = 0.2.

f\

18

Theoretical methods A close connection can be shown to exist between the response function X(co) and the absorption (or dissipation) of energy from the fictitious force / . A clear and concise general description of this property is given by Landau and Lifshitz (1969). To illustrate this point, we continue with the example of the damped harmonic oscillator and take the case where the driving force has the simple form f(t)=f0 cos cot (1.44) Using (1.38), (1.40) and (1.42), we may then prove "(0 = i/oCxM exp(-iart) + x*(co) exp(icof)] (1.45) With the interaction Hamiltonian taking the form given in (1.36), the average energy dissipation Q per unit time due to the force f(t) is

= - i i / o { * M [ l -exp(-2kor)] - * * M [ 1 -exp(2icor)]} (1.46) using (1.44) and (1.45). The terms proportional to exp(2kor) and exp( — 2icot) give zero when averaged over the period 2n/a> of the driving force, and the energy dissipation becomes 6 = i/S©Im[z(a>)] (1.47) It can be checked directly from (1.41), or from Fig. 1.8, that a> Im[x(a>)] ^ 0 so that Q ^ 0 as required on physical grounds. Another connection with the dissipative properties comes from considering the mean square displacement <w2(f)>, where the angular brackets indicate a statistical mechanical average. The frequency Fourier transform <M2>W of this quantity is often referred to as the power spectrum of the thermal fluctuations. It can be shown (e.g. see Landau and Lifshitz 1969) that (u2s>a> *s directly related to the energy dissipation Q per unit time, and so it follows from (1.47) that there is a relationship between <M2>CO and Im[/(a>)]. In the 'classical' (or high-temperature) regime of kBT»h(o the result is <"2>«, = —Im[z(o>)] (1.48) nco This is an example of the fluctuation-dissipation theorem; its general quantum-mechanical form is proved in §A.3 of the Appendix, and in the present application becomes <"2>c = - [>M + 1] Im[x(c»)] n where n(co) is the Bose-Einstein thermal factor: n(a>) = [cxp(tUo/kBT) - 1] ~x

(1.49) (1.50) 19

Introduction Equation (1.49) reduces to give (1.48) in the limit of kBT»ha). We may note that, in the classical limit appropriate to (1.48), (u2}^ is a symmetric function of a>, but this is not the case for (1.49) because n(-co)+l

= -n(co)

(1.51)

This behaviour is illustrated for two different temperatures in Fig. 1.9 using the response function #(o>) of (1.41). In the context of excitations in solids the preceding mathematical example can be generalised as described in the Appendix. Briefly, the stimulus (or 'force') could be applied at one position and the response calculated at another position. This would yield a position-dependent response function (or Green function) by analogy with the relationship (1.40). The calculation will generally involve solving inhomogeneous equations (finite-difference equations or differential equations), whereas evaluating the dispersion relations as discussed in §1.3.1 corresponded to the simpler task of solving homogeneous equations. The response functions exhibit a resonant behaviour corresponding to any bulk or surface excitations, and this property enables the dispersion relations to be deduced from them. Also, from the response functions together with the fluctuation-dissipation theorem, the power spectra of the various excitations can be calculated. An example of this type of calculation occurs in §2.4 where response functions (Green functions) are derived for acoustic waves in a semi-infinite elastic medium. Fig. 1.9 The power spectrum of <M2>W versus co for two different values of the ratio kBT/hcoo, taking T/coo = 0.2.

A kBT

20

Experimental methods 1.4 Experimental methods It is appropriate now to describe some of the experimental methods used for studying the properties of excitations in the vicinity of surfaces and interfaces. We begin with a brief account in §1.4.1 of some of the techniques used to analyse the surface crystallographic structure and composition. Then a more detailed consideration is given of experimental methods for studying the dynamics of surfaces, i.e. the excitations themselves. Certain techniques are specific to particular types of excitation and are best introduced in the context of the relevant later chapter, so at this stage we discuss only some of the methods of more general applicability. We describe in §1.4.2 inelastic light scattering, which has proved to be an extremely sensitive probe of long-wavelength surface excitations, and this is followed by an account of various types of inelastic particle scattering in §1.4.3. Other more specific methods, to be described later in this book, include measurements of thermodynamic properties, such as the vibrational or magnetic contributions to the surface specific heat. When such quantities are calculated (e.g. see Chapter 3) they may be found to possess a characteristic temperature dependence different from that of the corresponding bulk effect. This is due to localised surface excitations and/or surface perturbations in the density of states of the bulk excitations. Measurements of these temperature-dependent effects can, in appropriate cases, yield information about the excitations. Surface excitations that generate electromagnetic fields can be investigated with optical techniques utilising evanescent wave coupling between the juxtaposed surfaces of two media of different refractive index. This is the principle of attenuated total reflection (ATR). It has been extensively applied to surface plasmons and surface polaritons, and we reserve discussion of the method until Chapter 6 on surface polaritons. Magnetic resonance techniques can be used to study the excitations (magnons) in ordered magnetic materials. In particular, spin wave resonance (SWR) in thin films can provide information about surface magnetic properties. The technique depends on applying an oscillating magnetic field to excite selectively magnons with small but nonzero wavevector (see Chapter 3). 1.4.1 Analysis of surface structure We have already mentioned in §1.2 the techniques of low-energy electron diffraction (LEED) and Auger electron spectroscopy (AES) for characterising the static surface properties. The former method provides 21

Introduction an accurate determination of the surface crystallography, while the latter is sensitive to the chemical composition. In LEED a beam of monoenergetic electrons (typically with energies 20-500 eV) is directed at the surface of a sample under ultra-high vacuum (UHV) conditions, and a diffraction pattern is produced. The high atomic scattering cross sections for electrons in this energy range makes LEED extremely sensitive to surface atomic arrangements (the penetration depth is typically less than 1 nm). A variation of the technique is reflection high-energy electron diffraction (RHEED) in which electrons with energies between 1 and 10 keV are used. In order to keep the penetration depth small in this case, thereby retaining surface sensitivity, it is necessary to use grazing incidence and emergence. The diffraction patterns from both LEED and RHEED can be interpreted by standard methods, e.g. using the Ewald sphere construction, but a proper theory of the intensities is extremely complicated. The basic principle of AES is rather different and can be understood by reference to Fig. 1.10. When an atom is ionised by the production of a core hole in level A, typically by incident electrons of sufficient energy (~ 1.5-5 keV), the ion may lose some of its potential energy by filling this core with an electron from a shallower level (B) together with the emission of energy. This energy may appear either as a photon, as in Fig. 1.10(a), Fig. 1.10 Energy level diagrams showing the filling of a core hole in level A, giving rise to (a) X-ray photon emission or {b) Auger electron emission. The levels are labelled with their one-electron binding energies [after Woodruff and Delchar 1986]. KE

-Ec

(a)

22

(b)

Experimental methods or alternatively as kinetic energy given to another shallowly bound electron (C), as in Fig. 1.10(fr). The latter process is known as the Auger effect and is allowed energetically provided EA >> EB + Ec. The kinetic energy of the emitted electron (which is roughly equal to EA — EB — Ec) is characteristic of the atom from which it originates. The Auger electrons have a short mean-free-path, and so their detection outside the sample provides a surface-sensitive probe of chemical composition. The Auger effect also plays a role in the technique of surface extended X-ray absorption fine structure (SEXAFS), in which surface-specific features of the X-ray absorption coefficient in thin films are measured. Another spectroscopic technique is photoelectron spectroscopy, in which the surface is bombarded with photons of sufficient energy to ionise an electronic shell so that an electron is ejected into the vacuum. The emission energies are characteristic of the surface atoms, and the angular dependence gives information about the surface structure. Usually either ultraviolet photons are employed (ultraviolet photoelectron spectroscopy, or UPS) or X-ray photons (X-ray photoelectron spectroscopy, or XPS). The above techniques, together with other methods such as electron microscopy, that are used mainly to study the static properties of surfaces are extensively reviewed elsewhere, e.g. see the books by Prutton (1983) and Woodruff and Delchar (1986). 1.4.2 Inelastic light scattering Raman and Brillouin scattering of light by dense media were first demonstrated in the 1920s and 1930s, but it was not until the advent of the laser, together with other technical developments, that these methods were widely applied to bulk excitations in solids and liquids. In recent years they have been used to study surface excitations, particularly in relatively opaque materials where the light penetrates only to a surface region. General references for light scattering are the books by Hayes and Loudon (1978) and Cottam and Lockwood (1986), and there have been numerous review articles emphasising the applications to surface excitations (e.g. Sandercock 1982; Nizzoli 1986). The essential distinction between the techniques of Raman and Brillouin scattering lies in the frequency analysis of the scattered light. In Raman scattering this is achieved by use of a grating spectrometer, typically with shifts in the wavenumber of the light in the range 5-4000 cm" 1 . Conversions to other frequency- and energy-related units are given in Table 1.1. In Brillouin scattering a Fabry-Perot interferometer is used and the wavenumber shifts are typically in a range up to about 5 cm"*. The instrumental resolutions obtainable are of order 1 cm" 1 in the case 23

Introduction Table 1.1. Conversion factors between frequency- and energy-related units Frequency (GHz) lGHz lcm"1 lmeV 1 K

29.979 241.80 20.836

Wavenumber (cm" 1)

Electron volts (meV)

Temperature (K)

0.033356

0.0041357 0.12399

0.047992 1.4388 11.605

8.0655 0.69503

0.086173

The conversion factor is obtained by looking along the appropriate row to the column giving the required units After Cottam and Lockwood (1986).

of Raman scattering and several orders of magnitude less for Brillouin scattering. As an example, we show in Fig. 1.11 a schematic arrangement for Brillouin scattering off the surface of an opaque sample. It was the development of the high-contrast multipass Fabry-Perot interferometer (Sandercock 1970) that made such experiments possible, and shortly after this the first measurements on Si and Ge surfaces were reported (see Chapter 2). We consider first the kinematics of light scattering in a bulk (effectively infinite) transparent medium. The simplest processes involve the incident light of frequency co, and wavevector k, creating or absorbing a single excitation of frequency a> and wavevector q, thereby scattering into light of frequency a>s and wavevector k s . These are represented in Figs. 1.12(a) and (b), and are known as the Stokes and anti-Stokes processes respectively. Conservation of energy and momentum imply that hcox = h(os±h(D

(1.52)

hkl = hks±hq

(1.53)

where the upper and lower signs refer to Stokes (cos < co,) and anti-Stokes (cos > a>x) scattering respectively. In principle an additional term #Q, where Q denotes a reciprocal lattice vector, could appear in (1.53) corresponding to Umklapp processes. However, as seen later, |k,| and |k s| are sufficiently small compared with Brillouin-zone boundary wavevectors that we need consider only Q = 0 in light scattering. In many simple cases involving bulk media, the ratio of intensities for anti-Stokes and Stokes scattering is given by the thermal factor n(co)

— — = exp P \ [n(©)+l]

(

ha>\

(1.54) kBTj

[

}

where n(co) is defined in (1.50). Hence anti-Stokes scattering is less intense, in general, than Stokes scattering from the same excitation. However as 24

Experimental methods emphasised by Loudon (1978), (1.54) holds only provided the lightscattering mechanism has certain symmetry properties under time reversal, and exceptions to (1.54) have been observed (e.g. in certain bulk magnetic systems). The conservation conditions (1.52) and (1.53) impose limitations on the wavevector q, so that only excitations near the centre of the Brillouin zone are detectable by light scattering. This follows from noting that the optical wavevectors are related to their frequencies by N = Wife

|ks| = rjscos/c

(1.55)

Fig. 1.11 Schematic experimental arrangement for Brillouin scattering off the surface of a sample. The detection system includes a photomultiplier (PM), discriminator (Discr.), stabiliser (Stab.) and multichannel analyser (MCA) [after Sandercock 1982]. Sample

Fig. 1.12 The (a) Stokes and (b) anti-Stokes scattering processes corresponding to creation and absorption, respectively, of an excitation (wavy line). (a)

V

kr

a), q

25

Introduction where r\x and rjs are the refractive indices corresponding to frequencies cox and s , and that typically co«(o l (see Problem 1.7). We now turn to backscattering of light from the surface of a semi-infinite medium, taken as before to be in the half-space z ^ 0. A scattering geometry with the incident and scattered light beams in the same vertical plane (the xz plane) is assumed, as shown in Fig. 1.13. Medium 1 has real constant relative permittivity ex (typically e1 = l corresponding to vacuum or air) and medium 2 (the scattering medium) has relative permittivity e 2(co) which may be frequency dependent. The incident light beam of wavevector k n in medium 1 is partially transmitted through the surface to medium 2 where the wavevector becomes k 2I . For the wavevector components perpendicular to the surface we have (1.56)

c

(1.57)

— c

while the property of translational in variance parallel to the surface implies 2I, ||

—

I

C1-58)

All

There are similar expressions for wavevectors k l s and k 2S of the scattered light beam, but with co, and 0, replaced by a>s and 9S respectively. The light-scattering process is subject to the energy conservation condition (1.52) as in a bulk medium, but (1.53) is modified to

%„=«*,„ ±«q,,

(1.59)

which determines q ||. The other wavevector component qz of the excitation is not fixed because the semi-infinite system does not possess translational Fig. 1.13 Assumed geometry for light scattering from the surface of a semi-infinite medium (medium 2). The incident and scattered light beams in medium 1 are indicated. Scattered light Incident light Medium 1 (vacuum)

>„ k i r

i I Medium 2 (the scattering medium) I

26

Experimental methods in variance symmetry in the z direction. Consequently there is a spread of values of the qz wavevector component, and this will lead to a broadened peak being observed in the light-scattering spectrum if the excitation frequency depends explicitly on qz. This effect is called opacity broadening since it depends on the optical absorption in the medium, and examples are given in Chapters 2 and 4. The connection between theory and experiment is provided by the scattering cross section a, which is defined as the rate at which energy is removed from the incident light beam by the scattering, divided by the power flow in the incident beam. A proper calculation of o for a semi-infinite scattering medium must take account of the possibility of scattering from either bulk or surface excitations and of the transmission of the incident and scattered light beams through the surface at z = 0. The formalism is described in detail by Mills et al. (1970) and Bennett et al. (1972); here we outline a derivation following Nkoma and Loudon (1975), Cottam (1976) and Tilley (1980). If A is the area of the sample surface through which the scattered beam emerges, the beam cross-sectional area is A cos 8S and from the definition we have aJa,y

W

cos 6S<\ES1\>> |En|2

Here the factor a>,/cos allows for the change in quantum energy of the scattered photons compared with the incident photons, and E n and ES1 are the incident and scattered electric fields, respectively, in medium 1 (z > 0) where the measurements are made. If we write for the scattered electric field E S i=ZE s l (Q)exp[i(k s , r r | | -/cl 1 z)]

(1.61)

Q

where r(| = (x, y) and the summation is over all wavevectors Q of the thermal excitations, the cross section becomes a = ( M / | E n | 2 ) X cos

fls<Esi(Q)-Efi(Q')>M

(1.62)

Q,Q

The wavevectors Q and Q' can be split into components parallel and perpendicular to the surface as Q = (Qu, Qz) and Q' = (Q'(|, Q'z). As shown in Problem 1.8, the summations over Q(| and Q\\ may be carried out and the final result may be expressed as a differential cross section d2cr/dQ dcos, describing the scattering into an elementary solid angle dQ with scattered frequency between cos and cos + dcos: d 2 (T

EiCOiCQcAA COS 2 0c

^

27

Introduction where A is the surface area of the scattering medium, the 2D wave vector q || is given by the conservation condition (1.59), and <. . .>Ws denotes the contribution to the power spectrum at frequency cos:

< . . . > = fdco s <...> ws

(1.64)

The next stage in evaluating d2a/dQ dcos would be to relate the electric field variables in medium 1 to the corresponding quantities inside the scatterer (medium 2). The incident light transmitted to medium 2 can be regarded as interacting with a crystal excitation to produce a polarisation. It is the radiation of light by the polarisation that produces the scattered beam transmitted to medium 1. An example of such a calculation is given in Chapter 2 using linear response theory. In some cases there may be contributions to the light scattering due to a diffraction grating being ruled on the surface or due to a periodic corrugation ('surface ripple') of the surface caused by the propagating excitation. A recent experimental modification has been to use the evanescent wave coupling of the ATR method in conjunction with Raman scattering to observe surface excitations (see Mattei et al. 1982). 1.4.3 Inelastic particle scattering Although inelastic light scattering offers very high sensitivity and resolution for studying surface excitations, it is limited to the excitations with wavevectors very close to the Brillouin zone centre, as explained in §1.4.2. Under appropriate conditions the inelastic scattering of particles by crystal excitations can involve larger momentum changes, and hence this type of scattering may provide a probe of surface excitations at wavevectors extending throughout the Brillouin zone (although generally with less favourable resolution than in light scattering). Here we shall be concerned mainly with the scattering of electrons and neutral atoms. Electron energy loss spectroscopy (EELS) has proved to be successful for studying the dynamics of surfaces, i.e the elementary excitations. For the early work in the 1960s, which was applied principally to plasmons and electronic transitions, the energy losses in the inelastic scattering process were typically many electron volts and the resolution was of order 250 meV. The development of high-resolution EELS in the 1980s was motivated by studies of surface phonons and led to much improved sensitivity, corresponding to an energy resolution of ~ 7 meV (or ~55-60 cm" 1 in wavenumber units). A detailed review is given by Ibach and Mills (1982). The technique can nowadays be used to measure dispersion relations of excitations for wavevectors throughout the entire 2D Brillouin zone associated with the layers parallel to the surface. This 28

Experimental methods is exemplified by the work of Szeftel et al. (1984) for surface vibrational modes (phonons) on Ni. In their experiments an incoming electron of kinetic energy £, and wavevector kj impinges on the surface, emits or absorbs a surface phonon of frequency co and 2D wavevector q(|, and emerges with energy Es and wavevector k s (see Fig. 1.14). Conservation of energy and of momentum parallel to the surface give the conditions EY-Es±fico = 0

(1.65)

hk^sin 9X - sin 0S) = hql{

(1.66)

where the + or — sign in (1.65) refers to annihilation or creation of a phonon, respectively, which in turn corresponds in the EELS spectrum to a gain or a loss. Equation (1.66) has been approximated by taking into account that ha>« £, (typically hco might be of the order of tens of meV with EY ~ 100 eV), implying |ks| ~ |k,| = fe,. Also by omitting a reciprocal lattice vector term from (1.66) we have ignored the possibility of Umklapp processes. By varying the angles 0, or 0S, q n can be scanned over its whole Brillouin zone and the dispersion relation of co versus q 11 can be deduced. Some experimental results are given in Chapter 2. Next we discuss the inelastic scattering of heavier particles, which would make the study of excitations at large wavevectors more easily accessible than is the case with electron scattering. Inelastic neutron scattering has been widely applied to investigate bulk excitations, but in the context of surface studies it has the disadvantage that neutrons interact only weakly with the atoms in the sample. Hence the penetration depth of the neutrons is large (several metres) and the technique is relatively insensitive to surface effects. There are two obvious ways in which the neutron intensity scattered from surfaces and interfaces can be enhanced: the first is to increase the amount of viewed surface, and the second is to select a favourable scattering geometry. Multilayer systems such as superlattices are an ideal way of achieving a large amount of surface or interface area. As regards Fig. 1.14 Geometry for inelastic particle scattering, used in the discussion of EELS and the scattering of atoms.

/VWNAA/*-

hn>, q,

29

Introduction the geometry, a method of enhancing the surface signal is to send the neutron beam at grazing incidence to the surface, and to measure the intensity of the Fresnel reflected beam. Progress along these lines, particularly for magnetic materials, has been reviewed by Felcher (1985). A technique that has proved more useful for studying surface excitations is inelastic scattering of light neutral atoms, such as He atoms. At suitably low energies (~20meV) the He atoms are scattered only by the first monolayer at the surface, and also their de Broglie wavelength is comparable to the lattice dimensions so that the entire Brillouin zone can be probed. It was the development of a He nozzle beam with good velocity resolution (Av/v < 0.01) by Brusdeylins et a\. (1980,1981) that first enabled time-of-flight (TOF) spectra to be obtained for atomic beam scattering from surface phonons. The apparatus consists of a beam source, a target chamber and a mass spectrometer detector at a fixed angle of 90° to the incident beam (i.e. 0, + 0S = 90° in Fig. 1.14). The beam is chopped into pulses before impinging on the target and producing a scattered signal. For the kinematics we employ conservation of energy and in-plane momentum, as in the preceding discussion of EELS. On putting £ r = ft2k?/2M and Es = h2kl/2M for the incident and scattered kinetic energies (M = mass of He atom), the energy conservation condition for a scattering process in which an excitation is created becomes — (k?-ki) +ftco= 0 (1.67) 2M Conservation of in-plane momentum gives (for the 90° scattering geometry), h{kx sin 6X - ks cos 9J = /zq,,

(1.68)

where we have again, for simplicity, ignored Umklapp processes. Note that, since hco is not necessarily small compared with Ex in atomic beam scattering, we cannot put fcs~fc, as was done in (1.66) for EELS. Eliminating fes from (1.67) and (1.68) yields the following relationship between co and q^: co—-

h2

K

—^— + tan0,J - 1 fc,cos0,

7

(1.69) J

Hence a T O F spectrum provides a scan along a parabolic path in (co, q^) space, from which the dispersion relation of the surface excitation may be deduced. Examples are discussed in Chapter 2.

1.5 Problems 1.1 For non-coplanar vectors a 1? a 2 and a 3 in three dimensions, prove that (1.5) and (1.6) provide equivalent definitions of the basic vectors bl9 b 2

30

Problems and b 3 of the reciprocal lattice. By making use of (1.6) and (1.7), verify that (1.4) is satisfied. 1.2 Corresponding to each of the five Bravais lattices in two dimensions (see Fig. 1.3), deduce basic vectors b t and b 2 for the reciprocal lattice and hence find the Brillouin zone. 1.3 If a surface plane is specified by Miller indices (/i/c/), this means that h, k and / are the smallest three integers such that \&i\/h, |a2|/fc and |a 3 |// give the relative values of the intercepts of the plane on the three basic axes. Prove that, for a simple cubic material with a (hkl) surface, the distance apart of adjacent lattice planes parallel to the surface is

where a is the lattice constant. 1.4 Starting from the driven equation of motion (1.37) for a mechanical oscillator, verify that the response function #(co) is given by (1.41). 1.5 For the choice of f(t) given by (1.44), work through the steps leading to (1.47) for the energy dissipation Q per unit time. 1.6 Consider the example in §1.3.2 of linear response theory for a damped mechanical oscillator. From (1.41) for x(co) and (1.48) for the 'classical' fluctuation-dissipation theorem, show that r

nmj (co20-co2)2 + r2co2 If the oscillator is lightly damped (i.e. F «co 0 ), the right-hand side of the above result is nonzero only for co close to ±a> 0 . Use this to approximate the above result as

)lJ (co0 - \co\)2 + y2 where y = jT. Now show that the mean square value <w2) of the fluctuating displacement, as calculated from

<«2>=

W J -co

leads to <«2> ~ kBT/ma>l. This is just the result expected according to the principle of equipartition of energy. 1.7 The conservation rules governing Stokes scattering of light from a bulk transparent medium are given by equations (1.52) and (1.53) with the upper set of signs. Use these equations and (1.55) to show that the 31

Introduction magnitude of the excitation wavevector q is q = (rjicof + r]la>i — 2>7I77scoIcos cos

6)l/2/c

where 6 is the angle between the directions of the vectors k] and k s , and the other notation is defined in §1.4.2. For the special case of rj{ = r]s( = rj) for the refractive indices, prove that q = rj[co2 + 4Wlcos sin 2 (0/2)] 1/2 /c Most light-scattering experiments satisfy co «coh so that Hence estimate the range of accessible g-values corresponding to typical parameter values r\ = 1.5 and wJ2n = 5 x 10 14 Hz. 1.8 The summations over Q and Q' in (1.62) can be split into summations over the parallel components Q ( | and Qy and over the perpendicular components Qz and Q'z. Show from the property of translational in variance parallel to the surface that the summation over Q|| gives a Kronecker delta <5Q Q \ The summation over Qy can be converted to an integral as

In terms of spherical polar coordinates we have d/c^ d/cgi = |k s l | 2 cos 0S sin 0S d0 s ds = (e^i/c2) cos 0S dQ where dO is the element of solid angle. Hence show that the cross section
J

By differentiating with respect to a>s and Q, verify (1.63) for the differential cross section. 1.9 Use equations (1.67) and (1.68) to verify (1.69).

32

Surface waves on elastic media and liquids

From the microscopic point of view, both bulk and surface elastic modes in a solid are described by lattice dynamics. This gives a description for all wavevectors q (for bulk modes) or q!( (for surface modes) in the appropriate Brillouin zone. Although the method is complete, it does involve, however, considerable algebraic complexity. In the longwavelength limit, corresponding to |q|a « 1 or |q,| \a « 1, where a is a typical interatomic distance, the atomic displacements vary only slowly from one atomic site to the next. It is then usually convenient to use the macroscopic description given by continuum elasticity theory. The equations of elasticity theory can be obtained from lattice dynamics, but it is simpler to derive them independently. This chapter contains some elementary lattice dynamics and related experimental results, but wherever possible we use the simpler methods of elasticity. We begin with ID lattice dynamics, which is sufficient to introduce the techniques, to illustrate in principle how the long-wavelength limit is taken, and to introduce the simplest sort of surface mode. We then move on to classical elasticity theory and in particular its application to surface waves. We give a fairly detailed discussion of the Green functions of elasticity and their application to fluctuation phenomena and the theory of inelastic light scattering. We describe two main sorts of experiments: Brillouin scattering and inelastic particle scattering. As explained in §1.4.2, Brillouin scattering probes modes in the wavevector region | q j « « 1, so the results are described by elasticity theory. Inelastic particle scattering, on the other hand, involves substantial momentum transfer, so that the modes involved come from all parts of the Brillouin zone, and a lattice-dynamical treatment is necessary.

33

Surface waves on elastic media and liquids 2.1 One-dimensional lattice dynamics

2.1.1 Monatomic lattice The simplest model for lattice vibrations is the ID 'mass-andspring' arrangement shown in Fig. 2.1 (a). An infinite number of identical masses m are joined by springs of spring constant C. lfun is the longitudinal displacement of mass n from its equilibrium position, the equation of motion is

(2.1) For normal mode solutions, all the masses vibrate at the same angular frequency: un oc exp( — icot). Then (2.1) becomes -mo)2un

= C(un+l

+ M M _! - 2un)

(2.2)

This equation is the typical member of an infinite set of coupled difference equations. The set is solved by use of Bloch's theorem (see §1.1), according Fig. 2.1 The ID models discussed in §2.1: (a) monatomic lattice: masses m coupled by springs C; (b) interface between two monatomic lattices; (c) diatomic lattice; alternating masses 171^1112 coupled by springs C; (d) 'Surface' of a diatomic lattice (on left-hand side). The longitudinal displacement of each mass in the surface mode is proportional to the arrow shown. C

Cx »•

a

Q

^2

/vQ'WVWvQ'w

**

m

(c)

m1

Cm

id) 34

One-dimensional lattice dynamics to which un = u0 exp(inqa) (2.3) where, as shown in Fig. 2.1 (a), a is the equilibrium distance between the masses. The solution corresponds to a wave travelling along the ID array. Substitution of (2.3) into (2.2) gives co2 = (4C/m)sin2£qa)

(2.4)

This result is derived and discussed in all introductory books on solid-state physics. We pause only to note the limiting expressions co2 = (Ca2/m)q2

for qa « 1

AC co2 = — when qa = n (2.5) m The first of these shows that the mode is a sound wave with co = vq in the long-wavelength limit, and the second gives the frequency at the Brillouin-zone boundary q = n/a. Two further consequences of (2.1) may be noted. First, one can pass direct from (2.1) to a continuum equation which is applicable at long wavelengths, qa« 1. Assuming slow spatial variation, we write un + 1 =un±a — + \a 2 — (2.6) dz dz2 and (2.1) becomes m d2u d2u =Ca— (2.7) a dt2 dz2 This has the form of the ID wave equation with density p = m/a and elastic modulus K = Ca. It will be recalled that the velocity of sound is v = (K/p)1/2, so (2.7) is consistent with (2.5). For a second elaboration of (2.1) we consider the example of an interface, as depicted in Fig. 2.1(b). We take all masses as equal, but the spring constant changes abruptly from C x to C2 at one of them, say u0. The equation of motion for the mass at the interface is

To lowest order in the continuum aproximation, (2.6), this becomes

F

uz

Cl~

(2 9)

-

OZ

where w_, u+ are the continuum displacements to left and right of the interface mass. In terms of continuum parameters, (2.9) is n

d2u

- - K

du

-

_L^

8u+

n

im

35

Surface waves on elastic media and liquids In terms of frequency co and wavevector q, the left side of (2.10) is of order co2 while the right side is of order q. Since co and q are proportional in the acoustic mode under discussion, the left side is of second order in the small quantity qa and can therefore be neglected. Thus the equation of motion of the interface mass yields the continuum boundary condition du-

du+

It will be seen in §2.2 and §2.3 how relations like this are used in discussing surface effects within elasticity theory. 2.7.2 Diatomic lattice For reasons which we discuss later, there are no surface-related vibrations associated with the monatomic crystal of Fig. 2.1 (a). In fact, the simplest model in which surface effects arise is the diatomic crystal of Fig. 2.1(c). Here alternating masses mx and m2 are joined by springs of constant C; note that the lattice vector is 2a and the first Brillouin-zone edge is therefore at 7c/2a. To begin with, we consider an infinite medium. We take the equilibrium positions as (2n + \)a for masses mx and 2na for masses m2, where n is an integer. The equations of motion of adjacent masses are

m2 - ^ = C(u2n+1

+ u2n_x - 2u2n)

(2.13)

Following Bloch's theorem, we write solutions "2n+i = Ux exp{i[(2n+l)qa-a>i]} U

w2w = 2 exp[i(2nga - cot)]

(2.14) (2.15)

Substitution into (2.12) and (2.13) gives two alternative expressions for the ratio UJU2: l/i = 2C cos(gfl) U2~2C-m1(D2

=

2C - m2co2 2Ccos(qa)

Equating these two expressions we find a quadratic equation for co2, whose solution leads to the dispersion relation as it is usually quoted: co2 = C[-

+— )±C||— +— ) - ' " " "~'|

(2.17)

The graph relating co to q is shown in Fig. 2.2, where it is assumed that mx >m2. 36

One-dimensional lattice dynamics Like (2.4) for the monatomic lattice, (2.17) and Fig. 2.2 are discussed in solid-state texts. The acoustic branch occupies the frequency interval co ^ (2C/m 1 ) 1/2 and the optic branch occupies (2C/m 2 ) 1/2 ^ co ^ [2C(mi + m 2 )/m 1 rn 2 ] 1/2 . The frequency interval (2C/mx)1/2 < co < (2C/m2)1/2 is a stop band for bulk modes. In order to understand surface modes we need to go beyond the above discussion and probe further the nature of the stop band; for later reference we also look at the high-frequency region co>[2C(m1 +m2)/m1m2Y/2. Equation (2.16) can be reorganised as an equation for qa: AC2 sm2{qa) = 2C(m1 + m2)co2 - m1m2co4r

(2.18)

The solutions for real q are those shown in Fig. 2.2. However, in the stop band and in the high-frequency region (2.18) admits solutions for complex q. In the former range (2.18) gives sin2(qa)>l while in the latter sin2 (qa) < 0. The corresponding expressions for q are n/2a + \y and iy, where y is real. These results are summarised in Table 2.1. Although they are at first sight of only formal significance, they serve to introduce the idea that the complex value of q in the stop band is associated with a surface excitation. The results we shall derive were first found by Wallis (1957), using a somewhat different method. Fig. 2.2 Dispersion graph, co versus q, for the diatomic lattice of Fig. 2. l(c). Mass values are m2 = 0.5m!. The surface mode frequency cos = Cll2(m^1 + m^1)1'2 is marked.

2.0 -

1.5

37

Surface waves on elastic media and liquids Table 2.1. Nature of solutions of the dispersion equation (2.18) for a diatomic lattice Frequency range

Form of q

«^(2C/m1)1/2 (2C/m 1 ) 1/2 < co < (2C/m 2 ) 1/2 (2C/m 2 ) 1/2 ^ co ^ [2C(ml + m2)/m1m2]1/2 [2C(m1 + w2)Aw1m2]1/2 < co

q real q = n/2a + \y q real g = iy

We now consider a semi-infinite diatomic medium, terminated at its left-hand end by an atom of the lighter mass m2, as shown in Fig. 2A(d). The equations of motion (2.12) and (2.13) apply to all the atoms except the end one, for which m2 —-^ = C(ux — u0) (2.19) dr Suppose we now substitute the trial solutions (2.14) and (2.15) into (2.12), (2.13) and (2.19). All the equations except (2.19) are satisfied provided UJU2 is given by (2.16). The condition for (2.19) to hold is -m2co2U2

= CIU1 exp(iqa) - U2~]

(2.20)

With the use of (2.16) for UJU2 this becomes 2(C - m2co2) cos(qa) = (2C - m2co2) exp(iqa)

(2.21)

This equation and the second equality of (2.16) are simultaneous equations for co2 and q. It is clear from (2.21) that they cannot be satisfied for real q. There is, however, a solution for real co and complex q. It is an easy matter to eliminate exp(iga) between (2.16) and (2.21); the resulting equation for co2 is co2(m1m2co2 - Cml - Cm2) = 0

(2.22)

2

The first solution is co = 0; it is seen from (2.16) that this corresponds to q = 0 and from (2.14) and (2.15) that all the un are equal. That is, this solution corresponds to uniform translation of the lattice. The second solution is co = ± cos, where co^Cim^1

+m2-1)

(2.23)

This frequency cos lies in the stop band, as shown in Fig. 2.2. To understand the nature of the mode at this frequency, we recall that within the stop band the wavenumber q is complex, q = n/2a + \y. Substitution of this into (2.16) gives sinh2(ya) = (m1 — m2)2/4m1m2 38

(2.24)

One-dimensional lattice dynamics Equations (2.14) to (2.16) give u2n+i u2n

=

(2C - m2co2) exp(-ya) 2C sinh(ya)

(2.25)

and u2n + 2

2C sinh(ya) exp( - yd) — m2co

2

(2.26)

These relations exhibit the relative phases of the motions of successive masses, and they show that the amplitude of the motion decreases exponentially with distance away from the surface. The amplitude and phase relations are shown schematically in Fig. 2A(d). Because of the decreasing amplitudes it is natural to refer to the mode under discussion as a surface mode, in accordance with §1.1.2. The above calculation illustrates clearly how a surface mode can arise within a stop band of a bulk dispersion equation. The surface mode we have found occurs when the lighter mass m2 is the end atom. A repetition of the calculation shows that there is no surface mode when the specimen terminates at the heavier mass mx. It can be seen also that, as mentioned earlier, there is no surface mode in a monatomic lattice. In fact, the diatomic lattice becomes monatomic if the masses are put equal, m l = m2. In that case, it is seen from (2.24) that y = 0, so that the wavevector q is purely real and the mode at frequency cos is not localised at the surface. In terms of the bulk dispersion curve (Fig. 2.2) there is no stop band at q = n/2a for mi=m2. This follows from (2.17), which gives at2 = oo2 for both branches in this case. Thus for mx = m2 the surface mode degenerates into one of the bulk modes. As it stands, the calculation we have outlined leaves one major question unanswered. Suppose the specimen also terminates at the right-hand end, and consists of N masses in all. There must be N normal modes, but in (2.22) we have identified only two of them. Thus we have 'lost' N — 2 modes, which looks like carelessness. A careful discussion of the AT-mass problem is given by Wallis (1957), who shows that the missing modes occur in the acoustic and optic mode regions of Fig. 2.2, and as N -• oo they essentially become the bulk modes. For the semi-infinite medium, we can see that the missing modes are in fact reflected bulk waves. Since the bulk dispersion equation (2.17) is even in q, (2.12) and (2.13) are satisfied by the solutions U

2n+i = £/ 1 {v4exp[i(2n+ \)qa~] + £exp[-i(2rc + \)qd]} exp(-icot)

u2n = U2{A exp(i2nqa) + B exp( — \2nqa)} exp( — icot)

(2.27) (2.28)

provided the ratio UJU2 is given by (2.16) and co and q are related by 39

Surface waves on elastic media and liquids (2.17). Substitution of these expressions into the equation of motion of the surface mass, (2.19), gives the ratio A/B, that is, the reflection coefficient for a bulk wave incident on the surface. The details are left for Problem 2.2. It was explained in §1.1.2 that because of translational invariance parallel to the surface, a surface mode is characterised by a wavevector q{l in the surface plane. The ID structures that we have been concerned with so far can only describe modes that have no variation in the surface plane, that is, modes for which q,, = 0 . We could in principle extend the previous discussion by considering the lattice dynamics of a surface on a 3D crystal; this was done for the first time by Wallis (1959), and later theoretical literature can be traced from Mazur and Maradudin (1981). As might be expected, the calculations become fairly laborious, since we are dealing with 3D displacements of masses in a 3D crystal structure. We do not pursue the matter further here. As will be seen below, surface modes with q(| near the centre of the 2D Brillouin zone have been studied experimentally by inelastic light scattering for some time. However, the investigation of surface modes for all values of qu in the Brillouin zone was a purely theoretical matter until about 1980. After that date, the experimental techniques of inelastic neutralatom scattering and inelastic electron scattering became sufficiently well developed to yield data on surface modes. The experimental techniques were described in Chapter 1; we defer discussion of the results until §2.7, since it is convenient to deal first with the surface modes predicted by elasticity theory. 2.2 Bulk elasticity theory 2.2.1 General formalism and wave equations In the previous section, we showed for the ID monatomic crystal that in the long-wavelength limit the difference equations for the lattice displacements go over into the elastic wave equation. Furthermore, the equation of motion of an interface atom generates the boundary condition of elasticity theory. In a similar way, for long wavelengths the difference equations of motion of masses in a 3D crystal become the elastic equations of motion, and the equations of motion of surface masses yield the same boundary conditions as can be derived from elasticity theory. The interested reader is referred to Maradudin et al. (1971) for the details. Here we follow an easier path by starting from the standard equations of an elastic medium. It should be noted that, like (2.7), the equations of elasticity apply only to the small-g (long-wavelength) linear part of the dispersion curve in the acoustic-mode branch. Thus we shall find surface modes that are related to the acoustic branches, and not the analogue of 40

Bulk elasticity theory the surface mode at frequency cos. Such analogues will be found, however, in various subsequent chapters. In this preliminary section we review briefly some standard equations of elasticity theory. The notation is that of Landau and Lifshitz (1970), to which the reader may turn for further details. The state of an elastic medium is characterised by two symmetric second-rank tensors namely the strain utj and the stress o^. With the vector u(r) denoting the displacement from equilibrium at position r, the strain is defined by

S) The definition of the stress is that on a surface of elementary area d/ whose normal is the unit vector n, component i of the force is

*i = X M * d /

(2.30)

k

The basic assumption of linear elasticity theory is that the stress and strain are linearly related, that is to say, it is assumed that a generalised Hooke's Law holds. It is written

This introduces the elasticity tensor Xijkl. It is of fourth rank, and since (Tij and ukl are symmetric tensors, kijkl is symmetric on interchange of i and j and of k and /. It also satisfies kijkl = kklij. Given these symmetries, it can be seen that Xijkl may have up to 21 independent components. However, in a crystal the point-group symmetry generally reduces this number; the form of kijkl for all the different crystal classes is given by Nye (1985). We shall consider only isotropic media, for which there are just two independent components. These are conventionally taken as Young's modulus E and Poisson's ratio cr, or sometimes as the Lame parameters X and fi. The stress-strain relations are

X + 2fi)uxx + k(uyy + uzz) E _ —— uxy = 2/auxy

(2.32) (2.33)

with relations for ayy, ayz, etc. being found by appropriate permutation 41

Surface waves on elastic media and liquids of suffixes. The alternative sets of parameters are related by

"-ma

(2 35)

-

Poisson's ratio satisfies the inequality — 1 ^ o ^ \ for reasons of hydrostatic stability, and in practice the stronger inequality 0 ^ a ^ \ is satisfied. The form of propagating waves in a bulk elastic medium is found from the equation of a mass element, namely ^

(2.36)

With the use of (2.32) and (2.33) this becomes

It can be seen from this that longitudinal and transverse waves propagate with different velocities. A longitudinal wave satisfies V x uL = 0, since for a plane wave uL oc exp(iq«r) this implies that q x uL = 0, that is, uL is parallel to the propagation vector q. For a longitudinal wave, then V x ( V x u L ) = 0, so that V(V-uL) = V2uL and (2.37) reduces to the ordinary wave equation ^

= ^V 2 u L

(2.38)

where the longitudinal velocity vL is given by

For a transverse wave, V • uT = 0, since for a plane wave this gives q • uT = 0. The second term on the right-hand side of (2.37) then vanishes, so that the transverse velocity vT is given by v$ = , ^ , (2.40) 2p(l+(r) Equations (2.39) and (2.40), together with the inequality satisfied by a, show that vL^21/2vT (2.41) 2.2.2 Reflection of acoustic waves at a free surface As a preliminary to the discussion of surface modes in the next section, we now deal with the reflection of bulk acoustic waves at a free 42

Bulk elasticity theory surface. We take the elastic medium to be in the half-space z < 0, with the surface as the plane z = 0. The surface plane is assumed to be stress-free, which in view of the definition of the stress tensor means oxz = ayz = <7zz = 0. With the use of the relations (2.32) and (2.33), these may be written in terms of the strain tensor as uxz = 0

(2.42)

uyz = 0

(2.43)

a(uxx + uyy) + (1 - c)uzz = 0

(2.44)

Consider now a wave incident on the surface at some angle 8 to the normal. The propagation vector q and the normal to the surface define the plane of incidence, which as shown in Fig. 2.3 we take as the xz plane. To begin with, we assume that the wave is transverse, with displacement u in the y direction; this is sometimes referred to as s-polarisation. The incident wave is u = u 0 cxp[i(qxx -f qzz — cot)]

(2.45)

where qx/q2 = tan 6

(2.46)

2

ql + ql = co /v$

(2.47)

u0 = (0, a, 0)

(2.48)

and The reflected wave is u = ux Qxp[i(qxx — qzz — cot)]

(2.49)

with ux = (0, b, 0). The same qx appears in (2.49) as in (2.45) because the boundary condition of (2.43) must be satisfied for all x. The fact that the qz values are the same follows from (2.47), and (2.46) then shows that Fig. 2.3 To illustrate reflection of acoustic waves at a free surface. Left: s-polarised transverse wave incident. Right: p-polarised transverse wave incident.

43

Surface waves on elastic media and liquids the angles of incidence and reflection are the same, as indicated in Fig. 2.3. Finally, application of the boundary condition, (2.43), shows that b = a for the amplitudes. Matters become more complicated if we turn to an incident transverse wave polarised in the xz plane, known as p-polarisation. Equations (2.45) to (2.47) continue to describe the incident wave, but now u o = (a,0, - a t a n 0 )

(2.50)

We may attempt to describe the reflection by means of a reflected transverse wave given by (2.49) with u1 = (b, 0, b tan 0). However, this introduces only one amplitude b, whereas two boundary conditions, (2.42) and (2.44), have to be satisfied. This problem is resolved by observing that a reflected longitudinal wave is generated in addition to the reflected transverse wave, as sketched on the right of Fig. 2.3. In order that the boundary conditions may be satisfied for all x, this wave must have the same qx value as the incident wave. It is therefore described by u = u 2 exp[i(qxx

— qLz — cot)~]

(2.51)

with q2x + q2L = cj2/v2L

(2.52)

u2 = (c,0, - c c o t 0 L )

(2.53)

and Here the angle of reflection 0L is given by qx/qL = tan 0L

(2.54)

Since vL > vT, (2.46), (2.47), (2.52) and (2.54) imply that 0L > 0, as drawn in Fig. 2.3. These equations show in fact that

^ 4 sin 2 0

Vj

»*>

With the introduction of the longitudinal wave with independent amplitude c, it is possible to apply the boundary conditions of (2.42) and (2.44) to find the reflection coefficients b/a and c/a; the details are left to Problem 2.3. One consequence of (2.55) should be noted. If the angle of incidence 0 is sufficiently large such that 0 > 9C where sin 2 6C = v\/vl

(2.56)

2

then sin 0L > 1. The meaning of this can be seen from (2.52). For 0 > 0C, ql > co2/vl, and (2.52) can only be satisfied if ql is negative. Thus we put qL = i/cL, and the longitudinal wave of (2.51) becomes u = u 2 exp[i(qxx — cot)] exp(/cLz) 44

(2.57)

Surface elastic waves This is a wave travelling along the surface, with amplitude decaying exponentially with distance into the medium. It is accompanied by the reflected transverse wave, and the combination is called a pseudo-surface wave. 2.3 Surface elastic waves 2.3.1 Rayleigh waves The free surface of an isotropic elastic medium supports a surface mode known as the Rayleigh wave. It results from a mixture of waves with p-transverse and longitudinal polarisations and its properties follow from equations of the previous section. We look for a solution of the wave equations, (2.38) and the corresponding equation for the transverse component, together with the boundary conditions, (2.42) and (2.44). The displacement is written as a sum of longitudinal and transverse parts, u = uL + uT, each part satisfying the wave equation with the appropriate velocity. In order to find a surface mode, we draw on the form of the pseudo-surface wave and assume that both parts are localised at the surface: UL,T OC exp[i(4xx - cot)] exp(>cL/rz)

(2.58)

Both components have the same value of qx so that the boundary conditions can be satisfied for all x. Since the components separately satisfy the appropriate wave equation, the constants KL, KT are given by KL,T = ( ^ - « 2 / < T ) 1 / 2

(2.59)

The x and z components of uL and uT are related through the equations V x uL = 0 and V • uT = 0, which imply iWLz-KL«L* = 0 U

(2.60)

K U

i
(2.62)

uT = b(KT, 0, -iqx) exp[i(qxx - cot)] exp(fcTz)

(2.63)

With these forms substituted, the first boundary condition, (2.42), gives 2KLqxa + (q2x + K2T)b = 0

(2.64)

It is convenient to write (2.44) in terms of vL and vT. It is seen from (2.39) and (2.40) that <x/(l _ a) = 1 - 2v2/v2

(2.65)

The boundary condition (2.44) then yields 2v2TKTqxb +

[V2(K2

- q2x) + 2v2q2x~\a = 0

(2.66) 45

Surface waves on elastic media and liquids Here K[ can be eliminated in favour of K\ with the help of (2.59): 2KTqxb + (q2x + K2)a = 0

(2.67)

Equations (2.64) and (2.67) give two independent expressions for the ratio a/b. Equating these expressions, we obtain the dispersion relation for the Rayleigh wave: 4lC L IC^ = ( ^

+

|C 2)2

( 2 6 g )

Squaring this and substituting for KI and K\ we find the more explicit form ^qt(q2x

- co2/vi)(q2x - co2/v2) = (2q2x -
(2.69)

This equation is homogeneous in a> and qx, so the solution must have the property that co is proportional to qx9 and we write co = £vTqx

(2.70)

It is clear from (2.69) that £ is determined by the ratio v?/vl, so in view of (2.65) £ depends on Poisson's ratio a and not on Young's modulus E. The dependence of £ on a is shown in Fig. 2.4; as is seen it varies only a little around the value 0.9. Equation (2.70) shows that the Rayleigh wave is like an ordinary acoustic wave, but with a velocity £vT that is slower than either of the bulk velocities vT and vL since £ < 1. It is sometimes convenient to represent the dispersion equation as in Fig. 2.5. This answers the question: What modes propagate for a given value of the in-plane wavevector qxl The lowest frequency mode is the Rayleigh wave. At a somewhat higher Fig. 2.4 Dependence on Poisson's ratio a of the Rayleigh-wave parameter { of (2.70).

0.94 -

0.90

0.86 0.1

46

0.2

0.3

0.4

0.5

Surface elastic waves frequency transverse bulk modes given by co = vT(ql + q2)112 can occur. However, on the diagram only qx is given and qz can take any value. Thus the transverse modes occupy a bulk continuum defined by co ^ vTqx. At higher frequencies still is the longitudinal bulk continuum, co ^ vLqx. The bulk continua are represented by shaded regions in Fig. 2.5. It is of interest to see how the above calculation is modified in an elastic slab with a second free surface, at a distance L say from the first. It may be expected intuitively, and it is true, that for large L (which means KLL » 1 and KTL» 1) there is a Rayleigh wave on each surface.t For a given qx these are degenerate in frequency. If, as a thought experiment, we let L decrease, the displacements of the Rayleigh waves overlap and the degeneracy is lifted. In order to make this idea precise, we now solve the elastic equations of motion with the boundary conditions of (2.42) and (2.44) applied at z= —L as well as at z = 0. In place of (2.62) and (2.63) we take trial solutions uL = [*ifa*, 0, -i/c L ) exp(fcLz) + a2(qx9 0, IKL) exp(-/c L z)] x exp[i(qxx - cot)]

(2.71)

uT = [b^Kj, 0, -iqx) exp(fcTz) 4- b2(-KT, 0, -iqx) exp(-/c T z)] x cxp[i(qxx — cot)]

(2.72)

t The analogous property is, however, not true for all types of surface mode. For example, it does not hold for the surface magnetostatic mode of a ferromagnet, to be discussed in Chapter 4. Fig. 2.5 The dispersion equation for Rayleigh waves. The transverse and longitudinal bulk continua are shown shaded. Also shown (broken line) for later reference is an approximation to the frequency scan in a Brillouin-scattering experiment.

Transverse continuum

Rayleigh wave

47

Surface waves on elastic media and liquids The terms in exp( — KLZ) and exp( — KTZ) do not appear in (2.62) and (2.63) because they diverge as z-> — oo. However, the specimen now extends only to z = — L, so they must be included. Equations (2.71) and (2.72) introduce four amplitudes al9 a 2 , bx and b2. The boundary conditions give four homogeneous linear equations in these amplitudes, and the solvability condition is the dispersion equation. It is

A=

2qxKL

q2x + K*

4* + *T

2^x/cT

2qxKL e x p ( - K L L )

{q2x + K\) exp{-KTL)

KLL)

-2qxKLexp(-KLL) {ql + K$)exp(-KLL)

2qxKrexp(-KTL)

(q2x + icJ)exp(-ic T L) -2qxKLexp{-KTL)

~^XKL

<1X + KT -2<1XKT

<7X + * T

(2.73)

For fairly large L, this has terms of zero order and terms of first order in the small quantities exp( — KLL) and exp( — KTL). In fact, since v\^^2v\, it is seen from (2.59) that KT < /cL, so that the most significant higher-order terms are those involving exp( — KTL). With only the lowest-order terms in this quantity retained the expansion of A is A = [4q2xKLKT - (q2x +

K2)2]2

- [4q2xKLKT + (q2x +

K2)2]2

exp(-2* T L) (2.74)

As L - • oo, the equation A = 0 gives (2.68) twice, once for the uncoupled Rayleigh wave on each surface. For finite L, the second term mixes the Rayleigh waves and lifts the degeneracy, as stated earlier. 2.3.2 Other slab, surface and interface modes The coupled Rayleigh waves just described are not the only modes that can travel along an elastic slab. For a localised mode on a semi-infinite medium occupying z < 0 it is necessary to have a value of qz in the lower half of the complex plane, for example qz = - k as in (2.58) or qz = n/2a — \y by analogy with the surface mode discussed in §2.1. For a slab, however, (2.71) and (2.72), with imaginary qz, are not the only possible solutions of the wave equations and boundary conditions. There are also solutions with qz real; these are often referred to as slab modes. To see how they arise, we note that we could write down solutions like (2.71) and (2.72) with factors exp(±ig L z), exp(±ig T z) replacing exp(±/c L z), exp(±/c T z). As can be seen from our discussion of the pseudo-surface mode, there are also combinations of co and qx for which qTz is real while qLz is imaginary. Whether the qz values are real or imaginary, there are four amplitudes involved, and the solvability condition for the boundary conditions gives a dispersion equation of the general form of (2.73). The above discussion was concerned with the coupled p-transverse and longitudinal modes of an elastic slab. As we have implied, there are no 48

Surface elastic waves surface modes in s-polarisation. However, there are slab modes with real qz. The calculation of their properties is easily outlined. The appropriate trial function is u = \_ax exp(i^z) + a2 exp( - \qzz)~\ exp[i(^ x x - cot)]

(2.75)

2

with ql + ql = co jv\. Application of the boundary condition, (2.43), at each interface gives two homogeneous equations in ax and a2 which are readily solved for the dispersion equation. All the elastic modes of a slab are collectively called Lamb waves. We have not given details of the calculations; the interested reader is referred to the definitive account by Meeker and Meitzler (1964). Two further ramifications are worth mentioning. The introduction of one or more interfaces modifies the mode spectrum and therefore, at least in principle, changes the specific heat. Burt (1973) uses a very neat Green-function method to derive the specific heat of a semi-infinite medium. Second, just as with bulk elastic waves, it is necessary to have a quantum theory of the acoustic field for some problems. The quantisation procedure for a semi-infinite medium is given by Oliveros and Tilley (1983). The Rayleigh wave and Lamb waves are connected with stress-free surfaces. In the presence of interfaces between different elastic media other modes may occur. The analogue of the Rayleigh wave for a single interface between two media is called the Stoneley wave, which however only exists provided the elastic constants of the two media satisfy certain conditions (Scholte 1947). When it does exist, the Stoneley wave has the dispersion equation co = vsqx

(2.76)

where vs is given by an equation similar to, but more complicated than, (2.69) for the Rayleigh-wave velocity; this equation is quoted by Albuquerque (1980), for example. The simplest surface-related wave in s-polarisation is the Love wave, which occurs in a film of thickness L on a semi-infinite substrate provided the transverse acoustic velocity v'T of the substrate exceeds the transverse acoustic velocity vT of the film. For a wave travelling in the x direction, with common x and t dependence exp[i(qxx — cot)] in the film and substrate, the transverse (y) displacement is f(z) = A cxp(iqzz) + B exp( - iqzz)

(2.77)

in the film, and /(z) = frexp(/c;z)

(2.78)

in the substrate, which is assumed to occupy the region z < —L. Here ql=
(2.79) 49

Surface waves on elastic media and liquids and K'z2=q2x-co2/v'T2

(2.80)

In order for Love waves to exist, both qz and KZ must be real, so that the displacement is an oscillatory function of z in the film and a decaying function in the substrate. This is somewhat similar to the behaviour of the electromagnetic field in a waveguide, which decays with distance into the waveguide walls but oscillates with distance across the dielectric medium within the waveguide. Thus Love waves are the first example we have met of what are generically called guided waves. Equations (2.77) and (2.78) must satisfy three boundary conditions: zero stress at the top surface of the film, equal displacements and stress at the film-substrate interface. These three boundary conditions give three homogeneous equations for the amplitudes A, B and B\ and the solvability condition is the dispersion equation. As shown by Albuquerque et al. (1980), for example, it is ten(qzL) = ii'Kz/fiqz

(2.81)

where \i and \i! are the Lame parameters of the film and substrate respectively. Equation (2.81) is similar to the one that gives the bound states in a quantum-mechanical square well. A graphical analysis similar to that applied in quantum mechanics shows that as the frequency increases the number of possible Love waves also increases, the higher waves oscillating more rapidly within the film. An example of the dispersion curves is shown in Fig. 2.6. 2.4 Acoustic Green functions 2.4.1 Infinite medium The previous section was concerned with solutions of the homogeneous wave equations and appropriate boundary conditions. The most important information obtained is the dispersion equation, co versus q, where q is a 3D wavevector for bulk waves and a 2D wavevector for surface waves. Considerably more information is available from the calculation of a Green function, which can be obtained as the solution of an inhomogeneous wave equation. The formal theory of Green functions, as far as it is needed for this book, is given in the Appendix. In particular it is shown there that an appropriately defined linear-response function is the same as a Green function, and that it may be used to calculate the power spectrum of thermally excited fluctuations. It was seen in §1.4.2 that inelastic light scattering is governed by the fluctuation spectrum, so that an evaluation of the relevant Green function is central to the theory of any given light-scattering experiment. In this section we present 50

Acoustic Green functions relatively simple examples of the calculation of acoustic Green functions; a much more comprehensive account is given by Cottam and Maradudin (1984). Let us start with the equation for a ID longitudinal wave travelling in the z direction (i.e. (2.38) with V2 -* d2/dz2) and we write the displacement simply as u. We augment the equation with a periodic point force of frequency co at position z'\ F = (f/A) exp(-i(ot)d{z - z')

(2.82)

where A is the surface area of the specimen in the xy plane. The energy of interaction of this force with the elastic medium is Hx=-\

f

F(z)u(z) dx dy dz

= -w(z / )/exp(-icot)

(2.83)

since the integral over x and y gives A. The linear-response formalism of §A.2 then shows that the displacement at position z is related to the amplitude / by the retarded Green function: i/(z) = « W (z);i/(z')*>X>/ This is a specific illustration of the relation (A.20).

(2-84)

Fig. 2.6 Love wave (full curves) dispersion for vT = O.515v'T and \JL — 5/z. The bulk continuum is shaded [after Albuquerque et al. 1980].

51

Surface waves on elastic media and liquids With the inclusion of the driving force, the wave equation becomes dt2

T

dt

Here a damping term has been added. The displacement u must be proportional to exp( — icot), so (2.85) reduces to 7T + 42w= dz

f

—Abi?-z>)

(2.86)

where q2 = (co2 +

(2.87)

ICO/T)/VI

A consequence of (2.87) is that the root with Re(g) > 0 also has lm(q) > 0. Thus, for example, the solution exp(igz) of the homogeneous wave equation travels in the +z direction with an attenuating amplitude. The solution of (2.86) in an infinite medium is V

pvlqA

™/:-l-

_'K

(2.88)

This is easily verified as follows. Away from z = z', the form is exp[ + ig(z — z')], which satisfies the homogeneous equation. As z -> ± oo, u -• 0 because Im(g) > 0. Finally, the first derivative du/dz has a step discontinuity at z = z', so the second derivative is proportional to S(z — z'). The reader is invited, in Problem 2.6, to check this last step and to confirm that the coefficient of the delta-function is correct. Equations (2.84) and (2.88) now give for the Green function «w(z); u(z')*»«> = (k/2po>2A) exp(i*|z - z'\) 2

(2.89)

2

Here co + ico/r has been replaced by co in the denominator. This assumes small damping, COT » 1, and is well satisfied in subsequent applications to light scattering. It may be helpful to illustrate some general properties of Green functions with the aid of this simple example. First, the power spectrum of thermally excited acoustic waves is (u(z)u(zT>co = (kBT/nco) Im«ii(z); w(z')*»«, = (kBT/2np(D3A)lq1 cos(|z - z'\) - q2 s i n ^ J z - z'|)]

xexp(-92|z-z'|) (2.90) where we have written q = qi+iq2 and employed (A.33). Thus the correlations between the thermally excited field amplitudes at z and z' depend only on z — z' (as follows from translational invariance), and have the form of damped oscillations. 52

Acoustic Green functions The wavevector Fourier transform of (2.89) is G(Q9a>)= f°° cxp[-ie(z-z')]«u(z);II(Z')*»«, d(z-z')

(2.91)

It gives the response of the medium to a driving force proportional to exp(igz — icot). The integral is readily evaluated when the range is split into (—oo,0) and (0, oo); the contributions at infinity vanish because lm(q) > 0. The result is 1 ^ pco2A(Q2-q2)

(2.92)

In the absence of damping, T -» oo, this may be written 1 1 2 pvlA (Q - (o2/vl) It is seen that the pole of G(Q, a>) is at co = vLQ, the excitation energy of the system.

