Metal Surface Electron Physics

Metal Surface Electron Physics

A. Kiejna and K.F. Wojciechowski University of Wroctaw, Poland

Metal Surface Electron Physics

A. Kiejna and K.F. Wojciechowski University of Wroctaw, Poland

U.K. U.S.A. JAPAN

Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, U.K. Elsevier Science Inc., 660 White Plains Road, Tarrytown, New York 10591-5153, U.S.A. Elsevier Science Japan, Tsunashima Building Annex, 3-20-12 Yushima, Bunkyo-ku, Tokyo 113, Japan

Copyright 91996 Elsevier Science Ltd

All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publisher. First Edition 1996

Library of Congress Cataloging in Publication Data Kiejna, A. Metal surface electron physics/A. Kiejna and K. F. Wojciechowski p. cm. Includes index. 1. Surfaces (Physics). 2. Electronic structure. 3. Metals--Surfaces. 4. Interfaces (Physical sciences). I. Wojciechowski, Kazimierz, prof. nadzw, dr hab. I1. Title. QC173.4.S94K54 1996 530.4'17--dc20 95-45019

British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library

ISBN 0 08 042675 1 Hardcover

Printed and Bound in Great Britain by Alden Press, Oxford

Contents Preface

I

......................................

Classical Description of Metal Surface

1 The geometry of metal crystals and surfaces 1.1 Bravais lattices and metal structures . . . . . . . . . . . . . . 1.2 ................................ 1.3 Crystallographic notations . . . . . . . . . . . . . . . . . . . . . . 1.4 Some features of the geometrical structure . . . . . . . . . . . 1.5 Two-dimensional lattices . . . . . . . . . . . . . . . . . . . . . . . 1.6 Notations of the real surface structure . . . . . . . . . . . . .

vii

1 3

. . . . . 3 ... ... 7 . . . . . 10 ... 13 . . . . . 16

2 The surface of real metals 2.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Lattice relaxation and reconstruction of surfaces . . . . . . . . . . . . 2.3 Vibrations of surface atoms and the temperature . . . . . . . .

19 19 21 27

3 Thermodynamics of the surface of crystal 33 3.1 Basicnotions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 Equilibrium shape of crystalline particles . . . . . . . . . . . . . . . . 36 3.3 Thermodynamics of microscopic single crystals . . . . . . . . . . . . . 41 3.4 Surface energy, surface tension and surface stress . . . . . . . . . . . . 45

Quantum Theory of Metal Surface

51

4 Electrons in metals 53 4.1 model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.2 Infinite and finite potential well . . . . . . . . . . . . . . . . . . . . . . 58 4.3 Jellium model and electrons near metal surface . . . . . . . . . . . . . 62 4.4 Electron gas in the Hartree-Fock approximation . . . . . . . . . . . . . 65 4.5 Exchange and correlation energy . . . . . . . . . . . . . . . . . . . . . 68 4.6 Fermi hole and the origin of image force . . . . . . . . . . . . . . . . . 69 4.7 Stability of jellium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.8 Surface energy of semi-infinite free-electron gas . . . . . . . . . . . . . 73

...

111

5 Electron density functional theory

5.1 Thomas-Fermi method and its extensions . . . . . . . . . . . . . . . . 5.2 Hohenberg-Kohn theory . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Kohn-Sham equations . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Electron gas near the metal surface 6.1 Thomas-Fermi electron density profile . . . . . . . . . . . . . 6.2 Self-consistent Lang-Kohn method . . . . . . . . . . . . . . . . . 6.3 Effective potential . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 The local density of states . . . . . . . . . . . . . . . . . . . . . .

77 78 80 82 85

. . . . . 85

... ... ...

88 89 91 95

7 Sum rules and rigorous theorems for jellium surface 7.1 The phase-shift sum rules . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Budd-Vannimenus theorems . . . . . . . . . . . . . . . . . . . . . . . . 7.3 The virial theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95 97 100

8

103

Surface energy and surface stress 8.1 Surface energy components . . . . . . . . . . . . . . . . . . . . . . 8.2 Surface energy of jellium . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Reintroduction of the discrete lattice of ions . . . . . . . . . . . 8.4 Variational treatment of lattice effects . . . . . . . . . . . . . . 8.5 Structureless pseudopotential model . . . . . . . . . . . . . . . 8.6 Surface stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.. 103 .. 104 . . . . 107 . . . . 112 . . . . 115 .. 119

9 Work function

123 9.1 The definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 9.2 Work function of semi-infinite jellium . . . . . . . . . . . . . . . . . . . 124 9.3 Discrete-lattice corrections to the work function . . . . . . . . . . . . . 128

10 Work function of simple metals: relation between theory and experiment 10.1 Jellium part of the work function . a role of the correlation energy . . 10.2 Work function of the metal bounded by the flat surface . . . . . 10.3 Face-dependent part of work function . . . . . . . . . . . . . . . . . . 10.4 Polycrystalline and face-dependent work functions . . . . . . . . . . . 10.5 Relation between theory and experiment . . . . . . . . . . . . . . . . . 11 Variational electron density profiles: trial functions 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Conditions satisfied by various exact electron density profiles . . . . . 11.3 Examples of the trial electron density profiles . . . . . . . . . . . . . . 11.4 Smoluchowski’s density profile and different contributions to the energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iv

131 131 133 134 135 137 141 141 143 144 146

Image potential and image plane 12.1 Limitations of the classical picture. Image plane position . . . . . . . . 12.2 Linear response of electron system to static perturbing charges . . . . 12.3 Response of metal surface to a perturbing charge . . . . . . . . . . . . 12.4 The exchange (Fermi) hole near the metal surface . . . . . . . . . . . . 12.5 Origin of the image potential . . . . . . . . . . . . . . . . . . . . . . .

153 153 157 159 161 165

13 Metal surface in a strong external electric field 13.1 Electrostatic field at the surface . . . . . . . . . . . . . . . . . . . . . . 13.2 Linear and non-linear contributions to the response . . . . . . . . . . . 13.3 Effect of the ionic lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Field induced relaxation and field evaporation . . . . . . . . . . . . . .

171 171 177 179 181

14 Alloy surfaces 14.1 The Vegard law and the volume of formation of an alloy . . . . . . . . 14.2 Semi-empirical theory of alloy formation . . . . . . . . . . . . . . . . . 14.3 Surface properties of alkali metal alloys . . . . . . . . . . . . . . . . . 14.4 Work function of ordered alloys . . . . . . . . . . . . . . . . . . . . . . 14.5 Surface segregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

187 187 189 192 194 198

15 Quantum size effect and small metallic particles 15.1 The notion of size effect . . . . . . . . . . . . . . . . . . . . 15.2 The non-oscillatory QSE . . . . . . . . . . . . . . . . . . . . 15.3 Oscillatory quantum size effect . . . . . . . . . . . . . . . . 15.4 Small metallic particles . . . . . . . . . . . . . . . . . . . . 15.5 Magic numbers . . . . . . . . . . . . . . . . . . . . . . . . .

203 203 204 206 213 218

Metal Surface in Contact with Other Bodies

..... ..... ..... ..... .....

. . . . .

221

16 Adsorption of alkali atoms on metal surface 223 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 16.2 Work function changes due to alkali metal adsorption . Classical 224 picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Density-functional calculations . . . . . . . . . . . . . . . . . . . . . . 227 16.3.1 The model of . . . . . . . . . . . . . . . . . . 227 16.3.2 The adsorption of single alkali atoms on metallic substrate . . 232 16.4 Relation between theory and experiment . . . . . . . . . . . . . . . . . 237 16.5 Sum rules for a metal with an adlayer . . . . . . . . . . . . . . . . . . 240 16.5.1 Phase-shift sum rule . . . . . . . . . . . . . . . . . . . . . . . . . 240 16.5.2 Budd-Vannimenus theorem for a system . . . . . 241 16.6 Analytical density profiles for jellium-alkali adlayer system . . . . . . . 242 V

17 Adhesion between metal surfaces 17.1 General considerations . . . . . . . . . . . . . . . . . . 17.2 Adhesion of semi-infinite metallic slabs . . . . . . . . . 17.3 Exact relations for bimetallic interfaces . . . . . . . . 17.4 The force between metal surfaces at small separations

. . . . . .

. . .. .. ....

... . . .. . ... . . . . .

245 . 245 . 247 . 253

. 256

18 Universal scaling of binding energies 263 18.1 Scaling of adhesive binding energies . . . . . . . . . . . . . . . . . . . 263 18.2 Universal binding energy curves . . . . . . . . . . . . . . . . . . . . . . 266

Appendices A

2 73 A . l Fundamental constants . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 A.2 Atomic units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 73 A.3 The quantities characteristic for the electron gas and screening . . . . 275

B Planar average of the potential difference

277

C Surface correlation energy for the Ceperley-Alder parameterization

281

D Linear potential approximation for a metal surface

283

E Finite linear potential model

287

References

289

Index

299

vi

Preface During the last thirty years metal surface physics, or generally surface science, has gone a long way from infancy to a mature life. This was possible due to the development of vacuum technology and the new surface sensitive probes on the experimental side and new methods and powerful computational techniques on the theoretical side. The aim of this book is to introduce the reader into the essential theoretical aspects of the atomic and electronic structure of metal surfaces and interfaces. Our primary idea was to give some theoretical background to students of experimental and theoretical physics to allow further exploration into research in metal surface physics. Since we have included also many important experimental results it is not a book on the theory of metal surfaces but rather on the physics of metal surfaces. It should be useful for graduate and advanced undergraduate students of physics and materials science, physicists, theoretically oriented chemists and metallurgists interested in fundamental aspects of metal surface physics. The field of surface physics has grown so much that some selection of topics has to be made. We tried to survey a certain range of metal surface physics phenomena and to describe them in a unified way. The major part of this book covers the electronic properties of surfaces. The presentation is based on the real-space approach to the problem. This was enabled by making use of the density functional theory and the jellium model with its extensions. We demonstrate the utility of simple jellium model in discussing the fundamental quantities that govern energetics of metal surfaces and as a starting point in explaining more complicated processes occurring at the surface. The book shows how this description can be improved by introducing the discrete lattice effects. Basing on the jellium model allows to keep the mathematical complexity at the level that is easily understood by advanced undergraduate students. The book consists of three parts. The first part is devoted to classical description of geometry and structure of metal crystals and their surfaces and surface thermodynamics including properties of small metallic particles. Part two deals with quantummechanical description of electronic properties of simple metals. It starts from the free electron gas description and introduces the many body effects in the framework of the density functional theory, in order to discuss the basic surface electronic properties of simple metals. This part outlines also properties of alloy surfaces, the quantum size effect and small metal clusters. Part three gives a succinct description of metal surface in contact with foreign atoms and surfaces. It treats the work functi.on changes due to alkali metal adsorption on metals, adhesion between metals and discusses the universal aspects of the binding energy curves. In each case we aimed to give references to a most representative literature. vii

Some part of the material presented in this book is covered by the authors of textbooks which have appeared in recent years and are given in our list of references. They differ however, in the way of presentation and our intention is to complement them rather than to compete. We are grateful to Dr. Marek T. Michalewicz for the time he spent proof-reading the greater part of the manuscript and correcting our English. We express our thanks to the American Institute of Physics, Woodbury, the American Physical Society, College Park, Editions de Physique, Les Ulis, Elsevier Science, Amsterdam and Oxford and the Institute of Physics Publishing, Bristol, as well as all the authors to whom we addressed, for their kind permission to reproduce or redraw the figures from the relevant publications.

Adam Kiejna Kazimierz F. Wojciechowski July, 1995

viii

Part I

Classical Description of Metal Surface

This Page Intentionally Left Blank

Chapter 1

The geometry of metal crystals and surfaces 1.1

B r a v a i s l a t t i c e s and m e t a l s t r u c t u r e s

In crystalline solids, like metals, atoms are arranged in a regular manner. An ideal single crystal 1 is defined as a body of atoms (ions) stacked to form a three-dimensional net which is determined by the translation vectors al, a2, a3. From the definition a single crystal is free of lattice imperfections. A characteristic feature of the translational s y m m e t r y of a lattice is that an arrangement of atoms about a given lattice point, determined by the vector R, is identical with that observed from any other point that can be reached by the simple transformation R ~ = R + trial -+- n2a2 -}- n3a3,

(1.1)

where al, a2 and a3 are any non-coplanar vectors and ni = 0, +1, i 2 , + 3 , . . . ,

i = 1,2,3.

(1.2)

A three-dimensional lattice produced by all the points whose locations are defined by the vectors R ~ is called a Bravais lattice (Bravais, 1866). There are a total of fourteen three-dimensional Bravais lattices which are discussed in the textbooks on solid state physics (cf. Ashcroft and Mermin, 1976; Kittel, 1967) or crystallography. The majority of single crystals of metals is characterized by one of the following close-packed structures (see Fig. 1.1): - A1 structure of the face-centered cubic (fcc) lattice; - A2 structure of the body-centered cubic (bcc) lattice; - A3 or the hexagonal close-packed (hcp) structure which can be viewed as created from two mutually shifted simple hexagonal lattices. 1This word originates from the Greek krystaUos which means ice.

C H A P T E R 1. G E O M E T R Y OF C R Y S T A L S A N D SURFACES

A

9

I

v

{Q}

w

{b} {c}

Fig. 1.1. Conventional unit cell of: (a) the A1 structure of face-centered cubic lattice; (b) the A2 structure of body-centered cubic lattice; (c) the A3 or hexagonal close-packed structure. The latter unit cell can be depicted by combining two simple hexagonal (Bravais) lattices.

From the point of view of metallurgists, the metal (or intermetallic) structures which do not fall under one of these categories are classified as complex. If a metal has a variety of polymorphic forms, then at least one of the forms adopts one of the structures listed above. There is a connection between the position of an atom in a periodic table and the crystal structure: the elements belonging to the same group in general have the same type of structure. Metals which crystallize in A1, A2 and A3 structures are listed in Tables 1.1-1.3.

1.2

U n i t cell

The parallelepiped based on the translation vectors al, a2, a3 introduced in the former Section constitutes a unit cell. The choice of the unit cell is by no means unique. If the translation vectors ai are chosen in such a way that ]ai ] - min,

i = 1, 2, 3,

(1.3)

then these vectors are called the base translation vectors or primitive lattice vectors and they determine a primitive unit cell, or a simple elementary cell of the lattice. For instance, in a simple cubic (sc) crystal lattice of Fig. 1.2, one cannot choose vectors ai of the length smaller than l al I=1 a2 I=1 a3 I-- a. However, this is possible for example, for the fcc lattice. Thus, the unit cell shown in Fig. 1.2a is a simple elementary cell. The unit cell and the primitive unit cell of the fcc lattice are depicted in Fig. 1.2b. As it is seen there is only one atom per primitive unit cell. In practice, for symmetry reason, it is more convenient to deal with the conventional unit cell (or the Bravais elementary cell). Conventional unit cell is a three-

1.2. UNIT CELL

5

Table 1.1 Selected metMs crystMizing in the face-centred cubic lattice structure A1 and their lattice constant a. Data from Wyckoff (1974).

Metal

a [A]

Metal

a [A]

A1 Ag Au Ca Cu

4.05 4.09 4.08 5.58 3.61

Ni Pb Pd Pt Sr

3.52 4.95 3.89 3.92 6.08

dimensional region with which the whole space could be filled by means of translations belonging to a certain subset of all the translation vectors of a given Bravais lattice. Simply, these are the cells shown in Fig. 1.1. Note that, in general, each of these cells contains more than one atom. In the following, wherever we speak about a unit cell we will mean such a conventional unit cell as defined above. Similarly, the length a of the unit cell side in the cases of the A1 and A2 lattices, and the lengths a and c for the A3 lattice will be called the lattice parameters or lattice constants. The characteristic lattice parameters of metals are given in Tables 1.1-1.3. The volume f~ of a primitive unit cell is defined by the mixed product of the base lattice vectors ai ft = a l - ( a 2 • a3), (1.4) and is generally smaller than the volume of unit cell. For example, it is equal to a3/4,

A

Y

_/1

j f w

w

(Q)

(b)

Fig. 1.2. Primitive unit cell of the simple cubic lattice (a), and the face-centered cubic lattice (b).

CHAPTER 1. GEOMETRY OF CRYSTALS AND SURFACES Table 1.2 Selected metals crystalizing in the body-centred cubic lattice structure A2. The values of the lattice constant a, taken from Wyckoff (1974), are measured in the room temperature unless otherwise indicated.

Metal

a [/~1

Metal

a [h]

Ba Cr Cs Fe K Li Mo

5.02 2.88 6.05 (5K) 2.87 5.23 (5K) 3.49 (78K) 3.15

Na Nb ab Ta Wl V W

4.23 (5K) 3.30 5.59 (5K) 3.31 3.88 3.03 3.16

a3/2 and to (x/~/2)a2c for the primitive unit cells of the A1, A2 and A3 structures, respectively. In many cases it is useful to represent the crystal structure by the Wigner-Seitz primitive unit cell (or simply the Wigner-Seitz cell). This cell is a smallest volume bounded by planes which divide in half all the lines connecting the nearby lattice points and are perpendicular to the lines. In this way only one atom is associated with each Wigner-Seitz cell. The example of such cell for the bcc lattice is given in Fig. 1.3. The noticeable high symmetry features of the Wigner-Seitz cell which is also called a symmetric cell, and particularly of its counterpart in the reciprocal lattice space, are widely exploited in calculations of the electronic structure of solids.

I I I I I

I I I I I

Fig. 1.3. The Wigner-Seitz cell of the body-centered cubic lattice.

1.3. C R Y S T A L L O G R A P H I C N O T A T I O N S Table 1.3

Selected metals crystalizing in hexagonal dose-packed lattice structure A3 and their lattice parameters a and c. Data from Wyckoff (1974).

1.3

Metal

a [h]

c [h]

c/a

Be Cd Co Gd Mg Re Ti Zn Ideal A3 structure

2.27 2.98 2.51 3.63 3.21 2.76 2.95 2.66

3.59 5.62 4.07 5.78 5.21 4.46 4.68 4.95

1.58 1.89 1.62 1.59 1.62 1.62 1.59 1.86

-

-

1.63

Crystallographic notations

The three-dimensional lattice may be thought of as created of various sets of parallel planes. Each set of planes has a particular orientation in space. The space position of any crystallographic plane is determined by three lattice points not lying on the same straight line. Drawing such a plane we can find its intersections with the three crystal axes defined by the directions of the translation vectors ai. The intercepts of a crystallographic plane with the crystal axes will occur at the integral multiple of the lattice parameters a, b and c, along the principal axes i.e., 0x - la, Oy - m b and 0z = nc (where l, m, n are integers). The numbers la, mb and nc can be used to define the Miller indices of a given crystallographic plane. For this purpose one takes the inverse of the numbers i.e., the (/a) -1, (rob) -1, (nc) -1 and reduces these to the smallest integers (including zero) which have a common ratio. The result is given in the form of three numbers with no common factor greater than 1 and are denoted by h, k, 1. These three numbers, written in parentheses, determine the given set of crystallographic planes and are called Miller indices (hkl). As an example let us consider two planes of the A2 structure (Fig. 1.4). Plane 1 intersects the z-axis at a distance of 0z = a from the origin of coordinate system 1 = ~1 , oy 1 _ 1 and and is parallel to the xy-plane, thus Ox - oc and Oy = oc, so, 0-7 !0 z = 1a " Multiplication of these numbers by a gives three numbers h = 0, k - O, and 1 - 1. Thus, plane 1 of the lattice of interest is denoted by indices (001). In a similar way one gets the Miller indices of plane 2 to be (011). For the fcc and bcc structures, Miller indices given in an arbitrary sequence, describe the same type of crystallographic planes of identical symmetry: (hkl) = (khl) =

CHAPTER 1. G E O M E T R Y OF CRYSTALS AND SURFACES

zT

Fig. 1.4. Two different lattice planes of the body-centered cubic lattice.

(lhk). Similarly, the negative values of Miller indices describe the same type of (hkl) planes: (hkl) - ( - h - kl) = (hkl). In such a way, plane 2 of Fig. 1.4 can be denoted equivalently by (110), (101) or by (ii0). In the A3 structure (hcp) the sequence of indices cannot be, in general, arbitrary. Besides, a four-index notation (hkil) of Miller-Bravais is often used for convenience with the fourth index, i = - ( h + k), inserted between k and I. For cubic crystals a direction which is perpendicular to a (hkl) plane is denoted by giving corresponding Miller indices in square brackets, i.e. in the form [hkl]. For other structures to denote crystallographic directions one introduces three indices u, v, w, instead of h, k, l. A set of crystallographic planes (hkl) which are equivalent in respect of symmetry is denoted by using braces {hkl}. Similarly, a set of crystallographically equivalent directions [hkl] is denoted by broken brackets (hkl).

0 I

I/ (100)

(110)

I\ (111)

Fig. 1.5. The arrangement of lattice points in the most densely packed planes of the facecentered cubic lattice.

1.3.

CRYSTALLOGRAPHIC

NOTATIONS

Examples of the arrangement of lattice points (which, in the following, will be conventionally called atoms) in several low-index planes of the n l (fcc), A2 (bcc) and A3 (hcp) lattices are given in Figs 1.5-1.7. The interplanar distance, or the spacing between two subsequent parallel lattice planes is, in general, given by the formula 21q"

dhkz = I Chkz I

(1.5)

where G hkl is the vector of the reciprocal lattice (Hilton, 1963) (1.6)

G hkZ- ha~ + ka~ + la~ spanned by the three primitive reciprocal lattice vectors:

a7 -

27r

27r

--ff a2 x a3,

a:~ -

--~ a3 x a l ,

27r a~ -

- ~ a l x a2,

(1.7)

where ~t is given by Eq. (1.4). For particular stuctures dhkl can be expressed by the lattice parameter and the Miller indices of the plane to give: For the fcc and bcc lattice,

a

d~kz = Q s ( h 2 + k 2 +/2)1/2,

8 - fcc, bcc,

(1.S)

where a is the lattice parameter, and

Qfcc -

Qbcc -

1, 2, 1, 2,

if h, k, 1 are all odd numbers, if h, k, 1 are of mixed parity,

(1.9)

if h + k + 1 is an even number, if h + k + 1 is an odd number.

(1.10)

A

I

( 110 )

( 111 )

( 210 )

Fig. 1.6. The arrangement of lattice points in the (110), (111) and (210) planes of the bcc lattice.

CHAPTER 1. GEOMETRY OF CRYSTALS AND SURFACES

10

I

/

I

/

(0001)

{1120)

Fig. 1.7. The arrangement of atoms in the (0001) and (1120) planes of the hcp structure.

For the hcp lattice

dhCP _

c~/'~ hkZ -- [4r2(h 2 + hk + k 2) + 3/2 ]1/2'

(1.11)

where r - c/a, a and c are lattice constants (cf. Fig. 1.1). The interplanar distances for a specific metal can be easily obtained using the above formulas and the values of lattice parameters given in Tables 1.1-1.3. Since the set of parallel lattice planes (hkl) is determined uneqivocally by the reciprocal lattice vector G hkl perpendicular to the set of planes, taking the scalar product of reciprocal lattice vectors corresponding to two different planes one gets the angle a between the planes, (hlklll) and (h2k212), of a crystal. For the A1 and A2 structures this angle can be expressed by Miller indices and is immediately given by

{ = arccos

hlh2+klk2+lll2

[(h2 + k2 + 12)(h2 + k2 +/22)]1/2

} .

(1.12)

For example, the angle between the (11 1) and (1 10~ planes of fcc and bcc lattices calculated from formula (1.12) is equal to arccos V/2/3 __ 35~ '.

1.4

Some features of the geometrical structure

In each crystal lattice 2 a given lattice point is surrounded by a certain number of neighbouring points which are away from it by the same distance Ri. Let Rmin denote the minimum distance among Ri. Then the coordination number can be defined by z(Rmin) --- z(R1N),

(1.13)

2In the following we will use alternatively this term for a crystal structure or a type of structure.

1.4. FEATURES OF GEOMETRICAL STRUCTURE

11

Table 1.4

Number of neighbours Z(RiN), beginning from the second (i = 2, 3, ...8) for the facecentred cubic lattice structure (AI). i

2

3

4

5

6

7

8

Z(RiN)

6

24

12

24

8

48

6

where R1N denotes the distance between the given point (or atom) and its nearest neighbours in the meaning given above. The number defined by (1.13) is commonly called the number of nearest neighbours. Consequently, the number of atoms next to the nearest neighbours (second nearest neighbours) will be denoted by z(R2g), etc. The number of subsequent neighbours is meaningful, e.g. in calculation of interactions between atoms both in the bulk of a crystal and for external (foreign) atoms on a crystal surface (as during adsorption). Table 1.4 gives typical values of the number of next nearest neighbours z(Rig), i = 2 , 3 , . . . , 8 for the A1 lattice. Coordination numbers z(R1g) and other characteristics for the A1, A2 and A3 lattices are collected in Table 1.5. Considering interactions between atoms, quite often, we restrict our considerations to the first nearest neighbours only. It follows from Table 1.4 that in the fcc lattice this might be insufficient. In this case third nearest neighbours are meaningful as well since in spite of their greater distance they are numerous and in the presence of a slow-varying interaction potential their contribution to the total energy could be comparable with that, say, of the second nearest neighbours. Another characteristic of the geometrical structure of metals is the fraction of space filling or packing density, P. Suppose that a crystal is built up of atoms represented by rigid balls of the radius equal half the distance between the nearest neighbours, this quantity is defined as the ratio P =

volume of atoms in the cell volume of the cell '

(1.14)

where the cell means a conventional unit cell. Atoms in metals tend to a der_.sest packing in a crystal lattice. Both the fcc and the hcp structures correspond to the densest possible packings of hard spheres with P = 0.74. The essential feature of a metal structure is its dense packing; details of the structure itself do not play so important role. For example, although a melted metal completely loses its crystalline order, its rather close packed configuration is maintained. Even at a complete vanishing of a crystalline structure the electrical properties of the metal remain almost unchanged. A characteristic of the filling of a crystallographic plane (hkl) is the surface density of atoms, •hkl, which corresponds to the number of lattice points per planar unit cell ~ k z = d~kl ~,

(1

15)

12

C H A P T E R 1. G E O M E T R Y OF C R Y S T A L S A N D S U R F A C E S

Table 1.5 Some characteristic features of the geometric structure of metals.

Structure Number of atoms Coordination type per unit cell number

A1 (fcc)

4

12

Packing density

6

= 0.74

Representative metals

A1, Ag, Au, Cu, Ni, Pb

A2 (bcc)

2

8

8

- 0.68

Fea, K, Lia, Mo, Na, Nb

A3 (hcp)

2

12

6

= 0.74 Cd, Cos, Li~, Mg, Zn

where the d~k z for the related structures are given by formulae (1.5-1.11) and ft is a volume of the unit cell. Low-index planes have a high density of atoms lying in a plane. The following sequence shows the crystal planes of the A1 structure having the highest surface density of atoms (cf. Fig. 1.5): (111) with a surface density of 2.31/a 2, (001) with 2/a 2, (110) with v/2/a 2. The most densely packed planes in the A2 structure are the following: (110) with a surface density of 0.707/a 2, (001) with 0.5/a 2, (210) with 0.408/a 2. The (111) face in the latter structure has a much smaller density of packing which is equal to 0.289/a 2 (see Fig. 1.6). The small surface density is manifested by a relative instability of the (111) plane which tends to achieve a more stable structure. In the A3 (hcp) crystal structure the highest surface density of atoms is ascribed to the (0001) or (001) face, and it is equal to 1.15/a 2. This face is shown in Fig. 1.7 and as it can be seen its atomic arrangement is identical with that for the (111) face

13

1.5. TWO-DIMENSIONAL LATTICES

C12

a2

~

Cl2

A

& w

v

w

a ~ a2pT=90 ~

a~ r a21 ~'=90 ~

p - rectangutar

c-rectangutar

C12

a, =a~ =ap ~-=90 square

C12

,w

w

a, r a2p ~'~,go 0

obtique

a~=a2=ap ~r=120o

hexagonat

Fig. 1.8. Five types of the two-dimensional Bravais lattices.

of the A1 (fcc) structure (Fig. 1.5). However, both of the faces differ in the stacking sequence of the successive layers of atoms. If the atomic sites in the topmost layer of a crystal of the A1 structure are marked by a letter A and those of the second layer by B, then atoms of the third layer will occupy positions C. Such stacking of the successive (111) layers in this structure may be denoted in the form ABCABCAB ..., whereas the stacking sequence of the (001) atomic layers for the A3 structure runs according to the pattern ABABAB ....

1.5

Two-dimensional

lattices

In comparison to fourteen three-dimensional lattices in 3D space there exist only five two-dimensional Bravais lattices: oblique, hexagonal, rectangular (orthorhombic), rectangular centered and quadratic (regular) (see Fig. 1.8). These follow from the restrictions imposed by symmetry operations (such as translations, rotations by an angle of 27r/n, where n = 1, 2, 3, 4 and 6, and reflections). The lattices are determined by the primitive translation vectors al and a2. Thus, a complete vector of translation takes the form Rm - mlal + m2a2 (1.16) An example of the oblique lattice is the pattern of the arrangement of atoms on the (210) face of an Al-structure crystal. Similarly, we are dealing with the twodimensional hexagonal lattice when crystals of the A1 or A2 structures have been

14

C H A P T E R 1. G E O M E T R Y OF C R Y S T A L S A N D SURFACES

Fig. 1.9. The atomic rows on the (111) face of tungsten (bcc lattice).

cleaved along a surface parallel to the (111) crystallographic plane. The same is observed at the surface of a metal of A3 structure obtained by cleaving parallel to the (0001) plane. A primitive, two-dimensional rectangular lattice is observed on the (110) plane of an Al-structure, whereas the rectangular centered lattice is revealed on the (110) face of an A2 metal. The quadratic lattice can be illustrated by the (100) face of an A2-structure. Sites of surface atoms do not necessarily coincide with points of the ideal lattice. A certain atom or a group of atoms that constitute the base, may be assigned to each point of the translation lattice. For a given translation lattice and the positions of the base atoms (and its composition) we can say that the atomic structure of a surface is fully determined. The position of base atoms must satisfy the requirements of point s y m m e t r y operations (rotations, mirror reflection) which transform a crystal lattice into itself with one point fixed. The point symmetry transformations constitute ten two-dimensional point groups or symmetry classes. Combination of the two-dimensional point groups of the basis and of five translation lattice types determines the complete s y m m e t r y of a Bravais lattice and results in 17 two-dimensional space groups. In order to determine the position and/or orientation of atoms on a surface only two Miller indices (hk) are needed. They are defined in the same way as the indices (hkl) for a space lattice. The Miller indices can be applied to determination of the distances dhk, between rows of atoms with h and k indices on the particular Bravaislattice plane. An example of such atomic rows on the W(112) plane 3 is shown in Fig. 1.9. In general, we have

dhk --

27r

I ghkl'

(1.17)

3In the following we will denote a (hkl) face of a given metal, M, by M(hkl); so in the above example we are dealing with the W(l12) face.

1.5.

TWO-DIMENSIONAL

LATTICES

15

where ghk is the two-dimensional vector of reciprocal lattice. In the way analogous to the three-dimensional case the vector of the two-dimensional reciprocal lattice can be defined as follows ghk -- ha~ + kay, (1.18) a~ and a~ are primitive translation vectors of the reciprocal lattice, which can be expressed by the primitive translation vectors of the primary lattice in the following way

a2xfi aI

--

2~

al"

( a 2 • ilL)'

(1.19) IA1 • a l a2

--

2~-

a2" (1:1 x al)"

The vector fi is a unit vector normal to the plane. It follows from these relations that ai .a~ = 2~5ij

(1.20)

where 5ij is the Kronecker delta function I,

5~j-

i--j,

o, iCj.

(1.21)

The spacing of the atomic rows for the particular lattice types is defined more explicitly by the following formulae: Oblique lattice: 1 d~k

=

h2 a 2 sin 2 "~

+

k2 2hk cos -y . a 2 sin 2 ~' a la2 sin 2 "y

(1.22)

Hexagonal lattice: 1 h 2 + hk + k 2 d~ k a2 -

-

(1.23)

Rectangular lattice (simple, p, and centered, c): 1 d2hk

h2 k2 a2 + a22

(1.24)

1 h2 +k 2 d~ k a2

(1.25)

Quadratic lattice:

Dealing with the atomic arrangement on a crystal surface we can speak about atomically smooth or rough faces of a crystal. The relief of the host face of a crystal plays an important role in the processes of adsorption i.e., when foreign atoms or

16

C H A P T E R 1. G E O M E T R Y OF C R Y S T A L S A N D SURFACES

molecules are deposited onto a clean surface of a metal (compare Chap. 16). From the point of view of the adsorption phenomenon the smoothness of a crystal face is a relative notion; it depends both on the structure of the face and on the size of the adsorbate atom (ion). For instance, for the adsorption of a cesium atom with its large radius, the (112) face of tungsten (Fig. 1.9) may be considered as smooth. However, the same plane may be regarded as a rough one for the adsorption of lithium or copper atoms whose atomic radius is small in comparison with the atomic or even ionic radius of tungsten. Usually, low-index faces are thought to be smooth and the ones with high Miller indices are regarded as rough. Although conventional, such a classification is, of course, arbitrary.

1.6

N o t a t i o n s of t h e real s u r f a c e s t r u c t u r e

Owing to reconstruction of the surface layer of atoms as well as adsorption occurring on a surface, the structure of the surface net may differ (and usually it does) from the structure of the set of parallel planes lying beneath, in the metal interior. Structure of the surface is specified by taking the unperturbed net of the substrate plane with its known structure and Miller indices as the reference lattice. If the base translation vectors of the plane lattice in the bulk of crystal are denoted by al and a2, then the translation vectors bl and b2 of the surface lattice, that has been modified by a reconstruction of the surface or is due to an adsorbate monolayer, may be expressed in the form bl = m 1 1 a l + m12a2, (1.26) b2 - m21al

+ m22a2.

(1.27)

This can be written equivalently by means of the transformation matrix M:

(bl)) (roll 5 m21 2 )( al )-M( al m12

m22

a2

a2

(1.28)

where mij are integers. It is easy to see that the surface area of unit cell of the lattice (bl, b2) is equal to the product of the determinant value of the matrix M, det M, and the surface area of unit cell of the lattice (al, a2). The value of det M allows to classify the resulting structures in the following way: (i) Determinant of the matrix M is an integer: the structures of the surface layer and of the substrate are simply related, and the structure of a system constituted by the overlayer and the substrate is called simple. (ii) If the value of det M is a rational number, then the lattices (bl, b2) and (al, a2) are rationally related; the structure of the system constituted by the outer layer and the substrate is named the coincidence-site structure and the outer lattice is called commensurate. (iii) If the value of det M is an irrational number, then the lattices (bl, b2) and (al, a2) are irrationally related, and the structure on the outer lattice is called incommensurate.

1.6. NOTATIONS OF SURFACE STRUCTURE

17

In the matrix notation the entire system, substrate-surface-layer, can be denoted by the following formula S(hkl)-M-~TA (1.29) where S(hkl) means the crystallographic orientation of the substrate S, M denotes the matrix of the transformation and A is the chemical stoichiometry of fl various atoms that constitute the base of the unit cell of the surface layer. In the case when the angles between the base translation vectors of the lattice (bl, b2) and (al,a2) are equal, a notation proposed by Elisabeth Wood (1964)is commonly used. This notation expresses the relation between the length of the base translation vectors (hi, b2) of the surface lattice and that of the (al, a2) of the reference lattice in the form

S(hkl)-(JjbllxaiJjb2a2I)R~

(1.30)

where a is the angle by which the lattice has been twisted in relation to the reference lattice as a result of rotation R. If a = 0, the factor R a standing in the second term may be omitted. A letter p may precede the parenthesis when the unit cell of the (new) surface lattice is primitive or a letter c, when the unit cell is centered. In

oToo]o

0 0 0 00 0~0 0 0

0~3~)~0

oio o1%o o.o

o

0 0 00e<}--O00 0 0 0 0 0 0 0 0 0 0 0 0 0

0

~

0 0 0

(a)

0

0

{b)

0

0

ooooo

0

oTo o1"o o olo'olo o

0 0 0 0~0 0~0 0

(c)

000

0

0/0,,0

0

0 0

o ~ o D o o o

o o~o c~ o o o o-

o

0 0 0 0 0 0 0

{d)

Fig. 1.10. Examples of two-dimensional surface structures denoted according to the Wood nomenclature: (a) p ( l x l ) ; (b) p(2• (c) c(2• (d) (x/2 x x/2)R45 ~ In each case the possible manners of filling the lattice sites of a structure by adatoms are indicated by black circles.

18

C H A P T E R 1. G E O M E T R Y OF C R Y S T A L S A N D SURFACES

the simplest case, when the surface structure is identical with that of the underlying layers, the structure of the surface is denoted as (1 x 1). Illustrations of some surface structures are given in Fig. 1.10.

Chapter 2

T h e surface of real m e t a l s 2.1

General remarks

Speaking about a surface of a solid one usually thinks about the topmost layer of atoms only. However, in general the term surface means a few of last atomic layers of the solid, whose geometrical or electron structure has been disturbed by breaking of the translational symmetry of the crystal in the direction normal to the surface. Surfaces can be divided into the idealized perfect ones, which do not exhibit any lattice defects (vacancies, impurities or intrusions, stacking faults, etc.) and the real, imperfect ones such as those in common technological use, where all the defect types are present. Usually a perfect surface is thought of as a lattice plane revealed by the ideal cleavage of single crystal into two parts, parallel to the crystallographic plane, without changing atomic configuration in either resulting parts. In practice, the use of single crystals enables to eliminate point defects but total elimination of dislocations is not possible. In order to avoid impurities the measurement should be carried out under ultrahigh vacuum conditions i.e., below a residual gas pressure of 10 -1~ Torr. 1 To illustrate the importance of this restriction one may note that at a pressure of 10 -6 Torr a monolayer of oxygen is formed on a metal surface (~ 1015 atoms per cm 2) in one second, under the assumption that all the molecules approaching the surface would stick to it and form a single adlayer. Under the vacuum conditions of 10 -9 Torr this time extends to one hour. The knowledge and understanding of the properties of perfect surfaces and their link with the properties of the bulk crystal is of great importance in understanding of the properties of real surfaces. In general, under usual experimental conditions, such perfect surfaces cannot be obtained. In some cases, however, a carefully prepared surface may be considered as a perfect one. The density of dislocation in metals, which typically amounts to l0 s per cm 2, can be reduced by one or two orders of magnitude. This value is lower by several orders than the density of metal surface atoms which amounts to 1015 atoms/cm 2, whereas the atomic density in the bulk of metal is s o m e 10 22 atoms/cm 3. A comparison of the two latter numbers demonstrates that any information about surface atoms must 1The unit 1 Torr - 1 mm Hg - 133.32 Pa 19

20

CHAPTER 2. REAL METALS SURFACE

Fig. 2.1. Schematic view of a surface atom and its nearest atomic surrounding.

be selected and separated from the sea of information about the bulk atoms that is dominating most of all. Structure of a surface plane of the perfect single crystal should coincide with t h a t of the inner lattice planes parallel to the plane of interest. However, any atom at the surface of a truncated crystal has quite a different spatial environment than atom in the bulk. The number of its neighbouring atoms is considerably less than t h a t of an interior atom (Fig. 2.1). Thus, the total energies of interaction are different for a surface atom and the one in the bulk. The interaction of an inner atom is characterized by a stronger binding than for a surface one. Each bulk atom of a metal with the fcc (A1) structure is surrounded by twelve nearest neighbour atoms which are distant by ax/~, where a is the lattice constant. For the same structure, an atom lying in the (111) plane has only 9 nearest neighbours whereas on the (100) face the number is 8 and on the (110) it has only 7 nearest neighbours. Thus, thinking in terms of the bonds, on these faces there are three, four and five bonds less, with respect to bulk environment. 2 As a consequence, positions of the atoms in the outermost plane, and often in the two or three last layers, are modified. The phenomenon leading to a rigid change of the distance between surface planes, without changing the two-dimensional unit cell is called surface relaxation of the lattice. The relaxation a n d / o r other extrinsic agents may be followed by the reconstruction of the surface which is manifested by a change in the atom arrangements or in the periodicity in the direction parallel to the surface. Also, segregation of one of the alloy-forming elements may occur on a surface of a single crystal of alloy. Moreover, owing to adsorption of foreign atoms onto a surface a new surface structure, quite different from that of the substrate, may be formed. The above surface processes are schematically illustrated in Fig. 2.2.

2This consideration has only a very qualitative meaning. In the case of metals we cannot speak about the orientational bonding.

21

2.2. LATTICE RELAXATION AND RECONSTRUCTION

0 00 000 0 0 0 0 0 0

O0 O0 O0 0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0

00

0 0 0 0 0 0

Retaxation

Reconstruction

0 0 0 0 0 0

00

0

0

00

0

0

00

0

Segregation

0

000

000

00

000

0

Adsorption

Fig. 2.2. Schematic classification of surface processes which modify the structure of the surface lattice.

2.2

Lattice r e l a x a t i o n and r e c o n s t r u c t i o n of surfaces

The phenomenon of surface relaxation has been revealed experimentally by the low energy electron diffraction (LEED) studies of surface structures. Later investigations by the ion beam and atom beam scattering techniques have confirmed that the topmost layer of ions in the metal is slightly shifted (by 1 - 10%) inwards or outwards the metal (with no effect on the two-dimensional lattice). Although, in some cases, different methods yield results which are 3 0 - 50% in variance, in general the qualitative agreement between the results of measurements is good. Selected measured relaxations for low-index faces of several metals are collected in Table 2.1. From a comparison of numbers given in the table follows that majority of the low-index faces shows a contraction of the interplanar spacing. It can be seen as well that smaller relaxations are observed on close-packed faces such as the fcc(111) and (100), bcc(110) and hcp(0001). Relatively large changes in the interplanar spacings (contractions) are observed on the most open low-index faces of metals. This concerns the (110) face of the fcc structure and the (100) or (111) face of the bcc structure. This seems to indicate an increase of relaxation with decreasing density of packing in crystallographic planes or, in other words, with increasing roughness of the planes. Now, we may define surface roughness as the ratio of the area of surface plane to the area occupied in this plane by intersections of atoms of the radii equal to one half the bulk nearest-neighbour distance. The magnitude of relaxations for the six faces of an iron single crystal versus the roughness (Sokolov et al., 1984) presented in Fig. 2.3, clearly shows that relaxation enlarges with increasing roughness of the face. The shortening of the distance between the first and second surface atomic layers

22

CHAPTER 2. R E A L ME TA LS SURFACE

Table 2.1 Relaxation of the outermost layer of the ions for low-index planes of different metal

~t~uctu~e~. D~t~ from: J o ~ (1978), I~gle~eld (~985) ~ d ~

Structure

Metal

A1 (fcc) A1 Ag Cu Ir Ni Pb Pd Pt Rh

Relaxation [%] at crystal plane

(111)

(100)

(110)

+1 + +2.5 0 0 + -4 -2 -1

0 0 0

-5+ -15 -6 + -10 -6 + -10 -7 -5 +-9 -17

~0++1 ~ 0

(11o)

A2 (bcc) Fe Mo Na V W

0++1

- 3 + +2 +2 0 0

-3

(lOO)

(111)

-1 -9 + -11

-15

,,~0 ~ 0

-7 -4 + -11

(0001)

A1 (hcp) Be Cd Co Gd Ti Zn

de~ Veen (~985).

0 0 0 -3.5 -2 -2

,,~ 0

2.2. L A T T I C E R E L A X A T I O N A N D R E C O N S T R U C T I O N

-30

I

I

I

I

23

'{

I

Fe TOP LAYERRELAXATION

-25

{111}

-2O "-" -15 {'4 ,v"O

13 '<:1

-10 -5

{110}' {li0}

+

0

I

I

i

I

i

I

I

05

1.0

1.5

2.0

2.5

3.0

3.5

4.0

ROUGHNESS Fig. 2.3. Top-layer relaxation versus roughness at iron-crystal faces. Redrawn with permission from Sokolov et al. (1984). @1984 Elsevier Science Ltd, UK.

in simple metals 3 may be explained in the framework of the model given by Finnis and Heine (1974). A main role in this model is played by electrostatic forces exerted on the surface atoms. The force that is acting on a surface ion core equals the electrostatic force of the interaction with all the other ions and with the electronic charge density. In the case of simple metals we can assume that (in a zeroth approximation) the electronic density is uniformly spread out over the volume of the crystal and sharply cut off at its geometrical surface (Fig. 2.4). As can be seen from this figure, ions of the surface layer are located in asymmetric Wigner-Seitz cells. The positive and negative charges in each cell must neutralize themselves mutually. Thus, an electrostatic force, oriented towards the bulk of metal, will be exerted on each surface ion (along the direction of the electrostatic centre of gravity of each cell). Consequently, the ions dwelling the surface cells will be shifted towards the bulk and the interplanar distance between the first and the second ionic layers will be shortened. This simple model leads to a good estimation of the largest contraction of the spacing for the roughest faces of the crystal. This is related to the fact that, the more asymmetric a cell is the greater is the force which acts on the ion to displace it towards the gravity centre of the cell. Complete calculations within the framework of the Finnis-Heine model should take into account the modification of the electronic density distribution, due to the displacement of the ions. Thus, full selfconsistency in 3By simple metals we understand the s-p bonded metals where the d and f electrons do not play an important role.

24

C H A P T E R 2. R E A L M E T A L S SURFACE

Fig. 2.4. Asymmetric Wigner-Seitz cells at a surface.

the calculation of surface charge density is required. The Finnis-Heine model is not applicable to explain relaxations at noble or transition metals surfaces. To account for the effect of localized d-electrons on the shortening of interplanar spacing at transition metal surfaces the first principles total energy calculations have to be performed (Fu et al., 1984; Bohnen and Ho, 1990). More recent experimental data on the surface relaxation revealed the existence of damped-oscillatory multilayer relaxations. Such variations (of the longitudinal type contraction-extension) of the interplanar spacing occur within a few (~ three) outer layers of low-index metal surfaces (cf. Table 2.2). On rough faces, besides the oscillatory relaxation, also displacements of the surface atomic layer in the direction parallel to the surface are observed (not necessarily accompanied by the change in the twodimensional lattice). For instance, for the Fe(211) face, the outer ionic layer is shifted relatively to the neighbouring one by 0.24/~. The active forces of the displacement tend to place the atoms in more symmetrical sites with respect to the position of atoms of the layer lying beneath, i.e. in those sites where a surface atom may have a highest number of first nearest neighbours. It is reasonable to expect that such sites correspond to the minimum of energy. The measured magnitudes of the oscillatory surface relaxation are in quite a good agreement with theoretical predictions within the electrostatic model accounting for the three-dimensional screening response

2.2. L A T T I C E R E L A X A T I O N

AND RECONSTRUCTION

25

Table 2.2 Multilayer relaxation at metM surfaces, di,i+l denotes the relaxation between i-th and (i + 1)-st layer.

Metal (face)

Relaxation [%]

Al(ll0) Ag(ll0) Cu(ll0) Fe(211)

Reference

d12

d23

d34

-9(-8.4) -9.5 -8.5 -10.5

+5 (4.9) +6 +2.3 +5.1

-2(-1.6) -3.5 -0.9 -1.7

a b c d

Data from: (a) Nielsen et al. (1982), Andersen et al. (1984). (b) Holub-Krappe et al. (1987). (c) Adams et al. (1983). (d) Sokolov et al. (1984).

of the electrons to the presence of relaxed ions (Barnett et al., 1983). Such calculations based on the total energy minimization, as well as the calculations within the effective-medium theory (Jacobsen et al., 1987) show that every ion in a metal will tend to occupy a position where the surrounding electron density allows to minimize its energy. Thus a surface ion in the topmost layer surrounded by lower than optimum electron density will be shifted inward. The more rough the plane is, the lower density surrounds ions and the larger displacement is required to reach the optimum density region. The shortening of the topmost layer will cause an increase of the electron density around the second-layer atoms above the optimum value. As a consequence, to compensate for this, the ions of second layer shift outwards expanding the distance between the second and the third layer (Stensgaard, 1992). An extension of the Finnis-Heine model, which enables to describe both the multilayer relaxation and the displacement of the outermost layer in parallel to the surface, was made by a semi-empirical electrostatic model of Jiang et al. (1986). Among the energy terms contributing to the total energy of the crystal it is mainly the electrostatic energy which is sensitive to the variation of the geometrical structure. Thus looking for the minimum of the total energy to a first approximation, we may limit ourselves to the minimum of the electrostatic energy of metal. Dividing the metal into electrically neutral layers (Fig. 2.5) of the width d, the electrostatic energy, E, of a semi-infinite crystal of metal may be written as a sum of the two terms (X)

E-

CEEI(R~)+ j=l

(X)

E j,k>j=l

E2(Rj,Rk)

(2.1)

CHAPTER 2. REAL METALS SURFACE

26

X

,nterna[ tayers

V

~

surface tayers

Fig. 2.5. Illustration of the method of summation over the surface cells.

where C is an empirical factor, and

2~Z2 j - [R~ l A - d(

E1 (R~) -

-~) d

(2.2)

is the energy of the interaction of the ions with the charge Z in the j - t h occcupancy sites of the two-dimensional lattice, with the same magnitude of electronic charge uniformly spread out in a layer of the width d. The vector R j determines the initial position of the reference atom in the j-th lattice; the R~- is its component perpendicular to the surface and R~ is the parallel component, respectively. A denotes the area of the surface unit cell of the plane lattice. The energy E1 attains its minimum when the ionic lattice coincides with the centre of the layer. The second term

E2(Rj, Rk) -

2rZ 2 A

exp[iKh. (R~ - R~) h

Kh (R-~ - Rf)]

(2.3)

Kh

represents the interaction between the j-th and the k-th layers. The reciprocal lattice vector Kh is equal to Kh -- 27r(hla~ + h2a~),

(2.4)

where a~ and a~, are the base translation vectors of the reciprocal lattice and h l and h2 are integer numbers. The summation in (2.3) proceeds over all the values of hi and h2 except hi = h2 = 0. Thus, the factor C appearing in (2.1) weighs the contribution of symmetrical force which is exerted on the ion and represented by the terms E1 in relation to the asymmetrical forces originating from the presence of the surface and represented by the E2 terms. The value of the factor C can be adjusted for each metal to fit best to the experimental data. At the values of Khd fixed by the geometrical structure of the bulk of metal, the structure of a relaxed surface is found

2.3. VIBRATIONS OF SURFACE ATOMS

27

by calculating the minimum of E with regard to the coordinates of the R j layer for all j's. The model, by using the empirical parameter once adjusted for a given metal, demonstrates a good accordance with experiment for almost all the observed surface relaxations. Another crystallographic problem being the subject of intensive studies is surface reconstruction. Geometrical structure of a reconstructed surface plane differs from the bulk ones by arrangement of atoms which are displaced less or more from their equilibrium positions or even removed in part. In other words, in a surface layer, a two-dimensional atomic lattice is formed which has different periodicity than the planes lying beneath, in the bulk. It has been shown by LEED analyses on the W(001) face of the crystal, which has the (1 • 1) structure at a temperature exceeding the room temperature, that upon cooling down the crystal to the room temperature the structure modifies into the (V~ • x/2)R 45 ~ The process is reversible and upon elevating the temperature the original (1 • 1) structure is recovered - so, we are dealing with a reversible phase transition. A similar type of structural transitions is observed also on the clean Mo(001). The structure (1 • 1) on the (001) and (011) faces of Ir, Pt and Au single crystals is metastable and can be reconstructed to the (1 • 5) and (1 x 2) structures. Besides these clean-surface reconstructions, the adsorption-induced reconstructions are also observed. In this case the periodicity of the surface net may be determined either by the periodicity of adsorbate atoms only, or by the periodicity of the reconstructed substrate. The adsorption of foreign atoms may also lead to the removal of already existing reconstructions on clean surface. It should be noted that the question why a crystal surface is subject to reconstruction or, in other words, why the reconstructed surfaces are energetically more favorable, can be answered only by the complete calculations of the surface electronic structure accounting for stresses existing on metals surface (cf. Chapter 8).

