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 — 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 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> |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

=

W J -co

leads to ~ 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 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

iX>/ 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 . 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|/ 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) = (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!
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 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 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 -• 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(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- 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

(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 ) 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 (