2.4.2 Semi-infinite medium and film Having dealt with the Green function for a bulk, ID medium, we now turn to a semi-infinite ID medium, which as before we take to occupy the half-space z < 0. We still seek a solution of (2.86), the wave equation in the presence of a point driving force, but the boundary conditions are different. Whereas (2.88) is the solution of (2.86) that is bounded as z -^ oo and as z -> — oo, we now require that the solution is bounded as z -^ — oo and satisfies the condition of zero stress across the plane z = 0. It is seen from (2.44) that this latter condition is simply ^ =0 atz = 0 (2.94) dz In order to satisfy (2.94) we add to (2.88) the complementary function Cexp( — iqz), which is the solution of the homogeneous wave equation that is bounded as z-• — oo. The amplitude C is determined by the requirement that the complete solution satisfies (2.94); the resulting expression for u is u = (iqf/2pco2A){exp(iq\z - z'|) + exp[-iqr(z + z')]} (2.95) where again weak damping, COT » 1, has been assumed. The surface-related Green function <<w(z); uiz')*))^ is the coefficient of/ on the right-hand side of this equation. The result just derived, that the Green function has a term in z — z' and a term in z + z\ is typical of Green functions in a semi-infinite medium. 53

Surface waves on elastic media and liquids

These dependences can be viewed pictorially as illustrated in Fig. 2.7: the disturbance originating at the source point z' can propagate to the observation point z either directly or via a reflection off the surface. The calculation of the Green function for a semi-infinite medium can be extended without undue difficulty to a film of thickness L. In that case the zero-stress boundary condition, (2.94), is applied at both z = 0 and z=—L, and the complementary function includes an additional term D exp(igz). Coefficients C and D are determined by the two boundary conditions. The details are left for a problem; the exponential factors that appear may be illustrated in a similar way to Fig. 2.7. Fluctuation spectra can be derived from the Green function of (2.95) in the way that (2.90) was derived from (2.89) for an infinite medium. Note, however, that the presence of the surface at z = 0 destroys translational invariance in the z direction, and consequently the fluctuation spectra depend on z and z' separately, not simply on z — z'. Examples of fluctuation spectra <|M(Z)|2>CO, with z = z', are given by Loudon (1978a) for a film of thickness L. The wavevector Fourier transform of the Green function may be expected to be of physical interest, and indeed it was seen in §1.4.2 that the Brillouin-scattering cross section is closely related to this Fourier transform. To be precise, for Brillouin scattering we shall need the Green function involving the displacement derivatives uzz(z) = duz(z)/dz, where the subscript in uz(z) is inserted to emphasise that we are dealing with longitudinal modes whose displacement is in the z direction. We define

«"„(*); «„(*')*»«> = ^ £ ; «u z(z)l ux(z')*» m

(2.96)

Differentiating the Green function from (2.95) yields «u zz(z); usz(z')*»
|z-z'| 54

z + z'

Acoustic Green functions Because of the loss of translational invariance, the Fourier transform involves two wave vectors Qz and Qz separately; it is defined by o

««2Z(6Z); "„(&)*»„= Um ~[

[dzdz'«uzz(z);wzz(zT»M -Lo

x exp(-iQ 2 z) exp(iQ>') (2.98) Here a macroscopically large thickness L o has been introduced for formal reasons. The integrals in (2.98) are elementary, and evaluation gives pco2ALl

(Q'z - q)(Qz + q)(Qz - Q'z)

where the limit L o -• oo has already been taken in the exponential factors that arise. Equation (2.99) will be used in the discussion of Brillouin scattering in the next section, and in particular it will be seen there that the factor L\ in the denominator eventually cancels out from the expression for the cross section. The results derived so far in this section are for the ID wave equation in which the amplitude of the wave depends only on the coordinate z normal to the surface. Consequently these Green functions contain no information about surface modes, which have a plane-wave dependence upon distance along the xy, or surface, plane. In order to incorporate surface modes it is necessary to generalise the Green functions to include dependence on x and y. As usual, the x axis is chosen to lie along the propagation direction. The natural generalisation of a Green function such as that of (2.96) is then «w y (z); ukl(zf)*yyQxy(O. Here i, j , k and / can be x, y or z, and the Green function depends on in-plane wavevector Qx as well as on a>. The quantities utj{z) are displacement derivatives: utj(z) = dujdxj (2.100) as distinct from the strain components utj defined in (2.29). These Green functions are found by solving the wave equation in the presence of a force F = (f/A) exp((iQxx - icot)d(z - z1)

(2.101)

which is an obvious generalisation of (2.82). The mathematical steps are no different from those that have been described for the ID case, although the details are obviously somewhat more tedious. The most important property of these more general Green functions is that appropriate ones have poles corresponding to the presence of surface modes. For example, for a semi-infinite medium the Green functions «M O -(Z); ukl(z')*)}Qxt(a with i, j , k, I all equal to x or z all contain the denominator DK = 4v*Q2xq^qTz

+ (a>2 - 2v2TQ2x)2

(2.102) 55

Surface waves on elastic media and liquids where

Q2x + qL = (o2/vt

(2.103)

and Q2x + q2TZ = v2/v2T (2.104) This result was first derived by Loudon (1978b). It is easily verified that the equation DR = 0 is satisfied only when both qLz and qTz are imaginary, and that it is then equivalent to the dispersion equation for the Rayleigh wave, (2.69). Thus the Green functions have a pole when the Rayleigh-wave condition is satisfied. On the other hand, when i or k is equal to y the Green function has no pole; this corresponds to the absence of s-polarised surface modes. We have given an elementary method for calculating Green functions for systems with planar interfaces. A more advanced method, called surface Green-function matching (SGFM), has been developed by Garcia-Moliner and collaborators; the method has found applications in a range of problems, including in particular the theory of electronic surface states. Detailed accounts of the SGFM technique are given by Garcia-Moliner (1977) and by Garcia-Moliner and Flores (1979). 2.5 Normal-incidence Brillouin scattering 2.5.1 Some experimental results As explained in §1.4.2, the application of Brillouin scattering to acoustic (and other) surface modes stems from the development by Sandercock of the multipass Fabry-Perot interferometer as a practical spectroscopic instrument. The results first published (Sandercock 1972a,b) were obtained with a single multipass interferometer, while many of the later results (e.g. Sandercock 1978) involved the use of a tandem multipass interferometer. The latter instrument has the advantage of a larger free spectral range. The experimental techniques and results are reviewed in Sandercock (1982). It is convenient to start our discussion with the first results, which were for Brillouin scattering from silicon and germanium surfaces. These are reproduced in Fig. 2.8, together with some later data for Ge. At all the wavelengths used, the refractive indices of Ge and Si are high. Thus although outside the specimen the incident and scattered light propagate in directions at right angles, inside the specimen both incident and scattered light propagate very nearly along the normal to the surface. We can therefore simplify the discussion of these results by using the theory obtained for normal-incidence backscattering, in which both incident and scattered light propagate along the normal to the specimen. 56

Normal-incidence Brillouin scattering The features of the experimental results that call for comment and explanation are the following. First, although the Stokes and anti-Stokes peaks are at approximately the positions that would be predicted by simple 'bulk-type' kinematics, they are substantially broadened compared with typical Brillouin spectra of transparent media. Second, as the frequency of the incident light increases, approaching the band-gap frequency, the optical absorption coefficient increases and with it the experimentally observed line broadening. We shall see that the line broadening is indeed due to absorption of light in the specimen, and for that reason it is referred to as opacity broadening. The third feature of interest is that the broadening is asymmetric. 2.5.2 Theoretical formulation In light scattering it is necessary to determine the nature of the coupling between the incident light and the scattered light and the excitations of the system that are responsible for the scattering. In the present case of scattering by acoustic phonons, two mechanisms may be involved. First, thermally excited acousticfluctuationswithin the specimen Fig. 2.8 Brillouin backscattering from Si and Ge surfaces. The optical absorption coefficients are about 106 m " 1 in (a), 2 x 107 m " 1 in (b) and 6 x 107 m " l in (c) and (d). Spectra (a) to (c) were taken with a single multipass interferometer, while (d) was taken with a tandem interferometer which has a larger free spectral range [after Sandercock 1982].

uuJ -7 -6 (a)

I

L

L

5

-5

IAA

A = 632.8 nm

L T R T L*

6

7

-6-4-2

0

2 4

6

Ge[100] A = 488 nm

A = 514.5 nm

R

-8-6-4-2 0 2 4 6 8 'C'

m

Si[100] A = 488 nm

Frequency shifts (cm"1)

- 6 - 4 - 2 0 2 4 6 ^ ^ Frequency shifts (cm"1)

57

Surface waves on elastic media and liquids frequency-modulate the optical dielectric function. The coupling is described by the acousto-optic (or Pockels) tensor. It is this coupling that governs Brillouin scattering in a bulk specimen. In the second mechanism, the acoustic fluctuations modulate the surface of the specimen, and light is scattered off the moving surface with a frequency shift. This surfaceripple mechanism is generally dominant in highly reflective specimens, whereas the acousto-optic mechanism dominates in relatively transparent specimens, like those used for the results of Fig. 2.8. We choose to give the theory in detail for acousto-optic coupling, ignoring surface-ripple coupling; the effects of the latter mechanism are described later. In order to develop the theory, we must consider the transmission of the incident and scattered light across the surface of the specimen. We use essentially the method of Loudon (1978a), which in turn is a development of the macroscopic formulation of light scattering by bulk specimens (see, e.g. Hayes and Loudon 1978). The notation and sign convention for optical wavevectors are given in Fig. 2.9, where as before we take the specimen to occupy the half-space z < 0. We deal explicitly only with Stokes scattering, so that cos < cox and a phonon is created in the scattering process. The incident-light wavevectors are given by fcj\ = e^/c2

(2.105)

2 2

k?2 = e2((ol)co /c

(2.106)

are

where ex and £2(^1) the dielectric constants at frequency co, of the upper and lower media; usually the upper medium is vacuum, and s1 = l. Similar relations hold for fcsl and feS2. It is emphasised in (2.106) that s2 is Fig. 2.9 Notation for wavevectors of incident light, with frequency coh and scattered light, with frequency cos, for normal-incidence backscattering. I

I'

*S1 '

Si

Vacuum z= 0

////////////A «, I 2

(

I,

58

Medium

s,

Normal-incidence Brillouin scattering generally frequency-dependent, and it should be regarded as complex. For most cases, e2 varies sufficiently slowly with frequency that we may take £2(a>s) = M^i)- The exception is when the specimen is a semiconductor with col close to the band-gap frequency; this is known as resonant Brillouin scattering and will not be discussed here. With the sign convention of Fig. 2.9, the incident light in the medium contains a propagation exponential exp( — ikl2z). In order that this should correspond to a wave propagating in the —z direction, as is the case for the incident wave, it is necessary to take the square root with Re(/cI2) > 0. The usual restrictions on complex dielectric functions then ensure that Im(/cI2) > 0. Thus exp(-ifc I2 z) contains a factor exp[Im(/c I2)z], so that the incident wave decays with distance into the specimen. This enables us to define the electromagnetic skin depth 5. If the decrease of amplitude with distance into the specimen is written as exp( — |z|/<5) it is clear that d = l/Im(/cI2). A similar argument to that just given shows that Re(/cS2) > 0 and Im(fc S2 )>0. The first step in developing the theory of Brillouin scattering is to relate the incident-light amplitude inside the medium to the amplitude outside. This uses standard electromagnetic boundary conditions, and gives El2 = 2knEn/(kn+kl2)

= fEl

(2.107)

which defines the Fresnel factor / for later use. Coupling to thermally-excited acoustic excitations within the medium is conventionally described in terms of displacement derivatives utj rather than strain: uij = dui/dxj

(2.108)

It is convenient to start by considering a single Fourier component "ij = M Q i ) exp(iQ^ ~ ™t)

(2.109)

The coupling between the displacement derivative and the incident light is described by the Pockels tensor pafiyt1. Its effect is to produce a polarisation PI at frequency cos = a>l — co within the medium: P"s = P"soexp(iK0z-icost)

(2.110)

where £

"(Q'z)*

(2.111)

and K0=-kl2-Q'z

(2.112)

The appearance in (2.111) of the factor — e2, where s2 is the opticalfrequency dielectric constant, results from the definition of the Pockels 59

Surface waves on elastic media and liquids tensor. For Stokes scattering, it is the complex conjugate amplitude w* that appears in (2.111); as seen from (2.109), w* has frequency dependence exp(iart)> a n d since E, carries a factor exp( —ico,£) the right-hand side of (2.110) varies with time as exp( — ia>,£ + iart) = exp( — icost) with cos = co, — co. Equation (2.111) already contains the polarisation selection rule for this scattering process. For example, with plane-polarised incident light on an isotropic specimen, we may take the x axis along the E vector, so that E I2 = (£,2, 0,0), or p = x. The only non-vanishing component of the Pockels tensor is then pxxzz. Thus P s o , and ultimately the scattered light ES1, are also x-polarised, and scattering is solely by the longitudinal component uzz. For simplicity, we now assume that we are dealing with such scattering; the derivation is easily generalised. The polarisation P s of (2.110) extends from the surface some distance into the specimen. Its variation with z depends partly on the z-dependence of the thermally-excited acoustic field w™, but the appearance of the factor exp(iXoz) ensures that P s is confined to a thickness of the order of the optical skin depth at the surface. Maxwell's equations may be written in such a way that P s appears as a driving term on the wave equation in the specimen. Explicitly, with x-polarised light throughout, the amplitude E S2 satisfies d2ExS2/dz2 + ki2ExS2 = - ((oi/e0c2)PxS0 exp(iX o z)

(2.113)

Details of the derivation are left to Problem 2.9. This equation shows that P s acts as a source within the medium for radiation of frequency a>s. The radiated light propagates to the surface at z = 0, where some of it is reflected and some is transmitted as a field E S1. It is this field E S1 of frequency cos that is ultimately detected in the Brillouin spectrometer. In order to find an expression for it, we solve (2.113) together with standard boundary conditions at z = 0. In z < 0, the solution of (2.113) consists of a particular integral plus a complementary function: £s2 = AOPXSO exp(iXoz) + A2 exp(-i/c S2 z)

z< 0

(2.114)

while for z > 0 there is only the complementary function: Ex1=A1Qxp(ikslz)

z>0

(2.115)

Since Im(/cS2) > 0 and Im(/csl) > 0, the complementary functions in (2.114) and (2.115) tend to zero as z -• — oo and z -• + oo respectively. The amplitude Ao in (2.114) is found by direct substitution in (2.113). Amplitudes A1 and A2 are then found from the boundary conditions that Ex and Hy are continuous at z = 0; they are both proportional to the driving amplitude Px0. The expression for Al9 together with (2.111) for 60

Normal-incidence Brillouin scattering Px0 and (2.107) relating El2 to En, enables one to find the scattered amplitude £ S 1 in terms of the incident amplitude En: EUQ'z) = (
where p is shorthand for p g = ksJ(ksl

(2.116)

and / is defined in (2.107). The factor g is

+ kS2)

(2.117)

In the second line of (2.116) the notation h(Q!z) is introduced for later use. It is convenient to pause at this point and see where we have got to. We are viewing the light scattering as a three-stage process. In the first stage incident light of frequency co, is transmitted across the interface into the medium. This is described by (2.107). In the second stage, the incident light within the medium mixes with the thermally-excited acoustic field uzz to produce a driving polarisation P s at frequency cos, as described by (2.110) and (2.111). In the third stage, P s radiates light at frequency cos whose amplitude in the upper medium is given by (2.116). The experimentally measured quantity is the differential cross section 2 d a/dQ dcos. As shown in §1.4.2, this is given by

where A is the surface area of the specimen and A is the area illuminated by the incident light. Equation (2.118) is derived from the more general (1.63) by the simplifications el = l, 9s = 0 and q^\ = 0 , which apply since we are dealing with light incident from vacuum at normal incidence. Equation (2.116) enables us to relate the power spectrum appearing in (2.118) directly to the acoustic field:

X <Ei1(Qz)Esl(Qfz)>(as=\En\2 z <^(ez)^(e;Kz(ezKz(e;)*>a,

Qz,Q'z

ll2

Qz,Q'z

Z i/*(e)Me;)«w22(ej;w"(G;)*»w

(2.119)

The steps taken here deserve comment. First, the factor En in (2.116) is at frequency coh so that the power spectrum at frequency cos on the left side of (2.119) is proportional to a power spectrum at frequency a) = co, — cos on the right side. Second, the fluctuation-dissipation theorem, derived in §A.3, has been used to relate the power spectrum to the imaginary part of the corresponding Green function. The product h*h is taken outside the Green function since it is just a number, and the classical limit kBT»hco is used, as is appropriate for Brillouin scattering. Equations (2.118) and (2.119) give a direct relation between the cross section and the phonon Green function; they are the generalisation to 61

Surface waves on elastic media and liquids surface scattering of the relation used for bulk scattering. Using the definition of h(Q'z) in (2.116), we may write the cross section more explicitly:

where It remains to evaluate the sum over Qz and Qz appearing in (2.120). The details are left for Problem 2.11; the result is Im

(2.122)

The symbols 0, p, / , q and K are defined in (2.117), (2.111) with p = pxxzz, (2.107), (2.87) and (2.121) respectively. Very often, as for the Si results of Fig. 2.8, the phonon damping is negligible; in that case q is the purely real quantity co/vL. It is then easy to show that Im

(K + q)(K* - q)(K - K*) = (q2 - K\ + K2)2 + AK\K\ ( where Kx and K2 are the real and imaginary parts of K; K = K±+ \K2. Since K is independent of frequency, the Brillouin lineshape is given by the g-dependence of (2.123). This lineshape was first derived by Dervisch and Loudon (1976) by a more elementary argument than we have used here. The function is shown in Fig. 2.10; as indicated there its peak is at qm = (K\ + K22)1/2 and it has width Aq = 2K2. It is clear from (2.123) and Fig. 2.10 that the Dervisch-Loudon lineshape is qualitatively in agreement with Sandercock's experimental results of Fig. 2.8. For small X 2 , the Stokes peak is at qm = K1= kl2 + /cS2, as predicted by simple kinematics; as K2 increases the peak position moves to higher frequency. The peak width Aq is directly proportional to the imaginary part K2 = Im(fe,2 + fcS2); it therefore increases with the opacity of the specimen. An elementary 'uncertainty principle' argument for this effect is that momentum conservation in the z direction fails by Sq ~ 1/8, where 8 is the optical skin depth, so one therefore expects a linewidth of this order. Finally, the function has the characteristic skew shape that is seen in the experimental results. The quantitative agreement with the Ge data is good (Dervisch and Loudon 1976). As mentioned at the beginning of this section, the theory described so far has been for scattering by the acousto-optic mechanism, whereas in highly reflective specimens, particularly metals, the surface-ripple mechanism makes an important contribution and can be dominant. The 62

Normal-incidence Brillouin scattering

scattering amounts to a Doppler shift of the incident light as it is reflected from the moving surface. At time t, the position of the surface is shifted from its equilibrium position z = 0 to z = uz(z = 0, t). The theory involves expanding the reflected light amplitude up to first order in the small quantity uz(z = 0, t), so that ES1 oc uz(z = 0, t). A formula similar to (2.118) applies for the cross section, and since ES1 is linear in uz(z = 0) it transpires that the cross section is proportional to the power spectrum <\uz(z = 0)|2>w. This quantity is readily obtained by means of the fluctuation-dissipation theorem from the Green function of (2.96). Thus this Green function is the fundamental quantity for surface-ripple scattering as well as for acousto-optic scattering. The detailed theory of surface-ripple scattering was given by Loudon (1978c), Rowell and Stegeman (1978a) and Bortolani et al. (1978). We gave the theory of the cross section for normal-incidence backscattering by the acousto-optic mechanism because it is the simplest example. Also it has the advantage that it contains the essential physics of all such calculations. For more complicated geometries, such as oblique-incidence scattering, we shall therefore give a largely qualitative discussion of the experimental results, and the reader may take it that the theory has been worked out along the lines we have already shown. It should be emphasised that a number of different but equivalent formulations were given in the literature at about the same time, and we Fig. 2.10 Dervisch-Loudon lineshape of equation (2.123).

I U

63

Surface waves on elastic media and liquids

have naturally chosen to give the one with which we are most familiar. Alternative accounts, including in some cases surface-ripple as well as acousto-optic mechanisms, were given by Subbaswamy and Maradudin (1978), Rowell and Stegeman (1978b) and Marvin et al. (1980a, b). The most complete theoretical account of Brillouin scattering by phonons is that given by Bortolani et al. (1983); it is sufficiently general to describe scattering by unsupported films and by films on substrates as well as scattering by semi-infinite media, and it includes both surface-ripple and acousto-optic mechanisms. 2.6 Oblique-incidence Brillouin scattering

We now review results obtained with more general scattering geometries than normal-incidence backscattering. By 'oblique-incidence' we mean the use of a geometry and instrumentation such that the excitations responsible for the light scattering have a significant wavevector component qx parallel to the surface. More detailed reviews, to which the reader may refer, are given by Sandercock (1982), by Stegeman and Nizzoli (1984) and by Nizzoli (1986). The experimental scan over scattered frequency is closely approximated by a scan over co atfixedqx in the relevant dispersion curve. This is shown for the Rayleigh-wave spectrum in Fig. 2.5. Since a surface wave, such as the Rayleigh wave, has a definite frequency for a given qx9 there should therefore be a sharp peak in the scattering cross section at this frequency; this peak is broadened only by acoustic damping. By contrast, where the scan line passes through a bulk continuum, the cross section shows opacity-broadened structure like that discussed in detail in the previous section. This is clearly illustrated by the results for a Si surface shown in Fig. 2.11. The sharp lines marked R are the Rayleigh-wave peaks, while the lines T and L are due to opacity-broadened scattering by transverse and longitudinal bulk phonons respectively. The scattering is dominated by the acousto-optic mechanism, and the results have the same general form as early theoretical predictions (Loudon 1978b). The data of Fig. 2.11 show a strong feature, called the central peak, centred at zero frequency. Its width exceeds the instrumental width, so it is not simply the broadened laser line that is usually seen. By analogy with scattering by bulk liquids, it was assumed for a long time that the central peak is due to scattering by entropy fluctuations. However, a more detailed investigation by Lindsay et al. (1985) showed that at least under the conditions of their experiment it is in fact due to a two-phonon scattering process in which one optic phonon is absorbed and one is created. 64

Oblique-incidence Brillouin scattering In the theoretical analysis of oblique-incidence scattering, the cross section is still determined essentially by the imaginary part of an appropriate Green function, typically G(QX, co) = «w o (g 2 ); ukl{Qfz)*}}Qxf(O. As mentioned at the end of §2.4, this function has a pole in the complex co plane when co and Qx are related by the dispersion relation for the Rayleigh wave. If damping is small, this pole is very near to the real axis. In the vicinity of this pole, then, the frequency-dependence of the Green function may be written G(QX, co) = G0(co)lco - coR(Qx) -irjT1

(2.124)

where the Rayleigh-wave frequency is written coR(Qx) + irj9 the small imaginary part rj representing the damping. G0(co) is a function of frequency which has no singularities near coK(Qx). With the use of the symbolic identity (A.28) of the Appendix, (2.124) gives for the imaginary part Im G(QX, co) = nG0(coK)d[co - coR(Qxft (2.125) It is this delta-function that corresponds to the sharp Rayleigh peak observed in the experiments. Scattering from metal surfaces depends on the surface-ripple mechanism. This is illustrated in Fig. 2.12, which shows data taken from the surface of polycrystalline Al and theoretical results in which the specimen is approximated as an isotropic elastic medium with a = \. As for Si, Fig. 2.11, the Rayleigh wave shows as a sharp peak, while at higher frequency shifts there is broadened structure. The theoretical expressions for surface-ripple scattering involve the optical and acoustic parameters, Fig. 2.11 Brillouin spectrum of Si surface in oblique-incidence backscattering [after Sandercock 1978].

1.5 R

B 1.0

Si (001)

5

-

4

-

3

-

2

-

1

0

1

2

3

45

65

Surface waves on elastic media and liquids

but not the acousto-optic coefficients. The former quantities are known for many materials and the theoretical expressions can therefore be calculated without free parameters; the theoretical curve in Fig. 2.12 is an example. For many semiconductors both acousto-optic and ripple mechanisms contribute to the cross section. Usually, not all the parameters entering the theory are known and, in particular, the acousto-optic parameters may not be available. In that case, as shown by Marvin et al. (1980b) for example, the acousto-optic parameters may be determined by afitof theory to experiment. The results of such a fit are shown in Fig. 2.13. The lowest-frequency peak is due to the Rayleigh surface wave. The peak at just under 0.6 cm" * is due to a pseudo-surface wave of the type discussed in §2.3, while as for Si, Fig. 2.11, the peaks at around 1.3 cm" 1 and 3 cm" 1 are due to bulk transverse and longitudinal modes. The results depicted in Figs 2.11 to 2.13 involve the single surface of a semi-infinite specimen. More complicated specimens involve thin films Fig. 2.12 Oblique-incidence Brillouin backscattering from the surface of polycrystalline Al. The points are the experimental data, and the line the theory for surface-ripple scattering [after Loudon and Sandercock 1980].

1 -

66

Oblique-incidence Brillouin scattering

deposited on substrates, for example Alfilmson semiconductor substrates as discussed by Bortolani et al. (1979). Accounts of these experiments and the relevant theory are given in the review articles mentioned at the beginning of this section. The theoretical and experimental investigations described in this and the previous section took place over a period of about ten years, and effectively established surface Brillouin scattering as a spectroscopic tool. Interest subsequently shifted to various applications of this new form of spectroscopy. For example, Harley and Fleury (1979) applied the technique to layered transition-metal dichalcogenides like TaSe2, while Kueny et al. (1982) investigated the Rayleigh-wave velocity on artificiallyprepared Cu/Nb superlattices. The surface-mode peaks shown in Figs 2.11 to 2.13 show only instrumental broadening, because the surfacewave damping is very small. However, as shown in Fig. 2.14, Vacher et al. (1980) were able to measure a non-instrumental linewidth in measurements on an amorphous Si film. The lines shown are for three different specimens. The first is a single-crystal specimen, with (100) propagation on a [001] surface. The second is a single-crystal specimen, 'amorphised' to a depth of about 1 pcm by bombardment with Si ions. The third is an amorphous film of about 0.5 fim thickness with a rather Fig. 2.13 Experimental and theoretical Brillouin cross section for the (110) surface of GaAs with qx along [110]. The theory, including ripple and acousto-optic contributions, is fitted by choice of the acousto-optic parameters [after Marvin et al. 19806].

o

1

2 Frequency shift (cm"1)

67

Surface waves on elastic media and liquids

high (10%) hydrogen content, prepared by rf sputtering on to a singlecrystal substrate. It is seen that the second specimen differs from the first in having a somewhat lower Rayleigh-wave velocity, while the third has a lower velocity still and in addition shows an intrinsic linewidth. This linewidth is attributed to attenuation of the surface wave, and Fig. 2.14 is the first example of the use of BriUouin scattering to measure surface-wave attenuation. The experiments described so far may be regarded as conventional light scattering in which the light, whose properties are well understood, is used to probe the acoustic properties of the specimen. An interesting variant is the application of BriUouin scattering to study exciton-polaritons in semiconductors. An account of this topic is given in §6.1.2. For the moment it is sufficient to point out that light is transmitted within the semiconductor as an exciton-polariton, and the exciton-polariton can be investigated as long as the acoustic properties are understood. 2.7 Inelastic particle scattering

As for bulk specimens, inelastic light scattering has the advantage of high precision, but the disadvantage that it can only probe excitations near the centre of the BriUouin zone. Thus the modes investigated by BriUouin scattering, as described in the previous two sections, are of long Fig. 2.14 Lineshapes for BriUouin scattering off the Rayleigh surface wave in three Si surfaces: (a) crystalline (v = 4810 ms" 1 , v = 0.6504 cm" 1 , F = 1.17 x 10~ 3 cm" 1 ); (b) amorphised by Si ion bombardment (i? = 4290ms~ 1> v = 0.5754 cm" 1 , r = 1 . 0 0 x l 0 ~ 3 c m " 1 ) ; (c) rf sputtered (v = 3420ms" 1 , v = 0.4580cm" 1 , F = 3.87 x 10" 3 cm~ 1 ). In each case, v is the Rayleigh-wave velocity measured from the line-centre frequency v, and F is the linewidth. The instrumental width is w [after Vacher et al. 1980]. (b)

(a)

t

w

A

1 \

~ 1\ \

Count

2

/

\

I

A^ \

-5

•

i

0

i

— L—4 j—x

5

-

Frequency (10~3 cm"1)

68

\

/

—i—i—L_i—i—|

-5

(c)

5

1

\ 1

1

1—

Inelastic particle scattering

wavelength, and an account can be given entirely within continuum elasticity theory. In order to carry out experiments on surface modes of shorter wavelength, which would be described by the microscopic phonon dynamics to which an introduction was given in §2.1, it is necessary to use probe particles carrying greater momentum than photons. Neutron scattering, which has been extensively applied in the study of bulk phonons, cannot be used for surface phonons as discussed in §1.4.3. The two techniques that have been applied with success are inelastic scattering of 4He atoms and inelastic electron scattering. Fig. 2.15 TOF spectra for <112> azimuth inelastic 4 He scattering from Ag(lll) surface. Values of the angle of incidence 6\ are marked. The incoherent elastic peak attributed to crystal imperfections is shown by an arrow at the bottom. The inelastic peaks are shown by arrows at the top [after Doak et al. 1983].

c •e

1

0.8

1.0

1.2

1.4

1.3

1.5

1.7

1.9

Time-of-flight (ms)

69

Surface waves on elastic media and liquids

Some experimental details of 4He scattering were given in §1.4.3. It was also shown there by means of elementary kinematics (energy and wavevector conservation) that a time-of-flight (TOF) spectrum gives a scan along a parabolic path in (co, q||) space. These scans can be projected on to the (co, g(|) plane to give phonon dispersion curves. The first results published were for ionic crystals, particularly LiF (Brusdeylins et al. 1980; later references are given by Brusdeylins et al. 1985), and for metals (references given by Doak et al. 1983). As an example, Fig. 2.15 shows TOF spectra for a Ag(lll) surface (Doak et al. 1983). The angle between the incident and scattered beam was fixed at 90°, so that 0, = 45° gives specular reflection. The data are therefore paired in angles of incidence equidistant from 45°; the further the angle is from 45°, the greater the corresponding wavenumber. The peak positions can be plotted in the Brillouin zone, as is done in Fig. 2.16. The lower set of data points and theoretical curve in Fig. 2.16 are the extension across the Brillouin zone of the Rayleigh-wave dispersion curve shown in Fig. 2.5; it will be recalled that the Rayleigh wave figured prominently in the experimental results for inelastic light scattering. Although theory and experiment are in good agreement for the Rayleigh wave, this is not the case for the upper set of data points, which fall away from the appropriate theoretical curve. Bortolani et al. (1984) gave the full dynamical theory of the scattering, and in particular by choice of surface force constants they were able to

Fig. 2.16 Reduced-zone plot of data points in inelastic 4 He scattering from Ag(lll). Full circle, solid crystal; open circle, epitaxial crystal. The bulk continuum is shaded. Surface dispersion curves calculated by Armand and by Bortolani et al. The broken curves are pseudo-surface modes [after Doak et al. 1983].

70

Inelastic particle scattering

fit the data points and also the relative magnitudes of the peaks in the TOF spectra. The full theory of atomic scattering is demanding, since both the force constants of the phonon dynamics and the interaction potential between the surface and the incident atoms are involved. A review is given by Bortolani et al. (1986). Like those shown in Figs 2.15 and 2.16, the early results concerned scattering by acoustic modes. The cross section for scattering of atoms by optic modes is smaller, but even so Brusdeylins et al. (1985) were able to detect optic-mode scattering in NaF. Fig. 2.17 Inelastic electron-loss spectra for Ni(100) surface. Beam energy 180 eV. As indicated in Fig. 2.18, the scattered beam was collected at a fixed angle, and the angle of incidence was varied to vary q\\. The full lines are the calculated spectral densities for the vertical motion of atoms in the first layer [after Lehwald et al. 1983]. I | ?ii = 4 nm" 1

1

1

- 6 4 cm"1

—

o

°

o

o

o

200 -

o

o

o

o

o <

o o


V ,J

0 ?n = 12.6 nm

\ 1

(b)

-l

50 —

o°

o

o

40 -139 cm

20 __

o o

10 0

1

o

o

o

°r

o

o

o o

o o

"^139 c m

0

o

O

°o

o

-200

WmaX

°°\

o

A J—TTf Vl

1

o

o

"

-

o

o

o

X

_

o

o

100

(a)

o

0

300

1

54 cm"1

o

400

30

1

OCXOCCDOO

jt- x l O 1

0

200

400

0) (cm" 1 )

71

Surface waves on elastic media and liquids Inelastic electron scattering was established as a viable technique for observation of surface phonons by Lehwald et al. (1983). The essential improvement in experimental technique was the application of a highresolution spectrometer for relatively large incident energies of order some hundreds of electron volts. Experimental data for two different values of surface-wavevector transfer are shown in Fig. 2.17. As for other forms of scattering, the kinematic analysis (§1.4.3) permits translation of the peak positions into a dispersion curve in the (co, q{l) plane. This dispersion curve is shown in Fig. 2.18. As for the earlier results obtained by 4 He scattering, the dispersion curve is the extension of the Rayleigh wave of elasticity theory across the Brillouin zone. The fit to a simple theory is remarkably good.

Fig. 2.18 Loss-spectrum peak positions of Fig. 2.17 versus q\\. Full line: dispersion curve from nearest-neighbour central-force model, with force constant chosen to fit bulk phonon spectrum. Broken line: 20% increase in force constant between first and second layers [after Lehwald et al. 1983].

150

THz

O Eo = 322 eV D £ f t = 180 eV

10

72

Surface waves on liquids

The theoretical curves in Fig. 2.17 are spectral densities, and do not represent a full theory since the interaction between the electron beam and the surface ions is not included. The full theory is given by Xu et al. (1985), and compared with experimental data in which further surfacephonon branches are observed on Ni(OOl). The first observation of a surface optic phonon by inelastic electron scattering was reported by Oshima et al. (1984) on the TaC(lOO) surface. This is a metallic compound, so the conductivity screens out the Coulomb interaction, which would otherwise lead to a surface polariton (see Chapter 6) that might obscure the microscopic optic phonon. As Fig. 2.19 shows, the scattering peaks lead to a dispersion curve in which co is almost independent ofq^. TaC has the simple rocksalt crystal structure, and in a nearest-neighbour central-interaction model the theory of the surface mode on the (100) surface reduces exactly to the theory of the diatomic Fig. 2.19 Experimental dispersion curve for surface optic phonons on TaC(100). The theoretical value of equation (2.23) is marked [after Oshima et al. 1984].

80

A\\\\\\\\\\\\\\\\\\\\\\\\V ,\\\ Optical continuum \ \ \ bulk band

60 . O-0-O-O.-O--Q0O

40

\

o

.

* Nearest-neighbour central interaction model

73

Surface waves on elastic media and liquids linear chain, given in §2.1. The surface-mode frequency of (2.23) is marked on Fig. 2.19. The relatively small difference between this and the experimental result may be due to modified force constants near the surface (surface relaxation) or to the effect of second- and third-neighbour forces. In this section we have given only a brief account of inelastic particle scattering. A much fuller review is to be found in Toennies (1987). 2.8 Surface waves on liquids There are a number of important similarities between surface waves on liquids and the elastic surface modes on solids which form the subject of the earlier part of this chapter. We therefore devote a final section to liquid surfaces. In §2.8.1 we sketch the derivation of the wave equation in a bulk isotropic liquid and discuss the properties of the bulk waves; this is parallel to §2.2.1. In §2.8.2 we discuss the surface modes; although there are some analogies there are also important differences between these and the surface modes on an elastic solid that were the subject of §2.3.1. In §2.8.3 we outline briefly results that have been obtained for Green functions and fluctuation spectra. The formalism used is essentially that explained in §2.4. Calculations of Green functions were motivated to a considerable extent by the need for a theory of Brillouin scattering from liquid surfaces, and in §2.8.4 we describe some of the experimental results. The general formalism of hydrodynamics is given in a large number of texts; all the results we shall need are found in Landau and Lifshitz (1959). A very clear review of liquid surface modes is given by Loudon (1984), and further details and developments may be found there. 2.8.1 Bulk liquids and the wave equation We start by finding the equation of motion for the displacement u in a viscous, compressible liquid. Since we are concerned primarily with wave motion, we take the equation of motion in a linearised form. As in §2.2.1, u(r) is the displacement from the equilibrium of the fluid element at position r. In order to describe the viscous forces, we also need the velocity field v(r) = du(r)/dt. For plane waves with harmonic time dependence exp( — icof), v(r) and u(r) are related by v(r)=-icou(r) (2.126) The equation of motion of a mass element is given by the analogue of (2.36), which was used in elasticity theory. We write it in the form jt{pvi) = Y.^tJ^u 74

(2.127)

Surface waves on liquids where the left-hand side is the rate of change of momentum, and the right-hand side is the net force on the element. Now we must find the form of the stress tensor aik appropriate for a liquid. The compressibility means that the volume element resists compression or dilation, and since the liquid is isotropic the resulting force is in the direction of the normal to the surface on which it is measured. Thus this part of aik has only diagonal elements GXX etc., and we put ax°xmp = CV-u

(2.128)

since V*u measures the increase in the volume of the element. In addition to the compressibility part, there is a contribution to aik from the viscosity of the liquid. The presence of viscosity means that there is a stress induced by spatial variation of v, that is, by dvk/dxt. The most general expression is the analogue of (2.31), namely (2.129) As in elasticity, the fact that the medium is isotropic means that the fourth-rank tensor Xijkl has only two independent components. The reduced form of (2.129) can be written analogously to (2.32) and (2.33):

The particular form of the constants in these equations simply follows the convention used in hydrodynamics; £ and n are the two coefficients of viscosity. Equations (2.130) and (2.131) may be combined as ^

v

-

v

(2 i32)

-

The stress tensor, given by the sum of (2.128) and (2.132), is now substituted into the equation of motion (2.127). The result is i ( p ! ; i ) = C ^ - V . u + (C + i i / ) ^ - V . v + i/V2t?I.

(2.133)

This may be simplified and brought into a more useful form. We are dealing with a linearised theory, and since v is a first-order quantity the density p can be taken as the constant equilibrium value. Thus — (pvj = pdvjdt = —icopVi = —p(D2Ui (2.134) ot ot Second, the term in V2vt can be rewritten with use of the vector identity 75

Surface waves on elastic media and liquids V x (V x v) = V(V • v) - V2v. With these changes, and with the use of (2.126) to eliminate v, we rewrite (2.133) as -pco2u - [C - io(£ + fiy)]V(V-u) - iconV x (V x u) = 0

(2.135)

This is our first objective, the equation governing wave propagation in a viscous, compressible fluid. As in the elastic solid, we distinguish between longitudinal and transverse waves. For longitudinal waves, V x u = 0, and the last term in (2.135) vanishes. Substitution of the plane-wave form wocexp(iq«r) in (2.135) yields the dispersion relation pco2 = [ C - ico(£ + ^)~]q2

(2.136)

since u and q are colinear. Equation (2.136) describes ordinary longitudinal sound. If the viscosity is neglected, the propagation is undamped since the imaginary term on the right-hand side vanishes. Comparison with the simple form co = vLq shows that the longitudinal velocity vL is given by the well-known result vl = C/p

(2.137)

The reintroduction of damping changes (2.136) into an equation between two complex variables co and q. However, the experimental conditions may be such that one of these variables is purely real; for example, the wave may be initiated by a source of real frequency co. The wavenumber q is then complex, q = qx + iq 2, and the plane-wave form becomes (2.138) where the z axis is taken along q. It is seen that the imaginary part q2 describes the decay of the amplitude with distance from the source. In practice, as seen in Problem 2.12, the values of the viscosity coefficients are such that #2 <<:#i> and longitudinal waves are only lightly damped. It is commonly stated that since a liquid has no restoring force against a shear distortion, transverse waves do not propagate. This is true, broadly speaking, but we can use (2.135) to make a more refined statement. In a transverse wave, V-u = 0, and for the plane-wave form exp(iq«r) we then haveq x ( q x u ) = —q 2u. Thus (2.135) reduces to the dispersion equation pco2 = -icorjq2

(2.139)

If co is t a k e n as real, a n d q = qx+ iq2, this gives q1=q2

= (pco/2rj)1/2

'

(2.140)

The transverse 'waves' are therefore damped, with decay length comparable to the wavelength. 76

Surface waves on liquids 2.8.2 Surface ripples Surface waves on liquids are familiar in everyday experience, ranging from long ocean waves to ripples on a cup of tea. The restoring force for ocean waves is the gravitational attraction on the mass of water in the wave crests. At sufficiently short wavelengths a second restoring force due to surface tension comes into play. Surface tension, which has its microscopic origin in the different balance of forces on molecules near the surface of the liquid, appears macroscopically as an energy per unit area of surface. The surface resists stretching, and thus there is a restoring force on a wave due to the curvature of the surface. We now give a derivation of the dispersion relation for surface waves, including both gravitational and surface-tension restoring forces. The derivation is close in spirit to the discussion of the Rayleigh wave in §2.3.1; it includes compressibility but neglects viscosity. It will be seen, however, that the compressibility has negligible effect. The notation is indicated in Fig. 2.20. The liquid is in the half-space z < 0, with the undisturbed surface forming the plane z = 0. We deal with a wave travelling in the x direction, and u(x) denotes the displacement of the surface from equilibrium. Time dependence is taken in the singlefrequency form exp( — icot). We have to solve the wave equation (2.135) with the correct boundary conditions. The first two of these are the same as for the Rayleigh wave, namely zero shear stress: c7yz = axz = 0

(2.141)

With viscous damping neglected, these are automatically satisfied, as is seen from (2.131). The boundary condition on the longitudinal stress is ozz = OLd2uJdx2-pguz

(2.142)

Here a is the surface tension (surface energy per unit area) and g is the acceleration due to gravity, so that the first term is the surface-tension force due to curvature of the surface, measured by d2ujdx2, and the second

Fig. 2.20 Notation for the surface-ripple calculations for liquids.

z = 0^

-^

^—

^^

77

Surface waves on elastic media and liquids is the gravitational restoring force. The sum of these is balanced by the normal component of stress, given by (2.128). The important difference between liquids and solids is that for the latter the shear-stress conditions (2.141) must be taken into account and lead to the mixing of longitudinal and p-transverse waves that is characteristic of solid surfaces. In liquids, however, only (2.142) is required, and the surface wave involves only longitudinal displacements. We therefore start with a trial solution in the form of (2.62), namely u = a(q, 0, —\K) exp([i(gx — cot}] exp(/cz)

(2.143)

This satisfies the condition q x u = 0 for a longitudinal wave, and the choice of imaginary wavevector in the z direction means that the displacement is localised near the surface. The wave equation in the form of (2.135), with viscosity neglected, gives the relation between q and K: q2-K2

= co2/vl

(2.144)

The longitudinal stress component ozz is found from (2.128); applied to (2.143) this gives ozz = iCa{q2 - K2) = iCaw2/vl

(2.145)

where (2.144) has been used and the exponential factors are implicit. The right-hand side of (2.142) can be found immediately from (2.143): ocd2uz/dx2 — pguz = (\oiKq2 + \pgK)a

(2.146)

Substitution of (2.145) and (2.146) in (2.142) then yields the dispersion relation in the form pco2 = cucq2 + PQK

(2.147)

which with use of (2.144) may also be written pco2 = (q2 - c» W W

+ 99)

(2.148)

Equation (2.148) simplifies if the compressibility of the liquid can be neglected which is equivalent to the condition q»co/vL

(2.149)

In that case (2.148) becomes pa>2 = (xq3 + pgq

(2.150)

which is the form that is usually quoted. It is also seen from (2.144) that K~q

(2.151)

Equation (2.150) is illustrated for water in Fig. 2.21. The two terms on the right-hand side of (2.150) are equal for q/2n = 58.4 m~* or wavelength X = 1.71 cm, so as indicated this may be taken as the intermediate point between the long-wavelength gravity waves and the short-wavelength surface-tension ripples. The velocity of longitudinal sound, vL, is 78

Surface waves on liquids approximately 1500 m s"* for water, so it is easily checked that inequality (2.149) is well satisfied for the co and q ranges of Fig. 2.21; in fact the line co = vLq is indistinguishable from the vertical axis. Since for large q (2.150) has the behaviour co oc g 3/2 , the dispersion curve and the line co = vLq must eventually cross. However, for water this is at wavenumber q/2n % 4.9 x 109 m " *, may orders of magnitude larger than the scale of Fig. 2.21, and also larger than values that are accessible in light scattering. In fact, for such large values of q damping of the surface ripples becomes important, so a much more complicated derivation with viscosity terms retained must be performed.

Fig. 2.21 Surface-ripple dispersion a = 72.8x 1 0 - 3 J m " 2 .

5.0

2.0

curve

A (cm) 1.5 1.2

for

water.

The

surface

tension

is

1

150

79

Surface waves on elastic media and liquids The derivation of (2.150) given here is close to that used for solid surfaces, as we remarked. The conventional derivation is somewhat different, and assumes from the outset that the liquid is incompressible. Details are given elsewhere, for example in Landau and Lifshitz (1959) or Barber and Loudon (1989). The calculation of Green functions for acoustic problems by the method of linear response theory was explained in §2.4. One application of the results is to the calculation of thermal fluctuation spectra by means of the fluctuation-dissipation theorem. With the aid of further analysis, given in §2.5.2, the Green functions can be used in the evaluation of light-scattering cross sections. The techniques can be taken over for liquid surfaces. In the calculation of the Green function a driving force, for example of the form (2.101), is added to the equation of motion (2.135). This is solved, with the appropriate boundary conditions, such as zero stress for a free surface, to give the linear response function. Detailed results are given by Loudon (1984), who deals with the rather general case of two semi-infinite liquids separated by a thin elastic film; the Green functions of a semi-infinite liquid with a free surface are obtained as a special case. The linear response to the driving force (2.101) gives the Green function <<wt(z); uf(z')yyqt(O9 where q is an in-plane wavevector. From this, the power spectrum P(q, a>) = <|nr(0)|2>«,© of thermal fluctuations of the free surface is obtained from (A.33). An example is shown in Fig. 2.22, on which, for a fixed value of q there are two very different frequency regimes of interest. At low frequencies co ~ (<xq3/p)1/2 the spectrum is dominated by surface ripples, while at frequencies a> ~ vLq the bulk longitudinal modes are seen. 2.8.3 Light-scattering experiments The two different frequency regimes that are a feature of Fig. 2.22 mean that different experimental techniques are required for the surfaceripple and bulk-wave regions. The frequencies involved in the surface ripples are too low to be detectable as frequency shifts in Brillouin scattering. However, a number of studies cited by Loudon (1984), have been concerned with the surface-ripple spectrum, particularly of water. Byrne and Earnshaw (1979a) applied the technique of photon-correlation spectroscopy to measure the very small frequency shifts involved in inelastic scattering of obliquely incident light. In this technique the time-dependence of the photon-photon correlation function, after correction for instrumental effects, gives the evolution with time of the scattering excitation at the wavevector q determined by the scattering geometry. The time-dependence corresponding to a Lorentzian in the 80

Surface waves on liquids

frequency domain is a simple damped oscillation: f(t) =f0 cos(av) e x p ( - n ) (2.152) As shown in Fig. 2.23, Byrne and Earnshaw present their results in the form of the ^-dependence of a>0 and F. The figure shows fairly good agreement with approximate calculations using accepted bulk values of the viscosity. Later experimental work (Earnshaw and McGivern 1987)' covering a wider range of q values gives results that are in excellent agreement with those predicted from bulk viscosity values. Viscosity is a response function, and like other response functions, for example the dielectric function of a gas to be discussed in §5.3.2, it may be different from its bulk form in the vicinity of a surface. However, these experiments on water show no evidence of such an effect.

Fig. 2.22 Thermal fluctuation spectrum P(q,co) of the free surface of Hg for fixed in-plane wavevector q = 107 m" 1 : (a) surface-tension region, (b) longitudinal-mode region [after Loudon 1984]. (a)

(b)

81

Surface waves on elastic media and liquids

One reason for the interest in the surface of water is that it serves as a suitable substrate for layers that provide models of biological membranes whose elastic properties are of considerable interest. The application of photon-correlation spectroscopy to monolayers of fatty acids is described by Byrne and Earnshaw (1979b). Fig. 2.23 Wavevector dependence of surface-ripple parameters of water, (a) coo and (b) V. The lines are theoretical, using published values for bulk water [after Byrne and Earnshaw 1979a]. 105 (a)

104

. I 105

103L_ 104 1

q (m )

10V

104

103

102 4 10

82

, i

q (m"1)

105

Problems

It is seen from Fig. 2.22 that the fluctuation intensity in the bulk-wave region is some orders of magnitude less than in the surface-ripple region. Nevertheless, some measurements have been successfully performed, and the Brillouin-scattering spectrum for the surface of Hg is shown in Fig. 2.24. The spectrum is seen to be of similar shape to the bulk-wave part of Fig. 2.22. 2.9 Problems 2.1 Derive (2.23) for the surface-mode frequency in the mass-and-spring model of Fig. 2.1 (d). Confirm (2.25) and (2.26) for the relative amplitudes. 2.2 In (2.27) and (2.28) for a ID diatomic lattice, amplitudes A and B are introduced. Show that the reflection coefficient is A

Cexp( — iqa) — (C — m2a)2) Qxp(iqa)

B (C — m2co2)exp( — iqa) — C Qxp(iqa) 2.3 Use the theory of §2.2.2 to find the reflection coefficients c/a and b/a for the incident p-polarised transverse wave shown on the right of Fig. 2.3. 2.4 When a longitudinal elastic wave is incident on a stress-free surface, reflected longitudinal and p-polarised transverse waves are generated (see §2.2.2). Find the reflection coefficients for this process. 2.5 The boundary conditions for the Love-wave displacements of (2.77) and (2.78) are

Fig. 2.24 Bulk-wave Brillouin cross section for scattering off the surface of Hg. The inset shows the scattering geometry. [Figure due to J. G. Dil, reproduced by Loudon (1984).]

1k ! 76.5°

>s section

_

"77T

U

0.11

0.22

(jj/2nc (cm"1)

83

Surface waves on elastic media and liquids duy/dz = 0

at z = 0

\x duy/dz = \i! <

Confirm that application of these leads to the dispersion equation (2.81). 2.6 Equation (2.88) for the solution to the driven wave equation in an infinite medium may be written if fexp[ig(z - z')] for z>z' ~ pv[qA jexp[ - iq(z - z')] for z z'. 2.8 Generalise the arguments of §2.4.2 to find the acoustic Green function <
(Oliveira et al. 1980). Combine (2.120) and (2.99) to obtain (2.122). 2.12 Consider (2.136) for the propagation of longitudinal sound in a liquid. Typical values of the parameters are quoted by Loudon (1984) in SI units as Ar:

p = 1.41 x 103; vL = 0.85 x 103; rj = 2.8 x 10~4; f = 1.9 x 10" 4 .

Hg:

p = 13.6 x 103; vL = 1.5 x 103; r\ = 15.6 x 10~ 4 ; C = 7.02 x 10" 4 .

Confirm that for both those liquids damping of longitudinal sound is small, q2 « q x with a> real.

84

Surface magnons

Magnons (or spin waves) are the low-lying excitations that occur in ordered magnetic materials. The concept of spin waves was introduced by Bloch (1930), who envisaged some of the spins as deviating slightly from their ground-state alignment and this disturbance propagating with a wave-like behaviour through the solid. Because the spins are properly described by quantum-mechanical operators, the spin waves are also quantised with the basic quantum being referred to as the magnon (by analogy with the phonon for quantised lattice vibrations). Magnons can be studied through their contribution to thermodynamic properties (e.g. specific heat) or more directly by techniques such as light scattering, neutron scattering and magnetic resonance. We begin this chapter by giving an introductory account of the theoretical models required to describe magnons in simple ferromagnets and antiferromagnets. Using the Heisenberg model we then present the theory of bulk magnons in infinite ferromagnets, as a preliminary to generalising the theory to semi-infinite ferromagnets and to ferromagnetic films. In the last two cases we show that surface magnons may occur localised near the surface(s) and that bulk-magnon properties are modified. After extending the theory to bulk magnons and surface magnons in Heisenberg antiferromagnets, we give a review of experimental results for surface magnons in Heisenberg magnetic materials. Finally we discuss surface magnons in Ising ferromagnets. This is of interest not only for certain real magnetic materials, but also because the Ising model is widely employed to describe the excitations in order-disorder (hydrogen-bonded) ferroelectrics. 3.1 The magnetic Hamiltonian Ferromagnets are characterised by an interaction between neighbouring electronic spins that gives rise to a parallel alignment at low 85

Surface magnons

enough temperatures. In simple antiferromagnets the sign of the interaction is such that an antiparallel ordering of spins is favoured. As the temperature is raised the long-range magnetic order in either material decreases, until eventually there is a phase transition to a paramagnetic state. The critical temperature at which this occurs is known as the Curie temperature Tc and the Neel temperature TN in ferromagnets and antiferromagnets respectively. The nature of the interaction that is responsible for the magnetic ordering was first explained by Heisenberg (1928), who was able to show that it is electrostatic in origin and due to quantum-mechanical exchange. Details are given in most textbooks on quantum mechanics or magnetism (e.g. Schiff 1955; Mattis 1965; Wagner 1972), but we may summarise the essential steps of the argument as follows. We consider two neighbouring magnetic ions, which each have one electron. The electrons have spin \ and obey Fermi-Dirac statistics. Therefore their total wavefunction, which may be expressed as the product of a spatial part {// and a spin part #s, must be antisymmetric with respect to interchange of the electrons. The two possible spin states, which are the symmetric Xs=i anc* ^ e ant i~ symmetric / s = 0 with S denoting the total spin quantum number, have to be combined respectively with the antisymmetric (i// _) and symmetric (\l/ + ) spatial wavefunctions. Expressions for i//+ and ^_ can be written down in the Heitler-London theory, and are different provided there is appreciable overlap of individual electronic wavefunctions. Consequently when the total energy is calculated from first-order perturbation theory, there is a difference of energy between the S = 1 and S = 0 states. With an appropriate choice of origin this can be represented by the operator — J S i ^ , where Sx and S2 are spin operators for the two electrons. The quantity J is known as the exchange interaction, and its magnitude is a measure of the degree of overlap of the electronic wavefunctions. If J > 0 the S = 1 state will be favoured, corresponding to parallel spins, as in ferromagnetism. Similarly the situation J < 0 would be associated with antiferromagnetism. When the above simplified argument is generalised to the whole system of spins in a magnetic medium we arrive at the Heisenberg Hamiltonian * = -

I

JtfifSj

(3.1)

where the summation is over all distinct pairs i and j . The exchange interaction Jtj is short range, and it is often sufficient to consider only nearest-neighbour sites i and j . A proper treatment of exchange is much more complicated than indicated here: the subject has been reviewed by 86

The magnetic Hamiltonian Anderson (1963) and by Herring (1966) for magnetic insulator and metals respectively. More generally, in some magnetic materials the Heisenberg Hamiltonian may take the anisotropic form ^ = - 1

Jy(fi:e5fSJ + 6^SJ + 6rSfS;)

(3.2)

with dimensionless parameters ex, ey, ez. Of particular interest are the cases where (i) sx = ey = sz=l, which is just the isotropic Heisenberg Hamiltonian, and (ii) sx = ey = 09 £ z = l , which is known as the Ising Hamiltonian. The latter case is important for discussion of order-disorder ferroelectrics using a pseudo-spin approach (see §3.7). In addition to the Heisenberg exchange interaction there are the magnetic dipole-dipole interactions. Each spin S; has magnetic moment gfiBSh and the classical dipole-dipole interaction between these moments gives a contribution to the energy of (e.g., see Panofsky and Phillips 1962)

where r 0 is the vector joining sites i and j . Here fi0 is the permeability of free space, g is the Lande g-factor and /iB is the Bohr magneton. Typically the dipole-dipole interactions are much weaker than the exchange interactions (since usually iiog2\i\jr\3« J tj for neighbouring ions), and they are unimportant in determining the static magnetic properties such as the magnetisation. Similarly they produce only a small perturbation in the magnon frequency for most wave vectors q in the Brillouin zone. However, in the limit of small q (i.e. long wavelengths) the influence of dipole-dipole interactions on the magnon frequency becomes important and eventually dominates. This is essentially because the exchange is short range whilst the dipole-dipole interactions are long range. Typically exchange interactions are dominant for magnons with |q|>10 8 m~ 1 , compared with about lO^m" 1 for a Brillouin zone boundary wavevector. For |q| < 10 7 m" 1 the dipole-dipole interactions are dominant, and in the intermediate regime (107 m" 1 ^ |q| ^ 108 m" 1 ) the two types of interaction may be comparable. The above numbers are approximate and may vary for different magnetic materials. The characteristics of the surface magnons will be shown to depend on which of the three wavevector regimes is applicable. In this chapter we are concerned with the case where exchange interactions are dominant and we neglect the dipole-dipole interactions. The effects of dipole-dipole interactions are the subject of Chapter 4. 87

Surface magnons 3.2 Magnons in bulk Heisenberg ferromagnets As the simplest illustration we begin by considering the bulk magnons in an isotropic Heisenberg ferromagnet which is assumed to be infinite in extent. The Hamiltonian can be expressed as •**= - i I

J S S

ij i' J ~

MBB0

Z Sf

(3.4)

where the factor of \ in the first term has been included to allow for the fact that each pair of sites will be counted twice in the summation over i and j . The second term describes the effect of an applied magnetic field Bo in the z direction, which is also the direction of average spin alignment. The magnon excitations can be calculated from (3.4) by a variety of different techniques (e.g. see Mattis 1965), many of which involve transforming from spin operators to other operators such as boson creation and annihilation operators. However, in the present case it is convenient to work directly in terms of the spin operators since this will allow us to introduce temperature-dependent effects in a straightforward manner. We define operators S/ and S,~ at site j by S?=S1±iS>j

(3.5)

and the commutation relations satisfied by the spin operators can then be written as IS?, Sr] = ISfaj

[S?, Sf ] = ± S ? Stj

(3.6)

in units such that h = 1. The notation is that [A, B] is the commutator (AB — BA) between any two operators A and B. We also require the equation of motion for any operator A, which from elementary quantum mechanics (e.g. Schiff 1955) has the form i ^ = [A,^] 0=1) (3.7) at where Jf is the Hamiltonian. Taking the case of A = S / , and using (3.4) and (3.6), we obtain i A 5 / = gpJIoS]- + 1 J y (SfS; - S*S +)

(3.8)

The product of spin operators in (3.8) may be simplified by means of the Random-Phase Approximation (RPA):

(Sfs; - s*s?) - <sf >s; - <sj>s+

(3.9)

where the angular brackets <.. .> denote a thermal average. Equation (3.9) represents a 'decoupling' of the product of operators, whereby each Sz is replaced by its nonzero thermal average. It should be a satisfactory approximation provided the spins are fairly well aligned in the z direction.

Magnons in bulk Heisenberg ferromagnets If we now define a wavevector Fourier transform by

/

^

X

,

)

(3.10)

q

where N is the number of sites, and we use the property that <Sf> = <SZ> = <SZ> due to the equivalence of lattice sites, the approximated (or linearised) equation of motion becomes i^tSZ=coB(q)S+

(3.11)

This describes a simple harmonic oscillator with angular frequency coB(q) where (3.12) coB(q) = GUBBO + <SZ>D/(O) - J(q)] The quantity J(q) is defined by J

tj = N~1Y J(q) exp(iq. (r, - r,)]

(3.13)

q

Typically it is found (as in the example below) that J(q) decreases with increasing |q|. Consequently a>B(q), which represents the bulk-magnon frequency, has its minimum value giiBB0 at q = 0 and increases with |q| (for a fixed direction of q). The precise form of J(q) depends on the range of exchange interactions and on the crystal symmetry. In the case of a simple cubic lattice (with lattice parameter a) and exchange coupling J to the six nearest neighbours only, it is easily verified that J(q) = 2J[cos(qxa) + cos(qya) + cos(g z a)] For small values of # = |q| (such that qa«l) relation for cubic systems takes the form

(3.14) the magnon dispersion

coB(q) = GUBBO + Dq2 + 0(q4)

(3.15) z

2

where expansion of the cosine functions shows that D = (S }Ja in the case of (3.14). As the temperature T is increased, the thermal average <SZ> (which is proportional to the magnetisation) decreases monotonically. The temperature dependence of <<SZ> can be estimated by mean-field theory (also known as molecular-field theory), which was first proposed using classical arguments by Weiss (1907). If spin fluctuation effects are ignored in (3.4) for J^ the mean-field Hamiltonian is obtained as ^MF

= -g/JLB I [*o + * E ( 0 ] S ?

(3-16)

i

where BE(i) is the average effective field acting on spin S; due to the exchange interactions:

89

Surface magnons There are no mean-field terms in (3.16) proportional to Sf or Syt because <S*> and <SJ> vanish for the Hamiltonian (3.4). In the present case BE(i) is independent of site i and equal to <SZ> J(0)/#/iB. For a system with spin S = \ the calculation of <SZ> is now straightforward. Corresponding to the eigenvalues +\ of the Sz operator, the mean-field energies from (3.16) are + ^g^B(^o + #E)> a n d it is easy to show that (see Problem 3.1) <SZ> = i t a n h [ ^ B ( B 0 + BE)/2fcBT]

(3.18)

2

Since BE is proportional to <5 > the above equation can in principle be solved self-consistently (by numerical methods if necessary), to obtain <SZ> for any values of Bo and T. For general values of S, the right-hand side of (3.18) would be written in terms of the Brillouin function for spin S (e.g. see Mattis 1965). This approach gives the prediction that <5Z> -• S as T - > 0 K , and <SZ> decreases as T is increased, falling to zero at Tc = S(S + l)J(O)/3/cB if the applied field Bo is zero. Although we have followed a quantum-mechanical approach here, magnons can also be interpreted semi-classically in terms of precessing spins, as represented schematically in Fig. 3.1. For bulk magnons the spins precess with constant amplitude s but varying phase angle , leading to a spin wave disturbance propagating through the crystal. 3.3 Semi-infinite Heisenberg ferromagnets We now generalise the theory of the preceding section to the case of a semi-infinite Heisenberg ferromagnet occupying the half-space z ^ 0.

Fig. 3.1 Semi-classical representation of magnons in a ferromagnet: (a) the ground state; (b) a bulk magnon, showing the precessing spin vectors viewed in perspective and from above.

(a)

90

Semi-infinite Heisenberg ferromagnets The system is assumed to be effectively infinite in the x and y directions, so that there is only one surface (the plane z = 0) to consider. The Hamiltonian will be taken to have the same form as in (3.4), except that the exchange J(j near the surface may differ in general from the value in the bulk. This difference can occur because the exchange interaction is related to overlap integrals between electronic wavefunctions, as explained in §3.1, and the electronic wavefunctions and/or the lattice parameters may be perturbed near a surface. 3.3.1 Microscopic theory As a specific example, we consider a simple cubic lattice with a (001) surface. The exchange interactions Jtj will be assumed to couple nearest neighbours only, having the value Js if both i and j are in the surface layer and the bulk value J otherwise. This is a model first considered by Fillipov (1967) and it is represented schematically in Fig. 3.2. The direction of average spin alignment will be taken along the z axis, arid to determine the equilibrium configuration we again use mean-field theory. Equations (3.16) and (3.17) still apply, but BE(i) will now depend on site i. We introduce an index n (= 1, 2, 3 , . . .) to label the layers parallel to the surface; n is related to the coordinate z by z = -(n - \)a. From symmetry considerations it is clear that BE(i) and <S?> depend on position only through the layer index n. For n = 1, each spin has four neighbours in the surface and one in layer 2, and so l) = 4Js{Sz}1 + J(SZ)2 (3.19) Fig. 3.2 The (001) surface of a semi-infinite ferromagnet with simple-cubic structure, indicating the nearest-neighbour exchange constants J and Js-

n= 1 --

Js

\ J J

n= 2— J

J J

J

n= 3—

J

J

J

J

J J

J J

J J J

J J

X

J

J

J J

J J

J

\

J

91

Surface magnons Similarly for n ^ 2, there are six neighbours and gfiBBE(n) = J((Sz}n.1

+ 4<Sz>n + <S z>,I+1)

(3.20)

In the case of S = \ the mean-field equations are (Sz}n = itanh{0/i B [£ o + BE(n)~\/2kBT}

(n ^ 1)

(3.21)

by analogy with (3.18). When (3.19) and (3.20) are substituted into (3.21) we have a set of recurrence relationships satisfied by the spin averages (Sz)n for n = 1, 2 , . . . . In general they have to be solved numerically, but we note that for T -> 0 there is the analytic solution that <Sz>n -• \ for all n. An example of the numerical solution of (3.21) using an iterative approach is given in Fig.3.3, taking Bo = 0 and Js/J = 0.5. We next evaluate the frequencies of the magnetic excitations, showing that they consist of localised surface magnons in addition to bulk magnons. We restrict attention to the low-temperature region T« T c in order to avoid complications due to surface reconstruction (i.e. the variation of <Sz>n near the surface). In the case of spin S this enables each <Sz>n to be replaced by S. The linearised equation of motion, obtained using (3.8) and (3.9), is d (3.22) i - 5 / = gfiBB0S; + S X JtJ(S; - S + The dispersion relations for bulk and surface magnons can now be deduced from this equation, as follows. Fig. 3.3 The spin average <Sz>n as a function of temperature for several values of layer index n in a semi-infinite Heisenberg ferromagnet with simple-cubic structure, taking

£ o = 0and JS/J = 0.5.