2.3

V i b r a t i o n s of surface a t o m s and the Debye temperature

As a result of the broken translational symmetry in the direction normal to the crystal surface and, consequently, the lowered number of neighbouring atoms, the energy of surface atoms is higher than that of the ones in the bulk of the crystal. This is manifested, among others, by an increased amplitude of thermal vibration of surface atoms. The experimental method of low energy (20-500 eV) electron diffraction (LEED) is particularly useful to investigate the vibration of surface atoms. The LEED pattern which, in fact, is a representation of the reciprocal lattice of the surface structure can be realized only when the Laue condition for diffraction (or the equivalent Bragg's condition) is fulfilled. Let Rn be the vector of a plane Bravais lattice in the form of equation (1.16), kp the wave vector of the incident electron and kr that of the scattered electron. Then the Laue condition has the form (kp - kr)" Rn = 2 r n

(2.5)

CHAPTER 2. REAL METALS SURFACE

28

where n is an integer number. The intensity of the diffraction beam diminishes with increasing temperature of the crystal. Simultaneously, the intensity of the diffusive background pattern is increased, which can be explained as being due to the atomic vibration which in turn implies the diffraction condition not to be fully satisfied. Assuming that the atoms perform harmonic vibration, the intensity I, can be written as I(T) -- Ioe -2W, (2.6) where I0 is the intensity of the specular-reflected beam from the rigid lattice at the temperature T = 0, and W is the Debye-Waller factor which can be written as

w - !k 2

(u

cos r

(2.7)

In the above expression k is the electron wavenumber, r is the angle of incidence of the beam, and (u 2) is the mean square of the displacement of an atom from its equilibrium position. Assuming that the vibration of atoms in the metal is harmonic the mean square atomic displacement is given by

3h2T (u2) = MkBO~'

(2.8)

where M is the atomic mass of the metal, kB is the Boltzmann constant, and eb is the Debye temperature, i.e. the temperature which is related to the highest possible frequency of the lattice vibration, equal to

Wm --(67r2n)l/3v and it can be defined as follows

ob-

(2.9)

(2.10)

h

where n is the atom concentration, v is the phase velocity (it is equal to the sound velocity for the acoustic branch in the dispersion relation of the w). Taking into account (2.7) and (2.8) in (2.6) we see that measurement of the temperature dependence of the intensity of the diffraction beam can provide information about the spatial anisotropy of thermal vibrations of atoms, whereas the Debye temperature can be determined from the slope of the plot In(I/Io) versus T. As it is seen from (2.8) the mean displacement of an atom from its equilibrium position, which is equal to the square root of the (u2), is proportional to x/~. On the other hand, from the theory of lattice vibration within the framework of the harmonic approximation follows that the amplitude of vibration can be expressed by the force constants, C, (proportionality constants) of interatomic interactions. Then, as in the case of the average potential energy of the classical harmonic oscillator, we have

(u l

~

I

-~ksT.

(2.11)

In order to estimate what is the effect of the surface on the amplitude of the vibration of atoms in a rough approximation we may assume that the force constant C

29

2.3. VIBRATIONS OF SURFACE ATOMS Table 2.3

Root-mean-square amplitudes of surface atom vibrations perpendicular to the surface (s) and in the bulk (b). 0 is the corresponding Debye temperature. Metal (face)

Ag Pb Pd Pt

(111) (111) (111) (111)

(U2_l_>1/2 [/~]

(u~• '/2 [A]

0.129 0.298 0.144 0.135

0.089 0.162 0.074 0.064

Os [K]

Ob [K]

155 49 140 111

225 90 274 234

Ref.

a b a c

Data from: (a) Goodman et al. (1968). (b) Goodman and Somorjai (1970). (c) Lyon and Somorjai (1966). is determined by a pairwise interaction. Let us notice that a surface atom possesses approximately half the number of nearest neighbours compared with an atom in the bulk. Therefore, the force constant Cs which is a characteristic of the surface atom vibration is equal to 1 C~ = ~Cb

(2 12)

where Cb is the force constant in the bulk of the crystal. This means that surface force constant is softer than the one in the bulk. Hence, it should be expected that amplitude of surface vibrations will be accordingly increased (u2•

"~ 2 ( u ~ ) ,

(2.13)

where the (_L) denotes vibration perpendicular to the surface (s). Since the mean square displacement is related to the Debye temperature by equation (2.11) one may expect that es ,--, o h / v 1 ,-,-,0.7lOb, (2.14) where Os can be defined similarly as in (2.10) and termed as the surface Debye temperature. The examples of measured root-mean-square (rms) amplitudes of surface atoms and the corresponding surface Debye temperatures at the (111) face of some fcc metals are given in Table 2.3. The surface Debye temperature, O s, for the perpendicular (_L) and parallel (11) vibrations of atoms depends on the orientation or on the Miller indices (hkl) of crystal plane. The ratios esv = --, =• II, (2.15) Ob determined theoretically are collected in Table 2.4 and compared with that measured for Ni. It is seen that theoretical predictions demonstrating only a weak dependence of

C H A P T E R 2. R E A L M E T A L S SURFACE

30

Table 2.4 Values of a ratio of the surface to the bulk Debye temperature for the vibrations of atoms perpendicular, T• and parallel, TII, to the surface for the fcc (110), (100) and (111) faces compared with the measurements for Ni (Mr6z et al., 1983).

Face

Theory

Experiment

AdW h

AdW ah

J

vD

Ni

T•

(III) (I00) (110)

0.733 0.731 0.739

0.536 0.570 0.529

0.572 0.562 0.562

I / x / 2 - 0.71 0.71 0.71

0.554 0.572 0.545

TII

(111) (100) (110)

0.878 0.683 0.816" 0.683 #

0.887 0.669 0.727* 0.447 #

0.986 0.931 0.948

0.94 0.87 0.87* 0.71 #

0.891 0.860 0.870

A d W - Allen and de Wette (1969); h - harmonic, a h - anharmonic, J - Jackson (1974), v D - van Delft (1991), *[001] direction; #[110] direction.

both the perpendicular and parallel component of surface Debye temperature on the crystal face are in a reasonable accordance with experiment, although neither surface relaxation nor reconstruction was taken into account. It follows from (2.8) that

r~ =
(2.16)

and since Ob > 08~ the following inequality is satisfied

> (U~ll>,

(2.17)

which is in agreement with measurements (of. Table 2.5). The Debye temperature is dependent both on the atomic structure of the surface and on the ratio of area of the surface to the volume of the crystal. Consequently, upon measuring the Debye temperature of smaller and smaller crystals we should have observed the size effect, 4 i.e. the dependence of the Debye temperature on the linear dimension, i = illS, of the sample (Wojciechowski, 1963) O(L) = 1 + ~const . O(L --+ c~) L 4 For a discussion of q u a n t u m size effect see Chap. 15.

(2.18)

2.3. V I B R A T I O N S OF SURFACE A T O M S

31

Table 2.5 The ratio of the surface and bulk root-mean-square (rms) amplitudes of an atom at the surface of Ni. The third column gives the ratio of rms amplitudes in the direction normal and parallel to the surface. Data from Grudniewski and Mr6z (1985).

Ni (face)

(u~• 1/2

(u~• 1/2

(U~A- )1/2

(U2[[/1/2

1.80 1.75 1.83

1.61 1.51 1.60

(111) (100) (110)

The above relation was confirmed experimentally by Viegers and Trooster (1977) (see also Matsushita and Matsubara (1978)). Finally, it should be noted that although the harmonic approximation works well for the bulk Debye's temperatures, for the surface atoms the anharmonic effects may become of great importance leading to substantial changes in the values of 08. In the above approximate analysis of surface vibration we have tacitly assumed that all observations are performed in the low temperature regime. Although at elevated temperatures, the anharmonic effects should be taken into account, it occurs that our approximate analysis enables to draw certain conclusion regarding the phenomenon of melting of metals. It is known that, according to the Lindemann criterion (Lindemann, 1910) a solid begins to melt when the amplitude of atomic vibrations is comparable with the nearest-neighbours distance R1N. In other words, a solid is melting if the ratio

(U2) 1/2

"7

=

(2.19)

R1N

reaches its critical value Vm. In Debye's approximation (when the Debye temperature is higher than the melting point of the crystal) using (2.8) we get the following relation between the critical value of '7m and O b "72 -

9h2Tm M k B O b2 R I2N

(2.20) "

To give a numerical example let us calculate "Tin for sodium. The melting point of sodium is 370 K and its Debye temperature equals 160 K. Taking the nearestneighbour distance R i g - 2r0, where r0 is the radius of a sphere surrounding each atom, we obtain "Tin ~ 0.12. Thus melting takes place when the root-mean-square amplitude of atomic vibration is roughly 0.12 of the nearest-neighbour distance. Since the amplitude of vibration of surface atoms is higher than that of the ones in the bulk of crystal, from formulae (2.19)-(2.20) it is clear that melting occurs first in

32

C H A P T E R 2. R E A L M E T A L S SURFACE

Pb (110) surface melting A

~- 25

(D >,, a O

o E

15

20

10

tD nr w .J Z t~J

uJ

n-15

O

<10

O rr Z

,<, s eD

(/) i

300

i

400

i

500

i

600

TEMPERATURE T (K) Fig. 2.6. Surface melting at Pb(110). The area of the surface peak of backscattered protons, expressed as the number of visible monolayers, as a function of temperature. The zero of the right-hand scale is defined at _~ 580 K. Redrawn with permission from van der Veen and Frenken (1986).

the outermost, surface layer and after it has been melted - in the subsequent atomic layers of the metal. In the case of surface melting the atoms are removed from their lattice sites what results in a reduction of the short-range order and the formation of a liquid-like surface region. The first demonstration of a reversible surface melting on Pb(100) was provided by the Rutherford-backscattering experiment of Frenken and van der Veen (1985). As shown in Fig. 2.6, at the temperature about 500 K, which is 100 K below the melting point of lead, the onset of partial disordering is observed. At ~580 K the outermost layer becomes fully disordered. Further increase of the temperature leads to a continuous increase in the thickness of disordered (molten) surface layer.

Chapter 3

Thermodynamics of the surface of crystal 3.1

Basic notions

The state of a given system is described I by its Helmholtz's free energy, F, which can be expressed by the internal energy, U, entropy, S, and temperature, T, of the system as follows F = U - TS, (3.1) where U and S are functions of temperature and of the so called external potentials Xi,

U = U ( T , xi),

S = S ( T , xi).

(3.2)

For a system with a variable number of particles the internal energy U alters in the following cases: (i) during the course of transferring the heat Q to the system; (ii) when the work L, is performed by the system; (iii) when the number of particles which constitute the system is changed. Thus, the first law of thermodynamics is formally expressed by U:-U1

=Q+L+M

(3.3)

where indices 1 and 2 denote the initial and final states of the system, respectively, and M is the energy required to transfer the unit mass. If the process of transition from state 1 to state 2 is infinitesimal, what may be denoted symbolically as: (1 --+ 2) = U --+ U + dU, 1For more details refer to Guggenheim (1967) and Kubo (1968).

33

34

CHAPTER 3. THERMODYNAMICS OF SURFACES

then

dU - dL + d Q + dM

(3.4)

where d means a change of a given quantity and not the complete differential of it. For example, the change of the external work performed over the system is equal to

d L = - p d V + E xjdXj J

(3.5)

where p denotes the pressure, V is the volume and Xj the external force of the j - t h type. 2 For example, when the potential energy of gravity is changed, dXj = dm, which is the mass change and xj is the gravitational potential Vgr; when the electrostatic potential energy alters, dXj -= dq which is the change of charge q and the xj is the electrostatic potential Vet. Analogously, also it can be written

dM= E#jdNj

(3.6)

where dNj denotes the change of the number of particles type of species j in the system, and #j is called the chemical potential. 3 Since there occurs the relation

dQ - T dS,

(3.7)

then, in the case of a system in which only the number of particles alters, we have

dU = - p d V + TdS + E #jdNj, J

(3.8)

or

d S-

dU p -~ + -~dY- E

#j ~dNJ"

(3.9)

J In the state of t h e r m o d y n a m i c equilibrium S = max.

(3.10)

Hence, if there are two systems, 1 and 2, which remain in t h e r m o d y n a m i c contact, in the state of equilibrium we have $1 + 5'2 = max,

(3.11)

2Note that in this Chapter, for a traditional reason, we will denote volume by V and not, as in the remaining chapters, by Ft. 3This name originates from Gibbs who used an analogy with the electric potential. Namely, in the case of the equal potentials on both ends of a conductor there is no current flow (i.e., the conductor is in electrical equilibrium), so in the case of mass transfer the equality of chemical potentials of the joth constituent imply the lack of the transfer of the constituent between the phases (which denotes the equilibrium of the chemical composition). When the free energy, F, of any homogeneous phase is regarded as a function of the independent variables T, V, N, the chemical potential ttj is the partial differential coefficient of F with respect to each Nj i.e., ttj - OF/ONj.

3.1. B A S I C N O T I O N S

35

so that (~(S1 --[- $2)

--

O,

(3.12) (~2 ($1 --[- ~2)

<

O,

where 5 denotes an infinitesimal variation. Let us assume that the joint system 1-2 which in the following will represent a crystal of metal (') and its saturated vapor (") is isolated, so that UI+U2

=

const,

VI+V2

=

const,

N~+N~'

=

coast,

(3.13a)

which implies that (Ul+V2)

=

0,

5(V1 + V2) = 5(N; + g;') -

O.

0,

(3.13b)

Based on conditions (3.13), equation (3.1) can be written in the form

(I.T11+.T221) (~UI+(Pl.~~.~ I 2p2)SVI_ E (#T1 PY)(~N;

(3.14)

2

where j refers to the species. Equation (3.14) implies the following possibilities for the thermodynamic equilibrium: 1. Thermal equilibrium (when systems 1 and 2 interact thermally) T1 = T2,

(3.15)

2. Mechanical equilibrium (for the systems interacting mechanically) 4 Pl -- P2,

(3.16)

3. Equilibrium of mass (when the systems exchange mass, i.e. particles) #j' - #j. "

(3 17)

In the state of thermodynamic equilibrium and in the case when the volume of the system is constant we have F = min. (3.18) 4The equality (3.16) holds only if surface dividing the systems 1 and 2 is planar. In the case when the b o u n d a r y surface between the phases is not a plane the condition can be written as Ap -Pl --P2 ~ 0 .

CHAPTER 3. THERMODYNAMICS OF SURFACES

36

~ ~

pnasel- vapor" ~

"~\~lXl

interface),~

N1+ N2- const, Vl+v2=const Fig. 3.1. A sketch of the two adjacent phases: saturated vapor and the solid body (crystal).

If two phases (in the case of interest saturated vapor of metallic atoms, phase 'prime', and the solid metal, phase 'bis') are bounded by each other, an interface region exists between the phases which occupies the area A (Fig. 3.1). The work needed to produce the interface can be expressed by the change of the free energy of the whole system F=aA, (3.19) where a denotes the work needed to form a surface of unit area. The quantity a is called the surface energy. The total free energy of the system consisting of phases (~) and (') and the interface is equal to F = F s + F' + F"

(3.20)

where F s is the surface free energy relative to the interface. We will come back to these problems in Section 3.4.

3.2

Equilibrium shape of crystalline particles

Now we will employ the condition of thermodynamic equilibrium of masses given by Eq. (3.17), and the general condition of the minimum free energy, Eq. (3.18), to discuss the equilibrium form of the single crystal. The problem of the crystal shape that emerges upon crystallization from the saturated vapor under conditions of thermodynamic equilibrium, has rather a long story and was attacked for the first time by Gibbs (1877). Then it was treated by P. Curie (1885) and Wulff (1901), and later by Landau (1950). A method of determination of the equilibrium form of crystal was proposed also by Stransky and Khaischev (compare Mutaftschiev, 1982). 5 5An excellent review of equilibrium and quasi-equilibrium properties of small particles was given by Nagaev (1992).

3.2. EQUILIBRIUM SHAPE OF CRYSTALLINE PARTICLES

37

The Gibbs-Wulff method is based on the conditions of thermodynamic equilibrium. The surface free energy, F s, of a single crystal bounded by the facets of the i-th kind 6 of surface areas Ai and characterized by their surface free energies ai, can be expressed by F ~ = ~ Aiai. (3.21) i The total free energy of the system saturated-vapor-crystal is given by formula (3.20). In the state of thermodynamic equilibrium the free energy has its minimum, therefore

3F

=

OF ~

~ai3Ai+ i +

T,V

-~OF T,u~V'+

5N' +

OV"

ON"

T,u

T,V

-- O.

(3.22)

Since the relations (3.13) are also fulfilled we can express the 5V" and 5N" by the 5V and 5N. Using in addition the known relations (3.1) and (3.4) we have

OF) - 0 ~ T,V

=

-

itJ

(3.23a)

-pJ

(3.23b)

where now j - ('), ("), and the equality (3.22) can be written in the form

5F = Z

a, hA, + ( i t ' - i t " ) h X ' - (p' - p " ) h V ' - O.

(3.24)

i Now we assume that the single crystal being formed from the vapor phase has a shape of convex polyhedron. The volume V' of such polyhedron can be expressed as the sum of the volumes of pyramids whose bases are the faces of the polyhedron with the area Ai, and the heights hi are perpendicular to the faces or to their extensions and are originated from the point O that joins the vertices of all the pyramids (Semenchenko, 1961). Thus the volume of the polyhedron is V' - -3 1Z

A, hi.

(3.25)

i

1(

Accordingly its infinitesimal change is

5V' - -~ ~

Aihhi + Z i

hihAi

)

9

(3.26)

i

An infinitesimal change of the volume 5V ~ can be also thought as a displacement of the surface area Ai by 5hi. This leads to another equality 7

5V' = ~

Aihh, + O(hA,).

(3.27)

i 6

Here the symbol i denotes a crystallographic face with given (all three) Miller indices.

70(in) denotes a function of order n which is smaller to higher order compared to preceding terms and may be omitted in further considerations.

CHAPTER 3. THERMODYNAMICS OF SURFACES

38

Comparing (3.27) and (3.26) and using again (3.27)we get 1

5V' = -~ ~ hiSAi.

(3.28)

i

Substituting

5V' from (3.28) into (3.24) finally we get E

ai- ~(p'-p")

5Ai + ( # ' - #")SN'- O.

(3.29)

Z

Since the variations 5Ai and 5N' are mutually independent, either of two terms on the left side of equation (3.29) must be equal to zero. Hence, the second term gives the condition of equilibrium of masses (Eq. (3.17)) while the first one gives following relation hi p, ai=--~(p'- ). (3.30) The surface of a single crystal is not a plane, therefore the inner pressures p' and p ' , in the phases (') and (") are not equal each to the other, s Their difference, for a given atomic species constituting both phases is constant. Therefore equation (3.30) can be rewritten in the form 2oi

hi

= const -- W,

i-

1,2,...

(3.31)

where W is the so called Wulff's constant; its physical meaning will be explained below. So in the state of thermodynamic equilibrium a given single crystal can be confined only by the facets (with their adequate Miller indices) that satisfy equality (3.31). This is the well-known Wulff's theorem. On the basis of the above theorem Wulff proposed a geometrical method of construction of the equilibrium form of a single crystal. This construction, illustrated in Fig. 3.2, may be formulated as follows. Through a fixed point O in the space, we draw straight lines which map crystallographic directions [hkl] of the given lattice. Starting from the point O, a set of vectors with their lengths proportional to a(hkl) is drawn along the straight lines. At the end of each vector the plane perpendicular to it is constructed. Then the inner envelope of the planes will determine the equilibrium form of the crystal (Fig. 3.2). The Wulff theorem is a realization of Gibbs' idea. According to the latter, the equilibrium form of a c r y s t a l - if the volume forces are not exerted - is determined by the minimum value of surface integral of the surface energy a(hkl) at a constant volume. Assuming the volume of the single crystal of interest to be V0, as in (3.25), we have A h, - 3v0. (3.32) i

Using equation (3.31) we can rewrite (3.32) in the form

E oiAi - 3WVo = const. i S C o m p a r e t h e f o o t n o t e on p. 35.

2

(3.33)

3.2. EQUILIBRIUM SHAPE OF CRYSTALLINE PARTICLES

\

\

\\

\

\ \

i\ \ ~\

39

\

Fig. 3.2. Two-dimensional Wulff's construction. It follows from (3.33) that at a fixed volume, V0, the sum of the products of the specific free energies by the surface areas of facets bounding the crystal in the state of thermodynamic equilibrium is invariant. Accordingly, if for a given crystallographic structure the fixed form of the single crystal is a regular polyhedron then, in general, the equilibrium form may be also bounded by facets which not necessarily belong to the polyhedron (i.e. the ones that would not be the facets of type 1). Such type 2 facets will have their total surface area of A2 - (A~ - A])a--L~, O'2

(3.34)

where A] denotes the surface area of the polyhedron facets when facets of type 2 have appeared. The form (3.33) of Wulff's law enables to interpret the physical meaning of the constant W. The left side of that equation is the work L s, needed to create the entire surface of the crystal. Thus accordingly W =

2L s 3Vo

2L = ~, vo

(3.35)

where L is the energy needed to create both the surface and the crystal which is confined by this surface. Thus it can be seen that the Wulff constant is simply related to the total energy required for crystal formation. A mathematically elegant method to determine the equilibrium shape of a crystal, on the basis of the Gibbs' idea, was given by Landau (1950) (see also Landau and

40

C H A P T E R 3. T H E R M O D Y N A M I C S OF SURFACES

Lifshitz (1970)). The surface free energy F s can be written in the form

F ~ - / a dA.

(3.36)

Let us assume that the equation of the crystal surface can be written in the general form z =

y).

(3.37)

Then

dA = V/1 + p2 + q2 dx dy, where p =

Hence

Oz Ox'

q =

Oz ay

(3.38)

(3.39)

f

F s - ~ a(p, q)V/1 + p2 + q2 dx dy.

(3.40)

The equilibrium form of a crystal is determined by the minimum of F s at a constant volume P

Vo = J z dx dy - const.

(3.41)

The above variational condition leads to the differential equation whose solution yields the envelope of the crystal in the state of equilibrium. The envelope satisfies Wulff's law and, moreover, its shape provides additional information about the crystal in the equilibrium state: the crystal is bounded not only by the fiat faces with low Miller indices but also by the small ones having high indices, whence the crystal is apparently rounded off. The conclusion given by Landau is as follows: "The equilibrium form (of a crystal) will be determined by few plain regions which, however, do not intersect at some angles but are connected by rounded surface regions." The Wulff theorem has essential meaning in the case of microscopic single crystals, i.e. the ones whose diameter is of the order of magnitude 10-7-10 -s cm. When the size of a crystal is macroscopic (~ 10 -5 cm), it cannot be in general believed that its form is in accordance with Wulff's theorem (Meyer, 1968; Linford, 1973). Atomic microcrystals have a reasonable chance to attain the equilibrium shape. In the case of large single crystals the time needed to obtain the equilibrium form is practically infinite. The beautiful experimental verification of the Wulff theorem was given by Heyraud and Metois (1983). These authors demonstrated that for micron sized crystalline particles equilibrated with their own vapor the shape of the particle was consistent with a Wulff construction independent of the actual size. If the orientation of the surface confining a crystal differs negligibly from that of the most stable plane, corresponding to the singularity in the a-surface (a closepacked plane), then we obtain a crystal surface consisting of several terraces of oneor two-atom height which continue the orientation of the close-packed, stable plane (Fig. 3.3). Since the orientation of such plane is vicinal to that of a close-packed (low-index) plane, such planes are called vicinal planes. They are characterized by high Miller indices.

3.3.

MICROSCOPIC

SINGLE

41

CRYSTALS

s

s

Fig. 3.3. Terraces at the solid surface. Let a be the angle constituted by a vicinal plane with the close-packed one. Then the surface energy a ( a ) of the terraced surface can be expressed by the magnitude of the surface energy of the close-packed plane, a(0), and by the contribution s of every terrace to the energy,

=

+

I l,

a

-

where a is the lattice parameter. Vicinal planes can be depicted as the fragments a of the surface near singularities (Nagaev, 1992).

3.3

Thermodynamics

of microscopic single crystals

Defay and Prigogine (1966) considering the equilibrium form of microscopic single crystals proved that, upon reducing mechanical effects in a macroscopic crystal to the hydrostatic pressure, the following relation for small crystals can be obtained 2r

A p = Pc - Pg =

hi '

(3.42)

where Pc is the pressure in the crystalline phase, P9 is that in the gaseous phase, and hi is the height of pyramid whose base is a crystallographic facet having the surface energy ai (similarly as in the previous Section). On the other hand, if the two phases, p r i m e (fluid) and bis (gaseous), remain in the state of thermodynamic equilibrium and are separated by a surface with the principal Gaussian radii of curvature pl and p2, then the well-known Laplace formula holds

(1 1)

Ap=p'-p"=a

--+ p,

(3.43)

Accordingly, if we consider a spherical drop of radius r - pl = p2, then from (3.43) follows that 2a ~p = --. (3.44) r

42

CHAPTER 3. THERMODYNAMICS OF SURFACES

Putting in relation (3.42) hi = r, we see that for the two different phase systems, gasliquid-drop and gas-small-crystal, remaining in equilibrium the pressure difference is expressed by the same relationship. This justifies the utility of the liquid-drop model in describing small crystals. From relation (3.42) it follows that, when hi = r, a very high pressure exists inside microscopic crystals of metal. Indeed, if we assume the magnitudes of the surface energy a = 1 0 2 - 104 atm as typical for metals and the 2h to be equal to 20-100 .&, then we obtain Ap = 102 - 103 dyne/cm. Such a high pressure causes the change in the lattice parameter at the surface of small crystals. If pr denotes the pressure of saturated vapor remaining in equilibrium with a droplet of radius r, and p ~ is the pressure of saturated vapor over an infinite plane surface (i.e., the conventional pressure of saturated vapor), then it can be shown that (2Via) Pr-p~exp

rkBT

'

(3.45)

where Vi denotes the specific volume per molecule of the liquid (V~ = Opi/Op). It follows from formula (3.45) that the pressure inside the drop (of the microscopic crystal) increases with its decreasing size (Fig. 3.4). The increased pressure inside the crystal will lead to a contraction of the lattice parameter with the decreasing particle size. In general, the lattice parameters of metallic particles are reported to contract although in some cases no changes are observed (Marks, 1994). The change in the lattice parameter with the size of the aluminum crystallite is displayed in Fig. 3.5. This is a illustration of the classical size effect, that is the effect of the specimen size on a physical quantity. The measurements of change in the lattice parameter as a function of a particle size allow to determine, in connection with Eq. (3.44), another important surface q u a n t i t y - surface stress which is discussed below.

~r

Fig. 3.4. Dependence of the pressure, pr, inside the microscopic crystal on its size.

43

3.3. M I C R O S C O P I C SINGLE C R Y S T A L S

0,406

bulk materia[

...--...

E

c

0.404

0 0

L..

(D

E

12I 0 121.. 0

0

0.402

0 0

O0 0

0

0

o fio O 0

0

0

0

4oO

0.400 O0

o

0

0.398

I

0

I

10 20 particle diameter (nm)

30

Fig. 3.5. Dependence of the m e a s u r e d lattice p a r a m e t e r on the size of particle. w i t h permission from Woltersdorf et al. (1981).

Redrawn

Another size effect of this type namely, the lowering of the melting temperature of small crystallites, 9 was predicted by Pawlow in 1909 (Pawlow, 1909). This phenomenon can be explained as follows. Let us consider a one-component system of the crystal remaining in the state of equilibrium with its saturated vapor. From equation (3.8) follows t t J - U j _ TSJ g J + pJV j ' J = ( ' ) ' )(,," (3.46) -

-

Using the condition of mass equilibrium (3.17) and relations (3.30) and (3.31) we get

#t =

U' - T S t + p'V' U t - TS' + pttVt 2aiV t Ni = Ni ~ h i N t.

(3.47)

The first term on the rhs of (3.47) represents the #' reduced to the same pressure p,t as that of phase (tt). Differentiating this equation with respect to temperature, pressure and the hi, and taking into account the relation (3.30), we obtain -~p ] T d P -[- (

0#" 0#" d T - 2aiV' dhi - (--~-p ) T dP + (--~-~-jp

(3.48)

where v' - V ' / N denotes the specific volume per atom (or molecule) of the phase ('). 9The effect, because of experimental difficulties, was observed as late as in 1954 by Takagi (1954).

C H A P T E R 3. T H E R M O D Y N A M I C S OF SURFACES

44 Since we have

=

v,

(3.49a)

=

-s,

(3.49b)

and s' - s" =

Tq '

(3.50)

where s - S I N and q is the heat (energy) of transition from the (') to (") phase. From the equality (3.48) we obtain the Clausius-Clapeyron equation generalized to the case of two phases coexisting in the state of equilibrium, and one of the phases is characterized by an excess energy (surface energy) of formation of a stable unit area of the interface between the phases, i.e.

(v' - v " ) d p - 2aiv' h~ dhi - ( s ' - s")dT = - TdT.

(3.51)

From the relation (3.51) we get

and

dp q 2aiv ~ dhi d--T = - T ( v ' - v") + h i2 (v' - v" ) dT

(3.52)

dT T(v' - v') 2Taiv t dhi = ~ dp q h2q dp'

(3.53)

-

-

At a fixed pressure p, both (3.52) and (3.53) yield

( d~hi)

p

- 2Taivt , h~q

(3.54)

which gives the dependence of the temperature of phase transition on the size of the sample. Assuming that q and v ~ are only weakly dependent on temperature, i.e. in the case when q _~ const and v ~ ~- const, we can calculate from (3.54), the dependence of the melting temperature Tm on the size hi of the crystal. Namely,

f~.

~ dT _ 2aiv ~ / i ~ dhi (o0) T q hi

(3.55)

whence ' Tin(hi) - Tm(c~) exp ( - 2aiV')hiq ~

(3.56)

where Tm(oC) denotes the temperature of melting point of a large sample. It follows from (3.56) that under the constant external pressure, the melting temperature decreases with decreasing size of a crystal. The experimental data illustrating a lowering of the melting point of gold are given in Fig. 3.6.

45

3.4. SURFACE ENERGY, TENSION AND STRESS

1400 butk --,...-..

,,,,. ~ID A / ~ ~''-' A_

1200

A

O I,.. Q~ f3.

E

(D

cZD c"

1000

(P

800

9

I

0

I

1

100

I

I

200

1

300

particle diometer (~,) Fig. 3.6. Dependence of the melting temperature on the particle size. Redrawn with permission from Borel (1981).

3.4

S u r f a c e energy surface stress

surface t e n s i o n a n d

In the literature the terms surface energy, a, and surface tension, V, are source of some confusion and are often used interchangeably. This is connected with a fact that for liquids the two quantities are equal. The process of formation of a surface of solid can be divided into two stages. First, the crystal is cleaved in two parts, which results in obtaining two new surfaces. The energy required to create unit area of a new surface is the surface energy. On cleaving the crystal the work needed to break a certain number of atomic bonds is performed. The breaking of the atomic bonds leads to the appearance of stresses near the surface, which cause the displacement of surface atoms to new positions. In order to attain that the new positions were equilibrium ones, some forces should be applied to the surface edges, that would equilibrate the action of the stresses existing within the crystal. The surface force acting on the unit length will be called a surface stress and denoted by the g. In the case of deformation or cleavage of a liquid drop the atoms or molecules inside the liquid change their positions and a new surface is formed. As the result, both stages of surface formation in a liquid merge into the one and the work performed during such process is given by formula (3.19). In order to distinguish between the surface energy, or, and the surface stress, g, let us try at first to determine the relation between a and the thermodynamic quanti-

46

CHAPTER

3.

THERMODYNAMICS

sotid

OF SURFACES

I I

vapor I ~ - - -

x

Fig. 3.7. Plot of the density of the one-component system (crystal-vapor) versus distance.

ties that characterize the system, in particular the ones characterizing the boundary between the phases. In the most common approach given by Gibbs, the phases in the bulk of crystal are regarded to be spatially homogeneous up to the interface. For a one-component system which consists of a solid remaining in equilibrium with its vapor, the distribution of density is presented in Fig. 3.7. The quantity X, which may represent e.g., S, V or N and is characteristic of the system consisting of two volume phases (') and (") alters on the surface by a certain v o l u m e c o n t r i b u t i o n X s, called the s u r f a c e e x c e s s to the quantity, so that X = X' + X" + X s. (3.57) Combining (3.4) and (3.8) we see that in the process for which the volume is not altered the surface excess of the internal energy d U s is equal to dU s = TdS s + #dN s + odA.

(3.58)

Integrating equation (3.58) we obtain U s = TS 8 + tiN s + aA.

(3.59)

For the interface, similarly to (3.1), we can define the surface free energy F s __ U s _ T S s.

(3.60)

Then equation (3.59) will take the form F s = tiN s + aA.

Hence O" -- f s __

ttF,

(3.61)

(3.62)

47

3.4. SURFACE ENERGY, TENSION AND STRESS

solid

I 1

vapor

surface', I I

0

~X

Fig. 3.8. Schematic outline of the surface phase, where the surface excess of particles equals zero. where f~ is the specific surface free energy (F~/A), and F is the number of excess particles per unit area of surface. The a defined in this manner does not depend on the choice of the dividing plane. As it is seen from (3.62), in most cases we have cr ~ f t . However, since the position of the phase boundary plane (i.e., of the surface phase) may be chosen arbitrarily, therefore in the one-component system of interest, which consists of particles of one type, the position of the boundary plane can be chosen in such a way that the surface excess of particles N 8 be equal to zero (Fig. 3.8). In this case we have a = f t . For this reason, sometimes also a is called the surface (specific) free energy. Moreover, in this case the change of the surface excess of the free energy can be written as dF ~ = - S ~ d T + adA. (3.63) Hence, taking into account (3.61), we get OT ,] A

(

- A \Oa

"

(3.64)

Therefore the surface part of the internal energy can be written in the form (3.65) It follows from equation (3.64) that S 8 could be determined by measuring the derivative Oa/OT. Since a(T) decreases with increasing T, the derivative Oa/OT is negative and S 8 increases with temperature. For most liquids the S ~ is a positive constant in a wide range of temperatures. This constant expressed in entropy units per molecule is close to the double value of the Boltzmann constant"

S s "~ 2kB.

(3.66)

From the above result, known as the EStv5s law, follows that disorder on a surface is greater than in the bulk of crystal.

48

CHAPTER

3.

THERMODYNAMICS

OF S U R F A C E S

Now consider the situation when tangential forces act on the already existing surface and cause stretching or contraction of the surface area by dA. In the general case of the anisotropic solid, the surface stress is strongly direction dependent. Let us consider a flat surface of a crystal with the area A and surface energy, a. In the theory of elasticity, the surface deformation can be expressed by an infinitesimal change d A of the elastic strain tensor eij d A = A deij 5ij,

(i, j = y, z),

(3.67)

where 5ij is the Kronecker delta function and the i, j numerate directions in the plane of the surface. Here we have adopted the usual convention that repeated index denotes summation over this index. The work which is needed to enlarge the surface area can be expressed as the work performed against the surface stress, gij, in an infinitesimal strain deij dL = Agij deij. (3.68) This work is equal to the increase in surface free energy and can be written in the form d(aA) - adA +

A d a - aASij deij + A ~ 0 ~ ) dQj. \ u~ij /

(3.69)

Equating (3.69) to (3.68) we get for any (arbitrary) component dc~j, the stress g{j, per unit length of the surface edge considered, acting along the direction j, perpendicular to the direction i 10(aA) (3.70) gij = A Oe~y " The surface stress, gij, and the stress-induced elastic strain, eij, are tensors related to each other by Oa gij = a~ij + - - . (3.71)

Os

Equation (3.71) determines the relation between the surface energy, a, and the surface stress, gij, for anisotropic solid. Positive gij means that the surface would prefer to contract within the surface plane. A more detailed derivation of equation (3.71) can be found in the work by Linford (1973). The surface tension, ~/, can be defined as an average of diagonal components of gij , i.e. 1 ~ / - -~(guu + gzz) (3.72) where direction x is assumed normal to the surface (there are no stresses in the xdirection at a fully relaxed surface). Surface tension is a scalar measure of the surface stress thus, it is given in the units of force per unit length. In the case of an isotropic solid the stress tensor gij has only diagonal components, i.e. gyy = gzz = g. (3.73) Likewise, from amongst the components of the elastic strain tensor eij - dlj/li, where li and lj are the sides of the surface element of interest, only the components eyy -

3.4.

SURFACE ENERGY,

TENSION AND STRESS

49

dly/ly and ez~ - dl~/lz will survive. Since lyl~ - A, then we have

cyy + e ~ =

lzdly + lydlz dA A = A"

(3.74)

Taking into account (3.74) and (3.73), in the case of isotropic surface, equation (3.71) will take the form do g = a + A d---~ . (3.75) When a liquid surface is stretched isothermally, additional atoms move out from the bulk into surface positions and the structure of the surface and hence a remain constant. Thus the last term in (3.75) and/or in (3.71) vanish and we set

For a rough estimation of the magnitude of the surface energy, a, we may make use of a definition which determines surface energy as the energy needed to cleave a crystal. This implies the breaking of a certain number of bonds of every surface atom with its neighbors. Thus, for the typical concentration of surface atoms, equal to about 1015 cm -2, and assuming the ratio of the number of broken surface bonds to that of the volume bonds to be ~ 0.3, in a metal whose binding energy in the bulk is 3 eV per one bond, we obtain a value of 0.3 • 3 • 1015 eV/cm 2 ~ 1400 erg/cm 2. This yields the correct order of magnitude of the measured surface energy. We will come back to the discussion of surface energy and of surface stress in Chap. 8.

This Page Intentionally Left Blank

P a r t II

Quantum Theory of Metal Surface

51

This Page Intentionally Left Blank

Chapter 4

E l e c t r o n s in m e t a l s 4.1

Sommerfeld's model

In 1928 Sommerfeld has proposed a simple model describing the motion of electrons confined in a metal. The great electrical conductivity of metals, as well as their high reflectivity suggest that they contain a large number of electrons which can move freely in the crystal skeleton build up from the positively charged ion-cores. This means, that the real electrostatic potential, V(r), which is periodic inside the crystal, may be replaced by a constant one. Moreover, since electrons are trapped by the metallic ions inside the metal, Sommerfeld (1928) represented this bounding by an infinite potential barrier placed at the metal surface. Thus, each of the free electrons of the metal (hereafter the region occupied by the metal will be denoted as region D) is described by the Schr5dinger equation of the form

h2 2mV2r + E r

-

0,

(4.1)

where it is assumed that the crystal potential V(r) = const = 0 in D, and V(r) = cr outside D. Taking the region D as the rectangular box of sides Lz, Ly and Lz, and of volume ~ = LxLyLz the normalized solution of Eq. (4.1) inside D, fulfilling the Dirichlet boundary condition r = 0 (4.2) on the surface S of the region D, has the form

r

(

= LxLyLz

)

1/2

(nzTrx) sin \(nyTry~ (nzTrz~]" -~y ]sin\gz

s i n \ L~

(4.3)

The eigenvalues, or permitted discrete values of E, are given by

2mE

_

7T 2

I n x2 53

ny2 + n z2 1

(4.4)

54

CHAPTER 4. ELECTRONS IN METALS

or by -

+

2+

2 =l

k

12

(4.5)

where kx, ky and kz are the components of vector k of length: k ~ - un~ L-T'

-

(4.6)

y' z)

with n~ = 1, 2, 3 .... The k~ can be interpreted as quantum numbers which define the state of the electron. The ground state energy (lowest energy) is equal to h271"2 / 1 1 1) 2m ~ + L--~+ ~z2

E(1, 1, 1 ) -

"

(4.7)

Taking L~ = Ly = L~ = L = 1 cm, we get E(1, 1, 1) _~ 10 -15 eV. Because the energy needed for transferring of one of the free electrons from a metal to infinity is of the order of several eV, one can see that for practical purposes we may put E(1, 1, 1) = 0. For a macroscopic sample (L __ 1 cm) the energy spectrum which follows from (4.4) is quasi continuous because the maximum energetic distance between two successive levels of this spectrum is of the order of 10 -15 eV. On the other hand the components of k-vectors defined by (4.6) can be represented by a lattice of points in a k-space. Thus in calculation of any quantity requiring summation over wave vector k one can replace a summation over k by integration, according to the transformation

2L3 /

E

--+ (27r)3

k

d3k'

(4.8)

where ( 2 r / L ) 3 is a volume occupied in momentum space by each k-state. The factor of two appearing in the nominator comes from the spin. According to the Pauli principle two one-electron states with opposite spins can be assigned to every k-point. In the ground state (at temperature T - 0 K), each of the states up to a maximum kmax is occupied by two electrons. Thus the points in the k-space fill a sphere of radius kmaz, called Fermi sphere. Consequently the maximum value kmax is called Fermi momentum or Fermi wave number and is denoted as kF. Hence for the system of N free electrons in the volume ~ = L 3, we have

N-

(47r/3)k3 2 (27r/L)3 ,

(4.9)

The average electronic density, ~, expressed in terms of kF is _

N

n-

f~ = 37r2.

k3

(4.10)

The energy of the highest occupied state at T - 0 K, is called Fermi energy

h2 E F - ~ m k 2 --

~2 ~m(37r2n)2/3

(4.11)

4.1. S O M M E R F E L D 'S M O D E L

55

The average kinetic energy per electron, ts, is given by t ~ =_

-~m k 2d3 k

d3 k =

Let us introduce now A(V), the number of eigenvalues (4.4) of equation (4.1)less than the bound V at the boundary condition (4.2), A(V) = E

1,

(4.13)

where

N = A [~/(Emax)]. (4.14) 2 One can expect that a priori A(7) has an asymptotic expansion (Balian and Bloch, 1970) given by -

-

~3/2

A(7) -

6~ 2 + a S v + bLO(Tn),

(4.15)

where a, b and 77 < 1 are constants and L is proportional to a typical linear dimension of the region D occupied by a metallic sample. Substituting this asymptotic expansion into (4.14), we obtain the following equation for 7: ~/3/2

S

L

= 3r 2 + 2 a ~ 7 + 2b O(Tn).

(4.16)

This equation shows that V, and thus obviously also EF, depend not only on the electron density but also on the size of the sample (cf. Chap. 15). In the bulk limit, as gt --+ c~, the last two terms on the rhs of Eq. (4.16) disappear and it turns into the familiar relation (4.10). Many physical quantities may be expressed in terms of the characteristic radius, called electron density parameter, r0, which gives the mean interelectronic distance in a metal and is defined by 47r ft - ~ r 30. --N(4.17) It is convenient to work with the quantity r~ expressed in units of the Bohr radius, ao (this is atomic unit of length called bohr, cf. Appendix A): r0 r~=--= ao

( 3 / 4 ~ ) 1/3 h2/me 2 "

(418)

The r s is called the Wigner-Seitz density parameter or Wigner-Seitz radius. Making use of Eqs (4.10-4.11), kF and EF can be expressed in terms of rs through the relations: (~_~) 1/31 ~F

--

--

rs

1.92 --

rs

~

a.u.

(4.19) EF

=

me4 ( 9 ~ ) 2 / 3 1 1.84 2h 2 7 r-~ = r 2 '

hartrees.

56

CHAPTER 4. ELECTRONS IN METALS

Similarly, the average kinetic energy of the noninteracting electron gas may be expressed as 1.105 t s = r2 hartrees. (4.20) In the metallic range of electron densities ~, the values of r8 and EF (See Table 4.1) will fall within the ranges: 1.6 _~ r~ < 6, 19 > EF >_ 1.5 eV. A word of caution should be said here. The term Wigner-Seitz radius, as applied to r s parameter, is somewhat misleading. In fact the Wigner-Seitz radius denotes the radius of the sphericalized Wigner-Seitz cell (compare Sec. 1.1), and in order to distinguish it from r8 we will denote it by rws. Thus the volume of the Wigner-Seitz cell is 47rr3 ~t -~- w s - --Noo' (4.21) where f~ is the metal volume and No the number of ions of valency Z. The number of electrons in a metal N - ZNo. Hence we see immediately that

rWS_ (._~)1/3 3 (__~0) 1/3 __ (~____~)1/3 (Z)

1/3

_zl/3rs.

(4.22)

It is clear that only for monovalent metals the rs and rws are equal. Since the distribution of energy states may be regarded as almost continuous, we may define the density of states Af(E) as the number of energy states lying between the energies E and E + dE. In the bulk limit (~t -+ co), taking into account (4.4), we obtain from (4.14)-(4.15),

Jkf(E)-2dA(E)-d~- 37r 2~t (2m)3/2dE3/2_~

dE

- 27r2~ \12m~3/2E1/2]h2

.

(4.23)

Since the occupation number of the state of energy E, is given by the Fermi-Dirac distribution function I(E) = {1 + exp[~(E - ~)]}-1, (4.24) then according to (4.23), the concentration of free electrons is given by the following equation

1 {2m'~3/2 f ~

- ~

~

)

E1/2dE

(4.25)

@(E-~)+I,

Here ~ = 1/(kBT), ks is the Boltzmann constant and T, temperature. The chemical potential, # ' fixed by the relation (4.11) is equal to EF 1 -- ~ (~_2_) EF of functions f(E) and d~/dE are shown in Fig. 4.1.

" The

graphs

4.1. S O M M E R F E L D ' S M O D E L

57

Table 4.1 Parameters of the free electron gas of some metals. Z is the valence, ~ is the electronic concentration, r s3 = 3/(4~'~) is the electron density parameter, EF is the Fermi energy, and kF = (31r2~,)1/3 is the Fermi wave number.

Metal

Z

Li Na K Rb Cs Cu Ag Au Be Mg Ca Sr Ba Fe Mn Zn Cd Hg A1 Ga In T1 Sn Pb Bi Sb Mo W

1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 4 4 5 5 6 6

~ [10 22 cm-3l

4.57 2.54 1.32 1.13 0.90 8.47 5.86 5.90 24.7 8.67 4.61 3.55 3.15 17.0 16.5 13.2 9.27 8.65 18.1 15.4 11.5 10.5 14.8 13.2 14.1 16.5 38.4 38.1

r~ [a.u.]

kg [10 8 cm -1]

EF [eV]

3.28 3.99 4.96 5.23 5.63 2.67 3.02 3.01 1.87 2.65 3.27 3.57 3.71 2.12 2.14 2.30 2.59 2.65 2.07 2.19 2.41 2.48 2.22 2.30 2.25 2.14 1.61 1.62

1.11 0.91 0.73 0.69 0.64 1.36 1.20 1.21 1.94 1.37 1.11 1.02 0.98 1.71 1.70 1.58 1.40 1.37 1.75 1.66 1.51 1.46 1.64 1.58 1.61 1.70 2.25 2.24

4.74 3.15 2.04 1.83 1.58 7.00 5.49 5.53 14.3 7.14 4.69 3.93 3.64 11.1 10.9 9.47 7.47 7.13 11.7 10.4 8.63 8.15 10.2 9.47 9.90 10.9 19.3 19.1

58

C H A P T E R 4. E L E C T R O N S I N M E T A L S

f(E)

I~k T I

~ksT l

l.IJ

1.0 \\

C 7O II

0.5

UJ Z |

EF

~

~

0

E

EF

{o}

Fig. 4.1.

....-

E

(b)

Plots of the Fermi-Dirac distribution function, f(E), and the density of states

dn/dE.

4.2

I n f i n i t e a n d finite p o t e n t i a l well

In the Sommerfeld model the free electrons are moving in the box limited by infinite potential barrier, which implies that they are described by the wave function of the form (4.3). The simplest way of accounting for the presence of a surface of metallic sample is to replace the three dimensional box by a slab of finite thickness L in xdirection, and extended infinitely in the y- and z-directions. Then we may assume that potential energy V(r) has the following form

V(r) = { O, (X),

for-c~_y,z___c~,

0_x___L, (4.26)

elsewhere,

i.e., we set infinitely high potential walls at x - 0 and x - L. For such an infinite barrier model (IBM) we can replace the actual infinite set of wave functions for free electron model by a finite number, imposing the Born-vonK a r m a n boundary conditions, characterized by a period L in the y- and z-directions, and fixed boundary at x - L (Bardeen, 1936; Sugiyama, 1960). Consequently, the appropriate wave functions will have the form (4.27)

Ck(X, y, Z) = Ae i(k~y+k'~) sin(k~x), with the normalization constant A = x/2x/~, and

ky

=

27my L'

kz - ~2~rnz L'

ny '

= 0, •

+2,..., (4.28)

kx

=

7rnx

L '

nx-

1,2,3 ....

4.2. INFINITE AND FINITE POTENTIAL WELL

3ettium ~dge, n/5

.-" 2 ; " ~ ' x

~ , ~ _ ~

59

.

1.0

_

\

"1

0.5

:',,1\ ,I

I

I

I

I

-14-12 -10 -8

-6

I

-4.

-2

0

2 2kFX

Fig. 4.2. Electron density profile for the infinite (dashed line) and finite-square-potential barrier model of metal surface.

The electron density distribution is given by a sum over occupied states in the k-space, occ

n(r) = ~

I Ck(r)12 9

(4.29)

k

To calculate the electron density at the surface it is useful to replace the summation over k by integration according to the prescription (4.8). Owing to the cylindrical symmetry of the problem the integral over k can be written in the form

2L3/d3 k = (27r)2 2L3j~okF (k2F- k2)dkz.

(27r)3

(4.30)

Thus, substituting the wave functions (4.27) into (4.29) we find =

n(=)

=

lf0k~(k2F - k 2) sin2(kzx)dkx

lr---ff

3 cos X 1+

X2

-

3 sin X ) X3 + . . . .

(4.31)

where fi is given by Eq. (4.10) and X = 2kFx. The electron density distribution (profile) near impenetrable infinite-barrier is illustrated in Fig. 4.2 (broken line). From this figure we observe that the density varies from its value ~ in the bulk metal and when sin(2kFx) = 0, i.e. over the distance lr/2kf, to zero in the location of the barrier. Deep in the interior of a metal density oscillates with a wavelength 7r/kF. This form of oscillations is called the Friedel oscillations.

60

C H A P T E R 4. E L E C T R O N S I N M E T A L S

W

Fig. 4.3. The finite-square-potential well. In the case of a real metal, thermionic or photo-emission of electrons is observed. Thus, a model of metal in order to be realistic, must give electrons possibility to leave a metal. To allow for this the motion of the free electrons of a metal can be simulated by the motion of electrons in the finite potential well (Fig. 4.3): 0,

in the region O1(~) - (0 _~ ~ _~ L~),

W,

in the region D2(~) - (0 > ~ > L~),

V(~) =

(4.32)

where ~ = x, y, z. For such a finite-square-potential barrier, the SchrSdinger equation can be separated into two sets of three equations for each of the regions D1 and D2: h 2 d2r 2m

d~ 2

h 2 d2r 2m

~- E~r

= 0,

+ (E~ - W)r

d~ 2

- 0,

(region D1),

(4.33a)

(region D2),

(4.33b)

with ~-~'~E~ - E. The proper wave functions are (Landau and Lifshitz, 1965)" _

where k~ given by

=

~ A sin(k~ + 5~),

in O1(~),

( Ce -~r , in D2 (~), / a~ = ~/2m(W-Ei) h~ , A and C being constants i•E 2

4,

(4.34) The phase shift 5~

5~ - arcsin[~ik~/(2mW) 1/2]

(4.35)

is determined from the requirement of continuity of the wave functions. The eigenvalues E~ of Eq. (4.33a) are given by the roots of the equation k~L~ = n ~ -

2~

(4.36)

61

4.2. INFINITE A N D FINITE P O T E N T I A L WELL Since we consider the case E~ E~ <_ EF < W, we can expand arcsin(~-~--~)1/2 series

x3 1.2 ~ arcsin(x) - x + ~ 4 2 . 4x. 5S +

....

into the (4.37)

Neglecting all but the first term in the expansion, from (4.36), we obtain h 27r2n~ Er = a(nr W) 2mn~ '

(4.38)

where 2h 1 + L~v/2m W

a(L~, W ) =

4h "~ 1 - L~V'2mW'

(4.39)

and n~ - 1, 2, 3, .... Putting Lx = Ly - Lz = L, i.e. replacing the rectangular shape of

sample by a cub~ for

have (Davydov, 1965)

<

h2~-2 2+ 2 E(L, W) = a(L, W ) 2 m L 2 (n 2 + ny nz),

(4.40)

where a(L, W) =

f (

1,

for W ~ oo, L = co nst.,

0,

for W --+

8h 2 m L 2"

(4.41)

From (4.40) and (4.41) follows that for the finite-square-potential barrier (W < oe) the Fermi energy, EF(L, W), is given by the equation

EF(L, W) - a(L, W ) E ~ ,

(4.42)

where 0 < a < 1 and E~ is determined by (4.11). We see, therefore, that in the case when the motion of an electron takes place inside the space bounded by the finitesquare-potential barrier, its energy levels, and thus also the Fermi level, depend on the size L of a sample. When L < 50/~, the so called quantum size effect appears. A more detailed discussion of this effect is given in Chapter 15. Usually, in surface physics applications, the three-dimensional finite-potential well is replaced by a potential well finite in the direction perpendicular to the surface only. In the other two directions the periodic boundary conditions are imposed (compare the next section.) At the absolute zero of temperature all energy states between E1 and EF are occupied. The relation between the barrier height W at the edge of a metal and the energetic width of region of the occupied energy levels is schematically illustrated in Fig. 4.4. The quantity q~ - W - EF, (4.43) is called the electron work function and is a measure of energy needed to remove an electron from a metal in the ground state to infinity i.e.,

= EN -- EN-1.

(4.44)

CHAPTER 4. ELECTRONS IN METALS

62

1r W

Fig. 4.4. Energy diagram for the metal-surface problem. Here EN and EN-1 denote the ground state energy of a metal with N and N - 1 electrons respectively. We have also assumed that potential energy of an electron at infinity is equal zero.

4.3

Jellium model and electrons near metal surface

A more realistic representation of the electron distribution at the surface of a simple metal is provided in the framework of the uniform positive background or jellium model. In this model the positive charge of point ions is smeared out uniformly over the volume of the crystal forming a sort of metallic jelly. 1 The replacement of the discrete ions in a metal by the uniform positive background leads to only small error in the energy of the real system consisting of the conducting electrons and of the metal ions. This may be seen from the following simple consideration. Assuming that the electrons are moving in the uniform positive background created by the metallic ions, the Coulomb interaction of an electron with the ions can be represented as

/ ~tdr2

Yei

-- -e

r12

On the other hand, the energy Ve~ of mutual interaction between electrons has the form ~-~gf dr2 I ~j(r2)1 2/r12. Thus representing ~ j ( r ) by the plane waves, we find that e

Vee =

~

~__N ( N - l ) e / d r 2

/ dr__22 ~_~1 r12 J

=

~

r12

Both terms Vei and V~ cancel each other to the accuracy ~-1 giving a contribution to the energy of about 10 -7 eV which is much smaller than the Fermi energy. The above argumement suggests that jellium model may be accepted as a rather good approximation for surface problems of large metallic samples, but its applicability to the fine particles must be considered with care. 1The term Jellium was introduced by Conyers Herring. It originates from two words: jelly and the Latin/English ending -urn, which characterizes the names of all simple metals (for example: natrium/sodium).

63

4.3. J E L L I U M M O D E L

|174174 |174174 ==~ . . . |174174 |174174

I

n§215 m

N

==>

0

•

Fig. 4.5. Positive charge distribution for the uniform background model (jellium).

For the metal with a surface the positive charge distribution n+ (r) is cut off sharply at the surface (Fig. 4.5) and may be represented (Bardeen, 1936) by a step function of the form {~, in a metal (4.45) n+(r) = 0, elsewhere, where ~ is given by Eq. (4.10). Metal as a whole must remain neutral. It means that the total charge (negative and positive) integrated over the whole space must be equal zero. Hence, we can write [n(r) - n~_ (r)]dr = O,

(4.46)

where n(r) denotes the electron density. In the case of the semi-infinite metal represented by jellium, with the x-axis oriented perpendicular to the surface, the Eq. (4.46) takes the form /In(x)

~ O ( - x ) ] d x - O,

(4.47)

oo

where n(x) denotes the electron density profile and e(x) is the Heaviside step function (equal 1 for x > 0, and zero for x < 0). In the above expression we have placed the jellium edge at x - 0, i.e. at the location of surface. Now, the question arises, in which position should we locate the jellium edge in respect to the position of the barrier ? In order to answer this question we should distinguish between the physical and geometric surface. The first one will denote the location of the barrier while the second, the place where the positive charge background terminates. To determine its relative position we will make use of the charge neutrality condition. The hard-wall infinite barrier situated at the geometric surface of the metal implies the electron charge distribution which is shown in Fig. 4.2. From this figure, one can see that in a surface region the electron density is small. Thus, the charge conservation requires that the position of the barrier has to be moved in respect to the geometric surface (Sugiyama, 1960). In order to calculate the shift of the barriers let us assume that metal, of volume ~ = LxLyLz, is represented by a jellium, confined between two parallel yz-planes distant by Lx one from the other. Assuming the periodicity conditions in the y- and z-directions, the finite square potential barrier of height W

64

CHAPTER

V=~

4.

ELECTRONS

IN METALS

J

W

-L+Q~.

0

-Q

x

Fig. 4.6. Illustration of the position, a, of the physical surface for the finite-square-potential barrier. at x = 0, and the infinitely high potential wall at x - - L (L > L=) (see Fig. 4.6) the wave function has the form 1

Ck(r) = ( n y n z ) l / 2

ei(kyy+kzz )

Ck=(x),

(4.48)

where, 2~x ) 1/2 ~=L + 1 sin~=e-~=='

x _ 0, (4.49)

1/2

~)k= (X) --

(2nx) n=L+l

sin(k=x-5=)

'

-L<x<0 -

'

Here a= and 5= have the same meaning as in Eq. (4.34), and k=L = n l r - 5=, where n - 1,2,3, .... For the wave functions (4.48) the electron density profile n ( x ) can be calculated from the expression (4.29). To fix the position, a (Fig. 4.6), the electron density profile is put into the charge neutrality condition of the form

~L

~ n T ( X ) d x -- 0

(4.50)

where the total charge density, a T ( X ) -- (electron d e n s i t y ) - (positive charge background)

(4.51)

equals now n ( x ) - ~ O ( - x - a ) O ( x + L - a ~ ) . For the infinite barrier model, the calculation (van Himbergen and Silbey, 1978) gives 37r a ~ = 8k g

+

71-2 8k2 L '

(4.52)

4.4. THE HARTREE-FOCK APPROXIMATION

65

while for finite-square-potential barrier of height W - li2p2/2m the result is a - a ~ - D,

(4.53)

where D = ~

()~- 1) 1/2 + (2 - )~)arcsin()~ -1/2)

(4.54)

with )~ = p2/k~. The shift of the jellium edge for the finite and infinite potential barrier is illustrated in Fig. 4.2. It follows from Eqs (4.53) and (4.54) that the position, a, of the physical surface of a metal varies from 37r/SkF (for L ~ ~), for a metal bounded by the infinite potential barrier, to zero for the finite square potential barrier of the height equal to the maximal energy of free electron in a metal i.e., to the Fermi energy ()~ = 1). We see, therefore, that the application of jellium model and the regard of the charge neutrality condition leads to the solution of the problem stated above: the physical and geometrical surfaces of real metal are shifted one relative to the other.

4.4

Electron gas in the Hartree-Fock approximation

Owing to the large difference between the electron and ion masses it is possible to separate the electrons and ions movement in a solid. This can be obtained by employment of the adiabatic (or Born-Oppenheimer) approximation. The adiabatic approximate Schr5dinger equation, for the electrons in a static lattice of ions, can be written in the form: E

-2-ram i - E I r i _ R

i--1

I

~+~

l~

[ri-rj "

I~ - E ~ '

(4.55)

"

where the successive terms describe kinetic energy of electrons, the electron-ion interaction and the Coulomb electron-electron interaction. The electronic wave function, ~, depends on the coordinates, r, of all N electrons and on their spins a

~I/ --

~/(rl(71, r 2 a 2 , . . . ,

rNCrN;

R),

(4.56)

and R determines the static position of the ions of valence, Z, in the Bravais lattice. The last term on the lhs of Eq. (4.55) prevents its separation into a set of singleelectron equations. It is posssible, however, to transform this many-electron problem of Eq. (4.55) into a one-electron form by making the interaction term a function of individual positions of the particles. Let us assume that a given i-th electron is moving in the electric field produced by the continuous distribution of all the other electrons, having the density n(r) - ~ Ir [2. Then the potential seen by the electron at ri will be: e2 /#~j 1 e2/#~j/ ] Cj(rj) [2 Vi(ri) = - ~ - . . ] r i - r j ] = = v 2 - ~ - . . drj ] r i - r j ]

2

fg

n(r')

Ir-r'[

= VZ(r) '

CHAPTER 4. ELECTRONS IN METALS

66

where we have replaced ri and rj by r and r', respectively. Thus, the one-electron Schr5dinger equation is - h2 V 2 + Vi~ 2m

VSZ(r)l r

= Er

(4.58)

where

Ze 2 (4.59) ViOl(r) - _ E l r_ R I R and the summation goes over all the positions of ions. The Pauli exclusion principle, generalized to the many-body case, demands from the wave function (4.56) to be antisymmetric in respect to the interchange of electron coordinates (including spin). Such a function for a system of N fermions was proposed by Slater (1930) in the form

1 E(_I)p ~I/= V/-~" P

N PH~i(xi),

(4.60)

i=i

where P denotes the permutations of ~i, being the product of orbital r and a spin function X~, and xi denotes both space and spin coordinates. The wave function of the form (4.60) constructed from spin orbitals (~n (X) is equivalent to the Slater determinant of the form 1

1

~l(Xl) ~1 (X2)

~2(Xl) ~2 (X2)

:

:

~01(XN)

~2 (XN)

"'" "'"

~N(X1)

"'"

det [~1 (x1)~2(x2)... ~N (x,~)]

(4.61)

The best choice of ~i's, in the sense of variational principle, leads to the Hartree-Fock equations (Seitz, 1940) for/-th spin orbital dx2]

U~i(x1)

--

"__

~PJ

r12 J

(4.62) where the operator U includes both kinetic and electron-ion part of the single-particle Hamiltonian, H(r~), and the functions ~i and ~j correspond to spins in the same direction. The second term in (4.62) represents the classical electrostatic interaction in which the summation goes over all N electrons. The third, exchange term in (4.62) accounts for the fact that electron does not interact with itself and has appeared in

4.4. THE HARTREE-FOCKAPPROXIMATION

67

the Hartree-Fock equations owing to the introduction of the Pauli principle to the theory. The corresponding integrals for ~i and ~j functions with opposite spins will disappear due to orthogonality of spin functions. Since in (4.62) spin does not appear explicitly we can rewrite it in the form Ur

e2 [~i[r

+

r12

-

j Cj(ri)

=

._.

spin II

(4.63) To see the physical sense of the exchange term we rewrite it as follows

.[/

e2 E

j=l

.r.]

r (r2)r

r12

Cy (rl)

=e2~.=i ~j(r2)+i(r2)~j(r1)dr2+i(r1)r12 N

~)i (rl) -- e x r

(4.64)

s p i n II

where sx is the exchangepotential energy. This can now clearly be interpreted as the potential energy of electron at rl due to the exchange chargewhose density at r2 is N

$

nx(rl, r2) -- e E Cj (r2)r162 j:l ~)i (rl)

(4.65)

Note that n~(rl, r2) depends on the orbital number i. Owing to the orthogonality and normalization of the one-electron wave functions r (r)r

dr = 5ij,

(4.66)

it is easily seen that the total exchange charge is

f

N e E / r (rl)r (r2)r (rl)r (r2) dr2

nx(rl, r2) dr2 -

j=l

r162

= e.

(4.67)

Since the exchange charge density n~, given by (4.65), varies not too strongly with i, we may replace it by an average nx(rl, r2) -- E Pi(rl)nx(rl, r2), i

(4.68)

68

CHAPTER 4. E L E C T R O N S IN M E T A L S

where Pi(rl)=

r

,

(4.69)

E r (rl)r (rl) J denotes the probability that an electron at r l will be in state i. We shall make use of Eq. (4.68) later in this Chapter.

4.5

E x c h a n g e and correlation energy

The motion of free electrons inside a metal may be described by the normalized plane wave functions Ok(r) = ~2-1/2eik'r. Using these functions, for a metal represented by jellium model one can easily calculate the exchange energy (Raimes, 1961) to get

k

(4.70)

3 e 2 N ( 9 ) 1/3 1 4 & r-~"

e~ (k) =

Thus the average exchange energy per particle is c~ - ~1 ~

)1/3 = 0.458

e ~ ( k ) - - ~3 ( 3 '~

k

hartrees.

(4.71)

rs

The total Hartree-Fock energy per particle for free electron gas is given by s

= ts -[- s

1.105

0.458

r2

rs

-~

hartrees.

(4.72)

Evidently, the electron interaction should be treated more accurately than in the Hartree-Fock approximation. Especially, a more detailed consideration of their electrostatic repulsion is needed, which accounts for the reduced probability of close encounters of electrons, particularly those of antiparallel spin. The difference between the exact energy of the ground state and the Hartree-Fock energy, E H F , is called correlation energy, Ec - E e x a c t -

This energy accounts for the correlated to their Coulomb repulsion. The first, of the correlation energy was given by for ~c = Ec/N, is still applied in many correlation energy (Pines, 1963) takes a ew=_

EHF.

(4.73)

motion of electrons of antiparallel spins due for a long time most successful, calculation Wigner (1934) whose interpolation formula calculations. The Wigner expression for the simple form

0.4______~4 hartrees. rs + 7.8

(4.74)

Recently, the correlation energy, for the wide range of electron densities, was determined by stochastic Monte-Carlo calculation of Ceperley and Alder (1980). The

4.6. FERMI HOLE AND THE IMAGE FORCE

69

numbers determined by Ceperley and Alder were very accurately fitted to simple functions of the electron density (Vosko et al., 1980; Perdew and Zunger, 1981). The Perdew-Zunger parameterization is given by the following function ec A = _

0.1423 1 + 1.0529r~/2 + 0.3334rs

hartrees,

(4.75)

which is accurate for the important densities corresponding to atoms, molecules and solids, and therefore is recommended for contemporary surface calculations.

4.6

Fermi hole and the origin of image force

For free electron metal, one can easily calculate the distribution of the exchange charge density given by (4.65) substituting for Ck(r) the normalized plane waves Ck(r) f~-l/2eik'r, to obtain (Raimes, 1961)

e (sinkFr12--kFr12coskFr12)e-ik.(r~-r2) nx(rl2) = ~

(4.76)

r132

Averaging the exchange charge over all electrons with one kind of spin, following Eq. (4.68) we get

9e~ ( sin t - t cos t ) 2 nx(rl2)- --~ t3 ,

(4.77)

where t - kFrx2. Adding to (4.77) the average density, - e ~ / 2 , of electrons with parallel spins, we find that the average charge density, at distance t/kF from electron

I

I

I

I

1.0 0.8 0.6 0.4 0.2 0.0 0

I

I

I

1r/2

it

31r/2

I

t

2~

Fig. 4.7. The exchange (Fermi) hole in the electron distribution of bulk metal.

C H A P T E R 4. E L E C T R O N S I N M E T A L S

70 at rl, is n(r12) - - - ~ -

1- 9

~5

= -2

F(t)"

(4.7s)

The graph of function F(t) is shown in Fig. 4.7. From this figure it is seen that if we would sit on one of the electrons we would see that the probability of finding another electron with the parallel spin in the surrounding region, 0 ~_ t ~_ kFrs, is nearly zero. Thus, one may conclude that each of free electrons in the bulk jellium is surrounded by the so called Fermi hole, of approximate radius r s, in the distribution of electrons with parallel spins. The spherical shape of the Fermi hole will be modified in the vicinity of the surface (Fig. 4.8). In order to answer the question, how the average Fermi hole will be changed near the metal surface, one may calculate nx(r12) using the wave functions (4.27). From the schematic picture of Fig. 4.8, it is seen that each electron is surrounded by a region depleted of electrons of the same parallel spins. The results of calculations have shown (Juretschke, 1953) that as the electron approaches the surface its exchange hole flattens and remains behind, in the region of high-charge density. When electron is far away from a metal surface, the exchange hole is flattening out to a thin planar charge deficit excluding in total a charge of - l el from the electron distribution. This hole together with its counterpart for the electrons with antiparallel spin, the correlation hole, gives an image charge distribution. As we will see a microscopic picture of the image force, which arises when electron is brought out of a metal, is determined by the electron exchange and correlation effects (see Chapter

12).

I

/ / / /

) (b)

~ (c)

/

/ / / /

Fig. 4.8. Schematic illustration of the exchange hole for an electron (dot) near the metal surface. The vertical dashed line shows the image plane position. The lower part shows the electron density profiles. Redrawn with permission from Serena et al. (1986). (~1986 The American Physical Society.

4.7. S T A B I L I T Y OF JELLIUM

4.7

71

Stability of jellium

Now, we are in position to calculate the total energy of a system consisting of moving electrons and the static uniform positive background. The total energy of this system will consist of the electrostatic, kinetic and exchange-correlation contributions. Let us consider each of the contributions to the total energy inside a single Wigner-Seitz cell.

for a monatomic, nearly close-packed Bravais lattice (fcc, hcp or bcc) the WignerSeitz polyhedron may be replaced by a sphere of radius r0 - Z1/3rs. The electrostatic (Coulomb) energy of a charged sphere will consist of three contributions: self-energy of the uniform electron gas, self-energy of the uniform positive background and the energy of interaction of electrons with the uniform positive background. To calculate the electrostatic self-energy of the uniform electron gas we determine first the electrostatic potential V(r) of the uniformly charged sphere (Fig. 4.9) V(r) = eQ(r) + e r

~

~o ~ dgt

(4.79)

r

where Q(r) = (47~r3/3)~e is the amount of charge inside the sphere of radius r, ~ is the average electron density and ~ is the volume. The integration gives

r0

2r 2

,

r < r0,

=

(4.80) Ze /,

The self-energy of the electrostatic interaction averaged over the Wigner-Seitz sphere is given by

~e 1 -

c~-

-

V(r)4~r2dr 4~r3o/3

3 Ze 2 =

5 ro

where the factor 1 takes care of counting each pair of electrons only once.

Fig. 4.9. Calculation of the potential of the uniformly charged sphere.

(4.81)

The

72

CHAPTER

4.

ELECTRONS

IN METALS

same result is obtained for a sphere charged uniformly with a positive charge. On the other hand these two positive contributions are compensated by the negative energy of electrons interaction with the uniform positive background. Thus the total electrostatic contribution is zero and the total energy per electron in the system is -

t (n) +

+

(4.s2)

where the terms standing on the right-hand side represent respectively the kinetic, exchange and correlation energy per electron. In order to be in mechanical equilibrium the total energy of the system must attain minimum which means that the derivative of eT(n) with respect to the equilibrium electron density must be equal zero = 0.

deT(~) = deT(rs) d~ drs

(4.83)

Substituting for the kinetic, exchange and correlation energy contributions in (4.82) the corresponding expressions (4.20), (4.71) and (4.74), the solution of the equation d (1.105._ 0.458 drs

r2

rs

0.44 / 7.8

rs +

=

0

(4.84)

gives the equilibrium density parameter r s ~- 4.2. The energy per particle in jellium given by (4.82), with the Ceperley-Alder correlation energy (4.75) employed, is plotted in Fig. 4.10 as the function of density parameter r s. A minimum in the energy curve for rs ~ 4 is clearly visible. It means that jellium is stable only at the equilibrium

0.25

(/) (D

0

.~

T

r--'-T---r

T

~

1

0.15

0.05

>, (D

c -0.05 w

-0.15

0

2

4 rs

6

8

10

(bohrs)

Fig. 4.10. Energy per electron in jellium versus density parameter rs.

4.8. SURFACE ENERGY OF FREE-ELECTRON GAS

73

electron density ( n e q - - 2.2 • 10 22 c m - 3 ) corresponding to rs ~ 4. At other densities the positive background exerts a pressure on the electrons to keep them at the given V s 9

The instability of jellium is manifested by the negative bulk modulus at low electron densities (rs ~- 6) and by the negative surface energies for the metals characterized by rs _< 2.5 (see Chap. 8). This pathological features of jellium can be rectified by introducing the discrete-lattice pseudopotential corrections or by the stabilized-jellium model (Chap. 8).

4.8

Surface

energy

free-electron

of semi-infinite gas

As noted in Chapter 3 the total free energy of a piece of metal is the sum of volume and surface term. The bulk energy term, proportional to the volume, can be determined for a large sample in which the distribution of electrons is uniform throughout. The surface energy term, proportional to the surface area, is determined in turn by the non-uniform distribution of electrons gas near the surface of the metal. This nonuniformity gives the total energy which is in excess of what would be expected if the uniform electron density was terminated abruptly at the surface. This excess is the surface energy of the electron gas. We can define surface energy, a, as the work per unit area required to split the crystal in two fragments along a plane. Let E be the energy of the electrons in the unsplit specimen and Es the energy of electrons in each half of the split one. Denoting by A the area of the newly exposed face of each half we have 1

=

(2E - E).

(4.85)

The main contribution to the surface energy is brought in by the change in the kinetic energy of electrons. For instance, in the Sommerfeld model of metal the surface kinetic energy is few times larger than the electrostatic energy contribution originating from the creation of surface double layer. As an instructive example we will sketch within this model, the calculation of the contribution to the surface energy arising from the sum of one-electron energies. Consider a macroscopic block of metal of dimensions 2Lx, Ly, L~, consisting of N electrons (Fig. 4.11). It is useful to impose the periodic boundary conditions in yand z-directions and the standing wave in the x-direction. Then for the solution of the SchrSdinger equation we can write

Ck(r)-

1 ei(k~y+kzz) sin(k~x) v/L~LyLz

(4.86)

where the wave numbers, ky and kz, have the forms 2~ny/Ly and 2~nz/Lz, and kx is equal to kx = n x r 2L~"

(4.87)

74

C H A P T E R 4. E L E C T R O N S IN M E T A L S

CLEAVAGE PLANE -7 //

I I

z~

I

I

.

0

.

.

.

Lz .

.

x

-a 0

x-

Lx

2Lx

Fig. 4.11. Splitting of the macroscopic block of a metal. The total energy, E, of the uncut block is equal to the sum over occupied states h2 k.

ky

kz

2

-

n=

k~

k~

(4.88)

kz

where the factor of 2 arises due to the spin degeneracy. The summation is carried over all occupied states up to the Fermi level and nmaz = 2kFLx/Tr. After splitting, the physical and geometric surfaces of the half of a metal will be subsequently shifted according to the charge neutrality condition (compare Section 4.3) and the finite-square-potential barrier will be placed at x = 0. Accordingly the electron wave functions will now have the form 1 ~i(k~y+a, ~).,. (x) C k ( r ) - v/LxLyLzC ~k=

(4.89)

where Ck: (x) is given by (4.49). Here, for simplicity in the normalization constant of (4.49) we have taken the limit of large L:. This allows to ignore the ace term on the second boundary of L: and the wave-vector dependent correction _~ 1/k:. This omission does not influence the results substantially. Now, ky and kz are the same as in (4.88) but, in consequence of the phase displacement at the surface, k: is equal to k= --

n x 7r - - (~x

L= + a ,

(4.90)

where n= is an integer number. Only the terms involving k 2 contribute to the change of the energy. For the half of the divided metal it will be given similarly to (4.87) by

Es-22m

n,=l

Ek~ Ek,

L=+a

+k 2+k

.

(4.91)

4.8.

SURFACE ENERGY

OF F R E E - E L E C T R O N

GAS

75

Substituting Eqs. (4.87) and (4.91) into (4.85) and noting that L y L z - A, after some algebra (Huntington, 1951) and integration over ky and kz, we obtain for the surface energy h 2 1 f0k~ (7r ) a = 2 m 7r2 ~ - 6x - k:r,a (k2F - k 2) kx dkx. (4.92) The dependence ~ - ~(k~) for the finite-square-potential is given explicitly by (4.35). Making use of this expression for ~ and of the expression (4.53) for the barrier displacement a, the integral in (4.91) is evaluated analytically to give h2k~

a = 16-0~-m [It

- 89(8 - 24A + 15A2) arcsin(A -1/2) q-~1 (15A - 14)(A _ 1) 1/2 ]

(4.93)

where, A, similarly as in (4.53-4.54), denotes the ratio of the barrier height, W = h2p2/2m, to the Fermi energy, EF. The first term in (4.93) gives the infinite-barrier result. In this case the surface energy is equal to the kinetic energy. For the finitebarrier potential a part of electrons leaks out beyond the location of the geometric surface (potential barrier location). Thus the change in their potential energy is given by

f0 aw (x)dx

(4.94)

where A W - EF. Integrating the exponential fall-off density profile resulting from (4.48)-(4.49) the Eq. (4.94) gives h2k~

32-~m

[A(4- 3A)arcsin(A -1/2) + (3A- 2)(A - 1) 1/2]

(4.95)

Subtracting this potential energy from (4.93) we get the surface kinetic energy for the finite-barrier potential h2k~ [ aKE = 16-~r2m [

1 +~(15A 2 - 16A - 8) arcsin(A -1/2) -F-3(2--5~)(,~--1) 1/2]

(4.96)

This expression appears to be very useful for testing more complicated models of the surface barrier (Mahan, 1975). The surface kinetic energy for the infinite barrier model, given by the first term in (4.96), is positive. Depending on A (the barrier height parameter), the kinetic energy term for the finite potential step, may become negative when the step is sufficiently low. To see this one may write the results (4.96) in atomic units as A(A) eKE--

r4

(4.97)

and plot the A dependence of the kinetic energy coefficient A(A) (Fig. 4.12). As is seen

76

C H A P T E R 4. E L E C T R O N S IN M E T A L S

0"03L''

I

'

-

-

I

'

I

;~

/

0.02F Q01 I-"~

o.o

"

-0.01

V -o.o3~

i I 2.0

,

I 4.0

,

I 6.0

I

80

Fig. 4.12. The kinetic energy coefficient, A(A), versus barrier height A. After Mahan (1975).

this function shows very dramatic changes. The realistic barrier heights determined from the variational procedure which included electron-interaction energies (Mahan, 1975) are given by A = 1 - 1.5, where the lower value corresponds to a lower rs. Thus we see that the infinite barrier model provides a poor description of a jellium surface.

Chapter 5

Electron density functional theory In this Chapter we outline the basic principles of the density functional theory which provides a suitable theoretical framework for calculation and discussion of various ground-state properties of metal. Application of this method to the metal surface problem proved very fruitful. The method avoids the problem of determination the ground-state wave function and the central role is played by the electronic density n(r). A complete and more rigorous account of the density functional theory can be found in a book by Parr and Yang (1989) The electron density in a metal is connected with the one-electron wave functions r by the following sum over the occupied states: n(r) -

(5.1) i--1

The functions r

satisfy one-electron SchrSdinger equation gr

cir

(5.2)

with the Hamiltonian H = - ~ 1V2 + V(r) ,

(5.3)

where V(r) is the assumed potential energy in which electrons move. From now on we will use atomic units unless otherwise stated. It is clear that r are determined from (5.2) once V(r) is known and consequently n(r) can be found from (5.1). It would be useful if one could express n(r) directly in terms of V(r), without calculating the one electron wave functions. Such a way is provided by the statistical approach of Thomas (1927) and Fermi (1928). The method of Thomas and Fermi although too crude in the treatment of the kinetic energy and not incorporating the effects of exchange and correlation, contains all the important ideas and serves as the starting point for development of density functional theory. 1 1A functional is a q u a n t i t y t h a t is d e p e n d e n t on a variable function and not on a n u m b e r of discrete i n d e p e n d e n t variables.

77

CHAPTER 5. DENSITY FUNCTIONAL THEORY

78

5.1

Thomas-Fermi its e x t e n s i o n s

method

and

In the Thomas-Fermi method (March, 1957; 1983) the inhomogeneous system of N electrons, in an external electrostatic potential Va(r), is characterized by its local density n(r). In other words it is assumed that in a small region of the phase space around position vector r, the external potential and the electron density are practically constant. As a consequence, locally for a given position in a metal, the inhomogeneous electron gas can be treated as a homogeneous one, and to construct total energy of the system the relations for the uniform free electron gas can be used. The kinetic energy per unit volume (energy density) in terms of the electron density n(r) (compare Chapter 4) may be written as:

ts[n] =

3

(31r2)2/3nb/3(r).

(5.4)

Hence the total kinetic energy is given by

To-Its[n]

d r = 3 (3~2)2/3 1 nb/3(r)dr.

(5.5)

The square brackets denote that ts is a functional of n(r). The potential energy of the electrons can be split in two parts; the energy in the external (ionic) potential Va(r) and the Coulomb repulsion energy in the potential V~(r) of a classical electronic charge distribution n(r), where

n(r') dr'. Ir - r ' l

(5.6)

Thus the total energy expressed in terms of the electron density may be written as

E[n]

1 j n(r)n(r') n(r) Va(r) dr + ~ I r - r' [ dr dr' 3 (3~2)2/3 1 +1-'0

n5/3 (r) dr.

(5.7)

As it is seen this functional does not contain exchange and correlation contribution. The ground-state density n(r) is determined variationally, by varying E with respect to n, subject to a subsidiary condition that the total number of electrons remains constant n(r) dr = N. (5.8) Then the variational principle gives the following Euler's equation (Elsgolc, 1961; Parr and Yang, 1989) for n(r) 5 {E[n(r)]- A/n(r)dr}

=0

(5.9)

79

5.1. T H O M A S - F E R M I M E T H O D

where A is the Langrange multiplier. Taking into account the subsidiary condition (5.8) the equation (5.9) implies that X = 5E[n]/~n, and from the definition of chemical potential # (Landau and Lifshitz, 1969), we conclude that X = p. If we now substitute Eq. (5.7) for E into Eq. (5.9)we find dr' + 71

i{V~

[n(r)]2/3 - # } 5n dr = O. (5.10)

Since this equation must be valid regardless of the variation 5n, this is possible only when n(r') dr'-+- 1 (37r2) 2/3 [n(r)] 2/3 ~._0. (5.11)

y:(r)+

SIr-r'l

7

The first two terms give the total electrostatic potential in the system and will be denoted r

n(r) r

- Va(r)4- S I r - r' [ dr''

(5.12)

Thus Eq. (5.11) will take the form: r

4-

71 (37r2)2/3 in(r)]2/3

-

#

(5.13)

which gives basic equation of the Thomas-Fermi method: 1

n(r) = ~ 2 [2(# - r

3/2.

(5.14)

This formula gives a relation we looked for: the density n(r) is expressed in terms of the external potential. The Eq. (5.14) is valid only when radicand is positive or equal zero. Otherwise we must take n(r) = 0. There exist several extensions of the Thomas-Fermi approach which try to rectify the deficiencies of the model, namely the neglect of exchange and correlation, and too crude treatment of the kinetic energy. The first extension comes from Dirac (1930) who included the exchange energy to the equation (5.7) for the total energy. On the ground of the similar argument like by derivation of the expression (5.4) for kinetic energy, the exchange energy density, e~, for homogeneous system can be written as s

-- - C x n413,

(5.15)

where the constant Cx = 3 (3)1/3 (cf. Chapter 4). The total exchange energy which should be added to (5.7) is E~

-

- C x i n4/3(r)dr.

(5.16)

Thus the variational procedure for the total energy leads to the following equation: r

4 c:[n(r)]ll3 = # 4- 7l(3r2)213[n(r)]213 - -3

(5.17)

80

CHAPTER 5. DENSITY FUNCTIONAL THEORY

This equation, quadratic in n 1/3, can be easily solved for n(r) as a function of the potential, r #, to give an analogue of Eq. (5.14) in the Thomas-Fermi-Dirac approximation. Proceeding one step further, by applying the same procedure and including the correlation energy (compare Chapter 4) one may derive the density-potential relation in the Thomas-Fermi-Dirac-Gombs approximation (GombgLs, 1943). Finally, as we have already mentioned, the validity of the expression (5.5) for the kinetic energy is limited to the case of very slowly varying electron density. In most interesting cases like in atoms and at metal surfaces, the electron density varies rapidly and corrections to the kinetic energy expression which take care of the inhomogeneity of the electron gas are needed. Such a correction was derived for the first time by von Weizsticker (1935) on the ground of simple argument for the system with one-level being occupied. If we forget for a while about the Pauli principle, in equilibrium all the electron will occupy lowest energy level. Then the electron density n is given by the square of the wave function r corresponding to this level

=fCf2,

(5.1s)

and the quantum-mechanical kinetic energy of electrons is given by

1/

T1 = ~

(Vr 2 dr.

(5.19)

Following (5.18) we have, r = n ~/2 which substituted to Eq. (5.19) yields the correction due to von Weizs~icker A / (Vn) 2 dr (5.20) Tl-g n ' where ~ : 1. Some later calculations have shown that A < 1, and for a slowly varying density the proper choice is ~ - 1 (Kirzhnits, 1957). The expression (5.5) for the kinetic energy is a consequence of the Pauli principle, so the full expression for the kinetic energy T of the inhomogeneous electron gas, which takes care of the Pauli principle, can be written as,

Tin] = T0[n] + T1 [n] where To and T1 are given by (5.5)and (5.20)respectively.

5.2

H o h e n b e r g - K o h n theory

The more general theory called density functional formalism has been formulated by Hohenberg, Kohn and Sham (Hohenberg and Kohn, 1964; Kohn and Sham, 1965). As we shall see the deficiencies of the Thomas-Fermi method encountered in the previous Section are surmounted by use of this formalism. Let us consider a system of N electrons moving under the influence of an external potential Va (r) and their mutual Coulomb interaction which have a unique, nondegenerate ground state ~. Then 9 is a unique functional of Va (r) and therefore the

5.2. HOHENBERG-KOHN THEORY

81

expectation value of the electron density is given by Eq. (5.1). It is clear that n(r) is a functional of the external potential Va(r). Hohenberg and Kohn proved that the converse is also true. The first of two theorems on which the density functional theory is based states that the ground state energy of a many-electron system in presence of an external potential Va (r) is a functional of the electron density n(r), and that it can be written as

E[n(r)] - ] n(r)Va(r) dr + Fin(r)].

(5.21)

Since the choice of Va(r) fixes the Hamiltonian H, also the many-body ground state energy E, is a unique functional of n(r). F[n] is an unknown but a universal functional of density n(r) alone, and does not depend on an external potential V~(r). The functional form of F[n] contains kinetic, Coulomb, exchange and correlation energies. Then, the variational (second Hohenberg-Kohn) theorem can be derived which states that, given the external potential V~(r), E[n] is minimized by the ground-state density, with the condition that the total number of electrons is conserved (see Eq. (5.8)) for all density distributions considered. The two theorems 2 allow to determine the groundstate density n(r) from the variational equation (5.9). It is convenient to separate out the long-range classical Coulomb interaction in F[n], and write 1 / n(r)n(r') F[n] - -~ i r _ r, I dr dr' + G[n] (5.22) where G[n] is a new universal functional of n(r), containing only kinetic, exchange and correlation terms. Inserting (5.22) into (5.21) and making use of the variational principle (5.9) we obtain the new, formally exact, Euler equation:

g a(r) +

n(r') dr' + ha[n] = #. I r - r'] 5n(r)

f

(5.23)

The first two terms give the total electrostatic potential in the system (the Hartree potential) which we have denoted r (see Eq. (5.12)) so, Eq. (5.23) takes the form: r

5G[n]

+ 5n(r) = #"

(5.24)

If we knew the functional G[n] we would determine n(r) from the solution of this equation. This, however, is not the case and the difficulty of the problem lies in the determination of the functional G[n] for the inhomogeneous system. In practice one has to make approximation to G[n] and minimize the expression for the total energy with respect to n(r). A gradient expansion of the form G[n(r)] = / d r

[g0(n(r))+ g2(n(r))I Vn(r)12 + . . . ]

(5.25)

serves as one of such approximations. The coefficient go(n) is the energy density (of nonelectrostatic terms) of a uniform electron gas of density n, and can be written in 2A particularly simple proof of the Hohenberg-Kohn theorems is due to Levy (1979).

82

CHAPTER 5. DENSITY FUNCTIONAL THEORY

the form

go(n) -- [ts(n) + ex(n) -[- ec(n)] n.

(5.26)

Here ts, ex and e~ are respectively, the average kinetic, exchange and correlation energies per particle of the uniform electron gas (compare Chapter 4). The coefficient of the first gradient term for the kinetic energy is: 1 g2(n) = 72n"

(5.27)

If we put, in (5.25), g2(n) = 0 and only consider the kinetic energy in go, then we obtain Eq. (5.14) of the Thomas-Fermi method. The incorporation of the other terms in (5.26) gives the extensions of the Thomas-Fermi theory discussed in the previous paragraph. These various approximations can be obtained by putting untruncated series for G[n] into (5.24) to yield r

dg2(n) IW(r) l dn ~=n(r)

+ dgo(n)

dn

n=~(r)

2g2(n(r))V2n(r)-t- . . . .

#

(5.28)

The validity of the gradient expansion is limited to the cases where both the Fermi wave number kF, and the Thomas-Fermi screening constant kTF -- V/4kF/Trao, are greater then I V / n I. It appears, however, that even in the metal surface region, where this criterion is not well fulfilled, a density gradient approximation leads to surprisingly good results. This will be discussed in Chap. 8.

5.3

Kohn-Sham equations

One of the most useful form of the solution of density-functional equations was derived by Kohn and Sham. To present their idea we start from Eq. (5.22), by dividing Gin] in two parts as follows Gin] =_ Ts[n] + E~c[n]. (5.29) The first term describes the kinetic energy of a system of noninteracting electrons having density n(r). What remains is E~c[n], the total exchange and correlation energy contribution. This term includes also a part of the kinetic energy of the interacting system which would be difficult to calculate due to the many-body effects. The total energy of the system of interacting electrons can now be written in the form

E[n] =

I

1 / n(r)n(r') i r _ r, I dr dr' + Ts[n] + E~[n].

Va ( r ) n ( r ) d r + ~

(5.30)

Applying again the variational principle we find the Euler equation:

Va(r) +

n(r') dr'-1 5T~[n] ~ 5Ex~[n] = #. I r - r' I 5n(r) 5n(r)

(5.31)

5.3. KOHN-SHAM EQUATIONS

83

We can regard Eq. (5.31) as describing a system of noninteracting electrons, moving in an effective external potential V~ff(r)" Vess(r)

=

~n(r) ( E [ n ] -

Ts[n])

n(r') [ r - r' Idr' + V~(r)

Va(r) +

(5.32)

where 5Exc[n] 5n(r)

Vxc(r)-

(5.33)

defines the exchange-correlation potential. Thus Eq. (5.31) can be written in the form ~Ts [n]

Veil(r)+ 5 n ( r ) = # '

(5.34)

which is formally the same as the one for a system of noninteracting electrons, with the Hartree potential replaced by V~ff(r). This implies that n(r) can be expressed in terms of the one-electron wave functions ~i(r) as N

n(r) = ~-~ [ ~pi(r)[2,

(5.35)

i--1

where the functions ~i(r) satisfy the one-electron Schrhdinger equation - ~1V2 + Veil(r) 1 ~i(r) - r

(5.36)

From the above equation it is clear that V~/y(r) is a functional of the density n(r). The set of equations (5.32), (5.35) and (5.36) is called Kohn-Sham equations and must be solved in a self-consistent way. It means that one first assumes a trial V~yf (r), then finds the solutions ~i, and obtains an approximation to n(r). This is used next to calculate from Eq. (5.32) new Very, which is put back into (5.36) and the procedure is repeated until the wave functions calculated from the two subsequent cycles agree to the required accuracy (i.e., self-consistency is reached.) Needless to say, the one electron wave function ~i(r) should not be interpreted individually. Only the electron density and properties that can be calculated from the density, can be determined by this method. The ground state energy corresponding to the density determined from (5.32) and (5.35-5.36) may be found from Eqs (5.21-5.22)and (5.29) with N

Ts In] = ~ ~i - / Veyf (r)n(r) dr.

(5.37)

n=l

The complexity of the many-body problem is contained in the unknown exchangecorrelation functional E~c[n]. This can be circumvented, however, by expanding E~c[n]

84

CHAPTER 5. DENSITY FUNCTIONAL THEORY

in a series of density gradients, in a similar way like we did for the functional G[n] (see Sec. 5.2). In a simplest approximation we may omit all but zeroth gradient in the series, i.e. we have

E~[n] = I dr [e~(n(r)) + e~(n(r))] n(r).

(5.3s)

This is so called local density approximation (LDA) in which relations appropriate for a homogeneous electron system are applied locally.

Chapter 6 Electron gas near the metal surface 6.1

Thomas-Fermi

electron

density

profile

The density functional formalism introduced in Chapter 5 constitutes a good basis for a discussion of the metal surface properties. What we have to determine first is the electron density distribution. We will consider a metal surface represented by semi-infinite jellium. We begin our discussion with the Thomas-Fermi treatment of the problem. The Eq. (5.13) of the Thomas-Fermi method for the semi-infinite jellium takes the form 1 3/2

~(~) = 5~-~ [2(~- r

where the electrostatic potential r d2r

dx 2

,

(6.1)

obeys the Poisson equation

(6.2)

= -41r [n(x) - ~ e ( - x ) ] .

Rewriting (6.1) in the form

r

= -~

+ #,

(6.3)

and using this in the Poisson equation (6.2) we obtain 1 (3~2)2/3

2

d2[n2/3(x)] dx 2

= 4'n" [ n ( x ) - 'h,e(-x)],

(6.4)

which can be written as 1 (3~.2)2/3 dn2/3(x)

-2

dx

d [dn2/3(x)] dx 85

- 4~ [n(x) - ~O(-x)]

dn2/3(x).

(6.5)

CHAPTER 6. ELECTRON GAS NEAR THE SURFACE

86

Integration of this equation for x < 0 (inside the positive background) with the boundary condition n(x) --+ ~, and dn(x)ldx ~ 0 for x ~ -oo, gives

[

3}

l (3r2) 2/3 I dn2/3(x) dx ] - 41r 2nh/3(x)5 - nn2/a(x) + g~5/3 , x < 0. Integrating Eq. (6.5) for x > 0 with the boundary conditions for x -~ oo, we obtain

l (3~r2)2/3

dx

= -8rnh/a(x)5

,

(6.6)

n(x) ~ O, dn(x)/dx --+ 0

x>0.

(6.7)

Demanding continuity of the solution of Eq. (6.5) for x - 0, we may equate the right-hand sides of Eqs (6.6) and (6.7) to obtain n2/a(0) = g3fi 2/a 9

(6.8)

To solve (6.6) and (6.7) it is useful to rescale the variables in order to write the solution in the dimensionless form. We define ~ - - X/J, TF,

(6.9) ~_ n/ft,

where ATF is the Thomas-Fermi screening length

1 i~TF

=

(4kF) 112 -

,

-

ao

(6.10)

and a0 = h2/me 2 is the Bohr radius (see Appendix A). Using these variables and the result (6.8), Eqs (6.6) and (6.7) are easily transformed to the form

~._.(~)1/2

d~,

S (~)~i,

and

< 0,

(6.11a)

-6 ~(~) =

x+

~

,

~ > 0.

(6.11b)

Equation (6.11a) cannot be integrated to give fi in terms of elementary functions of (Su and Shiau, 1991). However, far inside the positive background, its asymptotic form is found to saturate exponentially toward constant bulk density. Outside the metal surface ~ decays with the inverse sixth power of the distance. The Thomas-Fermi density distribution is plotted in Fig. 6.1. In the same way we could look for the solutions of the extended Thomas-Fermi equations (see Chapter 5) which include the exchange and correlation contributions.

87

6.1. THOMAS-FERMI ELECTRON D E N S I T Y

1.0

G" C

0.5 rs- 2

0.0 -1.5

I

-1.0

I

-0.5

I

0.0

~ _1~---=..~_

0.5

I

1.0

x (2#/kF)

Fig. 6.1. Comparision of the Thomas-Fermi (solid line) and self-consistent electron density profile (dashed line) for rs = 2.

They are, not interesting for our purpose because they lead to an electron distribution with a sharp cut-off for a certain critical density outside the metal surface (Garcia-Moliner and Flores, 1979). This critical density corresponds to r8 = 4.86, for the Thomas-Fermi-Dirac approximation and rs = 4.46, for the Thomas-Fermi-DiracGombas theory. It means that for lower densities the solutions of these equations do not exist. The further extension of the Thomas-Fermi theory, which includes inhomogeneity terms into the kinetic energy, leads to a complicated equation whose asymptotic solution for x --+ ~ shows an exponential decay of the electron density. Instead of solving this equation for n(x), in the way discussed above for the ThomasFermi theory, it is more convenient to make use of the variational principle and the trial electron density profile (Smith, 1969). This way of determining the electron density distribution will be discussed in Chapter 11. Before we pass to the discussion of the self-consistent method of Lang and Kohn we should notice that neither the Thomas-Fermi method nor its extension are capable of reproducing the Friedel oscillations I of the density distribution inside the metal which we have encountered considering wave mechanically the electrons in a square-potential well. 1The oscillations of the electron density profile found in the Thomas-Fermi-von-Weizs/ickermethod (Utreras-Diaz, 1987) are different in nature from the Friedel oscillations (see Chapter 11).

88

6.2

C H A P T E R 6. E L E C T R O N G A S N E A R T H E S U R F A C E

Self-consistent Lang-Kohn m e t h o d

As we have seen, discussing the density functional formalism, the method provides a self-consistent field procedure which allows to energy functional exactly. We will apply this method to the metal For the jellium model the total energy as functional of the electron be formally written as

1/

Ejr

= Ts[n] + E~[n] + -~

where r

dr r

f n(r I) r) = ]

n+(r !)

Ir- 'l

Kohn and Sham treat the kinetic surface problem. density n(r) can

r ) [ n ( r ) - n+(r)],

(6.12)

dr'

(6.13)

is the electrostatic potential. For the jellium model, the effective potential Veff in Eq. (5.36) varies only in the direction perpendicular to the surface and can be written in the form Yell(n; x) = r

+ Vxc(n; x)

(6.14)

where r is the electrostatic (Hartree) potential satisfying the Poisson equation (6.2) and Vxc is the exchange-correlation potential given by (5.33). If all gradient terms in Ezc[n] functional are omitted we obtain the local density approximation (LDA), for the exchange and correlation functional (Eq. (5.38)) and Vxc(n; x) - v.LDA(n" X) xc ~ '

-

-

d [nexc(n)] dn "

(6.15)

Here exc(n) is the sum of the average exchange and correlation energies per particle in the uniform electron gas of density n. The most popular forms for ex and er are given by (4.71) and (4.74), respectively. Integrating the Poisson equation (6.2) the electrostatic potential can be expressed (Byron and Fuller, 1974) in the form r

= -4~

dx'

dx" [n(x") - n+ (x")] + r

(6.16)

dx'

where r is an arbitrary constant. With the VeII given by (6.14) the one-electron Schrhdinger equation (5.36) can be written as --~ + V~.fl (r) - ei~i(r). (6.17) The effective potential (6.14) far inside the metal approaches a constant value. Thus the wave functions satisfying (6.17) have the form ~oi(r) - ei(k"Y+k~z)~k(x)

(6.18)

Tk(x) --+ s i n [ k x - 5(k)],

(6.19)

where, for x --+ - c ~ ,

6.3. EFFECTIVE POTENTIAL

89

(compare the discussion of the solution of this type in Chap. 4). Here, k - kx and 5(k) = ~(k~) is the phase shift. Substitution of (6.18) into (6.17) yields the following equation of the one-dimensional form

2dx 2 FVe/f(n;x) ~k(x)=

~--~

where the eigenvalues ~ are given by 1

2

2

Y~c(~)+ -~(k2 + ky + kz).