0.1

0.6

92

T/Tc

0.8

Semi-infinite Heisenberg ferromagnets The magnons correspond to wave-like solutions for 5 / of the form S/ = sn(q,|) exp(iqn -pj) exp(-icot)

(3.23)

where we are considering a Fourier component with angular frequency co and 2D wavevector q,( = (qx, qy) parallel to the surface. The factor exp(iqn«p7) in (3.23), where p = (x,y), is in accordance with Bloch's theorem and the translational symmetry of the system in the xy plane. Because of the surface, there is no such factor for the z direction and sM(q||) depends on z through the layer index n. From (3.22) and (3.23) we obtain the infinite series of coupled equations for sB(qu): {co -

gfiBB0

-SJ-

45J S [1 - y(q (| )]} 5l + SJs2 = 0

(3.24) (n>2)

(3.25)

where y(q,l) = [cos(^a) + cos(^a)]/2

(3.26)

Equations (3.24) and (3.25) can be solved by a variety of techniques to obtain the frequencies co of the various magnon modes, as discussed in the review article by Wolfram and Dewames (1972). However, a simple approach is to note that the bulk magnons correspond to solutions for sn(qj|) made up of two waves (one incident and one reflected): sn(<\\\) = v4(q,|) e x p [ - i ^ ( n - l)a] + B(q,,) exp[i^(n - l)a] (3.27) When this is substituted into (3.25) we obtain the dispersion relation that co = a>B(q), where q = (q\\9qz) is a three-dimensional wavevector and coB(q) = gi*BB0 + 2SJ[3 - 2y(qM) - cos(^2a)] (3.28) This is equivalent to the bulk-magnon dispersion relation in an infinite simple-cubic ferromagnet at T« T c , as can be seen from (3.12) and (3.14). The ratio B/A9 which is a reflection coefficient for the bulk magnon at the surface z = 0, can be determined using (3.24). It is easily shown that B/A is a complex number with modulus unity and a phase angle which depends on the surface parameter J s . The surface magnons may be found from the ansatz (as discussed in §1.3) of seeking attenuated solutions for sw(qM): *»(qil) = C(qi,)exp(-icna) From (3.24) and (3.25) this leads to co = cos(qn), where «s(qn) = G^Bo + 4SJ[1 - y(q,,)] - SJ[A - 1]2/A

(3.29) (3.30)

and

j ^ ^ V J

'

(3.31) 93

Surface magnons is the fractional decrease per layer in the magnon amplitude. Equation (3.30) represents a localised surface mode only if |A| < 1

(3.32)

or equivalently Re(/c) > 0, and this may be satisfied in two ways. One possibility is 0 < A < l , which occurs for JS<J and any q[} / 0 . The spin deviations on adjacent layers are then in phase, and the mode is called an acoustic surface magnon (see Fig. 3A(a) for a semi-classical representation): its frequency is less than that of the corresponding bulk magnon, i.e. cos(q^) < coB( Qz) f° r a n v Qz- The other case is — 1 < A < 0, which implies a phase change of 180° between the spin deviations on adjacent layers (see Fig. 3A(b)). This mode is known as an optic surface magnon, and its frequency is such that tus(qn) > coB(Qn > Qz)- I t c a n be shown from (3.31) that a necessary condition for optic surface magnons to exist in some part of the 2D Brillouin zone for q is J S > | J and that these modes occur only for | q j above some nonzero critical value. In Fig. 3.5 we give some numerical examples of the magnon dispersion relations as predicted by this simple model. We plot frequency co against q II for wavevector q|( in the (100) direction. The bulk magnons appear as a continuum, with lower and upper edges corresponding to qz = 0 and qz= ± n/a respectively. Curves A and B are examples of acoustic and optic surface magnons, respectively, as given by (3.30). Fig. 3.4 Semi-classical representation of surface magnons in a ferromagnet: (a) an acoustic surface magnon; (b) an optic surface magnon. Comparison should be made with Fig. 3.1(6) for a bulk magnon.

0

Q

0

0

O

O

0

0

0

o

(*)

94

Semi-infinite Heisenberg ferromagnets Further discussion of the dispersion relations has been given by various authors, and we mention in particular the review articles by Wolfram and Dewames (1972) and Mills (1984). We briefly quote here some extensions and generalisations to the results derived above for the surface-magnon frequencies. An additional effect is surface anisotropy (also known as pinning), which can often be represented by an extra term in the Hamiltonian (3.4) of the form i

The anisotropy may arise because the crystalline fields at the surface have a lower symmetry than in the bulk or because of surface impurities (e.g. see Levy 1981). In the simplest case the effective anisotropy field may be taken to have a constant value BAS for spins in the surface layer and to be zero otherwise. When the theory is modified to take this into account, one finds that the surface-magnon frequency is still given formally by (3.30) but the definition of A becomes 1 A^exp(-^) = j l - ^SJ^ + 4fl-^V -V(qil)]l ' (3-34) \ JJ '^"'-•J

If BAS < 0 (i.e the anisotropy field is in the opposite direction to the applied magnetic field), the existence condition (3.32) for a surface magnon can be satisfied for certain q{| values event if JS = J, unlike in the previous Fig. 3.5 A plot of magnon frequency (in units of SJ) against \q{]a\ in a semi-infinite Heisenberg ferromagnet for propagation wavevector q| = (^||,0). The bulkmagnon continuum is shown, together with three surface-magnon branches corresponding to: A, Js/J = 0.5, BAS = 0; B, Js/J = 1.8, BAS = 0; C, Js/J = 0.5, = - 0 . 8 . The applied magnetic field is such that gnBB0/SJ = 0.8.

95

Surface magnons

situation of BAS = 0. An example of an acoustic surface magnon in the presence of surface anisotropy has been included in Fig. 3.5 (curve C): note that at qM = 0 it is split off below the lower edge of the bulk-magnon continuum. Calculations of the surface-magnon spectrum for a variety of other models are to be found in the literature. A minor modification, which leaves the results for the magnon frequencies unaltered in the present case, is to take the applied field, the anisotropy field and the average spin alignment to be colinear in a direction parallel to the surface. The assumptions concerning the exchange interactions are, however, more significant in their effect. Another choice would be to take all exchange interactions (including those involving spins at the surface) to have their bulk values, denoted by J for nearest neighbours and j for next-nearest neighbours (e.g. see Wallis et al. 1967). For a simple cubic lattice with a (001) surface and in the absence of pinning, this leads to an acoustic surface magnon for all q |t / 0 provided J > 4 7 > 0 . The more general situation in which there may be nearest and next-nearest exchange interactions, together with several exchange parameters near the surface different from their bulk values, is extremely complicated. Several branches may be predicted for the surface-magnon spectrum, and in certain cases the surface spins may orient in a different direction from those in the bulk. We refer the reader to the article by Wolfram and Dewames (1972) for a comprehensive account. The existence conditions and frequencies of the surface magnons are sensitive to the choice of lattice structure and/or crystallographic orientation of the surface, and it is straightforward to generalise the calculations earlier in this section to fee and bec lattices and to (Oil) surfaces.

33.2 Continuum theory Although the preceding microscopic treatment is relatively straightforward for the semi-infinite Heisenberg model, this is not necessarily the case for more complicated materials and geometries. It is therefore helpful to outline an alternative approach based on using a continuum approximation for the ferromagnetic medium. It is applicable provided the wavelength of any excitation is large compared with the lattice spacing a, i.e. q^a« 1 and qa«l. In fact, the formalism as described here also requires that \KO\« 1, and consequently it is restricted to acoustic-type surface modes. By contrast with §3.3.1, we choose to follow a classical derivation in which the rate of change of the spin angular momentum is equated to the 96

Semi-infinite Heisenberg ferromagnets torque acting upon it: (3.35) where Bf, which represents an instantaneous effective field acting on Si9 has components derived from the Hamiltonian (3.4) by B? = - dJf/dSt

(a = x, y, z)

(3.36)

Equation (3.35) can be regarded as the classical equivalent of the quantum-mechanical equation (3.7). Either form of the equation of motion may be employed and, after the linear-magnon approximation has been made, they give the same magnon dispersion relations. This can be demonstrated by reference to Problem 3.4. The classical equation is sometimes preferred in a macrosopic treatment because, if required, it can readily be augmented by terms that describe a phenomenological damping (e.g. see Kittel 1986). At low temperatures T« Tc we write S; = Sz + ^ exp(-iart)

(3.37)

where z is a unit vector in the z direction and /if denotes the fluctuating component of Sf at frequency a>. A similar separation of B, into a static part and a fluctuating part can be made: B; = (g/JLBB0 + S £ JtAt + b{ exp(-iatf)

(3.38)

and the connection between b, and /if is provided by j

On substituting (3.37) and (3.38) into (3.35) and making the usual linearmagnon approximation and neglecting terms that are of second order in fit (because |/i f |« S) we obtain -icofit = zx\sb zx\sbt-t-

(gfi (gfiBBBB0 0 + + SS X X Jijjtit Jijjtit

(3.40)

Next we go over to a continuum representation in which /if and bt at magnetic site i are replaced by position-dependent functions /i(r) and b(r). Formally this can be achieved by writing

Mr) = Z ^ ( r - r f )

(3.41)

and averaging on the microscopic scale (over distances of order a few times a). We assume as before (see Fig. 3.2) a simple cubic ferromagnet with exchange coupling J to the six nearest neighbours, except in the surface layer where each spin has only five neighbours and the exchange may be different. On using (3.39) and making a Taylor-series 97

Surface magnons expansion appropriate to the continuum representation we obtain for z < 0 (e.g. see Phillips and Rosenberg 1966) b(r) = 6J/t(r) + a2J2V2/i(r) + . . .

(3.42)

The above result is proved in a way that is analogous to the treatment in §2.1 of vibrational modes in the ID monatomic lattice in the continuum limit (c.f. equation (2.6)). With (3.40) it leads to icofi(r) = z x (gfiBB0 - SJa2V2)fi(r)

(3.43)

which must be satisfied for all z < 0. Similarly, by taking the continuum limit for a spin in the surface layer, we obtain the boundary condition to be satisfied at z = 0. After some manipulation this becomes [dMr)/dz-<^f(r)L = o = 0

(3.44)

where t; depends on the surface characteristics:

and we have introduced a surface anisotropy field J5AS as in §3.3.1. The boundary condition (3.44) has the form introduced by Rado and Weertman (1959). Often it is postulated phenomenologically, but here it has been derived from the continuum limit. It can be considered as an analogue of the boundary condition (2.9) for the ID monatomic lattice. We now use the property of translational invariance parallel to the surface to write ji(r) = /i(z)exp(iq r p)

(3.46)

and by seeking travelling-wave and attenuated solutions of (3.43) and (3.44) we deduce the bulk-magnon and surface-magnon frequencies, respectively. The results are + SJa2(q2\ + q2)

(3.47)

2

(3.48)

SJa (ql - ?)

These are, as expected, just the long-wavelength (small |q| and |qj) approximations of the previous results (3.28) and (3.30). An attenuated solution of (3.43) is possible only if £ > 0, and this becomes the existence condition for a surface magnon. In fact, by using (3.45) and approximating (3.31) for the case of aq^ « 1, we can identify £, as being equivalent to K of the microscopic theory (see Problem 3.5). The frequency cos(qu) in (3.48) corresponds to an acoustic surface-magnon branch below the bulkmagnon continuum. Because the present analysis has been carried out for long wavelengths, no optic surface magnons are predicted. In summary, the continuum approach can be a useful simplification when we are concerned only with long-wavelength excitations. For 98

Semi-infinite Heisenberg ferromagnets example, the low-temperature contributions of the bulk magnons and acoustic surface magnons to the magnetisation are dominated by the long-wavelength region (where the excitations have lower frequencies), and similarly it is these excitations that participate in a one-magnon light-scattering process or in magnetic resonance. 3.3.3 Green functions and applications So far we have concentrated on evaluating the frequencies of the bulk and surface magnons, but more generally we may wish to know the intensities associated with the excitations. This information can be obtained from spin-dependent Green functions of the form <<S + ; 5">>w. These quantities contain a full description of the dynamic behaviour of the magnetic system, and can be employed to evaluate the thermodynamic properties of the magnons or the intensities of the magnons as measured, for example, by light scattering. Explicit forms of the Green functions for a semi-infinite Heisenberg ferromagnet have been derived by Cottam (1976a) using a microscopic theory and by Moul and Cottam (1979) using a long-wavelength continuum approach. We shall discuss only the latter case, in which linear-response theory can be employed in an analogous manner to the treatment of acoustic phonons in Chapter 2. Briefly, the calculation may be performed by taking an additional term, representing an external driving field bex(r), to be included on the right-hand side of (3.38). This produces an additional term in the linearised equation of motion (3.40), and by calculating the response of |i to bex we may deduce the required Green functions. This is in accordance with the general formalism described in the Appendix. The final result can be expressed as

(3 49)

' where A is the (macroscopically large) surface area of the ferromagnet, Q is a complex quantity which satisfies
QIIBB0

+ SJa\q\

+ Q2)

(3.50)

with Im(<2) ^ 0, and the other notation is as before. The bulk magnons correspond to Q taking the value of a real wavevector component qz in the z direction, and it can be seen that (3.50) then reproduces the bulk-magnon dispersion relation (3.47). The vanishing of the denominator in the second term of (3.49) occurs for <2 = i£, provided £ > 0 , and the 99

Surface magnons substitution of this into (3.50) gives the surface-magnon dispersion relation (3.48). The first term in (3.49) depends on the position labels z and z' only through \z — z'\ and may be identified as the Green function for the infinite crystal, while the second term, which depends on (z + z'), contains the effects due to the surface. The occurrence of two exponential terms with these position dependences is a typical feature of Green functions for excitations in semi-infinite media, and is seen also in (2.97) for a semi-infinite elastic medium. It is a straightforward application of the fluctuation-dissipation theorem (see the Appendix) to deduce from (3.49) the correlation function <5~(zr)5+(z)>co. Taking the case of z' = z we now evaluate F(co,z) = <|S + (z)|2>tt

(3.51)

as a function of co and z for a fixed value of the wavevector q M. From (3.47) the bulk magnons correspond to co > Q o , where &o = gHBBo + SJa2q\

(3.52)

and in this frequency range we obtain using (A.30) nAJaqz 1 2^

2 )

U&

&

( ? z ) + qzi

(^2)]|

(3.53)

This is nonzero at the surface of the ferromagnet (z = 0) where it takes the value

and away from the surface it oscillates as a function of z. In (3.53) and (3.54) CD is given by (3.50) with Q real and equal to qz, and n(co) is the Bose-Einstein factor defined in (1.50). There is also a surface-magnon contribution to F(co, z), occurring at co < Q o . It corresponds to the pole at Q = i{ in (3.49), and from (A.28) and (A.30) we obtain F(co, z) = {4[n(cos) + 1M/S/1} exp(2£z)S(co - cos)

(3.55)

The strength of this delta-function contribution at co = cos decreases with distance — z from the surface, as expected. The behaviour of F(co, 0) as a function of co, as given by (3.54) and (3.55), is represented schematically in Fig. 3.6. The integrated intensities / B (z) and / s (z), associated with the bulk and surface magnons respectively, can be obtained by integrating (3.53) and (3.55) over the appropriate ranges of co. For the surface-mode contribution this gives / s (z) as simply the strength of the delta-function in (3.55), but the evaluation of the bulk-mode contribution is much more 100

Semi-infinite Heisenberg ferromagnets complicated. There is a simplification, however, for temperatures such that fcBT»fi 0. In this case the total intensity, which is equal to [/B(z) + / S (z)], can be found using (3.49) and (A.36), and we eventually deduce for z = 0 that /s(0)//B(0) = 2{Sjyi2a?;\Ciy2 + (SJ)1/2a£]/cos

(3.56)

This ratio increases with increasing £, which corresponds to the surface magnon becoming more localised near z = 0 and lower in frequency. The Green function <<S+(z); S~(z')yy(O in (3.49) may also be employed in a number of other physical applications, and we give three examples here. One application is to calculate the average spin deviation [5 — <Sz(z)>] in the semi-infinite sample. This quantity will be nonzero due to thermal effects of the bulk and surface magnons even at T« T c. The connection with the Green-function formalism can be achieved by using the commutation property S+S~ -S~S+

=2SZ

(3.57)

for operators at the same site, and taking for simplicity S = \ we also have the identity S+S~ +S~S+ = 1

(3.58)

which follows from the properties of Pauli spin matrices. It is easy to show that the average spin deviation in this case is SS(z) E E \ - <S*(z)> = <S"(z)S + (z)>

(3.59)

The correlation function on the right-hand side can be obtained from the Green function (3.49) by using the fluctuation-dissipation theorem and integrating over co and q(|. Details of such a calculation are given by Cottam (1976a) and Cottam and Maradudin (1984). A partial cancellation Fig. 3.6 Schematic plot of the correlation function F(co, 0) for a semi-infinite Heisenberg ferromagnetic as a function of frequency [after Cottam and Maradudin 1984]. /fy>, 0)

101

Surface magnons takes place between bulk-magnon and surface-magnon contributions, leading to the overall approximate result:

SS(z) = Q M I d 3 q[l + cos(2^zz)]n[o;B(q)]

(3.60)

which was first derived by Mills and Maradudin (1967) using a different method. It follows from (3.60) that for small enough \z\ (such that cos(2gzz) ~ 1 throughout the dominant region of integration) then SS(z) is twice the corresponding value in an infinite crystal. The condition for this is |z| « z 0 , where z0 = aTc/T when Bo = BA = 0, while for \z\ » z 0 it can be shown that SS(z) is equal to the infinite-crystal result plus a correction term of order aT/|z|T c . Another application of the Green function is to light scattering, where the calculation closely parallels that in Chapter 2 for Brillouin scattering from acoustic phonons. It was established in §1.4.2 that the light-scattering cross section is related to a correlation function involving components of the scattered electric field. This can be related to a polarisation in the scattering medium, which can then be expanded in terms of the appropriate variables describing the excitation. For light scattering from vibrational modes in §2.5 this led to a description in terms of atomic displacement derivatives. For light scattering from magnons the polarisation must be expanded in powers of the spin operators S or, equivalently, the fluctuating magnetisation (e.g. see Hayes and Loudon 1978). In this way, and with the aid of the fluctuation-dissipation theorem, one may relate the cross section d2<j/dQda>s for scattering from the magnons in a Heisenberg ferromagnet to <<S + (z); S~(zr)yym in lowest order. Details of the formalism and results are to be found elsewhere (see Cottam 1976b, 1978b; Moul and Cottam 1979). It is found that the contribution to the cross section for Stokes scattering from the bulk magnons in an absorptive medium has the form d2a \ = F[n(a>) dQ dcos/bulk qz

(K-qz)

(3.61)

where co is the frequency shift of the scattered light, K is related to the wavevector components of the incident and scattered light perpendicular to the surface (defined as in (2.121)), and

tf)

(3.62)

The overall factor F depends on the scattering geometry and on the polarisations of the incident and scattered light; it is effectively a constant across a given spectrum. The predicted bulk-magnon spectrum consists of a continuum at co>Q o as illustrated in Fig. 3.7. The main peak 102

Semi-infinite Heisenberg ferromagnets

corresponds approximately to the frequency at which qz = Re(X) and its width is proportional to Im(K). The features are qualitatively similar to the case of light scattering from acoustic phonons considered in §2.5 and §2.6. The quantity c/)(qz), defined in (3.62), can be regarded as a complex reflection coefficient for bulk magnons at the surface. If £ > 0 there is also a contribution to d2 = a>s (
Fig. 3.7 The calculated lineshape for the cross section versus frequency difference (co — Qo) for light scattering at normal incidence from bulk magnons in an optically absorptive ferromagnet. The case of Stokes scattering in a low-temperature limit has been assumed, together with lm{K)/RQ(K) = 0.25 for the optical parameters and large pinning.

= -1

103

Surface magnons agreement with a result due to Mills and Beal-Monod (1974b), while the surface-magnon contribution is proportional to T2 (Cottam 1979). 3.4 Heisenberg ferromagnetic films The semi-infinite limit treated in §3.3 is a satisfactory approximation if the thickness of the ferromagnet is large compared with the surface mode attenuation length or any other characteristic length (e.g. the optical penetration depth in a light-scattering experiment). If this condition is not fulfilled, then it becomes necessary to treat the ferromagnet as a film (or slab) of finite thickness and to take account explicitly of the two surfaces. The same methods of calculation can be employed as before, but the imposition of two sets of boundary conditions makes the results algebraically more complicated. We shall show that the main effects of finite thickness on the magnons in Heisenberg ferromagnets may typically include (a) a splitting of the surface-magnon branch into two parts for appropriate film thicknesses and (b) an effective 'quantisation' of the bulk-magnon frequencies. The film thicknesses of interest may vary considerably depending on the particular ferromagnetic material and on the wavevector regime, but typically they might lie within the range of a few nanometres up to a few micrometres. Review accounts of bulk and surface magnons in Heisenberg thin films have been given by Puszkarskii (1979) and Levy (1981). We first consider a microscopic approach, generalising directly the calculation given in §3.3.1 for a semi-infinite ferromagnet. It is now assumed that there are N layers and that the nearest-neighbour exchange parameters Js and J's, acting between spins in the upper and lower surfaces respectively, may be different from one another and from the bulk exchange J (see Fig. 3.8). In analysing the magnon frequencies we again restrict attention to the low temperature region T«TC so that (Sz}n~S in each layer. Instead of obtaining the infinite series of coupled equations represented by (3.24) and (3.25), we now have the finite set: {co - gfiBB0 -SJ-

4SJ S [1 - yfaii)]^ + SJs2 = 0

(3.64)

{co - gfiBB0 - 2SJ - 4SJ[1 - y(q,,)]}sn + SJ(s n _ x + sn+ x) = 0 ( l < w < J V ) (3.65) {co - gfiBB0 -SJ-

4SJ'sl\ - y(q,, )]}sN + SJN + 1

(3.66)

The frequencies of the magnons can be deduced as before. For the bulk modes we seek travelling-wave solutions for the coefficients sn as in (3.27). It is easily verified from (3.65) that the dispersion relation (3.28) is still applicable in the thin-film case. However, the boundary conditions (3.64) and (3.66) each lead to restrictions on the ratio B/A of coefficients in 104

Heisenberg ferromagnetic films

(3.27) and a consistency condition then follows that tan(0. The term proportional to C'(qu) was absent in (3.29) for the semi-infinite limit, because the corresponding exponential term would increase in magnitude without bound as n -»oo. In the present case n is finite, and so both terms occur. On substituting (3.68) into (3.64)-(3.66), we eventually obtain co = cos(q^) where + 45J[1 - y(q,,)] + SJ[2 - exp(fca) - exp(-ica)] (3.69) Fig. 3.8 Model for the finite thickness ferromagnetic film in the case of simple cubic structure and (001) surfaces. The nearest-neighbour exchange constants Js, J's and J are indicated.

n= 1—

J

j

J n= N —j

J

,

\

J

J s

'

i,

J

1

J

J J's

1i

J J s

'

i

105

Surface magnons and K satisfies tanh(?cL) = [A — exp(fca)][A' — exp(faj)] — [A — exp( — faz)][A' — exp( — KO)~] (3.70) [A — exp(/o*)][A' — exp(/aj)] + [A — exp( —jaz)][A' — exp( —/ca)] In the limit of large film thickness (L -• oo) we have tanh(zcL) -• 1, and (3.70) has two solutions for K corresponding to exp(-/ca) = A and exp( — Ka) = A', provided |A| < 1 and |A'| < 1. These correspond to surface magnons of the type discussed in §3.3, one localised at the upper surface (z = 0) and one at the lower surface (z = — L). More generally it can be shown that there are either zero, one or two surface-magnon solutions, depending on the values of L, A and A' (see Puszkarskii 1970, 1979; Cottam and Kontos 1980). For a more detailed discussion of the bulk and surface magnons in a thin film we restrict attention to the case of long wavelengths, since the results simplify to some extent. The dispersion relations are obtained either by taking \q\\a\« 1, \q za\« 1 and |foz|« 1 in (3.67), (3.69) and (3.70), and approximating accordingly, or directly from a continuum approach as in §3.3.2. For the bulk magnons the frequencies are given by (3.47) where qz must now satisfy tan(4 z L) = qz(i + £')/(«' - ql) (3.71) The surface parameter £ referring to z = 0 has been defined in (3.45), and £' for the surface at z = — L is given by a similar expression. In the special case of £ = £' = 0 the above condition becomes simply tan(g z L) = 0, and hence qz = mn/L

with m = 0,1,2, etc.

(3.72)

However, if { and £' are sufficiently large that \£€'\ » ql for the wavevectors of interest, then (3.71) approximates to tan(qzL) = qzL0

(3.73)

where L o is a characteristic length defined by (3.74)

Provided £ ~ £', the above inequality implies \qzL0\ « 1 and so tan(g z L) ^ 0 as before. Thus for large pinning (large £ and £,') we again obtain the approximate quantised solutions for qz as in (3.72). Each of the above limiting cases corresponds to fitting an integral number of half-wavelengths into the film thickness L, with antinodes at the surfaces in the zero-pinning case and nodes at the surface in the large-pinning case, as illustrated in Fig. 3.9. For general magnitudes of £ and £' the solutions of (3.71) can 106

Heisenberg ferromagnetic films be investigated graphically by plotting each side of the equation as a function of qz and looking for the intersections. An example of this is given in Fig. 3.10 where we have taken £ and £' to be both positive. The quantisation of the bulk magnons has been discussed quite extensively in the literature, including cases where \qza\ is not necessarily small, and a good review account is given by Puszkarskii (1979). For the surface magnons the long-wavelength limits of (3.69) and (3.70) yield ^s(qn) = g/JLBBo + SJa2(qj - K2)

(3.75)

and tanh(KL) = K(S + £')/(«' + K2)

(3.76)"

where the physical solutions correspond to K> 0. Clearly, in the limit of Fig. 3.9 The quantisation of the bulk-magnon wavevectors in a Heisenberg ferromagnetic film for (a) zero pinning and (b) large pinning.

(a)

(b)

Fig. 3.10 Schematic form of the solution for qz of equation (3.71) by a graphical approach, taking £ and £' both positive. The full and broken curves represent the left- and right-hand sides of (3.71) respectively.

107

Surface magnons L -» oo, we have tanh(zcL) -• 1 and (3.76) may be rearranged as a quadratic equation in K with the two solutions K = £ and K — £'. Hence we recover the dispersion relation (3.48) of the semi-infinite case for a surface mode localised at z = 0, whilst the corresponding result involving £' refers to the surface mode at z = — L. The existence conditions for these modes are £, > 0 and £' > 0. In the case of film thickness L large but finite (such that KL» 1) the approximate solutions of (3.76) for K may be obtained by a simple iterative procedure. The results are (see Moul and Cottam 1983)

K ~ ^ 1 + 2 ( | ^ | ) exp(-2£L)l (-2£L)l

(3.77)

K * A(|^|) AI -

(3.78)

and

J

in the case of an asymmetric film (£ ^ £'), and K*ZLl±2exp(-ZL)]

(3.79)

if £, = £'. Equations (3.77)-(3.79) hold provided L » L o , where L o is defined in (3.74). For general values of L the nature and existence of the surface magnons may be investigated by graphical methods (see Puszkarskii 1972; Moul and Cottam (1983). For example, in the special case of symmetric boundary conditions (^ = £') we note that (3.76) may be rearranged and factorised into two parts to give either K = £ tanh(/cL/2)

(3.80)

K = £ coth(/cL/2)

(3.81)

or The graphical solution of these equations then leads to the following conclusions. First, if ^ 0 there are no surface-magnon solutions. Second, if £ > 0 there are two solutions (one from each of (3.80) and (3.81)) if L > 2/f and only one solution (from (3.81)) if L ^ 2/£. Note that 2/£ is just the value of L o for the case of £ = {'. In Fig. 3.11a numerical example is given for the variation of the surface-magnon frequencies with film thickness. For large L the two branches converge to the semi-infinite result. Calculations of Green functions of the form <<S + (z); S'iz')}}^ for a finite-thickness film enable the spectral intensities of the magnons to be discussed, just as in the semi-infinite limit. This has been done using a microscopic approach by Cottam and Kontos (1980) and using a longwavelength continuum theory by Moul and Cottam (1983). The results are algebraically much more complicated than for the preceding semiinfinite limit (in §3.3.3), and the reader is referred to the above papers 108

Heisenberg antiferromagnets

for details. The spectral function F(co, z), defined as in (3.51), now takes a different form from Fig. 3.6. There may be up to two surface-magnon peaks (occurring at a>< Qo), and the bulk-magnon spectrum no longer forms a smooth continuum but consists of a series of sharp resonance peaks corresponding to the 'quantised' bulk magnons. The contribution of the bulk magnons to F(a>, z) at z = 0 is illustrated in Fig. 3.12 for three different film thicknesses. The Green-function results have also been applied to calculate the spectra for light scattering and spin wave resonance. We discuss these results in §3.6. 3.5 Heisenberg antiferromagnets

In Heisenberg antiferromagnets the sign of the exchange interaction is such that an antiparallel ordering of neighbouring spins is favoured at temperatures below the Neel temperature TN. The system can be regarded as two interpenetrating sublattices corresponding to each spin type (loosely referred to as 'spin-up' and 'spin-down'). There will be no overall magnetisation (in zero applied magnetic field) because the contributions from each sublattice cancel one another, but there will be long-range magnetic ordering below TN. The Heisenberg Hamiltonian may be expressed as (3.82) S; - gfiBBA £ Sf + gfiBBA £ S) JT = Fig. 3.11 Calculated dependence of the acoustic surface-magnon frequencies (labelled a>sl and on COS2) thickness L for a symmetric ferromagnetic film. The assumed parameters 2 KTm" are O 0 = 2 cm" 1 , SJa 2 = 3.7 x 10" 15 c m ^ m 2 and

1.8 -

200

100 L(nm)

109

Surface magnons

where i and j refer to sites on sublattice 1 (spin-up) and sublattice 2 (spin-down) respectively, and Jtj is the corresponding exchange interaction. The last two terms in (3.82) describe the effect of an anisotropy field £ A , taken to have the same magnitude at each spin site but to be oppositely directed on each sublattice. The role of the anisotropy is important in antiferromagnets (particularly those of non-cubic structure) and its origin is usually due to crystal field effects (e.g. see Wagner 1972). In (3.82) we are ignoring, for the sake of simplicity, the effect of exchange interactions between sites on the same sublattice and the effect of an applied field Bo. Before discussing surface effects in antiferromagnets, it is instructive to derive the bulk-magnon dispersion relation in an infinite medium. The equation of motion for a S+ operator can be constructed as in §3.2, except that the Hamiltonian (3.82) must now be employed and there will be an equation of motion for each sublattice. After using the RPA decoupling,

Fig. 3.12 Calculated frequency dependence of the bulk-magnon contribution to F(co,0) for Heisenberg ferromagnetic films of different thicknesses L: (a) 110 nm; {b) 190 nm; and (c) 1000 nm. The assumed parameters are SJa2 = 3.1 x 10~15< { = {' = 4 x 107 m" l [after Moul and Cottam 1983].

no

Heisenberg antiferromagnets as in (3.9), we have (3.83) jt

+ 2q

- <S*>2J(q)S1+q

(3.84)

where wavevector Fourier transforms have been defined by analogy with (3.10) with the introduction of an additional subscript (1 or 2) as a sublattice label. From the symmetry between the sublattices, we have the property that <SZ>X = — <SZ>2 = <SZ>. The excitation frequencies may now be found from the coupled equations (3.83) and (3.84) by seeking solutions with a time dependence exp( — icot). This leads t o c o = ±o>B(q) where *>n(q) = {[<WiB*A + <Sz>J(0)T - <S z > 2 J 2 (q)} 1/2 For wavevector q = 0 the result can be written as

(3.85)

coB(0) = LcoA(2coE + o>A)]1/2

(3.86) an

Z

where we introduce the convenient notation coA = #/%BA d coE = <S > J(0) for the anisotropy and exchange frequencies respectively. Equation (3.86) gives the well-known antiferromagnetic resonance frequency; for antiferromagnets with coA«coE it approximates to (2coAcoE)1/2. We next consider surface and bulk magnons in semi-infinite Heisenberg antiferromagnets. The calculations can be carried out in a similar manner to those of §3.3 for semi-infinite ferromagnets and can be generalised, if required, to finite-thickness films. We take the antiferromagnet to occupy the half-space z < 0 and the directions of average spin alignment to be parallel and antiparallel to the z axis for sublattices 1 and 2 respectively. For simplicity we assume that the exchange couples only nearestneighbour sites on opposite sublattices and that all exchange interactions are the same as in the bulk. Likewise we assume that the anisotropy is the same at the surface as in the bulk. These assumptions regarding the exchange and anisotropy may readily be generalised (see, for example, the review by Wolfram and Dewames 1972). The results for the surface magnons in antiferromagnets depend sensitively on the lattice structure, and we shall consider two cases here: body-centred tetragonal antiferromagnets and simple cubic antiferromagnets. 3.5.1 Body-centred tetragonal antiferromagnets Antiferromagnets such as FeF 2 and MnF 2 have a rutile crystal structure in which the magnetic ions lie on a body-centred tetragonal lattice. A view of a semi-infinite sample with a (001) surface is shown schematically in Fig. 3.13. Note that each layer (labelled with index n) 111

Surface magnons parallel to the surface contains magnetic sites of one sublattice type only and the nearest-neighbour exchange coupling is to sites in adjacent layers only. The surface layer in Fig. 3.13 is occupied by spins of sublattice 1 (spins 'up'), and the surface therefore has the effect of removing the equivalence between the two sublattices that would exist in an infinite crystal. The linearised equations of motion satisfied by the spin operators in each layer may be found just as in the ferromagnetic case of §3.3.1, except that we now employ the Hamiltonian (3.82). The infinite set of coupled equations for the spin amplitudes sn(q ^), defined as in (3.23), is obtained as (co -

- 4JS)Sl

(co -

- SJS)s2m

(3.87)

- 4JSy o (q,,)s 2 = 0 +1

=0

- 4JSy o (qn )(s2m

(3.88) (co +

SJS)s2m + 4J5y o (q (| )(s2m.1

+s2m+1)

=

(3.89)

Here we have denoted (3.90) Fig. 3.13 The (001) surface of a semi-infinite antiferromagnetic with body-centred tetragonal structure: (a) perspective view; (b) side view. (a)

t

i

n= 1

\ /z = 3

i-—

(b)

112

n=4

(001) z

Heisenberg antiferromagnets and we have assumed for simplicity that the spin averages <SZ> may be approximated by the values S and —S for sites on sublattices 1 and 2, respectively, at low temperatures T«T N. The above finite difference equations may be solved to obtain the dispersion relations of bulk and surface magnons, as we now indicate. Following the same approach as for the semi-infinite ferromagnet, we investigate the bulk magnons by seeking travelling-wave solutions for sn of the form (3.27), but with different coefficients for the two sublattices. This leads to co = ± coB(q) where o>B(q) = {[0A*B*A + 8JS] 2 - [8JS 7 o ( q | | ) cos(k 2 c)] 2 } 1 / 2 (3.91) This is equivalent to the bulk-magnon dispersion relation in an infinite body-centred tetragonal antiferromagnet at T«T N, as can be deduced from (3.85). For wavevector q = 0 the result can again be written as in (3.86) provided we now define coA = gfiBBA and coE = SJS. The surface magnons correspond to attenuated solutions in which sn takes a form analogous to (3.29). It can be shown from (3.87)-(3.89) that the attenuation factor K of any surface magnon must satisfy exp( — KC) = {2coA + coE — 2o))/co

E

(3.92)

where c is the lattice parameter corresponding to the z direction in the body-centred tetragonal lattice. The only solution for the surface-magnon frequency that satisfies the localisation condition Re(jc) > 0 corresponds to co = cos(q||), where 1/2

(3.93)

This is the surface-magnon branch first predicted by Mills and Saslow (1968). It is an acoustic mode occurring at frequencies below those of the corresponding bulk magnons with wavevectors q = (q,,, qz), as illustrated by the numerical example in Fig. 3.14. At wavevector q(| = 0 we have 7o(q||) = 1 a n d the expression (3.93) reduces to cos(0) = [coA(coE + co A )] 1 / 2

(3.94)

Since taA«co E in many rutile-structure antiferromagnets (e.g. coAJcoE is approximately 0.015 for M n F 2 ) , it follows in such cases that the ratio cos(0)/coB(0) is close to 1/^/2. This difference between cos(0) a n d coE(0) should be advantageous for experimental studies of surface modes by techniques that involves small-wavevector excitations (such as antiferromagnetic resonance and inelastic light scattering). The effects of allowing the anisotropy field a n d the exchange parameter to have different values at the surface compared with the bulk have been calculated by Dewames a n d Wolfram (1969). They concluded that 113

Surface magnons (Ds(0)/coB(0) is still close to 1/^/2 when coA«co E, even when the surface parameters are changed significantly from the bulk values. However, for nonzero q^ the surface parameters have a larger effect leading to the prediction, in certain cases, of two surface-magnon branches that may occur either below or above the bulk-magnon continuum. 3.5.2 Simple cubic antiferromagnets Antiferromagnets such as KNiF 3 and RbMnF3 have a crystal structure in which the magnetic ions lie on a simple cubic lattice, as sketched in Fig. 3.15 for a semi-infinite sample with a (001) surface. Note in this case that each layer (labelled by index n as before) contains equal numbers of magnetic sites from both sublattices and that the nearestneighbour exchange coupling is to sites in the same layer as well as to adjacent layers. Hence, by contrast with body-centred tetragonal antiferromagnets, the occurrence of a (001) surface in a simple cubic antiferromagnet does not remove the equivalence between the two sublattices. It is this property that causes differences between the surface-magnon characteristics in the two structures. Fig. 3.14 A plot of magnon frequency against |qp| in a semi-infinite Heisenberg antiferromagnet with body-centred tetragonal structure, taking q| = (g||,0) and (oA/coE = 0.2. The bulk-magnon continuum and an acoustic surface-magnon branch are shown. 1.4 -i

1.4

114

Heisenberg antiferromagnets An infinite set of coupled equations satisfied by the spin amplitudes sn(q{l) at T « TN can be derived by analogy with (3.87)-(3-89). The results are (see Wolfram and Dewames 1969, 1972) (co - gfiBBA - 5JS)s[1] - 4JSy(q{l)s[2) - JSs^ = 0 2)

(3.95)

1)

(co + gnBBA + 5JS)s[ + 4JSy(q{] )s[ + JSs^ = 0 (co -

1

gfiBBA

2)

(3.96)

2

- 6JS)s< > - 4JSy(q „ )s|, - JS(s[ l 1+s^1) =0 (n>2) (3.97)

(co + gfiBBA + 6JS)s™ + 4 J5 7 ( q | 1 )s^ + J 5 ( s ^ , + s ^ i) = 0 ( H ^ 2 ) (3.98) The superscripts 1 and 2 in the spin wave amplitudes refer to the sublattice type, and y(q\\) is the geometrical factor defined in (3.26). The bulk-magnon dispersion relation can easily be found from (3.95)-(3.98) by seeking travelling-wave solutions as before. The result is co = ±coB(q) with coB(q) = {lgfiBBA + 6SJ] 2 - 4S 2 J 2 [2y( q | | ) + c o s ^ a ) ] 2 } 1 ' 2

(3.99)

As expected, this is equivalent to the bulk-magnon dispersion relation Fig. 3.15 The (001) surface of a semi-infinite antiferromagnet with simple cubic structure: (a) perspective view; (b) side view. (a)

A

V /

A /

n= 1

7

(001) z

n= 2

A A A A

1

i

J J

J J

J

J

J

\ J

J

J

J

1\

J

i L

-

n= 2

J J

x

J

\

—

n = 3

115

Surface magnons (3.85) for an infinite antiferromagnet, taking the special case of simple cubic structure and T«TN. The bulk-magnon frequency at q = 0 can again be expressed formally as (3.86) provided we now define the exchange frequency by coE = 65J. The surface-magnon solutions of (3.95)-(3.98) are more complicated to obtain than for the examples discussed earlier in this chapter. This is because, except in certain limiting cases, the surface magnons in simple cubic antiferromagnets cannot be characterised by a single attenuation factor (Wolfram and Dewames 1969, 1972). Hence the simple ansatz of (3.29) for sw(qn) will not be applicable in general, and this is a consequence of the fact that each layer n in the semi-infinite simple cubic structure contains sites from both sublattices. For the case considered here of the surface exchange and anisotropy parameters being the same as in the bulk, Wolfram and Dewames (1969) calculate a single surface-magnon branch of the acoustic type with frequency cos(qn) given by E + q>A)[4pj(l - y2(q|,)) + 12coEcoA

jy/2

At q = 0 and for coA « coE this approximates to cos(0)^(2coAcoE + ico 2 ) 1/2

(3.101)

By comparing this with (3.86) we see that o>s(0)/a>B(0) is approximately one when coA «coE (compared with the value 1/^/2 obtained in the case of a body-centred tetragonal antiferromagnet). Generalisations of (3.100) and (3.101) to situations with surface anisotropy fields and varying surface exchange parameters have been made by Wolfram and Dewames (1969). The results become very complicated and include the possibilities of various truncated acoustic and optic surface magnon branches. The effect of an applied magnetic field Bo on the surface magnons can also be studied (e.g. see Mills 1968). When Bo is applied along the z direction in an infinite antiferromagnet, the degeneracy in the bulkmagnon frequencies is lifted. Instead of <x> = + coB(q) when Bo = 0, where coB(q) is given by (3.85), we find co±=gfiBB0±coB(q)

(3.102)

when Bo ^ 0 and T« TN. As Bo is increased, co~ -• 0 for some value of wavevector q (normally at q = 0) and this represents an instability in the system. The spins reorient from alignment parallel and antiparallel to the z axis to new directions almost at right angles to Bo. This is the so-called spin-flop phase transition which has been observed in a number of antiferromagnets. Because the zero-wavevector surface magnons may have a 116

Experimental studies frequency lower than that of the q = 0 bulk magnons, the spin-flop transition may occur at the surface of a semi-infinite antiferromagnet at a smaller applied field Bo than for the bulk (Mills 1968). Apart from studies of the magnon dispersion relations, there have been some calculations of thermodynamic properties. In particular, Mills and Saslow (1968) have evaluated the sublattice magnetisation and the specific heat for a semi-infinite antiferromagnet with body-centred tetragonal structure. It is the long-wavelength magnons that influence most strongly the thermodynamic properties at T«TN and, as shown earlier in this section, the surface magnons in antiferromagnets of this structure are well split off in frequency below the bulk magnons. This should lead to an enhancement in the surface-magnon contribution to the thermodynamic properties, by contrast with the situation for ferromagnets or simple cubic antiferromagnets. This is confirmed in calculations by Mills and Saslow (1968), and we refer the reader to this paper for details. Further theoretical work for semi-infinite antiferromagnets has been concerned with evaluating the spin-dependent Green functions (see Cottam 1978a; Dos Santos and Cottam 1980). These results were applied to predict the lineshapes and integrated intensities for Raman scattering from the surface and bulk magnons. 3.6 Experimental studies

Although there is now an extensive theoretical literature for magnons at the surfaces of Heisenberg ferromagnets or antiferromagnets, as we have seen from the preceding sections of this chapter, the experimental data for exchange-dominated surface magnons are comparatively sparse to date. This situation is likely to change in the near future, particularly with improvements in techniques for spin-polarised particle scattering. The clearest direct observation of an exchange-dominated surface magnon has come from spin wave resonance (SWR) in measurements by Yu et al. (1975) for films of yttrium iron garnet (YIG) deposited on a nonmagnetic substrate. We have already referred to SWR in §1.4. To elaborate, rf radiation is applied to the magnetic sample such that typically the rf magnetic-field component is perpendicular to the spin-ordering direction. Resonance absorption occurs when the frequency of the rf field matches that of a magnon excitation. In thin-film samples the technique can be used to excite selectively magnons with small but nonzero wavevector q. For bulk magnons this corresponds to having a standing-wave pattern of magnons across the thickness of the film, as illustrated schematically in Fig. 3.9. If the film thickness is equal to an odd 111

Surface magnons

number of half-wavelengths of a q ^ 0 bulk magnon, then there will be a net magnetic moment to couple with the applied rffield,and resonance absorption may be observed. The technique has been applied to bulk magnons following early work by Jarrett and Waring (1958) and Seavey and Tannenwald (1958). The technique should also be applicable to surface magnons, and in the experiments of Yu et al. (1975) the observed surface magnon was found to be bound to the YIG/substrate interface and split off in frequency below the bulk-magnon region. The latter behaviour is as expected from theory if YIG behaves similarly to a ferromagnetic film (see §3.4) at low temperatures T« T c. A full theoretical analysis has been given by Harada et al. (1977). The light-scattering experiments to date have not provided evidence of exchange-dominated surface magnons. This is because, in most magnetic materials, the long wavelengths probed by light scattering (see §1.4) correspond to a regime where the magnetic dipole-dipole interactions are either dominant or comparable with exchange effects. In such cases the surface magnons have a different character, as we discuss in Chapter 4. However, light scattering experiments have demonstrated the role of exchange for the standing bulk magnons in thin ferromagnetic films, and some examples are given in §4.6. Two techniques that have been successfully used to study the surface magnetisation and other magnetic structure are spin-polarised electron photoemission and spin-polarised LEED. Reviews have been given, for example, by Alvarado et al. (1986) and Pierce (1986). In spin-polarised photoemission, the incident electromagnetic radiation (e.g. from a synchrotron source) produces photoelectrons whose dependence on angle, energy and spin polarisation can be measured to deduce the surface magnetic structure. An application has been to europium chalcogenides (such as the ferromagnets EuO and EuS) to show that in certain circumstances there may be surface spin instabilities corresponding to surface reconstruction in the outermost one or two atomic layers (see Campagna et al. 1974). Spin-polarised LEED is similar to the LEED experiments described in §1.4.1 but involves using a spin-polarised incident electron beam and/or measuring the polarisation of the diffracted beam. In this way Celotta et al. (1979) were able to measure the temperature dependence of the surface magnetisation in a Ni crystal with a (110) surface. A more thorough investigation has been carried out recently for the ferromagnetic glass Ni0 4 Fe 0 4 B 0 2 at low temperatures 7 « T c, and some data are shown in Fig. 3.16. The reduced magnetisation M(T)/M(0) is found to decrease according to the power law M(T)/M(0)= 1 - bT3/2 with increasing 118

Magnons in Ising ferromagnets temperature. The coefficient b is greater (by a factor of about 3) for the surface layer compared with the bulk. From theory we would predict ^Surface/^Buik = 2 for a simple cubic ferromagnet with uniform nearestneighbour exchange coupling (see §3.3.3). Since Ni 0 4 Fe 0 4 B 0 2 needs to be described by a more complicated theoretical model, a value of about 3 for the ratio of the b coefficients is not unreasonable. Measurements of the surface magnetisation in the antiferromagnet NiO have been reported using conventional LEED by Palmberg et al. (1968) and Hayakawa et al. (1971). In this case the magnetic unit cell of the NiO is larger than the spatial unit cell (because of the antiferromagnetic ordering), so there are specific magnetic Bragg reflections and it is unnecessary to employ spin-polarised beams. Finally we mention that there have been measurements of the surface specific heat in various magnetic materials due to Henderson et al. (1969, 1971). The considerable difficulties involved in comparing the data with theory have been discussed by Mills (1984). 3.7 Magnons in Ising ferromagnets In §3.1 we introduced the Ising model as the special case of the anisotropic exchange Hamiltonian (3.2) in which sx = ey = 0 and sz = 1. If there is an additional applied magnetic field in the x direction the resulting Fig. 3.16 The temperature dependence of the surface magnetisation (crosses) measured by spin-polarised LEED in Ni 0 4 Fe 0 4 B 0 2 compared with that of the bulk (full line) determined by conventional methods. The broken line is a best fit to the experimental data for the surface layer using the expression M(T)/M(0) = 1 - bT3/2 [after Pierce 1986]. 1.0 Bulk

0.9

0.8

0.7

50

100

150 200 Temperature (K)

250

300

119

Surface magnons Hamiltonian has the form ij

i

Here Jtj is the exchange interaction as before, and Q, represents a transverse field. Although this Hamiltonian provides a good description of some real anisotropic magnetic materials when subjected to a transverse field, it is of much wider applicability as a simple pseudo-spin model for hydrogen-bonded ferroelectrics as now discussed. Ferroelectrics are materials that exhibit a phase transition into a lowtemperature state with a spontaneous electric dipole moment. Two good books dealing with the bulk properties of such systems are by Blinc and Zeks (1974) and Lines and Glass (1979). The hydrogen-bonded ferroelectrics are materials such as KDP (potassium di-hydrogen phosphate) in which the occurrence of the electric dipole moment is associated with the ordering of protons in the hydrogen bonds in the solid. Essentially each proton is considered as being in a double potential well, as represented schematically in Fig. 3.17, where there are two positions A and B of stable equilibrium. For temperature T above a critical temperature T c the protons in the solid are distributed at random. However, for T
120

Magnons in Ising ferromagnets transition can be analysed using mean-field theory in an analogous fashion to the calculation in §3.2 for a Heisenberg ferromagnet. The critical temperature Tc satisfies (see Blinc and Zeks 1974) tanh(Q/2feBTc) = 2Q/J(0)

(3.104)

where the exchange term J(0) is defined as in (3.13). For T Tc it lies along the x direction. In the pseudo-spin model of a ferroelectric the electric dipole moment is proportional to <SZ>, so Tc represents the critical temperature between the ferroelectric (T < Tc) and paraelectric (T^TC) phases. Equation (3.104) holds only if 2 Q ^ J ( 0 ) , otherwise the paraelectric phase exists at all temperatures. The bulk excitations (magnons) can be calculated by various techniques, but it is particularly convenient to employ the torque equation of motion (3.35). The instantaneous field B, is found using (3.36) and (3.103). In the infinite medium case the thermal average <Sf> is independent of site i and will be denoted by R = (Rx, 0, Rz). According to mean-field theory for S = \ (see Blinc and Zeks 1974) R = (B°/2|B°|) tanh(|B°|/2/cBT)

(3.105)

where B° = (Q, 0, RzJ(0)) is the static part of the field B,. From this expression it is seen that Rx is constant for T Tc, while Rz has the property of being zero for T^TC (see Fig. 3.18). The equations of motion, derived from (3.35), (3.36) and (3.103), can be linearised by first writing S- = R + ^(q) exp(iq-r< - icot)

(3.106)

where \i is the fluctuating component at wave vector q and frequency co. Then expanding to first order in /i we obtain in component form -icofix = RzJ(0)fiy x

= [Q - R J(qW

(3.107) z

x

- R J(0)fi

y

=-Qfi

(3.108) (3.109)

The solution for a> of the above equations is easily shown to be co = ± a)B(q) where c*B(q) = {Q2 - QR*J(q) + [RV(0)] 2 } 1 / 2

(3.110)

This is the standard expression for the bulk excitations of the Ising model in a transverse field. 3.7.2 Theory for a semi-infinite medium We now give the generalisation of the preceding theory to the case of a semi-infinite medium described by the Hamiltonian (3.103). 121

Surface magnons Further details can be found in the original papers by Cottam (1983) and Cottam et al. (1984). The surface reconstruction equation giving the thermal average <Sf> = Rf in mean-field theory becomes Rf = (B?/2|B?|) tanh(|B?|/2/cBT)

(3.111)

with B? denoting the static part of the instantaneous field B f: (3.112) By analogy with §3.7.1 we obtain the linearised equations of motions by putting S. = R. + /f. Qxp(-icot)

(3.113)

and expanding to first order in /i£:

- itotf = Orf - Rf X Jitf - Mf I J|/Rj j

(3.115)

j

(3.116) Solving the above equations is not straightforward because Rf and Rzt depend on i in a complicated manner given by (3.111). The general Fig. 3.18 The temperature dependence of the spin averages Rx and Rz, calculated using mean-field theory, for an infinite bulk sample described by the Ising model in a transverse field. We have assumed the ratio Q/J(0) = 0.24. The zero-temperature values of Rz and R* are 0.44 and 0.24, corresponding to {{ - [Q/J(0)] 2 } 1/2 and Q/J(0) respectively. 0.5 r

122

Magnons in Ising ferromagnets solutions for Rf and Rf (and hence co) may be investigated numerically (see Cottam et al. 1984); we examine here a case in which an analytic solution is obtainable. We take temperature T> T c , whereupon (3.111) and (3.112) simplify to give Rf = \ tanh(Q f /2k B T)

R\ = 0

(3.117)

Equation (3.114) then has co = 0 as a solution, while (3.115) and (3.116) combine to give (co2 - Q2)/zf + QtRf X Jijtf = 0

(3.118)

j

To illustrate the solution of (3.118), which yields the bulk- and surfacemode frequencies, we take the case of the (001) surface of a simple cubic lattice. To allow for variations of Qt and Jtj at the surface we make the following simple assumptions. We take Qt = Q s if i is in the surface layer (z = 0) and Qt = Q (the bulk value) otherwise; we take Jtj to couple nearest neighbours only, with J(j = Js if both spins are in the surface layer and J 0 = J otherwise. From (3.117) the values of Rf in this model are i tanh(Qs/2/cBT) = R% if i is in the surface layer and \ tanh(Q/2/cBT) = Rx otherwise. Writing \i\ = )Un(q||)exp(iq|| «p), where n is a layer index as before with n = 1 being the surface layer and p = (x, y), we can re-express (3.118) as [co2 - Of + 4 Q s ^ J s y ( q | | f l ^ + QsRxsJfi2 = 0 2

LOJ -Q

2

x

+ 4QR Jy(q]l)K

+ OR'JQJL,^

+»n+1) = 0

(3.119) (n^2)

(3.120)

where y(q|() is defined in (3.26). The above finite-difference equations can be solved by seeking travelling-wave and attenuated solutions in an analogous manner to the solution of (3.24) and (3.25) for the semi-infinite Heisenberg ferromagnet. The bulk modes are found to have the dispersion relation coB(q) = {Q2 — 2Qi^xJ[2y(qu) + cos(g z a)]} 1/2

(3.121)

which is equivalent to (3.110) in the high-temperature phase (where JR2 = 0) and for a simple cubic structure. The frequency co of any surface mode must satisfy co2 - Q 2 + 4QRxJ{4y(q]]) + exp[-/c(co)a] + exp[?c(co)a]} = 0

(3.122)

provided Re[/c(co)] > 0 for the frequency-dependent attenuation factor, where q,,)

(3.123) 123

Surface magnons For the particular case of Q s = Q (and therefore R$ = Rx) the solutions of (3.122) are co = ±<ws(qn), where 4Jsy(q,|)

J

(3.124)

and we have denoted a = (Js/J) — 1. The attenuation factor K is now given by e x p ( - ^ ) = [4(ry(q||)]-1 (3.125) and the surface mode exists only if a and qt| are such that Re(/c) > 0. This leads to the possibility of 'acoustic' and 'optic' surface modes, just as for the semi-infinite Heisenberg ferromagnet in §3.3.1. In this case the conditions for acoustic and optic modes to exist in some part of the 2D Brillouin zone for q|( are J s > £j and Js < fJ respectively. A numerical illustration is given in Fig. 3.19. The basic theory has been extended to include damping of the modes (Cottam 1983) and the case of T < Tc (Cottam et a\. 1984). In the latter paper the authors also examine other theoretical models for surface modes in ferroelectrics, apart from the pseudo-spin approach. These are based on the Landau theory of phase transitions and on polariton theory. The Landau theory had previously been applied by Binder (1981) to discuss surface reconstruction in ferroelectrics; it involves an expansion of the free energy in terms of the order parameter, in this case the polarisation of Fig. 3.19 A plot of frequency (in units of Q) against |q|| (in reduced units such that |q|| = 1 at the Brillouin zone boundary) for the excitations of the Ising model in a transverse field. We have assumed q| = (q\\,0)9 T= 1.5TC, and J/Q = 0.5. The bulk-magnon continuum is shown together with three surface-magnon branches corresponding to Js/J values of: A, 1.75; B, 1.5; C, 0.25 [after Cottam et al. 1984]. 1.4r 1.2 1.0 0.8 0.6 0.4 0.2 0

124

0.2

0.4

IqJ

0.6

0.8

1.0

Problems

the medium. It is a macroscopic continuum method, which has some analogy with the continuum approach employed in §3.3.2 for the Heisenberg model. The second of the above methods, namely that based on polariton theory, is mentioned in Chapter 6. 3.8 Problems 3.1 By writing the Heisenberg Hamiltonian (3.4) out in full in terms of x, y, z components of the spin and by considering the average effective field acting on a particular spin Sh prove that the mean-field Hamiltonian is given by (3.16) and (3.17). For a system of spin S = % there are just two energy levels of the mean-field Hamiltonian, corresponding to eigenvalues ±j of the Sz operator. Use this to verify (3.18) and (3.21), respectively, for infinite and semi-infinite ferromagnets. 3.2 Show that the commutator [ 5 / , Jf], where Jf is the Hamiltonian (3.4) for a ferromagnet, can be expressed in the form

(You may find it helpful to recall that Jtj couples only different sites i and j). Hence verify the equation of motion (3.8). After linearising the equation of motion (as in (3.22)), prove that the set of coupled equations (3.24)-(3.25) is obtained for the semi-infinite ferromagnet at T« Tc. 3.3 The theory in §3.3.1 for magnons in a semi-infinite ferromagnet with simple cubic structure and (001) surface was made assuming nearestneighbour exchange only. Consider now an alternative assumption of taking all Jtj (including those involving spins at the surface) to have their bulk values, equal to J for nearest neighbours and j for next-nearest neighbours. Derive the analogous set of equations to (3.24) and (3.25). Use the ansatz (3.29) to deduce that a surface magnon must satisfy J + 2j[cos(qxa) + cos(qya)~]

in the absence of a surface anisotropy field. Hence find the existence condition and frequency of a surface magnon in this model. 3.4 In the continuum theory of §3.3.2 we employed for convenience the classical equation of motion (3.35). Verify that the linearised equations (3.43) and (3.44) can also be obtained from a quantum-mechanical approach. (This involves using the commutator equation (3.7) and the RPA to derive finite difference equations similar to (3.24) and (3.25). The long-wavelength continuum approximation can then be made.) 3.5 Show that for small wavevectors (a^y « 1) the geometrical factor y(q\\) defined in (3.26) has the form 125

Surface magnons

Hence find an approximate expression for the attenuation factor K of a surface magnon at small q^ by substituting for y(q,|) in (3.34). Show that, in this limit, K is equivalent to the pinning parameter £ defined in (3.45). 3.6 Verify that, for a symmetric ferromagnetic film (with £' = £), (3.76) can be factorised into the two equations (3.80) and (3.81). Check the statements made in the text concerning the graphical solution of (3.80) and (3.81). (For each equation in turn you should plot both sides as a function of K, and look for the number of intersections at K > 0.) 3.7 By substituting travelling-wave solutions of the form (3.27) into (3.64) and (3.66), obtain two expressions for the ratio B/A and hence derive (3.67). 3.8 Equations (3.75) and (3.76) for a ferromagnetic film were quoted in the text as being the long-wavelength limit of the microscopic result. Verify that they can also be derived directly using the macroscopic continuum theory which was introduced in §3.3.2. (This involves using (3.43) for 0 > z > — L , the boundary condition (3.44) at z = 0, and a similar boundary condition involving the parameter £' at z = — L.) 3.9 Using equations (3.92) and (3.94) prove that the attenuation factor K corresponding to the q n = 0 surface magnon in a body-centred tetragonal antiferromagnet with coA «co E is approximately

3.10 By eliminating \ix and /j? from the coupled equations (3.114)—(3.116) for the Ising model in a transverse field, prove that \iz satisfies j

\j

/

provided co^O. (This is a generalisation of equation (3.118) to temperatures above or below Tc.)