- r

(6.21)

It is convenient to shift the energy scale and put r

1 + Yxc(n) - - ~ k ~

(6.22)

Then Eq. (6.20) may be written in the form

]

1

2 dx 2 F V~//(n; x) ~k(x) = ~(k 2 - k2F)~k(x).

(6.23)

To determine the electron density distribution given by

n(x) - -~ 1 f0 kF (k2F- k2) lcflk(x)12 dk

(6.24)

the Eqs (6.23-6.24) have to be solved self-consistently with the potentials (6.14-6.16). Fig. 6.2 shows the electron density profiles computed for two r8 values in the local density approximation. The electron density decays exponentially in the vacuum. Deep in the metals interior the density behaves like

{3cos[2(kFx--5(kF))] n(x) ,,~ ~ 1 + (2kFx) 2

+0

(1)}

-~

(6.25)

exhibiting Friedel oscillations. The amplitude of the oscillations decays toward the metal interior as x -2. For low density metals the amplitude and the extent of oscillations is much greater then for the higher density metals. The electron density profile calculated self-consistently for rs = 2 is compared with the one determined by Thomas-Fermi method, in Fig. 6.1

6.3

Effective potential

An example of the self-consistent effective potential and its components, i.e. electrostatic and exchange-correlation parts is plotted in Fig. 6.3. As is seen the dominant contribution comes from the exchange and correlation potential. The electrostatic contribution is quite small and exhibits considerable Friedel type oscillations for larger rs. These oscillation are present also in Vxc-part of the total potential. Far in the metal

90

CHAPTER 6. ELECTRON GAS NEAR THE SURFACE

I

I

rs-5

I

I

1.0

POSITIVE BACKGROUND

ic

"~ 0.5 r

,

1.0

,

0.5

I

0

~~--, .... 0.5

1.0

x (21T/kF) Fig. 6.2. The electron density profile computed selfconsistently for the jellium surface. Redrawn with permission from Lang (1970). @1970 The American Physical Society.

interior the oscillations in both potentials cancel out exactly what asymptotically leads to a flat form of Veff. Outside the metal surface, the exchange-correlation potential shows an exponential decay. Actually, as the electron escapes from the metal, the exchange-correlation hole accompanying electron stays in the metal giving rise to an image potential. So asymptotically, outside the metal surface, one should expect that V=c behaves like x-1 rather than exponentially. This non-local feature of the V=c is not reproduced by the local density approximation (LDA). We will come back to this point in Chapter 11. The jellium effective potential (6.14) is a face-independent one, thus it generates face-independent electron density profile n(x) and consequently also isotropic other surface properties of metals. From the experiment, however, it is well known that many physical characteristics of metals, as for instance the work function, are strongly face-dependent. The face-dependence may be taken into account by using in the KohnSham equations the effective potential including the discrete-lattice pseudopotential correction (Monnier and Perdew, 1978). This problem is discussed in Chapter 8. Generally speaking, the effective potential of real metal should have asymptotic (x --+ c~) image-like behavior and has to be a face-dependent one.

6.4.

91

L O C A L D E N S I T Y OF S T A T E S

I

0.2

I

I

I 9

0.1

03 (D ID -~

0 cv

-

I

i

://

Fermi level

0.0

/

l

-0.1

-

-0.2

-

//

i II/

(D

i,i - 0 . 3 -0.4

-

-0.5 -1.5

VXC

j

Veff I

-I .0

/ r~=2

) I

-0.5

I

0.0

I

0.5

1.0

_

1.5

x (2"rr/kF-)

Fig. 6.3. Self-consistent effective potential at the surface of jellium of rs - 2. The metal occupies left half-space.

6.4

The local density of states

The local density of states is a very useful concept relevant in surface spectroscopies where we are interested in the contribution of electrons from different surface layers or atoms to the measured spectra. The density of states which we have introduced in Chap. 4 applies to an infinite perfect crystals. The local density of states for the given energy E, is defined by Af(E, r ) = E

]r

5 ( E - Ek).

(6.26)

k

Following this definition a total density of states JV'(E), is simply recovered by JV'(E) = / A/'(E, r) dr = E or

(6.27)

6(E - Ek).

k

For a macroscopic sample of large dimensions Lx = Ly = Lz = L, the components of wave number k~, ky and k~ are quasi-continuous and one may replace the summation in (6.26) by integration. Assuming a constant potential extending to 4-oo in the plane parallel to the surface and the potential barrier in the vicinity of x = 0, the wave functions of the type (6.16)-(6.17) allow to write (6.26) in the form

jV'(E, x)

=

(2--~-)3i]3

ICk(x)

5

E

-~ -

dkll dk

CHAPTER 6. E L E C T R O N GAS N E A R THE SURFACE

92

1/ Ir

~.~

o

E-

(6.28)

dk

where k = kx and kll = (ky, kz). Note that the electron charge density is given by

n(x) = 2

~oEFAf (E, x) dE.

(6.29)

where the factor two accounts for spin degeneracy. Inserting (6.28) into (6.29) and making use of the following relation for the step function O(x)"

O(x) dx - xO(x) we have

n(x)

'/

ICk(x) l2

x~-~xdx = xO(x)

( ")( EF-V

0

=

7r-~

=

--~ 1 ~ok~ I Ck(x) 12 (k,~ - k 2) dk.

(6.30)

,,. dk

EF--~-

(6.31)

We have already used the above formula in Section 4.2.

~=1.7~ 1.0 0.5 0.1

21f/kF

1,&,

x

v

Fig. 6.4. Local density of states as a function of x, for self-consistent effective potential of jellium (rs - 4). Different curves correspond to different energies, a = 1 corresponding to EF. Redrawn with permission from Werner et al. (1975).

6.4. LOCAL DENSITY OF STATES

93

V~ff(c~) -

If we put the zero of energy at the vacuum level, i.e. model of metal we get Af(E, x) - ~-~ 1 /0 k~ Ir

0, for the jellium

dk

(6.32)

where kE V / 2 [ E - Vefi(-c~)], and V~ff(x) is the effective potential. To discuss the asymptotic behavior of JV'(E, x) in the metal interior it is useful to employ the model of finite square-potential-barrier of Sec. 4.3. This corresponds to a replacement of Veyf(x) by a constant potential W < 0 and by V~ff(x) - O, for x > 0. Evaluation of (6.28) for the wave functions of the type (4.48), in the limit of x --+ - c o , yields -

A f ( E , x ) - ~kE ( 1 _ sin2[(kEx--6(kE)])2kEx

,

(6.33)

where kE -- V / 2 ( E - W). As can be expected, in the bulk limit the local density of states shows Friedel oscillations which, however, are damped more weakly compared to the oscillations in the electron density (Eqs (4.31) and (6.25)). Considering [ xkE I<< 1 and x _ 0 we obtain another interesting limiting case corresponding to the energies near the band edge W, for x _< 0 and to all energies E, at x - 0 Af(E, x) = [2(E - W)] 3/2 iwi

.

(6.34)

Now we can use these analytical formulas to the analysis of result of self-consistent

/ u~

21T/kF I

-0.

EF

E

Fig. 6.5. Local density of states (normalized by electron density n(x)) as a function of energy for several distances from the surface of jellium (rs = 4). Redrawn with permission from Werner et al. (1975).

94

CHAPTER

6. E L E C T R O N

GAS NEAR

THE SURFACE

calculations for the jellium surface of rs = 4 (Werner et hi., 1975). Fig. 6.4 shows the local density of states as a function of distance from the surface for several energies E. All curves show a strong Friedel oscillations of a period and amplitude diminishing with an increase of energy. In Fig. 6.5 the local density of states (normalized by the electron density) is plotted as a function of energy for different distances from the surface. The Af(E, x) rapidly approaches the parabolic bulk density of states (cf. Eq. (4.23)) as we move in from the surface. At the surface (x = 0), following (6.33) we should have A / ( E , x ) ( E - Ve/y(-c~)) 3/2, a relation which is approximately fulfilled for a self-consistent effective potential (instead of 3/2 for all rs, the exponent varies between 1.1 for rs = 6 and 1.6, for rs = 2). An E 3/2 variation of the density of states at the band edge results also from the tight-binding calculations.

Chapter

7

Sum rules and rigorous theorems for jellium surface There exist several sum rules and rigorous theorems for a jellium surface which can be derived as a special cases of the general theorems valid for many-body systems. These theorems provide a link between the bulk and surface quantities and may be used to test the internal consistency of calculation of surface properties. In this Chapter we will present some of these relations based on the charge neutrality of the system, the Hellmann-Feynman theorem and the virial theorem. Here we will limit our discussion to the case of a single surface of semi-infinite crystal. The further similar rules for metals covered with jellium and for two metallic slabs in an adhesive contact are discussed in Chapters 16 and 17.

7.1

The

phase-shift

sum

rules

Asymptotically, in the bulk metal, for x -~ - c ~ , the wave function in the direction perpendicular to the surface has a form

(7.~)

Ck~ (x) ~ s i n ( k z x - 5(kx))

where 5(kx), is the phase shift. This form of the wave function is general for any (y and z independent) surface potential which approaches a constant value as x -+ -c~. Thus, in the bulk, the whole information on the surface potential shape is contained in 5(k~:). The requirement of charge neutrality of the system implies the following sum rule for the phase shift of the wave function

jfOkF ~(k~)- 88k~k~- o

or

2_ k~ ~0

(7.2) 71"

= (~(k~))F - ~

~" 95

(7.3)

96

C H A P T E R 7. S U M R U L E S A N D R I G O R O U S T H E O R E M S

where the brackets represent the Fermi-surface average. This result, analogous to the Friedel sum rule for imperfections in metals, was derived originally by Sugiyama (1960), and later a rigorous proof of it was given by Langreth (1972). The fulfillment of the above phase-shift rule is the necessary condition for the potential to be selfconsistent. The phase-shift rule (7.3) can be used to determine the square-potential-barrier position relative to the jellium edge. Taking the position of the jellium edge at x - 0, the finite potential barrier will be placed at a. In reference to the jellium edge the phase shift is now equal to kxa + 5(k,:), where 5(k~) is given by (4.36). Inserting this into (7.2) after straightforward integration we get the result (4.53). The phase-shift may also serve to determine the surface density of states Af s (E). The total density of states is given by Af(E) = AI'b(E) + Aft(E)

(r.4)

where Afb(E) is the bulk density of states given by (4.23). Since the number N ( k ) of electrons in a semi-infinite metal laying inside a sphere of radius k is (Kenner and Allen, 1975) N(k) - ~ we have

+ 77

a ( k x ) - ~ kx dk~.

~2 dN k2 + A = ek =

(7.5)

,71.-1

r

4] k

(7.6)

where gt is the volume and A is the area of the surface. Hence the density of states function is AI'(E) - .IV'(k) dk _ a (2E)l/2 + 5(E)(7.7) d E - '-~ r ~ --4 where E(k) = k2/2. Comparing (7.7) with (7.4) we may write A/'(E) = ~2rib(E) + Ari s (E)

(r.s)

where rib(E) is the bulk density of states per unit volume, riB(E) is the surface density of states per unit surface and is given by

1[

riB(E)=75

5(E)-~

.

(7.9)

Using this quantity in Eq. (7.2) the phase-shift rule may be written (Paasch and Wonn, 1975) in the form: oEF ris (E) dE = 0.

(7.10)

This sum rule for the surface density of states similarly like (7.3) is a consequence of the requirement of metals charge neutrality. It is worth noting that Eqs (7.6) or (7.7) allow to extract the surface contribution to the quantities which depend linearly on the density of states. An example, for the electronic heat capacity and for the spin susceptibility, is given by Kenner and Allen (1975). General expression for the surface contribution to an arbitrary quantity which can be written as a sum over electron or phonon states is derived by Allen (1975).

7.2. B UDD- VANNIMENUS T H E O R E M S

7.2

Budd-Vannimenus

97

theorems

Due to its basic simplicity the general Hellmann-Feynman theorem (Cohen-Tannoudji et al., 1977) has been applied successfully to the calculations for the many-body systems. Among many applications, a special case of this theorem, the electrostatic theorem, has been proved to be extremely useful in surface physics problems. For a system of N electrons moving in a positive charge density n+ (r) the generalized electrostatic theorem (van Himbergen and Silbey, 1979) says that the variation of the total energy of the system E[ n, n+], upon a small arbitrary change, 5n+, in the positive charge distribution is given by 5E[n,n+] = -r 5n+(r)

(7.11)

where

r

[ n ( r ' ) - n+(r') dr'

Ir-r'l

J

(7.12)

is the electrostatic potential and n(r) is the electron distribution. One can rewrite (7.11) in the integral form to yield A E = - / 5n+ (r)r

(7.13)

It means that if we change the charge distribution by 5n+, then the total charge distribution nT, changes by 5nT -- 5n+ + ~n, where 5n is a readjustment of the electronic charge distribution, but the energy changes to first order in 5n+ only. The electronic part does not change (to first order) the value of the energy. This fact makes the electrostatic Hellmann-Feynman theorem very useful in the applications because the electron density which is much more difficult to calculate does not appear in the integrand of (7.13). We shall now apply this theorem to derive sum rules related to the jellium model of metal (Budd and Vannimenus, 1973; Vannimenus and Budd, 1974). This derivation follows closely the one given by Vannimenus and Budd (1974) (see also Lang (1983)). Consider a system of N electrons in the external potential created by a jellium slab of uniform positive charge density. The slab extends from x = - L to x = 0 along the x-axis normal to the surfaces of area A, so we have N = A L ~ and the system is neutral (Fig. 7.1a). Now we stretch infinitesimally the slab along the x-axis so that it extends from x - - L to x = 5L, keeping the metal neutral and the surface areas A, constant. Stretching of the slab can be thought (Fig.7.1b) as the addition of a perturbing charge of density -5~,

-L<x<0,

5n+(x) =

(7.14) - 5~,

0 < x < 5L,

where in order to keep the metal neutral we should have LS~ = ( ~ . - 5fi)SL. Then according to the Hellmann-Feynman theorem (7.13) the change in energy caused by

98

CHAPTER

7. S U M R U L E S A N D R I G O R O U S

(o)

n ~'/////////////////A m

-L

0

THEOREMS

~L

X

(b) ~5 v

X

Fig. 7.1. Stretching of the jellium slab. the stretch is ~E

=

E ( ~ + ~ ) - E(~)

=

-A]

/ , 8L

[r

(7.15)

r

J- L

where we have put r for the reference potential in the bulk. Making use of Eq. (7.14) this can be rewritten as (~E = - A L ~ t

r

- r

- ~

L/2 [r

-- r

dx

.

(7.16)

In the limit of large L the total energy can be expressed as the sum of a bulk and a surface contribution as, E(~) = ~ f ( ~ ) + 2Aa(~) = Ne.T(ft) + 2Aa(~)

(7.17)

where eT(~) is the bulk energy per particle and a(~) is the surface energy. Because neither the number of electrons nor the surface area change during the stretching of the slab, the energy change is given by doT

~E = - N - ~ - ~ -

2A

~.

(7.18)

Noting that ~tAL = N , we have ~E - - A L ~ t

_ de.T 2 da "1 ~n-~n + L~n ~ "

(7.19)

7.2. B UDD- VANNIMENUS THEOREMS

99

Table 7.1

Comparison of numerical calculation of electrostatic potential (in hartrees) with the exact numbers resulting from the Budd-Vannimenus theorem.

rs

r

2 3 4 5 6

_ dcT

- r

n d--~-

0.10474 0.02725 0.00366 -0.00554 -0.00960

0.10478 0.02719 0.00370 -0.00501 -0.01008

We can now compare the equations (7.16) and (7.19) for ~E. Identifying the volume energy terms and noting that for L --+ oc, r tends to its asymptotic value r we obtain r

- r

doT

= ~ d--fi-"

(7.20)

The above sum rule giving exact relation between the surface electrostatic potential and bulk energy is called the Budd-Vannimenus theorem. This theorem may be generalized (Monnier and Perdew, 1978) to the case when a jellium slab of macroscopic thickness, L, is subject to an external potential

V(x) = CO(-x + X)O(x + L + X),

(7.21)

where C is a constant, and X is some microscopic distance. Application of the Hellman-Feynman theorem yields the generalized Budd-Vannimenus theorem r

_doT

- r

n(X)

= n - ~ n + C--=---.n

(7.22)

In the case when positive background n+(x) is a non-negative, symmetric function changing only in the neighborhood of the background boundaries (x - +l) and n+(• = 0, the above generalization of Budd-Vannimenus theorem reads (Peisert and Wojciechowski, 1994)

1// -r176

- n

dn+(x)

oo ~ r dx

dx-~

deT(~) d~ + c n~("X )

(7.23)

As we have already mentioned, the Budd-Vannimenus theorem may be used as a criterion of the accuracy of self-consistent numerical calculations. In Table 7.1 some typical values of r - r calculated by Lang and Kohn (1970) are compared with ~dcT/d~. The agreement is very good at high densities and is slightly worse at

100

C H A P T E R 7. SUM RULES A N D R I G O R O U S T H E O R E M S

the lower densities. To obtain a relation for the derivative of surface energy we have to assume that r tends to its asymptotic value faster than 1/x. Thus taking the limit L --+ oc, and comparing the surface energy terms in (7.16) and (7.19) we obtain the VannimenusBudd sum rule: d--~ = -

[r

- r

dx.

(7.24)

oo

Expressing the bulk energy per particle and the density, ~, through the Wigner-Seitz density parameter rs, the Eq. (7.20) in atomic units, may be written in the form r

1

-- r

-- ~

rs 2

1 9r 471" - 4

--

1 dec r s l - - -~rs dr s

(7.25)

where cc is correlation energy per particle in the bulk electron gas. This equation provides very useful self-consistency criterion for any sophisticated numerical jellium surface calculation. For the most popular, Wigner expression for the correlation energy, Eq. (7.25) takes the form r

- r

= 0.7366r~-2

1 0.1467r~ 0.1527r~- - (rs + 7.8) 2.

-

(7.26)

Eq. (7.24) may be written alternatively as

da 9 /~) drs = 4rr 4 [r

- r

dx.

(7.27)

oo

It is seen that Vannimenus-Budd sum rule, Eq. (7.24) or (7.27), provides a direct way of calculating surface energy from a knowledge of the electrostatic potential. Another useful sum rule was derived for the derivative of the work function with respect to the average electronic density in the bulk. Using the linear response theory, Budd and Vannimenus (1975) have shown that the induced charge distribution (in(x) in a metal, due to a weak external electric field, is related to the work function change through the following equation

-

f

;

(7.2s)

where (I) is the work function.

7.3

The

virial

theorem

Consider the system of N electrons moving in the potential due to a fixed distribution of positive charge with density n+(r) such that f n+(r)d3r = N. If we replace the ground-state (many-body) wave function by a scaled trial wave function, dependent on the parameter A, and use the extremal property of the ground state (Vannimenus and Budd, 1977) we obtain the virial theorem of the form 2(T) + (V) -

0 /l 0A

dr dr' n + ( ~ , r ) [n+(r') - n(r')], i r - r' I

(7.29)

7.3. T H E V I R I A L

THEOREM

101

where we have made use of the fact that the total energy is minimum for A = 1. Here, (T) and (V) are the average total kinetic and potential energies of a uniform electron gas and n+ (A, r) = A-3n+ (r/A). The expression on the rhs of (7.29) is related to the pressure exerted on the positive charges. Thus for the uniform electron gas, in the absence of any surface effects, Eq. (7.29) can be written in the well-known (March et al., 1967) form OE dE 2(T) + (V) - - 3 ~ - ~ = -r8 dr--~

(7.30)

-O"~)NOE

where gt denotes the volume of the jellium, - ( is the pressure and the total energy E = (T) + (V). If the whole system is in equilibrium the term on the rhs of (7.30) vanishes and this equation reduces to the classic-mechanical relation between the kinetic and potential energies. It is clear that Eq. (7.30) provides a self-consistency condition which must be satisfied for the kinetic and potential energies determined separately in a given calculation. To derive virial theorem for the jellium bounded by the surface it is useful to write the total energy of the system as a sum of the volume and surface terms (Vannimenus and Budd, 1977) E = f~f(~) + Aa(~) = N ~ T ( 5 ) + Aa(5)

(7.31)

where A is the surface area, CT is the bulk energy per particle and a, is the surface energy for the uniform electron gas of density ~. Noting that f~ ( O/OFt )N = - ~ d / d ~ and cOA/Of~ = 2A/3f~, the Eq. (7.30), including surface contribution, gives 2(T) + (V) = 3 N ~ z ~

+ A (3~Z~n - 2) a.

(7.32)

Similarly as in the total energy in Eq. (7.31) we split the kinetic and interaction energies per particle into bulk and surface terms to give = (V) =

Nt~(5)+ Aak(~)+..., NE~(~) + Aa~(~) + . . . ,

(7.33a) (7.33b)

with eT(~Z) = ts(fi) + exc(fi) and a(fi) = ak(fi) + av(fi) being the bulk energy per particle of the inhomogeneous electron gas and the surface energy of the semi-infinite electron gas. Comparing the Eqs (7.32) and (7.33) we see that the bulk energies obey the relation 2ts(fi) + exc(fi) = 3~z deT(~t) de " (7.34) The same comparison for the surface terms gives the relation 2ak(~)+av(~)=

(3~ ~fi -- 2) a,

(7.35)

which we will call the surface virial theorem. Combining (7.35) with a = ak + av , we obtain ak -3(1 ~~) l d(aras) (7.36a) = a= r2 dr s

102

CHAPTER 7. SUM RULES AND RIGOROUS THEOREMS

and av=

d )

4-3~n n

a=

1 d (ar 4) r3 dr---~"

(7.36b)

The above relations (7.36) enable to obtain the kinetic and/or potential energy contribution to the surface energy a, if a(~) or a(rs) is known. A more careful treatment of the boundary condition in the derivation of the virial theorem is given by Heinrichs (1979). For a modification of the virial theorem for special geometries see Ziesche and Lehmann (1983) and Lehmann and Ziesche (1984).

Chapter 8

Surface energy and surface stress Surface energy and work function are the most important static physical quantities characterizing metal surface. We have already introduced these quantities in Chapter 4. Here we will discuss in more detail the different contributions to the surface energy and surface stress. Our considerations will be based on the density functional theory described in Chapter 5.

8.1

Surface energy c o m p o n e n t s

In Section 4.8 we have defined surface energy, a, as the amount of work per unit area required to split the crystal in two parts along the plane. Subsequently, we have evaluated the surface energy in the Sommerfeld model of metal as the sum of oneelectron energies. This approximate considerations however did not take into account the interelectronic interactions which contribute significantly to the total energy of the system. Denoting by n(r) the electron density distribution in a half of a metal and n'(r) in the unsplit crystal and recalling that the total energy of the system as a functional of the density is given by the sum of electrostatic, (non-interacting) kinetic and exchangecorrelation energy contributions we can write the surface energy as the sum of three terms a = a ~ + a~ + a ~ (8.1) where following (4.85) the individual terms can be written in the form: a~

=

1 2--A { 2 Z ~ [ n ] - E ~ [ n ' ] } ,

as

=

2A

1 {2T~[n]- T~[n']}

103

(8.2a) (8.2b)

C H A P T E R 8. S U R F A C E E N E R G Y A N D S U R F A C E S T R E S S

104

a~

1 {2Ezc[n]- Ezc[n']}

=

(8.2c)

2---A

The Ee8 is the total electrostatic energy of all positive and negative charge distributions, that includes also the self-energy of the positive charges

1 / n(r)n(r') / n+ (r')n(r) dr dr, ir_r, i drdr'ir_r,I

E~[n] =

1 / n+ (r)n+ (r') dr dr' +2 Ir_r,I

(8.3)

and can be written in the form 1_ [ [n(r') - n+ (r')] [n(r) - n+ (r)] dr dr' 2J Ir-r'l

E~,[~] =

11r

r)[n(r) - n+(r)] dr

(8.4)

[ n ( r ' ) - n+(r') dr'

(s.5)

where

r

r)

Ir-r'!

J

defines the total electrostatic potential energy of an electron. The more explicit general expressions for the kinetic and exchange-correlation energy contributions to surface energy cannot be written since the exact forms of Ts[n] and E~c[n] are unknown. As we will see in the next section more explicit form of these expressions can be given for the jellium model of metal.

8.2

S u r f a c e e n e r g y of j e l l i u m

For the jellium model of metal the electrostatic energy can be written as

iF

ae8 = -~

r

- n+(x)] dx

(8.6)

cx~

where r is the electrostatic potential satisfying Poisson's equation (6.2) and x points the direction normal to the surface. As already discussed in Chapter 5, the exact form of nonelectrostatic-energy functional, Gin] = Ts[n] + E~c[n], is unknown and one of the methods of approximation of G[n] is to expand it in gradients of electron density n(r) (cf. (5.25)). On the other hand in the Kohn-Sham method the kinetic energy contribution to Gin] can be determined accurately and only Exc[n] part is determined approximately. For the jellium model of metal the gradient expansion (Hodges, 1973) allows to write Eq. (8.2b) in the form ~ 0- 8 ~ oo

1 / _ ~ IVn(x)] 2 {gs[n(x)l - gs[~O(-x)]} dx + -~ n(x) dx r

8.2. SURFACE E N E R G Y OF JELLIUM

105

1 / _ ~ z/3 [ 1 (d2n(x)12 +~ n (x) n2(x ) dx 2 (x)

11 8 n3(x)

dx 2

3 n4(x)

dx

4] dx

dx... (s.7)

where

gs[n] = ts(n) n

(8.8)

is the kinetic energy of the uniform electron gas and the second and third integral are the first and second density gradient corrections obtained quantum mechanically. The numerical constant C - 540(3~2) 2/3. It turns out that incorporation of the gradient corrections is necessary for the accurate treatment of the kinetic energy in the metal surface problem (Ma and Sahni, 1979). Making use of (8.8) and of the charge neutrality condition the first term on the rhs of (8.7) can be written in the form

/?

{ts[n(x)]n(x) - ts[~]~O(-x)} dx =

oo

/?

{ts[n(z)] - ts[~]} n(x) dx

(8.9)

oo

Substituting the explicit form for ts[n] given by (4.12), the first term in (8.7) is given by

/?

10

cr

The exchange-correlation contribution is usually treated within the local density approximation. It means that all gradient terms are omitted and the exchangecorrelation surface energy component may be written as

~,= =

Z

{g=[~(~)] - g=[~o(-~)l}

d~

(S.ll)

oo

where g~[n] = ne~(n) and e~ = e~ + e~ is the sum of the average exchange and correlation energies per particle for the homogeneous electron gas. Similarly to (8.9), Eq. (8.11) can be written as

a~c

=

/? /?

{e~c[n(x)] n(x) - e~[g] ~ O ( - x ) } dx

(x)

=

{ ~ = [ ~ ( ~ ) ] - ~=[~1} n(~) ax

(s.12)

A more explicit form of o'~ can be given after specification of the form of exchangecorrelation energy density e~c. The surface kinetic energy, as, can be exactly evaluated in the Kohn-Sham method by summing of the energy parameters of the individual equations and substracting the

CHAPTER 8. SURFACE ENERGY AND SURFACE STRESS

106

potential energy (cf. Eq. (5.37)). The derivation of as term, for any potential barrier shape, given by Lang (1973) is sketched below. Similarly as in Section 4.8, consider a macroscopic slab of crystal of dimensions 2L~:, Ly, L~ (Ly, Lz >> Lx) consisting of N electrons and of uniformly distributed positive charges of density ~ = N/(2LxLuL~ ). The crystal is split in two identical fragments each of thickness Lx - L and with the electron distribution n(x) varying only perpendicular to the two surfaces located at x = 0 and x = - L , each of area A = LyL~. Note that the plane x = - L is at the center of the unsplit crystal. Following (5.37) the total kinetic energy of the fragment is g

oo

Ts[n] - E c i - 2A /_ Veyd[n, x] n(x) dx i=l

(8.13)

L

where ci is the i-th eigenvalue of (5.36) (cf. also Eqs (6.17-6.20)). To obtain the surface kinetic energy we have to subtract from (8.13) the kinetic energy of bulk electrons N

(x3

_bulk _ 2A /_ Vr i=1

where

n'(x)

-L]n'(x) dx

(8.14)

L

= ~ O ( - x ) , and

2 Eocc

N E _ b u5lik

__

[12 (k2 + ku2 + k2) + V~H[n'; _L]1

(8.15)

kx,ku,kz

i=1

Thus

0" s

=

2A

+

_bulk

-

- -

i

1

-

v

i=1

ss

xj

xj

L

/?

(8.16)

L

Since

F

ae(-x)dx

=

L

5

(s,17)

n(x)dx

L

we get

iNE ( c i

as = ~

i=1

_bulk

- ci

)-

F [VesS(x) -

VesS(-L)ln(x)dx.

(8.18)

L

The fist term may be evaluated by the method of Huntington (1951) and with L --+ co, Eq. (8.18) can be written in the following form

1 fok~'[r_6(k)](k2 as - 27r2 -~ - k 2)k dk - / /

[Vess(x) oo

Vess(-c~)]n(x) dx (8.19)

8.3. D I S C R E T E L A T T I C E EFFECTS

107

Table 8.1

Surface energy au of the uniform background model of metal surface and its electrostatic, aes, exchange-correlation, axc, and kinetic energy, as, components (in ergs/cm 2) as computed self-consistently by Lang and Kohn (1970).

r s

2 4 6

(T e s

1330 40 10

(T x c

(:r s

3260 265 55

-5600 -145 -5

(7 u

-1010 160 60

where 6(k) is the phase shift. The calculated surface energies of semi-infinite jellium are plotted in Fig. 8.1 and the numerical examples are given in Table 8.1. An inspection of the results shows that the dominant contribution to the surface energy comes from two terms: The kinetic energy at high densities, and from the exchange-correlation at low densities. The electrostatic energy is a relatively small term and less significant than the others. Contrary to the results for infinite potential barrier the kinetic energy is negative due to a pronounced effect of electron spill-out. The other two surface energy components are positive and compensate the negative kinetic energy for most of the metallic electron densities. However, for the high density metals (rs <_ 2.3) the kinetic energy prevails what results in an unphysical negative surface energy which would suggest that metal would cleave from itself.

8.3

R e i n t r o d u c t i o n of the discrete lattice of ions

In the jellium model of metal the real ion-lattice potential has been replaced by the potential of the uniform positive background. This implies that calculated surface energies and work function were isotropic. However, it is well known from the experiment that for a given metal these quantities depend on the exposed crystal plane. Thus in order to consider the anisotropy of surface quantities we have to reintroduce the discrete lattice. On the other hand the account for the discrete ions allows, as we will see, to rectify a discovered drawback of the jellium model - negative surface energy for high density metals. To describe the effect of the ionic lattice we have to replace the potential r due to the uniform positive background, by the sum of the ionic potentials v(r). In other words we have to add, to the electrostatic potential of jellium, the difference 6v(r) between the discrete-ions potential and the potential of jellium. If this difference were small we could apply a perturbation theory. The straight representation of the potential of the ions by the Coulomb potentials will lead to a very strong potential at the ion positions. However, as we know, the Coulomb potential of the ion can

CHAPTER 8. SURFACE ENERGY AND SURFACE STRESS

108

4000

I

I

I

'\ \

~" E

\

2000

\ \(Txc

0

L.

-"

0

L.

/

(lJ r"

/

-2000

0 0

/

/%

! !

tD

-4000 J!

_

!

-6000

2

l 3

l 4

rs

1 5

6

Fig. 8.1. Surface energy and its components for the jellium (rs = 2) surface. After Lang (1983). be replaced (Heine, 1970; Heine and Weaire, 1970) by the weaker pseudopotential, Wps(r), that corresponds to the same eigenenergy for the conduction electron. The ionic pseudopotential can be written as consisting of a long-range attractive part and a repulsive part wR, acting at short distances Z Wp~(r) = - - - + wR(r).

(8.20)

r

For simple metals Wps(r) may be represented by a simple form due to Ashcroft (1966) for which wR(r) compensates - Z / r term within a radius r~ around the nucleus (emptycore model potential)

~R(r) = _z e ( ~ - l~ i).

(8.21)

r

This potential has proven to provide a good description of the bulk properties of simple metals. The average value of this repulsive or non Coulombic part of the pseudopotential is

~R- ~

er~(r)-

~

e~ 4~r ~-~ = 2 ~ r

(s.22)

109

8.3. D I S C R E T E L A T T I C E E F F E C T S

N

"~

v

w

v

v

v

F'-dhkt~

_

w

v

v

_

_

i

Fig. 8.2. The process of formation of the surface. Now 5v(r) representing a perturbation to the uniform positive background

Wps ([ r - Ri [) - r

5v(r) = ~

(8.23)

i

is assumed to be small and the effect of the replacement of the jellium by a lattice of ions can be calculated using the standard perturbation theory. In the perturbative treatment the difference in the surface energies of the lattice model and that of jellium continuum is entirely due to the differences in electrostatic interaction of all negative and positive charges. In order to separate out the different contributions to the change in electrostatic part of surface energy it is convenient (Lang and Kohn, 1970) to visualize the process of formation of the surface in two steps (Fig. 8.2): In the first step the crystal is cut in two and the two halves are removed one from another while the electron density is held frozen at its bulk value. A contribution to the surface energy coming from this step consists of the electrostatic self-energy of the positive charge configuration (jellium continuum or lattice of ions) and is called classical cleavage energy (surface Madelung energy). In the second step the electron charge density is allowed to relax from the step-function form to its actual surface distribution. A change in surface energy due to interaction of relaxed electron dis-

110

C H A P T E R 8. SURFACE E N E R G Y AND SURFACE S T R E S S

Table 8.2 The calculated cleavage constant, a, for low-index faces of cubic and hcp lattice structures (Paasch and Hietschold, 1977; Monnier and Perdew, 1978).

Lattice

Face

a

SC

(100) (110) (111)

0.00395 0.02465 0.03120

fcc

(111) (100) (110)

0.00325 0.01434 0.04407

bcc

(110) (100) (111)

0.00563 0.03100 0.06295

hcp (c/a = ideal)

(0001)

0.00314

= Mg)

0.00327

(c/a = Zn)

0.00162

tribution with the discrete lattice correction 5v(r) is called pseudopotential surface energy and will be denoted here by aps. The calculation of the classical cleavage energy, acz, involves lattice-summation methods and is rather complicated. However, we may estimate crcz on the basis of a dimensional arguments. Suppose the cleavage plane does not cut through the core regions of the ions. The change in the electrostatic self-energy of the ions, distant by the lattice constant a, per unit area A, is of the order

A

a a2

where Z is the valence. Since the average electron density ~ ~ Z / a 3, we may write

ad = a Z ~

(8.24)

where the dimensionless constant a depends only on the ions conffiguration i.e., on the crystal structure and on the orientation of the crystal plane under consideration. The numerical results of cleavage constant a, for some planes of cubic and hcp lattices, are given in Table 8.2. As it is seen the classical cleavage energy is positive. Note that the classical cleavage energy is calculated accurately (not perturbatively).

8.3. DISCRETE LATTICE EFFECTS

111

The electrostatic relaxation energy or pseudopotential surface energy ap~, thanks to the stationarity of energy functional (5.21), may be evaluated within the first-order perturbation in 5v, using unperturbed electron distribution of jellium. The first order correction to the energy of the system in the presence of discrete ions, can be written as

AE[n] - ] 5v(r)n(r)dr

(8.25)

where n(r) is the electron density of jellium. In order to reduce the increased dimensionality of the problem caused by the replacement of jellium by a lattice of pseudoions we may average 5v(r) over the yz-plane parallel to the surface to get

5v(x) = /A ~v(r) dy dZ / /A dy dz

(8.26)

/ where A is the plane area. In this averaging the ion-lattice planes parallel to the surface are treated as thin, charged sheets. The explicit expression for 5v(x) evaluated with the Ashcroft form of the pseudopotential is given in the Appendix B. Using this form, the average of 5v(r) over the volume of the semi-infinite crystal (Monnier and Perdew, 1978) is

((~V)av

_ 7r~d2 - d1 Jf-d dx 5v(x) wR - 2d 6 '

(8.27)

where d is the distance between crystal planes. Following (4.85) and (8.25) the pseudopotential surface energy, to first order, can be expressed as I .

ap~

] 5v(x)[n(x) -

~O(-x)]

dx.

(8.28)

Numerical calculations of aps show that it is positive and of the same order of magnitude as a~z. This causes that the surface energy of the real metal, i.e. for jellium corrected for the presence of ions, is given by

a = au + aps + a~z

(8.29)

where a~ is the surface energy of jellium, is positive for all electron densities. Numerical values of different components of a, for most densely packed faces of selected metals, are given in Table 8.3. In evaluation of Eq. (8.28) it is tacitly assumed that ion cores do not overlap. For some of more loosely packed planes, however, for example for (111) plane of bcc metals, this condition is not satisfied, i.e. we have rc > d/2, where rc is the Ashcroft pseudopotential core-radius and d is the interplanar spacing. This can be accounted for in a slightly different form for 5v(x) as described in the Appedix B. This case is usually recognized by appearance of an extra, profile independent term erR, where

a R - -;r~2d ( r c - ~d)20 ( r c - 2d) ,

(8.30)

C H A P T E R 8. SURFACE E N E R G Y AND SURFACE S T R E S S

112

Table 8.3 Surface energy, a (in erg//cm 2) of most densely packed faces of simple metals calculated in the perturbational self-consistent scheme (Lang and Kohn, 1970) as the sum of jellium contribution, au, and two lattice contributions: pseudopotential, aps, and classical cleavage energy, acl.

Metal

Face

A1 Mg Na K Cs

(111) (0001) (110) (110) (110)

rs

2.07 2.65 3.99 4.96 5.63

a~

aps

act

a

-730 110 160 100 70

1050 300 35 25 20

408 130 33 17 12

730 540 230 140 100

is added to the rhs of a in (8.29). The important conclusion is that incorporation of the discrete lattice of ions corrects the serious drawback of the jellium model - the calculated surface energies are positive in the whole range of metallic densities.

8.4

Variational

treatment

of lattice effects

As we see the introduction of discrete lattice by first-order perturbation in (iv(r) leads to a very large change in surface energy especially for high density metals. As far as the electron density distribution is not much affected by the perturbation this is not contradictory. It occurs however, that even in the metals like A1, 5v(r) can be much greater than the value which would justify the application of first-order perturbation theory. Besides, for more open faces the average value of 5v over the volume of the semi-infinite metal gives a large fraction of the free-electron Fermi energy and of the potential barrier height at the jellium surface. For instance (hV)av equals to 3.0 eV for Al(ll0) and 1.8 eV for C s ( l l l ) (Monnier and Perdew, 1978). This large potential does not influence the electron density variation in the bulk, but its abrupt termination at the surface will affect the electron density variation in the surface region and it will give rise to an extra surface-dipole moment. These facts suggest that 5v should be treated more accurately than in the first-order perturbation theory. A complete description of the ground-state energy of a system of interacting electrons in an external potential arising from the interaction with ions represented by a local pseudopotential can be expressed as a following functional of the electronic density n(r):

1/

E[n] = Ts[n] + Exc[n] + -~

dr dr'

ir-r'l

8.4. VARIATIONAL TREATMENT OF LATTICE EFFECTS

S

+

dr

~ ~(I r -

1

I)n(r) + 7 ~

Ri

i

,

113

Z 2

I R~- Rj I"

i,j

(8.31) The first three terms in (8.31) represent the kinetic energy of noninteracting system, exchange-correlation energy and the electrostatic energy, respectively. The last two terms describe the interaction of electrons with the ions at sites Ri via a pseudopotential w(r), and the Coulomb interaction between ions. The prime in the last sum means that i = j term is excluded from the summation. In order to improve the convergence of the individual Coulomb terms appearing in (8.31) it is convenient to add and subtract a neutralizing positive background of jellium i.e., n+(r) = ~O(-x). With the ionic pseudopotential of the form (8.20) the energy functional can now be written in the form

E[n]-

1/"

Ts[n I + Exc[n] + 7

dr r

r)In(r) - n+(r)]

+ i dr5v(r)In(r)-n+(r)]+ S dr EwR(ir- R~ i)n+(r)

1S drdr' n+(r)n+(r') l.rEI r - R"~ l Ir-r'l i

+7

n+(r)

l

1

+~~ I R , . . z~3

Z2

- RjI'

(8.az)

with r

r)-Sdr

n(r[!- n+(r')

'

,

_r,

and 5v(r)=Ew(Ir-R

,

[

(8.33)

I

n+(r')

i I)+jdr'lr_r,

I

(8.34)

i

Now, according to the density-functional theory (cf. Chapter 5), the correct electronic density n(r) can be obtained by minimizing the energy functional (8.32). Since only the first four terms in (8.32) depend on n(r) we can consider only the expression

E[n] -

11dr

Ts[n] + Exc[n] + -~ f

+ j dr ~v(r)[~@) -

n+

r

r)[n(r) - n+(r)]

(r)]

(8.35)

which differs from the (6.12) functional for jellium by the presence of last term. Thus the self-consistent electron density may be constructed from auxiliary wave functions which satisfy Schr5dinger equation

--~

]

+ Velf(n; r) ~oi(r) = ei~oi(r)

(8.36)

114

CHAPTER 8. SURFACE ENERGY AND SURFACE STRESS

with the effective potential

Veff(n; r) = r

r ) + 5v(r) + V~(n; r)

(8.37)

where V~(n; r) = 5E~/hn(r). The self-consistent solution of Eq. (8.36) is a formidable task due to the three-dimensional spatial variation of Ve//. But the dimensionality of the problem can be reduced by application of the uniform background model. Now the electron density varies in the direction perpendicular to the surface only. In a simplest approach the electron density profile n(x) can be parameterized directly and the parameters are determined from the minimum of energy. Since the bulk density is held fixed during the variation, only the surface energy can be minimized. This approach, however, is restricted to using gradient expansion for the kinetic energy. Different classes of the trial density profiles are discussed in Chapter 11. Another possibility (Monnier and Perdew, 1978) is to replace the 5v(r) term in (8.37) by a simple parameterized function Y(x). Then, the variational wave-functions are generated from the solution of the Schrhdinger equation with the effective potential

Veil(n; x) = r

x) + V(x) + Vxc(n; x)

(s.38)

where r and V~c are obtained from (6.16) and (6.15). Note that for V(x) --+ O, equation (8.38) reduces to the effective potential for jellium [EQ. (6.14)]. It turns out that it is sufficient to take V(x) in the step-function form

v(x) =

(s.39)

The optimum value of C = Cm minimizes the surface energy a[n] and allows to determine the exact electron density profile n(x, Gin) - n(x). In contrast to the electron density profiles of the jellium model the variational density profiles demonstrate quite noticeable face-dependence (Fig. 8.3). For positive C the electron density is more spread out into the vacuum region compared with that of jellium, while for negative C the opposite effect is observed. This of course has important consequences for the calculated surface energies of particular crystal planes. Through the modified electron density, the effect of the ionic lattice will be felt also by the kinetic and exchange-correlation parts of surface energy. The surface energies calculated by this non-perturbative method generally agree, within 20%, with those of perturbational treatment of Section 8.3. The agreement is the best when (~V)av is small compared to Fermi energy EF(~Z) and to the potential barrier height at the surface. The numerical results of variational self-consistent calculation for A1 and Na are given in Table 8.4. As it is seen the surface energies show a strong anisotropy especially for A1. The calculated values are in accordance with bond-breaking arguments that the more open faces will have the higher surface energy. The predicted very strong face-dependence of surface energy, which for the ratio of the energy of the (110) face to the energy of the most densely packed (111) face gives 4.5, disagrees with the experimental findings. A zero temperature extrapolation of measured surface energy differences between various faces of lead shows that this ratio does not exceed 1.1 (Heyraud and Metois, 1983).

8.5. STRUCTURELESS PSEUDOPOTENTIAL

I

'

'

'

'

I

115

'

1.0

'

At

'

'

i

fcc

_

If: x c

0.5 Illl )

.......

(100}

. . . .

(110)

I

-0.5

,

,

~'..~,".~

~ : X" ~ , ,,%

-

~ . ~ ,

,

i

0

,

,

,'"

L

0.5

x ( 2'IT/kF } Fig. 8.3. Variational self-consistent electron density profiles at low-index faces of A1. Redrawn with permission from Monnier et al. 11978). Q1978 The American Physical Society.

8.5

Structureless pseudopotential m o d e l

The averaging of the potential difference 5v(r) over the plane parallel to the surface allowed us to remain within the one-dimensional problem. However, the crystallinity effects have introduced some additional parameters to the model: the metal valence Z, crystal structure and the pseudopotential core radius. It means that the price we had to pay for accounting for these effects was a loss of simplicity and universality of the jellium model. These features of the model can be recovered, however, in the structureless pseudopotential or stabilized jellium model (Perdew et al., 1990) which evolves from the variational self-consistent method of the previous section. 1 Let us start from a discussion of bulk and surface contributions to (SV}av expressed by (8.27). Consider crystal assembled from the sphericalized Wigner-Seitz cells, of radius r0 = zl/3rs, c u t along the exposed boundaries of the cells. In each cell the ion is smeared into a positive background n+(r) n+(r) -- ~ e ( r )

(8.40)

where O(r) is the step function, equal 1 inside a metal and zero outside. Let ~vws(r) be the sum of the pseudopotential from the ion in one of such cells and the electrostatic potential of the uniform electronic density contained within it (with the convention that ~vws(r) from this cell vanishes far outside it). Making use of Eqs (4.80) and 1The similar model called "ideal metal" which has similar surface properties was developed independently by Shore and Rose (1991).

CHAPTER 8. SURFACE E N E R G Y AND SURFACE STRESS

116

Table 8.4 Anisotropy of surface energies calculated in the variational self-consistent scheme (Monnier and Perdew, 1978).

Metal

Face

Surface energy (erg/cm 2)

A1

(111) (100) (110)

643 1460 2870

Na

(110) (100) (111)

227 245 321

(8.20) we find

5vws(r)

~-

(

Z 3Z r + wn(r) -4 2 ro

Zr2 I 2r ] O(ro - r).

(8.41)

Calculating the average of 5vws(r), over the volume of Wigner-Seitz spherical cell, for the first term in (8.41) we have l forO (Z) -~ dr 41rr2~,

3Z - 2r0

(8.42)

which will cancell the third term in (8.41). Making use of expressions (8.22) and (4.81) for the average value of the remaining terms finally we get ((~VWS ) av "" "WR --

3Z _ 27r~r2 lOr---o

3Z lOro"

(8.43)

Noting that the Madelung energy s for the Wigner-Seitz sphere containing point ion of charge Z and a uniform negative background ~ is

eM =

3Z + ees = 2ro

9Z 10ro

(8.44)

Where ees - -3Z/(5ro) is derived in Chapter 4 [Eq. (4.81)]. Eq. (8.43) can now be written as ( ~ V W S ) a v ~'~ W R "4- s "Jr ~es. (8.45) We see that (Svws)av, being the bulk contribution to (SV)av, is given by the sum of repulsive part of pseudopotential, average (electrostatic) energy of a collection of point

8.5. STRUCTURELESS PSEUDOPOTENTIAL

117

ions embedded in a uniform negative background of density ~, and the electrostatic energy of self-repulsion of this background inside the sphere of radius r0. It follows from the above considerations that in order to transform jellium into a real metal one has to subtract from the jellium energy functional (6.12) the electrostatic self-repulsion energy of positive background in each cell i.e., ~e8 / dr n+(r)

(8.46)

and to add the interaction between the electrons and the potential ~v(r) dr 5v(r)n(r).

(8.47)

In order to model 5v(r) we require that: (i) It tends to a constant value deep in the metal, (ii) It yields correct bulk energy and chemical potential. Thus, the simplest model for 5v(r) is (Svws)av. In this approximation the total energy of the real-metal model, as a functional of electron density is

Eps[n, n+]

= Ejeu[ n, n+] + (s

-[- WR) j dr n+(r)

+(Svws)a, J dr e ( r ) [n(r) - n+r)]

(8.48)

where Ejeu is the standard energy functional for the jellium given by (6.12). Following Eq. (5.32) we can define the effective one-electron potential

Yeff[n(r)] = r

r) + Yxc[n(r)] + (Svws)avO(r).

(8.49)

For a given rc the (Svws)av is determined by (8.43) or from the following relation

(, vws)o

=

d

+

(8.50)

However, we can go one step further and write the stability condition for the jellium, Eq. (4.83), corrected for a presence of discrete ions d

d---~[ts(~) + e=c(~,) + s

~- WR] - - -

(8.51)

0.

Solving (8.50), we find that at the equilibrium 2

2 4 dec

1Z2/3r2

(8.52)

From Eqs (8.50-8.51) it follows that if the pseudopotential has the Ashcroft form (8.20-8.21), a t the equilibrium density we have ( (~V W S ) a v "- - - ~ " ~ nd

[ts(~)+

sx c

(~)]

(8.53)

118

CHAPTER 8. SURFACE ENERGY AND SURFACE STRESS

1200

1000 E (9

f

AL

'

I

,

I

i

I

~

_

Zn

_

~(D

Mg

800

b >, C:D L_ (1) (.-

600

I,I

o

_

:"

400

200

F

I 'Oe,,,un

0I 2

Ii

I 3

_ i

rs

I i 4 (bohrs)

I 5

i

6

Fig. 8.4. Surface energies calculated for jellium and structureless pseudopotential model (Kiejna, 1993) compared with measured liquid-metal surface tensions - dots (Tyson and Miller, 1977) and results for stabilized jellium with structure - crosses (Wojciechowski and BogdanSw, 1994).

or taking into account (8.52),

(~VWS)av

--- - - 5

rs2 + ~

-T

1

rsl + -~rsdr---~.

It means that (SVWS>a v depends upon rs solely. As we see, by correcting the jellium model for the existence of discrete ions, we have transformed it into the structureless pseudopotential or stabilized jellium model. W h a t we have gained is the possibility of accounting for the crystallinity effects without any loss of simplicity of the model. The second term on the rhs of (8.48) is a correction to the jellium bulk energy while the third one gives a correction to the jellium surface energy. Following Eqs (4.85) and (8.1) the surface energy of stabilized jellium can be written as

f where ajeu is the standard jellium surface energy functional (see Sec. 8.1 and 8.2). The calculated surface energies for stabilized jellium 2 as function of r s (Kiejna, 2The self-consistency of stabilized jellium calculations can be controled by the sum rules (Kiejna

et al., 1993) similar to those presented in Chap. 7 for the ordinary jellium.

8.6. SURFACE STRESS

119

1993) are plotted in Fig. 8.4 as a solid line. In the same figure the surface energies calculated variationally from (8.55) but with (Sv)ws given by (8.50), not by (8.53) are also marked (Wojciechowski and BogdanSw, 1994) (crosses). Their comparison with calculated values for the ordinary jellium shows much improved agreement with measured liquid-metal surface tensions extrapolated to zero temperature (Tyson and Miller, 1977).

8.6

Surface stress

In Chapter 3 we have introduced the thermodynamic concept of surface stress. Recently this quantity has been calculated for several different metals and faces (cf. Needs et al. (1991) and references therein). The results of first-principle studies and calculations using semi-empirical potentials have suggested that surface stresses play a rSle in driving reconstruction of surfaces (Needs, 1987; Needs et al., 1991; Needs, 1993; Wolf, 1993; Cammarata, 1994). Residual surface stresses which act within the surface plane result from the reduced coordination number for the surface atoms in comparison to atoms in the metals interior. The reduced number of neighbors leads to a lower, than optimal for the bulk, electron density around the surface atoms. This will cause that surface atoms would try to reduce interatomic distances in order to increase average electron density. A shortening of the interatomic distance perpendicular to the surface (inward relaxation) will be stopped because of the electrostatic repulsion between atoms. Thus in order to increase the density to the optimal value there will be a tendency to reduce interatomic distances in surface plane. This will result in a positive or tensile stress. Theoretical and experimental results suggest that this type of stress is displayed at the surface of most metals. A negative or compressive stress seems to be characteristic for materials with more complicated crystal structure. In order to discuss the different energy contributions to the surface stress it is useful to employ the local density functional theory and to write the total energy of the system in the form E-

E

/

(1V2) dr C n - ~ Cn

lff +~-~

lgv2 / dr~ (r)+ dr n(r)cxc[n(r)]

(8.56)

n

where $(r) is the electric field produced by the electrons and ions and the electron density n(r) = ~--~n I Cn 12 is calculated from the electron wave functions Cn. Using the variational ansatz for the wave functions of the strained system, Cn = Cn(eij, r) where eij denotes the elastic strain tensor, the corresponding stress is given (Nielsen and Martin, 1985) by OE

Tij = Ocij =

E

/ drCnViVjCn + ~1 / [dr

1

$ i ( r ) e j ( r ) - -~Sij$2(r)

]

n

+

pxc(n(r))]

(8.57)

120

CHAPTER 8. SURFACE E N E R G Y AND SURFACE STRESS

where #xc is the exchange-correlation potential. We shall call the terms appearing on the rhs of Eq. (8.57) as the kinetic, electrostatic and exchange-correlation contributions to the stress. It should be emphasized that a one-to-one correspondence between the terms in (8.56) and (8.57) does not mean that the particular terms in Eq. (8.57) are simply the strain derivatives of the corresponding terms in energy (8.56). This can be seen considering the total energy to be a function of the strain and the wave functions (which also depend on the strain), i.e. to be of the form E(eij, Cn(cij)]. The stress can be written as

dE OE deij = Oeij + E

f OE 0r j OCn Oeij dr.

(8.58)

n

Because the total energy, E, satisfies variational principle the second term on the rhs will vanish. For a particular contributions to E, such as for example from kinetic energy, the second term, generally, will not be equal zero. These correction terms must cancel one another when calculating E. An example of the system for which these terms will vanish is the bulk jellium. When the positive background is homogeneously strained, the electron wave functions (i.e., plane waves) will be simply stretched, so the second terms will be identically zero. But for the surface quantities these terms will not be zero even for jellium. In order to understand the origin and different contributions to surface stress, it is useful to consider the jellium model. In this model the change of the strain de, is given by dfl de - - (8.59) f~ where fl = ~ r r 3 is the volume per one electron. Accounting for this in Eq. (3.69) or (3.71) one gets a simple formula for surface stress of jellium g =

rs da 3 drs

+---

(8.60)

where a is the surface energy. Since a is determined self-consistently for a number of r s values (see preceding Sections) one can evaluate the strain derivative represented by the second term in (8.60) and consequently g. The plots of g and a, as functions of density parameter rs, are drawn in Fig. 8.5. The values of surface stress, g, and surface energy, a, for rs -- 2.06 as well as the different energy contributions to these quantities are given in Table 5.5 and compared with the corresponding values for the A l ( l l l ) surface. Note that these contributions are not given simply by, for example, ak~ + 3 dake/drs, where the subscript "ke" denotes the kinetic energy part. There are, however, some relations between the individual contributions to g which were used to obtain the numbers given in Table 8.5. The first one is simply

g = gke + gxc + ges.

(8.61)

The electrostatic contribution to the surface stress, given by the second term in Eq. (8.57) is known as the Maxwell stress (Jackson, 1975). At a flat jellium surface the electric field must be pointed normal to the surface and there is no electric

121

8.6. SURFACE S T R E S S

1.25

~

T

~

T

I

1.00

% o~ oO

o.Ts 050

I,_

o'1 "0 C 0

>,

I..,.,.

0.25 0.00

r

r

-0.25

0

-050

"-I..,.

-0.75

//~~~t~_l__.............J

2

3

4

5

6

rs (a.u.)

Fig. 8.5. Calculated surface energy, a, and surface stress, g, for jellium as functions of the density parameter. Redrawn with permission from Needs and Godfrey (1990). @1990 The American Physical Society.

field in the bulk. Thus from the form of the Maxwell stress follows immediately that the electrostatic surface stress is negative and is given by =

(8.62)

where a ~ is the electrostatic part of the surface energy. Another relation can be obtained taking the three dimensional trace of Eq. (8.57). Calculating the difference between the volume averaged surface and bulk stresses one gets, per unit area 10(AT) = - 2 a k e - aes + 3gxc. (8.63) A Os The derivative on the left hand side can be calculated by noting that under a strain 1 e~j, A and rs become (1 + exx + %y)A and [1 + 5(ex~ + %y + ezz)]rs respectively, to yield da 2a + rs dr~ - -2ak~ - a e ~ + 3g~c. (8.64)

122

C H A P T E R 8. SURFACE E N E R G Y AND SURFACE S T R E S S

Table 8.5 The surface energy, a, surface stress, g, and their components for jellium of the average electron density corresponding to aluminium (rs = 2.06) compared with the results of ab initio calculation for A1(111) surface. All entries are in eVA -2. After Needs et al.

(1991).

Total Kinetic Exchange-correlation Electrostatic

Surface energy Jellium Al(lll)

Surface stress Jellium Al(lll)

-0.048 -0.303 +0.183 +0.072

+0.108 +0.234 -0.054 -0.072

+0.060 -0.258 +0.166 +0.152

+0.078 +0.216 -0.051 -0.087

Making use of Eq. (8.60) we obtain 3g - a = -2ake - aes + 3gxc.

(8.65)

Note the analogy of this equation with the virial theorem for the jellium surface [Eq. (7.35)]. Using the results for a calculated by Lang and Kohn (1970) and the relations (8.61), (8.62) and (8.65), allow to obtain the individual contributions to the surface stress of jellium (Table 8.5). The comparison of results of Table 8.5 shows that there is a good agreement between jellium and ab initio calculations of the kinetic and exchange-correlation contributions to the surface energy and stress for aluminum. It is also seen that the dominant contribution to the surface stress, similarly as for surface energy, arises from the kinetic energy. The neglect of the discrete ionic structure in the jellium model manifests in a poor agreement with ab initio calculations for the electrostatic contribution to the surface energy and surface stress. Nevertheless the results presented in this table demonstrate that the jellium model provides a good starting point for a qualitative discussion of the origins of surface stress. The presence of large enough stresses may lead to surface instabilities. It has been suggested that the strain derivative of surface energy, being the difference between surface stress and energy, g - a = da/de, is the driving force for surface reconstruction (Wolf, 1993) or the change of surface density of atoms (surface densification). Surface atoms tend to reconstruct towards a thermodynamically more favorable state for which g = a. A reconstructed surface structure is no longer in perfect registry with the underlying lattice. The results of theoretical studies of this problem lead to the conclusion that for the positive strain derivative of surface energy, the density of surface atoms tends to increase whereas for negative da/de, the density of surface atoms decreases. The calculations of surface stress for metal (A1, Au, Ir, Pt) surfaces show that in all cases the g - a is positive, thus suggesting that in most cases only first possibility of reconstruction, i.e. by an increase of surface atom density is likely.

Chapter 9

Work function 9.1

The definitions

Wigner and Bardeen (1935) have defined the work function as "the difference in energy between a lattice with an equal number of ions and electrons, and the lattice with the same number of ions, but with one electron removed. It is assumed, in both cases that the lowest electronic states are completely filled, so that the electron is removed from the highest energy state of the neutral metal." In order to use this general definition, let us consider the transferring of an electron from the crystal, bounded by planes (hkl) =_ i, to the point r outside the crystal. For such a transferring the following work is needed @i : A r

+ EN-1 -- EN : A r

AE,

(9.1)

where the rise, Ar in electrostatic potential across the /-th crystal surface is called the surface dipole barrier and A E = E N - EN-1. EN is the ground state energy of a metal with N electrons, whereas EN-1 denotes the ground state energy of metal from which one electron is removed. This work (I)i, is called work function, under assumption that the metallic sample with N electrons and N - 1 electrons contains the same number of ions. It is to be noted that though both r and E depend on the choice of energy zero, their difference does not. Assuming that the potential V(r) of the static distribution of ions does not change under transition from the metal sample with N electrons to the one with N - 1 electrons, we arrive at the equality AE=

~

Y(r),T

- #,

(9.2)

where # is the chemical potential. Thus Eq. (9.1) takes the form (I)~ - A r

#

(9.3)

Both terms that appear on the right-hand side depend on the arbitrarily chosen zerothpotential. Depending on the choice of this potential, the definition of the work function may be expressed in a different way, for instance: 123

124

CHAPTER 9. W O R K FUNCTION

Wigner-Seitz work function: The zeroth potential is taken as the electrostatic potential cWS, on the WignerSeitz sphere (compare Sec. 1.2) (Wigner and Seitz, 1933; 1934; Seitz, 1935), i.e. Ar Ws = Ar - c w s ,

(9.4)

and #ws

_ # _ r

(9.5)

thus o w s = AcWS _ # w s .

(9.6)

Lang-Kohn work function: The mean electrostatic potential inside the metal - ~

r

dr.

(9.7)

is taken (Lang and Kohn, 1971) as a reference potential, with ft being the metal volume. Thus ~LK = A~ i _ #, (9.8) where Ar -- Ar

r

-

(9.9)

and m

#=#-r

(9.10)

Here # is the chemical potential of the infinite system, relative to the mean electrostatic potential in this system.

9.2

W o r k f u n c t i o n of s e m i - i n f i n i t e j e l l i u m

For the uniform degenerate electron gas of density ~, from Eqs (5.24) and (5.28), putting r 0, we have _

#-

6G[n] _ 1 k2F + d [~exc(~)] 6n - -2 d~ '

(9.11)

where exc is the sum of the exchange and correlation energy per particle (compare Chap. 4). Therefore, using Eq. (9.8) we obtain 1 - a & - 5k -

d

(9.12)

For the jellium model we may omit the index i, of the surface dipole barrier because n(x) is independent of crystal's face. Thus finally the work function is given by the expression 1 d (I)g -- A r ~k 2 - ~ [ns Ar ' (9.13)

df~eT(~)df~

9.2. WORK FUNCTION OF JELLIUM

125

where the electrostatic surface dipole barrier Ar is obtained from (6.16) and can be written in the form Ar

-

r

-

r

-

xnT(x) dx,

4r//

(9.14)

oo

where

nT(X) -- n ( x ) - 'h,e(--x).

(9.15)

Equation (9.13) gives the total work function of the uniform background model, denoted by Lang and Kohn (1971) by (I)u. Very often a one-particle picture of work function is considered, and the work function is defined as the difference between the potential energy W, of an electron in the vacuum region, and the maximum energy of a one-particle state occupied at T - 0 K, i.e. the Fermi energy, EF (compare EQ. (4.43) and Fig. 4.4). From the many-electron point of view this is obviously an approximation and the question arises whether expressions (9.1) (thus also (9.13)) and (4.43) are identical. For a reasonable comparison of these two expressions the total energies in (9.1) have to be expressed by the one particle states under consideration. Actually Schulte (1977) has demonstrated that in the density-functional formalism the work function can be obtained from the single-particle Schrhdinger equations in the way suggested by the Sommerfeld model of Sec. 4.1. In the light of the above, the potential energy W, of an electron in the vacuum region according to (9.13) is

W - 4~r

xnT(x) d x - -~n [~exc(~)] 9

(9.16)

oo

The expression (4.43), with (9.16), is equivalent to (9.12) and is often called the gooproans work function, ~2g. This name comes from the Koopmans theorem (Koopmans, 1933) which states that if the Hartree-Fock eigen-functions in Slater determinants (4.61) are the same in the system with N, and in the one with N - 1 electrons, the eigenvalues can be interpreted as one-particle ionization energies. For further discussion see Seitz (1940). Another formula for the work function of semi-infinite jellium may be derived by making an alternative estimate of energy difference A E appearing in Eq. (9.1). Namely, denoting by nr. (r) the normalized to unity infinitesimal change in the electron density between two ground states EN and EN-1 , nr.(r) dr - 1

(9.17)

one can express the difference AE, as follows (Mahan and Schaich, 1974)

AE-

E N - EN-1 -- / dr nz(r) ( 0~ [neT(n(r))] + r

(9 18)

where eT(n) denotes the sum of mean kinetic, exchange and correlation energy per particle.

CHAPTER 9. WORK FUNCTION

126

Since after removal of one electron, the density readjusts so that it has the same average value deep inside the metal as before, the actual change in charge density is localized on the surface. Thus the charge density contracts a little near the surface, by a distance 5x, and the electron density profile n(r) changes into n ( r - ~Sx), where denotes the unit vector along the x-axis perpendicular to the surface. Therefore the change in the density is n~(r) = n(r - ~Sx) - n(r) --~ m--~x dn SX.

(9.19)

On the other hand, for a semi-infinite sample of surface area A, one can take n z ( r ) in the form 1 0n(r) nz(r)= ~A 0----~" (9.20) From Eqs (9.17), (9.19) and (9.20)we find that 5x ~ (~A) -1.

(9.21)

Substituting (9.19)with (9.21)into (9.18), we have AE

=

/

dr

(ldn)[d ~A dx

]

~ (neT) + r

1 [n(x)eT(n(x))]~-oo + / ~

1 dn(x)] dx r

dx

(9.22)

O0

Let us consider now the last two terms in Eq. (9.22). In the first one the electron density profile, n(x), appears which for the semi-infinite sample must satisfy the conditions: n(x) ~ ~, for x -+ - c ~ and, n(x) --+ 0 for x -~ +c~. From these conditions follows that 1 - n [n(x)cT(X)]~-c~ = CT(X --+ --C~). (9.23) where eT(X --+ --c~) means eT(n(x --+ --c~) = ~). Taking into account that the total charge density, nT(x) = n(x) -- ~O(--x), the second term in (9.22) can be rewritten in the form

/;[ oo

dx

]

1 dn(x) ~ dx r

/;

[

]

1 OUT(X) 5(X) - n Ox '

oodXr

(9.24)

where 5(x) is the Dirac function. Since = r

(9.25)

CX)

where r denotes the electrostatic potential at the surface (jellium edge), we must only calculate the integral

1 /_~ I=------_ dx r n

OUT(X) Ox "

9.2. WORK FUNCTION OF JELLIUM

127

Taking into account Eq. (6.16) for the electrostatic potential and integrating by parts, we get

I =

4~//

dxnT(x) / : c~

dx'nT(x'). oo

After interchanging the order of integration I

n

.__

oo

dx'nT(x')

dxnT(x),

we may add the last two expressions and divide by 2, to yield

47r ---~

I---~1

dx nT(X)

-- 0,

which vanishes because of charge neutrality condition (4.47). Thus, the identity (9.24) with (9.25) gives

// [ ~

dx

1 dn(x) ex

~

]

r

= r

(9.26)

r

(9.27)

Therefore, finally we get

AE-

~T(x -+ - ~ ) +

and, according to the definition (9.1), the Mahan-Schaich work function can be expressed in the form cMs

_ r

_ r

- ~.

(9.2s)

The work function calculated from the above expression and from that due to Koopmans are equivalent I if the electron densities are calculated self-consistently. Another expression for work function may be derived by analogy to the change in self-consistent field (ASCF) method for calculating ionization energies of atoms (Hedin and Johanson, 1969; and the references therein). As it is well known from elementary electrostatics the excess charge on a metal will be localized at the surface, thus the work function, starting from the definition (9.1), is given by the relation - lim

1 ( E [ n z ] - E[nr.=o]]

(9.29)

where n z ( r ) is the electron density for a state in which E electrons per unit area A have been carried off to rest at x -+ c~, and remaining electrons inside the metal have relaxed by formation of the surface charge density E (Monnier et al., 1978). Since 1F r o m t h e B u d d - V a n n i m e n u s

t h e o r e m (7.20) we h a v e deT

r

= r

thus t~ M S

--

r

-- r

--

+ ~ d~ '

deT dfi

Comparing this with (9.13) we find that 9K = ~MS

- ~

-

~r

d

-=-_( ~ ( ~ ) ) . a n - - -

CHAPTER 9. WORK FUNCTION

128

this relaxation process leaves the bulk electron density unchanged by the removal of an electron, we can write (9.29) in terms of the surface energy a alone, namely 2

OA

da

(9.30) E=O

In the case of jellium, when the class of variational density profiles admits a rigid displacement relative to the ionic background, a great simplification of the above work function expression occurs, i.e. formula (9.30) takes the form (9.28).

9.3

D i s c r e t e - l a t t i c e c o r r e c t i o n s to t h e w o r k f u n c t i o n

As it was mentioned above the work function of real metals is strongly face-dependent. For instance, the work functions measured for the (111), (100) and (110) planes of aluminum single-crystal are 4.26, 4.20 and 4.06 eV, respectively (Eastment and Mee, 1973). The effect of the actual ionic structure on the work function was considered by Lang and Kohn (1971). Similarly as in the calculation of the lattice effects on surface energy, in the first approximation the difference 5v(r), between the pseudopotentials of the lattice of ions and the electrostatic potential of the uniform positive background may be treated as weak and the first-order perturbation theory can be employed. Then, the ground-state energy of the neutral metal (with N electrons) in the presence of the perturbation is

Eg(~V)

-

E J + / 5v(r)ng(r)dr

(9.31)

where E J is the ground-state energy of the jellium and n g ( r ) is the electron density associated with the ground state in the presence of the perturbation. Now, consider a second system consisting of N - 1 electrons in the lowest state in the metal and one electron removed to infinity. Putting in the vacuum region 5v(c~) = 0, this energy can be written as

Ey_l(SV)

-

E J_l + / 5 v ( r ) n g - 1 ( r ) d r ' + r

(9.32)

where r is the value of electrostatic potential away from the surface (at infinity). The difference between the energies of these two systems defines the work function. Thus using (9.1) we find for the first-order correction, 5(I), to the work function

5O - - / 5v(r)[ng(r) -- aN-1 (r)] dr.

(9.33)

The difference of the densities appearing under the integral (9.33) describes a charge deficiency due to removal of one electron from the metal. This deficiency is localized in the surface region and can be denoted as an(r) = aN(r) -- nN-l(r) 2In Eq. (9.30) we use the index A instead of the a c ro n y m A S C F for the which is usually used in the literature.

field m e t h o d

(9.34)

change in self-consistent

9.3. DISCRETE-LATTICE CORRECTIONS

129

Table 9.1

Lattice corrections to the work function, 5~, and the face-dependent work function, 9, of A1 and Na computed by Lang and Kohn (1971). Metal

Face

5~ (eV)

9 (eV)

A1

(111) (100) (110)

0.19 0.32 -0.21

4.05 4.20 3.65

Na

(110) (100) (111)

0.03 -0.29 -0.39

3.10 2.75 2.65

where n~(r) satisfies the normalization condition (9.17). With this notation Eq. (9.33) can be written in the form

5~ - - / 5v(r)n~(r) dr.

(9.35)

Now, it remains to determine nr~(r). Since the missing electron rests at "infinity", we may smear out its charge over a thin sheet (parallel to the surface) and regard the charge deficiency nz(r), as induced by a uniform electric field directed perpendicular to the surface. Thus, nz(r) can be treated as a function of x only. Averaging 5v(r) over the yz-plane (compare Eq. (8.26)) and making use of (9.17) we may write

5~ - - A

/

5v(x)n~(x) dz -

oo

5v(x)nz(x) dx

//

!

nr~(z) dx,

(9.36)

where A, is the surface area and nE is the charge per unit area. An example of the magnitude of the corrections, 50, to the work function is given in Table 9.1. As is seen the corrections are of the order of few tenths of eV, as they should be, and have different signs thus giving the differences of the work function between particular faces reasonably comparable with the experimental data. As we have already discussed in Section 8.4, the assumption that 5v(r) is weak and can be treated by the first-order perturbation theory is not justified for many surfaces. In such cases a variational treatment is more appropriate. In a variational approach the face-dependence of the electron density profile n(x), and thus the corresponding dependence of work function Oi, may be taken into account by the use the effective potential in which r is replaced by r + 5v(x). Such a replacement in Eq. (9.24) gives

dx

1 dn(x)

[r

+ 5v(x)]

-

dx 5v(x)-~,~1dn(X)d___~.

r O0

(9.37)

CHAPTER 9. WORK FUNCTION

130

Taking this into account in Eq. (9.22) and using definition (9.1) finally we arrive at the formula 3 d -ni(x) (9.38) ~D : oMS zr / dx 5vi(x) d--x -~' (x)

where (~MS is given by Eq. (9.30) and index i, denotes that the density profile depends on the crystal face.

3In Eq. (9.38) we use the index D, for the displaced-profile change-in-self-consistent-fieldexpression instead of the acronym DPASCF.

Chapter 10

Work function of simple metals: relation between theory and experiment 10.1

J e l l i u m part of the work f u n c t i o n a role of the correlation e n e r g y

Treating the metallic sample as a semi-infinite jellium the work function may be expressed by Eq. (9.13), where the local density approximation (LDA) is accepted. This expression consists of two terms. The first term, the surface dipole barrier, depends on the electron density profile n(x), whereas the second one depends only on the mean electronic density ~. The self-consistent potentials and electron density profile at the surface can be determined from the solution of the Kohn-Sham equations described in Chapter 5. The electron density profile may be also calculated variationally accepting trial function as described in Chapter 11. Having n(x) and using Eq. (9.14) one can calculate the jellium part of work function ~ . In actucal calculations the exchange and correlation energy is approximated by the LDA. The results of self-consistent calculations of ~u done by Lang and Kohn (1970) as the function of the density parameters r8 are presented in Fig. 10.1. These values were computed using for the correlation energy the approximation due to Wigner. In the same figure we have also drawn the work functions recomputed with the CeperleyAlder form of the correlation energy. It is clearly seen that application of this modern form of the correlation energy yields the work functions lower by ~ 0 . 1 - 0.2 eV compared to Lang-Kohn, the difference being larger for low electron densities. This makes the curve for Ceperley-Alder correlation energy slightly more steep. It turns out, however, that these calculations based on the LDA, lead to the exchange-correlation potential which decays exponentially at large distances outside the jellium surface, instead of giving the correct asymptotics of the classical image-like potential. This seems not to have any significant effect on the work functions. The change in the work 131

132

C H A P T E R 10.

l

'

I

W O R K F U N C T I O N OF SIMPLE M E T A L S

'

'

I

I

'

Wigner

4.0 ""

Ceperley-Alder ..........

NLXC

~3.5 >

v

(D tO

-

\ \',xx

(9 t-

a_3.O -

\Nxx x

-

2.5

2.0

-j ,

2

I

3

J

rs

I

4 (bohrs)

,

I

5

i

6

Fig. 10.1. Work function calculated self-consistently within uniform positive background (jellium) model: Solid l i n e - the LDA with Wigner correlation energy; dashed l i n e - the same with the Ceperley-Alder correlation; dotted l i n e - nonlocal calculation of Alvarellos and Chac6n (1992).

function due to non-local gradient corrections is found to be typically about 3% of the local values (Rose et al., 1976) of Lang and Kohn. Alvarellos and ChacSn (1992) have performed non-local calculations of (I,,,(rs) in the weighted-density approximation obtaining larger values of work function for higher densities and smaller for lower ones in comparison with the Lang-Kohn values (Fig.10.1). The absolute change of the work function is _< 5%. In conclusion, the nonlocality of the exchange-correlation potential leads to steeper decrease of calculated ~u as function of r s. Comparing the theoretical curves for ~u with the experimental values taken from the Table 10.1, we see that they are in agreement only qualitatively. The inclusion of non-local corrections seems to improve this agreement. To rectify the remaining deficiencies of the model, as we have already discussed in Section 9.3, some other corrections to simple jellium calculations are needed.

10.2. W O R K FUNCTION OF THE ' R E A L ' M E T A L

133

Table 10.1

The experimental values of work function of polycrystalline simple metal samples recommended by Fomenko and Podchernyaeva (1975), F-P, and by Michaelson (1976), M. Work function (eV)

10.2

Monovalent metals

F-P

M

Li Na K Rb Cs

2.38 2.35 2.22 2.16 1.81

2.90 2.75 2.30 2.16 2.14

Polyvalent metals

F-P

M

A1 Pb Ca Sr Ba Zn Mg

4.25 4.00 2.80 2.35 2.49 4.24 3.64

4.28 4.25 2.87 2.59 2.70 4.33 3.66

W o r k f u n c t i o n o f t h e 'real' m e t a l b o u n d e d b y t h e fiat s u r f a c e

In Chapters 8 and 9 we have shown that one can improve jellium by introducing pseudopotential corrections to the model. The difference 5v(r) between pseudopotentials of the semi-infinite metal and of uniform positive background depends on the pseudopotential parameters and on the lattice spacing. One can ask what is the average influence of crystallinity on the work function ? In other words, how the averaged discrete-lattice effects will modify the jellium work function. It is expected that work function calculated in such jellium plus averaged pseudopotential model, which we will call pseudopotential fiat surface, will correspond to the work function measured for polycrystalline samples. In order to introduce the average pseudopotential contribution let us write the expression for the work function in the form

~fz = ~u(TZ) + ~ps(~z, re)

(10.1)

where (I)~ is the jellium contribution

e~ - 4~

/? ~[~(x)- ~+(~)]

ex-

[~F(~)+ .=(~)]

(lO.2)

and (~ps (Tt, rc) --~ - ((~V> a v

(10.3)

is the average (bulk) pseudopotential contribution, rc being the core radius of the Ashcroft pseudopotential.

134

CHAPTER 10. WORK FUNCTION OF SIMPLE METALS

From Sec. 8.5 we see that the transformation of the jellium to the pseudopotential model, which simulates the real metal, leads to a dependence of the face-independent part of work function on the individual metal specificity, which in the above picture of metal represents the core radius, re, of the Ashcroft pseudopotential usually fitted to some measured properties. It is to be noted that now we have used average value of 5v(r) over a Wigner-Seitz cell and confined the semi-infinite metal, bounded by the uncorrugated flat surface, to the region x _ 0. Therefore the work function is given by the following expression

Ofl

- - ~i~u -~- ( S V W S ) a v

f

oo

1 dn(x) dx-~ dx

(10.4)

Note that (Svws}av influences the electron density profile n(x) and thus, according to (10.2), also (I)u. If we will make use of the metal-stability condition (cf. Eq. (8.51)), (Svws}av can be expressed as a function of rs only i.e., independent on re, and we will get the expression for the work function in stabilized-jellium model.

10.3

Face-dependent

part of work function

As it was already said in Chapter 8 the work function of a real metal is strongly facedependent. According to (10.4) the face-dependent work function may be written in the form Oi = Oft + 5Oi, (10.5) where (I)fz is the face-independent part and 5(I)i, the face-dependent contribution to the work function. The face-dependent part, in the case of semi-infinite metal may be written in the form =

/?

x[n (x)

-

ex +

(10.6)

oo

where D~l is the contribution to the surface dipole barrier that arises from the distortion of the Wigner-Seitz cells which occurs at the classical cleavage of the crystal, n(x) is the electron distribution at the flat surface and ni(x) is the electron charge distribution which arises from the subsequent relaxation of the electron density. Since the charge neutrality condition has to be satisfied both for the semi-infinite jellium and for the semi-infinite real metal and because the main contribution to the value of the integral in (10.6) comes from the surface region in which the Friedel oscillations are large, one can state that the value of this integral is small and approximately, write 5~i "~ D~z. (10.7) Denoting the surface contribution to (SV}a, by -D~z, we can write ( a V ) a v "-- ( a V W S } a

v --

D~z,

(10.8)

135

10.4. P O L Y C R Y S T A L L I N E AND FACE-DEPENDENT WF

Table 10.2 The values of the factor F = 125 x 2, where x = di/ro, di being a distance between the two neighboring crystal planes and ro = zX/3rs.

fcc Face

(111) (100) (110)

bcc F

Face

0.21'73 0.7631

Face

0.337'6

(110) (100) (111)

1.5815

hcp F

(0001)Zn (0001)Mg

1.3687 2.0562

F

-0.1982 -0.1685

and employing Eqs (8.27) and (8.43) one obtains

D~z- 3Z [1-~2(di)21 -~o

10r0

==-

Z2/3

8r---~F(di/r~

(10.9)

where F(x) - ! ~ - x 2. Table 10.2 shows that F(x) is positive for fcc and bcc structures, and negative for hcp structure. Therefore we arrive at the conclusion that

for fcc and bcc structures, and ~i > ~ fl,

for i = (0001) face of the hcp structure.

10.4

Polycrystalline and face-dependent work functions

In experiment we can distinguish three "kinds" of work function: (i) the mean or polycrystaUine work function ~Poly obtained from measurement on polycrystalline samples, (ii) the total work function (I)T, determined by measurements of the total thermionic- or field-emission current from a single-crystal tip, and (iii) the work function of single-crystal face (I)i, measured for a particular crystal face with Miller indices (hkl) = i. The polycrystalline surface may be treated as a composition of patches (Herring and Nichols, 1949; Dobretsov and Gomoyunova, 1966; Sahni et al., 1981) each patch being a certain cleavage plane of crystals. Then, the ~pozy may be interpreted as an average of the work function of all exposed patches

~Polu = E fi~i // E f i i

i

(10.10)

136

CHAPTER 10. WORK FUNCTION OF SIMPLE METALS

where Oi is the work function of cleavage plane corresponding to the i-th patch and fi, is its weight. Taking simply fi = Ai, where Ai is the area of the i-th facet, 1 we have (Dobretsov and Gomoyunova, 1966)

9p o t y - E A , ~ , / E A , . i

(10.11)

i

When the surface of a metallic sample is clean enough, and the experiment is performed in an ultra-high vacuum, measurements of Ogozy and OT, within the limits of the accuracy, give the same value (Kiejna and Wojciechowski, 1982). Therefore, practically one may consider OPoly -- OT. However the comparison of Opozy with the work function calculated theoretically is not simple. According to the thermodynamics of crystal growth, the equilibrium shape of a crystal is governed by the Wulff theorem (see Sec. 3.2). On the polycrystalline surface (usually taken as a fraction of the total surface considered) the low-index crystal faces are exposed, which have a greater surface energy than the high-index crystal planes. Therefore, for the comparison of the theoretical calculations with the experimental data, it seems appropriate to compare OPoZywith the mean value of the work functions calculated for the three lowest-index crystal faces. It follows from the theory (see Chapter 9.3) that the face-dependent work function may be represented by a sum of two components O~ = Ofz + 50~

(10.12)

where Oft is the face-independent part of work function and 5Oi is face-dependent contribution to the work function. According to Eqs. (10.11) and (10.12) the mean work function of a polycrystalline surface can be written as

(Oi} = Oil + E AiSOi]/ E Ai i

(10.13)

i

In the case of metals one can suppose that the (111), (100) and (110) faces all have essentially the same surface energy 2 and therefore their areas Alll "~ A100 ~ All0 which, in view of the said above, leads to the following approximate expression for the average work function, 1 (5(i)111 + 5(i) 100 -~- 5Ol10) ,
(10 14)

giving the work function value that may be compared with the experimental polycrystalline data (Opozy). The face-dependent values of work function calculated from Eq. (10.12), can be directly compared with the experimental data of Oi, but the latter for clean metallic single crystals are very poor and uncertain, particularly in the case of alkali metals. 1A more accurate deefinition of weights fi was given by Sahni et al., (1981). 2A zero temperature extrapolation of measured surface energies of various faces of Pb shows t h a t the ratio of the largest energy to the smallest one does not exceed 1.1 (Heyraud and Metois, 1983).

137

10.5. R E L A T I O N B E T W E E N T H E O R Y AND E X P E R I M E N T

Table 10.3 Face-dependent values of work function for low-index crystal faces of aluminium measured by E tme t Mee (1973), Crepstd ( 976), Gas. A1 (face)

Work function (eV) E-M

GGS

(111) (100) (110)

4.26 4.20 4.06

4.24 4.41 4.28

Among the simple metals, the face-dependent experimental data of work function are available only for low-index faces of a crystal of aluminum. However, different groups reported different sequence of increasing of the values of r (see Table 10.3). The trend observed for the ~i values reported by Eastment and Mee (1973) is consistent with the argument given by Smoluchowski (1941) according to which the work function decreases with a decrease in the packing density of the crystal plane. The trend of the results reeported by Grepstad et al. (1976) values is in complete disagreement with the arguments provided by the above mentioned Smoluchowski rule. For alkali metals there is a lack of measured values of ~i from single crystals because of great difficulties in preparation of such samples. However, investigations of the epitaxial growth of these metals on tungsten have shown (Mlynczak and Niedermayer, 1975) that the monolayers of alkali metals grown epitaxially on a tungsten single crystal have the same structure as the underlying faces of tungsten and corresponding to that of a single crystal faces of alkali metals. From the discussion of the experimental data (Btaszczyszyn et al., 1975) follows that for sodium and potassium we observe ~110 > ~100 > ~111, which is in agreement with the Smoluchowski rule.Smoluchowski rule For cesium adsorbed on tungsten, this trend is very weak (Sidorski et al., 1969) and the average nearest-neighbor distance in the cesium-metal (fcc and bcc) monolayers for the saturation coverage, equals 4.92 A, whereas in the bulk of metal this distance amounts to 5.25/~ (Agura and Murata, 1989). As a consequence, one may conclude that cesium monolayers form densely packed hexagonal (or quasi-hexagonal) lattices (irrespective of the symmetry of the substrates) which do not appear in the bulk cesium. Therefore the mean work function (_ 1 . 9 - 2 eV) measured for cesium monolayers on metals cannot be referred to the work function value of any low-index crystal face of the bulk cesium.

10.5

Relation between theory and experiment

The comparison between flat surface calculations of work function and experimental data are shown in Fig. 10.2 and in Fig. 10.3 where the calculated (ffi) values together

C H A P T E R 10. W O R K F U N C T I O N OF SIMPLE M E T A L S

138

4.5 A,,A

4.0 >

3.5

E 0 (9

--

ZX

W--B

~x'~--

Z~

-

~ ~ ~

-

_~3.0

I,

0

2.5

2.0

1.5

AI Pb Mg ? t , ?

2

I

5

L_i Sr BaN a t ,t T t

rs

4

(bohrs)

,

K ~

5

Rb t

Cs ,?

I I

6

Fig. 10.2. Comparison of the work function values for a fiat surface of jellium - dashed line (Lang and Kohn, 1971), stabilized-jellium- solid line (Kiejna, 1993) and jellium plus averaged pseudopotential - triangles (Wojciechowski and Bogdan6w, 1994) with the experimental polycrystalline data (dots) of Table 10.1.

with the corresponding experimental data are presented. In Fig. 10.2 the jellium work function calculated self-consistently (Lang and Kohn, 1970) is represented by dashed line. Solid line represents the self-consistent calculation for the flat surface of the stabilized jellium (Kiejna, 1993) by the use of the condition (8.50). The experimental data of work function measurements for simple metals collected in Table 10.3 are plotted as dots connected in most cases by the vertical bars. As is seen from this table the lower end of the bar corresponds always to the value recommended by Fomenko and Podchernyaeva (1975) whereas the upper one to the value preferred by Michaelson (1977). They coincide only in the case of Rb. From this figure we observe that the work functions calculated in the framework of stabilized-jellium model are lower for large values of the density parameter r s and, greater for smaller rs, than the jellium values. Generally, a much better agreement of stabilized-jellium results with the experimental data is observed, especially for high and low electron density metals. For the intermediate electron densities both the

10.5.

RELATION

BETWEEN

THEORY

AND EXPERIMENT

139

stabilized jellium and ordinary jellium model give similar results. They differ from the measured values by ~ 0.5 eV for Li, Ca, Sr and Ba. Looking for the possible reason of these discrepancies one may suspect that they originate from the fact that the stabilized jellium correction, (SV)av, due to the use of the bulk stability condition is independent of the pseudopotential core radius, re, characterizing individual metal. In order to check what results from rejecting the assumption of stabilized jellium model that the core radius re is unique function of r s, Wojciechowski and Bogdan5w (1994) have performed variational calculations using the trial electron density profile proposed by Perdew (1980) (compare Chapter 11). The results represented in Fig. 10.2 by triangles show that the introduction of the average pseudopotential contribution to the work function calculations gives slightly better agreement with the experiment than stabilized jellium for Ca, Ba and Na. However, for Li, Sr and high density metals (A1, Pb, Mg) the agreement is less satisfactory. It seems that for low and intermediate

4.5 "~'~i" q

4.0 >

"

',

o o o

MPLW

u n n

SJ-AV

aaa

W-B

.. \',,

'~ \',,. ~_-~ ~., \ m

3.5

E ~O O E

Z~.,,,.~ ', "\,,. El,,

=3.0

I,

2.5

2.0 - Zn AI Pb

1.5 t t , t 2

Ca Mg

Li Sr B a N a i

3

t

,t

?

I

4rs (bohrs)

K I

Rb

11

5

t

Cs it

Fig. 10.3. The mean values of work function calculated using the results from variational selfconsistent (MPLW, squares), stabilized jellium (SJ-AV, circles) and variational - jellium plus pseudopotential (W-B, triangles) calculations compared with the experimental polycrystalline data (dots). The broken lines connecting the points are drawn to guide an eye only.

140

C H A P T E R 10. W O R K F U N C T I O N OF S I M P L E M E T A L S

electron density metals (r8 > 3) this model is equivalent to stabilized jellium. It fails, however, in the quantitative description of high density metals, A1, Pb and Mg. (Note the difference in r s values employed by Wojciechowski and BogdanSw and that their calculation is not self-consistent). In Fig. 10.3 the mean work functions calculated by the use of Eq. (10.14), are compared with the polycrystalline data. Three sets of theoretical values plotted in this figure are calculated using the data from: self-consistent variational calculations (MPLW) -squares (Monnier et al., 1978; Sahni et al., 1981), self-consistent stabilizedjellium calculation (SJ-AV) -circles (Kiejna, 1993), variational- jellium plus pseudopotential calculation (W-B) - triangles (Wojciechowski and BogdanSw, 1994). It may be seen from this figure that the averaging of face-dependent work functions from the latter two calculations gives the values which are in a closer agreement with experiment than the values for a specific crystal face. This result shows that the mean surface contribution to work function generally lowers the flat surface work function for fcc and bcc metals and rises it for hcp metals.

Chapter 11

Variational electron density profiles: trial functions II.I

Introduction

The shape of the electron density profile (EDP) at the surface plays a very important role in the description of surface properties of metals. The self-consistent electron density profiles, determined numerically and tabulated by Lang and Kohn (1970) are often used as an input for further calculations of surface quantities. In many cases, however, the tabulated form of the function is not convenient to handle. Therefore for practical purposes one tries to approximate or replace them by a simple function, which permits to perform primary analytic calculations. The most exploited trial function n(x), describing the surface EDP, is the one originally proposed by Smoluchowski (1941). Let us consider first what conditions have to be fulfilled by the trial function in order to become a "good" trial function. Let n ( x ) denotes EDP in the plane perpendicular to the surface located at x - 0. The trial function n ( x ) should be continuous with its first derivative in the whole region:-c~ _< x _< +c~. It is also clear, from our considerations in Chap. 6, that the electron density should take the following limiting values lim n ( x ) - ~ (11.1) x-~-

and

cx)

lim n ( x ) -- 0

x--++cx~

where, as usual, the metal occupies the the left half-space. The self-consistent jellium calculations have demonstrated surface n(0) < ~/2.

(11.2) that for a clean metal

(11.3)

Therefore a "good" trial function, simulating real EDP, has to fulfill the above conditions. Moreover, the electrostatic potential due to the total charge distribution should 141

142

CHAPTER 11. VARIATIONAL DENSITY PROFILES

satisfy the Budd-Vannimenus theorem (see Chapter 7). It is also desirable that E D P reproduces the first main Friedel oscillation for the metals characterized by the larger value of rs (Fig. 6.2). In order to comply with the above restrictions the trial function generally must depend on several variational parameters/~i (i = 1, 2 , . . . ) which have to be determined by the above conditions and, additionally, by the charge neutrality condition of the form

I / a T ( x ; ~i) dx

O,

(11.4)

oo

with

aT(X; ~i) = n(x; ~i) -- n+(x). (11.5) The parameterized trial density n(x; ~i), according to the density functional formalism, has to minimize the Hohenberg-Kohn ground state energy functional E[n(x; ~i)] of the inhomogeneous electron gas. Since the energy of the bulk crystal remains constant during the variation we can minimize the surface energy a only, instead of the total energy (Smith, 1969). Therefore we have =0,

(11.6)

where index j, numerates those parameters from the set {/~1,/~2,... ~n} - {~} which remained free after fulfilling the other conditions listed above. It is clear that in order the EDP to be flexible enough, for different densities of the positive background, the number of variational parameters has to be as small as possible. This can be achieved by imposing only some of these conditions. The surface energy as the functional of electron density, similarly as total energy of jellium model, is given by the sum of electrostatic, kinetic and exchange-correlation energy contributions. In an approach where the parameterized trial densities are employed, the kinetic energy functional is approximated by its expansion in a series of density gradients. In order to get calculated surface quantities to the accuracy comparable with those from self-consistent calculations, the density-gradient expansion of the non-interacting kinetic energy up to the fourth-order (compare Section 8.2) has to be used (Ma and Sahni, 1979). The similar expansion was suggested for the exchange-correlation contribution to the energy E[n]:

Exc[n] - / d3rn(r)exc(r) + / d3r Bx~[n(x)] IVn, 2 +

....

(11.7)

In this expression the first term gives the LDA contribution and only the first densitygradient correction 1 is considered, where

Cxc(rs)

Bxc = n4/3(r ) 9

(11.8)

lit is to be noted that the gradient corrections to the exchange and correlation energy are questionable because, as a rule, when the LDA fails the exchange-correlation hole extends over a greater space than the sphere with convergence radius of the expansion of the electron density into the series of the Taylor type (Fulde, 1991).

11.2. CONDITIONS FOR DENSITY PROFILES

143

For the Cxc(rs) coefficient in (11.8) the values found by Rasolt and Geldart (1975) as well as those of Gupta and Singwi (1977) are used. Further we shall use (11.8) with C~c(rs) given by Gupta and Singwi. Another convenient form for the non-local term is the nearly profile-independent wave-vector analysis correction (Perdew, 1980).

11.2

Conditions satisfied by various exact electron density profiles

Below we summarize the conditions satisfied by discussed in the previous chapters, different exact electron density profiles of semi infinite jellium. (a) A s y m p t o t i c s o l u t i o n s in t h e bulk: (i) From the Thomas-Fermi type calculations (Lang, 1973), for x --+ - o c results: v(x) =

_ n

_~ 1 - 0.621exp(x/ATE)

(11.9)

where ~TF = (4--~F)1/2 is the Thomas-Fermi screening length. (ii) In the Thomas-Fermi-von-Weizsgcker approach (Thomas-Fermi terms plus the first gradient correction to the kinetic energy) the electronic density profile, for x --+ -oo, approaches exponentially its bulk value ~, oscillating about it v(x) ~ 1 + Ae ~x cos(/~x + 6),

(11.10)

where A and 5 are constant, a and ~ are the real and imaginary parts of the momentum (Utreras-Diaz, 1987). These oscillations are found to be an effect of the von Weizsgcker term rather than arising from the exchange-correlation interaction, as one may have thought. The oscillations although similar in structure to the Friedel oscillations appearing in the self-consistent solution of Kohn-Sham equations (6.23-6.24), are not of the Friedel type because their wave number is not 2kF and their decay is exponential instead of 1/x 2. (b) T h e b u l k - s u r f a c e condition: The Budd-Vannimenus theorem (7.20) linking electrostatic potential at the location of surface and in the bulk with electron energies in the bulk should be satisfied. (c) T h e j e l l i u m edge x - 0" (i) In the high density limit, the correct electron density at the jellium edge results (Perdew, 1980) from the Thomas-Fermi model and following Eq. (6.8)is 3/3/2 n(O) - ~, g .

(11.11)

144

C H A P T E R 11.

VARIATIONAL DENSITY PROFILES

(ii) From the numerical analysis of tabulated Lang and Kohn (1970) density profiles follows that the ratio nLg(O)/~t -- vLg(rs), may be numerically fitted with high accuracy (Rogowska et al., 1990) to the function VLK(rs)

--

0.50306 - 0.0265357r~ + 0.0005357143r 2.

(d) O u t s i d e t h e m e t a l surface (x > 0): For the Thomas-Fermi model we have: ~(x) = 27000

x

+a

)-~

(11.12)

(11.13)

where a - (1500) 1/4 and ATE is the Thomas-Fermi screening length (compare (6.11b)). (e) T h e a s y m p t o t i c s in t h e v a c u u m (x --+ +c~): Away from the surface of jellium bounded by the square-potential barrier the reduced density, ~(x), behaves (Gupta and Singwi, 1977) as u(x)--+ ~

exp(-2 I x I x/'~- 1)

)~

and v(x)

9 ~

SX4,

for A - 1,

for/k ~= 1,

(11.14)

(11.15)

where )~ = Vo/EF, Vo being the effective height of the potential barrier at the jellium edge, and x is measured in units of kg 1.

11.3

Examples of the trial electron density profiles

The trial electron density profile which describes the real EDP for the metals characterized by the small values of rs(___ 3) quite well, is given by a simple one-parameter function proposed by Smoluchowski 1 ~x 1 - ~e ,

x < 0,

1 ~e -~x,

x > 0.

(11.16)

v(x) =

As is seen from Fig. 11.1 this function neither reproduces Friedel oscillations nor satisfies the condition v(0) < 1/2. The practical calculations showed, however, that the function (11.16) models the jellium surface well for rs < 3. Similar, but a little more sophisticated form of trial EDP which satisfies the conditions (a) and (d) of Section 11.2, has been proposed by Perdew (1980): 1 - be a(t-t~

t < to

ble -(t-t~

t > to,

(11.17)

=

11.3.

TRIAL

DENSITY

PROFILES:

145

EXAMPLES

~(x}

self- consist.

\ rs~3

l~2 t_ria~

0 Fig. 11.1. Schematic representation of the self-consistent electron density profile (dashed line) and the Smoluchowski trial function (solid line).

where (11.18)

t =~'X/ATF,

and 51 = 1 -- b =

,

a = 51/5,

to = 1/51 - 2.

(11.19)

-y is the variational parameter and/~TF is the Thomas-Fermi screening length. The function given by Eqs (11.17-11.19) is depicted in Fig. 11.2. The modified shape of this function was used by Perdew et al. (1990) for variational calculations in the framework of the stabilized-jellium model (see Section 8.5). The trial EDP which satisfies the Budd-Vannimenus theorem and the condition a(ii) of the previous section was proposed by Schmickler and Henderson (1984): 1 - Ae ~cos(~,x + 5),

x < 0,

Be -~

x > O.

v(x) =

(11.20) ,

The six parameters A, B, a, ~, ~/and 5 fulfill the following relations resulting fromThe continuity of (11.20) and its derivative at x = 0: 1 - Acos5 - B,

A(a cos 5 - ~/sin 5) - ~B.

(11.21)

The charge neutrality condition (11.4): A(a cos 5 + 7 sin 5)

B : --.

(11.22)

The Budd-Vannimenus theorem: 4r~A

(~ +

,y2)2 [(a2

_ 72) cos5 + 2a~sin 5] - h,

(11.23)

146

C H A P T E R 11. VARIATIONAL D E N S I T Y PROFILES

"~

Smoluchowski 89

Perdew

\

-2.0

' -1.5

- 1.0

-0 1.5

- x, 0

05'

' 1.0

15'

21:) x [ bohr]

Fig. 11.2. Schematic representation of the Smoluchowski (solid line) and Perdew's (dashed line) electron density profiles. where, for the Wigner formula for correlation energy, h is given by h-

~-[

0"0796r3 ] 0 . 4 - 0.0829r~- (r~ + 7.8) 2 "

(11 24) "

By choosing A and B, say, as free parameters, the system of Eqs (11.20-11.24) is easily transformed into a system of two nonlinear equations for a and ~/, which may be solved numerically. The remaining two parameters can be calculated directly from the others. The free parameters can be determined by minimizing the surface energy (see Eq. (11.6)). The electronic density at the jellium surface of rs = 3, generated by function (11.20), is illustrated in Fig. 11.3 (solid line). It is to be noted, however, that the set of equations (11.21)-(11.24) has a real solution only for rs <_ 4.3. For larger values of the density parameter rs the function h(rs), given by (11.24) becomes negative when the set of equations reduces to a cubic equation which has only imaginary roots.