126

Surface magnetostatic modes

In this chapter we are concerned with magnetic excitations in situations where the magnetic dipole-dipole interactions are no longer negligible, as they were assumed to be in Chapter 3. It was explained in §3.1 that the relative importance of the exchange and dipolar terms in the magnetic Hamiltonian is largely influenced by the wavevector q of the excitation, with the dipolar terms becoming significant for |q| ^ 108 m" 1 typically. For 107 m " 1 ^ |q| ^ 108 m " 1 the exchange and dipolar terms may be comparable, and this is known as the dipole-exchange region. For wavelengths that are sufficiently long that |q|<10 7 m~ 1 the dipolar terms are dominant: they may either be treated microscopically using the Hamiltonian (3.3) or, more conveniently, macroscopically using Maxwell's equations of electromagnetism. A useful simplification occurs if |q|»a>/c

(4.1)

where a> is the frequency and c is the velocity of light. This inequality implies that the effects of retardation (i.e. finiteness of the propagation of light) can be neglected, and this corresponds to the so-called magnetostatic region. The range of values of |q| is typically from about 3 x 103 m" 1 to 10 7 m~ 1 for ferromagnets. At smaller |q| we have the electromagnetic region where it is necessary to employ the full form of Maxwell's equations including retardation: this case is dealt with in Chapter 6, where there is a section on magnetic polaritons. These different regions of magnetic behaviour are summarised in Fig. 4.1. We begin this chapter by briefly considering the effect of dipole-dipole interactions on the magnons in bulk ferromagnetic samples. Then we focus attention on surface magnetostatic modes in ferromagnets for various geometries (semi-infinite medium, thin film, double-layer system). Surface magnetostatic modes were first predicted by Mercerau and Feynman 127

Surface magnetostatic modes

(1956), Walker (1957) and Damon and Eshbach (1961); they were subsequently studied experimentally by microwave techniques (see Brundle and Freedman 1968). We also discuss the more difficult problem of surface modes in the dipole-exchange region for ferromagnets. Magnetostatic theory can be generalised to apply to antiferromagnets and we summarise results for the surface and bulk modes. Finally we describe the experimental studies on magnetostatic modes and dipole-exchange modes in various surface geometries. These are now very extensive and include light scattering, microwave techniques and spin wave resonance. 4.1 Dipole-dipole interactions in bulk ferromagnets

Before considering specifically surface effects and surface modes, we discuss briefly the role of dipole-dipole interactions in bulk magnetic materials. For a ferromagnet the total Hamiltonian is made up of the exchange Hamiltonian (3.4) plus the dipolar contribution (3.3). The magnon dispersion relation can be investigated by following the same

Fig. 4.1 The different regions of magnetic behaviour in terms of the magnitude |q| of the excitation wavevector. The numbers are approximate and may vary for different materials. ^ Brillouin zone boundary wavevector 109

Exchange region

108 Dipole-exchange region 7

10

106 Magnetostatic region 105 104 103 102

128

Electromagnetic region

Dipole-dipole interactions in bulk ferromagnets operator equation-of-motion method as in §3.2. Taking A = S / in (3.7) and using the RPA as before, we obtain Sq

t/(q)S q + K(q)Sq

(4.2)

Wavevector Fourier transforms of the spin operators have been defined as in (3.10) and U(q) = gfiBB0 + <S*>[J(0) - J(q)] + <S z >[D"(0) + iD"(q)] (4.3) V(q) = KSzXDxx(q) - D»(q) - 2iD^(q)]

(4.4)

The dependence on exchange and dipole-dipole interactions comes from J(q) and DMV(q), respectively, where the latter quantities are defined by

D"(«> = ^ i ( ^ ) I ^ ^ P ^ exp(-iq-r)

(4.5)

The summation is over all lattice translation vectors r, excluding r = 0, and [i and v are Cartesian components. We see that the equation of motion (4.2) involves a coupling to S~, by contrast with (3.11) in the absence of dipole-dipole interactions. The equation of motion for S~ is easily shown to be i jtS~ = - U(q)S~ - K*(q)Sq+

(4.6)

and the normal mode solutions of (4.2) and (4.6) with time dependence like exp( — icot) correspond to co = ±coB(q) with « B (q) = [ t / 2 ( q ) - | n q ) | 2 ] 1 / 2 (4.7) This is the modified dispersion for bulk magnons in the presence of dipoledipole interactions. At low temperatures where <52> -» S in (4.3) and (4.4) we recover the result that can be derived using boson operators instead of the spin operators (e.g., see Keffer 1966). The summations in (4.5) are complicated to evaluate because of the long-range nature of the dipolar interactions, and the problem has been fully discussed by Cohen and Keffer (1955). For a cubic lattice and small wavevectors (such that L " 1 « | q | « a ~ 1 , where L is of the order of the sample dimensions and a is the lattice constant) the dispersion relation takes the form - sin2 0)] )] 1 / 2 x lg/JLBB0 + Dq2 - gwoMoNJ1'2

(4.8)

where D is an exchange factor defined as in (3.15), M o is the magnetisation, and Nz is the classical demagnetising factor of the sample. The magnon frequency depends not just on the magnitude of wavevector q but also on its direction through angle 0, which is the angle between q and the z axis. 129

Surface magnetostatic modes The magnetostatic limit, in which exchange effects become negligible, corresponds to Dq2 «gfi Bfi0M0 in (4.8). For example, if Nz = 0 as in the case of a ferromagnetic film magnetised in a direction parallel to the surface, then <^B(q) = [ ^ B ^ O ( ^ B ^ O + 0/WoM o sin2 0)] 1 / 2 (4.9) The minimum and maximum bulk-magnon frequencies correspond to 0 = 0 and 9 = 90°, respectively, in this case. 4.2 Magnetostatic modes for a ferromagnetic slab We now derive the surface and bulk magnetostatic modes for a ferromagnetic slab. It is more convenient for this case to obtain the results using a macroscopic approach based on Maxwell's equations (without retardation) rather than the microscopic method of the previous section. The standard configuration for surface magnetostatic modes is to take the magnetisation M o parallel to the surface. We therefore adopt the geometry shown in Fig. 4.2, with the z axis along M o and the x axis perpendicular to the slab surface. The 2D in-plane wavevector q|( has components given by qII = (qy9 qz) = 4||(sin (p, cos <£) (4.10) where ql{ = (q2 + ql)1'2 and (/> is the angle between q and M o . The theory for this case was first worked out by Damon and Eshbach (1961). There have also been various review articles covering aspects of magnetostatic theory and we mention, for example, Wolfram and Dewames (1972), Sarmento and Tilley (1982), Cottam and Maradudin (1984), Mills (1984) and Tilley (1986). The calculations will be described for general slab thickness L, so the surfaces are at x = 0 and x = — L. The cases of a semi-infinite medium (L-> oo) and a thin film will be obtained as special limits. Fig. 4.2 The assumed geometry and coordinate axes for calculating magnetostatic modes in a ferromagnetic slab.

130

Magnetostatic modes for a ferromagnetic slab In the magnetostatic region the wavelength of any excitation is very large compared with the lattice spacing, and a continuum approximation for the ferromagnet should be valid. The macroscopic field variables B and H are related by B = fio(H + M), and the total magnetisation M at position r is given in terms of the corresponding spin operators by M = vogfiBS where v0 is the number of spins per unit volume. The torque equation of motion is dM/dt = y ( M x B ) = y/xo(M x H) (4.11) where y( = gnB/h) is ^ e gyromagnetic ratio, and B and H represent total effective fields. We write M = M0\z + m(r) exp(-kot) (4.12) H0H = Boiz + jU0h(i*) exp( - icot) (4.13) where l z is a unit vector parallel to the static magnetisation M o and the static external field B o , both taken in the z direction, and m and h are the fluctuating components at frequency co. On substituting (4.12) and (4.13) into (4.11) and making the usual linear-magnon approximation of neglecting small terms that are of second order in m and h (because |m| « M o at T ^ T c ), we obtain -icom(r) = y\z x |> 0 M o h(r) - Bom(r)] (4.14) This can be re-expressed as a susceptibility relation involving the x and y components of the fluctuating fields: (4.15) where Xa = comco0/(col - co2) Xb

2

2

= comco/(co -co )

(4.16) (4.17)

and we have defined the frequencies co0 = yB0 com = yfi0M0 (4.18) From Maxwell's equations (without retardation) the fields m(r) and h(r) are also required to satisfy V x h(r) = 0 V- [h(r) + m(r)] = 0 (4.19) The first of the above equations is automatically satisfied if h is derived from a scalar potential ifr by writing h = Vi//

(4.20)

Using (4.15) and the second of (4.19) the equation for \j/ inside the ferromagnet is

131

Surface magnetostatic modes while outside the ferromagnet we have simply VV = 0

(4.22)

From the property of translational invariance in the y and z directions it follows that ^(r) must be of the form ^ ( x ) exp(iqn *p), where p = (y, z). For (4.21) and (4.22) to be satisfied and for \j/ to vanish at x = + oo, we have al exp(-gnx) exp(iqy -p)

x >0

(4.23)

! [>2 exp(i^x) + a 3 exp( - i^x)] exp(iq „ • p) a4 expo?,,*) exp(iq,| -p)

0> x > - L

x < -L

(4.24) (4.25)

The quantity qx can be real or imaginary, and is given by

(l+Z.)(«2 + «Ji)-ZA 2 = O

(4.26)

The coefficients a, (7= 1 , . . .,4) can be determined by applying the standard electromagnetic boundary conditions at x = 0 and x = — L. In the magnetostatic case these boundary conditions are that (i) ij/ must be continuous across a boundary, and (ii) (hx + mx) inside the ferromagnet at x = 0 and x= —L must be equal to /ix outside. They lead to four homogeneous linear equations for the four coefficients a,, and the condition for a solution is q\ + 2qnqx(l + Xa) cot(qxL) - q2x(l + Xa)2 - q2yll = 0 (4.27) When qx is substituted from (4.26), the above equation determines the dispersion relations of the modes. The solutions for the mode frequencies are particularly simple when qz = 0, i.e. angle = 90° (the Voigt configuration). First, we see that (4.26) is satisfied if %fl = — 1. From (4.16) this condition is satisfied for co = ±coB, where coB = l(o0(co0 + (om)y/2 (4.28) This represents the frequency of bulk magnetostatic modes in the slab, and it is in fact equivalent to the result in (4.9) with 6 = 90°. Note that (4.28) is independent of the wavevector components qx and qy. Secondly, equation (4.26) can be satisfied if g x = ± i g | | , and these possibilities correspond to surface modes since (4.24) has localised solutions rather than wave-like solutions for 0 > x > — L. The two surface solutions have a common frequency a>s(qn), which is found by substituting for qx in (4.27). On using (4.16) and (4.17) the result can be rearranged as ^s(q II ) = [ K + i^m) 2 " Wm exp( - 2q „ L)] 1 / 2 (4.29) It is important to note that the mode solutions (given in terms of the coefficients a, in (4.23)-(4.25)) are different for the two surface states. If qy>0 it can be shown that qx = iq\\ or qx= —iq\\ would correspond to surface states localised near the lower surface (at x = — L) or the upper 132

Magnetostatic modes for a ferromagnetic slab surface (at x = 0), respectively, and vice versa if qy < 0. This unusual property is an aspect of nonreciprocal propagation, and we shall discuss it in more detail shortly. From (4.28) and (4.29) it follows that a>s is greater than a>B, and the limiting surface mode frequencies are (q\\L«l) (q\\L»l)

(4.30) (4.31)

The latter of the above results corresponds to the surface magnetostatic mode for a semi-infinite ferromagnet. The behaviour of coB and cos as functions of q^L is sketched in Fig.4.3. When qz ^ 0 the surface and bulk magnetostatic modes can still be investigated using (4.26) and (4.27), although the results are more complicated. For the surface mode the imaginary values of qx are now

qx= ±ifi where 11/2

(4.32)

taking the quantity inside the square bracket to be positive. Hence the attenuation constant is changed from qy (when qz = 0) to /J. From (4.27)

Fig. 4.3 The dependences of the surface and bulk magnetostatic mode frequencies on for a ferromagnetic slab. We have taken cam/coo = 6 and the case of qz = 0. 5-1

4-

3-

2-

1 -

0.5

—i—

1.0

1.5

2.0

—i—

2.5

—i

3.0

133

Surface magnetostatic modes the surface mode frequency satisfies q\ + 2
(4.33)

Some of the features of the dispersion relations are represented in Fig. 4.4, where the frequencies are plotted against the components qy and qz of the in-plane wavevector q(|. For a general value of qz the spectrum of bulk magnetostatic modes consists of discrete sheets that become degenerate at co = Icooi^o + w m)] 1 / 2 m the limit of qz -• 0. These sheets correspond to solutions with qx real in (4.26) and in practice are sufficiently close together to give a quasi-continuum of bulk modes. The frequencies are coB(q) = l>o(<^o + o)m) - co0comq2z/q2]1/2 (4.34) which is equivalent to (4.9). Hence the bulk modes come within the frequency range from coo to [coo(coo + &>m)]1/2- Above the bulk region in Fig. 4.4 there is a single sheet corresponding to the surface modes (see shaded region). The surface mode frequency decreases as the propagation wavevector is rotated towards the qz axis: for example, if ^ | | L » 1 the surface mode frequency is rosfaii) = il>o/sin 0) + (o)0 + (»m) sin (/>]

(4.35)

Fig. 4.4 Schematic plot of the dispersion curves for the magnetostatic modes in a ferromagnetic slab as a function of qy and qz [after Damon and Eshbach 1961].

Bulk states

134

Magnetostatic modes for a ferromagnetic slab As q n moves away from the y direction (or, in other words, as <> / varies from 90°) the magnitude of a>s(0(a>0 + co m )] 1/2 corresponding to the top of the bulk magnon region (see Fig. 4.4). In fact, (4.35) holds only for cf)c<(f) < 180° — c9 where c is given by sin(/>c = [ o ; 0 / K + a; m )] 1/2

(4.36)

Hence the surface magnetostatic modes are predicted to have nonreciprocal propagation properties, in the sense that cos(qn) ^co s ( — q^). Indeed no surface mode is possible, localised at that same surface, when q II is reversed. The behaviour on reversing q)( is illustrated in Fig. 4.5 for the case of <j> = 90°. We suppose that for one direction of q,, there is a surface mode localised at the lower surface of the slab, as in Fig. 4.5(a). Then if q|t is reversed in direction, keeping Bo and M o fixed, the new surface mode is localised near the upper surface, as in Fig. 4.5(ft). The two modes are degenerate in frequency and have the same attenuation length q^1. A thorough explanation of why the surface magnetostatic modes may show nonreciprocity has been given by Mills (1984), who concludes that it requires consideration of the space-group symmetry of the dipolar system as well as the lack of time-reversal symmetry. Surface magnetostatic modes in ferromagnets typically have frequencies that correspond to the microwave region, and there are applications for signal processing (see Ishak and Chang 1985). The frequencies are also very suitable for experimental studies by Brillouin scattering. These techniques, amongst others, and the extent to which they provide verification of the foregoing theory, are discussed later in this chapter. For a full comparison of theory with Brillouin scattering data it is also necessary to have results for the magnetic Green functions, and these quantities have been derived for a ferromagetic slab in the magnetostatic limit by Cottam (1979). Note that in the preceding discussion we have considered only the geometry in which the magnetisation M o is parallel to the surfaces (see Fig. 4.2). It can be shown that, when M o is perpendicular to the surfaces Fig. 4.5 Illustration of nonreciprocal propagation of surface magnetostatic modes in a ferromagnetic slab. When q| is reversed the localisation of the surface mode (shaded region) switches from one surface to the other as in cases (a) and (b). M.B.

(a)

Wl/lllllllllIUl, (b)

135

Surface magnetostatic modes of the slab, no surface magnetostatic modes occur (see Wolfram and Dewames 1972). 4.3 Magnetostatic modes for a double-layer ferromagnet We now consider the magnetostatic theory for a ferromagnetic double-layer system as reresented in Fig. 4.6, where there are two magnetic layers of thicknesses L x and L 2 separated by a spacer layer of thickness d. Such systems are of interest because they give rise to coupled modes, which have been studied by Brillouin scattering. For simplicity we take the magnetic layers to be of the same material and to be magnetised in the same direction (along the z axis). Also we assume the spacer layer to be non-magnetic, so that the coupling between the two magnetic layers is due to the long-range dipolar fields, rather than the short-range exchange interactions. The improvements in recent years in the techniques for thin film preparation and deposition have made feasible the fabrication of such double-layer ferromagnets. For Brillouin scattering experiments the layer thicknesses might typically lie in the range of a few nanometres up to a few hundred nanometres, and suitable ferromagnets would be metals such as Fe, Ni and permalloy. The magnetostatic theory described in §4.2 for a single slab can straightforwardly be generalised to double-layer systems of the type shown in Fig. 4.6 (see Grunberg 1980 for details). Equations (4.21) and (4.22) are still applicable for the magnetostatic scalar potential \\i inside and outside, respectively, of the ferromagnetic media. The solutions for \// in the different media can be written down by analogy to (4.23)-(4.25), and the application of the boundary conditions at the interfaces x = 0, x = — Ll9x= — (L1-\- d) Fig. 4.6 The assumed geometry of a ferromagnetic double layer separated by a nonmagnetic spacer layer.

— Nonmagnetic spacer layer

136

Dipole-exchange modes for ferromagnets and x = — (Lx + L 2 + d) leads to expressions for the mode frequencies. We shall quote the results only for the special case of qz = 0 (i.e. propagation wavevector q(| perpendicular to M o ), when it is found that co = [(o;0 + i c o m ) 2 - i ^ a ] 1 ' 2

(4.37)

The quantity a is equal to 1 for a bulk mode, giving the same result as in (4.28) for caB, while for a surface mode a satisfies

|

|

|

|

e x p ( - 2 ^

|

|

L

2

) ] = 0

(4.38)

This is a quadratic equation for a, giving two solutions as expected for coupled surface modes in a double-layer system. In the case of large separation of the magnetic layers (d-> oo) the solutions for a from (4.38) simplify to become exp( — iq^L^ and exp( — 2#||L 2 ). The corresponding frequencies given by (4.37) are just the surface magnetostatic mode frequencies for single isolated slabs of thickness L t and L 2 , as can be seen by comparison with (4.29). In the opposite limit of separation d -* 0 the solutions of (4.38) become a = 1 and a = exp[ —2^11^! H-L 2 )]. The former of these solutions represents a mode that has become degenerate with the bulk magnons, while the latter corresponds to the surface mode for a slab of overall thickness (L1+L2). The general predictions of (4.37) and (4.38) for the mode frequencies are illustrated by numerical examples in Fig. 4.7. The frequencies are plotted against the spacer-layer thickness d for the two cases of Lx= L2 = 15 nm (full curves) and Lx = 15 nm and L 2 = 150 nm (broken curves). It has been assumed that Bo = 0.05 T and ii0M0 = 2.2 T, appropriate to Fe. The theory has been extended to cases where the magnetisations in the two magnetic slabs differ. Grlinberg (1981) took the magnetisations to be of different magnitudes and to be in directions parallel or antiparallel to one another and perpendicular to wavevector q (|. Camley and Maradudin (1982) gave the magnetostatic theory of interface modes between two semi-infinite media of different magnetisations (i.e., the case of d = 0 and infinitely large slab thicknesses). We discuss double-layer magnetic systems again in §4.6 in the context of Brillouin scattering experiments. The generalisation from double-layer systems to multilayer systems (e.g. superlattices) is described in Chapter 7. 4.4 Dipole-exchange modes for ferromagnets In discussing the bulk properties of ferromagnets in §4.1 we treated exchange interactions and dipole-dipole interactions together. We now turn to the more complicated problem of the dipole-exchange surface 137

Surface magnetostatic modes modes that result from incorporating exchange into the magnetostatic theory of the previous two sections. A thorough account is given in the review article by Wolfram and Dewames (1972). The dipole-exchange region corresponds to wavevector values that are involved in many Brillouin scattering experiments on ferromagnets (see §4.6). When exchange is included, the equation of motion (4.14) for the magnetisation generalises to - icom(r) = yiz x {^oMoh(r) - [B o - (D/y)V2]m(r)}

(4.39)

where parameter D depends on the exchange coupling (e.g., it is equal to SJa2 in a simple cubic ferromagnet with nearest-neighbour exchange J). The differential operator [B o - (D/y)V2] represents the effective field acting on the spins in the long-wavelength continuum limit, as may be seen by comparison with (3.43). The theory can now be developed in a similar fashion to the magnetostatic case of §4.2, and we adopt once again the slab geometry of Fig. 4.2. The analogous result to (4.21), which determined Fig. 4.7 The mode frequencies of ferromagnetic double-layer systems as a function of the spacer-layer thickness d for two combinations of the thicknesses Lx and L 2 of the magnetic layers. The crosses denote experimental points obtained by Brillouin scattering from Fe films with L1 = L2 = 15 nm separated by quartz [after Grunberg et al. 1982]. 1.2 U

0.9

£

Fe/Quartz/Fe Bn = 0.05 T

Lx = 15 nm, L2= 150 nm

0.6

0.3

200

100 (/(nm)

138

300

Dipole-exchange modes for ferromagnets the scalar potential \j/ within the slab, is easily shown to be

+ coj- co2)(^~ + ^j + (61 - co2) ^

=0

(4.40)

where the differential operator 0 is defined by S=coo-DV2

(4.41) 2

Outside the ferromagnet V ^ = 0 as before. The characteristic solutions for \jj take the same form as in the magnetostatic case of (4.23) and (4.25) for x > 0 and x < — L, but within the slab the previous solution (4.24) generalises to

0>x>-L

(4.42)

where q^ ( 7 = 1 , 2, 3) are the roots of Dq\ + Dql)2 + com(co0 + Dq\ + Dq2x) - co2] l+Dq2x) = 0

(4.43)

We illustrate the nature of the solutions by taking the case of qz = 0 fan 1 M o ). From (4.43) it follows that qx1)=±iqll

(4.44)

q^=±D-''2l(0J2^Wj''2-^m-co0-Dqiy'2

(4.45)

qO) = ± i D

- i/2 [(co 2 + 1 ^ 2 }i/2 + 1 ^ + ^

+

^IJ-,1,2

(4 46)

The root ^f^^ is pure imaginary and is the same as that found for the surface magnetostatic mode in §4.2 for a ferromagnetic slab. The other two roots depend on the exchange parameter D: qi2) is typically real (for the values of the frequency co of interest) and therefore it corresponds to a bulk-like mode, while qx3) is pure imaginary corresponding to a highly attenuated surface mode that has no analogue in magnetostatic theory. The resulting dipole-exchange modes have eigenstates that are mixtures of the three types of waves (see (4.42)). The degree of mixing is found by application of the boundary conditions. These will include the usual magnetostatic boundary conditions (see §4.2), but clearly we also require additional boundary conditions (or ABCs) in order to solve uniquely for the coefficients a{{] and a(3J) in (4.42). We may note that this situation is formally analogous to that encountered in spatial dispersion theory for exciton-polaritons (see Chapter 6). In the present case the ABCs can be found from the torque equation of motion for a spin in the surface layer, as discussed by Wolfram and Dewames (1972). A simplified form is (e.g., see Mills 1984) )L = o = 0

(4.47)

) ] x = _L = 0

(4.48) 139

Surface magnetostatic modes

where fi denotes x or y, and { and £' are surface anisotropy (or 'pinning') parameters at the surfaces x = 0 and x = — L respectively. We may remark that the above equations are analogous to (3.44) for the case of a Heisenberg ferromagnet. Wolfram and Dewames (1972) conclude that the mixing of the three types of waves in (4.42) is small for q^ < 106 m" 1 typically, but becomes large for ^u>10 8 m~ 1 . For qz = 0 and small q^ the solutions are approximately: (i) a magnetostatic surface mode with frequency given to a very good approximation by (4.29), and (ii) standing bulk magnons with q™ real and taking a series of discrete (or 'quantised') values. The latter case is analogous to the quantisation of the wavevector component for Heisenberg ferromagnetic films discussed in §3.4. A simplified Fig. 4.8 Calculated frequencies (full lines) of the surface mode and the standing bulk magnons as a function of film thickness L. The results are for Fe films with exchange effects included and taking fio = 0.1 T and q\\ = 1.82 x 107 m" 1 . The corresponding experimental points obtained from Brillouin scattering are shown (full circles, cos; open triangles, coB, n = 1; full triangles, coB, n = 2; open squares, coB, n = 3; full squares, w B , n = 4) [after Grunberg et al. 1982].

30

60

90 L (nm)

140

120

150

Dipole-exchange modes for ferromagnets treatment gives the quantised qi2) values as nn/L with n = 1, 2 , . . . , and it is then simple to show using (4.45) that the corresponding frequencies are

17

^

2

Dn2n2\(

= I ( G)0 + Dq\ + " ^ 2 - ))((

nVXV II

+

+ />
(4.49)

The admixture of the g<.3) wave is small for small q^9 and its main role is in enabling the boundary conditions at x = 0 and x = — L to be satisfied. We further illustrate these results for the case of qz = 0 by means of Figs 4.8 and 4.9, which both show the frequency versus L dependence for Fe films. For small L the frequencies of the standing bulk modes become large, as predicted from (4.49), and will eventually exceed the frequency of the surface mode. This leads to the behaviour shown in Fig. 4.8. Away from a 'crossover' region there is little interaction between the surface and bulk branches, but near a 'crossover' there is appreciable mixing of the modes and the occurrence of mode repulsion as shown by the more detailed plot in Fig. 4.9. For large film thickness the number of standing bulk magnons intersecting the surface mode branch becomes very large. Hence in the limit of L -• oo the surface branch is a virtual branch in the quasi-continuum of bulk-magnon states. This produces a damping of the surface modes, i.e. there is a radiative decay of the surface modes into bulk magnons of the same frequency. The term 'leaky' is sometimes used to describe this aspect of the surface-mode behaviour. Wolfram and Dewames (1970, 1972) calculate for qz = 0 and L -> oo that the radiative damping of the surface mode is proportional to q * while its frequency is Fig. 4.9 As in Fig. 4.8 but showing the effect of mode repulsion of the surface mode and the n = 1 standing bulk magnon. In this case the parameters are Bo = 0.02 T and 41 = 2.42 x 107 m" 1 [after Grunberg et al. 1982]. 40 Fe

O

30

-

20 20

m

• •

•

25

30 L (nm)

141

Surface magnetostatic modes modified to = Q)o + i^m + D'q\ (4.50) This modifies (4.31) for the case of q |( L » 1. Here D' is an effective exchange parameter, and according to Wolfram and Dewames it is related to the bulk exchange factor D by D' = ID. The magnitude of the damping term (when qz = 0) is such that the effect is negligible for ^ n < 1 0 7 m ~ 1 . However, the damping increases rapidly with q^ so that no well-defined surface mode of the magnetostatic type exists for q^ > 108 m " 1 . The case of qz ^ 0 for the dipole-exchange modes has been analysed in detail by Camley and Mills (1978) for a semi-infinite ferromagnet and by Camley et al. (1981) for a ferromagnetic film. In particular, they developed a linear-response formalism to calculate the spin-dependent Green functions, which were then applied to light scattering (see §4.6). They studied the effect of the exchange interactions on the non-reciprocal properties of the surface mode as the propagation angle (see Fig. 4.2 for its definition) is varied. For example, in the case of a semi-infinite ferromagnet, the authors found that as (/> decreases towards the critical angle c defined in (4.36) the damping of the mode increases dramatically until the surface mode is no longer well defined. This occurs over a finite interval of angle 0 near to $ c , with this interval being broader if Dq\ is large. This is illustrated in Fig. 4.10 by some numerical calculations for the spectral function Fy(q\\9co), defined as the Fourier transform with respect to qy and co of the spin correlation function <Sy(r, t)Sy(rf, t')} at the surface layer (x = 0). Plots of Fy(q^9 co) against co for various values of are shown, taking three cases of the ratio Dq\/co0 covering a range from weak to strong exchange effects. In Fig. 4.10(a) with Dq\/co0 = 0.01 the broadening of the spectrum is very small for most values of (/>, but it sets in fairly abruptly when 0 decreases below cj>c ~ 22°. The broadening is stronger in Fig.4.10(b) with D<^/coo = 0.1, but nevertheless the peaks are well defined for most cj> and the frequencies are close to those given by the magnetostatic theory. However, in Fig. 4.10 (c) with Dq\/co0 =1.0 the radiative damping is severe even for cj) appreciably larger than <j>c. The effects of exchange interactions in light-scattering experiments on EuO and Fe are discussed in §4.6: in the case of EuO the assumptions of the magnetostatic theory are well satisfied, but in Fe the stronger exchange can lead to significant corrections to the magnetostatic results. 4.5 Magnetostatic modes for antiferromagnets A theory of the long-wavelength surface magnetostatic modes in antiferromagnets was first given by Tarasenko and Kharitonov (1971) and extended by Camley (1980). A further analysis, correcting some 142

Magnetostatic modes for antiferromagnets inconsistencies in the earlier work, was made by Luthi et al. (1983). The above references were for semi-infinite antiferromagnetic samples; the extension to afinite-thicknessantiferromagnetic slab was given by Stamps and Camley (1984). The calculations can be carried out in a manner that closely parallels the treatment due to Damon and Eshbach (1961) for the ferromagnetic case (see §4.2). Therefore, in this section it will suffice to summarise only the main steps and to discuss the implications of the results; the interested reader is referred to the original papers for details. As well as the similarities, it should be noted that there are important differences between Fig. 4.10 The calculated frequency spectrum, Fy(q^, a>) versus OJ/CJ0, of spin fluctuations in the surface layer of a semi-infinite ferromagnet. Curves are shown for different values of the propagation angle , and each figure corresponds to a fixed Dqj/coo: (a) 0.01; (b) 0.1; (c) 1.0. It has been assumed that a>m = 7w 0 , which implies 0 C ~ 22° [after Camley and Mills 1978]. 75 60 45 30 15 0 0.0

1.0

2.0

3.0

4.0

5.0

to

20

63

1.0 0.8

15

° W\

- ™°4\

Fy 0.6 10 0.4

18° 0.2 0.0 2.0

3.0

4.0

5.0

^MVV^

1 1 * JIIJJJ , ,,

2.0

4.0

6.0

143

Surface magnetostatic modes

the ferromagnetic and antiferromagnetic cases. On the practical side, the frequencies of the long-wavelength modes in antiferromagnets are typically in the far infrared region of the spectrum, rather than in the microwave region as for ferromagnets This has consequences for their experimental study (see §4.7), and indeed it is only very recently that the existence of surface magnetostatic modes in antiferromagnets has been confirmed experimentally. On the theory side, another difference compared with ferromagnets is that antiferromagnets have two sublattices of spins and consequently two branches are expected to the bulk-magnon spectrum (e.g. see §3.5 for Heisenberg antiferromagnets). By analogy with the theory described in §4.2 for the ferromagnetic case, we now have a torque equation of motion of the form (4.11) for each of the sublattice magnetisations M p: ^

= yMMpxHp)

(P=l,2)

(4.51)

We take p = 1 and p = 2 to refer, respectively, to the sublattices with static magnetisations parallel and antiparallel to the z axis (the direction of the external field Bo). As before, we separate each of the total fields into their static and fluctuating parts: Mp=± Moiz + mp(r) exp( - icot) 4.52) fi0Up = [fl0 ± (BE + BA)]iz + / W r ) exp(-ic»0 (4.53) Here M o is the magnitude of the static sublattice magnetisation (taken to be the same for each sublattice), while BE and BA are effective fields describing the exchange and the uniaxial anisotropy acting on each sublattice. The upper and lower signs in (4.52) and (4.53) correspond to p = 1 and p = 2 respectively. We next substitute (4.52) and (4.53) into (4.51) and linearise the coupled equations of motion. The results are expressible in terms of a susceptibility relation as in (4.15): the appropriate relationship is the response of the overall fluctuating magnetisation m(r) = m1(r) + m2(r) to a field such that hx(r) = h2(r) = h(r). It may be shown (e.g., see Sarmento and Tilley 1982; Mills 1984) that the x and y components of m and h are connected by a matrix having the same gyromagnetic form as in (4.15), but the definitions of %a and Xbare changed to Xa = X++X~

(4.54)

+

Xb = X ~X~

(4.55)

with X± = 144

< AQ?m

° coA(2a;E + coA) -(co±

(4.56) CJ0)2

Magnetostatic modes for antiferromagnets The frequencies a>0 and com are defined as in (4.18), and in addition we have CDE = yBE and coA = yBA. The reader is invited to verify the steps leading to (4.54)-(4.56) in Problem 4.7. Note that in the special case of zero applied magnetic field (coo = 0) the susceptibility matrix is diagonal (i.e. Xb = 0)- I n addition to the above results, the fluctuating fields m(r) and h(r) must satisfy the unretarded Maxwell's equations (4.19). The theory then proceeds along the same lines as for the ferromagnetic slab in §4.2, with the only essential difference being the modified expressions for Xa and XbTaking first the special case of zero applied magnetic field, the frequencies co£ and co^ of the bulk magnetostatic modes are found to be G>B

(q) =

LOJA(2COE

+ ©A) + 2coAcom sin2 0] w

^B(q) = [<»A(2tt E + co A )] 1/2

(4.57) (4.58)

where 9 is the angle between the wave vector q and the direction of M o . Equations (4.57) and (4.58) for the semi-infinite medium are consistent, as expected, with bulk magnetostatic theory for a dipolar antiferromagnet (e.g., see Loudon and Pincus 1963). It should be noted that co£ T^COB, unlike the situation in §3.5 for Heisenberg antiferromagnets when Bo = 0, and this is a consequence of having now included the lower-symmetry dipolar effects in the calculations. One of the bulk modes (co^) is independent of q and is equal to the AFMR frequency (see (3.86)), while the other mode (a>^) depends on q only through the angle 0. The surface magnetostatic mode frequency cos in the case of Bo = 0 is found to be (e.g. see Camley 1980) 1 + sin z cp where is the angle between the in-plane 2D wavevector q{| and the direction of M o . When the frequencies of the surface and bulk magnetostatic modes are plotted against , it is found that the surface branch lies between the two bulk-mode regions (see Fig. 4.11). The operation of changing q(| —• —q (| (or, equivalently, (/>-• + 180°) can be seen to leave the right-hand side of (4.59) invariant. Hence cos(qn) = cos(~<1||)> a n d so the nonreciprocal propagation property of the surface magnetostatic modes in ferromagnets (see §4.2) does not occur for uniaxial antiferromagnets in zero applied field. However, in the presence of an applied field Bo the propagation of the surface magnetostatic mode in an antiferromagnet does become nonreciprocal (see Camley 1980). For example, if Bo is along the z direction and qj| is parallel or antiparallel to the y direction (corresponding to 145

Surface magnetostatic modes <j)—± 90°), the surface-mode frequency is + lcoA(2coE + o A ) + a) A o J 1 / 2

(4.60)

where we have denoted a>0 = y£ 0 as before. Thus a wave travelling in the + y direction has its frequency increased by an amount co0 whereas a wave in the — y direction has its frequency decreased by co0. This type of nonreciprocity is different from that in a semi-infinite ferromagnet where there is no surface wave propagation when q is reversed. The results for the case of general propagation angle and Bo # 0 are illustrated in Fig. 4.12. The surface modes do not propagate in all directions (by contrast with the situation in Fig. 4.11 when Bo = 0), and there are critical angles (/>c+ and c~ where the surface branches intersect the bulk-mode regions. 4.6 Brillouin scattering Brillouin scattering of light has proved to be a sensitive and successful experimental technique for studying the magnetostatic modes and dipole-exchange modes in layered ferromagnetic media. To a remarkable extent it has provided confirmation of the theoretical results described earlier in this chapter. Recent reviews of Brillouin scattering Fig. 4.11 The calculated frequencies of bulk and surface magnetostatic modes plotted against <j> for a semi-infinite antiferromagnet in zero applied field. Parameters appropriate to MnF 2 at low temperatures T
8.7

> (cm *)

1I

8.65 -

-90°

146

MnF2

Bo = 0

Surface mode

i

X^ ^

Surface mode

90°

Brillouin scattering

from surface and bulk modes in ferromagnets have been given by Sandercock (1982), Grunberg (1985) and Cottam and Lockwood (1986). The experiments have generally employed relatively opaque ferromagnetic materials (such as semiconductors and metals) in order to increase the importance of light scattering from the surface modes, since the light penetrates only near the surface. The high-contrast multipass Fabry-Perot interferometer, which made these measurements feasible, has been mentioned in §1.4.2. The first observation of surface magnetic modes by light scattering was in the ferromagnetic semiconductor EuO (Grunberg and Metawe 1977). The experiment was carried out in a backscattering geometry, so that the incident light wavevector k, and scattered light wavevector k s were antiparallel to one another and at oblique incidence to the surface of the sample. The applied field Bo and the magnetisation M o were both in a Fig. 4.12 The calculated frequencies of bulk and surface magnetostatic modes plotted against for a semi-infinite antiferromagnet with # 0 = 0.1T. Parameters appropriate to M n F 2 at low temperatures T
W WT

8.75

1

Surface mode

8.7 MnF 2 1?0 = 0.1 T

a) (cm"1) 8.65 -

8.6

Surface mode

8.55 -90°

90°

147

Surface magnetostatic modes

direction parallel to the surface and at right angles to the in-plane wavevector q,( (i.e. 0 = 90°).Some spectra for two different orientations of the sample, which can be regarded as semi-infinite in thickness, are shown in Fig. 4.13. The bulk-magnon peaks (labelled B) are seen to occur on both sides of the spectrum, but the surface mode peak (labelled S) occurs on one side of the spectrum only. Depending on the scattering geometry, it is observed either on the Stokes side or on the anti-Stokes side but never on both. The surface peak can be switched from one side to the other by either keeping Bo fixed and reversing the tilt of the sample (as in Fig. 4.13) or equivalently reversing Bo for afixedscattering geometry. The exchange interaction in EuO is relatively weak, and under the experimental conditions the assumptions of the magnetostatic limit are well realised. The frequency shifts associated with the bulk and surface modes agree closely with the theoretical expressions (4.28) and (4.31), respectively, for q^L»l. Also the property of the surface mode peak appearing on only one side of the spectrum is a consequence of the Fig. 4.13 Brillouin spectra for EuO for two different orientations of the crystal, showing scattering from the surface (S) and bulk (B) magnons. The applied field is Bo = 0.3 T [after Grunberg and Metawe 1977]. A

EuO

k

B

A, . vuJ

• kl

B

•A* 1

A/ i

1

i

I

0

i

-

1 1

Frequency shift (cm" )

148

-

2

Brillouin scattering

nonreciprocal propagation. In going from a Stokes to anti-Stokes scattering contribution, one effectively reverses the wavevector q N 9 and it has been shown in §4.2 that the surface magnetostatic mode with wavevector qy has no counterpart at the same surface with wavevector "q!h

Similar Brillouin scattering measurements have been made on thick samples of other materals, notably Fe and Ni (Sandercock and Wettling 1979). The optical penetration depth in these metals is much shorter than in EuO and also the effect of exchange interactions is more important. These two factors make the surface mode scattering more intense and also give a greater spread of the wavevector component perpendicular to the surface for the bulk magnons participating in the light scattering. Consequently the bulk-magnon peaks are broadened and become asymmetric in lineshape. This 'opacity broadening' is analogous to the phonon case discussed in §2.5. It is shown for the case of Fe in Fig. 4.14 (e.g., see the bulk peaks B in the spectrum for (/> = 90°). The effect of varying the propagation angle is also illustrated in Fig. 4.14. As (/> is decreased from Fig. 4.14 Brillouin spectra for Fe with Bo = 0.19 T, showing scattering from the surface and bulk magnons for different values of the propagation angle (f>. The bulk and surface peaks in the spectrum for <j> = 90° are indicated by B and S respectively [after Sandercock and Wettling 1979].

-1.0

-0.5

0 Frequency shift (cm"1)

0.5

1.0

149

Surface magnetostatic modes 90° the surface mode peak shifts to lower frequency and eventually merges with the bulk. This is in accordance with the theory of §4.2, and from (4.36) we estimate that
11 !

Fe 0 8 B,.,

200 -

£

B2

1

|| B,

I

mh

100 -

/v

/

1

i

-1.0

150

1/

\ B;

B

* /

1

1

0.5

1.0

1 -0.5 Frequency shift (cm"1)

Brillouin scattering

L = 120 nm, but when L = 5 nm the bulk modes all have frequencies outside the experimental range. The corresponding plots of the mode frequencies versus L have been given in Fig. 4.8, demonstrating good agreement between theory and experiment. Another important feature of Fig. 4.16 is that for sufficiently small values of L a surface mode peak is observed on both the Stokes and anti-Stokes side of the spectrum, by contrast with the case of a thick film. This effect can be understood in terms of the nonreciprocal propagation of the surface mode (see Fig. 4.5). If the direction of propagation in Fig. 4.5(a) corresponds to Stokes scattering, then the situation in Fig. 4.5(b) (where ql{ has been reversed) will correspond to anti-Stokes scattering. If the film thickness L is large compared with the optical penetration depth, it follows that no surface mode peak will be observed for Stokes scattering off the top surface because the light will not penetrate to the lower surface where the mode is localised. In this case there will only be anti-Stokes scattering from the surface. However, if L becomes comparable with or smaller than the optical penetration depth, the light will reach both surface regions and a surface peak will occur on both sides of the spectrum. The optical penetration depth in Fe is of order 15-20 nm. The mode repulsion effect, Fig. 4.16 Brillouin spectra for Fe films of various thicknesses deposited on a sapphire substrate. The applied field is Bo = 0.1 T [after Griinberg et al. 1982].

L

120 nm 60 nm

30 nm

15 nm •

I

I -1

I 0

1

Frequency shift (cm *)

151

Surface magnetostatic modes

that was mentioned in §4.4 (see also Fig. 4.9), between the surface mode and the standing bulk modes has been studied in detail by Brillouin scattering in Fe films (Kabos et al. 1984). An unexpected feature of the data in Fig. 4.16 for Fefilmson a sapphire substrate is that the anti-Stokes scattering from the surface mode is much more intense in very thin films than in thicker films. On the other hand this enhancement does not occur for thinfilmsof Fe on a silicon substrate (see Griinberg et al. 1982). A similar type of behaviour has been observed with other materials, and it has been explained by Cottam (1983) in terms of multiple internal reflections of the light within the film. The reflection coefficients are influenced by the optical parameters of the substrate material, as well as for the ferromagnet, and this can lead to an enhancement in appropriate cases (e.g. see Fig. 4.17). Finally we mention some Brillouin scattering measurements on double layer ferromagnetic systems (Griinberg et al. 1982). Some data for Fe films separated by a quartz spacer layer are shown in Fig. 4.7, where there is fair agreement with the magnetostatic theory. The discrepancies, which are particularly apparent for small spacer-layer thickness, are probably caused by the effects of exchange coupling.

Fig. 4.17 The intensity for anti-Stokes scattering from the surface mode in Fe films for two choices of substrate material, sapphire and silicon. The experimental points (open circles, Fe on sapphire; crosses, Fe on Si) are due to P. Griinberg and coworkers. The full lines are calculated including multiple reflections according to the theory of Cottam (1983) [after Cottam and Lockwood 1986].

20

152

40 L(nm)

60

80

Other experimental studies 4.7 Other experimental studies

Although Brillouin scattering has provided a wealth of experimental data concerning the magnetostatic modes and dipole-exchange modes in ferromagnets, other experimental techniques have also been applied to study the surface modes in ferromagnets and (very recently) in antiferromagnets. The surface magnetostatic modes in ferromagnets were first investigated by direct microwave excitation, and we briefly describe some early experiments by Brundle and Freedman (1968) on a rectangular slab of yttrium iron garnet (YIG). The sample dimensions and the experimental arrangement are indicated in Fig. 4.18. The microwave source is a coaxial line carrying an rf current at one end of the slab (y = 0), and the detector is a similar line at the other end (y = d). One type of experiment involved inducing a surface wave by a short pulse in the source wire. The surface wave propagates in the y direction, and the delay time for arrival at the Fig. 4.18 The geometrical arrangement for the microwave experiments of Brundle and Freedman (1968) on a YIG slab [after Wolfram and Dewames 1972]. Detector rf (out)

YIG slab

153

Surface magnetostatic modes detector is measured. The delay is simply d/vg, where vg is the group velocity of the surface wave. The group velocity can be calculated from the magnetostatic dispersion relation (4.29) using vg = dco/dql{9 and the result is (see Problem 4.9) vg = — [ K + ico

m)

2

-co s 2 ]

(4.61)

Since the surface modes satisfy the inequality cos < a>0 + ^com (see §4.2) the right-hand side of (4.61) is positive. Brundle and Freedman employed a fixed microwave frequency, but varied coo (by varying the applied field Bo), and they compared the delay times with those calculated from (4.61). The agreement was not particularly good, possibly because of the effect of finite slab dimensions in the y and z directions. Better results were obtained by Brundle and Freedman in a second type of experiment in which a continuous rf current is maintained in the source wire. The surface wave is assumed to travel on the top surface of the slab to the end y — d and then to return to y = 0 along the lower surface. When the total path length Id (neglecting the slab thickness L compared with d) is an integral multiple of the wavelength, there is constructive interference which can be observed as a series of geometrical resonances as the applied field is varied. This enables the surface mode frequency co s versus wavevector q,, to be deduced at a series of discrete ^n values, namely q^ = nn/d with n = 1, 2 , . . . . The resulting dispersion relation was found to be in reasonable agreement with (4.29). Subsequently other microwave experiments were made for the time delay and for the surface wave dispersion. Generally these employed much thinner films (e.g. L ~ 10 /mi) and led to better agreement with the magnetostatic theory. References are given in the review article by Wolfram and Dewames (1972). Another aspect that has been studied experimentally concerns the directional effects for propagation of the nonreciprocal surface modes. By appropriate rotation of the applied magnetic field one can essentially 'steer' a magnetic surface wave (e.g. see Wolfram and Dewames 1972). The existence of surface magnetic modes in the antiferromagnet MnF 2 has recently been established by optical reflectivity measurements, utilising the nonreciprocity of the modes in an applied magnetic field. The experimental conditions were such that retardation effects were of importance, and so the results are discussed in §6.7.3 on magnetic polaritons. 4.8 Problems 4.1 By writing down the x and y components of the vector equation (4.14), verify the susceptibility relationship (4.15) for a ferromagnet. Prove that, 154

Problems if new variables are defined by m± —mx± \my and h± =hx± \hy, then fcoj(co0 -co) 0

0 (OjJicoQ

This result shows how the matrix in (4.15) may be diagonalised. 4.2 By applying the magnetostatic boundary conditions (see §4.2) to the general solutions (4.23)-(4.25) for a ferromagnetic slab, obtain four homogeneous linear equations connecting the coefficients au a2, a3 and a4. Write down the condition for a solution to exist in terms of the vanishing of a 4 x 4 determinant, and hence verify (4.27) for general qx. Show that, when qx = ±ij9 (/? real) as in the case of a surface mode, (4.33) is satisfied. 4.3 The theory of the surface magnetostatic modes in a ferromagnetic slab of thickness L was described in §4.2. Consider now the calculation of cos for the limiting case of a semi-infinite ferromagnet (L -• oo) occupying the half-space x < 0. It is easy to see that (4.23)-(4.25) become replaced by exp(-g,|X)exp(iq r p) x>0 where the complex qx has lm(qx) ^ 0 and satisfies (4.26). Now take the case of qz = 0 and apply the magnetostatic boundary conditions (described in §4.2) at x = 0. Use this to show that the frequency of the surface magnetostatic mode (corresponding to qx= —iq\\) is COS = CO0 +

2Wm

as quoted in (4.31). 4.4 Generalise the previous question to verify that, in the case of qz ^ 0 for a semi-infinite ferromagnet, the frequency a>s on the surface magnetostatic mode is given by (4.35). 4.5 The frequencies for the magnetostatic modes of a double-layer ferromagnet are given by (4.37) and (4.38). Use these results to discuss the solutions for the surface magnetostatic mode frequencies of a ferromagnetic film (of thickness L) separated by a non-magnetic spacer layer (of thickness d) from a semi-infinite ferromagnet. For a fixed value of 4||, sketch the mode frequencies versus L and versus d. 4.6 In the dipole-exchange theory for a ferromagnetic slab, verify using (4.40)-(4.42) that the quantities q^ correspond to the roots of (4.43). In the special case of qz = 0, check that the solutions for q{j] are as quoted in (4.44)-(4.46). 4.7 Start from the torque equation of motion (4.51) for a two-sublattice antiferromagnet and substitute for the total fields Mp and Hp using 155

Surface magnetostatic modes (4.52) and (4.53). Linearise the resulting equations to first order in the fluctuating components. On putting m = 11^+m 2 and h = h 1 = h 2 , prove that the susceptibility relation between the x and y components of m and h is given by (4.15) and (4.54)-(4.56). 4.8 Using the susceptibility relations (4.54)-(4.56) and taking the case of zero applied magnetic field (co0 = 0), verify the expression (4.59) for the surface magnetic mode of a semi-infinite antiferromagnet. (Apart from the different susceptibility relations, the calculation closely parallels that for the ferromagnetic case.) 4.9 Use the dispersion relation (4.29) to calculate the group velocity vg = d(os/dq^ of surface magnetostatic waves in a ferromagnetic slab. Verify that the result can be expressed in the form quoted in (4.61).

156

Electronic surface states and dielectric functions

In earlier chapters of this book we have seen how in some circumstances a surface mode can arise at a frequency within a forbidden gap for the corresponding bulk mode. For example, the calculation of §2.1.2 for a semi-infinite diatomic lattice shows such a mode appearing in the frequency gap between the acoustic and optic branches. The first calculation of this kind for electronic structure was carried out by Tamm (1932), who considered a model of a simple ID system comprising a semi-infinite lattice of square wells. For about twenty years after his calculation was published, various other calculations for model systems were performed. Attention then shifted to the question of performing realistic calculations for actual surfaces, the theoretical methods being developed from those used for calculation of bulk electronic band structure. A full treatment of this later work would involve an extensive account of band-structure theory, and is beyond the scope of this book. Instead, we restrict attention in §5.1 to a brief account of model calculations together with a description of some relevant experimental results. The reader interested in more recent work on electronic surface states may refer to Ehrenreich et al. (1980), and in particular the article by Kelly (1980) contains a list of earlier reviews. The electron-band calculations are concerned with single-electron properties. Since electrons interact with one another and with the positive ions of the lattice via the long-range Coulomb interaction, collective plasma-like properties can occur. The two main aspects are screening of the Coulomb interaction by redistribution of mobile charge, and the appearance of plasma oscillations. These may be described in terms of a generalised wavevector- and frequency-dependent dielectric function e(q, co). The general properties of the dielectric function for any material are summarised in §5.2. We return to the collective properties of the 157

Electronic surface states and dielectric functions

electron gas in §5.3. As will be seen, screening and plasma oscillations can be generally and economically described by means of the many-body techniques developed in the Appendix. These techniques are applied to the derivation of e(q, co) for the electron gas in §§5.3.1 and 5.3.2. An alternative description, valid at long wavelengths, is given by quasiclassical hydrodynamic equations. This approach is outlined in §5.3.3, since its intuitive appeal can make it the preferred method in this wavelength region. In some circumstances, for example at the surface of liquid helium, in a suitable field-effect transistor structure or within the semiconductor superlattices that will be discussed in Chapter 7, thin sheets of mobile electrons or holes can be generated. These can be sufficiently thin that it is a good approximation to treat them as a 2D electron gas. In §5.3.4 we show how the techniques of §5.3.1 are modified to apply to the 2D case. Sections 5.2 and 5.3 establish the forms of the dielectric function that will be used in the discussion of plasmon-polaritons in Chapters 6 and 7. Polaritons are coupled modes of a photon and another excitation, so that the plasmon-polariton is a coupled photon-plasmon. The other main example for Chapters 6 and 7 is the phonon-polariton, which is the coupled mode involving a photon and an optic phonon. In §5.4, therefore, we digress from the main theme of this chapter in order to derive the form of e(q, co) appropriate for the reststrahl frequency region of an ionic crystal, in which the dielectric properties are governed by the optic phonons. 5.1 Single-electron surface states

5.1.1 Theory As mentioned, the first model calculation was that given for a ID semi-infinite lattice by Tamm (1932). Subsequently a number of related calculations appeared, notably those by Goodwin (1939) and Shockley (1939), who both dealt with finite rather than semi-infinite ID arrays of potentials. We summarise here the complete and illuminating discussion given by Shockley. The model considered is sketched in Fig. 5.1 (a). It consists of N repeat units of a general potential V(x). The potential outside the specimen has a fixed value Vo related to the work function of the material in question. Shockley draws attention to two differences between his model and that used by Goodwin, which is based on the tight-binding approximation. First, the potential within a cell is more realistic in Goodwin's model, Fig. 5.2(fc), since it corresponds more closely to an atomic potential. Second, the potential in the end cells within the specimen differs from that in all other cells. 158

Single-electron surface states Within one cell of Fig. 5.1 (a), centred at x = 0 say, the general solution of Schrodinger's equation may be written in terms of the two complex amplitudes a and b: 4,(x) = ag(x) + ibu(x)

(5.1)

where g(x) and u(x) are respectively even and odd about the mid-point of the cell. The phase factor i is inserted in the second term for later algebraic convenience. Use of Bloch's theorem shows that in the neighbouring cell, centred at x = d, the wavefunction is ij/(x) = Qxp(iqd)[ag(x -d) + ibu(x - d)]

(5.2)

Continuity of \j/ and its derivative xj/' at the point x = d/2 yields ag + ibu = exp(iqd)(ag — ibu)

(5.3)

ag' + ibu' = Qxp(iqd)( — ag' + ibu')

(5.4)

where the functions g, u and their derivatives g'9 u' are evaluated at x = d/2 and use has been made of the symmetry of g and u. Solvability of (5.3) and (5.4) for the amplitudes a and b requires XBH2(qd/2)=-g'u/gu'

(5.5)

In the limit of an infinite ID crystal (5.5) determines the band structure. For a given energy E the functions g(x) and u(x) appearing in (5.1) are determined. Thus the right-hand side of (5.5) is also determined. If it is Fig. 5.1 Potentials used by Shockley (a) and Goodwin (b) in surface-state calculations [after Shockley 1939].

iViViViViViVi

to V

H

i

i

i

i

i

i

<

YiYhrMYh/iYr (b)

159

Electronic surface states and dielectric functions positive then (5.5) has a solution in which q is real, so E lies within a pass band. Alternatively, if the right-hand side is negative, then (5.5) has no solution for real q, and E lies within a stop band. For determination of the energy-level spectrum and wavefunctions of the finite-length system sketched in Fig. 5.1 (a) it is necessary to match the Bloch functions to wavefunctions for the outside regions where the potential is Vo. In these regions the wavefunction is exponentially decaying away from the specimen. The ratio il/'/ij/ therefore has a definite sign, namely positive on the left of the specimen and negative on the right. The implications for bulk and surface states are discussed in detail by Shockley, and the main results may be described with reference to Fig. 5.2. In the spirit of the tight-binding approximation, that shows the energy bands derived from three of the energy levels of the isolated atom. The curves are drawn for a finite specimen comprising eight equally spaced atoms, so each 'band' is represented by eight relatively close curves. As d -> oo, the energy-level diagram simply shows three eight-fold degenerate eigenvalues at the positions of the atomic energy levels. For finite but relatively large d, for example the value dv marked, the usual broadening into tight-binding bands is seen. When d is sufficiently small, for example d2, two eigenvalues E S1 and ES2 appear between the two lower bands. These correspond to symmetric and antisymmetric combinations of states with amplitudes that are maxima at one or other surface. As would be expected Fig. 5.2 Energy E versus lattice parameter d for a ID specimen with eight atoms [after Shockley 1939].

160

Single-electron surface states

by analogy with the discussion of the Rayleigh waves in a finite elastic slab (§2.3.2), the separation in energy ES2 — ES1 decreases exponentially with increase in the total thickness L = Nd of the specimen. The above results are for a ID model. As for phonons, for example Figs. 2.5 and 2.6, or magnons, for example Fig. 3.5, in the 3D case the surface states form a band characterised by the dispersion relation E = £(q||), where qy is the wavevector in the plane of the surface. On an E versus qn plot, the bulk bands appear as a quasi-continuum in the usual way. In terms of the usual description of energy bands, the surface states of Fig. 5.2 arise within zone-boundary gaps, since they correspond to extreme values ofg'/g or u'/u. In calculations of 3D band structures, gaps between extrema can arise inside the Brillouin zone. That is, as sketched in Fig. 5.3, gaps of type B can arise as well as gaps of type A. The question then arises whether surface states can be associated with type B gaps. This is considered by Fortsmann (1970) and by Pendry and Gurman (1975), who conclude that surface modes do indeed appear in these gaps. As previously indicated, we have restricted attention to relatively simple Fig. 5.3 Zone-edge (A) and non-zone-edge (B) gaps in a hypothetical band structure [after Fortsmann 1970].

161

Electronic surface states and dielectric functions model calculations. Later work was concerned with realistic calculations for surface states within the general framework of electron-band theory. The techniques used were generally more advanced than the wavefunction matching described here. For example, the surface Green function matching (SGFM) technique mentioned in §2.4.2 has been widely applied in electron problems. A good introductory account is given by Heine (1980), and other articles in the same volume give more detail. The April 1975 issue of Physics Today is devoted to surface physics and, in particular, the article by Schrieffer and Soven gives an introduction to theoretical approaches and further references. 5.1.2 Some experimental results The electronic surface states described in §5.1.1 are referred to as intrinsic surface states, since they are a property of microscopically clean surfaces. The experimental study of these states involves working in ultrahigh vacuum, of order 10" 10 Torr, since at higher pressures contamination of the surface takes place very rapidly. In fact, a great deal of attention, both experimental and theoretical, has been given to the extrinsic surface states associated with adsorbed or absorbed molecules. Experimental work on both extrinsic and intrinsic states is reviewed by Eastman and Nathan (1975), Park (1975) and Plummer et al. (1975). Here we do no more than briefly review results obtained by photoelectron spectroscopy that give clear evidence of the presence of intrinsic surface states. As mentioned in §1.4.1, in the technique of photoelectron spectroscopy the surface under study is irradiated with monochromatic (fixed-energy) ultraviolet or X-ray photons. The spectrum of emitted electrons gives information about transitions within the specimen. Only electrons generated within an escape depth / of the surface are emitted; since / is restricted by strong electron-electron scattering to a value of typically 1 to 2 nm, photoelectron spectroscopy is very sensitive to surface states. Depending on its energy relative to the Fermi level, a surface band in a semiconductor can be either occupied or unoccupied. Results indicating the presence of a filled band on a Si surface are shown in Fig. 5.4. Curve b is taken after sufficient time for oxygen to be adsorbed to the surface; it is known that the chemical binding of the oxygen to the surface Si atoms removes the intrinsic surface states. The difference between curves a and b, namely curve c, therefore corresponds to photoemission from a filled surface band, situated at an energy just below the top of the bulk valence band. The technique used for Fig. 5.4 cannot be applied for an unoccupied surface band. Eastman and Freeouf (1974) overcame this difficulty by 162

General properties of dielectric functions

performing measurements at afixedenergy E of the emitted electrons but with the incident photon energy ho variable. The primary absorption of the photon involves excitation of an electron from an occupied 3d core state, either 3d5/2 or 3d3/2, into an unoccupied state. The core state is then repopulated by nonradiative, or Auger, transitions, and the Auger electrons generate the photoemitted electrons by means of electronelectron scattering. When the energy of the incident photon is resonant with the energy difference between an occupied and an unoccupied state, the photoemitted electron intensity is relatively large. Thus the emitted intensity reflects features of the photoabsorption spectrum. An example of such a spectrum is shown in Fig. 5.5. As in the previous example, the adlayer of Sb suppresses the intrinsic surface states. Curve a shows two shoulders at around hco = 29 eV; the separation between them corresponds to the spin-orbit splitting Aso between the d5/2 and d3/2 states, which also shows up on the difference curve (c). Further analysis shows that this unoccupied surface band falls in the gap between the bulk valence and conduction bands. 5.2 General properties of dielectric functions

The dielectric function determines the response of a system to an external electromagnetic field. It is therefore basic to the discussion of Fig. 5.4 Intensity of photoemitted electrons from Si(l 11) surface at incident photon energy of 12 eV. Curve a, freshly-cleaved surface; curve b, after 7 h exposure at 3 x 10~ 10 Torr; curve c, difference between a and b [after Eastman and Grobman 1972]. x3

3

1 -6

-5

-4

-3

-2

-1

Initial energy (eV)

163

Electronic surface states and dielectric functions

electromagnetically coupled modes that will follow in Chapters 6 and 7. We set out here some general properties before turning to collective properties of the electron gas in §5.3. Fuller treatments of the dielectric function are given in standard texts, such as Kittel (1986), Ashcroft and Mermin (1976), Elliott and Gibson (1978), Landau and Lifshitz (1971) and Born and Huang (1985). An extensive and illuminating account is given by Agranovich and Ginzburg (1984). In general, the dielectric function s determines the relation between the electric displacement D(r, t) and E(r', r'); for a medium with translational invariance this relation takes the form D(r, t) = £0

S(T -r',t-

(5.6)

t')E(r', t') d V At'

Several points should be noted about this equation. First s is a second-rank tensor, although for an isotropic or cubic medium it reduces to a scalar. Translational invariance means that s is a function of r — r', not of r and r' separately. A surface breaks translational invariance, so in principle s may change from its bulk form when r and r' are within a few atomic spacings of the surface. The way in which this arises for the semi-infinite Fig. 5.5 Photoemission from Ge(l 11) surface. Intensity is plotted versus photon energy ha> at fixed emission energy E = 4 eV. Curve a, clean surface; curve b, surface coated with a monolayer of Sb; curve c, difference between a and b (surface state contribution). EB(d5/2) = 29.1 eV; A so = 0.55 ± 0.05 eV; E* = 4 eV [after Eastman and Freeouf 1974].

•S3

c o •55 V3

1 28

164

30

32

34 (eV)

36

38

Collective properties of the electron gas electron gas will be described in §5.3.2. However, the modes with which we deal in subsequent chapters have decay lengths within the medium which are typically hundreds of interatomic spacings. Their properties are therefore determined by the bulk value of s, and there is usually no appreciable error in assuming that this bulk value describes the medium right up to the surface. Since £ is a function of r — r', it is convenient to Fourier transform (5.6): D(q, co) = soe(q, co)E(q, co)

(5.7)

Thus s is in general a function of q and co. The dependence on q is referred to as spatial dispersion, since its origin is seen in the dependence of s upon r — r' in (5.6). Spatial-dispersion contributions to the dielectric function of the electron gas are derived in §5.3. In the electron gas, however, and in most other dielectric media to be discussed in Chapters 6 and 7, the q dependence of s may be neglected for the long-wavelength (small q) modes that arise from coupling to the electromagnetic field. An exception arises in the case of the exciton-polaritons that form the subject of §6.1.2. Except for exciton-polaritons, then, we deal with media characterised by a dielectric function s(co) which is a frequency-dependent tensor. A further simplification is that in an isotropic or cubic medium the dielectric function reduces to a scalar function of frequency, e(co). An elementary consequence of the frequency dependence of £ is that the refractive index rj = s1/2 is also frequency-dependent; hence the name 'optical dispersion' for the frequency dependence of e. We should distinguish, in principle, between the longitudinal dielectric function £)((q, co) describing response to an electric field E)( satisfying the condition V x E(| = 0, or q x E|| = 0, and the transverse dielectric function E±(q, co) describing response to a transverse field E ± satisfying V-E ± = 0 or q*E ± = 0. (It will be recalled that a theorem in vector calculus states that any field E can be expressed as a sum of longitudinal and transverse parts, E = E|| + E ± . ) However, as is pointed out by Nozieres and Pines (1966), for example, 6|( and e± must coincide in the limit q -• 0, since the direction of q and with it the distinction between longitudinal and transverse then become undefined. With the neglect of spatial dispersion we therefore also neglect this distinction. 5.3 Collective properties of the electron gas 5.3.1 Density correlations and the Lindhard function We now discuss collective properties, particularly screening and plasma properties. In this section we use the formalism of second 165

Electronic surface states and dielectric functions

quantisation together with some of the properties of Green functions introduced in the Appendix. Within the random-phase approximation, the response is described by means of the Lindhard function. Expansions of this function for low frequencies and long wavelengths yield respectively expressions for quasi-static screening and plasma oscillations. In §5.3.3 it will be shown how plasma oscillations may also be treated within a simple hydrodynamic formalism. Readers who are unfamiliar with second quantisation may therefore pass on without too much loss to §5.3.3; alternatively, they will find introductory accounts of second quantisation in books on many-body theory, such as Abrikosov et al. (1963), Fetter and Walecka (1971) or Rickayzen (1980). In a metal, mobile electrons move in a background offixedpositive charges, namely the metallic ions. The simplest model that retains the essential collective properties is jellium, in which the electron gas is taken as isotropic and the ions are replaced by a continuous and uniform distribution of positive charge. This is the model that will be treated here. The Hamiltonian for the electrons in jellium is ^ = l ¥ X q,ff

+ 2 I q,q',k

t;(k)aq+X',^q'-k,^q+k,.

(5-8)


where (5.9) Here the single-electron states are labelled by the 3D wave vector q so that the Hamiltonian describes an infinite system, a is a subband index referring to the two spin states. aq a and aqa are the destruction and creation operators for electrons, defined to satisfy the usual anticommutation relations, for example q

For the case when the electrons interact via the Coulomb interaction, v(k) is the Fourier transform of the Coulomb potential e2/4ns0\r — r'| between electrons at r and r', namely v(k) = e2/e0Vk2

(5.11)

where V is the volume of the system. The derivation of (5.8) is given in detail by Fetter and Walecka (1971). It should be noted in particular that the term in q = 0 is omitted from (5.8); this omission takes account of the presence of the neutralising positive-charge background. In (5.8), box normalisation is assumed so that sums over k and q are converted to integrals in the usual way as in (1.19). Now suppose that the jellium interacts with an external scalar potential 166

Collective properties of the electron gas U(r91). The interaction Hamiltonian is jrx = -e |d 3 rn(r,r)L/(r,r)

(5.12)

where the operator n(r, t) describes the deviation of the electron-gas density from its equilibrium value. It is given by n(r, t) = il/ + (r, t)\j/(T, t)

(5.13)

where ij/(r, t) is a Heisenberg field operator,

iA(r, t) = exp(ijTt/h) X *Ak>K,. exp(-iJTt/«)

(5.14)

k,
From the formalism given in the Appendix it follows that the linear response of the density operator itself to l/(r, t) is <w(r, co)} == ee

3 ddV ^ ( r -- r', r'0(r r\ co)U(r\ co)

(5.15)

where 3){j — r', co) is of the form shown in (A. 19): r',o)) = «n(r);n(r')» < t t (5.16) Calculation of the dielectric response is therefore a matter offindingthe Green function Q). It is convenient to go over to the Fourier transform ®(q, co). With the use of (5.13), (5.14) with i / ^ r ) = V~1/2 exp(ik-r), and definitions given in the Appendix, it can be shown that ^(q,co)=K«pq;pq+»w

(5.17)

where k,<j

a

q,.

= P- q

(5.18)

The last step follows from a change of variables. For the sake of generality, we start with the Green function The Green function <2> satisfies the equation of motion given as (A.6) of the Appendix. We apply this initially to the Green function ^flkt^k+q,*; Pqt>w fr°m which ®aa, is derived by summation over k. For the noninteracting system, with the Hamiltonian given by the first term only of (5.8), the Green functions can be evaluated exactly. The details of the various commutators are left for Problem 5.2, and (A.6) together with those commutators gives A.+,.a;P^'»a,

(5.20)

where / k is a thermal occupation number: +

(5.21) 167

Electronic surface states and dielectric functions It is just the Fermi-Dirac distribution function [1 + exp(e k/fc BT)]~ 1, independent of o in the present example. Equation (5.20) can be solved exactly, and Q)aa. is then found by summation over q:

»2,.(q, co) = (In)-^...

I k

/fc/fc+

«

fico-ek+q + £k

= \baa,®\% co) (5.22)

The superscript 0 has been introduced to emphasise that this result applies for the noninteracting system, and ®°(q, co)is introduced for later use. The factor \ ensures that ®°(q, co)= J ^ , Q)%a. (q, co). With interactions included, the commutator with the second term of the Hamiltonian (5.8) must be added. It can be seen on general grounds, or from the specific result given in Problem 5.3, that the equation of motion now involves Green functions of the general form ((a+a+aa; p^>> w . These cannot be expressed exactly in terms of the starting Green function, which has the simpler form ((a+a; p ^ ) ) ^ . However, we can find an approximate solution by introducing the decoupling scheme a^a^a^a^ —• — a^a3 + a^a 4 . That is, pairs of operators are replaced by their thermal averages, and minus signs are introduced as required to account for changes of ordering of the fermion operators. For historical reasons, this decoupling scheme is known as the random-phase approximation, or RPA. With the RPA simplification of the relevant commutator, as given in Problem 5.3, it is found that the term "(q)(A-A+q) I

«flk% 1 % + *. 1 ;p q .a'»o,

(5.23)

k'.ffi

is added to the right-hand side of (5.20). In the present case of a spin-independent interaction v(q), all the Green functions with spin indices are proportional to baa. and we therefore introduce the notation % > k + q , , ; P q V»«, = ^ ( k , q,
(5.24)

Equation (5.20), including the interaction term (5.23), may then be written (hw - sq + k + £k)iF(k, q,
+ S*Mi)(A - A + q ) I W , q> o)) (5.25) k

Now divide this equation by (/ko - eq+k + ek) and sum over k and a. The result is ®(q, co) = @°(q,co) + 27r^°(q,a))v(q)@(q,(o) with the solution ^ W ) 1 - 27c^°(q, coMq) 168

(5.26)

(5.27)

Collective properties of the electron gas Thus in the RPA, the linear-response function ^(q, co) of the interacting system is expressed in terms of ^°(q, co), derived originally for the noninteracting system. ^°(q,co) is referred to as the Lindhard function, in reference to the original publication (Lindhard 1954). Equations (5.26) and (5.27) are more usually derived by the Feynman diagram techniques of quantum field theory; a clear account is given by Abrikosov et al. (1963). We go on to apply these results to the electron gas in jellium. It is worth noting in passing, however, that the expressions derived for ^(q, co) in (5.26) and (5.27) have other applications. Thus if v(q) is treated as a constant, appropriate for an uncharged Fermi liquid such as normal 3 He (Tilley and Tilley 1986), ®(q, co) can be used as the basis for a discussion of the collective mode known as zero sound; this is done by Abrikosov et al. (1963), for example. If i?(q) is spin-dependent, then the generalised ^ffff,(q, co) contains information on the collective modes, i.e. magnons, of an itinerant-electron ferromagnet such as Ni. 5.3.2 Dielectric function It is now convenient to introduce the dielectric function of the electron gas. We take the external potential l/(r, t) in (5.12) to be that of an external charge (7(r, t) = epeqxi exp(-icor) exp(iq-r)

(5.28)

This plane wave interacts with the single Fourier component p* of the density operator, with interaction Hamiltonian ^1 = -2nv(q)p^p^

(5.29)

By the usual linear-response formula, the density response is = 27r^(q,coMq)pqxt

(5.30)

We may relate this to the dielectric function as follows (Nozieres and Pines 1958). Maxwell's equations in vacuum give for the electric-field Fourier component Eq e o q.E q = e(p- t + < p q » (5-31) In the dielectric description of the continuous medium, the corresponding Maxwell equation gives the response to the external charge density: 80q-e{q9Q>)Eq = ep*«

(5.32)

Comparison of (5.31) and (5.32), with use of (5.28), establishes the relation 6(q, co) = [1 + 2nv(q)3f(q9 co)] ~ *

(5.33)

Substitution of (5.27) gives the useful alternative expression e(q, co)=l-

27n;(q)^0(q, co)

(5.34) 169

Electronic surface states and dielectric functions Equation (5.34) is the general RPA expression for the dielectric function of the 3D electron gas. It could be discussed at considerable length, but we restrict attention to the form that applies at relatively long wavelengths and low temperatures, q « qF and T«TF. Here qF is the Fermi wave vector and TF the Fermi temperature, given by kBTF = eF = h2qF/2m. These requirements are not too restrictive in a metal, for which qF ~ I/a, where a is the interatomic spacing, and TF >i 104 K. The appropriate limiting form of @° is given in Problem 5.5, and is repeated here for convenience: 2n<3°(q9 co) ~ nVq2/mco2

(5.35)

where n is the electron density. Equations (5.11), (5.34) and (5.35) now give s(q,co)=l-co2p/co2

(5.36)

col = ne2/som

(5.37)

with which will be recognised as the plasma frequency of the electron gas. Equation (5.36) shows that a(q, co) is negative for co < cop and positive for co > cop. As (5.33) shows, the zero of e(q, co) at co = cop corresponds to a pole of the Green function ®(q, co), which confirms that cop is an excitation frequency of the system. What has been done is hardly the simplest way of deriving (5.36), and a more straightforward method is given in §5.3.3. However, the results are fundamental, and our derivation may be used to bring out the nature of the approximations made in simpler approaches. It was assumed that the single-electron eigenfunctions were plane waves, so that strictly speaking the results apply for a bulk electron gas. It may be noted that in principle the expressions derived exhibit spatial dispersion, that is, ^-dependence of the dielectric function; it is only in the long-wavelength limit that (5.36) applies and spatial dispersion is absent. Equation (5.36) represents only the lowest-order term of a Taylor expansion in q/qF, the next term being of order (q/qF)2. In a coordinate-space formulation, this corresponds to the appearance of terms in V2Sn, where Sn is the deviation from equilibrium density. This term will be seen to arise naturally in the hydrodynamic description to be given in §5.3.3. In the long-wavelength limit on which (5.36) is based, Im[^°(q, co)] vanishes, as mentioned at the end of Problem 5.5. Thus in this lowest order the dielectric function is purely real and damping is absent. A simple way of introducing damping will be explained in §5.3.3. The functional dependence of (5.36) upon co is illustrated in Fig. 5.6. It may be noted for future reference that (5.36) applies to the electron or hole gas in a semiconductor such as InSb as well as to metals. For a semiconductor, cop depends on the doping since that determines the carrier density n. 170

Collective properties of the electron gas Typically the plasma frequency of a semiconductor is in the infrared, whereas the plasma frequency of a good metal is in the near ultraviolet. For example, for electrons in InSb with a density 10 23 m~ 3 , cop/2nc = 770 cm" 1 or fia)p = 96 meV, while ha>p = 6.2 eV for Na. The account given in §5.3.1 and in §5.3.2 up to this point applies to the infinite 3D electron gas, since plane-wave eigenfunctions are used. It is fairly easy to see in a qualitative way how the discussion is modified for the case when one or more boundaries are present, although the details become quite involved. Accounts are given by Dasgupta and Beck (1982), Flores and Garcia-Moliner (1984) and Del Sole (1986). For semi-infinite jellium, for example, the Hamiltonian is written in terms of electronic eigenfunctions of the semi-infinite medium rather than the plane waves appropriate for bulk jellium. The sums in (5.8) and in the subsequent derivation of the RPA dielectric function are then sums over the quantum numbers of these eigenfunctions. If the derivation is taken as far as the q2 terms that lead to V2dn terms in the bulk, the analysis yields boundary Fig. 5.6 Free-electron dielectric function of (5.36). The plasma frequency cop is given by (5.37) with the free-electron density of Ag. 0 10

20 ho) (eV)

-10-

-20-

171

Electronic surface states and dielectric functions

conditions on Sn. The need for these additional boundary conditions, usually referred to as ABCs, will be seen from a different point of view in §5.3.3. The use of ABCs was also a feature of the discussion of dipole-exchange modes of §4.4. In general, the calculations for surfaces yield a correction to the bulk value of the dielectric function that is significant only for the first few atomic layers. This leads to corrections to electromagnetic quantities, such as reflection coefficients, that are only a few percent of the values found from the Fresnel expressions with the bulk dielectric function. However, these corrections are large enough to be detected experimentally. As an example, Fig. 5.7 shows the calculated differential reflectivity for two polarisations from the surface of Si(lll)-2 x 1 (the notation is explained in §1.2). The wavefunctions used were based on the tight-binding approximation. Fig. 5.7 may be compared with the experimental results shown in Fig. 5.8. It is seen that in the frequency range /ko<0.8eV covered by the experiments the agreement is good, particularly as concerns the large anisotropy between the two different polarisations. There are some circumstances in which a thin charge sheet, which may be approximated as a 2D electron gas can be produced. A Lindhard type function may be used for the 2D electron gas, and a brief account is given in §5.3.4. The main use that will be made subsequently of (5.36) is in Chapter 6, where the properties of plasmon-polaritons will be discussed, and in Fig. 5.7 Calculated differential reflectivity {R - Rs)/Rs, where R s is the reflectivity of the 2 x 1 H covered surface, for Si(ll 1)—2 x 1. The light is normally incident, and two directions of polarisation are shown: full line, electric field parallel to [011]; broken line, electric field parallel to [211] [after Del Sole 1986]. 4.0

k

3.0 2.0 1.0 0 -1.0 -2.0

172

/is 1.0

A *^\ 2.0

** * * 3.0

^

4.0

v £(eV)

Collective properties of the electron gas

Chapter 7 for the analogous modes in superlattices. These are electromagnetically coupled modes of wavelength X comparable with that of visible light. For the bulk plasmon-polariton, therefore, the inequality q/qF« 1 applies and it is legitimate to use the lowest-order Taylor expansion that leads to (5.36). The simplest form of surface plasmonpolariton arises at the interface between a metal and vacuum, and is found for a range of frequencies below cop in which s(co) is negative. It is localised at the interface in the sense that its amplitude decays exponentially with distance away from the interface. The decay length is of the same order of magnitude as the wavelength X, so that the mode penetrates at least some hundreds of interatomic spacings into the metal. The modifications of £(co) from its bulk form due to the presence of the surface are of much shorter range than this. It is therefore usually sufficient to characterise Fig. 5.8 Experimental results for differential reflectivity of Si(lll)—2 x 1. The polarisations are the same as in Fig. 5.8 [after Chiraradia et al. 1984].

5.0-

4.0-

3.0

2.0-

1.0-

0.0

0.3

0.4

0.5 Energy (eV)

—i—

0.6

173

Electronic surface states and dielectric functions the metal as having the dielectric function (5.36) right up to the surface. It is this description that will form the basis of the discussion in Chapters 6 and 7. 5.3.3 Hydrodynamic description We give a very brief introducton to this topic; a full account is given by Boardman (1982). The electron gas is treated as a continuous fluid with density n(r, t) and velocity v(r, t)y with associated electric current density j(r, t)\ }=-ne\

(5.38)

where dependence on r and t is understood. The particle current density is HV, so n and v are related by the continuity equation dn/dt + V-(n\) = 0

(5.39)

The velocity is assumed to obey the acceleration equation m(

+ v . V )v = -e(E + v x B) - mv/r - (mP2/n)Wn \dt J

(5.40)

As in all hydrodynamics, the total derivative D/Dt = d/dt + vS7 (5.41) is used; the term v»Vv is nonlinear, and is usually negligible. The term — mv/r introduces the collision, or damping, time t. The final term results from the compressibility of the electron gas. The value of the parameter P is discussed by Boardman (1982). The compressibility term ultimately leads to a term in q2 in the dielectric function e(q, a>), and comparison with the q2 term found by expansion of (5.34) shows that p2 = 3i;pa>p/5co2, where vF = hq¥ /m is the Fermi velocity. In practice, the simpler form P2 = 3fp/5 is often assumed. It is a major advantage of the hydrodynamic method that the magnetic-field and damping terms can be included so readily. The dielectric function describes the linear response of the electron gas, and we therefore proceed by linearising (5.38) to (5.40). We treat the electric field as the driving force, so that E and v are first-order quantities, whereas n is written as a sum of zero- and first-order terms: n = n0 + n x (5.42) We consider a single-frequency field E exp( —icot), so d/dt is replaced by — ico. The linearised forms of (5.38) to (5.40) are then }=-noe\

(5.43)

-koHi-h noV-v = 0

(5.44) 2

icom\ = mv/r + eE + e\ xB + (mp /n0)Vn1 174

(5.45)

Collective properties of the electron gas Substitution of nx from (5.44) into (5.45) gives the basic equation for v: icom\ = mv/t + eE + e v x B - (im^2/co)V V • v = 0

(5.46)

We restrict attention for the moment to a bulk electron gas, for which it is sufficient to take E in the plane-wave form E exp(iq-r — icot). We also set the magnetic field equal to zero. The operator V is then replaced by iq, and (5.46) can be solved for v: v = (eE/m)(ico - 1/T - ij8 VA»)~ *

(5.47)

Equations (5.43) and (5.47) are what is needed to determine the dielectric function e(q, co). The relevant Maxwell equation is written in the alternative forms iqxH=j-ico£oE

(5.48)

and iq x H = - icoe0e(q, co)E (5.49) where H is the first-order magnetic field generated by the electron current j . In (5.48), j is substituted from (5.43) and (5.47); comparison of (5.48) and (5.49) then yields e(q, co) = 1 - colico2 + ico/i -

fi2q2)'l

( 5 - 50 )

where the plasma frequency cop was defined in (5.37). Equation (5.50) is the previous (5.36), generalised to include damping through T and spatial dispersion through the compressibility term P2q2. Although the method does not have the generality of the microscopic theory which was the subject of §§5.3.1 and 5.3.2 it has the characteristic advantages of a hydrodynamic approach that it is relatively simple and damping is easily included in a phenomenological way. The main restriction is that it is a long-wavelength method, valid for qa«\, where a is the interatomic spacing. At the end of §5.3.2, it was pointed out that the calculations presented there were specifically for a bulk material, since the electron states were labelled by wavevector q. In the present approach, (5.46) could be used for a finite specimen such as a semi-infinite medium or a film. However, the solution given in (5.47) holds only for the bulk medium, since it involves q. For a finite specimen, the presence of the term in VV- v means that (5.46) can be solved only if a boundary condition on v is applied at the surface or surfaces. For example, a description of the reflection of light from the surface of semi-infinite jellium involves solving Maxwell's equations together with (5.43) and (5.46) for the medium. The boundary conditions are continuity of tangential E and H together with the boundary condition on v. It is because the latter is additional to the electromagnetic conditions on E and H that it is referred to as an ABC. The question of 175

Electronic surface states and dielectric functions ABCs for the electron gas will not be discussed further here; the reader may refer to the reviews cited in §5.3.2. As mentioned in §5.3.2, for the applications to plasmon-polaritons to be made in Chapters 6 and 7, it is sufficient to describe the medium as having the dielectric function (5.36) right up to the surface. Since this dielectric function is free from spatial dispersion, the question of ABCs will not arise. However, for some applications, such as the calculation of attenuated total reflection (ATR), it is essential to retain the damping term. We therefore for later reference quote explicitly the form of (5.50) with damping but without spatial dispersion, namely 8(o)) = 1 - col(o)2 + ico/r)-1

(5.51)

The hydrodynamic method can be used to give a simple treatment of the effect of an applied static magnetic field B o . The term e\ x B o is retained in the equation of motion (5.46). A straightforward extension of the derivation we have outlined shows that the dielectric tensor becomes anisotropic, and takes the form £ = | ie2

ex

0 |

(5.52)

+ lM

(5.53)

where fn (m

(5.54) and £3 is given by (5.51). The z axis is taken in the magnetic field direction. The cyclotron frequency coc = eBJm has been introduced. In (5.52) damping is included but spatial dispersion has been neglected. It is seen from (5.52) that in the presence of the magnetic field B o the xy components of s take a gyrotropic form, similar to the magnetic susceptibility tensor of (4.15). 5.3.4 Two-dimensional electron gas In some circumstances it is possible to have an electron gas in the form of a charge sheet that is sufficiently thin that it can be treated as a 2D electron gas. Two important examples are a charge sheet at the surface of liquid helium and a charge sheet in a semiconductor. Liquid helium has a dielectric constant of about 1.06, which is sufficient for electrons to be trapped in the weak image potential at the surface. The areal density of the 2D electron gas trapped in this way can be varied by 176

Dielectric function of ionic crystals a static electric field applied normal to the surface. A review of the topic is given by Cole (1974). In semiconductors, a static electric field applied normal to the surface, as in a field-effect transistor, can attract a charge sheet of electrons or holes to the surface, and again the areal density can be varied by changing the field. In addition, electrons can be trapped at a heterojunction, like that between GaAs and A ^ G a ^ ^ A s for example. Further discussion of this form of trapping is deferred to Chapter 7. It was first pointed out by Stern (1967) that the RPA discussion of the 3D electron gas, which formed the subject of §§5.3.1 and 5.3.2, could be taken over with appropriate modifications for the 2D electron gas. In particular the Lindhard function @°(q\\, co) can be defined as a function of the 2D wavevector q^ and frequency co. It is given formally by the same expression as the 3D function, (5.22), but naturally the sum is over a 2D wave vector ky. The dielectric function continues to be given by (5.34). Stern gives the result of an explicit evaluation of @°(q\\, o). Fetter (1973) showed how the hydrodynamic approach, outlined in §5.3.3, can be applied for the 2D case. He included the compressibility term, analogous to the last term of (5.45), that leads to spatial dispersion. His paper includes calculations of various response functions of the 2D gas embedded in a 3D medium. Subsequently Karsono and Tilley (1977) generalised this work to the case where the embedding media are different on either side of the layer, and also discussed a line of mobile charge, i.e. the ID electron gas. In a later paper, Fetter (1974) discussed a periodic array of 2D electron-gas sheets; some results for a periodic array of charge lines are given by Karsono and Tilley (1977). Both Stern (1967) and Fetter (1973) discuss electromagnetically-coupled modes propagating parallel to the charge sheet and localised on the charge sheet. Stern exclusively, and Fetter mainly, treat the electromagnetic field within the electrostatic approximation, which is analogous to the magnetostatic approximation that formed the basis of Chapter 4. Rather than introduce the electrostatic approximation explicitly here, we defer further discussion to §6.2.4. There, retardation is included, so that the electromagnetic field is treated exactly. The results obtained by the electrostatic approximation are derived as a limiting case. 5.4 Dielectric function of ionic crystals The dielectric function of the electron gas, discussed in §5.3, will be central to the discussion of plasmon-polaritons in Chapter 6. The other main example that will feature there is the phonon-polariton, which arises from the coupling of light to the optic phonons of an ionic lattice. By way of preparation, we here derive the expression for the dielectric function 177

Electronic surface states and dielectric functions that will be needed. Our account is quite brief, and is based on the method introduced by Born and Huang (1985). Most solid-state textbooks deal with the topic, and Ashcroft and Mermin (1976) give a particularly full account. For simplicity, we restrict attention to an isotropic or cubic medium, in which all vector quantities, such as P and E, are parallel. We may then use the lattice dynamics of a ID diatomic lattice, given in §2.1.2, as our starting point, and what follows may be regarded as an extension to include coupling to the electric field. The discussion of anisotropic media does not involve any new points of principle, and is not given here. As in §2.1.2, we consider an infinite diatomic lattice in which masses m1 and m2 alternate. We now take the crystal to be ionic, so that the two sublattices are associated with opposite electric charges. The mode that couples to light is the long wavelength (q = 0) optic phonon at frequency coT, say. This follows from (2.16) for the relative displacement, which shows that the sublattices move in antiphase. Thus the optic phonon is associated with an oscillating dipole moment, and therefore couples to light. The relevant wavenumber is q ~ 2n/A, where k is the wavelength of the light; on the scale of the Brillouin-zone edge n/2a this is effectively q = 0. We denote by u the relative displacement of the two sublattices. The polarisation P contains a term proportional to u, as well as a term due to the electronic susceptibility rather than the ionic displacements: P = eo(au + xE)

(5.55)

As mentioned, for an isotropic medium P, u and E are parallel. In (5.55) E is the macroscopic electric field, the value found by averaging the local field Eloc over many unit cells. Evaluation of the constants of proportionality a and x depends upon details of the lattice dynamics and electronic polarisability, and need not concern us. The equation of motion for u is {-co2 - icoT)u = -cofu + j?Eloc

(5.56)

On the left-hand side a damping term with damping constant T is included; it arises from an assumed damping force — Tdu/dt. The first term on the right-hand side ensures that in the absence of coupling to the electric field the mode frequency is c%, and the second is the driving force due to the local field. In order to derive an expression for the dielectric function from (5.55) and (5.56) it is necessary to find the relation of E loc to E. This is discussed by Ashcroft and Mermin (1976), for example. What Born and Huang pointed out, however, is that this relationship must be linear. Equation (5.56) can therefore be replaced by an equation involving E rather than Eloc: (-co2 - icoT)u = -to\u + yE 178

(5.57)

Dielectric function of ionic crystals Equations (5.55) and (5.57) are readily solved for P : (5.58) —

CO —

10)1

The defining equation for the dielectric function is D = a0E + P = eoe(a>)E

(5.59)

Combination of (5.58) and (5.59) gives

+ \

2

COj — COZ —

)

(5.60)

ICOiJ

where £oo = l + Z

(5-61)

co£-Q)f = a y / ( l + x )

(5.62)

and Equation (5.60) is the conventional form; it is written in terms of e^, the high-frequency (electronic) dielectric constant, and coL, the LO (longitudinal-optical) phonon frequency. The zero-frequency value of £ is !

(5-63)

which can be recognised as the Lyddane-Sachs-Teller (LST) relation. The frequency dependence of (5.60) is illustrated in Fig. 5.9. For typical dielectrics, coT lies in the far infrared, generally with wT/2nc in the region 100 to 1000 cm" 1 . It was mentioned in §5.3.2 that in semiconductors the plasma frequency lies in the infrared. With suitable doping, it can be made close in value to the TO frequency coT, so the dielectric function includes both plasma and optic-phonon contributions. The total dielectric function can be found by combination of the derivations we have given. For completeness, we record the form quoted by Palik et al. (1976) for the dielectric tensor of a doped polar semiconductor in the presence of a static magnetic field B o in the z direction. The tensor has the form of (5.52), with co2 — CD2 — icoF

co[co2 — (co 4- i / t ) 2 ]

e 2 in the form of (5.54), a n d

•"•-0 +*^-*£*i> Following the implications of this relatively complicated expression would take us too far afield, however, and it will not be used subsequently. 179

Electronic surface states and dielectric functions 5.5 Problems 5.1 Give a proof of (5.15). 5.2 Prove the following commutation results, used in the derivation of (5.20).

where the a+ and a operators are fermion creation and annihilation operators satisfying (5.10).

Fig. 5.9 (a) Real and (b) imaginary parts of dielectric function e{co) for GaAs, which is accurately described by (5.60). Numerical values are coj/2nc = 269.2cm" 1 , coL/2nc = 292.8 cm" 1 , T/2nc = 2.5 cm" 1 and e^ = 10.9 (Kim and Spitzer 1979). In (b) the sharpest curve is drawn for the quoted value of F; the two broader curves, drawn for comparison purposes, correspond to values of 5 x 2 . 5 cm" 1 and 25 x 2.5 cm" 1 .

250

-200 -

-400 -

180

260

270 X 280

290 300 310 (i)/2nc (cm" 1 )

320

Problems 5.3 By extending the results of Problem 5.2, evaluate the commutator 0 k A + q.
Z

l7 k

( >qi.aiflq2.»2flq2-k'.ff2flqi+k'.
Show that on application of the RPA decoupling the expression becomes

Z "( 5.4 The Lindhard function ^°(q, co) is defined in (5.22). With the sum converted to an integral by means of (1.19), it is

_iv_ r (2TC)

J

At zero temperature the Fermi occupation factor is fk = 6(qF-\k\) where 6 is the step function, see (A.2) of the Appendix, and qF is the

6000 -

5000 Im(e) 4000 -

3000 -

2000 -

1000 -

267

268

269

270

271

181

Electronic surface states and dielectric functions Fermi wavevector. For the free electron gas, q% = 3n2n, where n is the density. Fetter and Walecka (1971) evaluate a zero-temperature Green function n° which is closely related to £#°(q,co). In fact, ^ ° is a retarded function (response function), so that as in (A.26) the correct form in the complex plane involves making the replacement co -* co + irj in (5.22). By contrast, Fetter and Walecka have the so-called causal function, in which the replacement is co -> co + irj sgn(co). This means that the real parts are the same, Re(^°) = Re(7i°), but the imaginary parts are different. Following Fetter and Walecka, prove that at zero temperature (q,a>)]=

2V

-9

f

d3/c%F-|q|) 1

1 2

(q«k + ^7 )

hco-

where 0> denotes the principal-value integral. Hence show that h2

(2TT)3 {

2K L

2K

L

\K

\K

2)

2/ J

In

1 + (V/K + ; 1 - (V/K: -f \K

where K: = g/gF and v = mco/hq\. 5.5 If the above expression in Problem 5.4 is applied to a description of plasma oscillations, the limit with v fixed as K —• 0 is required. Show that this limit is Re[^ u (q, co)]

mqF h2

V

2K2

as K -• 0 4TT3 3V2

that is, Re[^ u (q, co)] ~

nV q2

as q -»• 0 27rmco2 Note that it can also be proved that Im[^°(q, co)] = 0 in this limit.

182

6 Surface polaritons

In dealing with magnetic surface modes in Chapters 3 and 4 we found it useful to classify modes, to some extent, by the range of the dominant restoring force. Thus Fig. 4.1 shows that for large enough wavevector |qiJ >> 108 m " 1 , the short-range exchange forces determine the nature of the surface modes. Over a wide range of intermediate wavevectors, approximately 104 m " 1 < |c|M| < 107 m " 1 the dipole-dipole force is dominant, and the important magnetostatic approximation holds. The properties of the magnetostatic surface modes are found by solving Maxwell's equations without retardation in the presence of the frequency-dependent tensorial magnetic susceptibility % given by (4.15) to (4.17). Finally, for long-wavelength modes, |q | < 10 3 m~ 1 , retardation cannot be neglected, and it is necessary to solve the full form of Maxwell's equations with susceptibility %. This is the polariton, or electromagnetic, region, discussion of which has been deferred to this chapter. Most of this chapter is concerned with electromagnetic, or polariton, modes arising from frequency-dispersion in the dielectric function rather than the magnetic susceptibility. The origin of dispersion was discussed in Chapter 5. Generally it results from the presence of a resonant mode carrying a dipole moment, like the optical phonon in a polar crystal. It is possible to define an electrostatic region, analogous to the magnetostatic, in which properties of bulk and surface modes are found by solving Maxwell's equations without retardation. The calculations are outlined in §6.2.2, but in practice the electrostatic modes are not very important, and most of the chapter deals with genuinely electromagnetic modes. Furthermore, spatial dispersion will not be included except in the discussion of bulk and surface exciton-polaritons. In §§6.3 to 6.3 we discuss bulk and surface dielectric polariton modes. Section 6.4 is an introduction to the important topic of nonlinear effects. 183

Surface polaritons The experimental technique that has been most comprehensively applied is attenuated total reflection, or ATR, and this forms the subject of §6.5. Section 6.6 describes what has been done to investigate surface polaritons by some other experimental methods. Finally, §6.7 deals wih surface magnon-polari tons. The guiding of electromagnetic waves at interfaces, which is the general theme of this chapter, is a matter of considerable importance in applications, and there is a substantial engineering literature on the subject. The interested reader will find more about applications in Marcuse (1982) and in Stegeman and Wallis (1986). 6.1 Bulk polaritons 6.1.1 Phonon and plasmon polaritons The propagation of light through a bulk medium characterised by a frequency-dependent dielectric function is governed by Maxwell's equations. For a plane wave with all fields proportional to exp(iq-r — icot) propagating through a bulk isotropic medium, the equation V • D = 0 gives £(co)q-E = 0

(6.1)

This has the solutions that either e(co) = 0

(6.2)

q.E = 0

(6.3)

or In the absence of damping, the first of these gives co = coL for an ionic medium, with dielectric function given by (5.60), and co = cop for the plasma with the dielectric function of (5.36). These are the longitudinal modes; they have no surface counterpart, and will therefore not detain us.f The second solution, given by (6.3), is the transversality condition (E transverse to q). For this solution, the V x E and V x H equations yield in the usual way q2 = e(co)a)2/c2

(6.4)

which is the propagation equation for the transverse mode. If the damping term in s(a>) is neglected then it is an easy matter to draw the dispersion curve corresponding to (6.4). For the phonon case, with e(co) given by (5.60), the asymptotic and limiting forms are

f It may be noted that in a detailed treatment, as given by Pines and Nozieres (1966) for example, it emerges that the dielectric function for longitudinal modes is different from that for transverse modes.

184

Bulk polaritons q2~e(0)co2/c2 2

q -> oo

CO«CO T

co-+coT

2

q <0

COT<0)
q2 ~z^to2lc2

CD-* co

(6.5)

These are illustrated in Fig. 6.1. The reader will note in particular that for coT < co < coL, q is pure imaginary, and therefore this frequency interval appears as a stop band in Fig. 6.1. Comparison with the results derived for lattice dynamics in Chapter 2 makes it plausible that if a surface mode exists it will be within this stop band. It will be seen in §6.2 that there is indeed such a mode, namely the surface polariton. The dispersion curve drawn in Fig. 6.1 may be seen as resulting from the crossing of the phonon line co = coT and the photon line q = s1/2co/c, where e is imagined to change slowly from s(0) to e^. These modes interact strongly, and the crossover is therefore eliminated with repulsion of the dispersion curves, as shown schematically in the insert to Fig. 6.1. Thus the dispersion curves of Fig. 6.1 describe a mode of mixed phonon-photon character. Many years ago, the word polariton was coined to describe such a mixed mode, so one may say that Fig. 6.1 depicts the bulk polariton Fig. 6.1 Dispersion curve for bulk phonon-polariton in GaAs; numerical values quoted for Fig. 5.9. Asymptotic lines q ~ [8(0)]1/2
1.5-

0.5

1.0

2.0

3.0 cq/a)r

4.0

5.0

6.0

185

Surface polaritons dispersion curve. Nowadays polariton is used for any mixed mode involving a photon. The plasma dielectric function of (5.51) is the toT -+ 0 limit of the phonon function. The dielectric function, drawn in Fig. 5.6 for y = 0, and the plasmon-polariton curve, drawn as Fig. 6.2, can clearly be derived from the corresponding phonon curves by letting coT -• 0. For a doped polar semiconductor, s(co) has two resonances, one plasma-like at co = 0, and one phonon-like at co = coT. Correspondingly there are two branches in the polariton dispersion curve rather than the one of Fig. 6.2. This is illustrated for n-InSb in Fig. 6.3. The dispersion curves of Figs. 6.1 to 6.3 were drawn for the case when damping is neglected in the dielectric functions. Equation (6.4) is then a relation between two real variables q and co, and the dispersion curve is a natural representation. However, if damping is included, (6.4) relates two variables q and co which in principle are complex. There is then not much sense in discussing (6.4) in isolation, since the conditions of a given experiment will determine what trajectory is followed in the space of the two complex variables q and co. For example, in inelastic light (Raman or Brillouin) scattering in a bulk medium, the experimental constraint is that q is real, since it is the wavenumber difference between the incident and scattered light. The real part of co then appears as a line position, and the imaginary part as a linewidth. Conversely, it is sometimes the

Fig. 6.2 Bulk plasmon-polariton dispersion curve. 4.0

186

T

Bulk polaritons

case in attenuated-total-reflection (ATR) experiments that co is constrained to be real. With damping present, then, discussion of the dispersion relation without reference to experimental conditions is a rather sterile exercise, which we shall not undertake. In this section we have given a very brief account of bulk polaritons in the simple cases of an isotropic medium whose dielectric response is dominated by one or two resonances. These bulk modes were the subject of very active investigation, particularly by Raman scattering, from the mid-1960s. A good account of that work is found in various papers in the Conference Proceedings edited by Burstein and De Martini (1974). A number of extensions and generalisations of calculations outlined in this section were necessary. The theory of bulk polaritons in an anisotropic crystal with multiple resonances is given by Merten in Burstein and De Martini (1974), and several articles describe related experimental results. As we have seen in earlier chapters, for a theory of light-scattering intensities it is necessary to go beyond the solution of the homogeneous equations of motion, which in this case are Maxwell's equations together with the appropriate dielectric functions. Several articles in Burstein and De Martini (1974) give accounts of light-scattering theory; one approach is to use the linear-response method, which is summarised in §1.3.2, and in the Appendix, where additional references are quoted.

Fig. 6.3 Bulk polariton dispersion curve for n-InSb with doping 10 23 m 3 . The dielectric function e(a>) has parameters (Palik et al. 1976): coj/lnc = 180cm" 1 ; coL/2nc= 192 cm" 1 ; e^ = 15.7, m*/me = 0.022, leading to cop/2nc= 161cm" 1 . /

/

/ /

0)/(OT

/

1.5-

/ / / / /

/ /

J

1.0-

0 7-

1

.—

1

cq/a)r

10

187

Surface polaritons 6.1.2 Exciton-polaritons and resonant Brillouin scattering As mentioned, the exciton-polariton is the most important case in which spatial dispersion of the dielectric function must be taken into account. In this section we give an introduction to bulk exciton-polaritons. The corresponding surface modes are discussed in §6.2.5. An exciton is an electron-hole pair, bound by the Coulomb interaction. Excitons in their own right are a major field of study; introductory accounts are given by Kittel (1986) and Smith (1978), for example, and Rashba and Sturge (1982) give a comprehensive account. The type of exciton of interest here arises in a direct-gap, polar semiconductor, such as GaAs or ZnSe. The exciton is formed of an electron from near the bottom of the conduction band and a hole from near the top of the valence band. Since the valence band usually comprises a heavy-hole band, a light-hole band and a band split off by the spin-orbit coupling, and the bands are not strictly isotropic, the full theory of the exciton is quite complicated. We restrict attention to the simplest model, in which an electron of effective mass me is bound to a hole of mass mh. This model contains the basic physics, and is adequate for an account of much of the experimental data on exciton-polaritons. The exciton wavefunction spreads over many interatomic spacings, and the exciton is well described by the theory of the hydrogen atom. The bound-state energies are K=-^-—-

2

(6-6)

with Bohr radius aB given by aR = 4n£oshh2/iie2

(6.7)

where /x is the reduced mass: JU" 1

=m~1 H-m^1

(6.8)

In these expressions eb is the part of the dielectric constant arising from excitations other than the exciton itself. Typically eb is of order 10. In addition, the effective mass in semiconductors of interest is of order one-tenth of the free electron mass. Consequently the exciton Bohr radius, (6.7) is 5 nm or more, while the effective Rydberg constant (coefficient of — \jn2 in (6.6)) is at most some tens of millielectron volts. In particular, the ground-state binding energy |£ x | is much smaller than the semiconductor gap energy £ g . An exciton can move through the crystal as a bound complex. Addition of the kinetic energy of the centre-of-mass motion at wavevector q shows 188

Bulk polaritons that the whole complex has energy g

\ \

(6.9)

where M = me-hmh

(6.10)

In a polar semiconductor, the exciton is a dipole-active mode that couples strongly to light. One consequence is that the onset of optical absorption is at Eg — \EX\ rather than at the direct gap Er The absorbed photon creates an exciton and not a free electron-hole pair, and therefore the energy necessary is reduced by the exciton ground-state energy. In suitable cases such as Cu 2 O a sequence of absorption lines at energies Eg — |£B| for different n is seen on the shoulder of the main absorption threshold at Eg; an example is given in Kittel (1986). A dipole-active mode gives a contribution to the dielectric function. For the exciton, one can take over the classical or quantum-mechanical theory that gives (5.60) for the case of the TO phonon. The resulting expression contains a sum over the contributions from the excited states En in (6.9). The most important is the ground-state term Ex(q), and for the present introductory account we retain only this term. Ex(q) appears in place of the TO photon energy hcoT in (5.60), so the dielectric function is given by e(q, co) = £oo + S/(co2 + Dq2 -co2-

icoT)

(6.11)

whereftcoe= Eg — \EX\ and D = 1/2M. S is the dipole strength of the exciton resonance, and s^ the background dielectric constant. A phenomenological damping parameter F has been introduced. The most important feature of (6.11) is that spatial dispersion enters via the term Dq2; the physical origin of the spatial dispersion is the centre-of-mass contribution to the exciton energy in (6.9). The bulk exciton-polariton dispersion relation is found by solving Maxwell's equations with the use of the dielectric function of (6.11). The result is an obvious generalisation of results obtained in §6.1.1, namely *(q, co)lq2 - e(q9 co)w2/c2^ = 0

(6.12)

This gives a longitudinal mode if e(q, co) = 0 and a transverse mode if the second factor vanishes. For the moment we concentrate on the latter. As explained in §6.1.1, the dispersion curve is most meaningful when damping is neglected. The vanishing of the second factor in (6.12) then gives a quadratic equation in q2. Detailed investigation, left for Problem 6.2, shows that there is one real solution for q for oo< coL, but two for co > coL, where coL(>coe) is the generalisation of the corresponding phononpolariton parameter. The dispersion curve, as sketched in Fig. 6.4, 189

Surface polaritons

resembles that for a phonon-polariton, Fig. 6.1, except that the lower branch bends up for large q. As shown in the insert to Fig. 6.4, the curve may be viewed as resulting from the 'crossing' of the photon dispersion curve with the exciton dispersion of (6.9). An important practical difference is that exciton-polaritons occur near the gap frequency, typically in the visible or near infrared region, whereas phonon-polaritons occur in the far infrared. Fig. 6.4 brings out a major point of interest. Suppose s-polarised light of frequency a> > coL is incident from vacuum on the surface of an excitonic medium. As indicated, two transverse modes, of wavevectors q1 and q2, can propagate in the medium at this frequency. It should therefore be possible to evaluate two transmission coefficients Tx and T2 and one reflection coefficient R to give the amplitudes transmitted into the two propagating modes and reflected into the vacuum. However, Maxwell's equations yield only two boundary conditions. Thus, just as for the dipole-exchange modes discussed in §4.4, an additional boundary condition (ABC) is required. For the dipole-exchange mode the ABC can be found from the microscopic torque equations of motion, since the whole continuum theory may be derived from the microscopic theory. Similarly in the present case the ABC can in principle be derived from the quantum-mechanical equation of motion of the exciton in the presence of the surface. However, since the exciton is an electron-hole pair, what Fig. 6.4 Sketch of exciton-polariton dispersion curves (full lines). Inset shows origin of curve in crossing of photon and exciton curves. The two transverse modes propagating at frequency co > coL are indicated. The longitudinal mode is also shown (broken line).

190

Bulk polaritons is required is in effect the solution of a quantum-mechanical three-body problem. The exact theory appears to be quite intractable, and in practice various ansdtze are used. One popular form, analogous to (4.47) and (4.48), is dP/dz + £P = 0

(6.13)

where P is the excitonic polarisation. Expressions for the reflection and transmission coefficients with this ABC are given by Tilley (1980), for example. An alternative description of the surface is called the dead-layer model. It is assumed that P = 0 in a layer of thickness 2aB at the surface. The boundary conditions that are applied are the usual electromagnetic ones at the two boundaries of the dead layer, together with P = 0 at the interface between the dead layer and the rest of the medium. For p-polarised incident light the situation is somewhat more complicated. In addition to the two transverse modes, a longitudinal mode is excited; as seen from (6.12) its wavevector qL is given by s(qL, a>) = 0. The ABC, (6.13) or whatever else is used, then has two components, so that altogether there are four boundary conditions. These are sufficient to determine the reflection coefficient R, two 'transverse' transmission coefficients 7\ and T2 and a longitudinal' transmission coefficient TL. It might be thought that ordinary reflectivity measurements would discriminate between different ABCs. In practice, however, because of the number of free parameters and variability of specimens, experimental data can be fitted by a range of ABCs. Some discussion of the reflectivity experiments is given by Weisbuch and Ulbrich (1982). The use of Brillouin scattering to investigate exciton-polaritons was suggested by Brenig, Zeyher and Birman (1972), BZB for short. As shown in Fig. 6.4, light of frequency a> > a>L incident on the medium generates two transmitted waves, at wavevectors qx and q2 say. As a simplification, let us first ignore complications due to the absorption of light in the medium and reflection of phonons at the surface. The Brillouin scattering may then be described by the kinematics of simple bulk scattering, as described in §1.4.2. For normal-incidence backscattering, the Brillouin process must scatter a polariton from one of the + q branches onto one of the — q branches. Conservation of frequency and wavevector, (1.52) and (1.53), shows that the frequency shift for Stokes scattering is found from a line of slope vs, where vs is the relevant acoustic phonon velocity. Likewise, anti-Stokes scattering requires a line of slope —v s. These are illustrated in Fig. 6.5. It is seen that for incident frequency co > coL, both Stokes and anti-Stokes lines split into four, corresponding to the four scattering possibilities 1 -• 1', 1 -> 2', 2 -• 1' and 2 -> 2'. 191

Surface polaritons A description of the scattering that includes light absorption and phonon reflection can be given by an extension of the calculation for normalincidence Brillouin scattering outlined in §2.5.2. The details are found in Tilley (1980). It emerges that if the frequency dispersion of the dielectric function is not too great, each scattering peak has the Dervisch-Loudon lineshape. The theory also gives expressions for relative intensities of the different lines. Resonant Brillouin scattering has been observed in a wide range of polar semiconductors. An example is shown in Fig. 6.6. It should be noted that the differential cross section changes very rapidly with incident photon energy, and for this reason these experiments are always performed with a tunable dye laser as source. It is seen from Fig. 6.6 that, as expected from the kinematics of Fig. 6.5 only one Stokes peak is seen for a> < a>L, whereas a multiplet is observed for ftco = ha>L + 0.25 meV. For larger co, the multiplet is washed out by the increasing opacity broadening. The peak positions in results like those shown in Fig. 6.6 can be fitted Fig. 6.5 An illustration of the kinematics of Brillouin scattering via exciton-polaritons [after Weisbuch and Ulbrich 1982].

*r

192

Bulk polaritons

by the exciton-polariton model of Fig. 6.5, and by this means the polariton parameters are deduced. An example of such a fit for GaAs is shown in Fig. 6.7. Theoretical fits like that shown in Fig. 6.7 are given by the simple kinematics of Fig. 6.5; they do not test the full theory for lineshapes and relative intensities. The full theory has been tested to a limited extent by So et al. (1981). Since the incident and scattered photon energies are close to the band gap (that is the significance of the word resonant) the refractive index changes very rapidly with photon energy. Consequently the simple expression for the Stokes :anti-Stokes ratio given in (1.54) no longer applies. The results of Fig. 6.6 serve as an illustration. For a>L the anti-Stokes intensity is greater than the Stokes, and the ratio of the intensities clearly varies with photon energy. Thus the frequency dependence of the Stokes:anti-Stokes ratio may be used for comparison of theory and experiment, as shown in Fig. 6.8. The two theoretical curves shown correspond to slightly different ways of extending the theory of §2.5.2. In the space available here, it has not been possible to give more than a brief introduction to exciton-polaritons. A much fuller account is given by Weisbuch and Ulbrich (1982). Fig. 6.6 Resonant Brillouin scattering in GaAs at 12 K for incident light frequencies around coL (hcoL = 1.60 eV). Stokes lines are to the left [after Weisbuch and Ulbrich 1982^

AJ^^A^tr^^Wr

1|B^^^^J

Sk

+0.25

-0.30

-0.55

-^W -0.2

I 0

V^_ 1.40 +0.2

A£[meV]

193

Surface polaritons 6.2 Surface polaritons: single-interface modes 6.2.1 Basic properties

It has been known for many years that an electromagnetic wave can propagate along an interface between two media of which at least one is dispersive. In fact the calculation to be presented in this section for a mode at the interface between two isotropic media appears as a problem in Landau and Lifshitz (1971). Interest increased greatly from the late 1960s, when ATR was established as a viable technique for experimental study of surface polaritons. Detailed reviews of that work are given by Otto (1974 and 1976) as well as in several articles in Burstein and De Martini (1974). For a recent account of the whole subject, the reader may refer to Agranovich and Mills (1982). In this chapter, as elsewhere in the book, we shall give theory first, followed by a discussion of experimental results. Thus this section and the next are theoretical; the ATR experiments are the subject of §6.5. We consider the plane interface between two semi-infinite dielectric media; as shown in Fig. 6.9 we take the z axis normal to the interface and the x axis as the direction of propagation of the mode. Explicit Fig. 6.7 Measured Brillouin shifts as a function of incident photon energy in GaAs. The full lines are theoretical curves deduced from the model indicated in Fig. 6.5 [after Weisbuch and Ulbrich 1982].

+ 0.5

-0.2

0 AE (meV)

194

+0.2

Surface polaritons: single-interface modes calculation shows that there is no surface mode with the E field in the y direction, so we therefore take E in the xz plane: E = (Elx, 0, Elz) exp(iqxx - icot) exp(i
(6.14)

in medium 1, with a similar form in medium 2. We take a single frequency a>, and since boundary conditions will be applied on the whole plane z = 0 the wavevector qx must be the same in both media. Clearly qx is the propagation vector, and one of our aims is to find the dispersion equation
For the mode to be localised, E must decrease with distance from the interface. Thus lm(qlz) > 0 and lm(q2z) < 0. In order for (6.14) to satisfy Fig. 6.8 Stokes:anti-Stokes intensity ratio versus wavelength for resonant Brillouin scattering via exciton-polaritons in Cu 2 O. Full line, theory of So et al. (1981a); broken line, theory of Tilley (1980) [after So et al. 1981b]. 1.4-1

1.1-

\ 0.8-

0.5576.6

577.0 577.4 Wavelength (nm)

577.1

Fig. 6.9 Notation for surface-polariton calculation. Medium 1 occupies the half-space z > 0, and medium 2, z < 0.

z= 0

195

Surface polaritons Maxwell's equations in both media, the wavevectors must satisfy o)2/c2

i=l,2

(6.15)

In particular, this means that when sx and e2 are real, the localisation requirement is ql > e^/c2. The equation V • D = Ogives the ratios of the field amplitudes in (6.14): qxEiX + qizEiz = 0

1=1,2

(6.16)

The amplitudes in the two media are related by continuity of the tangential component of E and the normal component of D: Elx = E2x

(6.17)

£iElz = s2E2z

(6.18)

Equations (6.16) to (6.18) are four homogeneous equations in the four field amplitudes, and the solvability condition is the required dispersion relation. Equations (6.16) and (6.17) give Eiz in terms of Elx, then substitution in (6.18) gives the dispersion relation in the implicit form qiz

q2z

(6.19)

£2

Substitution from (6.15) leads to the explicit result

alA^

C2 £i + £ 2

(6-20)

As for bulk polaritons, we start by neglecting damping, so that e1 and e2 are both real. It then follows from (6.15) and the localisation requirement that qlz and q2z are both pure imaginary, and we write *iz = i*i (6-21) q2z=-iK2 (6.22) where the signs incorporate the localisation condition. Equation (6.19) now shows that for the surface polariton to exist, e1 and s2 must have opposite signs: £l£2<0

(6.23)

Since the right-hand side of (6.20) must be positive, we have the further condition e1+e2<0 (6.24) Equations (6.23) and (6.24) determine the frequency intervals in which the surface polaritons occur. In many experimental situations, one of the dielectric functions is frequency-independent and positive, as is the case if medium 1, say, is vacuum. In that case, (6.23) and (6.24) show that the other dielectric constant, s2 say, has to be negative, e2 < — £ t . Medium 2 is then called the surface-active medium. 196

Surface polaritons: single-interface modes As a concrete example, we consider the interface between vacuum, with ex = 1, and a polar medium described by the dielectric function of (5.60) (see also Fig. 5.9). Equation (6.24) then shows that the surface polariton occupies the frequency interval coT<(Ds

(6.25)

where cos is given by fi2(G>s)=-l

(6.26)

or explicitly ^s2 = (eoo^ + a)|)/( £oo + l)

(6.27)

As anticipated, the frequency interval is within the stop band for bulk polaritons. The dispersion curve is illustrated in Fig. 6.10; the asymptotic values are qx -• co/c

\qiz\ -• 0

a s co -* coT

(6.28)

qx -• oo

\q iz\ -• oo

a s co -+ cos

(6.29)

and These limiting values for \qiz\ show that the surface polariton is weakly localised at the low-frequency end, and strongly localised at the high-frequency end. For a second example, we take e1 = l and s2(co) in the plasma form of (5.51), again with damping neglected. The equations replacing (6.25) and (6.27) are 0 < co < cos

(6.30)

G>| = a # 2

(6.31)

and The dispersion curve is illustrated in Fig. 6.11; equations (6.28) and (6.29) apply on the understanding that coT is replaced by 0. The surface modes at the interface between a vacuum and a medium with dielectric properties specified by (5.52) are discussed by Palik et a\. (1976). Equations (6.16) to (6.19) may be used to find the relative amplitudes and phases of the electric-field components in the two media, and the magnetic-field components can be derived from Maxwell's equations in the usual way. Furthermore, Poynting's theorem can be applied to find the directions and magnitudes of the energy flow in the media; details are given by Nkoma et al. (1974). The results are illustrated in Fig. 6.12 for the case when 81>0. 197

Surface polaritons Fig. 6.10 (a) Surface polariton dispersion curve for a Ga As/vacuum interface. Numerical values for GaAs are quoted in Fig. 5.9; with these values cos/coT= 1.081. {b) Dispersion curve of (a) on an expanded scale. 2.00

1.08T

1.00

198

1.50

2.00

2.50

3.00

Surface polaritons: single-interface modes Fig. 6.11 (a) Surface plasmon-polariton dispersion curve for a metal/vacuum interface. The graph applies for any metal with the dielectric function of (5.51). (b) Dispersion curve of (a) on an expanded scale. 3.0

2.0

1.0

2.0

1.0

3.0

0.6-

0.4-

0.2-

1.0

2.0

3.0

199

Surface polaritons 6.2.2 Electrostatic approximation The frequency a>s for which qx -> oo in dispersion curves such as those of Figs. 6.10 and 6.11 is of particular interest. For example, in electron energy-loss experiments on metals the momentum transfer is relatively large, and a loss peak is observed at a>s. The condition qx » a>/c means that the wavelength 2n/qx of the wave along the surface is much smaller than the free-space wavelength Inc/co. Retardation effects, that is, the nonzero propagation time of an electromagnetic signal, are therefore unimportant, and the asymptotic wave with co ~ cos and qx » a>s/c is called an electrostatic surface wave. It is seen from (6.20) that the (fixed) frequency of the electrostatic surface wave is given by 81+£2=0

(6.32)

with explicit expressions for the vacuum-bounded polar dielectric and plasma given by (6.27) and (6.31) respectively. In view of the importance of the result, we now derive (6.32) by means of an explicit electrostatic approximation. The calculation is the analogue of that presented for the magnetostatic surface wave in §4.2. In the electrostatic approximation, media 1 and 2 are described by potentials Kx and V2 satisfying V 2V t = 0

i = 1,2

(6.33)

Fig. 6.12 Electric and magnetic field amplitudes and Poynting vector for an interface with Ei > 0 (upper medium) and s2 < 0 (lower medium), {a) Electric field pattern in the xz plane. The field magnitudes are oscillatory in the x direction but decrease exponentially away from the boundary, {b) An analogous diagram of the magnetic field pattern in the xy plane, (c) Poynting vector pattern close to the boundary. The energy flow in medium 1 is greater than that in medium 2, giving a net flow of energy to the right when qx is positive [after Nkoma et al. 1974].

— \

\

f

(a)

yi

t

t

t

t

!

t

(b)

I

I — \

\ 200

—

\

Surface polaritons: single-interface modes A plane wave localised at the interface and travelling in the x direction is described by Vx = V10 Qxp(-Kxz) exp[i(qxx - cot)~]

(6.34)

with a similar expression for V2. Substitution into (6.33), which involves only spatial derivatives, gives Kf

= ql

i=l,2

(6.35)

Thus the potentials are Vx = V10 cxp(-qxz)

exp[i(qxx - cot)]

V2 = V20 exp(qxz) exp[i(gxx - cot)]

(6.36) (6.37)

with fields E = — W given by E1 = ( - i « x , 0 , ^ ) F 1

(6.38)

E2 = ( - i ^ , 0 , - ^ ) K 2

(6.39)

Equations (6.36) and (6.37) have the property, which they share with magnetostatic waves in the Voigt configuration, that the decay constant normal to the interface is equal to the wave vector along the interface. Obviously this follows from the fact that the potential satisfies Laplace's equation (6.33). It is seen further from (6.38) and (6.39) that the field components Ex and Ez in either medium are equal in magnitude but 90° out of phase. This is a special case of the general surface-polariton field patterns illustrated in Fig. 6.12. In order to derive the frequency of the electrostatic surface wave, we apply standard boundary conditions to (6.38) and (6.39). Continuity of Ex gives V10 = V20

(6.40)

and then continuity of Dz gives ^(1x^10=

-^iClxV20

(6-41)

In view of (6.40), the above equation reduces, as expected, to the result previously derived in (6.32). The calculations we have presented in this section apply to linear media, in which the dielectric functions are independent of field strength. Laser sources of sufficient intensity are nowadays available for nonlinearities in the dielectric response to be important. The inclusion of nonlinearities leads to a number of new effects, for example the appearance of a 'self-guided' TE mode for incident power above a threshold intensity. Self guiding is more easily understood by reference to two-interface modes, which are the subject of the next section, so the discussion is deferred until §6.4. 201

Surface polaritons 6.2.3 Anisotropic media An important extension of the results so far derived is to the case when one of the media is anisotropic. A brief history with references is given by Mirlin (1982); convenient reviews are given by Borstel et al. (1974) and by Borstel and Falge (1977 and 1978). Here we shall simply quote a result that will be useful in later sections. We continue to follow the notation of Fig. 6.9, so that (x, y, z) are 'surface-related' axes. Now, however, we suppose that medium 1 is isotropic but medium 2 is anisotropic, so that its optical properties are characterised by a dielectric tensor s2(co). This tensor has principal axes, (x', / , z') say, and considerable algebraic complexities arise because in general these axes are at an arbitrary orientation with respect to the surface-related axes (x, y, z). As before, we consider a mode propagating in the x direction, q,, = (qx, 0). It is sufficient for a simple illustration to consider the special case when the y axis is a principal axis of s2(co), y — y' say. In this case the boundary conditions at the surface do not mix s- and p-polarisations, and we choose to deal with the latter, E = (Ex, 0, Ez). With these simplifications, it can be shown that the dispersion equation for the surface polariton is

rf-^%^ C

CyO.

(6-42)

£1

Here e'x and ez are the principal values of s2 in the x'z' plane, while ezz is the appropriate component of s2 in (x, y, z) axes. The reader is invited in Problem 6.4, to verify the special case of (6.42) when (x, y, z) and (x\ / , zf) axes coincide. One application of these results is to ferroelectrics. In §3.6 we dealt with surface magnons in the transverse Ising model, which describes hydrogenbonded ferroelectrics like KDP. For long wavelength modes, coupling to the electromagnetic field becomes important. Many ferroelectrics, both hydrogen-bonded and the other main class, displacive, can be characterised in the simplest approximation by a uniaxial dielectric constant of the form /e±

0

0'

£ = J 0 s± 0 I

(6.43)

\ 0 0 in terms of principal axes. The component £(| has a significant frequency and temperature dependence arising from the phase transition, while s± can be treated as constant. In a displacive ferroelectric, such as BaTiO 3 , the transition at a temperature Tc from paraelectric to ferroelectric phases consists of a relative displacement of the positive and negative ions in the 202

Surface polaritons: single-interface modes lattice. This displacement has the same symmetry as a long-wavelength optic phonon. Thus the restoring force for the optic phonon decreases as T -> Tc, and the phonon frequency coT decreases. This means that e^ can be represented by the TO phonon form of (5.60), with a temperaturedependent resonant frequency. Near to Tc, however, the damping becomes important and other mechanisms come into play with the effect that £|| has a more complicated form. It is clear that in principle the surface-polariton properties may be used to investigate the behaviour of g||. Some discussion of the possibilities is given by Cottam et al. (1984), but there have not yet been any experimental investigations.

6.2.4 Charge-sheet modes In §5.3.4 it was mentioned that in suitable circumstances very thin sheets of mobile electrons can be trapped at an interface, and that these sheets can be described as a 2D electron gas. As we now show, a surface-polariton-like mode is predicted to propagate parallel to the charge sheet, with the amplitudes localised at the position of the sheet. The derivation of the dispersion relation is a straightforward extension of that given in §6.2.1 for the ordinary surface polariton. We use the axes of Fig. 6.9 and the notation of §6.2.1, but we now suppose that a 2D charge sheet of particle density v m~ 2 is localised at the interface z = 0. Equations (6.14) to (6.17) still apply, but (6.18) must be replaced. It is equivalent to the continuity of the field component Hy; since there is now a sheet of mobile charge at the interface the correct form is H2y-Hly=jx

(6.44)

where j x is the current density in the charge sheet. As in the 3D plasma, j x is driven by the alternating electric field at the interface, Elx, which from (6.17) is equal to E2x. We ignore collisions in the electron gas, so we can write from the 2D equivalents of (5.43) and (5.47) j x = ive2Elx/mcc>

(6.45)

Substitution of X, from (6.45) in (6.44) gives the relation that replaces (6.19), namely ^1 + ^ = ^ Kx

K2

CO2

(6.46)

where KX and K2 are still given by (6.21) and (6.22). The frequency Q s is given by Q s = Ve2/eomc

(6.47) 203

Surface polaritons In the electrostatic limit, it follows from (6.15), (6.21) and (6.22) that KX and K2 are both replaced by qx. Equation (6.46) then simplifies to o) = inscqx/(si+s2)y/2 (6.48) The dispersion relation (6.46) was first derived by Nakayama (1974) by the method described here. Equation (6.48) was derived earlier by Stern (1967), who used linear-response theory rather than simple dynamics to describe the electron sheet. An important property of (6.46) and (6.48) is that a localised mode can propagate even when £x and e2 are both positive. This is in striking contrast to the ordinary polariton, for which e1 and e2 must have opposite signs, (6.23). The dispersion curve for ex = e2 is shown in Fig. 6.13. There has not been a great deal of experimental work on these plasmon modes of a single charge sheet. One exception is the study by Batke and Heitmann (1984) of space-charge layers produced at a semiconductorinsulator interface by application of a voltage to a superposed electrode. They couple far-infrared radiation to the plasmon by means of a grating, as will be discussed briefly in §6.5.1. Their transmission spectra show clear evidence of the charge-sheet plasmon. 6.2.5 Surface exciton-polaritons In §6.1.2 we saw how the centre-of-mass momentum of the exciton leads to significant spatial dispersion in the dielectric function. We now consider how the calculation of surface-polariton properties, as given in §6.2.1, is modified by spatial dispersion.