11.4

Smoluchowski's density profile and different contributions to the energy

This section gives a brief account of the calculations of the surface properties for a simple Smoluchowski's model of electron density profile. As it was shown (Kiejna, 1981) the use of such trial function and the incorporation of the gradient corrections

11.4. C O N T R I B U T I O N S TO THE E N E R G Y

147

1.2~ r---r~

Ir

0.8

\\

,,,-,... X E

0.4 0.0 -0.6

- 0./-,

-0.2

X/XF

0.0

0.2

Fig. 11.3. Comparison of the Schmickler-Henderson electron density profile for rs = 3 (solid line) with self-consistent results of Lang and Kohn. The distance is measured in units of the Fermi wavelength. for the kinetic and exchange-correlation energies (see Chapter 8) in the variational calculations leads to a good agreement of the calculated surface energies and work functions with the values calculated self-consistently by Lang and Kohn (1971) (see Fig. 11.4). The use of the Smoluchowski trial function for EDP gives a good description also of the other surface properties of metals such as, for example, the bimetallic adhesive energy (Kiejna and Zi~)ba, 1985), work function of alkali-metal alloys (Kiejna and Wojciechowski, 1983) or the effect of strong electric field on lattice relaxation at a metal surface (Kiejna, 1984) which will be discussed in further Chapters. Below we present the results of evaluation of the contributions to surface energy for the Smoluchowski electron density profile. Due to its simplicity this trial function allows to evaluate all energy integrals analytically and to present the results in a compact form. (a) Hartree (electrostatic) energy. The electrostatic potential due to the positive background (jellium) and electronic charge is: 2r~ x<0,

7'

r

(11.25)

-

4~

2~

~x

x>0,

Using this potential in Eq. (8.6) one gets for the surface electrostatic energy associated with the total charge: ~2 8.9525 • 10 -2 aes[n]- 2~ 3 ~3r6 9 (11.26) (b) Kinetic energy. The surface kinetic energy functional given by Eq. (8.7) can be written as a sum

148

C H A P T E R 11.

VARIATIONAL DENSITY PROFILES

of homogeneous electron gas and density-gradient contributions. The homogeneous electron gas contribution to as gives

(7!0) -- 1-6 9 (37r2)2/3n5/3{~ 0 (2-2/3-1) --~ -

+~1 ln[ 6 (1 -- 2-1/3)]

v / ~ arctan l ( 1 - 2 1(1/ 3+) }2-1/3 v/~ )

0.1509 =

~r 5

(11.27)

.

(c) Gradient expansion terms for the kinetic energy The first density gradient correction is:

a!l)

__. If__ ~ 7---2

=

(dn(x)) 2 ln2 dx d z - 13~ 7-2

I

o0 n ( x )

/32.2983 • 10 -3 r]

(11.28)

The second density gradient correction, given by the last integral on the rhs of (8.7), for n(x) of the form (11.16) yields a! 2) = (3~2n)1/3 (25/3 _ 1)/33 =

Z34.3267 • 10-4

3600~ 2

rs

(11.29)

(d) Exchange and correlation energy The local-density-approximation contribution to the surface exchange energy yields

9 (3)1/3~4/3{ 5 O" x

--7

~--

~(1

1 ln[ 6(1 - 2-1/3)] 2

--1/3 -

2

I

)

1 21J3 ]}

V~ arctan v/~ (1 + 2-1/3)

3.7075 x 10 -2

~r4

(11.3o)

For the Wigner interpolation formula (4.74) the LDA surface correlation energy can be written in the form ~W

= 0.168~ h (0.079/~ 1/3) - 4.011~rx 310 -2 h(rs/7.8)

where

h(p) -p{~(21/~- 1)- p(22,3_1)+ ~2ln(2 x/3+p~)

(11.31)

11.4.

C O N T R I B U T I O N S T O THE E N E R G Y

149

4.0 . , , " - ~ "--. " . ~ . ~--.\ _ "" ""., "'x~. "\,,,\x.~x -

&5 _

~

--'-

Wigner (serf-cons. L ~K) Wigner ---Ceperley- Alder

~,

NN N

z 0 FZ

&O

~

~.~. ~,~~~'~.

2.5

15

i

r

2

I

i

3

I

4

~

l

5

i

rs

6

Fig. 11.4. Work function versus rs computed by using the Smoluchowski trial function with Wigner's and Ceperley-Alder's correlation energy. Solid line represents results of selfconsistent calculations.

1 [# - v - p(# + v) - p5 In ( 2 -1/3 l+p4- p )]}

+ 1 + p3

(11.31a)

and ~1 In[ 6(1 - 2 -1/3 )] _~ 0.10667

#-

V= ~

1

arctan

I

1--2-1/3 1 V/.~(1 + 2_1/3) .

(11.315) (11.31c)

Similar expression can be derived for the Ceperley-Alder correlation energy 0.CA. In this case, however, the space integrals on the metal side cannot be integrated analytically. The corresponding expression for 0.cCA is given in the Appendix C. (e) Density gradient correction to the exchange-correlation energy The first density gradient correction with the coefficient of Gupta and Singwi (1977) yields: O.(1 )

xc

_

2.4628

X

r0"433

10 - 3 / ?

(dn/dx) 2 oo

n4/3(x ) dx

CHAPTER 11. VARIATIONAL DENSITY PROFILES

150

Table 11.1 The values of the variational parameter ~m minimizing the surface energy functional

with Wigner (W) or with Ceperley-Alder (CA) correlation energies. r8

~w

~CA

2.0 2.5 3.0 3.5 4.0 4.5 5.O 5.5 6.O

1.14 1.05 0.98 0.92 0.86 0.82 O.78 0.74 O.7O

1.15 1.06 0.99 0.93 0.87 0.82 O.78 0.74 O.7O

:

~ 2 / 3 9 (21/3

1) 2.4628 • 10 -3 _ ~ 1.1086 x 10 -3 -

-

r0.433

-

-

r2.433

9

(11.32) Summarizing we observe that the total surface energy of jellium can be written in the form O': ~O" i i

c~(~)Z -3 + [c~(~) + c 5 ( ~ ) + c6(~)] Z-~ + [c3(~) + c7(~)] Z + c 4 ( ~ ) Z 3

(11.33)

where ai denotes respective energy contributions listed in the order of appearance in Eqs (11.26-11.32) and Ci(r~), (i = 1 , . . . , 7) denote respective constants depending only on the density parameter r~. From Eq. (11.6) we obtain the following equation determining ~m (r~) which minimizes the energy functional

3c~(~)Z~ + [c~(~) + c ; ( ~ ) ] Z4 - [c~(~) + c ~ ( ~ ) + c~(~)] ~

- 3c~(~) = o.

(11.34)

The numerical solution of Eq. (11.34) for the Wigner (W) and Ceperley-Alder (CA) correlation energies gives the values of parameters ~m which are collected in Table 11.1. Both, ~w (rB) as well as ~CA(rB), are regular functions of rs and may be well

11.4. CONTRIBUTIONS TO THE E N E R G Y

151

Table 11.2 Coetticients an of the power expansion of the parameter ~m (rs) for Wigner (W) and Ceperley-Alder (CA) correlation energies.

n

0

1

2

3

4

5

aW

2.128

--1.044

0.467

--0.132

0.022

--0.002

aCnA

2.096

--0.986

0.441

--0.127

0.021

--0.002

approximated by the following power series: 5 ~W(rs)

--

~

Wn a n rs,

2 _< rs _< 6,

(11.35)

n--0 5 CA n a n rs ,

~mCA (7's) =

2 < rs _< 6,

(11.36)

n--0

where the coefficients a nw and a nCA are listed in Table 11.2. Employing the Smoluchowski trial function one can make some general conclusions concerning jellium approximation with respect to the surface properties. For instance, using the expression (11.33) for surface energy and substituting there ~'s as functions of the density parameter rs given by Eq. (11.35) or Eq. (11.36), after some algebra one may observe, that E(rs) has a maximum for r8 _< 2. These leads to the conclusion that the jellium model is unstable for r~ _ 2. Finally, the combination of the perturbational treatment of the ionic lattice effects in connection with simple analytic form (11.16) for electron density profile allows to write the pseudopotential part of surface energy ap~, defined by Eq. (8.28) in a simple analytic form (Paasch and Hietschold, 1975; Kiejna, 1982) 21r~2d ( ~ d

cosh(~rc) ) 2sinh(Zd/2) '

d rc < ~, (11.37)

aps --

21r~2d { 1 - ~d sinh[~(d-rc)]-exp(-~rc)} ~-2 Zd ~ 2 sinh(Zd/2)

'

d 3d 2 < r~ <

where, in evaluating the difference (~v(r) between pseudopotentials of the semi-infinite lattice of ions and the electrostatic potential of the uniform positive background the Ashcroft model potential is employed (Appendix C). The minimization of surface energy functional (8.29), including the gradient expansion for the kinetic energy up to the fourth order, with respect to ~ allows to determine the face-dependent electron density distribution, surface energy and work function in a very good agreement with the results of self-consistent calculations (Kiejna, 1982).

This Page Intentionally Left Blank

Chapter 12

Image potential and image plane 12.1

Limitations of the classical picture. Image plane position

The exact form of the potential experienced by an electron escaping from or approaching the metal surface is essential for many surface spectroscopies that involve charge transfer between a metal and the vacuum. In many practical cases the potential of interaction is approximated by the image potential. The form of the particle-surface interaction is also of essential interest in the context of interaction of ions and positrons with metal surfaces. If the point charge q, is located in the vacuum at the distance x, from the perfectly conducting semi-infinite metal (Fig. 12.1a), the screening charge (of opposite sign) of surface density E E-qx (X 2 _~_y2 + Z2)3/2 ' (12.1) is induced on the surface of the Jmetal. As it is well known from the classical electrodynamics the best wayjto analyze the electric potential for such problems is to apply the classical methpd ~)~fimages. In this method the induced screening charge may be replaced by the image charge, - q , of opposite sign, located at the distance - x , inside the metal (Fig. 12.1b) (Jackson, 1975). Thus the electrostatic force between the charge +q and its image -q, is simply given by -q2/(2x)2. Hence the screening energy (image charge energy) is equal to the work done in bringing the charge from infinity to - x -x q2 q2 qVim(X) - (2x)-----~ dx - -4-~" (12.2)

/c

This potential-energy function is called classical image potential. Because some questions arise when the external charge is not the classical test charge but is identified as an asymptotic electron i.e., a one that belongs to the 153

154

CHAPTER 12. IMAGE POTENTIAL AND IMAGE PLANE

Metal

Metat

Vacuum q X

@

Vacuum

-q @

q -X

(a)

X

(bl

Fig. 12.1. Point charge in front of the metal surface. The screening charge induced at the surface (a) is replaced by the image charge (b).

system of electrons considered, we first focus our attention on the case when the electron is treated as an external charged particle. The classical formula (12.2) is not realistic when the charge approaches the geometrical surface located at x = 0: it diverges to -c~. The reason for this is that at small distances from the surface the wave function of the external electron begins to overlap with the wave functions of the electrons in the metal. At such distances, the real quantum-mechanical interaction differs from the classical - 1 / 4 x form. The singularity of classical form (12.2) does not allow for the matching of image potential to the finite inner metallic potential. To avoid this difficulty Seitz (1940) has modified the image potential and proposed to write it in the form 1

Vim(X)=- (4 x+~l)'

(12.3)

where W is the potential depth in the metal. This expression, resulting from the quantum-mechanical considerations based on the infinite-barrier model, places the image plane into the metal and thus 1

= -a(x + x0)'

(12.4)

where x0 defines the image plane position. As it is known, however, from preceding chapters the potential barrier at the surface of a real metal has the limited height and the location of the image plane has to be determined with respect to the so called physical surface of metal appearing "outside" its geometrical surface. Thus the image potential should be of the form 1

(x)- -4(x-

155

12.1. LIMITATIONS OF CLASSICAL PICTURE

where the point x s is to be regarded as determining the effective location of the metal surface. There is no inconsistency in this confusing point because each theory is referencing the x-origin to a different point. The self-consistent theories place x = 0 either at the position of last plane of ions or at the jellium-step edge. Thus the effective image plane lies to the right with respect to this point, whereas in the model with electron density being a step function terminated at x = 0 (Newns, 1969), or assuming an infinite surface barrier (Gadzuk, 1970) at x - 0, the effective image plane is located on the negative side of x. In other words, in the derivation of the classical image potential (12.2) the metal surface is not located precisely. Its precise location will depend on the spilling out of the electronic charge distribution near the metal surface. If the external charge is an electron, it cannot be treated as a distinguishable from other electrons in the metal and it must be considered quantum mechanically. Theoretical considerations in this spirit were carried out over fifty years ago by Bardeen (1940) who, parameterizing the distance x, of one electron on the perfect metal surface, has shown that an external electron experiences potential of the form

V(x)=

1 a 4x ~ x 3'

(12.6)

where a is a constant. Thus it proves that the asymptotic behavior of V~m(X) is -1/4x. As it was mentioned in Section 4.6 the potential felt by electron outside the metal originates from the exchange-correlation effects. The image behavior ought to manifest itself in the effective potential V~ff[n(x)] of the Kohn-Sham one-electron equation for large distances from the surface. It means that asymptotically, as x --+ +ec, the effective potential should have the form (12.5). As it was shown by Lang and Kohn (1970) the image behavior is fully included in the V~ff, but not in the case when the exchange-correlation part of V~ff is given in the LDA. In the latter case the effective potential falls off exponentially as x -+ +c~. This is because the LDA is not valid in the surface region where the strong variations of the electron density take place. When the exchange-correlation energy is treated nonlocMly, for example in the weighted density approximation (Gunnarsson and Jones, 1980), the V~ff correctly approaches classical image potential (12.5) for larger distances from metal surface. The rigorous proof that the asymptotic behavior of the Kohn-Sham effective potential is indeed of the -1/4x form, was given independently by several groups of workers (Almbladh and yon Barth, 1985; Sham, 1985; Harbola and Sahni, 1989). The discussed limitations, both of the classical image potential theory and of the self-consistent LDA barriers, have led to the use of simple models describing the potential energy barrier at the metal-vacuum interface (Jennings and Jones, 1988). The most important features of such a surface barrier are included in the "saturated image barrier" (Jennings et al., 1984) of the form

4(x - x0)

{1 - exp[A(x - xo)]},

if x > x0,

=

(12.7)

-Y0 A e x p [ B ( x - x0)] + 1'

otherwise,

156

C H A P T E R 12. IMAGE P O T E N T I A L A N D I M A G E P L A N E

where constants, A and B, are determined from the continuity of V(x) and its derivative at point x = x0, which corresponds to the image plane position. The values of ~, x0 and V were determined by fitting the form (12.7) to the jellium potential and to the average effective potential resulting from the three-dimensional calculations. As is seen from Fig. 12.2 the potential (12.7) shows the proper image form in the vacuum. In the following sections we will try to look a little closer at the exchange-correlation hole at the metal surface and try to explain the origin of the image potential. Before that we will introduce the basic equations of the linear response theory.

I

!

I

I

f/

r,. 2

-0.5

t/

-

-1.0

_4

J 0000001

-0.5

I I ! I I

/ !

-0.5 I i

t

-10

-5

I

0

:

,

,

5

10

x (a.u.)

Fig. 12.2. The effective potentials at the surface of jellium with rs -- 2, 4 and 6. The open circles represent the results of Lang and Kohn (1970) and the solid line is the model potential of Eq. (12.7) The dashed curve is the classical image potential. Redrawn with permission from Jennings et al. (1988). @1988 The American Physical Society.

12.2. LINEAR RESPONSE TO PERTURBING CHARGES

12.2

157

Linear response of electron s y s t e m to static perturbing charges

For the sake of further discussion it is useful to introduce, in the framework of the density functional formalism, the basic equations of the linear response theory. Consider unperturbed electron gas of density n0(r) in an external potential V0(r) due to the positive ionic charge n+(r). We are interested in the change 5n(r) of the electronic density due to the presence of an additional (perturbing) small charge of density 5n+ (r) giving rise to a perturbing potential 5v(r) = -

/ (r_______~) 5n+ dr'. Ir-r']

(12.8)

The change in the total electrostatic potential r will be denoted by 5r and the shift in the chemical potential by 5#. Now, we linearize equation (5.24) and (5.12) i.e., we write r = r 5r n(r) = n0(r)+ 5n(r) etc., to get the following equations for the first order screening density (Ying et al., 1975): 5r

+

/[

52Gin] ] 5n(r')dr'=5# Lbn(r)bn(r') ~o

and 5r

-- 5v(r) +

5n(r) [ r - r' I dr''

(12.9)

(12.10)

Using the energy functional defined in equation (5.21), the change in the energy resulting from the interaction between the metal and the perturbing charge can be written as ~E = - ] n+(r)bv(r) dr + Ev+av[n0 + 5n] - Ev[n]. (12.11) I .

J

Making use of (5.30) and (12.10) this can be written to second order 1 / 5 n ( r ) b n ( r ' ) d r dr' 5v(r)bn(r) dr + ~ i r _ r, I

5E = +

r

1/ +2

- 5n+(r)] dr +

/ 5G[n]

6n(r) 5n(r)dr

52G[n] 5n(r)bn(r')dr dr'. 5n(r)bn(r')

(12.12)

The first three terms give the electrostatic components. Using (5.24), the fourth and fifth terms can be transformed into f r

+ # dN

CHAPTER 12. IMAGE P O T E N T I A L AND IMAGE P L A N E

158 and

1/

2 where 5 N 6E

1

5r

dr + ~ 5 # 5 N

f a n ( r ) d r . Thus the energy change to second order is

if

--

-~

6v(r) 6n(r)dr

-

j

(12.13)

(r) dr + #6N + ~I a p 6N.

r

The first term on the right-hand side of (12.13) gives the electrostatic energy of interaction between the perturbing charge and the induced charge distribution. For a point perturbing charge away from the surface, this term as we will see in the next section, has the image-potential form. The second term is the electrostatic energy of interaction between the perturbing charge distribution and the unperturbed metal. The third term is of no particular interest because it is a constant independent of the position of the perturbing charge. The fourth term will vanish for a spatially localized perturbing charge, since such a perturbation will not change the chemical potential of a semi-infinite system. The induced charge distribution can be obtained similarly as in the calculation of the lattice effects on the work function (Section 9.3).

0.3

6 xr

rs-2

0.2 0.1

(,4D

v

-20

-.=1 v

-15

'V

-10

V

-

V

/

x (e.u.) 0

Xo

5

10

Fig. 12.3. The charge density induced at the surface of jellium of rs = 2 by a weak external electric field. The density is normalized by the condition f-~oo 5n(x)dx - -1. Redrawn with permission from Lang and Kohn (1973). @1973 The American Physical Society.

12.3. RESPONSE TO A PERTURBING CHARGE

12.3

159

R e s p o n s e of metal surface to a perturbing charge

According to Eq. (12.13) of the preceding section the change in the energy of the system due to the interaction of an external point charge q with the charge distribution 5n(r) induced at the surface is given by

5 E - - ~1

f

av(r)an(r)dr

(12.14)

where we have put # - 0 and 5v(r) is the potential due to the point charge. To determine the induced charge density 5n(x) on the jellium surface we may consider first the change in the electron density induced by a weak uniform electric field, $~t, normal to the metal surface. In this case 5n(x) is obtained by calculating self-consistently the electron density distribution for a system with the external field included into the effective potential and by subtracting from it the electron distribution for the unperturbed metal surface. If the electric field is small then 5n(x) will correspond to the linear response regime. The induced charge density determined by such a procedure, for r~ = 2, is plotted in Fig. 12.3. The position of the centre of mass of the induced density distribution,

xo -

?

xSn(x) dx

oo

/

/;

5n(x) dx.

(12.15)

cx~

can be considered the effective location of the metal surface in the sense that, for a fixed total electrostatic potential inside the metal, well outside the surface the external field contribution to the potential is

s

xo).

(12.16)

As was demonstrated by Lang and Kohn (1973) the change of energy (12.14) can be written as q2 ( 1 ) (12.17) 5E--4(x_xo) +O (x-x0) 3 " It means that for $cxt -~ 0, x0 also determines the location of the image plane. Thus the image plane position may be determined either from Eq. (12.15) or, by fitting the effective potential Vcff(x) at the point x = xl _> x0, to the form (12.5). The positions of the centre of mass calculated for the jellium model, in the localdensity approximation for exchange and correlation, range from 1.6 bohrs for rs = 2 to 1.2 bohr for rs = 6 (Table 12.1). The position of the image plane depends on the method of calculation (Efrima, 1981) but usually lies about 1-1.5 bohrs outside the jellium edge for metallic electron densities. It is weakly dependent on the screening length, lying closer to the jellium edge as r~ increases. One can ask, how the reintroduction of the discrete lattice effects modifies the image plane position ? The recent calculations of x0 versus r~ for the stabilized jellium model, which accounts for the lattice effects in an average way, have revealed the opposite trend (Kiejna, 1993) to that observed for jellium. The position of x0 shifts more outwards the positive background edge with an increase of r~ (Fig. 12.4). This

CHAPTER 12. IMAGE POTENTIAL AND IMAGE PLANE

160

Table 12.1 Positions x0 of the image plane calculated for metallic densities in the LDA from

Eq. (12.15) (Serena et al., 1986).

r s (bohrs)

xo (bohrs)

2 3 4 5 6

1.57 1.35 1.25 1.17 1.10

indicates the importance of lattice effects in such calculations and suggests that one may expect that the image plane location will depend on the metal crystallographic plane. It is clear that when the difference potential 5v(r) (see Chapter 8) contributes to the metal effective potential it will modify the electron density distributions and the values of x0 calculated from (12.15) will be changed compared to the values for jellium (see Table 12.2). Moreover these positions which are determined relative the geometric surface or, uniform positive background edge (x = 0), are strongly facedependent (Serena et al., 1988; Kiejna, 1991). There is another possibility of choosing the reference plane. Instead of the jellium edge position (x = 0) the location of xo can be tied to the position of the outer atomic plane in a metal which is usually located at one half of the interplanar spacing. Then the effective surface in a metal is located

Table 12.2

Positions of the image plane (in bohrs) for the most densely packed planes of three metals, xo is the position relative to the jellium edge, while xo + d, is taken in reference to the location of the first lattice plane in a metal (Kiejna, 1991).

Metal

Face

rs

A1 Li Na

(111) (110) (110)

2.07 3.28 3.99

x0

x0 + d

1.12 0.83 1.49

3.32 3.18 4.35

12.4.

FERMI HOLE NEAR

1.7

161

THE SURFACE

I

I

I

I

I

I

I

1 /

1.5

O

1.3 O

1.1

Pb

A1 0.9 1' ? 2

Mg ,?

I 3

Li ? ,

Na 1'

Rb Cs 1'~ 1' ,1' .....

K

,

4 5 DENSITY PARAMETER r s

6

Fig. 12.4. The image plane position versus density parameter, rs, for the jellium and stabilized jellium model. After Kiejna (1993).

at d

x i m = xo + -~

(12.18)

in front of the first atomic layer, where d is the interplanar spacing. In Table 12.3 we give the image plane position for low-index planes of A1 and Na determined from the stabilized-jellium model and from the fully three-dimensional calculations. As is seen the x0 values differ by a factor of 1 . 5 - 2, with more densely packed planes having the smaller x0. This strong face dependence of x0 is greatly reduced for the Xim values.

12.4

T h e e x c h a n g e (Fermi) hole near t h e m e t a l surface

As we have discussed in Section 4.6, the account for the Pauli exclusion principle in the Hartree-Fock approximation results in the appearance of the exchange (Fermi) hole in the electron density distribution around each electron. On the other hand the other electrons with antiparallel spins, will also move away under the action of

CHAPTER 12. IMAGE POTENTIAL AND IMAGE PLANE

162 Table 12.3

The image-plane positions (in bohrs) for low-index faces of se/ected simple metals determined from the stabilized-jellium model (SJ) (Kiejna, 1993) and ab initio (AI) calculations (Inglesfield, 1987; Lain and Needs, 1993).

Metal

A1

Na

Face

xo

xo + d

SJ

AI

SJ

AI

(111)

1.17

0.95

3.37

3.16

(100)

.57

.10

3.48

3.01

(110)

2.11

1.51

3.46

2.87

(110) (100)

1.47

4.34

2.13

4.15

(110)

2.57

3.74

electrostatic Coulomb forces. It means that they will correlate their motion so as to create the positive correlation (Coulomb) hole around electron which will compensate the electron charge. So each electron in the interior of a metal is surrounded by spherically symmetric exchange-correlation (Fermi-Coulomb) hole. We have also seen that the radius of this hole is approximately r s. Now we are going to discuss the structure of this hole when the electron is localized in the surface region or moves through the metal surface to a point distant from it. Up to recently it was commonly accepted that when an electron is moving from the metal bulk to a distant point in the vacuum, the exchange-correlation hole begins to flatten at the surface region and remains localized at the surface when electron is removed to infinity (Boudreaux and Juretschke, 1973). Thus the exchange-correlation hole charge distribution localized at the surface forms the classical image charge. As we will see this picture has to be revised: the exchange hole is localized to the surface region only for electron positions close to the surface. When the electron is removed to infinity the exchange hole is completely delocalized and spread throughout the metal. Consequently, it is the correlation (Coulomb) hole that forms the image charge at a metal surface. The conclusions concerning the structure of the Fermi hole can be drawn on the basis of analysis of the Hartree-Fock equation for an electron in the jellium. Analysis of the structure of the Coulomb hole is much more complicated. Below, we will discuss only the results for the exchange hole. Let us see first how the exchange hole looks like near the surface of a metal bounded by an infinite potential barrier. Taking the wave functions of the form (4.48) the density of exchange charge can be written in an analytic form (Juretschke, 1950; Moore and March, 1976; Sahni and Bohnen, 1985). The expression (4.65) for the exchange charge density at r ~ for the wave function labeled by k-vector can be written

12.4. FERMI HOLE NEAR THE S URFA CE

163

as

nx(r, r'; k) = ~

¢~,(r')¢k(r')¢k,(r) Ck(r) "

(12.19)

Physically more interesting quantity is the average exchange charge density, fix(r, r~), given by (4.68-4.69). In the k-notation the exchange charge density averaged over all electrons with one kind of spin is given by E ~x(r, r')

-

¢~ (r)¢~, (r')¢k (r')¢k, (r)

k,k'

(12.20) E ¢~ (r)¢k (r) k

Now inserting for Ck(r) the functions of the form (4.27) we get the following analytical expression (Sahni and Bohnen, 1985) for the average exchange charge density nx near the surface of a metal bounded by an infinite potential wall

ft {j(kgR) - j[(kFR) 2 + 4yy']1/2} 2 O(-y)O(-y') 1 - j(2y)

~x(r, r') - ~

(12.21)

where R = [ r - r'[, y = kFx, y'= kFx ~ and

j(t)

3(sin t - t cos t) -

t3

3 = ~-jl(t),

(12.22)

where jl (t) is the first-order spherical Bessel function. It is interesting to trace position of the exchange hole with reference to the position of electron approaching metal surface. In Fig. 12.5 the section through the (normalized by ~/2) exchange hole is plotted for three different position of the electron in a metal. The ~x distribution of Eq. (12.21) is cut perpendicular to the surface through rll , rill = 0, where rll is the position vector in the plane parallel to the surface. It is seen that inside a metal, away from the surface, the hole is symmetric about the electron as in the uniform electron gas (Section 4.6). For electron at the jellium edge (Fig. 12.5b) and at the position close to the potential barrier edge (Fig. 12.5c) the hole becomes distinctly flatter as compared to the bulk. Instead of considering the slices through the fix(r, r !) it is more natural to study the planar averaged exchange charge density

nx(x, x') - / drll / dr'll fx(r, r' )

(12.23)

which is dictated by the symmetry of the problem. This quantity for the infinite barrier model is plotted on the right-hand side of Fig. 12.5 for the positions of the electron considered in the graph on the left-hand side of the figure. Both quantities give a similar picture. Some differences arise, however. There is more structure in the graphs on the rhs. The maximum of the curve on the left-hand side of Fig. 12.5a coincides with the position of electron whereas that on the rhs does not.

164

C H A P T E R 12. I M A G E P O T E N T I A L A N D I M A G E P L A N E

1.0 0.8

0.3

a)

ELECTRON AT y,,-4

i

}E .LIUIi / :DeE

0.1

0.2

0

0.0 1.0

\ L

--C'4 IC

/q

m

ELECTRON AT 3ELLIUMI

~..= 0.8 EDGE t

y=-31f

t= U-

0.6 0.4

0.0 1.0 0.8

0.0 0.3

E,ECT ON E,

(b)

EDGE y--- .

0.2

cO

~

..._.. )

d

\

~c 0.2 ,c

i 3ELLIUM p " EDGE

0.2

0.4

d ----

(a)

I I

0.6

=J

;

0.1 X

t

~ . , , ~

lJ

I

I

,c 0,0 0.3

I

I

I

I

I

(c)

ELECTRON AT y

ELECTRON AT y=-0.05

0.2

0.6 0.4

0.1

0.2 0.0

L

-10

-8

-6 - 4 . -2 y' (a.U.)

0.0

I

-10

I

-8

I

-6

I

I

-4 -2 y' (o.u.)

0

2

Fig. 12.5. Variation of the slice of the average exchange charge density fix(r, r') and of the planar averaged exchange charge density fix(y, y') versus y' = k f x ' for different positions, y, of the electron and for the infinite barrier model. Redrawn with permission from Sahni and Bohnen (1985). @1985 The American Physical Society.

12.5.

ORIGIN OF THE IMAGE

POTENTIAL

165

The infinite-barrier model does not permit electron to move beyond the potential barrier edge and thus it is not suited for investigating the asymptotic behavior of the exchange hole. The important informations about this behavior can be obtained from the study of the surface bounded by the linear-potential model (Appendix D). The wave functions generated by this model are very accurate. Moreover, electron density and other surface properties can be calculated semi-analytically as a functions of the slope parameter YF -- k F X F (see Appendix D) and are entirely equivalent to those of fully self-consistent calculations. The slope parameter YF is a measure of density variation: as YF increases the density varies more slowly; YF -- 0 corresponds to the infinite barrier model. A typical metallic density (r8 ~ 2.5) corresponds to YF -- 3. The planar average exchange charge density for this value of YF is plotted in Fig. 12.6. Note, that for the linear-potential model the natural spatial coordinates are yl = k F X l and Y2 - k F x 2 , and for YF -- 3 the jellium edge position determined from the charge neutrality condition is at y = 1.187. For the electron inside the metal up to the position of the jellium edge the exchange charge density distribution is like for the infinite-barrier model. When the electron is moved outside the surface some structure develops in the exchange charge distribution. This structure grows at the expense of the principal peak (hole) as the electron moves further away (Fig. 12.6c). Thus we observe, that instead of narrowing and increasing in magnitude, the principal peak diminishes and the hole spreads further into the metal. In the asymptotic limit the hole is spread throughout the crystal (Sahni, 1989). Summarizing, the exchange hole is localized at the surface only for electron positions close to the metal when its spatial extent is small. It means that the exchangehole contribution to the image charge and/or the exchange-correlation potential is limited only to distances close to the metal surface. Consequently, it must be the correlation (Coulomb) hole that is localized at the surface and which must be the image charge for all other electron positions.

12.5

Origin of the image potential

As we have learned the Fermi hole is delocalized at the surface. Now one can ask about the structure of the potential due to the unit charge of the delocalized exchange hole. Will it lead to an image potential ? A qualitative answer to this question can be given comparing the classical image charge with the quantum-mechanical exchange charge distribution. The classical image charge induced by an external point charge is of zero thickness in the direction perpendicular to the surface and is spread over the entire surface. On the other hand, as we have seen, the distribution of exchange charge is three-dimensional and extends into the metal. Thus, the classical and quantummechanical charge distributions differ significantly and consequently one would expect that the corresponding potentials would also be different. This problem can be explained considering the asymptotic structure of the Slater potential (Slater, 1951), V ~ tat~'(r) -

] ] r - r'p

" ~z (r, r ~)dr ~

(12.24)

where ~ ( r , r') is the average exchange charge density (compare (12.20)) at r' for an

166

CHAPTER 12. IMAGE POTENTIAL AND IMAGE PLANE

0.3I"

' ELECTRON AT y - - 2

i

(a)

I

0.2

LLIUM EDGE

0.1 0.0 --------rj 0.3

d

,..

I

~

I

ELECTRON AT ZIELLlUNf.,~

I

(b}

-,~ 0.2 "~

0.1

"s2150.0

0.3

I

I

I

I

ELECTRON AT y=5

(C

0.2 0.1 i

0.0 -8

-6

-4

-2

0

y'(a.u.)

2

4

6

Fig. 12.6. The planar averaged exchange charge density ~=(y, y') versus y' = ]~FX' for different positions, y, of the electron calculated for the infinite barrier model. Redrawn with permission from Sahni and Bohnen (1985). @1985 The American Physical Society.

12.5. ORIGIN OF THE IMAGE POTENTIAL

0.0

i I

3ELUUM i EDGE"" i -

i

-0.4 Slaler

Vx(y)

IMAGE POTENTIAL-,,,,~. . . . . . . . . . . . . . . . . . .

..I

I

-0.2

167

4"

./

.

y~.....-~S~.~-----'"'-

.~'q"

11"

,,,,~,."/

,,,,,, ,,,,..,,,,, .,-- " - "

/

~

~ ~

rs -- 4.0

I J

~

I_/ I

-

oHJ , i -2

0

I

2

,

I

,

I

4 6 y ( XF/21]" )

,

I

8

J

10

Fig. 12.7. Comparison of the image potential and the Slater potential (normalized to 3kF/27r) calculated for the wave functions generated by the finite-linear-potential model. Redrawn with permission from Sahni (1989).

electron at r. The behavior of the Slater potential (normalized to its bulk value 3kF/2r) calculated for the wave functions generated by the finite-linear-potential model (see Appendix E) at the surface of rs = 4 is plotted in Fig. 12.7 (Sahni, 1989). It is seen that, outside the metal surface, the ySlater(y) potential differs significantly from the image potential. It approaches asymptotically the function 1/y which in terms of original (unnormalized) variables is equal (3/2r)/x "~ 1/(2x). Thus asymptotically, the Slater potential due to the delocalized exchange hole has imagepotential-like form but with a coefficient which is approximately two times larger than the image potential coefficient of a 1/4. One can ask whether the Slater potential is the correct local exchange potential felt by the electrons. The answer would be yes but only in the case when the Fermi hole were static. However, the exchange charge distribution is dynamic and changes for each electron position. Thus in contrast to the electrostatic potential r which is determined by a static charge distribution, the exchange potential is determined by a charge distribution that depends on the position of the test electron. As such the latter potential can be determined in the following manner (Harbola and Sahni, 1989a; Sahni, 1989). Imagine any system of interacting electrons in which each electron is surrounded by its exchange-correlation hole charge distribution. The exchange-correlation energy E=c[n] may be thought as the energy of interaction between an electron at r and its exchange-correlation hole charge density n=c(r, r ~) at

168

CHAPTER 12. IMAGE POTENTIAL AND IMAGE PLANE

r ~, and consequently we may write

1//

E=[n] =

r') dr dr," I r - r' l

(12.25)

Basing on this definition of E~r the exchange-correlation potential seen by the electrons is the work done in bringing electron from infinity to its final position r in the electric field of its exchange-correlation hole charge density. According to Coulomb's law the electric field due to xc-hole charge distribution is $~(r)

_

f nxc(r, r ' ) ( r - r') dr'

J

ir_r, 13

(12.26)

and the work done against the force of this field is W~c(r) - -

E ~ . dl.

(12.27)

oo

Harbola and Sahni (1989) proposed to replace the functional derivative of E~[n] appearing in the effective potential of the Kohn-Sahni theory by the work Wxr which can now be determined directly from the exchange-correlation hole. Thus now the equation to be solved is,

[1

]

- ~ V 2 + Ves(r) + Wxc(r) Ck(r) - ekCk(r)

(12.28)

or with Wx~(r) = W~(r)+ We(r), where the work done to move an electron against the electric fields of the exchange and correlation hole charge distribution is separated. The total charge of the exchange-correlation hole is equal +e (i.e., in atomic units it equals to +1), . [ n ~ ( r , r')dr' - 1.

(12.29)

We may consider the exchange-correlation hole as being comprised of its exchange and correlation components, and to write n ~ ( r , r') = n~ (r, r') + n~(r, r').

(12.30)

We know (Eq. (4.67)) that the total charge of the exchange hole is also equal +e. Consequently, because of (12.29) the total charge due to the correlation hole must be equal zero f no(r, r')dr' - 0.

(12.31)

Thus, asymptotically far from this charge the contribution of the correlation hole to Wxc must vanish and the asymptotic structure of Wxc is that of Wx alone. Since the exchange hole is known explicitly in terms of the wave functions (see (12.19)) the potential W~ can be determined exactly. On the other hand the asymptotic form of W~ far from the metal surface must be the image potential - 1 / 4 x .

12.5. ORIGIN OF THE IMAGE POTENTIAL

169

To determine the asymptotic form of the potential Wx, instead of solving the KohnSham equations, one can employ the semianalytical wave-functions generated by the finite-linear-potential model (Harbola and Sahni, 1989). In Fig. 12.8 the universal function Wx(y) normalized by the Slater potential (3kF/2~) for the homogeneous electron gas is plotted for rs = 2 as a function of electron position y = kgx. In this figure the function -1/2y is also plotted for comparison. The latter function in terms of the original variables corresponds to 34 1 r sx ~ !4 x " It is evident that about y - 50, the -exchange potential W~, merges with -1/2y curve. This means that for the electron distant from the surface, W~ is the image potential. Thus, this result demonstrate that asymptotically the image potential arises solely due to the exchange interactions between electrons. The above conclusion, based on the very transparent approach, is in contrast to our understanding based on classical physics that the image potential is strictly a consequence of Coulomb correlation effects. Finally, it should be noted that although the same conclusions on this issue result from another, distinctly different, quantummechanical approach (Harbola and Sahni, 1993), the calculations based on the GW

0.00

-0.02 W•

13kF/21~) -0.04

/' " /

rs--2.0

-0.06

-0.08

I

10

I

20

I

I

30 40 y( ;kF/21~ )

50

Fig. 12.8. Plot of the function Wx(y) (normalized to 3kF/27r) representing the work done against the electric field due to the Fermi hole. Redrawn with permission from Harbola and Sahni (1989). @1989 The American Physical Society.

170

C H A P T E R 12. I M A G E P O T E N T I A L A N D I M A G E P L A N E

approximation (Hedin and Lundqvist, 1969) indicate clearly that the classical limit of the surface barrier is due to the Coulomb-correlation effect (Eguiluz et al., 1992). The ultimate reason for this discrepancy is not obvious. Thus, by closing this chapter, we should emphasize that although already much has been done, a fuller and more satisfactory understanding of these effects remains to be given in the future.

Chapter 13

M e t a l surface in a s t r o n g e x t e r n a l electric field 13.1

E l e c t r o s t a t i c field at t h e s u r f a c e

According to the classical electromagnetic theory the metal is treated as an ideal conductor and the electric field is perfectly screened in the metal. Thus an external electric field of magnitude F, applied normal to the metal surface, drops discontinuously to zero at the surface. The metal interior is screened from this field by the screening charge of magnitude E - F / 4 ~ . This excess charge may exist only on the metal surface which is considered as a mathematical plane. From the microscopic point of view, however, the screening charge distribution and also electric field vary smoothly over distances of a few atomic diameters in the direction perpendicular to the surface. Let us begin discussion of the microscopic screening at metal surfaces by considering a metal exposed to a uniform static electric field perpendicular to the surface i.e., F0 - F0~ where ~ is a versor. Such a field can, for example, be generated by a uniformly charged sheet parallel to the surface placed far away in the vacuum E0=

F0

2~"

(13.1)

This external charge will induce an oppositely charged sheet singular at the metal surface and equal ES(x) where 5(x) is the Dirac function. Thus the total electric field between the external and the induced charge planes is F = 4~E = -4~E0.

(13.2)

However, in the real situation, because of the smooth electron distribution at the metal surface the induced charge will not be an infinitely thin sheet of electrons at the geometrical surface. Instead, the induced charge must have a finite spread. Generally for more intense fields the electron density distribution can be written as the following 171

172

C H A P T E R 13. M E T A L I N A S T R O N G E L E C T R I C F I E L D

expansion aye(x) = n o ( x ) + E n l (x) + E 2 n 2 ( x ) + . . . ,

(13.3)

where no is the equilibrium density profile for a neutral surface and the induced densities nl and n2 represent the linear and nonlinear contributions to the response. For a weak external fields it is sufficient to consider only the linear response terms (see Section 12.2). The linear response, however, cannot appropriately account for the effect of electric fields of the order of 1 V/A which occur at the metal surface in the field emission experiments or at metal/electrolyte interfaces. The applied electric field, F, is characterized by the induced surface-charge density E=F

/

6n(x) dx

(13.4)

5n(x) = nr~(x) - no(x).

(13.5)

4-7 -

where 6n(x) is the induced charge density

The total charge neutrality condition requires that

/?

[nr~(x) - n+(x)] dx - E.

(13.6)

(x)

As we have learned in Chapter 7 this condition can be equivalently expressed by the Sugiyama phase-shift sum rule which for the charged surface takes the following form (Theophilou and Modinos, 1972) ~o kF kS(k)dk = ~ E F - ~2E, 4

(13.7)

where 6(k) is the wave-vector-dependent phase shift of the wave function. The electron density distribution n~(x) for the charged system can be obtained either variationally by employment of the trial function for the electron density (compare Chapter 11) or self- consistently by solving the Kohn-Sham equations discussed in Chapter 6. The equations for a charged surface are unaltered as compared to the neutral case, except of the boundary condition for the electrostatic potential in the vacuum region which now reads: r (+oc) = F, where prime denotes a derivative, and the modified charge neutrality condition (13.6). Fig. 13.1 shows typical results of self-consistent calculations (Gies and Gerhardts, 1986) for the effective potential and electron density at charged surface of jellium (rs = 3) and for the electric field varying from negative to positive values. Here we adopt the sign convention where positive field, F, corresponds to charging the metal positively. For large negative fields the electron density profile becomes steeper and is shifted into the bulk. This is clearly visible in Fig. 13.2 (Schreier and Rebentrost, 1987). It is also seen that for negative fields the Friedel oscillations are increased. For positive fields electrons are pulled out of the metal and the vacuum tail of the density is increased. For high enough fields the lowering of effective barrier would allow electrons to leak out from the metal. In Fig. 13.3 the induced charge density

13.1. E L E C T R O S T A T I C FIELD A T THE SURFACE

!

I

I

I

I

I

173

I

I

I

I

F-- 0.4 V/~

0

,I

Xo

F=O

IC X C F -- -0.8

%

l J_

U.I

0 s,

F = -2.1

2;

m

0

I

1

...... -

F--4

,

!

,,.,,.

'

_ m

0

-6

I

I

-4

I

I

-2

I

I

0

I

l

I

2

X-Xb {A) Fig. 13.1. Electron density profile n and effective potential, Veff, for different fields, F, applied to the jellium surface of rs - 3. x0 indicates the centroid of statically induced surface charge. The distance is measured relative to the position of jellium edge xb. Redrawn with permission from Gies and Gerhardts (1986).

174

C H A P T E R 13. M E T A L IN A S T R O N G E L E C T R I C FIELD

1.2

,

I

l ~ ' r

x

T

r

~

1.0 0.8 IC X

0.6

_

~i~

_

14 c"

0.4

7

X\k

0.2

0 -10

-6

-2 x (a.u.)

2

6

Fig. 13.2. A comparison of the electron densities, n~, at the surface of jellium of rs - 3 for different field strengths: F - 0 (chain curve), -0.87,-2.2 and -4.3 EF/e~,F. Redrawn with permission from Schreier and Rebentrost (1987).

(in(x) is plotted for several field strengths. For a weak external field the results are similar to the linear response curve. For the higher field strength the deviations from the linear response result are seen. The first peak of the Friedel oscillations increases and is shifted into the metal with increasing negative field. The induced charge density is characterized by its center of mass x0(Z) --

/

xSn(x) dx

//

(in(x) dx -- -~

xSn(x) dx.

(13.8)

/

As we remember, here, we consider the model where electric field is produced by the applied distant charge plane. The electrostatic force exerted by the induced charge on the plane is - 2 r E 2. It is equal and opposite to the force acting onto the jellium. It means that under influence of the electric field the Budd-Vannimenus theorem (7.20) will take the form (Theophilou, 1972)

deT

~-d-~

27r~ 2 -

r

r

-

-

~

~ ,

( 1 3 . 9 )

which implies (Budd and Vannimenus, 1975) that

~2 f

-~n =

xSn(x) dx. oo

(13.1o)

13.1. ELECTROSTATIC FIELD AT THE SURFACE

,4

'

I

I

'

I

175

i

I

'

0.3 x

~. 0.2 %

%

t

x 0.1

_

,

,

,

,

_

I,o

0

i

-10

i

,

-6

i

-2

,

x {a.u.)

i

2

i

6

Fig. 13.3. A comparision of the induced charge density, 5n(x), for field strengths: F = -0.43, -0.87 and -2.2 EF/eAF. The chain curve represents linear response density nl (see Sec. 13.2). Redrawn with permission from Schreier and Rebentrost (1987).

Thus, x0(E) can be alternatively calculated from the expression

1 9

LOO

xSn(x)dx

E

2n"

(13.11)

In the absence of external field (E = 0) the values x0(E = 0) determine the image plane position (see Table 12.1). As indicated in Fig. 13.1, for the positive electric field the center of mass, x0, of the induced charge shifts outwards whereas for negative fields it moves towards the jellium edge. The results for the centre of mass of the induced density at the jellium and stabilized-jellium surface are shown in Fig. 13.4. As we have already discussed in Chap. 12, for stabilized-jellium model the image plane position of high density metals lies closer to the positive background edge then for ordinary jellium whereas for low density metals the reverse trend is observed. Accordingly, the x0(E) values for stabilized jellium are lower for high density metals as compared to jellium (Kiejna, 1993a). With an increasing positive field, x0 moves towards the background edge and becomes negative faster for the stabilized jellium than for the ordinary jellium. For rs = 2, in the given range of fields (surface charge E = 1 a.u. corresponds to the field strength 51.4 V/A), the x0's calculated for both models differ by 0.5-0.25 bohrs.

176

C H A P T E R 13. M E T A L I N A S T R O N G E L E C T R I C F I E L D

I 1.6

--

I

'

I

I

I

0 @

1.2 -

rs=2

1

0.8 o

0.4

-

0.0

-

0

I

-0.4

I

-5

I

I

5

I

I

15

25

z (1O-Sa.u.)

i

1.6

o

I

i

I

i

I

m

o

(b)

o

1.2

I

rs=3

O.S o

0.4-

D

0.0 -0.4 -0.8

-2

i

I

2

I

I

6

I

I

10

I

I

14

P. ( 1 0 - S a . u . ) Fig. 13.4. Centre of mass of the induced charge density versus surface charge at the surface of jellium (open circles) and stabilized jellium of rs = 2 and 3. After Kiejna (1993a).

13.2.

13.2

177

CONTRIBUTIONS TO THE RESPONSE

Linear

and non-linear

contributions

to the re-

sponse As we have seen (Fig. 13.2) the response of the electrons to the more intense electric fields becomes non-linear. In general the response of the electron system to external field, F0(x), can be described by the response functions, X, according to hind(X)

--

n~(x) - no(x)

--

f J X1 (X, xl)Fo(x 1) dx' X2(x,x', x" )Fo(x')Fo(x")dx' dx" + . . .

+

(13.12)

where nind(X), is the induced charge density. Using F0 = 27rE and by comparison with Eq. (13.3) we may define the average response functions hi(X)

21r [ Xl(X, X I ) dx', J

=

(13.13) n2(x)

f

(2~)~ j x ~ ( z , ~,, x") d~' dx".

-

The functions nl(x) and n2(x) are plotted in Fig. 13.5. These are obtained in the following manner. Using the expansion (13.3) one has (Weber and Liebsch, 1987)

hi(X) and

1

-

-

~-~[n+(x)- n~(x)],

1

n2(x) = ~ - ~ [ n + ( x ) + n ~ ( x ) -

2n0(x)]

(13.14) (13.15)

where n + (x) and n~ (x) are self-consistent electron density profiles for a semi-infinite metal with a small positive or negative surface charge E. The linear relation (13.2) between the external field and the induced charge, which is fulfilled for arbitrary field strengths, imposes some constraints on the response functions nl and n2. Namely, by inserting (13.3) into the charge neutrality condition (13.6) one gets (13.16)

~ nl (x) dx c~

and (13.17)

~ n2(x) dx = O. oo

The centroid of the second-order induced density n2(x) x2 =

x2n2(x) dx

oo

/?

/

oo

xn2(x) dx

(13.1s)

178

C H A P T E R 13. M E T A L IN A S T R O N G E L E C T R I C FIELD

lies about 0.5 bohr farther away from the positive background edge than the image plane x0 (Weber and Liebsch, 1987). The linear part, n l, of the response can also be calculated explicitly within the first-order perturbation theory using the static limit of the time-dependent density functional theory. The calculational procedure is described by Dobson and Harris (1983), Liebsch (1987), Schreier and aebentrost (1987). The expansion (13.3) has proven to be very useful in studying optical second harmonic generation current originating when laser light is reflected from the metal surface, which is now probed experimentally (Song et al., 1988). As demonstrated 0.3

i

i

i

i

I

0.2 o

i

0.1

I

I

l

0

L]

-0.I

I

-20

N

I

!

-10 x (a.u.)

,

,

I

I

0

I0

4O

W fr

~" 20 cO 13. U'%

0 I_.

(p C

"-= 20 -

o z

-40 -20

I

I

I

-10

0

10

x (a.u.) Fig. 13.5. T h e linear and nonlinear response functions, n l (x) and n2(x), obtained from (13.3) for r s - 3. The broken curve shows n l calculated by neglecting exchange and correlation effects. Redrawn with permission from Schreier and Rebentrost (1987).

13.3.

IONIC

LATTICE

179

EFFECTS

by Weber and Liebsch (1987) for a weak electric field there is a direct connection between the static second-order density n2 and the low-frequency second-harmonic current density. The first moment P2 of second-order density n2(x) P2 -

-

F

xn2(x)dx

(13.19)

(3O

determines the dimensionless parameter a, which was introduced by Rudnick and Stern (1971) to characterize the second-harmonic surface current a - 4~p2.

(13.20)

For jellium surface the values of a vary between 28, for r~ - 2, and 7 for r~ - 5.