Fig. 6.13 Charge sheet dispersion curve corresponding to (6.46) for ex = e2, both taken positive. 10 r

Cf

100

204

200

Surface polaritons: single-interface modes In the absence of spatial dispersion, the surface polariton is found in some frequency range within the bulk-polariton stop band; examples are given in (6.25) and (6.30). As shown in Fig. 6.4, however, the bulk exciton-polariton has no stop band; the branch labelled 2 there has the large-g behaviour co ~ Dq2 because of the spatial dispersion. The situation is closely analogous to that described for dipole-exchange modes in §4.4. In the absence of exchange effects, pure magnetostatic surface modes propagate, as seen in §4.2. The inclusion of exchange, which is necessary for larger q values, means that the magnetostatic mode becomes 'leaky', and radiates into the bulk mode with which it is degenerate in frequency. Similarly in the present case there is no uncoupled surface excitonpolariton; instead we have a leaky surface mode. A theoretical treatment of the surface exciton-polariton analogous to that given in §6.2.1 for the phonon- or plasmon-polariton was given for the first time by Maradudin and Mills (1973). In their calculation the medium is described by the isotropic dielectric function of (6.11); we summarise it with reference to Fig. 6.9. It is assumed that the mode is p-polarised, so that in the non-surface-active medium the electric field is given by (6.14) with wavevector qlz satisfying (6.15). In the excitonic medium, however, q2z is found as a function of qx and co from (6.12). In the absence of damping, (6.12) has one real root for q as a function of co, namely branch 2, for co < coL, and three real roots for co > coL. There is no frequency range in which only imaginary roots are found. Likewise, when (6.12) is regarded as giving q2z as a function of qx and co, the root corresponding to branch 2 is always real. With damping present, this root is off the real axis, but for a realistic value of the damping it is still predominantly real. The other two roots for q2z are predominantly imaginary for small co, and real for larger co. As in the calculation of reflection and transmission of p-polarised light, mentioned in §6.1.2, the electric field in the excitonic medium has three components, so that the analogue of (6.14) is E= t

Plx, 0, E\z) exp(i
(6.49)

Here a labels the three solutions of (6.12) for q2z. The amplitude E2z is related to Ea2x by V • D = 0 for the transverse modes, and by V x E = 0 for the longitudinal mode. In all, then, four field amplitudes have been defined, namely Elx and Ea2x for a = 1, 2, 3. Four linear homogeneous equations in these amplitudes are found from the two electromagnetic boundary conditions and two ABCs resulting from (6.13) or whatever is used instead. The condition for these equations to have a solution gives the dispersion 205

Surface polaritons

relation qx((o). As anticipated, it is found that even in the absence of damping qx is complex, so that the surface wave is attenuated. The attenuation is due to energy leakage into the bulk branch 2. With the damping parameter in (6.11) included, the attenuation of the surface mode increases. As was seen in §6.1.1, the definition of a dispersion curve becomes ambiguous in the presence of attenuation, since the dispersion relation is then between two complex variables a> and qx. Nevertheless, for small attenuation a dispersion curve can be a useful guide to the eye, and an example of a calculated curve is shown in Fig. 6.14. The calculation outlined here assumed an isotropic medium; many excitonic media are anisotropic, and the calculation must be appropriately generalised. The experimental study of surface exciton-polaritons began with studies of the ZnO surface by Lagois and Fischer (1976) and De Martini et al. (1977); a detailed account of this and later work on ZnO is given by Lagois and Fischer (1982). The former group used the method of attenuated total reflection (ATR), and a discussion of their results is deferred to §6.5 which gives a detailed account of ATR. The latter group generated the surface exciton-polariton by means of frequency-doubling the light from an intense dye laser. The principle of the method is sketched in Fig. 6.15(a). ZnO Fig. 6.14 Calculated dispersion curve of surface exciton-polaritons for complex frequency GO and real wavenumber qx. The calculation was done for a ZnO exciton using the ABC P = 0, which is a special case of (6.13) [after Lagois and Fischer 1978],

206

Surface polaritons: single-interface modes does not have a centre of inversion symmetry, so the incoming light of frequency v and in-plane wavevector kx can generate a surface mode of frequency co = 2v and wavevector qx = 2kx. Detection of the surface mode was simple in principle. As will be seen at the end of §6.5 in a little more detail, roughness on the specimen surface destroys the in-plane translational invariance, and allows the surface mode to radiate. Wavevector qx can Fig. 6.15 (a) Sketch to show principle of nonlinear optical excitation of surface excitonpolariton on ZnO used by De Martini et al. (1977). (b) Experimental results for real and imaginary part of qx = q'x + \q"x obtained by De Martini et al. Curves are theoretical with parameters 6^ = 6.15, e(0) = 6.172, ha>T = 3.421 eV and fcr = 0.25 meV [after De Martini et al. 1977].

(a)

12.5

3.422

2.0

2.2 5

2.4

1

Wavevector ^ ( 1 0 cm" )

207

Surface polaritons

be varied by varying the angle of incidence 6. For given excitation frequency v and mode frequency a> = 2v the wavevector qx corresponding to peak intensity gives a point on the dispersion curve, while the linewidth in qx is a measure of I m ^ ) . The results obtained are shown in Fig. 6.15(fo). It may be noted that De Martini et al. analysed their results using a model that lacks spatial dispersion; the parameters quoted in the caption to Fig. 6.15 correspond to the theoretical expression of (5.60). Later, however, Fukui et al. (1980) showed that a good account can also be given when spatial dispersion is included. Most recent experimental and theoretical work on surface excitonpolaritons has been concerned with either ATR or nonlinear generation, although Brodin et al. (1984) carried out luminescence studies on ZnTe. Fukui and Tada (1982) proposed that as an extension of the method described in §6.1.2 surface exciton-polaritons could be probed via resonant Brillouin scattering. The principle is sketched in Fig. 6.16. Fukui and Tada give a detailed theoretical analysis, which is a generalisation of that outlined in §6.1.2, but no experimental results have yet been reported. 6.3 Surface polaritons: two-interface modes

We now extend the calculations of the previous section to a geometry with two plane interfaces. The notation is indicated in Fig. 6.17. Fig. 6.16 Principle of resonant Brillouin scattering experiment proposed by Fukui and Tada (1982). Possible scattering processes are indicated [after Fukui and Tada 1982].

208

Surface polaritons: two-interface modes

As before, we take the z axis normal to the interfaces, which are situated at z = 0 and z= —L. The x axis is taken along the direction of propagation. As was discussed in §2.3 for surface elastic waves, for sufficiently large L independent surface polaritons propagate on each interface. If we imagine L decreasing, thefieldsof the two surface polaritons propagating for a given qx start to overlap, and the degeneracy in frequency between them is lifted. Thus we expect to see two modes, essentially a bonding and an antibonding combination. The behaviour for magnon-polaritons is rather different (see §6.7), but even in the case of phonon- or plasmonpolaritons there are other possibilities from that just described. In the surface-polariton calculation of §6.2.1 q2z was constrained to be imaginary in order that the polariton field should tend to zero as z -• — oo. In the present case, qlz and q3z must be imaginary (in the absence of damping), but q2z can be either imaginary or real. Imaginary q2z corresponds to the description just given, where the surface polariton on one interface is perturbed by the presence of the other interface. For real q2z, on the other hand, the z-dependence of the fields in medium 2 is oscillatory, so that medium 2 is behaving like a waveguide. For this reason the mode is called a guided-wave polariton.

The previous paragraph was concerned, implicitly, with p-polarised modes. Once the possibility of guided-wave polaritons is recognised it is necessary to discuss also s-polarised guided-wave modes. It turns out that these can occur, and indeed so far they have been the modes of most importance in applications of dielectric waveguides. We first derive the governing equations for the modes, both surface-polariton and guided-wave type. The calculation is a straightforward extension of that given in the previous section, and some of the

Fig. 6.17 Notation for two-interface calculations.

z= 0

-L

209

Surface polaritons equations presented there still apply. We write the E fields in the three media as E = (Elx, 0, -qxEJqlz)

exp(iqxx - icot) exp(i^flzz)

z> 0

(6.50)

E = {a exp[i
0> z> —L

(6.51)

E = (£ 3 x , 0, - qxE3x/q3z) exp(iqxx - icot) exp[i
z< - L (6.52)

Here (6.16) has been applied to relate the z and x components in each medium. The localisation condition is Im(g lz ) > 0 and Im(g 3z ) < 0, which requires ql>efo2/c2

i=l,3

(6.53)

The four amplitudes Elx9 a, b and E3x are related by four boundary conditions, two at each interface. They take the form af+bf-'=Elx fi2(^-V1)/«2, qf'1-bf=E3x

(6.54) = Ci£i7«i.

(6-55) (6.56)

*2W~l ~ bf)/q2s = e3E3x/q3z

(6.57)

/=exp(i
(6.58)

where Equations (6.54) to (6.57) are four homogeneous equations in the amplitudes, and the solvability condition gives the dispersion equation. It is readily seen that the general form is

± ^

(6.59)

This is somewhat too general for comfort, and we therefore specialise to the symmetric geometry, sx =s3. The localisation condition becomes qlz= — q3z = \K1

wit\\K1>0

(6.60)

and (6.59) is seen to have two solutions exp(i4 2z L)= ±(elq2z + ie2K1)/(e1q2z-i£2Kl)

(6.61)

Equation (6.61) applies equally to surface-polariton and guided-wave modes. For surface-polariton modes, q2z is pure imaginary, g 2z = k 2 . The two solutions of (6.61) can then be reorganised into the forms

210

e2/e1 = —(K2/K1)

tanh(^K:2L)

e 2 / fil = -(KjKi)

coth(^c 2 L)

(6.62) (6.63)

Surface polaritons: two-interface modes As L -• oo, the hyperbolic functions in (6.62) and (6.63) tend to one, so that both equations are asymptotically the same as the dispersion relation for the single-interface mode in the form given by (6.19), (6.21) and (6.22). Thus, as expected, (6.62) and (6.63) do describe the bonding and antibonding combinations of the surface-polariton modes. The general form of the solutions is illustrated for two different thicknesses of LiF slab in Fig. 6.18. So far in this section it has been assumed that all three dielectric constants are real, so that as usual in the discussion of dispersion relations damping has been neglected. Fukui et al. (1979) and Sarid (1981) carried out numerical studies of the effect of including damping. In both cases, the starting point was the dispersion equation, (6.59) in general, or (6.62) and (6.63) for a symmetric case. Fukui et al. took qx real and calculated the imaginary part of co, while Sarid took co real and calculated lm(qx). These different assumptions correspond, ultimately, to different experimental arrangements. For example, real co corresponds to a wave launched along the film at that frequency, and 1/Im(gx) is then a measure of the decay length of the wave along the film. Both Fukui et al. and Sarid considered the particular case of metal films and plasmon-polaritons, so that the dielectric function of the film had the form of (5.51). The important result obtained by both groups concerns the higher-frequency mode. Sarid's

Fig. 6.18 Two-interface surface polaritons for a LiF slab bounded by vacuum. 1, qTL = 0.1 (L = 0.5 /an); 2, qTL = 0.2 (L=1.0/mi); 3, qTL>2 (L> 10 /mi). Here qT = coT/c. The full curves represent the calculation with retardation, the broken curves without retardation [after Bryksin et al. 1974].

1.5

0

1

2

10

211

Surface polaritons result is that its decay length increases as the film thickness decreases, and correspondingly Fukui et al. find that the lifetime 1/Im(a>) increases as the thickness decreases. The mode is therefore called the long-range surface plasmon (LRSP). In the LRSP for a film between identical dielectrics the electric-field pattern is antisymmetric across the film, while in the lower-frequency mode the electric-field pattern is symmetric. The long decay length of the LRSP may be seen as arising from the fact that as the film thickness decreases a smaller proportion of the mode energy is transported within the film. Experimental confirmation of these results has been obtained by a number of groups, one of the earliest to appear being that by Craig et al. (1983). A related experimental curve obtained by ATR is shown subsequently. In applications to nonlinear optics it is helpful to use long-range modes, so the LRSP is likely to be of considerable technical importance. As stated, the earlier work was concerned with surface plasmonpolaritons. The extension to surface phonon-polaritons was carried out by a number of authors; a good account is given by Wendler and Haupt (1986a), who have also discussed (Wendler and Haupt 1986b) the surface plasmon-phonon-polariton found in a doped semiconductor when the plasma frequency is close to the TO phonon frequency. We now turn to guided modes in p-polarisation. For real q2z, the two forms of (6.61) are fiz/fii = (
(6.64) (6.65)

as can also be seen directly from (6.62) and (6.63). The numerical, or graphical, task of solving these is similar to that of finding the odd and even parity modes in a quantum-mechanical square well, or of finding the dispersion curves for the Love waves mentioned in §2.3.1. For small L, there is only a small number of modes; the number increases as L increases until for large enough L the guided-wave modes form a quasi-continuum, which in the limit becomes the bulk continuum. The dispersion curves are illustrated for a moderate value of L in Fig. 6.19(a). Comparison with Fig. 6.10(a) shows how the curves occupy the parts of the a>qx plane that ultimately as L -• oo are occupied by the bulk continuum. The calculation for the s-polarised guided-wave modes is similar to that for the p-polarised modes, and the details are left for a problem. It emerges that the equations for p-polarisation can be converted to those for s-polarisation by the substitutions £2
(6.66)

Surface polaritons: two-interface modes (6-67) Thus the general result, taken over from (6.59), is (6.68) while for a symmetric geometry the equations are (6.69) (6.70) The corresponding dispersion curves are shown in Fig. 6.19(b). We should make an elementary comment which is implicit in the formalism presented. In applications, dielectric waveguides are usually fabricated from materials which all have positive dielectric functions. In the geometry discussed, Fig. 6.17, an essential condition for guided waves to propagate is that q2z should be real while qlz and q3z are imaginary.

Fig. 6.19 Dispersion curves of guided-wave polaritons in a film placed in vacuum, {a) ppolarised, (b) s-polarised [after Ushioda and Loudon 1982]. (a) TM modes

(D = Cqx/€2((i)Y'

(b) TE modes

213

Surface polaritons

In view of (6.15), this requires simply that s2 is the largest of the three dielectric constants, or in terms of the refractive indices ni = ej /2 , n 2 >max(n 1 ,n 3 ) (6.71) This is the waveguiding condition for the geometry of Fig. 6.17. The guided waves that have been described are of importance in engineering applications; further details are given in the relevant chapter of Marcuse (1982). It may be noted that engineers frequently use the notation qx =fico/c.Comparison with the equation q = rjco/c for a bulk wave in a medium of refractive index Y\ means that p is described as the effective refractive index. Theoretical and experimental results may be presented as a graph of /? versus a>; this amounts to a dispersion graph of the kind used here with the horizontal and vertical axes interchanged. 6.4 Surface polaritons: nonlinear effects

The results presented so far in this chapter have been derived on the assumption that the dielectric responses of all the media involved are linear. This assumption breaks down for sufficiently intense sources and media with strong nonlinear response. The inclusion of nonlinearities leads to a range of new physical effects, and in addition all methods of signal processing rely on the exploitation of nonlinearities. Thus both because of the physics involved and because of potential applications, the properties of nonlinear surface and guided waves are of great interest. At the time of writing the subject is in a very early stage of development, and the full range of possibilities is not mapped out. In this section, therefore, we give some introductory comments and present an exact calculation for the simplest case of self guiding. The interested reader may refer to the much fuller introduction to the subject given by Stegeman et al. (1986) and to the literature review contained in Wendler (1986). When the dielectric functions are linear the governing equations for the electromagnetic field in the presence of boundaries are a set of linear differential equations and boundary conditions. In that case a number of powerful and general results apply. In particular, the principle of superposition holds, so that the sum of two solutions is itself a solution. Furthermore, it is possible to find for a given frequency a complete set of solutions which have the property that an arbitrary solution at that frequency can be expressed as a linear combination of functions of the complete set. For the two-interface geometry of the previous section, a complete set comprises the surface-polariton modes and the guided-wave modes together with the 'radiation modes' in which the wavevector qz is real in media 1 and/or 3. In the presence of nonlinearity, these results no 214

Surface polaritons: nonlinear effects longer apply, so that a general discussion is much more difficult and we do not attempt anything of the sort. Nonlinear optics proceeds by expansion in a Taylor series of the nonlinear polarisation P N L induced by one or more intense electromagnetic fields. The lowest-order effect involves a susceptibility # (2):

^L2 = I * 8 W j,k

(6.72)

This generates wave-like solutions with sum and difference frequencies coa ± ojb, where a>a and cob are the frequencies of the fields E a and E b . The tensor x\]i *s °f third rank, and the number of nonvanishing elements depends in the usual way (Nye 1985) on the crystal symmetry of the medium. The next highest order involves three E components:

P?L3 = I zSi£J£j£f

(6.73)

j,k,l

One reason why effects depending on the fourth-rank tensor %(3) are of interest is that if all three amplitudes E a , E b , E c are at the same frequency co, then P N L 3 contains a component at frequency co, since co — co + co is among the frequencies produced. Thus %(3), in contrast to x{2\ can lead to nonlinear effects at a single frequency co. One such effect is the appearance of an intensity-dependent refractive index; in terms of the dielectric function we write E -• s + a|£| 2 . The example to be considered here is the self guiding of a TE mode at the interface between two media, one of which has such an intensity-dependent dielectric function. The original calculation was by Kaplan (1977). We use the notation and axes of Fig. 6.9, but it is assumed that ex is intensity-dependent: £1-»e 1+a1|E|2

(6.74)

while £2 is independent of intensity. We show that a TE mode can be 'self-guided' and propagate along the interface provided that OLX is positive and the intensity is sufficiently large. The physical reason follows from the comment made at the end of §6.3, that the waveguiding condition for a film is n2 >max(rc 1 , n3). A sufficiently intense wave propagating in medium 1 near the interface produces an increase in the refractive index near the interface because of the second term in (6.74). If this local' value is larger than the refractive index of medium 2, then the wave can be self guiding. Roughly, then, the intensity condition for guiding is Sx + OLi\E\2 > £2.

We assume we are dealing with a single-frequency mode propagating in the x direction, so that all amplitudes vary as exp(iqxx — icot), and as stated we consider a TE mode, so that the E field is in the y direction. 215

Surface polaritons The wave equation in medium 1 is ^

- (ql - cohjc2 - co^El\/c2)Ex = 0

If this is multiplied by dEJdz integral

(6.75)

it can be integrated to yield the energy

\ We are assuming here, as will be justified subsequently, that we can find a solution in which E1 is real. The constant of integration K has the same value for all z. However, we are seeking a solution localised at the interface, in which E1 -• 0 and dEJdz -* 0 as z -» oo. Substituting these asymptotic values, we see that K = 0. The solution of (6.76) is therefore

"kf

where, as in §6.2.1 K\

= q2x- co2ejc2

(6.78)

The integral can be evaluated in terms of hyperbolic functions, and the solution of (6.77) is Ex = (2/a1)1/2(cfc1/co) sechC/c^z - z 0 )]

(6.79)

Here z 0 is a constant of integration, which has yet to be determined. The lower medium is linear, and for a guided wave we have the solution familiar from §6.2.1 E2 = A exp(/c2z) exp(iqxx — icot)

(6.80)

where K2 is given by (6.78) with s2 in place of £x. The boundary conditions are that E and dE/dz are continuous, since the latter is proportional to Hx. Thus (2/a1)1/2(c/c1/co) sech^qzo) = A 1/2

(2/a1) (c/cf/co) tanh(K1z0) sech^Zo) = K2A

(6.81) (6.82)

Elimination of A yields jq tanh(fqz 0 ) = K2

(6.83)

Let us take stock of these equations. We may take the frequency co as given, so the unknowns are qx, A and z 0 . Unhappily we have only the two equations (6.81) and (6.82), so clearly something is missing. The clue is that we are dealing with a nonlinear system, and the form of the solution therefore depends in a nontrivial way on the wave intensity, or equivalently on the field amplitude. This is already clear from (6.79), where the amplitude has a value that depends on the other parameters *q and z 0 216

Surface polaritons: nonlinear effects as well as on the nonlinear coefficient oc1. We must therefore introduce the power explicitly by means of Poynting's theorem. The instantaneous power per unit length of y axis is

-I

(E x H) dz

(6.84)

and the intensity is given by the time-average < ^ > . Evaluation from the field expressions of (6.79) and (6.80) is elementary, if a little laborious. The instantaneous power &> has both a z and an x component. However, the former contains a factor cos(^x — cot) sin(qxx — cot) (recall that it is necessary to work in terms of real fields when applying Poynting's theorem) and therefore time-averages to zero. The z-directed power <^> is given by — sech 2(/c1z0) + 1 + tanhOqzo)

(6.85)

We now have the complete solution in principle, since (6.81), (6.82) and (6.85) determine qx, A and z 0 in terms of a; and <^>. A full solution requires numerical work, since all three equations are nonlinear. However, a number of qualitative features can be extracted from the solutions as they stand. First, since tanh(/c xz0) < 1, (6.83) shows that a solution is possible only if K2 < K1, or equivalently s2 > e^ Thus for self guiding to occur, the linear 'substrate' 2 must have a refractive index higher than the low-power refractive index e{/2 of the nonlinear 'cladding' 1. Second, (6.82) shows that z 0 must be positive, since from (6.81) A is positive. It is then helpful to sketch the right-hand side of (6.85), as is done in Fig. 6.20. The function plotted, namely the expression in square brackets in (6.85), has a minimum value of 2. This makes it plausible that there is a self-guided solution only if <^> exceeds some threshold value, as we anticipated on general grounds. Fig. 6.20 Sketch of (6.85) versus K^Q.

217

Surface polaritons Results of numerical calculations for the self-guided s-polarised mode discussed here are shown in Fig. 6.21. This shows both the real and imaginary parts of qx, the latter being evaluated by a perturbation method by use of Poynting's theorem to find the power loss per unit path length. The existence of a threshold power level shows clearly on the figure. Various extensions of the calculation reviewed in this section have been carried out. Wendler (1986) generalises the calculation to include the presence of a surface-active film between the two semi-infinite media. For a nonlinear film sandwiched between linear media, (6.76) still holds in the film, but the constant of integration K need not be zero since there is no boundedness condition. The field E in the film can then be expressed in terms of the Jacobi elliptic functions (Boardman and Egan 1985), of which (6.79) is a special case. As stated, further details and references are given by Stegeman et al. (1986). 6.5 Attenuated total reflection 6.5.1 Basic principles A basic property of the surface polariton discussed in §6.2 is that it is confined to the region of the coqx plane satisfying q\ > where, as in Fig. 6.9, ex is the dielectric constant of medium 1, occupying the half-space z > 0. This means that the surface mode cannot be excited by a light beam incident in the normal way in medium 1, since such a beam Fig. 6.21 Real and imaginary parts qf and q\ for a wave guided along the interface between a linear and a nonlinear dielectric. Parameters are e t = 2.4025-0.00 li, OLl/celel = 10" 9 m 2 W~ 1 , and e2 = 2.4336 or 2.56. Also shown are the field distributions for various q* values for s2 = 2.4336 [after Stegeman et al. 1986].

1.70

1.60

-I

-

-

—

—-5.

- 0.4 x 10-

- 0.3 xlO"3

s - 0.2x10"

(

\ = 2.4336

-—

Pi

1

i

50

100

.

Pn i 150

Power (mW mm"1)

218

/»,

i 200

0.1x10

Attenuated total reflection with qz real. For this reason the surface polariton satisfies ql + q2 = s^/c2 is called 'non-radiative'. The restriction to real qz is removed in the method of attenuated total reflection, ATR for short. Here we give an introductory account; more detail is to be found in review articles by Otto (1974, 1976), by Borstel et al. (1974) and by Abeles (1986). Three ways of coupling an incident beam in medium 1 to surface polaritons are illustrated in Fig. 6.22. In the grating method, Fig. 6.22(a), a diffraction grating is laid down on the surface of interest. The incident light has surface wavevector qx = s{/2(co/c) sin 0,. By Bloch's theorem, (see §1.1), this couples to modes with qx values qx = s\/2(co/c) sin 6} + mln/d

(6.86)

where d is the grating periodic distance and m is an integer. Clearly for appropriate values of m and d coupling to nonradiative modes is possible. Fig. 6.22 Techniques for coupling light to surface polaritons: (a) grating method; (b) ATR, Otto configuration; and (c) ATR, Raether-Kretschmann configuration. In each case, the surface-active medium is shown shaded.

(a)

to 219

Surface polaritons When co, 9X and the order m are such that a surface mode is excited, the intensity of the diffracted light in that order is reduced. These reduced intensities observed with a diffraction grating are known as Wood's anomalies (Wood 1935). As a method of observing surface modes, the grating method has the disadvantage that the presence of the grating perturbs the properties of the mode under study. It is therefore not described further. Two different versions of the ATR method are sketched in Figs. 6.22(b) and (c). Both methods involve, strictly speaking, a three-layer geometry, and in both the light is incident in a prism of dielectric constant s1. In the Otto configuration, Fig. 6.22(b), a spacing medium, typically vacuum, is adjacent to the prism, with the surface-active medium as medium 3. The prism is chosen with a relatively large refractive index, ex > 62, and the angle of incidence 0, is greater than the critical angle 8C for total internal reflection, 01>8c = sin~1(el/2/s\/2). Thus if medium 3 were absent, the incident light would be totally reflected. However, total reflection requires the presence of an evanescent mode with decreasing amplitude (imaginary qz) in medium 2. The tail of the evanescent mode can excite a surface mode on the 2/3 interface; this surface mode removes energy, and thus produces an attenuation of the total reflection. Put a little more precisely, the in-plane wavevector qx = e\/2{co/c) sin 9X

(6.87)

is the same in all three media. Clearly with sufficiently large £ x and 6Y this can be made to satisfy the condition qx > e\/2(o/c

(6.88)

necessary for excitation of a surface mode at the 2/3 interface. Then when the incident frequency co and qx lie on the dispersion curve of a surface mode the reflectivity is reduced below unity. The main technical problem in the application of the Otto configuration is the control and uniformity of the spacer thickness, i.e. the thickness of medium 2. There is a short discussion of the experimental techniques in Otto (1976). In the Raether-Kretschmann configuration, Fig. 6.22(c), the surfaceactive medium is deposited direct on the face of the prism. Thus the method is most readily applicable when the material of medium 2 is easily evaporated, and the configuration has been mostly used to study surface plasmon-polaritons on metals such as Al and Ag. In this case s2 is negative, and the electromagnetic field is automatically evanescent in medium 2. The surface mode of interest is at the 2/3 interface, and the necessary 220

Attenuated total reflection condition for excitation is qx>sl/2co/c (6.89) in place of (6.88). As in the Otto configuration, the reflectivity is reduced when co and qx lie on the dispersion curve. Two forms of experimental scan are possible in ATR in either geometry. First, there is a frequency scan with angle 0, fixed. It follows from (6.87) that in this case a straight-line trajectory is scanned in the coqx plane. Second, there is a scan over angle atfixedfrequency; the trajectory is then a straight line parallel to the qx axis. The two possibilities are shown schematically in Fig. 6.23. In either case, where the scan trajectory crosses the surface-mode dispersion curve a dip should occur in the reflection coefficient, since the surface mode is excited. As previously remarked, both the Otto and the Raether-Kretschmann configuration involve a three-medium geometry. Thus the properties of the surface mode at the 2/3 interface are perturbed by the presence of the 1/2 interface. If the thickness D of medium 2 is too small, then the system is 'overcoupled', and the properties of the isolated 2/3 interface will not be observed. If D is too large, on the other hand, the system is 'undercoupled', and the 2/3 interface mode is only weakly excited, so that only a small reflectivity dip is seen. Choice of optimum D is therefore an important part of experimental design. Fortunately, as we shall now see, the theoretical expression for the reflectivity is readily derived, and numerical simulations can be performed to give a guide to the choice of D value. Fig. 6.23 Schematic comparison of frequency scan and angular scan in ATR [after Otto 1974].

Surface polariton

221

Surface polaritons 6.5.2 Theory The theory of ATR in either configuration is formally quite similar to the theory of the two-interface polaritons given in §6.3. Since surface polaritons are p-polarised modes with the H field in the y direction and the E field in the xz plane, ATR occurs only with p-polarised light, in which the E field is similarly in the xz plane. In medium 1, the E field takes the form E = (£ 0 , 0, qxE0/qlz) exp(iqxx - icot)

exp(-iqlzz)

+ r(£ 0 , 0, - qxE0/qlz) exp(iqxx - icot) Qxp(iqlzz)

(6.90)

Here the first term describes the incident wave, and the constant Eo is regarded as given. The second term is the reflected wave; the object of the calculation is to find the complex amplitude reflection coefficient r. In media 2 and 3, the £ field can be written in the form of (6.51) and (6.52). The next step in the calculation is to apply the two electromagnetic boundary conditions at each interface; this gives four equations for the four coefficients r, a, b and E3x in terms of Eo. Solution of these equations is straightforward, and yields r= z) + exp(-2iq2zL)(s3q2z z) + exp(-2iq2zL)(e3q2z

- e2q3z)(e2qlz - s2q3z)(e2qlz (6.91)

Equation (6.91) is a kind of response function, and like other response functions it is most useful when damping is included. However, one or two comments are possible about its significance in the absence of damping. The condition for the numerator to vanish is formally identical to (6.59), while the condition for the denominator to vanish gives (6.59) with the sign of qlz reversed. This is because (6.59) is the condition for Maxwell's equations to have a solution with only one field term in medium 1, namely (6.50), whereas (6.91) is derived from (6.90), in which two field terms are present. Experimental work to date has mainly concentrated on the intensity reflection coefficient R = |r| 2 , although as pointed out by Abeles (1986), for example, by using the techniques of ellipsometry it is possible to extract more information by measuring both the amplitude and the phase of r. Our discussion will largely be concerned with R. 6.5.3 Experimental results ATR curves for an Ag specimen in the Otto configuration are shown in Fig. 6.24. The importance of choice of gap width L is quite 222

Attenuated total reflection clear. For large L, curve a, the system is undercoupled, and the reflectivity dip is small. For optimum L, curve b, the reflectivity drops almost to zero. For small L, curve c, the system is overcoupled. The ATR dip is then broadened and its minimum is shifted from the point found in optimum coupling. The Otto configuration has more often been applied in the infrared, usually to study surface phonon-polaritons. Experimental results for GaP are shown in Fig. 6.25; analysis of these results gives very good agreement with the theoretical curve for the GaP/vacuum surface polariton, as is seen in Fig. 6.26. Experimental results for an angle scan on Au in the RaetherKretschmann geometry are shown in Fig. 6.27. As may readily be checked from the value quoted for the dielectric constant of the prism, the critical angle for total internal reflection at a prism/vacuum interface is 6C = 33.8°. For 8 > 6C the experiment would give R = 1 were it not for two mechanisms: the small loss due to the imaginary part of e2, and the reflectivity dip due to excitation of the surface plasmon at the Au/vacuum interface. The first of these shows up in the plateau region around 34°, and the second of course is responsible for the main dip. It is clear from these results that the ATR method is capable of precise determination of Fig. 6.24 Experimental and theoretical spectra for Otto configuration with glass prism, vacuum spacer and Ag specimen; detailed geometry in inset. Cross, experimental points; full line, theoretical curves obtained by least-squares fit of (6.91). The curves correspond to an angle scan at He-Ne wavelength k = 633 nm. Curve a, gap width L = 951 nm, undercoupled; curve b, gap width L = 918 nm, optimum coupling; and curve c, gap width L = 581 nm, overcoupled. [Courtesy of J. R. Sambles and K. Welford.] 1.40

silver 0.00

38.00

38.60

39.20 39.80 Angle (degrees)

40.40

41.00

223

Surface polaritons

both the real and imaginary parts of the dielectric function without recourse to a Kramers-Kronig analysis. In §6.2.5 it was mentioned that ATR was one of the principal techniques used for study of surface exciton-polaritons. Comparison of §6.4.2 with §6.2.5 shows how the theory must be adapted for this case. For the Otto Fig. 6.25 ATR spectra (frequency scan) for GaP/air interface with Si prism. The curves are labelled by values of qx taken from (6.87) where k0 = co/c. Gap values were L = 40, 25, 12, 2.5 fim, increasing as qx increases [after Marschall and Fischer 1972].

f

1 (JL

100 -

N 1.023 A:0

qx =

V A i(l ;

/

\

80 -

-

1.056& 0

1 125 A:

Al

0,

2.00 k0

A

-

60 360

380 a)/2nc (cm"1)

400

Fig. 6.26 Dispersion curve of GaP/air interface. Theoretical (full line) and experimental (squares). GaP parameters as in Fig. 6.25 [after Marschall and Fischer 1972]. 0)

I

/

x=

0J C

/

400 I 1 1 / 1 / 1 I

(i)/2nc (cm"1)

l

380

360

/

i /

I

/

I

1.0

224

i

i

i

1.5

2.0

2.5

cqjo)r

Attenuated total reflection geometry, the surface-active medium is medium 3, and as in §6.2.5 it is described by three field amplitudes (two transverse, one longitudinal) rather than the one used in the derivation of (6.91). Corresponding to the two additional amplitudes, there are two extra boundary conditions, derived for example from (6.13). Thus it is still possible to find an expression for the reflected amplitude r, although the expression is more complicated than (6.91). The first calculation of this kind was reported by Maradudin and Mills (1973). It was pointed out in §6.5.1 that for the Raether-Kretschmann geometry to be used the specimen must be evaporated on to the coupling prism. Thus this geometry is not applicable for excitonic materials. The use of the Otto configuration is difficult, however, because the thickness of the space layer must be comparable to an optical wavelength, which now means typically a few hundred nanometres. Some discussion of experimental techniques is given by Tokura et al. (1981). Fig. 6.28 shows results obtained by these authors on ZnSe specimens with an evaporated MgF 2 spacer layer. It would take us too far to go into the details of these results or the discrepancy with the theoretical curves. It may be noted, however, that the authors divide their specimens into Type A and Type B, the former having freshly cleaved surfaces while the latter have surfaces that have been Fig. 6.27 ATR in Raether-Kretschmann geometry: angle scan on gold specimen at wavelength X = 700 nm. The theoretical curve is a least-squares fit of (6.91) to the data,yielding £i = 3.2217,e 2 = -16.781 + 1.3169i and L = 45.355 nm.ej and L are included in the fitting parameters since it can then be checked that the same values are obtained from fits to data obtained at different wavelengths. [Courtesy of R. A. Innes and J. R. Sambles.] 1.20

1.00

•t °-80 ^ 0.60-1 0.40 0.20 0.00

33.60

34.06

34.52 34.98 Angle (degrees)

35.44

35.90

225

Surface polaritons

exposed. The differences between the two types of spectra bring out the sensitivity of excitonic effects to surface conditions. Both bulk and surface exciton-polaritons are low-temperature excitations, since the exciton becomes ionised at moderate temperature. This is illustrated in Fig. 6.29, which shows the disappearance of Type B spectra with increasing temperature. As depicted in Fig. 6.22, a diffraction grating can be used to couple light into a nonradiative mode such as a surface plasmon, and equally a grating can couple light out. In a similar way, a rough surface couples light out. A rough surface can be regarded as a superposition of gratings Fig. 6.28 ATR spectra in p-polarisation for ZnSe in Otto geometry at 2 K. Gap distances 100 nm (Type A) and 90nm (Type B): critical angle of incidence 55°. The curves show frequency scans, with angles of incidence marked. Broken curves are theoretical, using form of ABC proposed by Rimbey and Mahan (1974) [after Tokura et al. 1981].

2.80

2 81

2.80

Energy (eV)

226

2.81

Attenuated total reflection of different periodicity, or equivalently one can say that a rough surface destroys translational invariance in the surface plane. Either way, it is clear that light can be emitted. The surface of an Ag film evaporated on quartz is already rough enough to produce significant coupling, and indeed light emission was observed in the earliest experiments (Kretschmann and Raether 1968). It was this radiative property of a rough surface that was exploited in the nonlinear-generation experiment of De Martini et al. (1977), sketched in Fig. 6.15(a). Much experimental and theoretical work has been carried out on rough surfaces; the reader may refer to the reviews by Raether (1982) and by Maradudin (1982) and to the more recent theoretical review by Maradudin (1986). In this section we have described the principle of the ATR method and presented experimental results obtained on three-medium geometries as illustrated in Figs. 6.22(ft) and (c) and analysed in (6.91). In fact the method has been applied widely to study more complicated specimens consisting of a number of layers. As an example, Fig. 6.30 shows an angle-scan ATR curve for a prism/MgF2/Ag/air geometry, that is, the Otto configuration with MgF2 as the spacer and Ag as the surface-active medium. Although the Ag film is bounded unsymmetrically, with MgF2 on one side and air on the other, nevertheless the higher-frequency mode behaves like the long-range mode (LRSP) that was described for a Fig. 6.29 Temperature dependence of Type B spectrum of Fig. 6.28 [after Tokura et al. 1981]. TypeB 6 = 65.4°

2.76

2.78 2.80 Energy (eV)

2.82

227

Surface polaritons

symmetric geometry in §6.3. This is clearly seen in Fig. 6.30, which shows that the high-frequency mode (lower angle) is a much sharper resonance than the low-frequency mode. ATR has also been extensively applied to the investigation of adsorbed molecular layers on Ag; a review is given by Abeles (1986). 6.6 Other experimental methods 6.6.1 Raman scattering

As mentioned at the end of §6.1.1, much of the experimental investigation of bulk polaritons was carried out by means of Raman spectroscopy. By contrast, compared with ATR spectroscopy, Raman scattering has yielded only a relatively small amount of information concerning surface polaritons. The combination of a small scattering volume with the need to employ near-forward scattering angles makes the experiments technically demanding. In fact the only results published to date are by Ushioda and co-workers. Some of their key results are presented in this section; a fuller account is given by Ushioda and Loudon (1982). It was thought at one time that surface polaritons might be seen in backscattering from the surface of a semi-infinite medium. This is true in principle, but the cross section is expected to be smaller by a factor of 102 or 103 than that for forward scattering from a slab specimen and, Fig. 6.30 ATR curve (angle scan) for a system comprising prism/MgF2/Ag/air at a wavelength of 632.8 nm. Cross, experimental points; full line, least-squares fit of theory, giving thicknesses 203 and 28.2 nm for MgF 2 and Ag films respectively. [Courtesy of J. R. Sambles and M. D. Tillin.] 1.20

0.00

228

48.2 55.! Angle (degrees)

63.4

71.0

Other experimental methods in fact, backscattering has never been observed. The reason was first explained in detail by Chen et al. (1975). The theory is analogous to that given for Brillouin scattering in §2.5, and in particular the backscattered field contains the factor (Qz + k\2 + kl2)~l seen in (2.116). Here fcf2,fc|2are the z components of the wavevectors of the incident and scattered light. It is seen that they add to generate a large denominator for backscattering. By contrast, for near-forward scattering the corresponding factor is (Qz + k\2 — k^)" 1 ; the z components of the optical wavevectors now subtract to produce a small denominator and consequently a larger cross section. Experimental results for the single-interface polariton were reported by Valdez and Ushioda (1977). The experiments were carried out on a GaP specimen of thickness L = 20 /mi. Correct choice of L was crucial. The argon-ion laser frequency is close to the GaP band gap, so that resonant enhancement of the cross section occurs (Hayes and Loudon 1978) and the specimen is fairly opaque; L is chosen large enough so that the upper- and lower-branch polariton curves effectively coincide on the single-interface curve, and small enough for some light to be transmitted. Results for the scattering intensity are shown in Fig. 6.31. The curves are Fig. 6.31 Raman spectra in a single-crystal slab of GaP with (111) faces. L= 20/mi. The curves are labelled by scattering angle 6 defined in inset [after Ushioda and Loudon 1982].

•a

e = 2.o° 6= 1.6° 390

395

400

405

410 1

Frequency shift (cm" )

229

Surface polaritons dominated by the large peak due to the LO phonon at a)hO/2nc = 403 cm" 1 . The surface-polariton scattering is the small feature marked by an arrow. It is seen that the frequency of this feature changes with #, corresponding to the dispersion of the surface-polariton curve. The peak position is plotted and compared with the theoretical dispersion curve in Fig. 6.32. The slight discrepancy between experiment and theory is thought to be due to surface roughness. The first experimental results to be published were in fact for the lower of the two modes of GaAs on sapphire in a two-interface geometry (Evans et al. 1973). In that experiment, the upper mode was not resolvable from the large peak due to scattering off the bulk LO phonon. Subsequently the technique was refined by Prieur and Ushioda (1975) so that the upper-mode peak could be detected. They made two essential changes from the earlier experiment. First instead of air/GaAs/sapphire they used benzene/GaAs/sapphire; the higher refractive index of benzene means that the upper-mode frequency is lowered away from the bulk LO frequency. Second, Prieur and Ushioda made use of the fact that the surface modes are not detected in backscattering, whereas the bulk-mode peaks are identical in forward and backward scattering. They therefore devised a subtraction technique to measure the difference between the two, which contains only surface-mode scattering. Scattering data are illustrated in Fig. 6.33, and the resulting dispersion curves are shown in Fig. 6.34. The possibility of observing guided-wave polaritons emerged from theoretical work by Mills et al. (1976) and by Subbaswamy and Mills (1978), who predicted that the scattering peaks should be comparable in intensity to those from surface-type modes, and therefore observable. Fig. 6.32 Surface-polariton peak positions of Fig. 6.31. Theoretical curve for air/GaP interface using dielectric function of (5.60) with coT/2nc = 367.3 cm" 1 , cojlnc = 403.0 cm" 1 and ^ =9.091 [after Ushioda and Loudon 1982]. 400 r-

r 398 396

394 2

3

Scattering angle (degrees)

230

Other experimental methods Fig. 6.33 Extraction of surface-mode Raman scattering peaks by the subtraction technique of Prieur and Ushioda (1975) [after Ushioda and Loudon 1982].

c

Frequency (cm l)

Fig. 6.34 Dispersion curves for upper and lower surface modes in Benzene/GaAs/Al 2 O 3 . Theoretical curves correspond to (6.69) [after Ushioda and Loudon 1982].

290

L

285

280

275

270

231

Surface polaritons Scattering data obtained on a 30 /mi single-crystal GaP slab are shown in Fig. 6.35, with the dispersion curves shown in Fig. 6.36. The publication of the first experimental results stimulated considerable theoretical effort. As was seen in the example treated in detail in §2.5.2, there are several steps in the development of a full theory that will account for intensities and lineshapes as well as peak positions. It is first necessary to find the equivalent of (2.111) describing coupling of the incident light amplitude E, at frequency cox to the field at frequency co to produce a polarisaton P s at frequency cos = co1 ±CO. AS with (2.111), the coupling mechanism is identical to that for scattering by the corresponding bulk modes. The polariton is a coupled mode containing both an electric field amplitude E and an optic phonon amplitude W. The polarisation P s contains contributions from both amplitudes: PI = a*fiEf2W* + bapyE?2Ey* (6.92) where summation over repeated indices is implied. The coupling tensors a and b are discussed by Hayes and Loudon (1978). Fig. 635 Near-forward Raman scattering by guided-wave polaritons in GaP slab. Curves are labelled by scattering angles [after Ushioda and Loudon 1982]. GaP 30fim

250

232

(111)

300 350 Frequency (cm"1)

400

Other experimental methods

The calculations for the transmission of the incident light into the specimen and the radiation of scattered light by the polarisation P s are identical to those outlined in §§2.5 and 2.6. The cross section is proportional to the power spectrum of the scattered field, and therefore is ultimately determined by power spectra involving the fields W and E of the polariton. These are conveniently evaluated from Green functions such as <<£a(Q); ^ ( Q ' ) ) ) ^ by means of the fluctuation-dissipation theorem. Thus, as in all light-scattering calculations, the underlying problem can be expressed as the evaluation of appropriate Green functions of the scattering mode. Calculations along the lines indicated here were first presented in full by Nkoma and Loudon (1975), Nkoma (1975) and Mills et al. (1976); these and subsequent papers are reviewed by Cottam and Maradudin (1984). 6.6.2 Light-emitting tunnel junctions

A method of excitation of surface plasmons that has attracted attention is to use the tunnel current flowing between two metals. Under Fig. 636 Dispersion curves of guided-wave polaritons in GaP slab. Experimental points from Fig. 6.35 and similar data; theoretical curves from (6.64) and (6.65) [after Ushioda and Loudon 1982]. 0= 1.0° 1.3°

1.8° 2.0°

3.0°

233

Surface polaritons

some circumstances, for example if the interfaces are rough or if the substrate is a diffraction grating, the junction is seen to emit light, typically in the visible part of the spectrum. An example of light emission by a tunnel junction is shown in Fig. 6.37, with the specimen configuration shown schematically in the insert. The first layer deposited on the glass substrate is CaF 2 ; this has a rough top surface, and consequently all subsequent interfaces are rough. The tunnel junction itself is Al/Al2O3/Ag, the A12O3 tunnelling barrier being formed by oxidation of the Al film. In the experiment a dc bias voltage Vo is applied between the Al and the Ag, and a tunnel current / 0 flows through the A12O3 barrier. The spectrum of light emitted by the junction is seen to change from red to blue as Vo increases. The actual spectrum emitted in the direction of the specimen normal is shown in Fig. 6.37. It should be noted that the spectrum is unpolarised, and the maximum photon energy hcoQ, corresponding to the short-wavelength cut off of the spectrum, is given to a good approximation by ha>0 = eV0. Thus at the short-wavelength end virtually all of the energy of a tunnelling electron is converted into photon energy. Two other important forms of tunnel junction are first a junction deposited on a diffraction grating, and second a junction in which the top electrode consists of discrete metal particles. In the first case the spectrum is p-polarised and has a sharp peak whose position varies with angle of Fig. 6.37 Spectra of light emitted by rough Al/Al 2 O 3 /Ag tunnel junction in normal direction. Curves are labelled by tunnelling current / 0 . Inset shows specimen geometry [after Dawson et al. 1984]. Thickness (nm) 25-40 2-3 f50-70 100-140

125

100

•s

75

50

25

400

500

600 A (nm)

234

700

Surface magnon-polaritons emission relative to the specimen normal. In the second case the spectrum is predominantly p-polarised but broadband. In all three types of light-emitting tunnel junction the tunnel current excites a surface plasmon which then decays radiatively because of the roughness to produce the observed spectrum. The nature of the plasmon excited depends on the form of the junction; for a detailed discussion and further references the reader may turn to Dawson et al. (1984). 6.7 Surface magnon-polaritons In this final section we turn to polariton effects arising from frequency dispersion in the magnetic susceptibility. These have already been discussed in Chapter 4 within the magnetostatic approximation; this merited a separate chapter because of the extensive experimental results that have been obtained, particularly by light scattering. We now deal with the longer wavelength region where inclusion of retardation is important; that is, we are concerned with the electromagnetic region of Fig. 4.1. The results to be presented are mainly theoretical, since very few experimental results have been reported to date. However, there is reason to believe that more experimental results will soon be available. Our discussion is largely based on that in earlier reviews by Sarmento and Tilley (1982) and by Tilley (1986). As in Chapter 4, we consider a magnetic material (ferromagnet or antiferromagnet) with a static magnetic field Bo directed along the z axis. The magnetic susceptibility tensor in the xy plane takes the gyromagnetic form of (4.15) where the form of xa a n d Xb depends on the material under discussion. As demonstrated in Problem 4.1, the susceptibility is diagonalised by transformation to the rotating-wave representation, (6 93) m±=mx±imy h±=hx±ihy (6.94) + ± ± ± with susceptibilities x > X defined by m =x h . Expressions for the susceptibility components of a ferromagnet are given in (4.16), (4.17) and Problem 4.1; for an antiferromagnet the corresponding expressions are (4.54) to (4.56). 6.7.1 Bulk-magnon polaritons The bulk polariton modes were first discussed by Auld (1960) for a ferromagnet; later authors dealt with antiferromagnets (Manohar and Venkataraman 1972; Bose et al. 1975; Sarmento and Tilley 1977) and ferrimagnets (Oliveira et al. 1979). Maxwell's equations lead to q2h - q(q-h) = (sa)2/c2)(h + m)

(6.95) 235

Surface polaritons for a plane wave exp(iq-r — icot). Since the static magnetic field is in the z direction, there is no loss of generality in choosing qy = 0. Equation (6.95) then yields h+ =£1m+

-£2m+

(6.96)

JT = -f 2 m +<*!?*-

(6.97)

^ = ±(2eco2/c2 - ql)l{q2 - eco2/c2)

(6.98)

where 2

2

2

2

Z2 = h J(q -™ /c ) When combined with the susceptibility equations m±=x±h±

(6.99) (6.100)

(6.96) and (6.97) yield the polariton dispersion relation (i-Za+)(i-Za-) = Z22X+x(6.101) which is a general form. It is clear from (6.98) and (6.99) that, as might be expected on general grounds, the dispersion relation depends on the angle 6 between the propagation vector q and the z axis. Dispersion curves for a ferromagnet are shown in Fig. 6.38. It is seen that one solution of (6.101) shows photonmagnon mixing, while the other does not. The results presented so far are concerned with the solution of homogeneous equations of motion for the magnetic system coupled to the electromagnetic field. A fuller treatment involves finding the Green functions as the linear response to an applied rf field hext exp(iq»r — icot). This has been done for bulk ferromagnets (Sarmento and Tilley 1976) and antiferromagnets (Sarmento and Tilley 1977). 6.7.2 Surface magnon-polaritons: ferromagnets We now turn to the surface polaritons at the interface between vacuum and a magnetic medium. We concentrate on the Voigt configuration, in which B o is in the plane of the surface and propagation is perpendicular to B o , since most of the results that have been derived concern this geometry. As in Chapter 4, we take the z axis along B o and the x axis as the normal to the surface, so that the propagation vector qLi is in the ±y direction. The magnetic medium is in the half-space x < 0. Following Hartstein et al. (1973), who were the first to investigate this problem, we work in terms of a gyromagnetic permeability tensor (6.102)

236

Surface magnon-polaritons where = Ml + la)

(6.103) (6.104)

Vxy =

(6.105) This is derived as in §4.2 with the addition of a constant 'background permeability'. Such a term can only arise from the contribution of resonances at a higher frequency than those appearing explicitly as poles of X- It is therefore plausible, for example, for a ferrimagnet at microwave frequency where )U could be due to the exchange resonance. It is worth retaining \i explicitly, since as will be seen its presence leads to the appearance of a mode that is absent for JX = 1. For the magnetic medium, Maxwell's equations now give V2h - V(V-h) - (e/c2)fi d2h/dt2 = 0

(6.106)

and 0

(6.107)

Fig. 6.38 Dispersion curves for bulk polaritons propagating along the static magnetic field (0 = 0) and perpendicular to it (0 = 90°) in a bulk ferromagnet. Bo = fi0M0 = 0.175 T [after Auld I960].

237

Surface polaritons while in the vacuum V2h-(l/c2)d2h/dt2

=0

(6.108)

In order to find the dispersion relation of the surface mode, we substitute into (6.106) to (6.108) and the boundary conditions the ansdtze h = h2 exp(iq || y) exp(/c2x) exp( - icot) h = h : exp(ign j;) exp( — KXX) exp( — icot)

x<0 x>0

(6.109) (6.110)

Substitution of (6.109) into the z component of (6.106) gives h2z = 0, and the boundary condition on tangential h gives hlz = 0. It follows that the surface polariton is s-polarised with E in the z direction. The other two components of (6.106), with (6.109), yield K2-ql^(sco2/c2)fiy

=0

(6.111)

where fiy is called the Voigt permeability Vv = l*xx + t*ly/Hxx Equations (6.108) and (6.110) give K2-ql+CD2/c2 = 0

(6.112) (6.113)

Equations (6.111) and (6.113) are the analogues of (6.15) for the dielectric problem; they characterise the individual media with no reference to boundary conditions. The Voigt permeability is central to this geometry. For the particular case of a ferromagnet, substitution of (4.16) and (4.17) gives t**/l* = C K + ^m) 2 - ^ 2 ] / K +
+

'iq^Jii^

(6.115)

As is seen from (6.104), as long as the resonance modes are undamped jj,xy is pure imaginary, and (6.115) involves purely real quantities. Equations (6.111), (6.113) and (6.115) are simultaneous equations for the three quantities K19 K2 and q^. Elimination of K1 and K2 yields an equation, quoted explicitly by Hartstein et al. (1973), for the dispersion function q\\((o). The most important property of the dispersion curve can be seen from (6.115). Since q^ occurs there linearly, the directions -\-q^ and —q y are not equivalent. Thus, as with the magnetostatic modes of Chapter 4, the propagation is nonreciprocal. 238

Surface magnon-polaritons The dispersion curve for a ferromagnet found by numerical solution of (6.111), (6.113) and (6.115) is shown in Fig. 6.39. The +ql{ mode is the generalisation of the Damon-Eshbach mode to include retardation. For large q^ the curve coincides with the Damon-Eshbach one, being asymptotic to the surface frequency cos of (4.31), which with \x included becomes <*>s = l>o + Kco0 + comy]/(n + 1 ) (6.116) For small q^ the Damon-Eshbach curve has the unphysical property that it crosses the vacuum photon line and reaches q^ = 0 at the frequency a>B of (4.28). The calculation with retardation corrects this; the mode still terminates at coB, but at the nonzero value qy given by 1; it starts on the vacuum photon line at frequency a)_ = ai-l)- 1 / 2 [G>g + Aia>oK + fi>m)]1/2 (6.H8) and merges with the bulk-polariton dispersion curve at a finite frequency. Hartstein et al. (1973) give a brief qualitative account of the surface polariton as qy moves away from the Voigt direction, restricting attention to s= 1. As for the magnetostatic surface mode, the surface polariton exists only for (j> < 0 C, where c is given by (4.30). This is qualified by Fukui et al. (1984), who allow e ^ l . They find that there is a mode propagating along B o , the Faraday direction. Fig. 6.39 Surface polariton dispersion curves (full line) for the + # | and —q\\ modes on a ferromagnet in the Voigt geometry. Also shown is the bulk dispersion curve (chain line) [after Hartstein et al. 1973].

239

Surface polaritons To date there has not been any experimental observation of surface ferromagnetic polaritons. The frequencies are around co0 ~ 1 GHz, with wavenumbers of order CDO/C. In light scattering, therefore, the scattering angle would have to be impracticably small to get down from the magnetostatic into the polariton region. Matsuura et al. (1983) suggested that ATR might be applied, but since the wavelengths are of order 1 cm or greater, specimens would have to be very large.

6.7.3 Surface magnon-polaritons: antiferromagnets The difficulties of observation of surface magnetic polaritons are in principle reduced for antiferromagnets, since resonance frequencies are then in the far infrared and specimens need not be so large. For a uniaxial ferromagnet the results of the previous section apply as long as the appropriate expressions (4.54) and (4.55), are used for xa a n d Xb m (6.103) and (6.104). The properties of surface polaritons on an antiferromagnet were discussed by Camley and Mills (1982). One important result is that, as for the magnetostatic mode discussed in §4.5, in zero applied field the propagation is reciprocal, cos(q^) = cos( — ql{). In the presence of an applied field, however, the propagation is nonreciprocal. In addition to the dispersion curve, Camley and Mills present calculations and numerical results for ATR involving surface antiferromagnetic polaritons. They conclude that ATR is viable, but to date no experimental results have been reported. Optical reflection measurements on MnF 2 by Remer et al. (1986) give indirect evidence of the nonreciprocity of surface polariton propagation in the presence of an applied field. To understand how this comes about one may argue loosely as follows. Ordinary reflectivity probes the region to the left of the light line, whereas the surface-mode dispersion curve lies to the right of the light line. In the presence of damping, however, the dispersion curve becomes broadened and the properties of the surface mode are echoed, to some extent, in the reflectivity. Remer et al. substantiate this argument by calculating the reflectivity with a damping time T included in the susceptibilities xa a n d XbMnF 2 has a relatively low anisotropy field and therefore a low resonance frequency. For this reason, Remer et al. were able to use a microwave source at 88 GHz driving a third-harmonic resonator to produce a fixed-frequency source at 264 GHz. The results are presented as reflectivity versus magnetic field. Figure 6.40 shows calculated and measured reflectivity curves for two different directions of applied field B in the plane of the surface. The results clearly show the expected nonreciprocity. 240

Surface mag non-polar itons 6.7.4 Two-interface modes The magnetic polariton modes of a film of finite thickness L in vacuum were first investigated by Karsono and Tilley (1978) and by Marchand and Caille (1980). We use essentially the notation of Fig. 6.17, but with x and z axes interchanged. Analogously to (6.50) to (6.52), the magnetic field is taken in the form 11 = 11! exp(i^f|| y) exp( — icot) exp( —K XX)

X>0

(6.119)

H = exp(ix>-L H = H 3 exp(i^ny)exp( — icot) exp(/c3x)

x < —L

(6.120) (6.121)

where qx is real for guided modes and imaginary, qx = iK2

(6.122)

for surface modes. The wave equations and divergence equation, (6.106) to (6.108), continue to apply, and in consequence (6.111) for K2 (or qx) and (6.113) for KX are still valid. The dispersion equation is found by applying standard boundary conditions, and the result for e = 1 as quoted by Marchand and Caille (1980) is (KJri + K22 + q\\ilyl\i2xx) tanh(/c 2 L) + 2K 1 IC 2 JI V = 0

(6.123)

This is the generalisation of the Damon-Eshbach result for a ferromagnetic film, (4.27), to include retardation as well as the extension of the result of Hartstein et al. (1973) to a finite thickness slab. Fig. 640 Reflectivity at 264 GHz off the surface of MnF 2 at 28 K. (a) Calculated with G>AF= 1.24 THz and (COAF^) 1 =6.5 x 10~ 4 , where O>AF is zero-field resonance

frequency, (3.86). Damping time x is chosen to give best fit to experimental results. {b) Measured [after Remer et al. 1986].

R (arb. units)

R (arb. units) . . . B~ — B+

(a) 0.8 0.6 0.4 0.2

o B-

(«

1 MJ

A

J i

0.6

ii

y— -

1/

0.8

B(T)

Ik iii \\M

ill

i

0.4

I

I

0.6

I

0.8 B(T)

241

Surface polaritons The surface mode solutions are those for which K2 is real and qx imaginary. Dispersion curves for the +q^ mode, calculated numerically, are shown in Fig. 6.41. Like the magnetostatic mode and the surface mode on a semi-infinite medium, the mode starts at the frequency coB of (4.28) and is asymptotic for large q^ to the surface frequency cos of (6.116). For the larger value QB = 2 of applied field, Fig. 6.41 (a), there is considerable separation between the dispersion curves and the semi-infinite result even for \y\MoL/c = 0A5. Since the scaling distance c/|y|M 0 is quite large, the finite-thickness corrections are likely to be of importance. The dispersion

Fig. 6.41 Dispersion curves for a ferromagnet with n= 1.25. Full line, + qj retarded slab mode from (6.123); broken line, +
2.56

2.50

a

2.44

^ 3 II

10

15

1.0

1.5

0.5

a

0.5

242

Surface magnon-polaritons

curves are well separated from the magnetostatic modes for QB = 2 but rather close for QB = 0.005; thus the effects of retardation show up more in strong applied fields. As with the dielectric guided-wave polaritons of Fig. 6.19, there are two frequency 'windows' for the guided modes. Typical dispersion curves in these windows are shown in Fig. 6.42. The lower-frequency modes start Fig. 6.42 Guided-mode dispersion curves (full line), bulk magnetic polariton (chain line) and vacuum light line (dotted line), (a) Lower frequency window, ^t = 1.75, QB = B0/n0M0 — 2, \y\M0L/c=l. {b) Upper frequency window, /x=1.75, Q B = 10, \y\M0L/c = 0.5 [after Marchand and Caille 1980]. (a)

18

12 II

10

20

15

(b)

31

3 II

a 21

21

31

41

cqt/\y\M0

243

Surface polaritons

on the light line K1 = 0 and have the short-wavelength asymptote Q -» Qv. In the high-frequency window, the lowest mode starts on the bulk magnetic polariton curve, given by qx = 0, and the higher modes again start on the light line. At high frequencies all the modes are asymptotic to the bulk polariton curve. As for the surface polaritons on a semi-infinite specimen, there have as yet been no experimental studies of these retarded slab modes. Fukui et al. (1984) have calculated ATR spectra for the Karsono-Tilley type modes, but just as with a single-surface mode the long wavelength implies a specimen of very large surface area. The outlook is presumably more hopeful for antiferromagnets, but to date no theoretical work has been published on dispersion curves or ATR spectra for polariton modes in an antiferromagnetic film. Just as for the bulk magnetic polaritons, more information can be obtained on the surface modes if Green functions are evaluated. This poses very considerable algebraic problems, but the polariton Green functions for a ferromagnetic film and semi-infinite medium have in fact been derived by Lima and Oliveira (1986). 6.8 Problems 6.1 The semiconductor G a ^ ^ ^ A s may be viewed as derived from GaAs by substitution at random Ga sites of a fraction x of Al. It exhibits two TO phonon resonances, one near to that of GaAs and one near to that of AlAs. The expression co\ — o r2 — I gives a good fit to IR reflectivity data (Kim and Spitzer 1979). For x = 0.3, numerical values are e^ = 10.16, col/2nc = 265.2 cm" *, TJlnc = 7.2 cm" \St = 1.24,co2/2nc = 360.6 cm" \ V2/2nc = 10.8 cm" \ S 2 = 0.93. (a) Sketch the real and imaginary parts of s(co) as functions of co. (b) Sketch the bulk-polariton dispersion curve. (c) Sketch the surface-polariton dispersion curve for a vacuum interface. (d) Write computer programs to draw the curves of (a) to (c) accurately. 6.2 The bulk exciton-polariton dispersion curve is described by (6.12) and (6.11) (with F = 0). Confirm the statement made in the text that there is one solution for co < coL and two for co > coL, and find the expression for coL. 6.3 Sketch and/or draw accurately the surface-polariton dispersion curve for an interface between vacuum and n-InSb in zero magnetic field, with an isotropic dielectric function given by (5.65).

244

Problems 6.4 Equation (6.42) gives a dispersion equation for surface polaritons on an anisotropic medium. When the (x, y, z) and (x', / , z') axes coincide it reduces to CO2 £ 1 £ z (£ x — £ j )

2

c2

sxez - s2

Derive this result from Maxwell's equations and boundary conditions. 6.5 In §6.2.4 the dispersion relation (6.46) for surface polaritons guided by a charge sheet was derived. Now suppose the charge sheet is replaced by a layer that can be polarised in the plane, so that (6.45) is replaced by (This might be realised by a suitable organic layer.) Prove that (6.46) is replaced by Kt

K2

Discuss this for the particular case = Sco2/(co2-co2) (a) Prove that Nakayama's result, (6.46), is the formal limit as col -• 0 with Scol = constant. (b) Sketch the dispersion curve. X(co)

6.6 Equations (6.62) and (6.63) give the dispersion relations for the surface modes of a symmetric slab. (a) Determine in which of the modes the electric-field pattern is symmetric about the mid-point of the slab, and in which it is antisymmetric. (b) Find the dispersion equations of the corresponding electrostatic modes first by letting c -• oo in (6.62) and (6.63), and second by generalising the potential analysis given in §6.2.2. 6.7 This is concerned with the calculation of the dispersion equation for the s-polarised guided modes. In place of (6.50) to (6.52) the E fields, which are all in the y direction, are written E — Ex exp{iqxx — icot) exp(ig

lz z)

z> 0

E = \c exp iq2 Jz + - J + d exp -iq2z(z + -J j x exp(igxx — icot)

0> z> — L

£ = E3 exp[ig 3z(z + L)] exp(igxx — icot)

z < —L

The boundary conditions involve the tangential magnetic-field component Hx, which is derived from the Maxwell equation for V x E. Show that the boundary conditions produce four homogeneous equations in the 245

Surface polaritons four amplitudes Et, c, d, E3:

cf-1+df=E3 where / is defined in (6.58). Show that these equations may be derived from (6.54) to (6.57) by the substitutions (6.66) and (6.67). 6.8 The number of solutions to (6.69) and (6.70) for s-polarised guided modes of a film depends on the value of L. Consider the case when zx and e2 are both real and positive, with £2 > £ i- Write the equations s2(o2/c2-ql)lf2L]

(6.69)

and

S^/f

/C

2 ?1

= -cot[i(e2co2/c2 - q2xyi2L]

(6.70)

(e2w2/c2-q2x)112 Sketch the left- and right-hand sides of these equations versus qx for various L values with co fixed. Hence show that (a) Equation (6.69) has one solution for small L, and the number of solutions increases as L increases. (b) Equation (6.70) has no solutions for small L. Find the value of L c such that for L< Lc the film is single-moded, i.e there is one solution of (6.69) and none of (6.70). Use (6.64) and (6.65) to investigate the corresponding results for p-polarised guided modes. 6.9 Check the derivation of (6.85) for the time-averaged power in the self-guided mode. 6.10 Confirm (6.91) for the ATR reflection amplitude.