13.3

Effect o f t h e ionic l a t t i c e

There have been rather few self-consistent calculations of the three-dimensional potential and screening charge density at metal surfaces including the effects of applied electric fields. One of the first calculations of this type has been reported by Inglesfield (1987) who considered the screening of electric field at AI(001) surface. He found that the screening charge is not distributed uniformly over the surface, but tends to build up on top of the surface atoms. The similar effect was observed for other low-index faces of A1 (Lam and Needs, 1992). The planar average of the induced screening charge density (in for the Al(001) surface looks very similar to the results for jellium with rs - 2, but the centre of gravity x0 of the screening charge lies at 1.1(+0.2) bohrs, measured from the geometrical surface, i.e. 0.5 bohr closer than the jellium value 1.6 for rs - 2. The zero-field values of x0 calculated by Lam and Needs (1993) for the A I ( l l l ) and (110) surfaces amount 0.95 and 1.51 bohr respectively. It is seen that the atomic structure of the surface shifts the image plane or electrical surface inwards with respect to the jellium value. This indicates that the electronic charge distribution at the real metal surfaces is much stiffer than the corresponding distribution for jellium. The latter conjecture is corroborated by the results of calculations within the stabilized-jellium model (Kiejna, 1993a). This also suggests that for a real metal surface the quadratic effects are much smaller than for jellium. The field-dependence of the position of the centre of gravity of the screening charge density can be reasonably fitted by the quadratic relations (Lam and Needs, 1993) for the A I ( l l l ) surface x0 - 0.50 - 0.11F + 0.0085F 2,

(13.21)

and by the following linear relation x0 - 0.80 - 0.075F

(13.22)

for the Al(ll0) surface. In (13.21) and (13.22) x0 is measured i n / ~ and the field is measured in VA -1 up to 5 V/~ -1 (Fig. 13.6). Aers and Inglesfield (1989) have studied the screening of electric fields of strength varying between +0.04 a.u and -0.02 a.u. at the Ag(001) surface. Fig. 13.7 shows

180

CHAPTER 13. M E T A L IN A STRONG E L E C T R I C FIELD

the contour map of the spatial distribution of the screening charge for F = +0.01 a.u. and the corresponding change in potential for this field is plotted in Fig. 13.8. In the case of Ag atoms one observes the exclusion of the screening charge from the atoms. In the case of Ag(001) surface the field dependence of x0 is given by x0 - 0 . 9 7 - S.S3F

(13.23)

where both x0 and F are given in atomic units. The result for the image plane position Xo - 0.97 bohr is again smaller than the corresponding value xo = 1.4 bohr for jellium.

0.50

m

m

m

m

m

n

m

m

m

m

m

m

m

m

m

m

m

m

m

n

m

m

n

m

m

n

m

m

"

0.45

(a)

0.40

A

<9

:,,?

0.35 0.30 0.25 0.20 0.15

,

i

,

,

I

0

i

,

,

,

I

1

,

, i

,

I

2

,

i

,

i

l

3

i

l.i

IN~i

4.

i

i

i

5

F (VA-') 0.80

0.75

,

,

.

,

,

i

,

,

,

,

i

,

,

,

,

i

,

,

,

,

i

w

,

,

,

o

(b)

0.70 0.65

~9 o

x

0.60 0.55 0.50 0.45

i

0

J

l

l

i

l

l

1

l

l

l

J

l

l

l

2

i

J

i

l

l

i

3

l

l

l

l

l

4

5

F (VA-')

Fig. 13.6. The position of image plane, xo, as a function of the applied electric field for the AI(lll) and the AI(ll0) surfaces, xo is measured with reference to the geometrical surface. Redrawn with permission from Lam and Needs (1993).

13.4. FIELD INDUCED RELAXATION

13.4

181

Field induced relaxation and field evaporation

Classically, metal is impenetrable to the electric field which should vanish at the electrical surface (image plane). Thus the electric field should not have any effect on the ion cores of the metal. However, the results of this Chapter have shown that the screening charge distribution in the metal has a finite thickness of several atomic diameters and the strong electric field can penetrate into the metal over the

9

.

9 ,, ,,

. . . .

".

.." ............

."

- i .,

,.

..

.. "'-..

i

". " '. '.

' ,' ,' "

";

,,

.. .............

,"

9

9 '.. ",.(.===================== / ,; ."

-

./

d2)

_.J

.. ,," 9

,' i

.o . . . . . . . . . . . . . . . . . .

-.

................ ' d

-..

o~176 .................. "....

," ,

~,,

i" ,..

.,

..~ ". "..:

',, :.

5.5 a.u. Fig. 13.7. The contour map of the screening charge at Ag(001) surface for the applied field F = 0.01 a.u. Solid and dashed lines correspond to the decreased and increased density, respectively. Heavy dots mark the nuclei. Reprinted with permission from Aers and Inglesfield (1989).

182

CHAPTER 13. METAL IN A STRONG ELECTRIC FIELD

distance of a few atomic diameters thus, acting at least at the topmost layer of ions. One may expect that the electric field will induce changes in the layer spacing. For sufficiently strong fields of the order of several VA -1, as it occurs at field ion tips, the surface atoms will evaporate (Miiller and Tsong, 1969). The minimum field strength, exceeding which at low temperatures the metal tip evaporates, is called the critical or evaporation field strength. It varies from 3 v A -1 for Cu to 6 V/~ -1 for tungsten. Experimental observations suggest that field evaporation of metal atoms occurs most likely at steps, kinks and edges which increase the local field on the crystalographic planes. Self- consistent calculations of the electric field and the electron

,

.-

-...

.-

...

5.5 a.u.

Fig. 13.8. Change in potential at Ag(001) surface induced by the field F - 0.01 a.u. Reprinted with permission from Aers and Inglesfield (1989).

13.4. FIELD INDUCED RELAXATION

183

density distribution for such geometries are extremely difficult. Therefore we will limit our discussion to the calculation of this effect only for the planar geometry of the surface. Let us discuss first the relaxation of surface ion-layer. The surface charge, depending on its sign, will try to move the surface layer of atoms changing the layer spacing. Due to the screening effect we will assume that these changes are limited to the spacing between first and second layers only. Thus the interplanar distance between these two layers of atoms is d12 = (1 + ~)d (13.24) where d, is the bulk interplanar spacing and A is the dimensionless parameter being zero for the unrelaxed lattice. In order to find the equilibrium relaxation for a given external field one can look for the minimum in the surface energy with respect to the lattice displacement. With reference to the models discussed in this book this can be done in either of two ways. First, more straightforward but less accurate, is to apply the variational method with the trial electron density profile (Kiejna, 1984). The other method is the self-consistent solution of Kohn-Sham equations of Section 6.2 and minimization of surface energy (Perdew, 1982; McMullen et al., 1982). Below we will sketch the calculations within the second approach. The process of the lattice relaxation can be envisagioned as follows. In the first step, the bulk electron density, ~, is truncated at x - 0 and the first layer of ions, initially at -d/2, is allowed to shift rigidly along the x-axis by the distance Ad. In the next step the electron density is allowed to relax to form the actual density distribution n(r). The interaction of the relaxed ions with the slab of the homogeneous electron density of thickness d (step 1) will lead to a modification of cleavage energy term (8.25) and the aR term resulting from the repulsive part of the pseudopotential for the overlapping ion cores (Chap. 8). The relaxation of the electron density will be also reflected in the kinetic, exchange-correlation, electrostatic and pseudopotential energy contributions to the surface energy. In order to account for the discrete lattice effects, in the case of relaxation, one can replace V(x) in the effective potential (8.38) by the following one-dimensional form (Perdew and Monnier, 1980) V(x,

= ce(-x)

+ 5v

(x) - 5 (x)

(13.25)

where 5v), (x) is the planar average of the potential difference for the relaxed lattice, and the constant C is chosen to minimize the surface energy of the unrelaxed surface. The relaxation dipole potential, given by the difference of the last two terms in (13.25) can be evaluated using for 5v~,(x) the expression given in Appendix B. Lattice relaxation requires modification of the surface energy components which arise because of the lattice effects. The pseudopotential surface energy contribution (8.28) for the relaxed lattice is

ap~

-

F O0

5v~(x)[n(x)

-

~O(-x)] dx.

(13.26)

CHAPTER 13. METAL IN A STRONG ELECTRIC FIELD

184

I)}

The simplified form (Perdew, 1982) for the change in the cleavage energy due to the relaxation (Hietschold et hi., 1980) in the range - 1 / 2 < A < 1/2, is

aCz(A) = a~z(O)+ 2zr~2d3 {A 2 + Z '

1

ITS,l"t ~ m n

S~(e

-~-

e N~,~,~

--

where Stun = ~

e ( k - ~ ' ~ COS

1

]

( k - 1)(m + n) .

k=l

(13.27)

(13.28)

a~z(0) is the cleavage energy of the unrelaxed surface, d is the bulk interplanar spacing and Nd is the stacking period. (mn is the product of d with the projection of the three-dimensional lattice vector onto the surface plane. The summation goes over all integers m and n except those for which (ran = 0. Finally, the surface energy term due to the core repulsion energy of the metal will change, when the surface layer ion-cores extend into the vacuum region, and we obtain

dA d + r c ) aR=-Tr~2d(-~+

20 ( - ~d+ A d + r c ) .

(13.29)

Among the uniform background contributions to the surface energy, only the electrostatic part will depend explicitly on the external field. Replacing n(x) by nr~(x, A) + Eh(x - oc) in (8.6), where nz(x, A) is a distribution of electrons which remain in the metal and E are electrons per unit area which were removed to infinity, the electrostatic energy becomes

:f

aes(A, E; nr~) - ~

r

A)[nz(x, %) - n+(x)] dx + ~r

A, oc)E.

(13.30)

OO

The total surface energy functional in the presence of electric field and relaxations can be written as a[A, E; nz(x, A)] = 5[nr.(x, A)] + ~[A; nr,(x, A)] + aes[E; nz(x, A)]

(13.31)

where 5[nrJ denotes the contributions to the total energy which do not depend explicitly neither on external field nor relaxation (i.e., the kinetic energy and the exchangecorrelation energy). The second term 5[A; nrJ contains the contributions which do not depend explicitly on the electric field: the ion-ion energy and the electron-ion energy. The zero-field calculation within this model (Perdew, 1982) showed a minimum in surface energy curve for 1% expansion of the first interlayer spacing at A l ( l l l ) for which the method is best suited. This result is in a good agreement with the 2% expansion measured by LEED. The results of self-consistent calculations of surface energy from equation (13.31) for various fields and displacements parameter, 0.0 < A _ 0.5, are plotted in Fig. 13.9. The solid curves display the change in the surface energy with Aa(A, F) - a(A, F) - a(0, F)

(13.32)

13.4.

FIELD INDUCED RELAXATION

At ( 111 )

800 E o

185

600

---

self- consistent

- -

non-serf-consistent

s

I,_ ,........

-~

400,

200 t.u

z

t

0

f

~ s s

.,,. ~

s

J

I..U

-200 D Or)

z -400

%

I

%

I..iJ (_'.'.'3

% %

z -600

I (_2

-800

%

i

0.0

0.1

0.2

0.3

"

" 0.4

0.5

LAYER DISPLACEMENT PARAMETER

Fig. 13.9. Change in surface energy of A1(111) versus surface layer displacement for various fields. Redrawn with permission from McMullen et al. (1982). @1982 Elsevier Science Ltd, UK.

for a given field, F = 47r~E. For larger fields the equilibrium relaxation, which is given by minimum in the curves, moves outwards. The largest outward relaxation is ~ 20% and appears for applied field strength of _~ 4.5 V/~ -1. For larger fields there is no minimum in the curves - they drop monotonically indicating that the critical field has been exceeded and the surface layer is stripped off. Thus if other surface processes, which may change arrangements of surface atoms, are neglected such calculations determine the critical field required to evaporate metal ions from a flat single crystal surface. For A l ( l l l ) the calculated critical field is "-" 5.5 V/~ -1 which is more than observed experimental value of 3.3 VA -1 (Waugh et al., 1976). It should be noted, however, that the comparison with experiment is difficult because the evaporation fields are measured only for rounded metal tips which expose many crystal planes and have many defects such as terraces or kinks.

This Page Intentionally Left Blank

Chapter 14

Alloy surfaces The surface composition and energetics of alloy crystals and small particles is of great importance in determining physical properties and processes occurring at the surface. In this Chapter we shall address surface properties of alloys, especially of simple-metal constituents, such as surface energy, work function and surface segregation. The elements of information on volume or bulk properties of alloys are given in the book of Alonso and March (1989). A more detailed description of structural and electron physics of alloys can be found in the monograph of Haasen (1978) and in a review by Hafner (1985). Alloys formed by simple metals with equal or similar valence can be described by the nearly-free-electron approximation in the framework of the density functional formalism. Also the surface properties of such alloys may be described in a similar way like for pure metals. This is the picture adopted in this Chapter.

14.1

T h e V e g a r d law and t h e v o l u m e of f o r m a t i o n of an alloy

The fundamental problem in the theory of alloys is the prediction of the variation of the mean atomic volume or, interatomic distance with concentration of constituents. Under the assumption that the alloy is completely random, i.e. there is neither long-range nor short-range order we will accept Vegard's law (Vegard, 1921) which assumes that the lattice parameter changes linearly with concentration. The linear relation between the lattice parameter a and concentration suggested by Vegard may be extrapolated to hold for the dependence of alloy volume on the concentration. The latter is known as Zen's law (Zen, 1956). Thus, denoting by ~A and f~s, the atomic volumes of metals A and B, and their concentrations by c and 1 - c respectively, the atomic volume of AcBl-c alloy is fie = CflA + (1 -- c)flB. (14.1) Consequently the electron density parameter r s~ (Wigner-Seitz radius) for homovalent alloy is rsc = [cr3A + (1 -- c)r3s] 1/3. (14.2) 187

188

C H A P T E R 14. A L L O Y S U R F A C E S

For heterovalent alloy one can assume that the average valence, Z~, approximately is Zc = cZA + (1 - c)ZB.

(14.3)

It means that we substitute an AcBl-c alloy by an effective monatomic solid with characteristics intermediate between the two components. This is the essence of virtual crystal approximation. Ionic effects are also relevant in alloys of simple metals. In description of an alloy in the density-functional-pseudopotential formalism we have to deal with two core-radii, rcA and rcB, of alloy components. In such situation, to a good accuracy, one may employ the linear relation between the Wigner-Seitz radius, r0, and the core radius, r~, of Ashcroft pseudopotential of individual components of alloy (Alonso and Ifiiguez, 1981) ro = "~rc + 5, (14.4) where ~/and 5 are constants. The optimized r~ is calculated by requiring equilibrium volume f~eq to agree with the measured value. Another coefficients of linear relation between density parameter rs and r~, one may obtain considering alkali metals solely (Kiejna and Wojciechowski, 1983). Thus, making use of Eqs. (14.2) one can calculate from (14.4) the core radius, r~ =_ r~(c), for the pseudo-ion of the alloy as a function of components concentration c, density parameters rsi and core-radii rci, where i =A, B.

The pseudopotential core radius r~ may be determined alternatively by use of the stability condition of bulk alloy i.e., from the condition that the bulk total energy per electron CT, attains minimum at the value r s~ corresponding to a given concentration C

OcT(r~; c, r~) = 0. ~rsc

(14.5)

Examples in which Vegard's law is strictly obeyed are very scarce (see e.g. Pearson (1972)). A better estimation of the trends in the volume of formation of the spbonded metals may be obtained using first-order pseudopotential perturbation theory (Girifalco, 1976). Namely, the formation of an AB random alloy may be considered to consist of two essential steps: first, we expand one type of atomic cells (say those of the atoms A if the electron density is larger in A than in B) and compress the other until each cell is electrically neutral at a constant average electron density r s. In a second step electrons are allowed to flow between the cells to equilibrate the difference in chemical potentials. The total energy per atom of a pure metal which includes the discrete ions is ET = Z~jell ~t_ Zs (14.6) where ejeu denotes the contribution coming from the first two terms in Eq. (8.31) and is given by (4.82), whereas eion is a sum of the Madelung energy s (Eq. 8.44) and the average value of the repulsive part of the Ashcroft pseudopotential wR (Eq. 8.22). Thus, after the first step, the total energy per atom in the A B alloy is given by (Hafner, 1985a) EAB(c,Z,r~,rr

=

Zej~u(rs) + CAZAeAon(r~) + c s Z B Q o , ( r 8

14.2.

189

SEMI-EMPIRICAL THEORY

=

-

--CB~

"-'B

\ rs ]

(14.7)

where CA and CB = 1 --CA denote the concentrations of atoms A and B, respectively, and Z is given by Eq. (14.3). The equilibrium value of the density parameter rs is obtained by using the zero pressure condition p = - O E T / O r s = 0, where E AB is given by Eq. (14.7). This procedure leads to the following equation 2 CAZAr2cA + CBZBrcB

_

0.0035Zr 3

~ r ] 5 / 3 + r_ + 0.2[CAZ~A

r75/3 B

) "+"0.102Z ] r~2

- 0.491Zrs.

(14.8)

Applying zero-pressure condition to components A and B, one may express the rcA (for cB = 0) and rcB (for CA = 0), through the Wigner-Seitz radii, r~A and r~B, of individual components A and B to get the following equation for the equilibrium density of the alloy ~ r]5/3 0.21,CAZ~A

+

^ r ] 5 / 3 ) + 0.102Z ] r~2 - 0.491Zrs CB~B

--

CAZA [(0.2ZA/3 + 0.102)r2A --0.491rsA] CBZB [(0.2Z2/3 + 0.102)r2B- 0.491rsB] = 0.

(14.9) Note that the first, small term on the rhs of (14.8) resulting from the Nozi~res-Pines form of the correlation energy (see Chap. 4) is neglected in (14.9). Equation (14.9) determines the crystal electron density as a function of composition. Note that in general, it does not give the relation (14.2) between rs and concentration, c, and therefore does not have the form of Vegard's law. The volumes of alloy formation, calculated from this equations, are in a better agreement with the experimental ones than those calculated from (14.1) (Hafner, 1985a). However, Girifalco (1976)studying the mixing energy of alloys formed by mono- and divalent metals has shown that Vegard's law will hold if two metals form random solid solutions.

14.2

Semi-empirical

theory

of alloy

formation The success of the semi-empirical model of alloy formation developed by Miedema and coworkers (Miedema et al., 1973; 1980) has stimulated an interest in the surface

190

C H A P T E R 14. A L L O Y S U R F A C E S

electronic properties of alloys. In this Section we will discuss briefly how the work function and surface energy enter Miedema's description of alloy. A complete description of this approach is given by Alonso and March (1989). Here, we would like to write down most important results of his treatment only. The model is based on partitioning of the alloy into atomic cells and assumes that the electron density in the Wigner-Seitz cells, which are the building blocks of the pure metals, remains unchanged as the cell is removed from the metal in order to form the alloy. A new parameter which allows to characterize alloy is the electron density at the boundary of a Wigner-Seitz cell, n w s . When dissimilar cells of two metals, A and B, are brought into contact there will be discontinuity Anb in the electron density at the boundary. The elimination of this discontinuity and formation of the smooth electron density distributions will require energy which results in a change of the heat of formation of alloy. The latter is defined for substitutional alloy AcBl-c by A H = E(c) - c E A -- (1 -- c)EB

(14.10)

where E(c) is the energy per atom of the alloy and, EA and EB, are the energies per atom of the pure metals. A second parameter is the energy of dipole layer which arises from the transfer of electrons between cells in order to equalize the chemical potentials of constituents cells in the alloy, similarly as in contact potential difference for macroscopic pieces of metal. Thus, for an alloy of two non-transition metals one simply has A H A B -- - P ( A r

2 + Q(An~/3) 2

(14.11)

where P and Q, are positive constants and, Ar

- r

r

(14.12)

is the difference in the work function or more precisely in the electronegativities of two metals, /kn~/3 _ r~A1/3_ ~ , (14.13) nA and nB are electron densities at the boundaries of Wigner-Seitz cells. The concept of electronegativity, r of pure metal occupying an atomic cell of radius r0, can be introduced considering a variation of the total number of electrons inside a cell from its equilibrium value Z, to a value z remaining in a cell after the transfer of z - Z electron from/to neighboring cells

r

=

-

OET (to, z) OZ

(14.14)

ro,Z

where ET is the total energy of a single (neutral) cell. From this definition of electronegativity follows that r = -# (14.15) where, # is the chemical potential which plays a role of Lagrange multiplier in the Euler equation for the energy of the electron gas contained in an individual cell.

14.2. SEMI-EMPIRICAL THEORY

191

,

4000

1000

4

E

~,~o

400 o

o~

o

I00 ~o/ 40

4

/

o

/

/o9

/

I

10

I

I

40 100 vap/ 2/3 AH s /Vm {k3/m 2}

~)o

Fig. 14.1. Linear relationship between the heat of vaporization per molar volume and the surface energy, a, for solid metals at T = 0. Open circles and dots correspond to nontransition and transition metals respectively. Redrawn with permission from Miedema et al. (1980).

It is clear now that electronegativity r entering (14.12) is related to the work function (see considerations in Chapter 10) and the latter can be taken as a first approximation to r The asterisk in r reminds that the work function had to be slightly readjusted in order to become a parameter relevant to alloying behavior. The form of the equation (14.11) allows to derive a direct connection between surface energy and the heat of vaporization of pure metals to give

AHvap ~2/3

--~ C10"

(14.16)

where Ft is the molar volume and C1 is a constant. This relation is compared with experimental data in Fig. 14.1. From (14.11) it is evident that, provided P and Q are truly constant for any choice of metals A and B the ratio, A r

A

1/3

determines the sign of the heat of

alloy formation. Thus, plotting Ar versus An~/3 one gets a value of the constant (Q/p)l/2 as a straight line separating alloys with positive heat of formation from those with a negative one. Such a test of (14.11) for solid binary alloys consisting of a transition metal and a transition, noble, alkali or alkaline-earth metals (Miedema et al., 1980) is plotted in Fig. 14.2.

192

C H A P T E R 14. A L L O Y SURFACES

_-- _/ -

T

_

:/

_

-

-

-/

§

§

1

§ §

,,--...

>.

;

o

-

_

_ _

_

.,,:::!

-

!

.

_

.

2

.

--

.

.

.

.

.

.

§

_~-__

+

+,+ § _

+§

-

§

+

+

_ -

~_--

§

+

I. § 2 4 7

.

_ _ -:

_ --=-__ %. _

=

]

. _

- -

__

_ _

.

_

_

+

§247

§247

,

+

§

§

§ §

§

+ +

_-

+ + +

.

+

** §247 * ~ §

§247

§

§ *

§ §

§ -

§247

.4~§

§ .~

_

Oo

-

"

+

§247 § +

"

§

§

~.§ §

o;

113

Anws

(a.u.)

-----~

Fig. 14.2 Demonstration of the validity of Eq. (14.11) for solid binary alloy consisting of transition metal and a transition, noble, alkali or alkaline earth metal. Redrawn with permission from Miedema et al. (1980).

14.3

Surface properties of alkali metal alloys

We start from the discussion of random alloys. For description of surface properties of disordered alloy AcBl-c it is convenient to use the virtual-crystal approximation (Stroud and Ashcroft, 1971) where the ionic lattice of alloy is replaced by a periodic lattice of mean monatomic pseudopotentials of the two components. This greatly facilitates the alloy calculations and allows to perform them in a way formally identical to the pure-metal-type calculations. The continuous dependence of the pseudopotential core radius on concentration of constituents can be calculated using the linear relation (14.4). Within the density functional formalism (Chap. 5) one can consider the surface energy functional analogous to that of a pure metal, i.e. of the form (8.29-8.30) O" - - O"u -Jr- O'ps -t- O'cl --~ O'R

(14.17)

which includes the jellium contribution (kinetic, exchange-correlation and electrostatic terms) and the lattice terms with ions represented by the local Ashcroft pseudopoten-

14.3. A L K A L I M E T A L A L L O Y S

193

tial. In a simplest approach one may perform fully analytical calculations of surface energy assuming that the electron density profile is determined by the variational principle for the energy (see Chap. Ii) using a trial function given by Eq. 11.16). As we have seen (compare Chap. 5), the incorporation of the fourth-order density gradients in the kinetic energy functional allows to obtain results for surface energies and work functions of pure metals which are in a very good qualitative agreement with the results of self consistent calculations for jellium and real metal surfaces.The discrete lattice ions give rise to the last three terms in equation (14.17). Minimizing the surface energy functional (14.17) with respect to the variational parameter one can determine the electron density profile as described in Chap. II. Assuming that alloys of alkali metals have the same crystallographic structure as their components i.e., bcc structure and using the appropriate formulas of Chapter 11, one can calculate surface energy and work function of alloys. It should be noted that from all possible binary combinations of alkali metals only KRb, KCs and CsRb form continuous solid solutions. The other (NaK and NaCs) form incongruently melting

9

'

!

I

I

I

'

i

'1

I

I

(I~0) 3501- \ \

7

~

E

L i - Na

Li-Rb

u r

250

-

Q,I C a,l t,./ r wI,L ::3

w

K

Li-Cs

"~ ; ~ < . " ~ . ~

150 .

Rb

i ~ _ Cs_L_'_'_~ K-Rb

.~.~.

. . . . . . . . . . . . . . .

50

A

,

, 02

,

-_-~

, O.L,

,

".

. . . .

_-..-_-_..__ _ _ . _ _

= 0.6

,

. 08

,

I B

Fig. 14.3. Surface energy of binary alkali-metal alloys against concentration of constituents for the (110) face. After Kiejna and Wojciechowski (1983).

194

C H A P T E R 14. A L L O Y S U R F A C E S

compound or the system (NaRb) with simple eutectics (Pearson, 1959; 1972; Girifalco, 1976). Figure 14.3 shows the calculated surface energy of (110) face for different intermetallic composition of alkali metals. As one can see from this figure, the greater difference between the average electron densities in the metals creating the alloy the steeper is the variation of the surface energy with the atomic concentration of the component. For alloys such as KRb and RbCs composed of metals characterized by relatively smaller difference in bulk electron densities (the difference between the Wigner-Seitz radii r~, being about 0.4) the surface energy varies linearly with the concentration of the constituents. Unfortunately the calculated dependencies of the surface energy on concentration for binary alloys of alkali metals cannot be compared with measured values since relevant experiments are very difficult from the technical point of view. The concentration dependence of work function for three low-index planes of binary alkali-metal alloys is displayed in Figs 14.4 and 14.5.The results for jellium and experimental work function are also plotted for comparison as functions of concentration. Similar to the surface energies, work functions vary faster with concentration for the alloys having greater differences between the average electron densities of the pure constituents. The concentration dependence of the work function for KRb, RbCs and NaCs is almost linear. A detailed comparison between calculated and experimental dependencies of the work function on the composition is practically impossible because, in the literature, there exist only few such measurements and these are of uncertain reliability. This is caused by many difficulties connected with the preparation of the appropriate samples and with the experiment itself. In Figure 14.4a the measured work function change versus concentration for a thin film KRb alloy prepared by deposition of both constituents on a molybdenum substrate (van Oirschot et al., 1972) is displayed. In Fig. 14.4b the measurements of Malov et al. (1974) for solid RbCs alloys are presented. In both cases the concentration change of the work function of alloy is linear and as it is seen from these figures the agreement between theory and experiment is quite good. On the other hand the experimental curves for NaCs alloys (Malov et al., 1974) quoted in Fig. 14.5 have essentially different character from the theoretical ones. This may be the result of creation of the Na2Cs compound and the fact that the calculations were performed under the usual assumption that the metals considered form continuous solid solutions throughout the entire range of concentrations. Thus, such anomalies cannot be explained by the model presented in this Section.

14.4

Work function of ordered alloys

A simple model of the preceding Section, describing surface properties of disordered alkali-metal alloys, can be applied to discuss qualitatively the influence of alloy ordering on its work function. The change of the alloy work function with the ordering was predicted long time ago (Sokolov, 1951; Wojciechowski, 1958).

14.4.

W O R K FUNCTION OF ORDERED ALLOYS

2.6

r~

9

i

|

!

I

!

i

I

I

195

i

9

i

l

2.4 C 0 .m t.J

2.2

2.0

K

2.5

L

0.4

02

"i

i

I

I

0.6

i

9

I

l

I,

l

I

..-......

"-" {:::

9

0.8

Rb

I

---

,..,......._

...

2.3

0 .m ,.,l,,u :::} wl,-. L 0

21

"0~

0 ~ 0 ~

[111) 1.9

Rb

t

~

0.2

I,

1

0.~

l

~ I

0.6

"'~.~ l

08

t

Cs

Fig. 14.4. The work function as a function of composition for different planes of the alloy: (a) KRD, (b) RbCs. The broken curve denotes the work function of the jellium and the chain curve represents experimental results. After Kiejna and Wojciechowski (1983).

196

CHAPTER

14.

ALLOY SURFACES

2.8

(110)

c: o

(loo)

~ ,dlm,

c)

3

2.2

\

2.0

\

9

\

9

~) 9 k

@~ 9

9

1.8

.\/ ~o

,

0.2 ,

/

Ol+ ,

,

0.6 9

'

O.t8

t

Cs

X

Fig. 14.5. The work function as a function of composition for different planes of the alloy NaCs. The work function of jellium is given for comparison (broken curve). The chain curves represent the experimental results at different temperatures: (a) 183 K and (b), 298 K. After Kiejna and Wojciechowski (1983).

The ordering of alloys leads to the occurrence of super-structures (Pearson, 1972). A well known example of such an alloy is the ~-brass (CuZn) which has ordered superstructure consisting of two simple cubic lattices of Cu and Zn ions, mutually shifted by one half of the lattice spacing in the x, y, and z directions (Fig. 14.6). In order to apply the model of Sec. 14.3 one has to determine the required parameters: the valency Z, electron density parameter rs and the pseudopotential core radius re. For CuZn alloy, the ratio of the number of electrons to a number of atoms per unit volume is 1.48 (Wernick 1965). This figure is close to a number of conduction electrons per atom in a unit-cell volume in CuZn-~ alloy. Taking c - 0.5, i.e. equal contribution of Cu and Zn atoms to the alloy and Z c u - 1, ZZn = 2, one gets for the bulk valence gAB 1.5. For determination of the core radius rc for disordered alloy, similarly as in preceding Section, one may apply the virtual crystal approximation. Using experimental value of the lattice constant (Pearson, 1959) to determine the equilibrium density parameter rs and taking Z = ZAB, Eq. (14.5) gives the following =

14.4.

W O R K F U N C T I O N OF ORDERED A L L O Y S

la} i

197

(b] 0 Cu,,O Zn

@ CuZn

Fig. 14.6. (a) Unit cell of the ordered CuZn-/~ alloy. (b) Schematic representation of the unit cell of disordered CuZn alloy.

expression for the core radius: 1[-4.42r~

+ (0.916 + 1.792Z2/3)r 2 + 0.031r3] 1/2

(14.1s)

where the last term in the brackets results from the Nozier~s-Pines formula for the correlation energy. For electron density distribution one may assume the one-parameter, trial electrondensity profile given by Eq. (11.16). This again enables to calculate analytically all profile-dependent jellium and discrete lattice components of the energy functional as function of variational parameter. In order to evaluate the pseudopotential surface energy, defined by Eq. (8.28), which results from the incorporation of alloy's discrete lattice of ions, one has to calculate first the planar average of the difference (iv(x) between pseudopotentials of the semi-infinite lattice of ions and the electrostatic potential of jellium for the ordered ~-brass. This can be evaluated making use of the additivity of the potential, i.e. by taking

: ,dh

: 0

:

I

I

~ C)-'I~-~ ,dh

x X

OCu OZn Fig. 14.7. Schematic illustration of the calculation of the potential difference, 5v(x), for CuZn alloy; d is the interplanar distance, circles represent the positions of planes.

198

CHAPTER 14. A L L O Y SURFACES

Table 14.1

The values of the work function (in eV) for the ordered and disordered CuZn-~ alloy. Columns denoted by (Cu) and (Zn) correspond to the Cu and Zn atoms in the first layer respectively.

Surface plane

(II0) (100) (111)

Ordered Cu

Zn

3.48 3.67 3.49

3.48 3.32 3.41

5v(x) =

Disordered

+

3.62 3.44 3.48

(14.19)

as illustrated schematically in Fig. 14.7. Having determined parameters of the system, the work function can be calculated in the identical way as in Sec. 14.2. The results of calculations (Kiejna and Wojciechowski, 1986) for the most densely packed crystallographic planes of CuZn are given in Table 14.1. The two values of work function of ordered structure are given depending on the atoms occupying positions in the first layer. As is seen from this table, at the most densely packed (110)-plane the work function for the disordered alloy is higher compared with the respective values for ordered alloys. This plane consists of both Cu and Zn ions and therefore the values of the work function given in columns (Cu) and (Zn) of the Table 14.1 are the same. For the (100) and (111) planes, whose every second plane is occupied either by Cu or Zn ions solely the situation is different. For the case when the Cu ions are in the first surface layer, the work functions of the ordered alloy are higher than for the disordered alloy. The difference between the values of the work function is very distinctive at (100) plane and negligible at (111) plane. When the Zn ions are distributed in the first and further odd layers, the calculated work function for the ordered structure is lower as compared with the values for the disordered alloy. As is seen, although simplified, the model predicts a variation of the work function with the alloy ordering.

14.5

Surface

segregation

Gibbs predicted, about hundred years ago, that the equilibrium composition of an alloy surface is not necessarily identical with that in the bulk what means that one component of the alloy segregate to its surface (Adamson, 1967). It occurs that this phenomenon might be more observable than the properties of ideal alloy surfaces. With recent advances in experimental surface-science techniques, surface segregation

14.5.

SURFACE SEGREGATION

199

has been experimentally confirmed in many binary-alloy systems (Kelley and Ponec, 1981). A theoretical understanding of this phenomenon is important in many technological applications such as catalysis, corrosion, adhesion, etc. There is some similarity of this effect with adsorption or coadsorption of metallic atoms on metal surface (compare Chap. 16). The thermodynamical aspects of surface segregation are discussed in a review article by Kelley and Ponec (1981) and in the monograph of Alonso and March (1989). Here we present briefly a simple model of this phenomenon using the density-functional formalism. Among various possible reasons for surface segregation, the Gibbs-type adsorption, where the reduction in surface energy is the driving force may simply be the dominant mechanism of surface segregation in many alloys (Joshi, 1979). Following this idea the element with a lower surface energy will be enriched on the surface. It is observed that only the outermost layer of an alloy has a higher content of one type of atoms. Succeeding layers were found to have about the same composition as the bulk. Thus, it seems to be quite natural to employ two-step jellium model as a simple model describing the segregated surface. In this model the positive-charge distribution is ~, x_<0, n+(x) Us, 0 < z <_ D, (14.20)

{

O,

D < x,

where ~ is the average electron density in the bulk and n8 is the average electron density in segregated layer of thickness D. A similar model was originally used to describe adsorption, of alkali metals (see Chap. 16) on metallic substrate. To simulate the electron density profile at segregated surface one can take simple trial function given by Eq. (11.16). It is to be noted however, that such function describes quite well the segregation region if n~ << nb or if nb "" ns. In the intermediate region of ns values the functions given by Eqs (14.20) and (11.16) do not describe considered system well. Let us denote by E s the ground state energy, calculated as a function of the average charge densities, ~, in the unsegregated bulk and of ns in the segregated surface layer of thickness D, E~ = E~(ft, n~, D). (14.21) Now if we denote by E the ground-state energy of the same bulk but without surface segregation, given as function of the same average density ~, i.e. E = E(~),

(14.22)

then considering the energy difference AE = E 8 - E

(14.23)

one can now formulate (Yamauchi, 1985) the following simple criterion of segregation: A E > 0, A E < 0,

there is no segregation, there is segregation.

(14.24)

200

CHAPTER

el

I

I

|

I

I

14.

ALLOY

SURFACES

I

0.0

-0.1

-0.3

- O.

I

0.6 0.8

1.0

1.2

1.4

1.6

1.8 2.0

Fig. 14.8. Segregation criterion or segregation potential per unit area, ( A E / S ) * , with respect to the ratio of the Wigner-Seitz radii of solute and solvent, v = r s A / r s B . H e r e , r s B - - 2.67 (Cu) the valence ZA = Z B = 1 and the bulk composition cs = 0.95 are assumed. Redrawn with permission from Yamauchi (1985). (~)1985 The American Physical Society.

Expressing the energies E and E ~, as functionals of the ground state electron distribution by using modified Smoluchowski's electron density profile (11.16) and (14.20) for the positive charge distribution, the above segregation criterion can now be written as a function of ~, n~ and D or, CA, c s and D, where c s is the concentration of components in a segregated layer. The composition c s for the bulk solid solution must be given as an experimental parameter. The segregated surface should be in equilibrium. Thus, applying the principle of minimum energy, the segregation criterion yields

(14.25)

AE* - min [ A E ( c B , C~B D)] c~,D

'

'

and corresponding values of c~* and D* for given Zi, rsi (i - A, B) and CB. Figure 14.8 shows the segregation criterion AE*, divided by unit surface area S, in terms of the ratio, v = r s A / r s B , of the density parameters of solute atom A, and solvent atom B, for Cu-based solid solution with c s = 0.95. It is assumed that

14.5. S U R F A C E S E G R E G A T I O N

1.2

I

201

I

I

I

I

I

1.0 csl., 0.8 CA

0.60.4. 0.2 0.0 I 0.6 0.8

1.0

I

1.2

I

1.4

I

1.6

I

1.8 2.0

Fig. 14.9. Surface composition of solute c~*, with respect to the ratio of the Wigner-Seitz radii of solute and solvent, v. The same conditions as in Fig. 14.8 are assumed. Redrawn with permission from Yamauchi (1985). @1985 The American Physical Society.

ZA = ZB = 1, rsB = 2.67 and a solute composition CA of five atomic percent. It is seen that in this case A E * / S is always negative except v = 1, which indicates the occurrence of surface segregation. This type of segregation can be found out from Fig. 14.9 where c~4" is plotted with respect to ~. As it is seen c~4" - 0, for v < 1, and c~4" - 1.0, for 1 < ~ < 1.35, what results in solvent segregation and the solute segregation, respectively. For ~ > 1.35, c~4" decreases and asymptotically approaches the average solute concentration in the bulk (CA = 0.05).

The qualitative feature of the A E * / S v e r s u s - v curve given in Fig. 14.8 is common whatever the values of ZA, ZB, rsB and cs are. Thus, the only decisive parameter that determines whether a solute or solvent segregation occurs is the ratio v of the density parameters. This allows to formulate the following rule: atoms of the component which has the larger Wigner-Seitz radius segregate to the surface (Yamauchi, 1985). As we have learned (see Chap. 8) the larger the Wigner-Seitz parameter r8 of a metal, the lower its surface energy is. It follows that the alloy surface with segregated atoms of the component characterized by the larger density parameter r8 becomes lowest in energy. This conclusion agrees with the one resulting from slightly different model including the discrete ion-lattice effects (Kiejna and Wojciechowski, 1983; Kiejna, 1990). In this model the core radius of the Ashcroft pseudopotential of the ions in the outermost layer could vary between the values corresponding to each of the two constituents. The surface energy was minimized with respect to two parameters: the density decay parameter ~ and the core radius re1 of the ions in the last layer. An enrichment of surface composition by one of the constituents should result in a minimum in the

202

C H A P T E R 14. A L L O Y S U R F A C E S

\

E 110 .g Q;

100

b

90

o

8O

Lf)

70

I

2.4

I

I

2.5

I

I

2.6

Core rQdius

I

I

2.7

I

I

2.8

I

I

2.9

rc, (cl.u.)

Fig. 14.10. Surface energy of the (110) plane of KcCsi-c alloy plotted against the pseudopotential core radius, re1, in the outermost atomic layer for different concentrations, c, of the constituents. The broken curve shows surface energy as a function of the concentration of the alloy constituents for the case without segregation. After Kiejna (1990).

surface energy with respect to rci. A typical plot of surface energy versus core radius rci for different concentrations of KcCsl-c alloy is given in Fig. 14.10, where rci varies between 2.38 for K and 2.93 for Cs. As is seen the surface energy falls off monotonically showing no minimum. It attains the lowest value for r~i = 2.93 which equals to the core radius of the pure Cs ions. One can conclude that for this system it is always cesium atoms which segregate to the surface. The same trend is observed for other alkali-metal alloys where, in accordance with Yamauchi's criterion, the atoms of the constituents which has the larger Wigner-Seitz radius enrich the alloy surface. The similar conclusion results from the calculation of Digilov and Sozaev (1988) performed for NaK system within the model which combines the double-step jellium model with the ion-lattice model of Kiejna and Wojciechowski (1983).

Chapter 15

Quantum size effect and small metallic particles 15.1

T h e n o t i o n of size effect

Advances made in growth techniques allowed to control the thickness of metal films deposited on various types of substrates. The electronic properties of films of finite thickness will differ from these of the semi-infinite metal. Electrons confined in such films due to lowering of the dimensionality have a quantization of states different from that in a bulk or semi-infinite metal. The variations of physical quantities arising from the lowering of the dimensionality are called the quantum size effect. The effect of the size and shape of metallic or semiconductor samples on their physical properties is known for a long time (cf. Slater, 1967; Tavger and Demikhovskii, 1968; Wojciechowski, 1975). The size effect of the sample on electronic properties of metal was already mentioned in Chapter 4, and will be now described in more detail. Let us consider the energy difference A E between the energy levels (see Eq. (4.4)) of the lowest state E(1, 1, 1) and the neighboring one E(2, 1, 1)

AE=E(2,1,1)-E(1,1,1)

(15.1)

for two cubic samples with their linear sizes in each direction equal to: (a) L = 1 cm, and (b) L = 5 s = 9.452 bohrs. From EQ. (4.4) we obtain for AE: 9 • 10 -~5 eV in case (a) and 3.6 eV in case (b), respectively. Consequently, when the size of a metallic sample diminishes from the macroscopic size, say of dimension of 1 cm to the dimension of an order of magnitude of atoms (~ 1 ft.), the spectrum of energy levels passes from the continuous to the discrete one which is typical of an atom where the level spacing is of the order of electronvolts. This is schematically illustrated in Fig. 15.1 where we have marked also the energy levels, E(1, 1, 1) and E(1, 1, 2), when one of the microscopic dimensions of the sample is different from the others. The quantum size effect (QSE) can be divided into two types: non-oscillatory and oscillatory effect. The former is in fact an averaged effect of the latter. It is manifested 203

CHAPTER 15. QUANTUM SIZE EFFECT

204

'EF

E112

~---- ~_--:#~-~ ~-~-~-~--~(~ =...--..-:-::-:.-.--'.-'..--::::. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . .............................:

.................

L-- r

0

E 2~ = E ~2~

E 2~ Elll

Lx = L y = L z = 1,g,

{a}

(b)

Lx=Ly# Lz

Em

(c}

Fig. 15.1. Spectrum of energy levels of: (a) large (L --+ c~) sample; (b) microscopic cubeshaped, and (c) cubicoid-shaped samples. by non-monotonic alteration of physical quantities with variation of the sample size. Let us start from the discussion of the non-oscillatory effect.

15.2

The non-oscillatory QSE

As it was shown by Weyl (1911) the number A(~') of eigenvalues 7 of the Schr5dinger equation (4.1), which are lower than a given number 7ma=, does not depend on the shape of the domain D, in which the equation is satisfied when the volume, gt, of the domain asymptotically approaches infinity, and is equal to ~-~ 3/2

A(Vma=) -

'~max

6~ 2

(15.2)

Relation (15.2) is valid both for the Dirichlet boundary condition (Eq. (4.2)) and for the Neumann condition imposed on the electronic wave function r

ar 0n s

=0,

(15.3)

which means the normal derivative of wave function taken at the boundary S, of the domain and n is the normal unit vector oriented outwards S. According to the Weyl theorem applied to a quantum description of free electrons, the sample (domain D) is sufficiently large (Ft -+ c~) when the ratio of the surface area, S, of the sample to its volume is much greater than the minimum de Broglie wavelength of electron, i.e. when

S

kg.

(15.4)

where )~F denotes the Fermi wavelength. Thus, in order to calculate the dependence Emax = EF(L), where EF denotes the Fermi energy, for a sample whose size does not

15.2. THE NON-OSCILLATORY QSE

205

EF(N) E~ F

1.4 1.3

CYLINDER CUBE SPHER

1.2

1.0

I

I

102

I

10 3

104

I

10 5

N

Fig. 15.2. Curves of the dependence EF(N)/EF(cr versus the number N of electrons for the samples in the shape of a cylinder, cube and sphere. Redrawn with permission from Rogers et al. (1984).

satisfy inequality (15.4), we can use the formula (4.15) or (4.16) rather than (15.2). It can be shown (Rogers et al., 1984) that equation (4.15) under consideration, A(7) -

~73/2 67r2 + aS7 + bLO(7 n),

where a, b and ~? are constants, in the case of spheres, cubes, right circular cylinders and layers, has only one real root, 71/2 - k, which in the limit of large values of L')'max, approaches k(L -4 ~ ) = kF(L -4 ~ ) , where kF is the Fermi wavenumber. Fig. 15.2 shows plots of the ratio EF(L)/EF(L -4 co), for the above mentioned shapes of a sample, as a function of the number N of electrons, taking into account that the ratio is dependent only on the product kFL which is proportional to N. Energy levels of electrons in metal, in the free-electron approximation, show a continuous, though still discrete, distribution and are dependent on transverse dimensions of the sample. If one of the dimensions, L, is sufficiently small, then non-oscillatory QSE occurs. The critical magnitude of Lc, below which the effect occurs can be estimated in the following way. Let us assume that the sample size diminishes in the direction of the z-axis, then the change of momentum Akz observed in this direction will equal to 7r/L and the indeterminacy of the momentum in a system with the Fermi momentum kF will amount to Ak << HE, or

71"

<< kg

--(37r2~) 1/a.

(15.5)

Consequently, above the critical magnitude L~,

1/3 the quantum size effect will not appear.

(15.6)

206

15.3

C H A P T E R 15. Q U A N T U M SIZE E F F E C T

Oscillatory q u a n t u m size effect

When we consider the motion of a free electron in a potential well of size L and of infinitely high energy walls, then the solution of the SchrSdinger equation, will be of the form (4.3) whose eigenvalues are as follows (cf. nq. (4.4)) 1

Ekx,k~,kz -- ~[K 2 + k2n],

(15.7)

where K is the wavevector parallel to the surface K 2 - k y2 + k z 2; and

7rn k n --

L

(15.8a)

n - 1, 2 3

(15.8b)

Components ky and kz are not subject to size quantization, which is not the case of the component in the direction of x-axis that was denoted above by kn (see Fig. 15.3). L is the distance between the planes on which the boundary conditions are given. In the case of Dirichlet condition n = 1, 2, 3 . . . . It is seen from (15.8) that, as L increases, new level normal to the surface becomes occupied. Such a new level falls below Fermi energy, I

27r2

2--

)~F

EF-

:k 2-

every time when L increases by ,~F/2. Primary problem in calculation of EF(L) is to determine the distribution of eigenvalues An (or k2n) because, when the distribution is known, one can calculate the exact density of states

N(E) - Zgn

(15.9)

(E- En),

n

where gn denotes degeneracy of the n-th energy level, En - k2n/2, and 5(x) is the

kx n=5 kF

n=4 n-3 n=2 n-1

k:

Fig. 15.3. Classification of eigenvalues E(kx,ky, kz) in the momentum space.

15.3. OSCILLATORY QUANTUM SIZE EFFECT

207

Nosc

L2L3 41T v k

L2L9 4~

Fig. 15.4. Plot of the oscillatory term of density of states in the case of a cubicoid sample ( L 1 << L2, L3).

Dirac delta function. In the case of a cubicoid with edges of length Ly = L2 -+ oo, Lz = L3 -~ c~ and Lx = L1 << cx~, i.e. for a finite slab represented by the infinite barrier model, the expression for Af(k) will take the form (Rogers et al., 1984)

Af(k2)

=

~t 2~2 E

ml

~k --

sin2kmlL1 2miLl

L2L3

27r2 -+- 47r2 E

L2L3 4~

sin 2km LI ml 1

L2L3 _

4~

"

(15.10)

ml

The first term corresponds to the asymptotic expression for the bulk density of states given by (4.23). It is seen that the middle term of the second line in (15.10) expresses the Fourier expansion of the oscillations of the period ~/kF with respect to L1 and of the period ~/L1 with respect to k. The latter are represented by the sawtooth curve in Fig. 15.4. The last term is a non-oscillatory correction. Thus the curve of the density of states versus slab thickness consists of a monotonic component upon which is superposed a small oscillatory component of the sawtooth form. The oscillations of the density of states curves lead to the appearance of similar, although less pronounced, oscillations in the Fermi energy versus thickness curves. The explicit expressions for the functional dependence of the Fermi energy and of the density of states for the finite square potential barrier are given by Rogers et al. (1986). The oscillatory QSE was treated self-consistently for the first time by Schulte (1976) using the jellium model and the density functional formalism. The object under consideration was a thin metallic layer (Lx = L, Ly, Lz -+ co). In such case, the Lang-Kohn system of equations (Sec. 6.2) can be applied too. However, in order to avoid the improper asymptotics of the LDA exchange-correlation potential outside

CHAPTER 15. QUANTUM SIZE EFFECT

208

{(eV) 4.5

4.0

x rs-2

3.5

x

3

3.0

x

i

•

5

•

6

2.5

2.~t '

i

'

~

'

~

4

L/x~

Fig. 15.5. Dependence of work function on the thickness of the layer, L / ~ F , f o r rs = 2, 3 , . . . , 6. )~F is the Fermi wavelength. Crosses denote the values calculated by Lang and Kohn (1970) for semi-infinite metal (L ~ c~). Redrawn with permission from Schulte (1976).

the slab, Schulte assumed that ~

,

X ~

--Xl,

X+Xo

yon(x)- r

v~(x)-

r

+ vLDA(x),

- - ~ ~ X - - X0

IX I~_ Xl,

(15.11)

X ~ Xl~

where the positions x0 and x l (0 < x0 < x l) are determined from the continuity condition for Ve//(x) at the above points, and Vxc(x) is given by the formula (6.15). Introducing the expression + l / 4 ( x + x0) yields a proper asymptotics of V e / / f o r x --+ +c~ (i.e., the asymptotics of the image potential) instead of the incorrect exponential decay which results from the LDA (cf. Section 12.1). The result of the work function dependence on the thickness, L, of a metallic layer of non-uniform electron density, ~, is shown in Fig. 15.5. From the figure is clearly seen the oscillatory character of the work function, ~, as a function of film thickness, L, around of the value ~(rs, L --+ c~) which denotes the one calculated self-consistently

15.3. OSCILLATORY QUANTUM SIZE EFFECT

209

by Lang and Kohn for a semi-infinite metallic sample. As is seen the oscillations are bigger for higher-electron-density metals and have a period that equals approximately )~F/2, or Ir/kF, where /~F is the Fermi wavelength. This means that QSE oscillations are easily distinguishable from the Friedel type oscillations, discussed in Chap. 6, that increase for lower-electron density metals and show a different period. To give a numerical example for the period w of the oscillations: for rs - 2, one gets W -- 1.73 /~ a n d for rs - - 6, T = 5.2 A. A slightly more t r a n s p a r e n t self-consistent calculation for a t h i n film of jellium, which avoids some difficulties of Schulte's t r e a t m e n t , has been p e r f o r m e d by Mola

>oo Z LU n

Z 0 rr

!

0.5

(D w ..J w

I

...--..