246

7 Layered structures and superlattices

A number of crystal growth techniques have been developed for the production of specimens which have the form of a succession of layers. In molecular-beam epitaxy (MBE) beams of atomic or molecular species passing through ultra-high vacuum impinge on a single-crystal substrate and in the right conditions crystal growth occurs epitaxially, that is, with the crystal structures in register. In metallo-organic chemical vapour-phase deposition (MOCVD) growth occurs by deposition from a flowing vapour. Both MBE and MOCVD are used to produce semiconductor specimens in which, for example, single-crystal layers of GaAs and Al^Ga^^As alternate. Metallic specimens, including those in which one or both constituents may be magnetic, are made by sputtering. A review of MBE is given by Joyce (1985). Parker (1985) and Chang and Ploog (1985) include detailed chapters on MBE as well as on the physics of the resulting specimens, while the latter also contains a chapter on MOCVD. The growth techniques can be used to prepare specimens consisting of alternating layers of thickness dl of constituent 1 and thickness d2 of constituent 2. Specimens can be prepared so that dx and d2 have any value from two or three atomic spacings up to the order of 100 nm typically. We refer to them as periodically layered structures or superlattices;

the latter has a more specific meaning for electronic states in semiconductor structures, as we shall see, but it is often convenient to use it as a synonym for the former. Many of the physical properties are greatly modified by the existence of the long spatial period D = dl + d2. As mentioned already in §1.1.2, the most important general consequence is that a new Brillouin zone edge appears with wavevector component n/D perpendicular to the interfaces. This can be much smaller than the zone-edge wavevector n/a related to the lattice constant a. Dispersion curves, such as for acoustic phonons for example, develop band gaps at these new zone edges. 247

Layered structures and superlattices

Within the context of this book, it is natural to distinguish between the 'bulk' modes of an infinitely extended superlattice, and 'surface' or 'guided-wave' modes that may occur in specimens with one or two free surfaces. Since a 'bulk' specimen consists of an array of interfaces, an understanding of bulk modes uses many of the concepts developed in earlier chapters, and much of this chapter is therefore concerned with the bulk modes. As before, the emphasis will be placed on acoustic, optical and magnetic properties, although introductions to single-electron states and plasmons are given in §§7.4 and 7.5 respectively. The reader should be cautioned that up to now most of the interest in semiconductor superlattices has centred on the electronic properties because of the potential for device manufacture. The balance of topics is therefore consistent with the rest of the book but not representative of the literature. In §7.1 we introduce the main ideas with a detailed study of the continuum-acoustic modes of a bulk superlattice. Using a formalism very similar to that of Albuquerque et al. (1986a), we show that the system can be described by means of a transfer matrix T which relates the field amplitude at coordinate z + D to those at z, where z is measured perpendicular to the interfaces. The dispersion relation is simply expressed in terms of T. Subseqent sections are also largely based on the transfer-matrix formalism. In §7.2 we turn to the lattice dynamics of a simple ID model of a diatomic superlattice; despite its simplicity, this model has been widely employed in the analysis of experimental results. An introduction to optical properties, with emphasis on the reststrahl region, is given in §7.3. After a brief account of single-electron states in §7.4, we turn to collective-electron properties, that is, plasma effects, in §7.5, and finally §7.6 is an introduction to magnetic superlattices. As indicated, this chapter is concerned with semiconductor and metallic superlattices in which the long period is produced by variation of crystalgrowth conditions. Long periods also exist in some other types of material. Intercalated compounds, for example intercalated graphite, are basically layered materials in which long organic molecules are interposed to space out the layers of the starting material (see e.g. Dresselhaus 1986). Here the length of the organic molecules can be chosen to give a stipulated layer-to-layer spacing. Liquid crystals (see e.g. de Gennes 1974) containing chiral, that is screw-like, molecules can undergo a helical distortion in which the mean orientation of the molecules spirals round an axis in space; the two phases of most importance are the cholesteric and the smectic-C*. In these the pitch of the helix is a function of temperature and concentration of chiral molecules and can be varied in addition by 248

Continuum acoustics: folded acoustic modes

applied electric and magnetic fields; it can have any value from a few tens to many hundreds of nanometres. Clearly the properties of these long-period materials show analogies with those of superlattices. However, the analogies have not yet all been worked out, and it would take us too far afield to discuss these here. 7.1 Continuum acoustics: folded acoustic modes 7.1.1 Propagation in one dimension

We first discuss in detail the propagation of an acoustic wave, either longitudinal or transverse, in a direction normal to the interfaces of an infinite superlattice. The notation is shown in Fig. 7.1. Within the

Fig. 7.1 (a) Infinite superlattice. The layers are characterised by thickness dit density pt and elastic modulus (appropriate to longitudinal or transverse waves) Ct. The z axis is taken normal to the interfaces, (b) Acoustic-phonon dispersion curve for infinite superlattice when layers are impedance matched. Slopes of lines are u, where (a)

D

249

Layered structures and superlattices layers the displacement u satisfies the ID wave equation

We aim to solve these equations with the boundary conditions (see §2.1.1) that u and C du/dz are continuous at each interface. Since the system has translational periodicity D = dl+d2 we can introduce the wave vector component Q by means of Bloch's theorem: u(z + D) = Qxp(iQD)u(z) (7.2) The solutions of the wave equations in the two media at frequency co are a superposition of a forward- and a back ward-travelling wave. It is convenient to represent the solution within any one layer in two alternative forms, one with the phases referred to the left-hand end of the layer, and the other with the phases referred to the right-hand end. Thus we write u = a\ expCig^z - ID)~\ + b\ e x p C - i ^ z - ID)~] = a? e x p p ^ z - ID - dx)] + bf e x p E - i ^ z -IDdj\ lD^z^lD + dx

(7.3)

where q1=co/v1 The amplitudes are related by |«f> = Fi|«P> where we have denoted

(7.4) (7-5)

**>-(£)

(76)

A)

(7-7)

and the 2 x 2 matrix Fx is

SI

-

with / t = expOq^) In a layer of the second medium, the displacement is

(7.8)

u = d\ exp[ig2(z - ID - d j ] + e\ exp[-i^ 2 (z - ID - dj\ = df exp[i = F 2 K> with similar notations and definitions to (7.4) to (7.8). 250

Continuum acoustics: folded acoustic modes The boundary conditions at z = ID + d1 relate \uf} to |w^>: '1 1 where Z = Ciqi/C2q2 (7.12) is the ratio of the acoustic impedances of the media. Similarly the boundary conditions at z = (/ + \)D give

Note that the same matrices occur in (7.11) and (7.13) since the formalism presented so far is symmetric between +z and — z directions. The equations above combine to give a transfer matrix T defined by |ul+i> = 7 X > (7.14) where explicit evaluation gives Z-1)/152 -^i(Z-Z-1)f;1s2 \ 1 )/^ f^c2-±i(Z + Z-i)f^s2) K' ] with the shorthand notation c2 = cos(q2d2) (7.16) s2 = sm(q2d2) (7.17) It is readily verified from (7.15) that the transfer matrix is unimodular, detT=l (7.18) as is required for conservation of energy. We now apply Bloch's theorem. It follows from (7.2) that k + i > = exp(iQDK> (7.19) so that [T-exp(iGD)/]|ii{-> = 0 (7.20) where / is the unit 2 x 2 matrix. Equally from the equation relating \u^x)

to K>, [T-1-exp(-i6Z))/]^> = 0 (7.21) so that combining (7.20) and (7.21) [i(T + T~x) - I cos QD\\u\y = 0 (7.22) This holds for the amplitude \u[} in any cell /, and so the matrix must vanish. Hence we have / cos QD = i(T + T~x) = \l tr T (7.23) where the last step follows from (7.18) since T is a 2 x 2 matrix. Here 251

Layered structures and superlattices tr T denotes the trace of the matrix T. With (7.15) for T, this gives the explicit form of the dispersion equation: cos QD = cos qxdx cos q2d2 — [(1 + Z 2 )/2Z] sin qxdx sin q2d2

(7.24)

This equation is long established, having first been derived by Rytov (1956). Equation (7.24) has some elementary properties of interest. First, if Z = 1 the layers are impedance-matched, and the medium is effectively continuous acoustically. The equation then reduces to cos QD = cos(qldl + q2d2)

(7.25)

which has solutions co( -A + dA=±QD

+ nn

(7.26)

This is the equivalent of the 'empty-lattice' approximation of electron-band theory, since we have applied Bloch's theorem but without any reflections at the interfaces. As illustrated in Fig. 7.1(b), (7.26) corresponds to the folding of the acoustic-phonon dispersion curve into the reduced zone. Note, however, that the slope of the curve is an average of the velocities in the two media. For Z ^ l , w e have the inequality 1 + Z 2 > 2Z. If (7.24) is reorganised into a sum of two terms in cos(q1d1 ±q2d2) it is then readily seen that for some values of <x> the absolute value of the right-hand side is greater than 1. For these values of co there is no solution with real Q. Thus for Z ^ 1 there are some stop bands in the dispersion curve. It can also be seen that if qxdx and q2d2 are not commensurate, the dispersion curve is not, strictly speaking, periodic in co. That is, the magnitudes of the stop bands vary irregularly with co. These features are illustrated in Fig. 7.2. The above calculation, like the rest of this chapter, is concerned with superlattices in which the unit cell consists of two layers. The method can be extended to the case where the unit cell consists of N different layers. A range of theoretical results for this case has been presented by Djafari-Rouhani and Dobrzynski (1987). 7.1.2 Light scattering Inelastic light scattering has been used to investigate folded acoustic-phonon dispersion curves like those of Fig. 7.2. The first such observation was reported by Feldman et al. (1968), whose specimens were long-period SiC polytypes. We concentrate, however, on more recent results obtained on semiconductor superlattices. As always, inelastic light scattering essentially probes the dispersion relation along a vertical line 252

Continuum acoustics: folded acoustic modes

in the coq plane. By appropriate choice of specimen periodicity D and incident-light frequency the value of q can be chosen to be on a scale comparable with n/D, so that much of the superlattice Brillouin zone 0 < q < n/D is accessible to light scattering. Furthermore, substitution of typical values for D and the acoustic velocity vl shows that the excitation frequencies in Fig. 7.2 can correspond to tens of wavenumber units. This means that Raman scattering can be used, at least for the higher branches, rather than the more difficult technique of Brillouin scattering which is normally required for light scattering off acoustic modes. The first experimental results on the folded acoustic spectrum were obtained by Colvard et al. (1980). A good, more recent, example of a Raman scattering spectrum is shown in Fig. 7.3. The experiment was carried out in backscattering, and the polarisation selection rules are the same as those for backscattering off the surface of an ordinary specimen, as discussed in §2.5. Thus in (x, x) polarisation the LA phonons are seen, and there is no scattering by phonons in (x, y) polarisation. Jusserand et al. (1983, 1984a and b) carried out somewhat similar experiments. They used specimens with longer spatial periods of the order

Fig. 7.2 The folded-phonon dispersion curve of (7.24). The parameters are dl/d2 = 3/7, vjv2 = 0.856 and Z = 0.779. These are chosen arbitrarily to illustrate the stop bands and the nonperiodicity of the dispersion curve [after Babiker et al. 1985a] 30 -

20

10

B A

253

Layered structures and superlattices

of 20 nm, and they also varied the frequency of the incident light so as to probe a range of values of Q. By these means they were able to investigate much of the Brillouin zone. Positions of their scattering peaks, in comparison with the Rytov dispersion curve, are shown in Fig. 7.4. A general review of experimental results on folded acoustic phonons, and also on the folded optic phonons that form the subject of §7.2, is given by Klein (1986). The predominant experimental technique has been light scattering. 7.1.3 Green functions, oblique propagation and surface modes

Some developments of the Rytov-type theory presented above should be mentioned. By analogy with the detailed treatment of Brillouin scattering off isotropic specimens (§2.5), a full theory of lineshapes and strengths in light-scattering experiments should start with a derivation of the appropriate acoustic Green function. Babiker et al. (1985a) found the Green function for longitudinal or transverse phonons propagating normal to the interfaces in an infinitely extended superlattice. Their result was used in an approximate theory of the light-scattering cross section by Babiker and Tilley (1984) and Babiker et al. (1985b); the theory is approximate because of the use of the Green function for an infinite rather Fig. 73 Raman backscattering from a sample nominally consisting of a repeat unit of 4.2 nm GaAs and 0.8 nm Al 0 3 Ga 0 7 As. Inset shows dispersion curve corresponding to (7.24) for longitudinal acoustic phonons in Rytov's model, with crosses at experimental points. Arrows on main graph indicate peak frequencies corresponding to the superlattice period of 5.22 nm determined by X-ray diffraction [after Colvard et al. 1985]. 1000 •

800 • 600

a

80 120 Raman shift (cm 1 )

254

160

200

Continuum acoustics: folded acoustic modes

than a semi-infinite structure, because the optical propagation is treated in an 'average-medium' approximation rather than with the formalism to be presented in §7.3, and because possible ripple mechanisms for scattering were not included. The poles of the Green function are at the positions given by the Rytov model, (7.24), and therefore the scattering peaks are predicted to be at these positions. This is in agreement with experimental results such as those of Figs. 7.3 and 7.4. Apart from this more-or-less trivial result, the theory also predicts that line intensities should decrease with frequency, in general agreement with the experimental results. However, a detailed comparison of theory with experiment is not yet available. The theory of acoustic propagation at a general angle to the interface planes was given by Camley et al. (1983a) for s-transverse waves. The Fig. 7.4 Peak positions for Raman backscattering by folded LA phonons. Repeat unit: specimen (a), 5.6 nm GaAs/7.8 nm Al 0 7 7 Ga 0 23 As; specimen (b), 15 nm GaAs/10.7nm Al o . 36 Ga O64 As [after Jusserand et al. 1983].

i

5 -

0.2

0.4

0.6

0.8 6

1 1

Wavevector(10 cnr )

255

Layered structures and superlattices

derivation of the dispersion relation of the bulk modes is not much more complicated than was given above for normal incidence. Camley et al. go on to discuss surface and interface modes of a semi-infinite superlattice in contact with either vacuum or a bulk elastic medium. It is clear on general grounds that surface modes may occur at frequencies within the stop bands of dispersion curves like Fig. 7.2. It is found that they do indeed occur; for details the reader may refer to the original paper. Similar results are presented by Bulgakov (1985). It should be emphasised that in contrast to the Rayleigh and Stoneley waves discussed in §2.3 these surface modes occur in pure s-polarisation and are a consequence of the alternation of elastic properties depicted in Fig. 7.1. It may be noted that the authors also derive a Green function that in an appropriate limit should reduce to that given by Babiker et al. (1985a). The propagation of longitudinal plus p-transverse waves is discussed by Djafari-Rouhani et al. (1983). The elastic displacement in each layer is then described by four amplitudes rather than the two of (7.3) or (7.9), since there are two amplitudes for the longitudinal and two for the transverse component of the displacement. Consequently the transfer matrix is 4 x 4; the explicit form is given by Djafari-Rouhani et al. The authors present numerical illustrations for the Al/W superlattice. They show both the bulk modes of an infinite superlattice and the surface modes of a semi-infinite superlattice. Examples of their results are illustrated in Fig. 7.5. Although the theory has been given in detail for oblique propagation, experimental results are hardly available as yet. This is partly because of specimen shape: a semiconductor superlattice is essentially a thick composite film on a substrate. Because of the high refractive index, for the same reason as in the light-scattering experiments off ordinary semiconductors discussed in §2.5, experiments in which the incident and scattered light pass through the top face of the specimen are tantamount to normal incidence. It is possible for the incident light to be put in, or the scattered light taken out, parallel to the interfaces, but no results on acoustic modes in such a geometry have been reported. 7.2 Lattice dynamics: folded optic modes 7.2.1 One-dimensional model

The previous section was concerned with the folding of the acoustic-phonon dispersion curve into the mini-Brillouin zone resulting from the long period D. The specimens of most interest are based on III-V semiconductors such as GaAs. Since these materials are diatomic, they support optic phonons as well as acoustic phonons, and the optic 256

Lattice dynamics: folded optic modes

phonons in a superlattice are also folded. We begin this section with a detailed discussion of the simplest ID model, then discuss more realistic models and experimental results. The experiments, which are reviewed in more detail by Klein (1986), are again mainly Raman scattering. The model to be discussed is summarised in Fig. 7.6. It is assumed that the interatomic spacing a and elastic constant C have the same values in the two components of the superlattice, the difference between the two components being the change from mass mx to m2. The model may be seen as a simple description of a GaAs/A^Gax _xAs superlattice, m0 being the As mass. The dynamical behaviour of an infinite ID diatomic lattice was summarised in §2.1.2. The dispersion equation is given in (2.17), with further characterisation in Fig. 2.2 and Table 2.1. For the superlattice we take solutions in cell / of the form of (7.3) and (7.9). Now, of course, qx and q2 are related to co by (2.17) with appropriate mass values substituted. There is no need to restrict attention to frequency intervals in which qx and q2 are both real; it will be seen in fact that there are phonon modes

Fig. 7.5 Bulk continua and surface bands for longitudinal plus p-transverse modes in the Al/W superlattice (full line, Al on surface; broken line, W on surface): (a) d\\ = d^; (b) d&\ = 5d w . The surface-layer thickness is assumed to be the same as that of a corresponding bulk layer. The variables are dimensionless with cf(Al) denoting the transverse velocity of sound in Al, and D = d^\ + d\y [after Djafari-Rouhani et al. 1983].

Q

Layered structures and superlattices of the superlattice in which only one of ql and q2 is real. The algebra is the same whether or not q1 and q2 are real. The amplitudes at the left and right ends of a layer are still related by the matrices F1 and F2 of (7.7). In §7.1, the amplitudes in the two constituents of cell / were related by the two boundary conditions at the interface to yield (7.11) and (7.13). As was seen in §2.1, the latticedynamical equivalent is provided by the equations of motion of the atoms on either side of the interface. Thus the equations of motion of the masses indicated by arrows in Fig. 7.6 provide the analogues of (7.11) and (7.13). The results are

1

' W>-( '

(7.27) OC2S2

and 1

1 a2s2"1

1

a11s l

1 |wf> a 2 s 2i

(7.28)

where st = exp(i^a)

i = 1,2

(7.29)

and OL1 and a 2 are the ratios of the displacements of the two masses in the relevant component, given by (2.16): 2C — m0co2

a)

i=l,2

2C - m^2

(7.30)

Equations (7.7), (7.27) and (7.28) combine to give a transfer matrix T as defined in (7.14). The detailed algebra gives

Fig. 7.6 Microscopic model for a diatomic superlattice in the region occupied by unit cell /. The masses are coupled by nearest-neighbour forces with spring constant C.

2

X . — /-I

258

O

•

- 2nx a •

. 2n2a

Component 1

Component 2

O •

O •

O:x cell /

O x O

x

Lattice dynamics: folded optic modes

It can be confirmed from these expressions that det T = 1. The dispersion relation continues to be given by (7.23), which can be brought into the explicit form

x sinO?^) sin(g2d2) (7.33) This result was first given by Jusserand et al. (1984a), and the explicit derivation by a Green function technique is given by Djafari-Rouhani et al. (1985). Equation (7.33) contains a considerable amount of information, which we now explore. In the continuum limit qxa« 1 and q2a « 1 it reduces to (7.24), as it should. The result of a numerical solution of (7.33) for a specific choice of parameters is shown in Fig. l.l{b). In order to interpret this result, we include in Fig. 7J(a) the dispersion curves for the two bulk media. This enables us to divide the frequency axis into six regions, as indicated on the figure. As a development of Table 2.1, Table 7.1 shows the character of the bulk wavenumbers qx and q2 in each of these regions. Comparison with Fig. 1.1 (b) shows that where qx and q are both real, in regions I and IV, the superlattice dispersion curve shows broad pass bands and narrow stop bands. This is similar to what was found for the continuum description in §7.1. In regions III and V, however, where only one of qx and q2 is real, the pass bands are narrow and the stop bands broad. A qualitative reason for this can easily be seen from (7.33). In region V, for example, q1 = \y. The trigonometric functions then become hyperbolic: c o s ^ ^ ) = cosh();1d1), sm(qldl) = \smh(yld1), tan(^xa) = i tanh(};1a). Thus the right-hand side of (7.33) is essentially of the form Ax cos(q2d2) + A2 sm(q2d2), with |X 1 |»1 and |y4 2 |»l. This expression can be rewritten as B cos(q2d2 + 0), with B » 1. Thus as co moves up through region V (7.34) has solutions for Q only in the very narrow frequency bands where \cos(q2d2 + 0)|« !/#• A n analogous situation occurs for region III. The phonon modes in regions III and V have an oscillatory spatial dependence in the medium with real q and an exponentially decaying spatial dependence in the other medium. They are usually called confined phonon modes. The frequency varies only slightly with Q, and is therefore the same as would be found in a single slab of the real-g medium embedded in the other medium. The above model is the simplest possible for folded optic modes but we shall see that it is in, at least, qualitative agreement with Ramanscattering results. A much more complete theory using 3D lattice dynamics with realistic values for the semiconductor parameters was given by Yip 259

Layered structures and superlattices Table 7.1. Frequency intervals of Fig. 7.7 Frequency interval

Superlattice dispersion curve

I

R

R

II III

C R

C C

IV

R

R

V

I

R

VI

I

I

Broad pass bands, narrow stop bands No solutions Narrow pass bands, broad stop bands Broad pass bands, narrow stop bands Narrow pass bands, broad stop bands No solutions

R = real, C = (n/2a) + iy, I = pure imaginary.

and Chang (1984). Their results have the same general form as shown in Fig. 7.7. The most important physical feature that is omitted from the simple model calculation is the influence of the long-wavelength Coulomb Fig. 7.7 (a) Bulk dispersion curves, (2.17), for two constituents of a superlattice with m 1 /m 0 = 0.93, m 2 /m 0 = 0.70. Frequency axis is co/a)u where col = [2C(mQl +m 1 ~ 1 )] 1 / 2 is the zone-centre optic-phonon frequency in medium 1. (b) Dispersion curve for a superlattice of these constituents with nl=n2= 10, i.e. 10 unit cells in each layer [after Albuquerque et al. 1988].

0.2 0.4 0.6 0.8 1.0 1.2 1.4 n/2 qa {a)

260

0

0.5

1.0

1.5 QD (b)

2.0

3.0

Lattice dynamics: folded optic modes interaction on the optic modes. In a bulk polar medium, this interaction is responsible for the LO-TO splitting, and is crucial to the theory of the dielectric response. Although Yip and Chang include the Coulomb interaction by a perturbation method, there is as yet no full and clear account of the matter. Most experimental groups have so far made comparison with the simple model. 7.2.2 Light scattering As with the folded acoustic modes, the main experimental technique that has been applied to study the folded-optic mode spectrum is Raman scattering. The calculation presented above is concerned with phonons propagating normal to the interfaces, so the relevant experimental results are from 180° backscattering at normal incidence. The selection rule for this geometry is that only longitudinal modes contribute to the cross section, so the data involve folded LO phonons. Figure 7.8 shows Raman spectra obtained by Jusserand et al. (1984b, 1985) from a number of GaAs/Al0 3 Ga 0 7As superlattice specimens. The peaks labelled 1 to 4 Fig. 7.8 Raman spectra in the LO phonon region from four samples consisting of repeat units having n monolayers of GaAs and ri monolayers of Al 0 3 Ga 0 7 As. The (n, ri) values from top to bottom are: (6,4), (9,9), (12,7), (17,12) [after Jusserand et al. 1984b]. T = 8 0 K X,

514.5 nm

z(x,y)z

S2

300

295 290 285 280 Frequency shift (cm"1)

275

261

Layered structures and superlattices

are the highest four confined states; peak A is assigned to an AlGaAs LO mode. The peaks correspond to LO modes confined in GaAs by AlGaAs 'phonon barriers', so that their positions should depend on the GaAs monolayer number n but not significantly on the AlGaAs monolayer number. Figure 7.8 shows the variation with n of the peak positions compared with theoretical curves derived from (7.33). The theory depends on the value of the AlGaAs LO phonon frequency; two possibilities are shown on Fig. 7.9. Figure 7.9 shows that the line intensity decreases with increasing folding number, as it does for folded acoustic phonons, Fig. 7.3. A theoretical account involves evaluation of the intensities from the relevant Green function theory by S. R. P. Smith (unpublished), and the resulting intensities do indeed decrease with folding number in agreement with the experimental results. Raman-scattering results relevant to optic modes that are not propagating normal to the interfaces were published by Zucker et a\. (1984, 1985). They arranged that the superlattice specimen was an optical waveguide by surrounding it with AlGaAs cladding layers. By this means they were able to guide out the scattered light in the plane of the layers, so that they had a 90° scattering geometry. The positions and polarisation Fig. 7.9 LO phonon frequencies versus n for (GaAs)n(Al0 3 Ga 0 7As) superlattices. Circles give experimental results. Lines give results of calculations based on (7.33). Full line for AlGaAs LO frequency corresponding to 280 cm" 1 , broken line to 286 cm" 1 [after Jusserand et al. 1984a].

S 285 -

10

262

20

Optical properties dependence of their peaks were consistent with a simple model in which the specimen is treated as a set of isolated GaAs layers. Each layer will then sustain Kliewer-Fuchs modes as described by (6.62) and (6.63); Zucker et al. were able to fit their results by taking the expressions for a GaAs layer in vacuum. The theoretical significance of this simple result is not altogether clear. 7.2.3 Surface modes Equation (7.33) and the subsequent discussion were concerned with the bulk modes of an infinite superlattice. As in continuum acoustics, surface modes may occur in stop bands, such as those of Fig. 1 .l(b). A brief discussion of these surface modes for the ID model is given by Djafari-Rouhani et al. (1985). Their calculation is analogous to that of Wallis (1957) for the surface mode of a semi-infinite ID diatomic crystal, which was outlined in §2.1. 7.3 Optical properties 7.3.1 General results The complete formal theory of optical propagation in an infinite periodic layered medium in which the constituents are isotropic or cubic was given by Yeh et al. (1977) and Yariv and Yeh (1977). An account is also given by Yariv and Yeh (1984). The theory is very similar to the continuum acoustics of §7.1. There is, however, the simplification that in optics the separation between transverse and longitudinal modes is maintained in oblique incidence. Thus the theory for waves travelling in an arbitrary direction through the superlattice is not much harder than that for waves travelling normal to the interface, and we proceed straight away to the general case. As indicated in Fig. 7.10, the notation is similar to that used for the acoustics calculation. Each medium is characterised by an isotropic dielectric function e x, s2 or equivalently refractive index rjx = e} /2 , rj2 = s\12. The z axis is taken normal to the layers, as before, and the x axis is chosen so that the propagation vector lies in the xz plane. As usual, therefore, we can distinguish s-polarisation, with E in the y direction, from p-polarisation, with E in the xz plane. We start with s-polarisation, for which the formalism of §7.1 applies with minor modifications. All fields are taken to have x- and ^-dependences proportional to exp(ig xx — ia>t). The E field in a layer of medium 1 can then be written in either of the forms of (7.3), where now ql is given by q\ + ql = sl(a2lc2

(7.34) 263

Layered structures and superlattices Equation (7.5) relates the amplitudes at the two ends of the layer, with the phase matrix Fx given by (7.7) and (7.8). The E field in medium 2 is written as (7.9), with (7.10) relating the two forms. The boundary conditions at an interface are that Ey and Hx are continuous, that is, Ey and dEJdz are continuous. Thus (7.11) and (7.13) are replaced by

wf> = <

t>

(7.35)

and (7.36) where ' = 1,2

(7.37)

Equations (7.5), (7.10), (7.35) and (7.36) combine to give a transfer matrix T: T = Q;1Q2F2Q;iQ1Fl

(7.38)

Explicit evaluation of T requires patience rather than insight; for example, the diagonal elements are Tn = / i [ / 2 f a i +q2)2 - / 2 ' f o i -
(7.39)

and T22 =f;1U2-1(qi+q2)2 -fiiqi-qifV^qi (7.40) The application of Bloch's theorem as given in §7.1 now shows that the dispersion relation is cos(QD) = cos(q1d1)cos(q2d2) - gs s i n ^ ^ J sin(q2d2) Fig. 7.10 Notation for optical calculation.

264

(7.41)

Optical properties where ( 7 - 42 ) The calculation for p-polarisation is similar to that just given. As noted in §6.3, the boundary conditions for s-polarisation can be converted into those for p-polarisation by the replacement q2/qi ->£2<7i/ei42- Thus the dispersion relation is given by (7.41), with gs replaced by GP = i(e 2 0i/£i02 + £i42/£24i)

(7.43)

Equation (7.41) gives the dispersion relation co = co(qx, Q); qx and co enter via (7.34) for qx and q2. A very full discussion of the implications when sx and e2 are positive and independent of frequency is given by Yeh et al. (1977) and Yariv and Yeh (1977). As for the acoustic case, band gaps occur at the points QD = nn. It is convenient now to work in an extended-zone representation; a typical curve of co versus Q for either polarisation is sketched in Fig. 7.11 (a). An alternative representation consists of drawing contours of constant frequency in the qxQ plane, as shown in Fig. 7.1 l(b). The contours for s- and p-polarisation are different, except that they touch in the direction qx = 0. This corresponds to propagation normal to the interfaces, for which s- and p-polarisation are indistinguishable. The band gaps at Q = n/D have the effect that the contours in thefirstzone are pulled out towards the zone edge; by analogy with electron-band theory this is a consequence of the gaps shown in Fig. 7.11 (a). The contours may be compared with those drawn for a conventional uniaxial medium, as shown in most optics texts (see, e.g., Lipson and Lipson 1969). In that case the contours corresponding to the ordinary wave are circles, while those corresponding to the extraordinary wave are ellipses. For the superlattice neither set of contours is circular, so in a sense both polarisations correspond to extraordinary propagation. Furthermore, for a frequency and direction corresponding to a gap region, there can be none or only one propagating mode rather than the two that are always found in a conventional medium. Although the theoretical treatment for positive and constant ex and s2 was given in complete and definitive form by Yariv and coworkers, there has been little experimental investigation. The case when s 1 and e2 exhibit dispersion is also clearly of considerable interest, not least for semiconductor superlattices. The theoretical development, down to (7.41), continues to apply, but a full discussion of the implications is lacking as yet. Similarly, there is room for much more experimental work. A useful simplification of (7.41) for the very long-wavelength limit was given by Raj and Tilley (1985). At low frequencies, as for a semiconductor in the reststrahl region for example, the wavelength is much longer than 265

Layered structures and superlattices

the periodicity D. Thus the inequalties qxD«l, QD«1, qxD«\ and q2D « 1 may be expected to hold; in terms of Fig. l.l\(b) we are discussing contours well inside the first Brillouin zone. In that case all the trigonometric functions in (7.41) can be expanded. Retention of the first two Fig. 7.11 {a) Dispersion curve OJ versus Q for fixed qx for optical propagation in either polarisation. In an extended-zone scheme, band gaps appear at Q = nn/D. (b) Constant-frequency contours in the qxQ plane. {a) IDD/C

n/2

it)

266

Optical properties terms for the cosines and the first terms for the sines leads to the simplified equations ZxxiQ1 + <7x) = u>2/c2 ZxxQ

2

e

1(

2 2

+ zz ll = u> /c

s-polarisation

(7.44)

p-polarisation

(7.45)

where exx = (eidi+s2d2)/D e

l=

e 1(

(7.46)

E1

z~z ( r ^i + i d2)ID (7.47) The same results, with alternative derivations and different physical discussion, were obtained at about the same time by Agranovich and Kravtsov (1985) and by Liu et al. (1985). The argument presented by the former authors is particularly illuminating. At long wavelengths the boundary conditions that Ex and Dz are continuous means that these field components have the same values over many periods of the superlattice. Thus we write Exi= E

_J

i=l,2

(7.48)

where Ex, Dz are the constant values of the fields. The other field components are given by

I= 1 2

'

E"-1% ^zi

8

i

—

U

(749)

zi

The spatial averages Dx = (d1Dxl + d2Dx2)/D and Ez are then given by Dx = sxxEx _ (7.50) x * Ez

=

£

zz ^z

with the dielectric-tensor components given by (7.46) and (7.47). Equations (7.44) and (7.45) show that in the long-wavelength limit the superlattice behaves as a uniaxial medium with the averaged (or effective) dielectric function obtained from (7.46) and (7.47). Raj and Tilley (1985) discuss the reststrahl region of GaAs/Al^Ga! _ xAs as a particular example. The frequency dependence of sxx and szz is quite complicated since Al^Ga^^As exhibits two-mode behaviour (see Problem 6.1), while GaAs has a single TO resonance. For this system, (7.46) has been used by Maslin et al. (1986) to give an account of normal-incidence far-infrared reflectivity measurements. The experiments were carried out by dispersive Fourier transform spectroscopy (DFTS), which as explained by Birch and Parker (1979) is a technique for making simultaneous measurements of the amplitude and phase of the complex reflectivity. Experimental results and theory are compared in Fig. 7.12. The theoretical curves represent a leastsquares fit to the data. The theoretical expressions are given by the usual 267

Layered structures and superlattices

classical optics of a multilayer system, with the superlattice described by (7.46). The dielectric functions ex and e2 appearing there are taken in the forms of (5.60) and Problem 6.1, and the parameters in those expressions are found from the fit to experiment. Equations (7.46) and (7.47) can be used together with standard results for optical propagation in anisotropic media to give an account of bulkpolariton propagation in the long-wavelength limit. This is done by Raj et al. (1987). 7.3.2 Green functions

Linear-response theory has been used by Babiker et al. (1987a) to evaluate the electric-field Green functions for an infinitely-extended two-component dielectric superlattice, as depicted in Fig. 7.10, with retardation effects included. Briefly, the calculation involves using Maxwell's equations to write down expressions for the electricfieldvectors in each component layer of the superlattice in the presence of an externally applied polarisation field Pext. The standard electromagnetic boundary conditions are applied at each interface, just as in the corresponding Fig. 7.12 Far-infrared reflectance data for a specimen including a superlattice comprising 60 periods, each composed of 5.5 nm of GaAs and 17 nm of Al 0 3 5 Ga 0 65 As. The insert shows the specimen structure with thicknesses given in jim. Regions 1 and 3, Al 0 3 5 Ga 0 65 As; region 2, superlattice; region 4, GaAs substrate. In the main part of the figure, crosses are experimental points and the curves are theoretical [after Maslin et al. 1986]. 1.0

0.0

268

250

300 350 Wavenumber (cm 1 )

400

450

Optical properties

homogeneous calculation in §7.3.1, and the periodicity property of the superlattice is utilised through Bloch's theorem as in §7.1. The required Green functions, which are of the form <<£^(g); £v(Q'))>a>» are then obtained from the linear response of the electric-field components to the polarisation Pext, in accordance with general results in the Appendix. Here \i and v denote Cartesian components, and Q and Q' are components of wave vector in the z direction. For an infinitely extended superlattice the Green functions are in fact all proportional to S(Q — Q'). It is found that the Green functions with \i and v both equal to y describe s-polarised modes; they have poles corresponding to the dispersion relation (7.41). Likewise those Green functions with \i and v equal to either x or z describe p-polarised modes. The general Green function results are rather complicated and will not be given here. However, as shown by Babiker et al. (1986c, 1987a), they simplify considerably in the low-frequency limit where retardation effects become unimportant. Some specific applications to light scattering from plasma modes are quoted in §7.4. 7.3.3 Surface modes

It has been brought out in previous sections that surface modes can appear on a semi-infinite superlattice in the stop bands of the folded dispersion curve. A striking illustration occurs with s-transverse acoustic modes, where as mentioned in §7.1 surface modes are predicted on superlattices although they do not occur on bulk specimens. Similarly, surface optical modes can occur simply as a result of the periodic alternation of the refractive index, for example in s-polarisation and with real refractive indices. Surface modes of this kind were already discussed by Yeh et al. (1977). However, in optics surface modes can also occur because one or both component media has a negative dielectric constant, and thus is surface-active. The full range of possibilities arising from these two different mechanisms has not been fully discussed in any specific case. Surface polaritons of the second kind in the reststrahl frequency region have been discussed within the long-wavelength approximation of §7.3.1 by Raj and Tilley (1985) and Raj et al. (1987). Within that approximation the superlattice is described as a uniaxial medium with dielectric tensor components given by (7.46) and (7.47). The surface-polariton dispersion equation is therefore given by (6.42), with the simplification that the x and z direction are principal directions of the dielectric tensor. The calculated dispersion curve for a semi-infinite GaAs/Al^-Ga^^As superlattice bounded by vacuum is shown in Fig. 7.13. Surface polaritons of this kind must satisfy the localisation condition exx<0. In Fig. 7.13 a distinction is made, following Hartstein et al. (1973), between modes with 269

Layered structures and superlattices

szz < 0, called real-excitation surface polaritons, and modes with ezz > 0, called virtual-excitation surface polaritons. In general, modes of the former kind continue to exist as qx^> oo, while the latter terminate at a finite value of qx. In addition to the dispersion curve, Raj et al. present theoretical results for the ATR spectrum of a finite superlattice. The spectrum shows considerable structure, including guided-wave as well as surface polaritons. A range of general results for the p-polarised polaritons of a finite superlattice were given by Haupt and Wendler (1987). In addition to the dispersion curves, they present a perturbation calculation in which the damping of the surface modes is found to first order in the damping constants F appearing in the expressions for the dielectric functions of the constituent media. Their most important conclusion is that just as for the homogeneous film discussed in §6.3, the surface mode of higher frequency has a long propagation distance. Thus the superlattice supports a long-range surface polariton. 7.3.4 Limits of validity of theory The results and discussions of the previous parts of this section have been based on the assumption that each component of the superlattice Fig. 7.13 Surface-polariton dispersion curves a> versus qx, within the long-wavelength approximation, for the interface between a GaAs/Al 0 1 4 Ga 0 86 As superlattice and vacuum. The superlattice is described by (7.46) and (7.47) with dl=d2 and values for the reststrahl parameter taken from Maslin et al. (1986). Full line, realexcitation surface polaritons; broken line, virtual-excitation surface polaritons; dotted line, vacuum light line [after Raj et al. 1987]. 450 -i

400-

350u)/2nc (cm"1)

300-

250200-

250

300

350 qx/2n (cm-1)

270

400

450

500

Single-electron states is characterised by a dielectric constant £, and it is assumed either implicitly or explicitly that e has the same value as it would have in the corresponding bulk medium. However, as mentioned in §5.4, the theory of the bulk dielectric function, for example in the reststrahl region, is essentially the theory of the response of the macroscopic polarisation P to the macroscopic electric field Emac. In the course of the calculation, it is necessary to express the local electric field Eloc in terms of P and Emac. For superlattices with thick layers, as used for Fig. 7.12 for example, it is probably justified to use the bulk e since the expression for Eloc is little altered from the bulk. When the component layers are only a few monolayers thick, however, the expression for Eloc must be modified and the assumption of a bulk e is not justified. The calculation that is then needed is equivalent to the inclusion of the Coulomb interaction in the lattice dynamics of §7.2, and as mentioned there no clear account has yet been given. It is for this reason that although Maslin et al. (1986) present experimental data for a specimen with a spatial period of four monolayers, they do not give any comparison with theory. 7.4 Single-electron states 7.4.1 Envelope-function approximation As was made clear at the beginning of this chapter, the main reason for the very large investment in MBE technology has been to gain more control over electronic properties than is possible with bulk specimens. Consequently electron states have been much more extensively studied than phonons or optical properties. It is beyond our scope to go into much detail; this section is therefore restricted to an introduction to the main ideas, presented so as to bring out the similarities to acoustic and optic properties. The reader who wishes to pursue the topic further should look elsewhere, and could start with the chapters by Kroemer and Bastard in Chang and Ploog (1985). The simplest picture of a semiconductor is of a conduction band and a valence band separated by a gap £ g. Correspondingly, a useful picture for electrons in a superlattice is shown in Fig. 7.14. 'Wells' of width dx of material 1, which has gap El9 are separated by 'barriers' of width d2 of material 2, which has gap E2. This picture shows the bottom of the conduction band and the top of the valence band, with degeneracy neglected. It is appropriate to GaAs/A^Ga^^s with x<0.3, among other materials, since GaAs and AlxGax_xAs are both direct-gap semiconductors, with both valence-band maximum and conduction-band minimum at the F-point (the centre of the Brillouin zone). AlAs and Al/ja^^As with x >0.3, on the other hand, are indirect-gap materials, 271

Layered structures and superlattices with conduction-band minimum at the X-point. Electron transfer across the interfaces is a more complicated process in such materials. Of the quantities defined in Fig. 7.14, Ex and E2 are known, and dl and d2 are known from the growth conditions. However, the band offset, Ec say, is not known a priori and has to be determined for each superlattice system. In some systems, Ec is negative. We now concentrate on motion of a single electron in the potential described by the upper half of Fig. 7.14. Bastard (1981) showed that provided the individual components are more than a few monolayers thick the motion can be described by means of the envelope function F(r) that describes the modulation of the electron wavefunction from one atomic site to the next. To be specific, the wavefunction is written
(7.51)

where wo(r) is the Bloch wavefunction corresponding to the bottom of the conduction band, satisfying wo(r + a) = wo(r), where a is a unit vector of the lattice. Provided F(r) varies only slowly on the scale of a, it satisfies Schrodinger's equation for a particle with the effective mass m appropriate to the conduction band. Bastard (1981) further shows that in the approximation that the bands are parabolic, E = h2q2/2m, the boundary conditions at the interface between two component layers are F1=F2

(7.52)

x

l

m; dFJdz = m2 dFJdz

(7.53)

Equation (7.53) implies continuity across the interface of the probability current density J = (h/2im)(F*dF/dz - FdF*/dz)

(7.54)

Further discussion of the envelope-function approximation is given in the chapter by Bastard in Chang and Ploog (1985). 7.4.2 Transfer matrix and dispersion relation The description of the electron motion now reduces to solving the Kronig-Penney model with the boundary conditions (7.52) and (7.53). Fig. 7.14. Simple model for electronic properties of a superlattice.

A V

272

Single-electron states For convenience we simplify and redefine the notation to that shown in Fig. 7.15. The calculation is very similar to those given in previous sections. For electron energy E > Vo, the wavefunction is a sum of forward- and backward-running waves in each medium: F(z) = ax cxpliq^z — ID)] + bx exp[ — \qx(z — ID)] lD
+ d^

(7.55)

F(z) = c, exp[i<72(z - ID - dj] + dx exp[-i^f 2 (z - ID - dj] /D + d 1 < z < ( / + l ) Z ) (7.56) where E = h2q2l2m1

(7.57)

2 2

E-V0 = h q /2m2 (7.58) As in §7.1.1, the boundary conditions are used to find a transfer matrix T relating (al + ubl+1)to (ah bx). Application of Bloch's theorem then gives the dispersion relation in the form of (7.23), and substitution of the explicit form of T (Problem 7.4) gives the result cos(gD) = c o s ^ d j cos(q2d2) — \(C 4- C~x) s i n ^ ^ J sin(^f2rf2)

(7.59)

where C = m2qjm1q2

(7.60)

For electron energy E < Vo, the wavefunction in the barrier regions is a sum of real exponentials. Formally, q2 is replaced by i/c2, where Vo - E = h2K\l2m2

(7.61)

The dispersion relation becomes cos(gD) = c o s ^ ^ i ) cosh(K:2rf2) — \(B — B~l) s i n ^ ^ J sinh(K:2^2)

(7.62)

where B = m2ql/m1K2

(7.63)

The discussion of (7.59) and (7.62) proceeds along familiar lines, and the analogy with the lattice dynamics of §7.2.1 is particularly close. For E > Vo both qx and q2 are real; this is a region of wide pass bands and narrow stop bands. For E < Vo ql is real and q2 is imaginary. We now have bound states in the well regions 1 coupled by evanescent waves in Fig. 7.15. Notation for Kronig-Penney calculation. v — v r

—

r

1

2

n

v=o-

ID

273

Layered structures and superlattices

the barrier regions 2. For K2d2 large, the coupling is weak: this is the tight-binding limit, with almost-flat bands at the bound-state energies of the wells. These comments are illustrated by Fig. 7.16. For the well depth chosen, the isolated well has three bound states. The barrier width is such that the well states are only weakly coupled, so for E < Vo we have very flat bands. For E > Fo, on the other hand, the stop bands are very narrow. Figure 7.16 may be compared with Fig. 7.7, which shows similar effects for lattice dynamics. It was mentioned at the beginning of this chapter that the term 'superlattice' is often used with a more restricted meaning as far as electronic states are concerned. In this context 'superlattice' denotes a specimen in which the barriers are relatively thin so that there is significant overlap between the wavefunctions of bound states in adjacent wells, and the E-Q curves arising from bound states therefore show significant dispersion. Conversely, the term 'multiple quantum well' (MQW) usually denotes a specimen in which wavefunction overlap is negligible and the E-Q curve is flat. Thus Fig. 7.16 corresponds to the MQW case. Preliminary experimental results related to the band structure for electronic motion perpendicular to the layers have been reported by Duffield et al. (1986). As indicated in Fig. 7.17, they carried out cyclotron resonance experiments on a superlattice specimen with the magnetic field Fig. 7.16. Electron bands in the envelope-function approximation for d1/d2 = 0.2 and im^lVJh2 = 80. (a)m2/m1 = 1, (b)m2/m1 = 2.0, (c)m2/m1 = 0.5 [afterR. N. Philp unpublished]. 120r 100

80

r

V 60 40

20

0

274

0.5

1.0

1.5 QD

2.0

2.5

3.0

Single-electron states applied in the plane of the layers. As shown there, the electron orbit traverses the layers, and the cyclotron-resonance frequency is coc = eB/mc, where mc is given by the standard expression for an anisotropic material (see Kittel 1986): mc=(h2/2n)dS/dE

(7.64)

and the derivative is the rate of change with electron energy E of the orbit

to

120

•

—

.

—i

•

100

H

l

'

1

80

60

I

40

20 n

-

0

•

.

i

i

i

i

0.5

1.0

1.5 QD

2.0

2.5

3.0

I

275

Layered structures and superlattices

area S in g-space. Loosely speaking mc is the geometric mean of the effective masses parallel and perpendicular to the layers. In subsidiary experiments in which B is perpendicular to the layers, the effective mass for motion in the plane is measured. Combination of the two measurements therefore yields the effective mass for motion perpendicular to the interfaces, which is determined by the curvature of the appropriate E-q curve in dispersion graphs like Fig. 7.16. The experimental method consisted of measurement of far-infrared transmission through the specimen by means of DFTS spectroscopy, which was mentioned in §7.3.2. Typical results are shown in Fig. 7.18. As explained, the combination of resonance frequencies for the two orientations of magnetic field yields a value for the effective mass perpendicular to the interfaces. Duffield et al. determine this mass for five different values of concentration x, obtaining results that are in satisfactory agreement with the envelope-function approximation. 7.5 Plasmons

This section is concerned with the coupled plasma oscillations of the electrons (or holes) in superlattices and MQWs. A simple theoretical Fig. 7.17 Magnetic-field orientation and electron orbit in the cyclotron-resonance experiment of Duffield et al. (1986).

276

Plasmons

description, which is applicable if the well thickness (d1 in the notation of Fig. 7.15) is sufficiently small, is to treat the charge layers as being 2D sheets separated by a dielectric medium of thickness d2. This approach is discussed in §7.5.1. Then in §7.5.2 we present the theory based on taking the charge layers to be of finite, rather than infinitesimal, thickness. In both cases, the calculations may be carried out either microscopically (e.g. by starting from a Hamiltonian and using RPA) or macroscopically (e.g. by hydrodynamic theory or by a description based on solving Maxwell's equation); this is a similar situation to the case of plasmons in a single medium as discussed in §5.3. For simplicity the account given here will emphasise the macroscopic approach. Light-scattering studies have provided verification of many of the theoretical results for the plasmon modes of a superlattice (or MQW), and we give an account of this work in §7.5.3. 7.5.1 Two-dimensional charge-sheets model

As discussed already in §5.3, the bulk 3D plasmon frequency at long wavelengths has a constant value cop, while the plasmon frequency Fig. 7.18 Inverse of transmission coefficient in cyclotron resonance experiments on an n-type specimen at 70-80 K. The curves peak (minimum transmission) at the resonance frequency. Applied field values are marked [after Duffield et al. 1986].

1.25-

1.00

1.25 -

1.00 50 100 Frequency (cm"1)

277

Layered structures and superlattices

of a single 2D layer is proportional to the square root of the wavevector 4l|. In the case of an infinitely extended periodic array of equally spaced 2D electron layers, the plasmon mode wasfirstcalculated by Fetter (1974) and subsequently studied by Das Sarma and Quinn (1982) and Bloss and Brody (1982), indicating a behaviour that lies between the above two extremes. To rederive their results in a form appropriate to long wavelengths we employ the model and notation of Fig. 7.19. Maxwell's equations may be utilised within each dielectric medium (A) subject to the standard electromagnetic boundary conditions at the charge-layer interfaces. The transverse electric field within the dielectric medium of cell n may be written as £?>(*, z) = exp[i(^x - (ot)]{a(n) exp[iqz(z - nD)] + bin) exp[-i^ z (z - nD)-]} (7.65) where ain) and bin) are the amplitudes for the forward- and backwardtravelling waves, and the condition for the electric field to satisfy the wave equation gives qz = (eco2/c2-q2x)^2

(7.66)

Without loss of generality we have taken the in-plane wavevector qy to be in the x direction. Using V x H = £oe(dE/dt), the magnetic field in medium A of cell n is obtained as

H{;\x, z) = (eoea>/qz) exppfayc - cot)]{a(n) exp[iqz(z - nD)] -b^Qxpl-iqz(z-nD)]} (7.67) The usual boundary conditions, that the electric field Ex is continuous across any interface and the discontinuity in the magneticfieldHy is equal to the current density at any interface, can now be applied at the interface z = nD. The complete solution is obtained noting that Bloch's theorem Fig. 7.19 Schematic illustration of a superlattice of equally spaced 2D charge layers separated by a dielectric medium (A). z= (n-l)D

n-\

278

Unit cell

D

nD

«+l

n+2

Plasmons implies (e.g., see (7.2)) a(n+1) = Qxp(iQD)ain)

b(n + 1) = exp(iQD)b{n)

(7.68)

where Q is the component of the excitation wavevector in the z direction. This leads to two linear homogeneous equations involving a(n) and b{n\ and the solvability condition is obtained as (see Problem 7.5) 2co2[cos(QD) - cos(qzD)] - Q2qzD sin(qzD) = 0

(7.69)

Here Q denotes a characteristic frequency parameter defined by Q = (noe2/sosm*D)1/2

(7.70)

where n0 is the areal density of electrons (of effective mass m*) in each 2D sheet. Equation (7.69) could equally well have been obtained by transfer matrix methods. The dispersion relation (7.69) simplifies in the limit q2. »sa>2/c2, which corresponds to neglecting retardation effects, because we then have qz = iqx. Substitution into (7.69) gives

_ j

qxDsmh(qxD) \^2 [2[cosh(^D)cos(eD)]J

l

'

}

in agreement with calculations by Fetter (1974), Das Sarma and Quinn (1982) and Bloss and Brody (1982) using different methods. It is easily verified from (7.71) that, except when cos(gD) = 1, we have co proportional to qx at long wavelengths (such that qxD«l). The behaviour of the plasmon frequency as a function of qx in the nonretarded regime is illustrated in Fig. 7.20 for three different values of QD. There are also additional solutions of (7.69) at higher frequencies such that eco2/c2 >; ql, when retardation effects become important. These have the characteristic of the perturbed photon line folded back in a reduced zone scheme (e.g., see Constantinou and Cottam 1986). Next we consider the plasmon modes of a semi-infinite superlattice, having the same charge layer structure as in Fig. 7.19 but occupying just the half-space z ^ 0. The region z > 0 is assumed to be filled by a medium of dielectric constant es, and we define the ratio rs = sjs. The surface z = 0 between the outermost cell of the superlattice and the external medium is different from all the other interfaces within the superlattice, and so the electron density nOs associated with its charge layer may be different from n0 (or even zero). On defining /as = nOs/no, we have two surface parameters rs and /*s as in Constantinou and Cottam (1986). In earlier calculations for semi-infinite arrays of 2D charge layers (Giuliani and Quinn 1983; Giuliani et al. 1984; Jain and Allen 1985a) a more restricted boundary condition was employed in which the only parameter was rs, whilst fis was arbitrarily taken equal to one. 279

Layered structures and superlattices The calculation of bulk and surface plasmon modes can be made by a direct generalisation of the infinite superlattice case. The electric and magnetic fields in each region of space, including z > 0, can be written down as in (7.65)-(7.67). The usual boundary conditions can then be applied at each charge-layer interface to provide relationships between the amplitude coefficients. In the case of bulk-plasmon modes the Bloch ansatz (7.68) still applies, and the same frequencies are obtained as for the infinitely extended superlattice. For the localised surface plasmons, which are characterised by a decaying amplitude with distance from the surface plane z = 0, equation (7.68) holds but with Q replaced by U, where Re(^,) > 0 for the attenuation factor. It is found (see Constantinou and Cottam 1986) that the frequency co of any surface plasmon must satisfy the quadratic equation /*,(//, - 1) smh(qxD)(qxDco)2 + Q[r s (l - 2*0 sinh(qxD) + cosh(qxD)]qxD(o + Q 2 (r 2 - 1) sinh(qxD) = 0

(7.72)

where for simplicity we have ignored retardation effects. The attenuation factor is found from exp( - W) = [rs -

sinh(q

(7.73)

Fig. 7.20 Numerical example of the bulk plasmon dispersion relation (7.71) in the absence of retardation according to the charge sheets model. The plots are in terms of dimensionless variables, co/Q. versus qxD, for three different values of QD: A, 0; B, 7i/2; C, 7i.

280

Plasmons Solutions for co may be written down from (7.72), and they represent surface plasmons only if co is real and |exp( — AZ>)| < 1. When fis = 1 for the surface charge density, (7.72) simplifies to give the solution CD = Q{qxDlcosh(qxD) - rs sinh(qxD)W

- r,2) sinh(qxD)} ^

(7.74)

provided |exp(- XD)\ = \cosh(qxD) - rs sinh(^ x D)|" x < 1

(7.75)

This dispersion relation is consistent with results derived by Giuliani and Quinn (1983). It corresponds to a surface branch above or below the bulk-plasmon continuum if rs < 1 or rs > 1 respectively, and the localisation condition (7.75) implies that there is a cut-off wavevector corresponding to qxD > ln|(l + rs)/(l - rs)|. The more general case of/i s / 1 can be studied using (7.72) and (7.73), and some numerical examples are shown in Fig. 7.21. As fis is reduced from unity it is predicted that each surface branch moves to a lower frequency and the cut-off value for qx is modified. The calculations described so far in this section refer only to intrasubband plasmons, because the excitations involve just one electron band. The 2D

Fig. 7.21 Numerical example of the surface plasmon dispersion relation, calculated from (7.72) and (7.73), for several values of the surface parameters; W, r s = 0.08, fis = 1; X, r s = 0.08, ^ s = 0.75; Y, r s = 4, // s = 1; Z, r s = 4, /xs = 0.75. The shaded region is the bulk plasmon continuum. The other parameters are D = 40 nm and e = 12.9 [after Constantinou and Cottam 1986].

0.5

0

1.0

2.0

3.0

281

Layered structures and superlattices charge sheets model has also been extended to the case of several electron bands, thus providing a description of intersubband plasmons. The formalism becomes much more complicated, and we refer to the work of Tsellis and Quinn (1984) and Eliasson et al. (1987) for details. Other studies have been directed to evaluating Green functions for the superlattice of 2D charge sheets (see, e.g., Jain and Allen 1985a,b,c; Hawrylak et al. 1985; Katayama and Ando 1985; Babiker et al 1986a,b; Eliasson et al. 1987); this has been mainly in the context of light scattering, as we discuss in §7.5.3. The bulk- and surface-plasmon frequencies of charge-sheet superlattices with an alternating structure have also been calculated. For example, Qin et al. (1983) considered superlattices with alternating electron and hole charge layers, while Constantinou and Cottam (1986) investigated the case of superlattices with electron charge layers separated alternately by media A and B that may have different thicknesses and dielectric constants.

7.5.2 Alternating bulk dielectric media A different theoretical model, which allows for finite thickness of the charge layers, is to treat the superlattice as alternating bulk layers, as in §7.3.1 and Fig. 7.10, with an appropriate choice for the dielectric functions s^co) and £2(tt0- For example, we could take sj = fij/>[l- (to?'/co)2]

7=1,2

(7.76)

corresponding to a superlattice in which the two components have charge densities rc, and bulk plasma frequencies < e(b7)

= (nje2/s08^mJ)1/2

(7.77)

and mf are, respectively, the background dielectric constant and Here the effective electronic mass in medium j 0 = 1 , 2). This model may be realised in metallic superlattices or semiconductor superlattices, where the layer thicknesses are sufficiently large that the bulk-like expressions (7.76) for the dielectric function provide a good approximation. In certain cases of interest (e.g. GaAs/A^Ga^^As structures) the electrons may be confined to just one of the media, and we then have the simplification of either c o J ^ O or co{p2) = 0. The excitations are coupled plasmon-polaritons, and their dispersion relations (for s- and p-polarisations) are given by the general results obtained earlier in (7.41)-(7.43). We are principally concerned with low-frequency modes (such that e/o 2 /c 2 « g 2), corresponding to the neglect of retardation effects, because this is the situation for the experiments to be discussed in §7.5.3. In the nonretarded limit there are solutions only 282

Plasmons

for the case of p-modes and the dispersion relation becomes 1 /e, £ 2 \ cos(QD) = c o s h ^ d j cosh(qxd2) + -1 — I — s i n h ^ d j sinh(g 2 \£ 2

ei/

xd2)

(7.78)

This can be alternatively expressed as (e.g. Camley and Mills 1984) -=-jS±O?2-l)1/2

(7.79)

where _ c o s h ^ d j cosh(qxd2) — cos(gD) sinh^Jsinh^^) When ex and £2 have the bulk-plasma response form of (7.76), there are two solutions co± for the frequencies of the coupled plasmon modes: (7.81) where we have introduced the ratios r^co^/ca™ and s = ei^)/ei^). The lower, or 'acoustic', branch with frequency co~ is just the analogue of the superlattice plasmon obtained in the 2D sheets limit. The occurrence of the upper, or 'optic', branch can be associated with finite charge-layer thickness. The dependence of co+ and co~ on qx is illustrated numerically in Fig. 7.22 for GaAs/A^Ga^^As, taking medium 2 to be the charge layers (GaAs) and medium 1 to have no charges (A^Ga^^As), corresponding to r = 0 in (7.81). We have considered afixedsuperlattice period together with various electron-layer thicknesses d2. Other numerical parameters have been assigned in accordance with the Raman measurements of Olego et al. (1982) for their Sample 2, i.e. D = 89 nm, 32 m * = 6.37x 10" kg, e ^ - e j ^ - n . O and no = n2d2 with no = 7.3x 15 2 10 m~ . The experimental sample corresponded to d2/D = 0.29, and we discuss comparison with the Raman data (crosses) in §7.5.3. The above calculations, which were for the dispersion relations of the plasmon-polaritons infinite-thicknesscharge layers, have been generalised to the evaluation of electric-field Green functions by Babiker et al. (1986c, 1987a,b) using the approach discussed in §7.3.2. A limitation of these calculations for the dispersion relations and Green functions is that they employ a scalar dielectric function for the charge layers with the bulk cop value. This would be an unsatisfactory approximation for thin enough layers, and some recent calculations by King-Smith and Inkson (1986, 1987) for this case using a microscopic theory imply that the scalar £ is effectively replaced by a tensor with ezz different from sxx and eyy. Another microscopic theory, also based on RPA, has been put forward by Wasserman and Lee (1985). 283

Layered structures and superlattices 7.5.3 Light scattering

The experimental work on Raman scattering from the plasma oscillations in semiconductor superlattices has been reviewed by Pinczuk (1984). Attention has mostly focussed on the GaAs/A^Ga^^As system with the experiments being carried out under resonance conditions in order to enhance the scattered intensity. The first experiments to demonstrate clearly Raman scattering from plasmons in a GaAs/A^Ga^^As MQW structure were due to Olego et al. (1982). Their light-scattering geometry is sketched in Fig. 7.23, where the incident and scattered light beams are in the same vertical plane (labelled as the xz plane). A right-angle configuration was employed, so 0 + $ = 90° for the angles as measured outside the sample. However, it should be noted that because the refractive index rj of the semiconductor is fairly large (rj ~ 3.6) the incident and scattered beams inside the sample Fig. 7.22 Dispersion curves for the frequencies of the acoustic plasmons (full curves) and the optic plasmons (broken curves) as a function of qxD (for fixed QD). The assumed values of the parameters are given in the text. The corresponding experimental results (Olego et al. 1982) for d2/D = 0.29 are shown (crosses). The theory curves are given for the following values of d2/D: V, 0.0; W, 0.29; Y, 0.6; Z, 1.0 [after Babiker et al. 1986c].

284

Plasmons are near to being antiparallel and parallel, respectively, to the z axis. In this configuration the light-scattering wavevector has components qx = (2ic/X)(sin 9 - cos 0) 2

2

qz = (2n/X)l(rj - sin 0)

1/2

(7.82) 2

2

+ (rj - cos 0)

1/2

]

2

^(2n/X)2rj(l-l/4rj )

(7.83)

where I is the wavelength of light in vacuo and the last line of (7.83) follows using rj»1 (see Problem 7.8). Hence by varying the angle 0, qx may be varied whilst qz is kept effectively constant. Raman spectra due to Olego et al (1982) at three different angles 6 and with X ~ 780 nm are shown in Fig. 7.24. Similar measurements have been reported by Fasol et al. (1985) and Sooryakumar et al (1985). The observed frequencies of the Raman peaks correspond well to the plasmon dispersion relations predicted in §7.5.1 and §7.5.2. This may be seen by reference to Fig. 7.22, where the crosses show data due to Olego et al (1982). Curve V (with d2/D = 0) corresponds to the 2D charge-sheets model and it is seen that theory and experiment are fairly close. However, slightly better agreement is achieved when the finite thickness of the charge layer is taken into account, and the full curve W (with d2/D = 0.29) is appropriate to the experimental sample. The optic plasmon branch for this value of d2/D (see broken curve W) occurs for hco ^ 20 meV. This is close to an intense peak in the Raman spectrum as measured by Fig. 7.23 The geometry for light scattering from plasmons in a superlattice with electron-gas layers.

Incident light

Scattered light

, Dielectric capping layer Electron-gas layers

285

Layered structures and superlattices

Sooryakamur et al. (1985), although those authors attributed it to a coupling between LO phonons and intersubband plasmons. Apart from the peak frequencies, a complete theory of Raman scattering from the plasmons should predict the spectral width and lineshape and the integrated intensity, and also the dependence of these quantities on the scattering geometry and the polarisations of the light beams. Light-scattering calculations have been carried out by several authors (e.g. Hawrylak et al. 1985; Jain and Allen 1985a,b,c; Katayama and Ando 1985) using an approach based on density-density correlation functions or Green functions. In particular, they dealt either with effects in semiinfinite or finite superlattices, where localised surface plasmons are predicted, or with effects of interactions between plasmon and phonons; they found results for the spectral linewidth and shape of the bulk-plasmon and surface-plasmon resonances. A different approach, based on general methods for calculating light scattering from polaritons in superlattices, was adopted by Babiker et al (1986a,b,c; 1987b) for both the ID sheets Fig. 7.24 Raman spectra for scattering from plasmons in a GaAs/Al^Ga^^As MQW structure for three different angles 6 in a right-angle scattering geometry [after Olego etal. 1982].

1

3 Stokes shift (meV)

286

5

Magnetic properties and thefinite-thicknesscharge-layer models. The formalism incorporated the dependence on the scattering geometry and the polarisation directions. For a fuller understanding of light scattering from plasmons, it would be helpful to have more experimental data, particularly for the integrated intensities and for systems other than GaAs/Al^Ga^^s. More work, both experimental and theoretical, should be directed towards the superlattice surface plasmon. 7.6 Magnetic properties We finally give an introductory account of superlattices in which one or both of the component materials are magnetic. The nature of the magnon excitations depends crucially on whether the long-range dipoledipole interactions or the short-range exchange interactions dominate, just as discussed in Chapters 3 and 4 for single magnetic layers, and this in turn depends largely on the excitation wavevector. We treat the case where dipolar terms are dominant (i.e. the magnetos tatic and magnetic polariton regimes) in §7.6.1, and then deal with the exchange coupling in §§7.6.2 and 7.6.3. It is towards the former case that most experiments to date have been directed, with the main results coming from light scattering and microwave absorption. 7.6.1 Magnetostatic regime and magnetic polaritons The simplest case is a superlattice in which ferromagnetic and nonmagnetic layers alternate, as represented in Fig. 7.25. The magnetostatic modes for this case can be calculated by a direct extension of the theory for a double-layer ferromagnet (see §4.3) and so we shall quote only the final results. The usual geometry corresponds to the applied field and magnetisation directions being parallel to the interfaces as in Fig. 7.25. The theory was first worked out by Camley et al. (1983b) and Griinberg and Mika (1983). The former authors calculated the magnetic Green functions as well as the dispersion relations, while the latter authors concentrated on the dispersion relations but included effects of a finite number of layers in the superlattice. For the special case of the in-plane wavevector q^ perpendicular to M o (the Voigt configuration), the results for the magnon frequencies may be summarised as follows: the frequencies can be conveniently expressed in terms of a parameter a, as in (4.37) for magnetic double layers, where the modes of the superlattice correspond to: (i) a band of bulk modes for which 2sinh(^||rf1)sinh((?l|rf2) cosh(4,|D)-cos(gxD) where q = (qx9 qy) is the 3D wavevector, and

(

}

287

Layered structures and superlattices (ii) a surface-magnon branch that has (for a semi-infinite superlattice) a=0

(7.85)

and exists only if d2>d1. As a numerical example we show in Fig. 7.26 the values of a for bulk and surface modes of the superlattice plotted against d1/d2 for fixed values of d2 and q^. It can be shown that the bulk-magnon mode of the superlattice is made up of a linear superposition of surface waves within each magnetic layer; the net effect is a bulk mode since the amplitudes and phase factors are such that Bloch's theorem is satisfied (see Camley et al. 1983b). The bulk modes always lie in the frequency range from [(O0(co0 + co m )] 1/2 to (co0 + icom), in the notation of Chapter 4. The surface mode of the superlattice is also a linear superposition of surface waves within each magnetic layer, but they are combined with an envelope function that decays exponentially with distance into the superlattice. This mode, which has the property of existing only if the magnetic-layer thickness d2 is greater than the spacer-layer thickness dl9 has frequency (co0 + jcom) as can be deduced from (4.37) and (7.85). This frequency is

Fig. 7.25 The assumed geometry for a semi-infinite magnetic superlattice composed of alternating ferromagnetic and nonmagnetic layers.

Ferromagnetic layer Nonmagnetic spacer

288

Magnetic properties

the same as that of the surface mode in a semi-infinite ferromagnet (see §4.2). Brillouin spectroscopy has been used to test experimentally the above prediction for the surface mode of the superlattice (Grimsditch et al. 1983; Kueny et al. 1984). In Fig. 121(a) and (b) we show examples of spectra obtained for Ni/Mo superlattices with d2 > dl and d2 dl9 and of only one anti-Stokes line in Fig. 7.27(5) when d2
Bulk modes of superlattice

Surface mode of superlattice -0.2

Fig. 7.27 Brillouin spectra for Ni/Mo superlattices in an applied magnetic field of 0.093 T: (a) d2 = 24.9 nm, d1 = 8.3 nm; {b) d2 = 10 nm, dx = 30 nm [after Grimsditch et al. 1983] 4 2 (a) (b)

I 0 0.4 -0.4 Frequency shift (cm"

! -0.4

0.4

289

Layered structures and superlattices

confirmation of the prediction concerning the surface magnon of the superlattice. Magnetostatic theory has also been applied to alternating ferromagnetic/ nonmagnetic superlattices for the case of the magnetisation Mo perpendicular to the interfaces (Camley and Cottam 1987). We already remarked in §4.2 that, for a single ferromagnetic slab in this perpendicular magnetisation case, no surface magnetostatic modes occur. However, the above authors showed that, when a superlattice of these magnetic layers is formed, the coupling between layers is such as to allow a sequence of surface magnetic modes in appropriate cases. Calculations were presented for both ferromagnets and antiferromagnets. A few theoretical papers have appeared generalising the above results to include retardation. Barnas (1987) deals with ferromagnetic/nonmagnetic superlattices, while a later paper (Barnas 1988) uses the transfer-matrix method to derive general formal results for both bulk and surface modes in magnetostatic, polariton and exchange-dominated regimes. The polariton regime was also discussed by Raj and Tilley (1987). All the results derived are for the Voigt configuration. As might be expected, the full results are quite complicated, but fortunately they can be greatly simplified in the long-wavelength limit. As shown by Raj and Tilley, the method due to Agranovich and Kravtsov (1985) can be applied to a magnetic superlattice as follows. We consider essentially the geometry of Fig. 7.25, except that both constituents are now taken to be magnetic. For propagation in the xy plane (Voigt geometry) each constituent is characterised by a gyrotropic permeability tensor of the form of (6.102). We consider an rf magnetic field in the xy plane; the wavelength is taken to be much longer than the superlattice period so that the B and H fields are the same in successive layers of component 1 and in successive layers of component 2. The boundary conditions are that Hy and Bx are continuous across interfaces, so these components are everywhere equal to their average values Hy and Bx. In medium 1 the field components are related by (7.86)

(7.SI) so that B ^

= fii'% + (fi^/n^BJfio x

Hx = bixyii™)Ry + (l//JLx i)EJii0

(7.88) (7.89)

where /xv = fixx + \i\yl\ixx is the Voigt permeability. Similar equations hold in component 2, and combined with (7.88) and (7.89) give expressions 290

Magnetic properties + d2B(y2))/(d1 + d2) and Hx:

for the average fields By = (d^

(7.90) (dx + d2)Hx = (dii/^/Hx1* + d2fiixy/^ix2i)Ry + (di/n^x + dJfi^BJfiQ Reorganisation gives relations in terms of the effective permeability tensor: /***

(7.91) medium

fixy\f

- fixy fiyyj \Hyy

with " -

^yy

/i

(7.93)

.

i

w

i

r?\

.

i

nu

('•"^•;

(7 95)

-

This derivation was given independently by Almeida and Mills (1988). The form of (7.92) corresponds to a kind of magnetic anisotropy, induced by the superlattice structure itself. It is straightforward to find the dispersion equation for wave propagation in a medium with this kind of magnetic anisotropy. For a TE mode, the relevant component of the dielectric tensor in the effective-medium description is syy: syy = (e1d1+e2d2)/(d1+d2)

(7.96)

This dispersion equation is Vyy + VxylVxx

Vxx + PxylPyy

Raj and Tilley (1987) show that this is the same form as can be obtained by a long-wavelength expansion of the general dispersion equation, derived by the transfer-matrix formalism. 7.6.2 Exchange-dominated properties: reconstruction Some exploratory theoretical work has been carried out for situations in which the exchange interaction is dominant and dipole-dipole effects can be neglected. It is important to recognise at the outset that even the nature of the ground state depends on the specific details of the system under consideration, and that in the presence of external fields and with variation of temperature many forms of spin reconstruction are possible. This is nicely illustrated by the hypothetical ferromagnetic/ antiferromagnetic superlattice discussed by Hinchey and Mills (1986a,b). 291

Layered structures and superlattices The system consists of alternating ferromagnetic and antiferromagnetic components, with a ferromagnetic coupling at the interface. The antiferromagnetic structure is one of alternating layers, successively spin-up and spin-down. It can be seen that the nature of the ground state depends crucially on whether the number NAF of layers in the antiferromagnet is even or odd. This is illustrated in Fig. 7.28. For NAF even, Fig. 7.28(#), successive ferromagnetic layers are aligned oppositely, and the ground state has no net magnetic moment, while for NAF odd, Fig. 7.28(c), the ferromagnetic layers are all aligned, and the ground state does have a magnetic moment. Even at zero temperature, the phase diagram in an applied external field of the system in Fig. 7.28 is quite complicated. Figure 1.2$(b) shows a possible state of the NAF-even superlattice in which all the ferromagnetic spins are aligned with the applied field. The Zeeman energy is lower than in the configuration of 7.28(a), but the interface-exchange energy is higher. As the number of spin layers Ns = NF + NAF in one period increases the relative importance of the interface-exchange energy decreases, so it is to be expected that the critical field Bc for a phase transition out of the state of Fig. 1.2$(a) decreases as Ns increases. Hinchey and Mills find Bc by a 'soft-mode' method, that is, they find Bc as the field for which a spin-wave Fig. 7.28 Ferromagnetic/antiferromagnetic superlattice. (a) Zero-field ground state when number of antiferromagnetic layers NAF is even, (b) Possible high-field state, (c) zero-field ground state when NAF is odd [after Hinchey and Mills 1986].

it 11 tit i i lit m i l t i ttj -j - «!

J

!

^

!

dx

*t-

d2

It mint i 292

!

AF

i

+~—

:

F

dY

i

i

AF

i

I

Magnetic properties frequency becomes zero. Typical results are shown in Fig. 7.29; the field units are such that the spin-flop transition of the bulk antiferromagnet corresponds to Bo = 1. As expected, for even N AF the zero-field ground state is rather unstable, with Bc -+0 as N -> oo, while for odd NAF the zero-field ground state is rather stable. The transition out of the ground state at the critical field Bc in Fig. 7.29(a) is not straight into the 'Zeeman state' of Fig. 7.28(ft); in fact what occurs is a 'twisted state' in which the spins gain a y component, so that they remain in the plane of the layers but are no longer aligned with the magnetic field. A detailed discussion of the twisted states and phase diagrams is given by Hinchey and Mills. Spin reconstruction is also discussed by Camley (1987) and Camley and Tilley (1988). They are concerned with the Fe/Gd system, which has Fig. 7.29 Critical fields Bc for a transition out of the ground state of the ferromagnetic/ antiferromagnetic superlattice as functions of N = iVp. (a) NAF even, corresponding to Fig. 7.28(a), (b) JVAF odd,corresponding to Fig. 7.28(6). The units of field are chosen so that the bulk AF spin-flop field is equal to unity [after Hinchey and Mills 1986]. 0.20

0.15

0.10

0.05

0.00

H

1

h

1.50

1.00

Bulk AF spin-flop field

0.50

0.00

10

12

14

293

Layered structures and superlattices

the interesting property, as demonstrated by Taborelli et al. (1986), that although Fe and Gd are ferromagnets, the exchange interaction at an interface is antiferromagnetic. The ground state of a Fe/Gd superlattice is therefore as depicted in Fig. 7.30(a). This may be expected to be unstable in modest applied fields, and indeed Camley and Tilley find that a phase transition occurs to a twisted state as shown in Fig. 7.30(b). Numerical exploration of the mean-field expression for the free energy can be used to find the phase diagram in the B0T plane. The detailed results, presented by Camley and Tilley (1988), show that for small values of n1 and n2 the phase diagram is sensitive to the precise number of spins in each layer. A formal development that should be of value in discussing reconstruction is the use of Landau-type expansions, as proposed by Fishman et al. (1987) and by Camley and Tilley (1987). The method applies only for the case when nx and n2 are large, and when the average spin value varies only slowly from site to site. The mean spin <Sf> at site i may then be replaced by a continuous variable M(x). The basic idea of Landau theory, as expounded in Landau and Lifshitz (1969) for example, is to expand the free energy as a Taylor series in the invariants of the system.

Fig. 7.30 Magnetic superlattice of Fe/Gd. (a) Ground state in absence of magnetic field, {b) Twisted state developed in applied field Bo [after Camley and Tilley 1988].

layers Fe

layers Gd

(a)

294

Magnetic properties For a slab of Fe, extending from x= —d1 to x — 0, the expression is 1==

f00

f°

dydz

J - oo

J -di

,

]}

(7.98)

Here a l 5 b x and c1 are constants, which may be related to the microscopic parameters for Fe, or may be determined later from comparison with experiment. It is assumed that M is confined to the yz plane, corresponding to the twisted state of Fig. 7.30(b). A similar expression for a term F2 in terms of a variable N, say, holds for the other component of the unit cell, namely a slab of Gd extending from x = 0 to x = d2. Finally it is necessary to include a free energy term for the interface. The lowest order invariants are (M 2 + M 2 ) x = 0 , (AT2 + N2)x = 0 and (MyNy + MZNZ)X=O, so that the interface free energy is

-I"

•N2)x

= 0

(7.99)

J -c

This introduces three more parameters 8l9 b2 and a. Within this model, the equilibrium state is now found as that which minimises F1 + F2 + FY. The minimum is found, in principle, from the Euler-Lagrange equations for M and N. However, these are quite complicated, and we simplify the problem by use of the constant-amplitude approximation: (My, Mz) = M(sin 6, cos 0)

(7.100)

(Ny, Nz) = N(sin 0, cos )

(7.101)

This approximation should be good at low temperatures, where the reconstruction is described in terms of spins of constant magnitude inclined at angles 0 and (j) to the field direction. Within this approximation, the free energy reduces to — B0M cos 0 + \c^

f

*<)

Jo + <xMN cos(0 - 4>)\x=o + aMN cos(0 -
(7.102)

Here certain constant terms have been omitted, and it is assumed that the equilibrium state has period d1+d2, so that (7.102) contains contributions from the two interfaces in the unit cell. A is the specimen area in the yz plane. 295

Layered structures and superlattices The Euler-Lagrange equations for minimum F are c xM — - - B dx 2

o

sin 0 = 0

(7.103)

and c2N —^ - B o sin 0 = 0 dx with boundary conditions at interfaces d6 c,M — -OLN sin(0 -) = 0 dx

(7.104)

(7.105)

c2AT-^ - aM sin(0 - 0) = 0 (7.106) dx In general the solutions of (7.103) to (7.106) may be given in terms of elliptic functions, as shown by Camley and Tilley (1988). For the particular case of two semi-infinite media in contact, the elliptic functions reduce to hyperbolic functions, and the solutions are cos \Q = tanhCA^j^! - y)] cos \4> = tanh[2 2 (j/ - y2)]

(7.107) (7.108)

where X\ = BolClM

(7.109)

with a similar expression for X\. yx and y2 are constants of integration, found from the boundary conditions, or equivalently from minimisation of the free energy. The details are left for Problem 7.10, and the predicted reconstruction for the case of two semi-infinite media in contact is shown in Fig. 7.31. It can be seen that, as might be expected, the width of the reconstructed region decreases as the applied field Bo increases. The Landau theory described here should in principle be applicable to ferroelectric superlattices, which are conceivable, although to date no examples have been reported. 7.6.3 Exchange-dominated properties: magnons In principle the magnon spectrum must be derived for each of the equilibrium states found for a particular superlattice, and in fact Hinchey and Mills (1986a,b) do give an extensive discussion for the ferromagnet/antiferromagnet superlattices with which they are concerned. The dispersion relation can be found by application of the transfer-matrix method to spin equations of motion, as used in Chapter 3. We give here an illustration for what is probably the simplest case, the ferromagnet/ ferromagnet superlattice with ferromagnetic exchange at the interfaces, so 296

Magnetic properties

that reconstruction need not concern us. This has been discussed by Albuquerque et al. (1986b) and by Dobrzynski et al. (1986). To be specific, following Albuquerque et al. (1986b), we consider alternating simple-cubic ferromagnets in which the interfaces are (001) planes. It is assumed that each component is described by the nearestneighbour Heisenberg Hamiltonian including an external field, (3.4), and the exchange constants are taken as Jt in one component, J2 in the other, and / across the interface. The structure is shown in Fig. 7.32. The problem is essentially one-dimensional, and the transfer matrix may be constructed exactly as described in §§7.1.1 and 7.2.1. Spin equations of motion for the operators S* are written down, and simplified by using RPA (see §3.2). Within each component the solution for S* is written as a sum of forward- and backward-travelling waves, as in (7.3). The equations of motion of spins adjacent to the interfaces, labelled a, /?, y, S in Fig. 7.32, relate amplitudes in adjacent layers, and the transfer matrix is built up Fig. 7.31 Variation of 6 and (j> with position x for a single interface between Fe and Gd. Magnetic field varies from Bo = 0.2 T (broken line) to 1 T (chain line) in steps of 0.2 T. Horizontal axis scaled by AO = 5.85 x 10 7 m~ 1 [after Camley and Tilley 1988].

10.0

-120

297

Layered structures and superlattices as in §7.2.1. Finally, the dispersion equation takes the standard form of (7.23). An example of the dispersion curves obtained for this model is shown in Fig. 7.33. Here the bulk-medium dispersion curves of Fig. 7.33(a) are drawn from (3.12), and the Brillouin-zone edge frequencies are denoted co1 and co2- The resemblance to Fig. 7.7 for diatomic-lattice phonons is striking. For a> < CD1 the bulk wavevector q1 and q2 in the two components are real, and the superlattice dispersion curve consists of broad pass bands with narrow stop bands. For (O1
II

o o o o o o o o o o O

O

O

y $

IJ

Or O/

o o o o o o o o o o n, a > < a > << (l-l)na

o

o

o

o

o

o

o o o o

Ina

Fig. 7.33 Bulk and superlattice magnon dispersion graphs for (100) propagation with parameters g^iBB0/4SJl = 1, J2/J1 = 2,1/J^ = 1.5. Both components are taken to have the same values of gyro magnetic ratio g and spin S. Relevant frequencies are hcoJgnBBo = 1 + 4SJJgnBB0 = 2, hco2/gnBB0 = 1 + 4SJ2/gnBB0 = 3. (a) Bulkmagnon dispersion curves for components 1 (lower curve) and 2 (upper curve), (b) Superlattice magnon dispersion curves for n1=n2= 10 [after Albuquerque et al. 1986b]. 0)9 = 3 r

&3

CO

I"

qta

298

Problems the superlattice dispersion curve then consists of narrow pass bands and broad stop bands. Barnas (1987b) generalises the formal results for a ferromagnet/ ferromagnet superlattice in a number of ways. First, he considers the case when the repeat unit consists of N different ferromagnetic layers; second, he includes uniaxial anisotropy; third, he discusses the surface magnons of a semi-infinite specimen as well as the bulk magnons. Particularly in view of the wide variety of possible combinations of ferromagnetic and antiferromagnetic exchange it is clear that the theoretical study of reconstruction and exchange-dominated magnons is still at an early stage. Likewise, at the time of writing most of the experimental work remains to be done. 7.7 Problems 7.1 In §7.4, the lattice dynamic calculations are given for a diatomic superlattice. Perform the equivalent calculations for a monatomic superlattice. Check that appropriate limits are found, in particular that the continuumacoustics results of §7.3 are found as the long-wavelength limit of your results. 7.2 Equations (7.39) and (7.40) give the diagonal elements 7\ , and T22 of the transfer matrix T for s-polarisation. Evaluate T completely, and confirm that T is unimodular, det T = 1. 7.3 Evaluate the transfer matrix T for p-polarised optical propagation. Confirm that T is unimodular, and that the dispersion relation is given by (7.41) and (7.43). 7.4 Show that the transfer matrix for the electron amplitudes in (7.55), defined by

\bl + l)

\bt

has elements

T2i=kfxy(s-l-s) where A

= =

t2-t1

299

Layered structures and superlattices tj =ik1/m1

t2 = ik2/m2

f=exp(ikldl) s = Qxp(ik2d2) Hence prove that T is unimodular, det T = 1, and derive the dispersion relation (7.62). 7.5 Starting from the expressions (7.65) and (7.67) for the electric and magnetic fields within the dielectric media of a 2D charge-sheets superlattice, prove the dispersion relation quoted in (7.69). You should apply the boundary conditions that

at the interface z = nD, as well as Bloch's theorem in the form (7.68). Here a is the 2D conductivity given by a = in0e2/m*co

in the notation of §7.5.1. From the solvability condition for the simultaneous equations relating coefficients ain) and b(n) you should obtain (7.69). 7.6 Prove that for small enough qx (such that qxD « 1) the dispersion relation (7.71) implies that w is proportional to qy (except when cos(QD)= 1). Find the constant of proportionality, and investigate the case where cos(QD)=\. What is the approximate dependence of co on qx for the opposite limit of qxD»V. 1.1 Rearrange (7.78) to verify the alternative form of the dispersion relation for nonretarded p-modes given by (7.79) and (7.80). Hence obtain the dispersion relation for acoustic and optic plasmons given by (7.81). 7.8 Prove the last step in (7.83) for the light-scattering wavevector by approximating the square roots using rj» 1. 7.9 Confirm that within the constant-amplitude approximation the Landau free energy Fx + F2 + F, reduces to (7.102), and that the conditions for a minimum in (7.102) are those given by (7.103) to (7.106). 7.10 This is concerned with the solution of the constant-amplitude equations (7.103) to (7.106) for two semi-infinite media in contact, (a) Prove that (7.103) has the first integral

and note that the boundary conditions 0-»O as x —> — oo 300

and

Problems d0/dx->O as x-> -oo imply that l\ = X\. (This simplification does not occur for the superlattice.) (b) Hence confirm (7.107) and (7.108) for 9 and . (c) Prove that upon substitution of (7.107) and (7.108) the free energy (7.102) may be written F-FA = (4BOM/A1)(1 -tt) + (4BONM2)(1 - t2) + *MN{(2t\ - l)(2t| - 1) - 4 ^ ( 1 - t\)l'2(\ - ti)1'2} where r2 = tanh(-/l2>;2) and FA is the free energy of a hypothetical state with 9 = § = 0 and a = 0. Minimisation of F with respect to tx and t2 determines yx and y2, and hence enables the curves of Fig. 7.31 to be drawn.

301

8 Concluding remarks

In the preceding chapters we have presented a survey of the dynamical properties of planar surfaces and interfaces. The selection of surface excitations, with which we illustrated the general principles, was wide ranging but inevitably incomplete. Our choice was guided by the existence of appropriate theoretical models, by the availability of data to compare theory and experiment, and in particular by recent advances in fabricating low-dimensional solid structures. Several topics relevant to the study of surface excitations have been omitted, or mentioned only in passing, in the previous chapters. It is useful now to provide some brief comments on these topics, together with references to one or two examples in each case. 8.1 Mixed excitations Apart from polaritons, which are excitations formed when a crystal excitation couples to a photon, and which were extensively discussed in Chapter 6, we have not considered 'mixed' modes of two excitations. These may be important for appropriate frequency and wavevector values, and as a relatively straightforward example we take the case of coupled magnetic and vibrational modes. In the long-wavelength limit, where the solid can be treated as an elastic continuum, these modes are known as magnetoelastic waves. For a ferromagnetic solid with simplecubic lattice structure and static magnetisation in the z direction, the energy of the magnetoelastic coupling for the magnon regime where only terms linear in the magnetisation M are retained takes the form (Kittel 1958) b

[Mx(uxz + uzx) + My(uyz + uzy)2 Mo' 302

(8.1)

Nonlinear effects and interactions between excitations Here b is a phenomenological coupling constant, Mo is the static magnetisation, and the u^ are elements of the strain tensor (see §2.2.1). In the absence of magnetoelastic coupling (b = 0) the excitations of a semi-infinite medium consist of (i) the bulk and surface elastic waves obtained from the equation of motion (2.36) with appropriate boundary conditions as discussed in §§2.2 and 2.3, and (ii) the bulk and surface magnons obtained from the torque equation (3.35) or (4.11) as discussed in §§3.3 and 4.2, respectively, for different magnetic models. When b ^ 0 the equations of motion must be supplemented by additional terms due to the coupling (8.1). This has been discussed, for example, by Scott and Mills (1977). These authors concentrated on a theoretical study of surface acoustic-type waves propagating in a semi-infinite ferromagnetic medium, taking the long-wavelength regime where exchange effects can be ignored (i.e. the magnetostatic limit). In particular, they presented calculations for two types of surface acoustic wave, one being in essence a Rayleigh wave (see §2.3.1) modified by the magnetoelastic coupling and the other a shear-like magnetoelastic surface mode that exists only because of the presence of the magnetoelastic coupling. These waves have nonreciprocal propagation characteristics, a consequence of their being formed by coupling to the Damon-Eshbach surface magnetostatic mode which is nonreciprocal as was discussed in §4.2. Scott and Mills conclude that these magnetoelastic surface waves can provide a useful and sensitive probe of the bulk and surface magnetostatic modes, especially at larger values of the in-plane wavevector qy. Other references to magnetoelastic surface waves in ferromagnets are to be found in the review articles by Maradudin (1981) and Cottam and Maradudin (1984). Calculations have also been performed for Rayleigh waves in paramagnetic rare-earth systems (Lingner and Luthi 1981; Camley and Fulde 1981). In this case the magnetoelastic interactions describe the coupling of the strain and the rotational part of the deformation tensor to the unfilled 4f electron shell. Paramagnetic crystals are of interest because their elastic moduli typically exhibit a strong dependence on temperature and applied magnetic field, leading in turn to strong temperature and magnetic field dependences of the Rayleigh wave propagation. Experiments to measure the Rayleigh wave velocity have been reported for CeAl2 and SmSb (Lingner and Liithi 1981), and are in good agreement with the theoretical predictions. 8.2 Nonlinear effects and interactions between excitations In practice all the excitations discussed in this book may exhibit nonlinearities to some degree. The nonlinearities lead to interactions between the otherwise independent solutions (normal modes) of the 303

Concluding remarks equations of motion for the system. Important consequences are that the mode frequencies become modified (or 'renormalised') due to the interactions and also the modes are 'damped', i.e. they acquire a finite (rather than infinite) lifetime. Other nonlinear effects include second (and higher) harmonic generation of surface and bulk excitations and nonlinear mixing of two (or more) excitations. An example of nonlinearity involving electromagnetic waves has already been discussed in §6.4, and we now give some additional examples involving vibrational and magnetic excitations. In the case of vibrational modes in semi-infinite elastic media, there have been extensive studies of second-harmonic generation and parametric mixing of Rayleigh surface waves. According to nonlinear elasticity theory (see Wallace 1970), there are forces created within the medium of the form

NL

c

/* = I ^^tlt

(8 2)

-

and surface stresses of the form 7^NL_

y [ivpa

(S)

<^<^p CXV

(8 -x\

OXa

In (8.2) and (8.3) the quantities c(b) and c(s) are combinations of third order elastic moduli; {u^} and {x^} denote Cartesian components of the displacement vector and position vector respectively. It is the occurrence of nonlinearities in the boundary conditions (resulting from (8.3)) as well as in the equations of motion that makes the analysis particularly complicated. The first theoretical work on second-harmonic generation of Rayleigh waves was reported by Viktorev (1963), and the first experiments were due to Rischbieter (1965, 1967) for Al and steel surfaces. Later work has been reviewed by Maradudin (1981) and Stegeman and Nizzoli (1984). As a further example we mention some calculations for semi-infinite Fig. 8.1 The four-magnon interaction dominant at low temperatures T
304

Excitations in wedges and at edges

Heisenberg ferromagnets where higher-order magnon-magnon interactions have been taken into account by using better approximations than the simple RPA discussed in Chapter 3. At low temperatures T«T C the dominant interaction is a four-magnon process in which two incoming magnons scatter to produce two outgoing magnons (see Fig. 8.1). The magnons may be surface or bulk modes in any combination. The damping (i.e. /?/T, where T is the lifetime) of surface magnons was calculated by Tarasenko and Kharitonov (1974). They found that in the absence of an applied magneticfieldand anisotropyfield,the damping of long-wavelength surface magnons is approximately proportional to q\T2 at T« Tc. This contribution comes from interactions of the surface magnon with three other surface magnons, and the result takes this form provided T/Tc« a2q\ « 1, where a denotes the lattice constant. Calculations for the renormalisation of the surface-magnon frequencies due to magnon-magnon interactions have been given by Kontos and Cottam (1984, 1986) and Mazur and Mills (1984); these results depend sensitively on the lattice structure and on the surface values of the exchange and anisotropy parameters. 8.3 Excitations in wedges and at edges

An interesting generalisation of the parallel-sided slab geometry, which we have discussed in many applications, is to the case where two planar surfaces meet at an angle 0 to form a wedge-shaped sample. The surfaces are now such as to destroy translational symmetry in two different directions, with obvious consequences for the symmetry of the excitations. In the geometry of Fig. 8.2 the apex of the wedge corresponds to the line x = y = 0. There is a translational invariance along the z direction Fig. 8.2 A wedge of angle 6 showing the assumed orientation of coordinate axes.

305

Concluding remarks

(assuming the wedge dimensions to be effectively infinite in this direction) but not along the x or y directions. This means that, under appropriate conditions, there may be localised excitations known as wedge modes that propagate in a wave-like fashion along the apex of the wedge but decay with increasing distance into the wedge from its apex and faces. The case of a right-angled wedge (0 = 90°) often leads to simplifications, particularly in microscopic calculations involving cubic materials, and we refer to the localised excitations in this situation as edge modes. However, the terms 'wedge modes' and 'edge modes' are often used interchangeably in the literature. The first calculations were for acoustic waves in wedges and edges formed by isotropic, elastic media (Lagasse 1972; Maradudin et al. 1972). The wedge modes have certain convenient properties, such as being essentially nondispersive, which make them of interest for waveguide applications. The number of wedge modes, and their speed of propagation, can be controlled by varying the wedge angle 0. In general, as 0 is decreased the number of modes increases and their propagation speeds become lower than that of Rayleigh waves. This is illustrated by a numerical example in Fig. 8.3. Apart from the ideal infinite wedge, calculations have been Fig. 8.3 The square of the speed of propagation of wedge modes (in units of the speed of bulk transverse acoustic waves) of T 2 symmetry as a function of wedge angle 6. The speed corresponding to Rayleigh waves is denotes by QR [after Moss et al. 1973].

306

Excitations in spheroidal and cylindrical samples

applied to more realistic waveguide structures, e.g. where the wedge-tip is rounded rather than sharp, or where the waveguide consists of a wedgeshape ridge on a substrate material. A thorough review has been given by Maradudin (1981). Electromagnetic wedge modes have also attracted much theoretical attention (see Maradudin 1981). Most calculations have employed the electrostatic approximation (see §6.2.2) since this considerably simplifies the analysis. In an infinite dielectric wedge surrounded by vacuum, one may then solve Laplace's equation by the method of separation of variables to obtain a scalar potential proportional to exp(igz) and localised about the apex of the wedge. Details are given by Dobrzynski and Maradudin (1972). When retardation effects are included, one is faced with the much harder task of solving the full set of Maxwell's equations in the wedge geometry; an elegant calculation of retarded wedge modes in a parabolic wedge has been given by Boardman et al. (1981). The magnons localised at a 90° edge of a Heisenberg ferromagnet with simple-cubic structure have been studied by Sharon and Maradudin (1973) using a long-wavelength continuum method and extended to shorter wavelengths by Maradudin et al. (1977) using a microscopic theory. They used a model in which all exchange interactions (including those involving spins at the surface) have their bulk values, denoted by J for nearest neighbour and j for next-nearest neighbours. At long wavelengths and in the absence of surface anisotropy fields, the magnon amplitudes of the edge modes have the approximate dependence exp[ - K(X + yj] exp[i(qz - cor)] (8.4) where the attenuation constant K is related to the propagation wavevector q by (8.5) provided K > 0. This is an acoustic-type branch occurring split off below the surface magnon, which is itself split off below the bulk continuum. This behaviour is illustrated in Fig. 8.4. 8.4 Excitations in spheroidal and cylindrical samples

Throughout this book we have idealised the surfaces and interfaces as being planar and smooth. Some generalisations to curved samples (in particular, spheroids and cylinders) are now presented, and the topic of surface roughness is treated in §8.5. We start by considering magnetostatic modes in ferromagnetic samples with curved surfaces. The case of spherical and spheroidal samples was 307

Concluding remarks worked out by Mercereau and Feynman (1956) and Walker (1957), followed a few years later by the case of a cylinder (Fletcher and Kittel 1960; Joseph and Schlomann 1961). The results are reviewed by Wolfram and Dewames (1972). We briefly consider the calculation for a long circular cylinder magnetised along its axis of symmetry (the z direction). A scalar potential i// may be introduced as in §4.2, and this must satisfy (4.21) and (4.22) inside and outside the sample respectively. After transforming to cylindrical coordinates (r, 9, z) in the usual way, one seeks separable solutions of the form iA(r, 0, z) = p(r) exppfoz + m0)]

(8.6)

Here m is a positive or negative integer, qz is the propagation wavevector in the z direction, and the radial part p(r) of the potential has to satisfy q2zp = 0

(8.7)

in the interior and a similar equation with xa = 0 in the exterior. The acceptable solutions of (8.7), which are regular at r = 0 and vanish as r -• oo, can be written down in terms of Bessel functions. It then remains Fig. 8.4 Calculation of dispersion relations for bulk magnons (full line), surface magnons (short dash) and edge magnons (long dash) in a simple-cubic Heisenberg ferromagnet with a 90° edge. The bulk dispersion curve refers to the lower edge of the continuum [after Maradudin et al. 1977]. 12.0 -

J'/J = 0 . 5

9.0

(D/JS

6.0

3.0

0.0

308

0.0

0.5

1.0

Surface roughness

to apply the boundary conditions at r = R, where R is the radius of the cylinder; these take the form of continuity of \j/ and continuity of the radial component of the magnetic flux density b at r = R. The resulting dispersion relation has a sequence of solutions, characterised by the positive index p (p = 1, 2, 3 , . . .), for the surface modes. In a long-wavelength limit (qzR« 1) the frequencies simplify to p=\ (8 8) p>i ' These surface magnons all have frequencies above those of the bulk manifold, i.e. cos > [coo(a)o + com)]1/2. I*1 general, the localisation at the surface of the cylinder increases with increasing qz; details are to be found in the original references. The magnetostatic calculation for a spherical or spheroidal ferromagnet can be carried out by an analogous generalisation of §4.2; the solutions for the potential ij/ involve Legendre functions instead of the Bessel functions of the cylindrical case. The theory of elastic surface waves in homogeneous isotropic materials, as discussed in §2.3 for a parallel-sided slab geometry, can be extended relatively straightforwardly to cylindrical and spherical geometries. The propagation of Rayleigh-type surface modes can then be studied in appropriate cases. The first calculation was due to Hudson (1943) for elastic waves travelling along the axis of a long circular cylinder; the considerable amount of subsequent work has been reviewed by Maradudin (1981). Another example is provided by surface and bulk polaritons at curved surfaces. The polariton dispersion relations and the associated electromagnetfielddistributions have been calculated for spheres and for circular cylinders using similar methods to those in Chapter 6. For example, phonon-polaritons in the absence of retardation have been considered by Englman and Ruppin (1968a); the theory was subsequently generalised by the same authors to include retardation (Ruppin and Englman 1968; Englman and Ruppin 1968b). Further references are contained in the review articles by Economou and Ngai (1974) and Kliewer and Fuchs (1974).

8.5 Surface roughness

All real solid surfaces are rough to some degree. This may be because of the way in which the sample has been prepared, e.g. by vapour deposition on to a substrate material, or because of subsequent treatment of the surface, e.g. polishing of the surface by an abrasive substance of a particular particle size. It is therefore relevant to determine how this roughness affects the properties of the excitations and how these properties 309

Concluding remarks

differ from those in samples an an 'ideal' plane surface. The use of a rough surface to couple external radiation out of the surface plasmonpolariton generated in a tunnel junction was mentioned in §6.6.2. In many cases the presence of surface roughness acts simply as a small perturbation on effects that already occur with plane surfaces, e.g. the frequency and/or damping of an excitation may be perturbed due to the roughness. For example, the modifications of Rayleigh surface waves propagating along a randomly rough surface have been studied by Urazakov and Falkovsii (1972) and by Maradudin and coworkers (see Maradudin 1981). The latter calculations were carried out by a perturbation method correct to order (d/b)2, where d is the root-mean-square departure from flatness and b is the mean distance between consecutive peaks and valleys on the surface. In other cases the random roughness can produce qualitative changes; an example is the roughness-induced splitting of the surface-plasmon dispersion curve observed experimentally (e.g. Palmer and Schnatterly 1971) and studied theoretically (e.g. Kretschmann et al. 1979). It can also produce large quantitative effects, such as the giant enhancement (by five orders of magnitude or more) of the Raman intensity for scattering from molecules adsorbed on certain metal surfaces. Although there are various mechanisms proposed for this enhancement, it is claimed that surface roughness can give rise to a significant part of it (Burstein et al. 1982). This is supported by the observation by Ushioda et al. (1979) that the Raman intensity for scattering from surface polaritons in GaP increased by a factor of four tofivewhen the surface of the sample was roughened. Apart from the random roughness referred to in the preceding paragraphs, there can be periodically corrugated surfaces, e.g. as in the case of a diffraction grating ruled on the surface. The use of a grating to couple external light to a surface polariton was explained briefly in §6.6.1. Indeed, grating surfaces are of considerable technological interest because of their role in optical surface acoustic wave devices, for example as surface mode to bulk wave transducers. Also, from the theoretical point of view, they allow a treatment of surface roughness in situations where a perturbative approach would fail. Examples of calculations for surface excitations propagating across the grooves of large-amplitude gratings have been given by Laks et al. (1981) for surface polaritons, Glass and Maradudin (1981) for surface plasmons, and Glass et al. (1981) for Rayleigh waves. For further details of the topic of surface roughness, the reader is referred to appropriate sections of the review articles by Maradudin (1981) and Cottam and Maradudin (1984). 310

Appendix Green functions and linearresponse theory

Here we summarise some basic results concerning Green functions and linear-response functions, and the connection between these quantities. Comprehensive treatments of Green functions are to be found in the numerous text-books on many-body theory; for example, Fetter and Walecka (1971) and Rickayzen (1980). General accounts of the linearresponse method have been given, for example, by Landau and Lifshitz (1969), Barker and Loudon (1972) and Forster (1975), while its specific application to surface problems has been reviewed by Cottam and Maradudin (1984). A.I Basic properties of Green functions For any two quantum-mechanical operators A and B we shall define the Green functions ((A(t); £(£')>>, with time labels t and t\ by «A(t); B(t')}) = i6(t - ?KlA(t), B(f')]e>

(A.I)

Here 6(t — t') is the unit step function corresponding to

*-«•>-{; t
and the operators are in the Heisenberg representation, i.e. A(t) = exp(i^fot)/4 exp(-Uf o r)

(A.3)

where Jf0 is the Hamiltonian describing the system. The angular brackets in (A.I) denote a thermal average, evaluated according to equilibrium statistical mechanics. Finally lA(t)9 B{t')\ = A(t)B(t') - eB(t')A(t)

(A.4)

where e is a parameter which can be chosen as either 1 or — 1. The conventional choice in boson and fermion problems is to take e = 1 and e = — 1, respectively, but this is not essential. 311

Appendix: Green functions and linear-response theory The definition in (A.I) is the same, apart from the overall sign, as that employed by Zubarev (1960) for retarded Green functions. Advanced and causal Green functions can also be introduced but we shall not require them here. One of the simplest properties of < l ( 0 ; * ( * ' ) » to be defined by

-j:

(A.5)

For most cases of practical interest the Green function {{A; !!}}„ can only be evaluated by making approximations, and several different approaches are available. One method is based on writing down the equation of motion of the Green function (Zubarev 1960): (A.6) This generates another Green function, whose equation of motion can itself be written down. After a certain number of stages of this process a decoupling approximation is used to simplify the Green function in the final equation and to yield a closed set of equations which can be solved for the required Green function. An example occurs in §5.3. Another method involves using perturbation theory in a diagrammatic expansion (Feynman diagrams). Details are to be found in the references given earlier. For most of the applications considered in this book we employ yet another approach, namely linear-response theory, which has a more direct physical appeal and enables appropriate Green functions to be deduced using macroscopic arguments. A.2 Linear-response theory A simple illustrative example of the method is given in §1.3.2, and we now present the formal theory. We make use of the concept of the density matrix p (for a review see ter Haar 1961), which may be defined by X™1 I

\

/

p = 2^ \m)pm(jn\

I

/ A *7\

(A.7)

m

where pm is the probability of the system being in the state denoted by |m> (assumed to be orthogonalised and normalised), and the summation is over all states. The density matrix is useful for applications to nonequilibrium (time-dependent) situations. 312

Linear-response theory We consider a system with total Hamiltonian J f given by

j r = JTo + Jfi(0

(A.8)

where Jfo is independent of time and Mf^t) is an external perturbation taking the form jTl(t)=-Bf(t)

(A.9)

Here f(t) denotes an external field which couples linearly with a system variable represented by the operator B, giving the interaction term (A.9). For example, if B represents an atomic displacement then f(t) would be a mechanical force (as in §2.4). We suppose that at time t = — oo the system is in equilibrium, described by the Hamiltonian 3tif0 and the corresponding density matrix Po

= exp(-Jf o /fe B r)/tr[exp(-jr o /* B r)]

(A.10)

where tr denotes the trace of the operators. The perturbation J^x is then increased adiabatically from zero, so that at a later time t the density matrix is equal to P = Po + Pi

(A.11)

The density matrix has the equation of motion (see ter Haar 1961) i dp/dt = tJT, p]

0=1)

(A. 12)

which can be proved from the definition (A.7). If (A.8) and (A.ll) are substituted into (A. 12) and only the terms which are linear in the perturbing field f(t) are retained we find i d P l /dr = [jf0, P l ] - [5, P o ]/(r)

(A.13)

where we have used the property that 3t?0 and p 0 commute. Equation (A.13) can be employed to show that - {exp(iJT0OPi exp(-iJf o r)} = i exp(iJTo0[*, Po] exp(-iJf o ')/(0

(A.14)

On integrating both sides of (A.14) with respect to t and rearranging, the formal solution for px is obtained as

< From the above expression, which is linear in / , we may deduce the response of the system to the perturbation (A.9). This response can be expressed in terms of the change A(t) that it produces in the mean value of any system variable denoted by the operator A. From the definition (A.7) it can be shown that the mean value of A is simply given by tr(p^4). The reader not familiar with this property is invited to work through a 313

Appendix: Green functions and linear-response theory proof in Problem A.4. Hence we have in the present case A(t) = tr(PlA)

(A.16)

On substituting (A.15) into (A.16) and using the cyclic invariance property for products of operators within the trace, we may express A(t) as

where <...> = t r ( P o . . . )

(A.18)

denotes a thermal average with respect to the unperturbed system. When use is made of (A.I), taking the case of a commutator Green function (s= 1), (A.17) becomes

-J:

A{t)=

«A(t);B{t'))>f(t')dt'

(A.19)

J -oo

In terms of frequency Fourier components the above result is simply Io = « A ; B» w F(co)

(A.20)

where {{A; -B>>w is given by (A.5), and f(t) exp(iart) At

(A.21)

0

with a similar definition for I w as the Fourier transform of A(t). Hence by explicitly calculating the response Am for a given externally applied field F(a>) we have a convenient method of deducing from the relationship (A.20) the required Green function <<X; £>>«,. Apart from the frequency the Green function will typically depend on other quantities such as position or wavevector labels, depending on the nature of the problem. For example, in surface problems it is often convenient to assume that the external field takes the form corresponding to a delta-function stimulus f(t)d(r-rf) at position r' within the system. The interaction energy with a position-dependent operator B(T) then becomes •#i = - f B(r)S(r - r') d 3 r/(0 = - B(r')f(t)

(A.22)

From the linear response in another operator A(r) at position r we are able to deduce, by analogy with (A.20), the Green function <> w . Examples for specific systems are given in the text (e.g. in §2.4). A.3 Thefluctuation-dissipationtheorem An important property of Green functions (response functions) is that they give the power spectrum of the excitations of the system. This is 314

The fluctuation-dissipation theorem achieved by application of the fluctuation-dissipation theorem, which relates the mean-square fluctuations in the excitation amplitude to the imaginary part of an appropriately defined Green function. We now outline the quantum-mechanical derivation of this result: for an alternative classical formulation (valid at high temperatures) we refer to Landau and Lifshitz (1969). The Green-function definition in (A. 1) involves the correlation functions CAB(t-t') = (A(t)B(t')) CBA(t-t') = (B(t')A(t)) Their dependence on (t — t') can be proved in the same way as for the Green function. It is easily shown, using the expression (A. 18) for the thermal average and the cyclic invariance property of operators under the trace, that the two correlation functions are related to one another by CAB{t-t') = CBA{t-t' + \p)

(A.24)

where /? denotes l/feBT. If frequency Fourier transforms {AB}^ and {BA}^ are defined for CAB(t — t') and CBA(t — t') respectively, as in (A.21) for F(a>), it is easy to show that (A.24) implies (BA}(O = cxp(M
(A.25)

When both sides of (A.I) are Fourier-transformed to the frequency representation, making use of (A.5) and (A.25), the result is [.-exp(-^)]d,-/ (o-o) +irj

(A.26)

where rj is a positive infinitesimal quantity (rj -* 0). In obtaining the above result it is helpful to use the identity (see Zubarev 1960) 2TTJ_00

(x + irj)

which can be proved by contour integration as in Problem A.5. Equation (A.26) can be separated into its real and imaginary parts by applying the symbolic mathematical relationship 1

)-ind(co-co') (A.28) co — co + irj \co — co' where 0* denotes that the principal value is taken in any integration over co'. Provided (BAy^ is a real quantity (e.g. as in the case when B = A""), we may take the imaginary part of each side of (A.26) to obtain f

Im<04; £ » w = n{\ - exp(-/to>)}<*>0«,

(A.29)

On rearrangement of the terms, this becomes = (l/n){n(co) + 1} Im«X; £ » „

(A.30) 315

Appendix: Green functions and linear-response theory where n(a>) is the Bose-Einstein factor Using (A.25) we also have the result {AB}^ = (l/n)n(a>) lm«A;

B»

w

(A.32)

Equations (A.30) and (A.32) are the standard forms of the flucutationdissipation theorem. At high temperatures (such that /Ja>« 1) they both reduce to the classical limit «

m

= <£M>W = (kBT/nco) I m « > l ; £ » „

(A.33)

By taking the real part of each side of (A.26) we obtain the additional relationship

Re«,;

J-

(©' - c o )

If {1 —exp( —/?«')} can be approximated by /?co' in the classical limit (|j?a/| « 1) we have for the special case of a> = 0 Re
ra=o

=0 J — oo

a.dat'

(A.35)

The integral on the right-hand side of (A.35) represents the total intensity (BA}tot associated with this correlation function, and so we arrive at the simple result that <J5A>tot = kBT R e « 4 ; B)} 0)

=0

(A.36)

Equation (A.36) holds provided the contribution from the region of large \co'\ (such that |/ta/| >> 1) can be neglected for the integral in (A.34). A.4 Problems A.I Derive the Green-function equation of motion in the form quoted in equation (A.6). You should start by differentiating both sides of (A.I) with respect to t, then use the operator equation of motion

and the definition (A.5). A.2 By starting from the definition given in equation (A.7), then differentiating both sides with respect to t and using the time-dependent Schrodinger equation, verify the equation of motion (A. 12) for the density matrix p. A.3 Check that equation (A.14) follows from (A.13). A.4 Prove the stated property (in §A.2) that the mean value of any operator A is equal to tr(pA) where p is the density matrix. You may find it helpful 316

Problems to rewrite tr(pv4) as

I <<>4> i

where the summation is over the complete set of states |i>, and then to employ the definition (A.7) for p. A.5 Derive equation (A.27) by evaluating the integral on the right-hand side and showing that it is equal to 0(t — t') as defined in (A.2). It is convenient to use contour integration, noting that there is a simple pole at x = —iy and the contour of integration may be taken as an infinite semicircle, completed either in the upper or lower half-plane depending on the sign of t-t' (see Fig. A.I). A.6 Verify the algebraic steps leading to equation (A.26).

Fig. A.I The complex x plane and the choice of contours of integration for Problem A.5.

^^\

Contour >v

y / ^^^^^

for / < t'

Contour for / > /'

317

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328

Index

ABCs, see additional boundary conditions acoustic phonons; see Phonons acousto-optic tensor 58ff, 66ff additional boundary conditions electron-gas 172, 175f excitonic 190ff magnetic 139 AES, see Auger electron spectroscopy Ag 69, 70, 171,223,228, 234 Al 66 antiferromagnets bulk magnetostatic modes 145 bulk magnons llOff, 115 bulk polaritons 236 Hamiltonian 109 spin-flop transition 116 surface magnetostatic modes 142ff surface magnons 113ff, 116 surface polaritons 240 antiferromagnetic resonance 111, 145 anti-Stokes scattering definition 24 magnons 152, 148, 150, 289 via exciton-polaritons 193 see also Brillouin scattering; Raman scattering attenuated total reflection (ATR): angle scan 221 basic principles 218ff frequency scan 221 Otto configuration 219ff, 223 Raether-Kretschmann configuration 219ff superlattices 270 surface exciton-polaritons 225f theory 222ff attenuation length 9, 12 Au 225 Auger effect 23, 163 Auger electron spectroscopy 12, 22f

band offset 272 basis 2 Bloch's theorem 4,9,34,36,93,159,219, 251, 264, 273, 278 boundary conditions acoustic 36, 43, 250 cyclic 6 electromagnetic 196, 203, 210, 264, 278 electron 159, 272 electrostatic 201 liquid surface 77 magnons 98 see also additional boundary conditions Bravais lattice 7 Brillouin scattering acoustic surface modes 56ff double layer ferromagnets 152 ferromagnetic films 150ff magnetic superlattices 289 principles 23ff resonant 188ff, 192ff Brillouin zone 4, 8 bulk continuum 47, 94, 146f causality 18 central peak 64 confined modes 260, 262, 274, 298 constant-amplitude approximation 295 correlation functions 312f Curie temperature 86 cyclic boundary conditions, see boundary conditions, cyclic cyclotron frequency 176, 274 cylindrical surfaces 3O7ff Damon-Eshbach modes, see magnetostatic modes, ferromagnetic dangling bonds 14 dead-layer model 191 decay length; see attenuation length

329

Index decoupling approximation 168, 312 see also random phase approximation demagnetising factor 129 density matrix 312 DFTS, see dispersive Fourier transform spectroscopy diatomic lattice, see lattice dynamics dielectric function definition 163f electron gas 169ff excitonic 189 intensity-dependent 215 ionic crystals 177ff longitudinal 165 RPA 170 semiconductors 179 superlattice 267 transverse 165 differential cross section definition 27 formula 32, 61 light scattering by magnons 102 dipole-dipole interaction 87, 128ff dipole-exchange modes, ferromagnets 137ff, 142 region 127 dispersive Fourier transform spectroscopy 267f, 276 displacement derivatives 59 edge modes 3O5ff EELS, see electron energy loss spectroscopy elasticity tensor 41 elasticity theory linear 40ff nonlinear 304 elastic waves bulk 10,42ff single interface 10, 45ff, 304 superlattices 249ff two-interface 47ff see also phonons, acoustic electromagnetic region 127 electron diffraction low-energy 12, 22, 118 reflection high-energy 22 spin-polarised 118 electron energy loss spectroscopy, principles 28f electron gas collective properties 165ff compressibility 174 dielectric function 169ff hydrodynamic description 174ff, 177 superlattices two-dimensional 176f, 203f, 276f electron states collective, see plasmons superlattice single-particle 27 Iff

330

surface single-particle 158ff electrostatic region 183, 200ff, 204 envelope function 272 EuO 148 evanescent mode 220 exchange interaction 86f, 96 excitons 188f extraordinary wave 265 Fabry-Perot interferometer 23 Faraday direction 239 Fe 140f, 149, 151ff FeF 2 111 ferroelectrics bulk modes 121 displacive 202 hydrogen-bonded 120 pseudo-spin model 120 surface modes 123f, 202 surface reconstruction 122ff ferromagnets bulk magnetostatic modes 130, 133 bulk magnons 88ff, 102, 104, 121, 123, 129, 147f bulk polaritons 236f continuum theory 96ff dipole-exchange modes 137ff double-layer system 136f, 152 films 104, 138, 140ff, 150 Hamiltonians 86f, 120 itinerant-electron 169 superlattice magnetostatic modes 287ff superlattice polaritons 290f superlattice reconstruction 29 Iff surface magnetostatic modes 130ff, 142, 153 surface magnons 93ff, 102, 106ff, 123f, 148 surface polaritons 236ff surface reconstruction 92, 122ff two-interface polaritons 24 Iff see also magnons field-effect transistor 177 Floquet's theorem 4 see also Bloch's theorem fluctuation-dissipation theorem 19, 61, 314ff folded modes acoustic phonon 249ff magnon 298f optic phonon 256ff frequency dispersion, see optical dispersion frequency-energy conversion factors 24 Fresnel factor 59 GaAs 67, 185, 193, 198, 230 see also superlattices: GaP 224, 229, 232f Ge 57, 164

Index grating coupling 219 grating spectrometer 23 gravity waves on liquid 78f Green functions acoustic 50ff, 254 basic properties 31 Iff dipole-exchange 142 jellium 167ff magnetic 99ff, 108 superlattices 254, 262, 268, 283 surface matching 56, 162 guided waves acoustic 50 see also polaritons Hamiltonian electron gas 166 Heisenberg antiferromagnet 109 Heisenberg ferromagnet 86 Ising ferromagnet 87, 120 mean-field for ferromagnet 89 harmonic oscillator 17ff Heisenberg model, see Hamiltonian Hg 81,83 Hooke's law 41 hydrodynamics electron gas 174ff, 203 liquids 74ff inelastic particle scattering 28ff, 68ff InSb 171, 187 intercalated compounds 248 interface modes 137 jellium 166 see also electron gas Kapitza conductance, magnetic 103 KDP (potassium di-hydrogen phosphate) 120 KNiF 3 114 Kronig-Penney model 272 Lamb waves 49 Lame parameters 41, 50 lattice dynamics one-dimensional 34ff superlattice 256ff three-dimensional 40, 69ff leaky modes 141, 205 LEED, see electron diffraction LiF 211 light-emitting tunnel junctions 233ff light scattering, see Brillouin scattering, Raman scattering Lindhard function 166, 169, 181 linear-response theory 16ff, 31, 312ff see also Green functions liquid crystals 248

liquids bulk waves 76 surface waves 77ff long-range surface plasmon (LRSP) 212, 227 Love waves 49ff Lyddane-Sachs-Teller (LST) relation 179 magnetic resonance, see antiferromagnetic resonance, spin wave resonance magnetisation, surface 92 magnetoelastic waves 302f magnetostatic modes antiferromagnets 142ff cylindrical surface 308 ferromagnetic double layer 136f ferromagnets 130ff spherical surface 309 superlattices 287ff magnetostatic region 127 magnons bulk Heisenberg antiferromagnet 1 lOff bulk Heisenberg ferromagnet 88ff bulk Ising, transverse field 121 continuum theory 96ff definition 85 edge modes 307f films 104ff magnon-magnon interactions 304 semi-classical 90 semi-infinite Heisenberg 90ff superlattice 296ff surface acoustic 94 surface antiferromagnetic 113,116 surface Ising, transverse field 124 surface optic 94 MBE, see molecular beam epitaxy mean-field theory 89, 92, 121 metallo-organic chemical vapour-phase deposition (MOCVD) 247 microwave experiments 153f Miller indices 31 mixed excitations 185, 302f see also magnetoelastic waves, polaritons MnF 2 111, 146f, 241 molecular-beam epitaxy 247 monatomic lattice, see lattice dynamics multiple quantum well (MQW) 274 Neel temperature 86 neutron scattering 29 Ni 72,118 nonlinear effects 214ff, 3O3f nonreciprocal propagation 133, 135, 142, 145, 149, 151, 238, 303 opacity broadening 27, 57, 63, 149 optic phonons, see phonons

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Index optical dispersion 165, 183 ordinary wave 265 Otto configuration, see attenuated total reflection particle scattering, see electron diffraction, inelastic particle scattering, neutron scattering phonons acoustic 34ff, 37, 249ff confined 259 cylinder 309 optic 37f, 73, 178f, 256ff sphere 309 see also elastic waves photoelectron spectroscopy 23, 118 photon-correlation spectroscopy 80f pinning 95 see also surface anisotropy, magnetic plasma frequency 170 plasmons superlattice 276ff superlattice intersubband 282 superlattice intrasubband 281, 286 superlattice surface 280 surface 197, 199 2D electron gas 203f 3D electron gas 157, 170, 175 see also polaritons Pockels tensor, see acousto-optic tensor Poisson's ratio 41 polaritons bulk exciton- 188ff bulk magnon- 23 5f bulk phonon- 184ff bulk plasmon- 184ff cylinder 309 guided-wave 209, 212ff, 230ff, 243, 245f Raman scattering 187, 228ff single-surface 194ff, 229 single-surface, anisotropic media 202f, 245, 270 single-surface, charge-sheet 203f single-surface, electrostatic 200ff single-surface, self-guided 215ff sphere 309 superlattice magnetic 290 superlattice surface 270 surface-exciton 204ff, 225f surface-magnon, antiferromagnetic 240 surface-magnon, ferromagnetic 236ff surface, nonlinear 214ff two-interface 208ff, 230 two-interface magnetic 24 Iff power spectrum 19 Poynting vector 200, 217f p-polarisation, acoustic 44 pseudo-spin model, see ferroelectrics pseudo-surface wave 45

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Raether-Kretschmann configuration, see attenuated total reflection Raman scattering bulk polaritons 187 folded acoustic modes 253ff folded optic modes 26If principles 23ff superlattice plasmons 284ff surface polaritons 228ff random-phase approximation (RPA) 88, 129, 166, 170, 177, 181 Rayleigh wave 45ff, 64ff, 68, 70, 303, 306 RbMnF 3 114 reciprocal lattice 3, 7 reconstruction, surface 1, 12ff, 92, 122, 124, 291ff reflection acoustic waves 42ff reflectivity, optical 154, 172f, 241 response function 16ff see also Green function, linear-response theory Reststrahl 177ff, 185, 197ff, 267f RHEED, see electron diffraction rough surfaces 234, 309f RPA, see random phase approximation scalar potential, magnetostatic 131 scattering atoms see inelastic particle scattering electrons, see electron diffraction, electron energy loss spectroscopy light, see Brillouin scattering, Raman scattering neutrons, see neutron scattering scattering cross section definition 27 see also differential cross section screening 157 selection rule, polarisation 60 selvedge 12 semiconductor superlattices growth 247 simple band model 27 Iff terminology for electron states 274 see also superlattices SEXAFS 23 SGFM, see Green function Si 57, 65, 68, 163, 172f skin depth, electromagnetic 59 slab modes 48f sound velocity longitudinal 42, 76 transverse 42 space lattice 2, 7 spatial dispersion 165, 170 specific heat, surface acoustic 49

Index magnetic 117, 119 spherical and spheroidal surfaces 307ff spin deviation lOlf spin-flop phase transition 116 spin wave resonance 117 spin waves, see magnons s-polarisation, acoustic 43 sputtering 247 Stokes scattering conservation rules 3If definition 24 magnons 102, 148, 150, 289 via exciton-polaritons 193 see also Brillouin scattering, Raman scattering Stoneley wave 49 stop band 37, 39, 161, 185, 256, 263, 269, 274 strain tensor 41 stress tensor 41,75 superlattices acoustics 249ff Al/W 257 Brillouin scattering 289 cyclotron resonance 274 definitions and terminology 11, 247 dielectric function 267 electron states 271ff ferromagnetic/antiferromagnetic 291 f Fe/Gd 294 GaAs/Al^Gai _xAs 254f, 261f, 267ff, 277, 286 Green functions 262, 268 growth 247 magnetic 287ff magnetic permeability 290f magnetic polaritons 290 magnetic reconstruction 29Iff magnetostatic modes 287ff magnons 296ff

Ni/Mo 289 optical properties 263ff phonons 249ff, 256ff plasmons 276f, 286 Raman scattering 252ff, 257, 261f, 284fif surface modes 256, 269f, 280, 286 surface-active medium 196 surface anisotropy, magnetic 95, 98 surface reconstruction, see reconstruction, surface surface ripple scattering 65fT surface tension 77 SWR, see spin wave resonance TaC 74 tight-binding approximation 160 time-of-flight (TOF) spectra 30, 69f torque equations, magnetic 97, 131, 144 transfer matrix 251f, 258, 264, 272f, 299f twisted state 293f uncertainty principle 62 unit cell 2 UPS, see photoelectron spectroscopy viscosity 75 Voigt configuration

132, 236, 287, 290

water 79, 82 waveguiding condition wedge modes 3O5ff Wigner-Seitz cell 3

214

XPS, see photoelectron spectroscopy Young's modulus 41 yttrium iron garnet (YIG)

117, 153

zero sound 169 ZnO 206f ZnSe 226f

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