!

|

'

-Ib

-5

-10

-5

'

b

x

I

I

)/5

g

I0

5

10

-0.1

EF

c D

O E o o

>-

-03

Z I.i..I

Veff { x )

I

I

I

x (otomic units) Fig. 15.6. Normalized electron density distribution in a thin film of thickness L = 10 bohrs and rs - 2 (upper part). Effective potential and its electrostatic and exchange-correlation components generating the density shown in the upper part. The Fermi level, EF, and quantized energy levels are also shown. Redrawn with permission from Mola and Vicente (1986).

210

CHAPTER 15. QUANTUM SIZE EFFECT

~'n (n) 6

5 4 3 2 n=l

0

5

10

15 x (a.u.)

Fig. 15.7. Eigenfunctions of the thin metallic film. Redrawn with permission from Schulte (1976).

and Vicente (1986). Fig. 15.6 shows a self-consistent electron density for a thin film of r~ = 2. Note that the Friedel oscillations which in a semi-infinite jellium case (Chap. 4) are relatively small are quite large and not reduced in amplitude at the center of the film. The corresponding effective potential and its electrostatic and exchange-correlation components are plotted in Fig. 15.6b. As is seen all potentials reveal considerable oscillations inside the film. The calculated quantized energy levels, Ek, of the electron motion in the direction perpendicular to the surface are also shown in this figure. The appearance of oscillations in ~(L) can be explained considering the variation of the surface dipole layer formed by the net charge density. Its variation with film thickness, L, can qualitatively be understood considering eigen functions corresponding to the energy levels of the film (see Fig. 15.7). It is seen that for higher energy levels the wave functions are only weakly bound in the film potential and show a long, exponentially decaying tail in the vacuum region. So when at the film thickness Ln, in accordance with (15.7-15.8), new level starts to contribute to the electron density (i.e., it falls below the Fermi level) the electrons occupying this level can be easily transferred into the vacuum region what will manifest in an increase of the dipole moment. For slightly increasing L, the level will slightly get lower on the energy scale and will acquire more electrons, thus enhancing the effect. But at further increase of

15.3. OSCILLATORY QUANTUM SIZE EFFECT

211

L, the level will get sufficiently deep and spatial extent of the wave function will be reduced. It will lead to the reduction of the dipole moment and to the appropriate decrease of the work function. The situation will repeat when the film thickness will increase by AF/2, again. In the jellium approximation a thickness of the metallic layer can be varied infinitesimally. In reality, the crystalline lattice should be considered and the film thickness can be changed by a discrete number of atomic layers only. Since the period of QSE oscillations for jellium is AF/2, therefore if this will hold for crystalline film too, one would expect that no oscillations will occur for a crystal film whose interplanar spacing (or the layer thickness) d, is equal nAF/2, where n is a positive integer. On the other hand the maximum oscillations will occur for d = ( 2 n - 1)AF/4. As estimated by Feibelman (1983), the effective Fermi wavelength taken as jellium value corresponding to the interstitial electron density in an augmented-plane-wave calculation (Moruzzi et al., 1978), is approximately equal to that representing the average valence-electron density. A second important point which appears when the discrete lattice effects are considered is the influence of surface relaxations. Looking for the minimum of a total energy for a crystalline film, the position of ions can be allowed to relax. This is different from the jellium model where the spatial distribution of the positive background does not vary. As we have seen in Chap. 2, lattice relaxation has also an oscillatory character, thus to some extent, it may effect the electronically induced QSE in a crystalline metal. The first systematic investigation of QSE in the real metallic films has been performed by Feibelman (1983). His self-consistent calculation of linear combination of atomic orbitals (LCAO) for a 1, 2, 3, 4, 6 layer thick A1(111) and 2, 3, 4, 5, 7 layer thick Mg(0001) slabs, have confirmed the appearance of the QSE for films thickness different from the effective )~F/2. Moreover, the QSE manifests as oscillations not only in the work function but also in surface energy as a function of number of layers. These findings were confirmed and extended to other properties by self-consistent pseudopotential calculations of Batra et al. (1986). The results for work function and surface energy (per surface unit cell of the film) calculated for the A1(111) films by different groups are compared in Fig. 15.8. All of these calculations predict the oscillatory behaviour of both work function and surface energy as a function of the the film thickness. Experimental observations of QSE are difficult because of many reasons. Kogan (1971) proposed a QSE observation using field emission of electrons. Such measurement were performed by Stark and Zwicknagl (1976). In order to experimentally observe the effect, a uniform layer, of mono- or multi-atomic thickness should be grown on a dielectric or insulating substrate, which is not a simple task (a serious obstacle will be the the formation of small islands impurities, the influence of the substrate, etc). Even if one has obtained such a thin layer it should be noted that the thickness of the layer is comparable with the oscillation period, say, of the work function, and consequently, the shape of the oscillations will be considerable distorted. In the case when oscillation period and the layer thickness are equal one to the other, the oscillatory QSE may vanish and not to be revealed by, for instance, the measurement of the work function. Nevertheless, there are reports of experimental evidence for the

212

CHAPTER 15. QUANTUM SIZE EFFECT !

- - -

>

I

I

5

r 0

,.I.,..*

0 c"

4

..9

~

..9

~

U...

9

._Y.

"'"'...

....0 -~ . .... .." ..... .."

.r-i

...... [ ] ......

I,...

"'"n ................

0

3

I

I

2

I

4

6

Number of Layers

06

I

I

.,, ..-.O

[] ................. --6...~

Q5 > (]J >,

04-

I._.

(D C LI.J (D 0 0 I_.

Q3-

I

% % \ \

/ \

/

/

I

I

I

02-

:3 V')

01-

O0

0

I

2

I

4

I

6

8

Number of Layers Fig. 15.8. Calculated work functions and surface energies, per surface unit cell, versus the number of layers for different slab calculations. Dots represent the results by Feibelman (1983), open circles denote results of Boettger et al. (1994) and results of Batra et al. (1986) are marked as squares. A horizontal line represents the experimental results. After Boettger et al. (1994).

15.4. SMALL METALLIC PARTICLES

213

oscillatory QSE (e.g., Stark and Zwicknagl, 1976; 1980; Schmidt-Ott and Burtscher, 1984; Nagaev, 1991; 1992).

15.4

Small metallic particles

Small metallic particles constitute entities that mediate the evolution of properties from single atoms to bulk metals. They might vary in size from the diatomic molecules, e.g. Na2 to a few thousands metal atoms. The interest in study of quantum finite size effects in small metallic aggregates and clusters has been stimulated through the availability of molecular beams and the discovery of electronic shell structure in simple metal clusters and their similarities to the nuclei (de Heer, 1993). They also exhibit electronic properties which are strongly dependent on cluster size distribution, and consequently they have useful applications Since for large clusters it is expected that the electron distribution becomes more and more spherical as the cluster increases, it is appealing to make a spherical approximation in determining the cluster structure. The electronic structure of small alkaline clusters can be reasonably well described by modeling the positive ionic charge of the spherical cluster as formed from the uniform positive charge background of density n+ (r) -

{ ~, 0,

r < R, r > R,

(15 12)

where ~ is the density of jellium defined as the average conduction-electron density of the bulk metal, and R is the radius of the cluster. As we know jellium is unstable for most of the densities characterizing bulk metals what manifest itself by negative surface energies for r~ _< 2.3 bohrs. This makes also problem when considering the stability of clusters. For example the aluminum cluster, treated as jellium (r~ = 2.07), is not stable against deformation and the energetically most stable shape for such a finite aluminum cluster is not a sphere but a foil. Despite of this pathology of jellium the model works surprisingly well in explaining the trends of many important properties of clusters (Brack, 1993) such as ionization potentials and electron affinities, dipole polarizabilities and photoabsorpti0n cross-sections and the most prominent spherical-shell closings or 'magic numbers' which are briefly discussed in the next section. For spherical cluster containing N conduction-electrons, the radius R is related to the electron (Wigner-Seitz) density parameter, r~, by

R-rsN

1/3.

(15.13)

For the cluster being the regular polyhedron with edge length a, the volume ~(a) of this polyhedron is related to N and r s in the following manner N 3 = ~t(a) 47rr~"

(15.14)

CHAPTER 15. QUANTUM SIZE EFFECT

214

The spill-out effect of conduction electrons (see Chap. 4) can be taken into account phenomenologically by increasing the value of r s used to determine the dimensions of the box (sphere or polyhedron). If the electron density is assumed to have spherical symmetry, the effective KohnSham potential (Section 5.3) is spherical and one can determine the single-particle states r from the spherically symmetric Kohn-Sham equation. The Kohn-Sham effective potential for a cluster can be written in the form

/ Ir-

n(r')

-

'l d3r' + V~c(r),

(15.15)

where a denotes the spin number (up $ or down $). The electrostatic potential due to the ionic positive background (15.12)is (compare (4.80)) a-

,

f o r r < R,

=

(15.16)

ZN r

for r > R.

The last term in (15.15) is the spin-dependent exchange-correlation potential. The total electronic density is given by

n ( r ) - E n~(r) a=%

(15.17)

where n~(r) are the spin-a components of the electron density. The spin densities n~(r) are obtained from self-consistent solution of the radial Kohn-Sham equation

[

0

where Ri Rnlma, is the radial wave function with quantum numbers n, l, m, a and energy eigenvalue Ei. In order to impose a spherical symmetry on the system, the spin component of the electron density is replaced by =

1

na(r) - ~ E

IRnzm~(r) 12"

(15.19)

nlm

Owing to the spherical symmetry of n+(r) the cluster is like a large atom and the electronic structure is straightforward to compute numerically. The equations (15.18)-(15.19) can be solved on a one-dimensional mesh in the variable r, using the codes employed in the atomic structure calculations (Beck, 1984; Ekardt, 1984; Brack, 1993). The most important physical properties of small metal clusters compared to an infinite flat surface is the ionization potential (IP) and electron affinity (EA). They

15.4. S M A L L M E T A L L I C P A R T I C L E S

215

are defined by

I P - EN-1 -- EN EA = EN-

(15.2o)

EN+I

where EN:i:q denotes the total energy of a cluster with N atoms and q excess electrons. In a simple electrostatic model of spherical jellium, the ionization potential can be defined (Seidl and Perdew, 1994) as the work needed to remove an electron from a neutral metallic sphere of radius R, 1 I P ( R ) - ~2 + ~ + O ( R - 2 ) ,

(15.21)

where ~ is the work function of flat surface of semi-infinite metal. Similarly, the electron affinity is the work needed to remove an excess electron from a spherical cluster 1 E A ( R ) - 9 2R + O(R-2)" (15.22) For very large clusters both quantities, I P and E A , would approach the work function of flat metal surface. The self-consistent calculations performed in this line for small metallic particles,

2S

3P

2F

.~

9

."

E 0 t3

3S

2D

1P I 1D

5

~0

2P

..o..

9

.

.

9

_. 9-

.

.

9

.

9.

o.

3 ..

2

..

r$

L_

= 4

""

"."

9 ..... 9

-.

.

.

.

.

.

.

.

.

.

I

5

.

.

.

.

.

.

.

.

.

,I

10

.

.

.

.

.

~ .

. ~ . . . . .

.

.

I

15

.

~

9.

9

ces

I,

20

R (a.u.)

Fig. 15.9. Variation of work function of small metallic particles with the particle radius, R, for rs - 4. (I)o~ is the work function of the flat surface of semi-infinite metal. The quantum numbers of the spherical potential well are shown to indicate the shell closure. The results for ls shell are not shown. A(I)~s is the electrostatic part of 4. Redrawn with permission from Elmrdt (1984). 9 The American Physical Society.

216

CHAPTER 15. Q U A N T U M SIZE E F F E C T

2

2 N=8 A

R =

L~. I c

8.000

-

I

~OiI

-0.2- .

-o.3-

i.. c

--

--0.1

8

10 12 1'4

_18--2:0--12 =

1'6

r (a.u)

--0.2

/

,- --0.5

-0.7

2

N= 92 R= 18.057 rs'4"O

Icc'~l J~--~'~s/'"~

N-lg8

"~.1 ~

~

R = 23.314 rs-4.0

--

2 Z 6 ~ 1"0 1"2 1~, 1'6 1~ ~2"0 2~)r'26r(o.u)2~} ----- "~_~_0.2 r - 0.1 [ ~ ~' ~ I~ 1'0 1'2 1~, 1'6

0.2.

..

,;

~/

/

~

-

0.4

-0.5.

- 0.5 - 0.6

- 0.6

-

0.7

-

-

0.8

0 . 7

I

8

'~ -o.5 ILl --0.7 --0.8

-0.8

P -0.3 - Q4.

!

4

--0.4

-0.6

w

!

2

>" --0.3 n"

-0.5

~>" - 01.

IB

(Q.u)

r~ -0.4..... Q;

N = 3/, R =12.958 ~-4.o

IE

.0

.,,.......

18 20

r (o.u)

,,/I"

-0.8

Fig. 15.10. Evolution of the electron density and energy levels of small metallic spheres (rs = 4) of the radius R, which contain N atoms. Redrawn with permission f r o m Ekardt (1984). (~1984 The American Physical Society.

15.4. S M A L L M E T A L L I C P A R T I C L E S

217

which consist of N atoms and can be approximated by spheres of the radius R(N), have shown (Ekardt, 1984) that the work function, (I)(R), runs in the way like the one plotted in Fig. 15.9. It shows a strong oscillatory behavior which can be attributed to the effect of shell structure. The spectrum of energy levels and the electron density along the sphere radius vary with a number of particles N, from the magnitude equal to that for an atom to that characteristic of the bulk metal (R --+ co). This is illustrated in a sequence of pictures shown in Fig. 15.10 in which the result of Ekardt's calculations for r s = 4 are shown. Note that the difference between the ionization potential and the electron affinity is independent of work function, thus providing information on the finite-size effects only. Fig. 15.11 shows experimental results for I P - E A versus inverse radius of A1 clusters compared with the values calculated within the spherical jellium model. As is seen, the oscillatory components disappeared and the curve is smooth (Seidl et al., 1991). The good agreement of calculated and measured values supports the adequacy of jellium model in description of small metallic clusters.

5

4 I-

.'.J

3 12_

i O

0.2

~

0.4

0.6

0.8

N -~/3

Fig. 15.11. Measured (dots) and calculated values of the difference between ionization potential (IP) and electron affinity (EA) of A1 clusters versus N -1/3. Redrawn with permission from Seidl et al. (1991).

218

15.5

C H A P T E R 15. Q U A N T U M SIZE E F F E C T

Magic numbers

Clusters of trivial case (Manninen, square-well

special stability occur with 2, 8, 20, 40, 58 and 92 atoms. Except of the of the dimer the other of listed clusters prefer to take a spherical shape 1986). These magic numbers are shell-closing numbers of a spherical potential confining the valence electrons inside the cluster.

Investigating sodium clusters containing between N = 2 and N = 100 sodium atoms per cluster, Knight et al. (1984) have found distinct regularities in the mass spectra of these clusters. The peaks or steps for certain masses, corresponding to N = 8, 20, 40, 58 and 92, are conspicuously large, especially compared with the peaks immediately following. Associating this main sequence N = 8, 20, 40, 58 and 92 with an electronic shell structure for sodium clusters Knight et al. have explained these magic numbers as follows. The effective one-electron potential inside the cluster is

2.0 1.6 1.2 0.8 0.4 0.0

(a) Li

-

!

-0.4 i J A 1.6 Ilpl ld ~ -'~ ---

.,--.,..

Z

,._,..


I,lllllJlll~,~l

If

12pl

lg

I 2dl

,

lh, 3s

1.2

I

(b) No

0.8

0.4 0.0

----

-0.4

1.2 0.9 0.6 0.3 0.0 --0.3

i

I Jl lll,~l,I

PPl le ~z~ if

12pl

llllillllllilllllJllllllllllillllIlilJ

8

20

34 40

lg

12dl

58

lh/3s

lliilill

68

N u m b e r of a t o m s per c l u s t e r ,

J

92

N

Fig. 15.12. The relative binding-energy change A E ( N ) (see text) versus N for (a) lithium (b) sodium and (c) potassium. The labels correspond to filled-shell orbitals. Redrawn with permission from Chou et al. (1984). @1984 Elsevier Science Ltd, UK.

15.5. MAGIC NUMBERS

219

simulated with a spherically symmetric rounded potential well I which has the form

u0

U(r) = - e x p [ ( r - Ro)/~ + 1]"

(15.23)

In (15.23), U0 denotes the sum of the Fermi energy (3.23 eV) and work function (2.7 eV) of the bulk, R0 is the effective radius of the cluster sphere and it is assumed to be rsN 1/3 (for sodium rs = 3.93). The parameter ~ determines the variation of the potential at the edge of the sphere (~ = 1.5 a.u). The effective potential near the surface of jellium of r~ = 4, calculated self-consistently by Lang and Kohn (1970) is close to a potential (15.23) with e = 1.8 a.u. and U0 = 6.18 eV. The appropriate SchrSdinger equation was solved numerically for each N which yielded discrete electronic levels characterized by the angular momentum quantum number 1 with degeneracy 2(2l + 1). The energy levels shift down slowly, and continuously as N increases. The difference A(N) = E ( N ) - E ( N - 1) (or A2(E) = E ( N + 1 ) + E ( N - 1 ) - 2E(N)) in electronic energy between adjacent clusters shows peaks when A ( N + 1) increases discontinuosly, as an energy level is just filled at certain N and the next orbital starts to be occupied in the cluster with N + 1 atoms (Fig. 15.12). The lowest orbital ls can hold two electrons (one spin up and one spin down), the next lowest, lp level, can hold six electrons, the ld ten electrons and the 28 two electrons. Accordingly we obtain the sequence of numbers N = 8, 18, 20, 34, 4 0 , . . . leading to the observed regulations in the mass analysis. Assuming that the contribution to the binding energy from the ion-ion electrostatic energy is a smooth function versus N one can conclude that the discontinuities in electron energy should remain in the binding energy of clusters. They are also well visible in Fig. 15.9, in the form of saw-tooth oscillations of the work function.

1In nuclear physics this form of the potential is known as the Woods-Saxon potential

This Page Intentionally Left Blank

P a r t III

M e t a l S u r f a c e in C o n t a c t w i t h Other Bodies

221

This Page Intentionally Left Blank

Chapter 16

A d s o r p t i o n of alkali a t o m s o n metal surface 16.1

Introduction

So far we have discussed the properties of clean metal surfaces i.e., of metals being "in contact" with a vacuum. Atoms of metal (or alloy) constituted a one-component system. Such a system is quite an idealized one from a viewpoint of our experience. Even in the ultra-high-vacuum environment there is a residual quantity of atoms or molecules which have a finite probability of arrival to and to stick to the surface of a sample. In this chapter we will consider a metal surface which is contiguous to the gas phase consisting of atoms of certain element (metal) which are dispersed in the vacuum. Some of these atoms will stick to the surface. The process of localization of foreign atoms onto the surface of a solid is called adsorption (Ponec et al., 1974). As a limiting case of adsorption one may consider interaction of an individual atom with the metal surface. The particular shape of the potential energy of such an interaction as a function of the distance from the surface i.e., the equilibrium distance and the magnitude of binding energy, depends both on the kind of atom and the substrate and the surface structure of the underlying metal. This Chapter is addressed to the discussion of adsorption of alkali metals which falls into the chemisorption regime. This type of adsorption involves significant modification of electronic structure of the adsorbate compared to its free atom state and is caused by overlap of atoms electrons with the valence electrons of the planar surface. All this Chapter is based on the microscopic or quantum-mechanical approach to submonolayer adsorption. A monolayer adsorption corresponds to complete atomic layer of adsorbate which covers a two-dimensional layer taking account of atomic size. The thermodynamics of adsorption is given in many textbooks and review papers (cf. Zangwill, 1988; Desjonqu~res and Spanjaard, 1994). Many aspects of the theory of chemisorption are discussed in the book edited by Smith (1980) and the experimental investigations of adsorption on metals are reviewed by Naumovets (1994). 223

224

16.2

C H A P T E R 16. A D S O R P T I O N OF A L K A L I A T O M S

Work function changes due to alkali metal adsorption. Classical picture

The adsorption of alkali-metal atoms on metal substrates has attracted significant interest in experimental research for more than 150 years. As a curious detail one can mention the Goethe's interest in such kind of adsorption (Mittasch, 1952). Although alkali metals is thought to be a simplest adsorption system, currently there is still some debate regarding the character of their bonding to the metallic substrate. The modern experimental investigations of alkali-metal adsorption on metallic substrates were initiated by Langmuir and his coworkers in the early thirties of this century (Langmuir, 1932; Taylor and Langmuir, 1933) and were stimulated by important applications of this phenomenon to the technology of efficient cathodes of high electron emission: the alkali metal, say cesium, adsorbed on the refractory metal surface reduces drastically its work function. Typical changes of the work function of metals, which are induced by the adsorption of cesium atoms on different planes of the same crystal, are shown in Fig. 16.1. For a small number of adsorbed atoms (adatoms) the work function drops rapidly with increasing number of adatoms and attains a distinct minimum. Then increases to saturate at the coverage corresponding to a full monolayer, and approaches a value approximately equal to the work function characteristic for the bulk alkali metal. From experimental data it follows that the dependence of work function on coverage is more sensitive to the kind of adsorbate than to the metal substrate and the type of exposed surface plane (Kiejna and Wojciechowski, 1981). According to Langmuir (1932) the described changes of work function at low coverage can be explained as follows. An atom approaching the surface is ionized while its conduction electrons are transferred to the metal conduction band. A positive ion induces its image in the metal and produces a dipole of the moment p. Denoting by ~n(r) the change in the metal charge density distribution, induced by the adsorbed atom, the dipole moment can be written as

p =/xSn(r)

dr.

(16.1)

If N atoms are adsorbed on the surface, the change in the surface dipole barrier and, following (9.3), in the work function, Ar is proportional to the number of adatoms per unit surface, i.e.

A~ - -4~pN.

(16.2)

By introducing the number Na of atoms per unit surface area, which is needed to obtain a monolayer coverage, we have N - NaO, where 0 _< 0 <_ 1 determines the degree of surface coverage. 1 lit should be noted that quite often a monolayer coverage is referred to the two-dimensional

substrate lattice. Since in most cases the sizes of adsorbed atoms differ from the dimensions of substrate atoms, a monolayer coverage referred to the adsorbate atoms corresponds to the fraction of that based on the substrate lattice. In such cases the coverage O, is normalized in such a way that for 0 - 1 there are as many adatoms as substrate atoms per unit area of clean, unperturbed (hkl) surface.

16.2.

W O R K F U N C T I O N CHANGES DUE TO A D S O R P T I O N

I

I

I

I

i

s

I

I

225

I I

c /w

4

,.e., 3

I I~~176'~W21 2L ~, ~'.

I :Q

0

1

I

2

I

3

tool / _~,~~-.~.~?-i I

4

I

5

I

6

I I

7

l

8

N (1014atoms/cm 2) Fig. 16.1. The experimental work function of the low-index planes of tungsten covered with Cs atoms versus adatom concentration.After Kiejna and Wojciechowski (1981).

Helmholtz's expression (16.2) describes the linear decrease of the work function, with increasing number of adatoms, for low coverages. However, as the number of adatoms increases, each dipole is depolarized by the adjacent dipoles. Denoting the initial dipole moment of each adatom by p0 and its polarizability by c~, the dipole moment of a single, i-th adatom in the presence of other adatoms can be expressed as pi = poi - c~gi,

(16.3)

where gi is the depolarization field on the i-th dipole, which is originating from the other dipoles. Taking the adlayer as uniform and ordered we can assume that g = Cp. Thus we have p0 P- 1 + aC (16.4) Parameter C is determined by the structure of the adsorbed layer and, usually, it is written on the basis of the Topping (1927) formula, which for small coverages is of the form: C = 9 N 3/2. Taking this into account in (16.4) and (16.2) we get: A~ - -

47rp0 1 + 9c~Na/2"

(16.5)

CHAPTER 16. ADSORPTION OF A L K A L I ATOMS

226

This simple example shows that depolarization which arises with increasing coverage, leads to deviation from linearity and to the appearance of minimum in the ~(0) curve. The mechanism of depolarization can be understood basing on the quantummechanical model proposed originally by Gurney (1935) and developed later by other authors. In this model, the appearance of particular bond in the process of adsorption is connected with mutual relations between such quantities as the ionization energy, I, and the electron affinity, A, of the atom and work function, ~, of the metal. The factor which determines type of bonding is also the change of the eigenstates of interacting systems. Consider an atom approaching the metal surface. Let the energy of valence s-level,

Ea, is close to the Fermi level of metal. In vicinity of the surface, the wave function of the valence electron of an atom and the wave functions of electrons in metal begin to overlap and the atomic level broadens. In the adatom equilibrium position, the atomic resonance level attains the width 2F, and its average energy is shifted b y / k E a (see Fig. 16.2) due to the image force and interaction of electrons with opposite spins (Schrieffer, 1972). This is an essence of the Gurney effect introduced to the theory of adsorption by Gadzuk (1967). The centre of atomic level is located above the Fermi level and the valence electrons can tunnel into the metal leaving on the surface an alkali ion with a net positive charge. The positively charged ion polarizes electron gas of the substrate and induces a screening charge. Consequently, the resulting dipole configuration reduces the surface barrier for electron emission. The adatom ionization is not complete. The degree of ionization that determines the magnitude of the dipole moment is given by the occupancy of the broadened atomic level. The occupancy of the level can be calculated if the local density of states

IA

&A

I

\

/

f

!

/

'~I, Q

EF ~

~

1

x=O

X=S

.....

il

Eo

Fig. 16.2. Schematic energy level diagram for metallic substrate-adsorbate system. I, denotes the ionization energy and, A, the electron affinity.

16.3. D E N S I T Y - F U N C T I O N A L

CALCULATIONS

Af(E, r) - E

[ 9i(r)12

5 ( E - Ei),

227

(16.6)

i

is known. Denoting the local density of states at the adatom position by A/'a(E), it can be written (Gadzuk, 1967; Newns, 1969) in the form of a Lorentzian

1

F

Afa(E) - ~ ( E - E~a)2 + F 2

(16.7)

where, A E a - E~a - Ea is the adatom energy level shift (Fig. 16.2). Both AEa and F, depend on the interaction matrix between adatom orbital and the Bloch-type states of the substrate. Since the electron states are filled up to the Fermi level, the occupancy of the atomic state will be given by

(Na)

-

~

F

Afa(E) dE.

(16.8)

The ionization degree of adatom, or its effective positive charge, is equal to 1 - (Na) and is determined by the halfwidth, F, and the position of the centre of gravity of the shifted level E~a. An increase in the coverage leads to a decrease of the energy, E'a , as a result of Coulomb interactions with the remaining adatoms and their screening charges. This in turn, according to (16.8), lowers the degree of adatom ionization and, consequently, its dipole moment, what leads to nonlinear changes in work function. The described modifications of Gurney's, or resonance, model provide a satisfactory explanation of the work function changes of metals due to alkali adsorption. However, in practice, calculation of the characteristic parameters of the model, i.e. the equilibrium position of the adatom, the energy E'a and the halfwidth F, is very complicated. This difficulties can be eliminated through the application of the density-functional formalism.

16.3

Density-functional calculations

There is a variety of density functional calculations of alkali metal adsorption on metallic substrate. They range from the models where the positive ions of the substrate are replaced by the uniform, semi-infinite jellium, whereas the adsorbate is represented either by a single isolated atom (Lang and Williams, 1978) or by a slab of the uniform jellium (Lang, 1971), to the cases with discrete-lattice treatment of the substrate and the alkali atoms treated either as jellium layer (Wu et al., 1988) or as a discrete atomic layer (c.f. Wimmer et al., 1983). We will discuss the results of the first two treatments. 16.3.1

The

model

ofjellium-on-jellium

A simple model which explains experimentally observed work function changes in the whole range of submonolayers coverage was given by Lang (1971). The model is natural extension of the jellium approach to adsorbate covered metal surfaces and is devoted to a special case of adlayer formed by alkali metal atoms. The substrate, characterized by a positive uniform background charge density, g, fills the half-space

228

C H A P T E R 16. A D S O R P T I O N OF A L K A L I A T O M S

na m

d

0

x

Fig. 16.3. Positive background charge density distribution for a metal of bulk density, fi, with adsorbed slab of density fla. x < 0, and the positive charges of adsorbed atom cores are replaced by a slab of homogeneous positive background of charge density ~a and thickness d. Thus, the adsorbate-substrate system is characterized by the following configuration of the positive charge density distribution (Fig. 16.3) ~ n + (X) --

~a

0

x<0, 0 < x _< d, x>d,

(16.9)

Since the alkali metals are monovalent we must have had - N , where N is the number of adsorbed atoms per unit area of the surface. In terms of the degree of surface coverage, 0, introduced in Sec. 16.2, we have ~ad - NaO, where 0 _< 0 <__ 1, and Na is the surface adatoms density at monolayer coverage 2 (0 = 1). Self-consistent solution of the Lang-Kohn equations of Sec. 6.2, with the positive background represented by (16.10), allows to determine electron density distribution n(x) for the metal-alkali-adlayer system. Fig. 16.4 compares the electron density profiles for a clean metal surface (rs - 2) and for the one covered with a monolayer of Na atoms. The charge neutrality condition which has to be taken into account gives now:

~ [n(x)

~ H ( - x ) ] dx

-

naOd,

(16.10)

OO

with H ( x ) being the Heaviside step function. In his calculations, Lang (1971) assumed that the adsorbate-slab thickness does not change substantially with a coverage. It was assumed to be equal to the interplanar spacing between the closest packed crystallographic planes in the bulk of alkali metal. The changes in the coverage were represented by the changes of na. The respective slab thicknesses for different alkali metals (in atomic units) are: dLi -- 4.68, 2 W h e n b o t h the s u b s t r a t e and a d s o r b a t e are r e p r e s e n t e d by jellium the two scales which are used to describe the a d s o r b a t e coverage are referred to the adsorbate. T h e scale of the n u m b e r of a d a t o m s per unit area, N = nad, where na is actual positive core-charge density at given coverage a n d t h e degree of coverage, 0 = na/~a = N / N a , where Na is the maximal n u m b e r of a d a t o m s per unit area of the monolayer.

16.3. DENSITY-FUNCTIONAL CALCULATIONS

1.0

f

229

r

!

SUBSTRATE

I\l" POSmVE I~ I

IC

x 0.5

c-

I'~ \\

0

-10

I

-5

0

BACKGROUND

,--- ADSORBATE \POSITIVE

"

ROUND

5

10

DISTANCE (atomic units) Fig. 16.4. Electron distribution at the bare jellium surface (dashed line) and for a substrateadsorbate system (solid line) with adsorbed monolayer of Na. The densities are normalized to the mean electron concentration in the bulk of rs - 2. Redrawn with permission from Lang (1971). @1971 The American Physical Society.

dNa = 5.73, dK -- 7.13, dRb -- 7.51 and des = 8.08. Since, among the work function components, only electrostatic surface dipole barrier depends on the electron density distribution then, given the electron density distribution, following (9.13) and (9.14), we can calculate the surface dipole barrier and the corresponding changes in work function induced by alkali metal adsorption. The calculated changes of work function versus the degree of coverage of Na and Cs atoms are presented in Fig. 16.5. The form of these curves corresponds well to that measured for the alkali adsorption on refractory metals (Fig. 16.1) and agrees with a recently measured curves for alkali adsorption on close packed planes of A1 (Fig. 16.6) and Mg. The most important fact about these curves is that they exhibit a minimum as the experimental curves do. Since the calculations were performed for a jellium substrate (of rs = 2) which is not adequate representation of transition or refractory metals, but for which the most complete experimental data are available, hence to make a quantitative comparison with experiment it is reasonable to consider only the position of minimum in the curves. The latter is only weakly dependent on the kind of substrate (Fig. 16.7). The calculations illustrate the experimentally observed decreasing trend along the alkali metal series, from Li to Cs, in both the minimum value of work function, ~m, and the corresponding coverage Nm. ~2m iS increasing function of d and is not strongly dependent on the work function of the substrate.

230

CHAPTER 16. ADSORPTION OF ALKALI ATOMS

I

I

I

A

-

~_~

0

I

1

d-dcs

I

2

I

3

N ( 1014Qtoms/cm2)

Fig. 16.5. Variation of work function ~ versus coverage, N, for Na and Cs adsorption. The adsorbate layer is taken to have a thickness, d, equal to the spacing between the closest packed planes of the bulk alkali. Redrawn with permission from Lang (1971). @1971 The American Physical Society.

The latter phenomenon is also observed experimentally (Kiejna and Wojciechowski, 1981). The uniform positive-background model of the metal-alkali adlayer system is physically justified for the adlayers which show distinct metallic character, i.e. for the coverages close to half a monolayer or greater. The partial ionization of alkali adatoms for low coverages causes that in this limit the jellium-on-jellium model may be questionable. But it is also appropriate for high enough coverages when the adatom wave functions have an appreciable overlap with each other. The reason why reasonable results for q)(0) are still obtained at lower coverages can be seen from analysis of the excess-charge distribution, 5n(x), originating from the presence of adsorbate on a metal surface (Fig. 16.8) where 5n(x) = n ( x ) - ha(X); n(x) and ha(X) represent the electron densities for the bare metal surface and the one covered with adsorbate. At low densities, the positive background slab of adsorbate acts very nearly as if it were simply the source of a uniform electric field (compare Chap. 13). Thus the negative excess charge, 5n(x), for low coverage is very similar to the charge density that would be induced in the substrate by such a field (Lang, 1989). The centre of gravity of the 5n(x), will be shifted towards the metal relative to the position of the centre of gravity

16.3.

DENSITY-FUNCTIONAL

231

CALCULATIONS

4.5

,

4.0

3.5 I- ~t

_

K / A t (111) ---cs/

X:,,. ' ,Os0.- 0 "0"0- 9

2.0

/

I

""

~0...0 0/O~h

1.5 1.0

0.0

0.2 '

0'. 4

0.6 '

08 i

1.0 '

1.2

Fig. 16.6. Work function q~ versus fractional coverage 0 for K and Cs adsorption on AI(lll). After Hohlfeld et el. (1987), Hohlfeld and Horn (1989).

of the positive charge. Thus a dipole layer will be formed. W i t h increasing coverage this shift decreases, which corresponds to the depolarization in Langmuir's model.

Now, returning to the curves of Fig. 16.6, on the basis of our argumentation, we can state that, in the range of low coverages, we deal with adsorption of ionic type which is characterized by a rapid fall of the work function with a number of adsorbed atoms. As the number of adatoms increases, more and more important role is played by their mutual depolarization which is responsible for the appearance of the minimum in the curve (I)(0). Coverage which corresponds to the work function minimum is a boundary coverage above which metallization of the adlayer occurs accompanied by an increase of the work function jointly with an increase of the electron density in the adlayer. At higher coverage the alkali atoms form metallic s-like conduction band. Evidence supporting the formation of such band at high coverage is obtained from the photoemission data of Wertheim et al. (1994) and from earlier experimental studies (Sidorski, 1972; 1984; Rhead, 1988). This is also in a good agreement with theoretical studies of Cs on W(100) (Wojciechowski, 1976; Wimmer et al., 1983; see also: Schemer et al., 1991; Serena and Garcia, 1991).

232

C H A P T E R 16. A D S O R P T I O N OF A L K A L I A T O M S

5.0

I

I

I

4.5 4.0 3.5

>

3.0

I

i

I

1 Cs/Re 2 Cs/Mo

I~

,<,,,

-

3 cs/w

-

-

,-e-,

2.5

-

.~X.k\X

2.0 -

\\.,"N\ \

3

1.5 1.0

i

0.5

I

I

I

I

1.0 1.5 2.0 2.5 N ( 1014atoms/cruZ)

I

3.0

3.5

Fig. 16.7. The measured work function dependence on adatom concentration for three different substrates covered by cesium atoms. After Kiejna and Wojciechowski (1981).

16.3.2

The adsorption of single alkali atoms metallic substrate

on

Let us discuss now the limit of single adatom on a metal surface. Use of a jellium substrate allows to take the advantage of cylindrical symmetry of the problem. Following the approach of Lang and Williams (1978), the single-particle solution of the Kohn-Sham equation for the atom-jellium system can be found from analysis of the Lippman-Schwinger equation of scattering theory corresponding to (5.36). The wave function representing metal-adatom system is expressed in terms of the Green function of the bare metal and the difference in effective potentials for the metal-adatom and the bare metal systems. The equilibrium position of adatom in front of the positive background edge is determined by the condition of minimum total energy of the system. Thus the only parameters which characterize the system are the electron density parameter of the substrate, r s, and the atomic number of the adatom. In the analysis of results it is convenient to define the difference in the density of

16.3. DENSITY-FUNCTIONAL CALCULATIONS

f~

2Ir

"-" x ~"

~ /,--.

1-

233

Futt Cs tayer 0.03of futt Cs I.oyer /-ADSORBATE

/POSITIVE /BACKGROUND

......

V•

(ATOMIC UNITS)

Fig. 16.8. Additional electron density 5n(x) associated with the presence of an adsorbed Cs layer. Redrawn with permission from Lang (1971). @1971 The American Physical Society.

states AAf(E)

- ff AAf(E, r) dr

(16.11)

where A N ( E , r) - Afraa (E, r) - Afro(E, r)

(16.12)

is a change in the local density of states between the systems: metal-adatom and the bare metal. The additional charge density associated with the presence of the adatom is: /_Er 5n(r) -A N ( E , r) dE. (16.13) O(3

Fig. 16.9 depicts the change in the density of states (16.12) caused by the adsorption of lithium atom on the surface of jellium (rs - 2). It is seen that adatom valence states broaden into resonances. For Li adatom, the middle of the resonance level is placed well above the Fermi level. This means that its occupancy is lower than that of a free atom. Consequently, a charge transfer occurs from the Li atom to the metallic substrate. This is corroborated by the calculated distribution of electron charge density near the adsorbed atom (Fig. 16.10). The calculated charge density contours clearly indicate that charge displacement from the adatom vicinity on the vacuum side towards the substrate occurs. Note that in the region of the adatom's core an opposite direction of polarization is observed, which causes an insignificant decrease of the dipole moment. It is clear that these results confirm Gurney's model. Recently Ishida (1988; 1990) extended a single-atom chemisorption approach to finite coverages of Na atoms on the semi-infinite jellium. He argued that the electron transfer is small and that the main effect in the alkali atom chemisorption is an

234

CHAPTER

1.0

I

16.

'

' ,-,-

'

I

II ',

LU

<:1 .

,I / Jl / ~l ,I /

~

/

I

\

/

I I I

.......

-15

ATOMS

.

/

.

,

I V

k

~, ~

~.

.

Si

I'

I I ..d " 0.5 _1

. _ . .

II

t.,I. !I

,

r-

OF A L K A L I

'

I

"~

ADSORPTION

~..~.----.~

-10

.' \I / x

~

,.-,..'

-

ENERGY RELATIVE TO VACUUM (eV)

Fig. 16.9. Changes of the density of states, AN(E), induced by the adsorption of single atoms on semi-infinite jellium substrate of rs = 2. Fermi level is indicated by a vertical full line. The vertical dashed line marks the bottom of the valence band. Redrawn with permission from Lang and Williams (1978). @1978 The American Physical Society.

internal polarization of the adatom. Consequently, the ionic picture of the alkali-atom bonding should be abjured and replaced by a covalent one. This conclusion seemed to be supported by the analyses of experimental studies too (Rifle et al., 1990). However the self-consistent surface-Green-function calculations of Bormet et al. (1994) for chemisorption of isolated Na atoms on the A1(111) have supported the low-coverage description of the alkali-metal adsorption given by Lang and Williams (1978). These calculations have provided clear physical interpretation for the charge residing between the Na adatom and the A1 substrate. Fig. 16.11 shows the difference density An(r) = n Na/Al ( r ) - n At ( r ) - n N~'f3~ (r)

(16.14)

where n N a / a t ( r ) and nAt(r) are the valence electron densities for the single Na atom on A I ( l l l ) , and for the bare A I ( l l l ) surface, respectively, nNa'/3s(r) is the electron density of a partially ionized atom whose occupancy of the valence level is given by the parameter fi. The difference density is plotted in Fig. 16.11a for a partly ionized 3s level fas = 15%. For this value the difference density on the vacuum side of the adatom is practically zero. On the substrate side of the adatom there is an increase of electron density. This charge was interpreted by Ishida (1990) as resulting from the polarization of the alkali adatom. Alternatively, it can be shown that An(r) can be associated with the metallic screening charge density. Replacing Na adatom in its position by a very small charge with q = -0.085e, where e is electronic charge, Bormet et al. (1994) calculated the screening charge induced on the surface. This is plotted in

16.3. D E N S I T Y - F U N C T I O N A L C A L C U L A T I O N S

il

I bohr

i

I I I I

I

235

I

J

vl',

I

I

I \

/ /

Fig. 16.10. Electron density contours for Li atom at the equilibrium distance from jellium (rs -- 2) substrate. The vertical line indicates the positive background edge. The upper part shows the contours of constant density. Lower part depicts the difference density: total electron density minus the superposition of atomic and bare-metal substrate densities. The adatom is indicated by +. Full lines encircle regions of increased electron density and dashed lines show a depletion region. Units are electrons/bohrs -3. Redrawn with permission from Lang and Williams (1978). Q1978 The American Physical Society.

236

C H A P T E R 16. A D S O R P T I O N

,'

,

9

,

,

.

.

.

.

,

.

..

,

o.15)

.

,

,

.

OF A L K A L I A T O M S

9

0.25

7.25

w

w

A

9

i

.

.

0.25 .

.

.

.

.

.

.

.

.

0.085 0.025

0.72~

F 9 -

0.025

"

Fig. 16.11. Difference density, An(r), for a 85~ ionized Na adatom (upper part) and the density of charge induced at the Al(lll) substrate by a negative external charge with q = -0.085e (lower part). Hatched regions correspond to an increased electron density and the cross-hatched regions show a depletion. Units are (10xbohrs) -3. Redrawn with permission from Bormet et al. (1994). @1994 The American Physical Society.

16.4. R E L A T I O N B E T W E E N

THEORY AND EXPERIMENT

237

Fig. 16.11b. Comparing both parts of the figure one observes remarkable agreement of the two density distributions. This provides argument that An(r) is a charge actuated by the metal surface in order to screen partially ionized alkali adsorbate.

16.4

Relation between theory and experiment

Recent measurements of the work function changes due to adsorption of alkali metals on the Al(lll) and the Mg(0001) surface performed by Hohlfeld and Horn (1987) allow for, quantitatively more detailed, verification of the model described in Sec. 16.3. Magnesium and particularly aluminum are typical examples of jellium metals and the work functions calculated self-consistently for clean surfaces of these metals are in a good agreement with measurements. Consequently, the experimental curves of the changes of work function of Al(lll) and Mg(0001) planes versus alkali coverage allow for verification of the theoretical models. As we have seen, with increasing coverage the ionic character of adsorption changes and becomes the metallic one for high coverages. Consequently, as follows from analysis of experimental data (Hohlfeld et al., 1987; Hohlfeld and Horn, 1989), the spacing between the topmost substrate layer of ions and the adsorbed alkali atoms varies between the ionic bond length, dion (twice the ionic radius) and the interplanar distance din, in bulk alkali metal, though, it is closer to dion. Accordingly, in the calculation, it is more appropriate to assume that the thickness of adlayer's slab varies linearly with the degree of coverage 0, d(O) = dion + (din - dion) O.

(16.15)

The self-consistent calculations of AO(0) for alkali adsorption in the jellium-on-jellium model of Sec. 16.3, but with d(O) allowing for adlayer relaxation accordingly to (16.15) have been performed by Serena et al. (1987). This model yields the better reproduced experimental behaviour of AO(0) than in the original Lang's model. It should be noted that the measurements were performed for single crystal planes of the substrate. So, in order to be more relevant to measurements, the model should reflect the real metal structure by accounting for the contribution of a discrete atomic structure of the substrate. To account for this an additional one-dimensional potential that mimics the lattice potential of the substrate can be included into the effective potential of the system (compare Sec. 8.3). This allows to preserve the one-dimensional character of jellium treatment and to include the lattice effects into self-consistent calculations from the very start (Kiejna, 1990a). The result of self-consistent calculations of changes in the work function AO versus coverage 0, for potassium and rubidium atoms on A1(111) are shown in Fig. 16.12. They are in good agreement with the, experimental curves. In particular, the magnitude of the work function minimum, ~m, and of the corresponding coverage, 0m, for K/Al(111) and Rb/Al(111) are in very good agreement with measured values. Similarly, the so called initial dipole moment, which corresponds to the slope of the linear segment of curve AO(0) for very low coverages 0, agrees well with the experiment. To give a numerical example for K/Al(111) system, with dm = 7.13 and dion = 5.03 bohrs, one obtains for A~(0) curve the work function minimum AOm = 2.39 eV at

238

C H A P T E R 16. A D S O R P T I O N OF A L K A L I A T O M S 0.0

~

-0.5 A

:~

-1.0

v

,e,

(a)

".

-1.5

\X ;

\ 9. ;

-2.0

-2.5

-

\ 9

.... \

"~

.o ~

\

%% ~

~ o

0.0

I

0.2

I

0.4

..." I ~ ..o 0"~" I ...oOO" . .0~

I

0.6

I

0.8

-05~

1.0

(b)

0.0

.-... -1.0 9- - -

-1.5

_

-2.0

~ ~ " "" ~ ""

-2.5 -3.0

0.0

0.2

0.4

0.6

0.8

1.0

Coverage 8 Fig. 16.12. Calculated work function changes for different coverages of K (a) and Rb atoms (b) on AI(lll) surface (solid curves). The dashed curve for K/AI(lll) was calculated for the second term (16.16) reduced by a factor 0.5. 0 - 1 corresponds to the number of atoms (per unit area) in a densest packed plane of alkali single crystal. The latter number is smaller than the number of atoms in the underlying AI(lll) plane. The dotted curves show the experimental data of Hohlfeld and Horn (1989). After Kiejna (1990).

Om ~- 0.5 which is in a good agreement with experimentally measured •m ---- 0.42 and A(I)m = 2.17 eV. The jellium-on-jellium model reproduces the main features of work function variation with the thickness d of the adsorbate layer. One can graph the calculated minimum work function which corresponds to a particular choice of slab thickness, d, for a given adsorbate (Fig. 16.13). It is seen that the minimum work function moves in the direction of increasing d. Thus if d is increasing function of the coverage, as for instance in Eq. (16.16), the minimum values of work function moves, for given value of d(O = 1) towards greater coverages. The slight increase with coverage, of the equilibrium distance between the last substrate plane to alkali-adlayer, was confirmed by the self-consistent calculations of Serena and Garcia (1991). It is to be noted, however, that simulation of adlayer by the homogeneous jellium slab can be regarded

16.4.

RELATION BETWEEN

THEORY AND EXPERIMENT

,'i

4o

-2

I

I

K Rb

239

I

R

L

E -3 -4 -5

-

YKX~

d (a.u.}

-

Fig. 16.13. The minimum of the work function change ACre(N) for the jellium (rs = 2) alkali metal adlayer system versus thickness, d, of the adlayer. Curves labeled as B, YK and R are obtained from the following trial density profile computations respectively: Bigun (1979), Yamauchi and Kawabe (1976) and Rogowska et al. (1991). The results of selfconsistent calculation represented by L, and the experimental points are taken from Lang (1971).

only as a very simplified picture of much more complicated structure of alkali-adlayers (Neugebauer and Schemer, 1992; Stampfl et al., 1994). Fig. 16.14 compares the measured work function change for the close-packed A I ( l l l ) surface with Na coverage at room temperature with Lang's calculations and recent theories. From this comparison it appears that the initial decreasing of work function may be explained in terms of the simple Langmuir-Gurney picture which provides a good qualitative but not quantitative agreement with experiment. The more sophisticated self-consistent calculations of Lang (1971) and of Ishida (1990) reproduce qualitatively the experimental ~(0) curve near its minimum but do not give the observed saturation at the monolayer coverage. In general, the Lang's model gives qualitatively a good agreement with experiment only for coverages less than monolayer. These facts indicate that the mechanism behind the change in work function curves with coverage are considerably more complex than this taken into consideration in the hitherto existing theories.

240

C H A P T E R 16. A D S O R P T I O N OF A L K A L I A T O M S

0.0

!

-0.4

..,-,,,,

>

"-" ,.~ <J

\\

-0.8

I

I

I

I

0I .2

013

0.4

015

\\\

-1.2 -1.6

0.0

0.1

Coverage 0

t

0.6

Fig. 16.14. Variation of the work function for Na/AI(lll). The experimental results (curve with filled circles) were obtained at room temperature (Hohlfeld and Horn, 1989) and the coverage is normalized such as for 0 = 1 there are as many Na adatoms as A1 atoms in a clean, undisturbed (111) surface. The theoretical results of Neugebauer and Schettter (1993), Lang (1971) and Ishida (1990) are shown as the dashed curve, the solid line and the continuous line with open circles, respectively. Redrawn with permission from Stampfl et al. (1994).

16.5

S u m r u l e s for a m e t a l w i t h an a d l a y e r

In Chap. 7 we have derived several sum rules for the bare metal surface. They are the useful checks of the self-consistency of the calculations. Here we briefly present the generalization of some of them for the case of metal covered with an adsorbed layer. 16.5.1

Phase-shift

sum

rule

For the jellium-on-jellium model the configuration of the positive charge background is represented by (16.9) as shown in Fig. 16.3. By adsorbing the jellium slab of thickness, d, on the bare surface we must compensate its positive charge by adding sufficient amount of electrons in order to fulfill the charge neutrality condition (16.10). The electron wave functions far inside substrate's interior will have the same asymptotic form (7.1) as for the bare surface, i.e. Ck~ (x) ~ s i n ( k x x - 6(kx))

(16.16)

where kx is the m o m e n t u m component perpendicular to the surface and 6(kx) =- 6x is the phase shift. The derivation of the phase-shift sum rule for the overlayer proceeds in the same way as for the bare surface (Langreth, 1972; Salmi and Apell, 1989). The neutrality

16.5. SUM R U L E S FOR A N A D L A Y E R

241

of the system to the right of a point x < 0 in Fig. 16.3, implies that (16.17)

N ( x ) = - ~ x + ~tad,

where N ( x ) is the number of electrons contained in a cylinder of unit area, ~ is the bulk electron density of the substrate and ~a is the corresponding density of the slab. Inserting the asymptotic form (16.16) into SchrSdinger equation, N ( x ) can be expressed (Langreth, 1972) as

X(x) -

+

2kx

sin(2k~x + 23z)] .

(16.18)

From the requirement of charge neutrality for x << - 1 , and from the equality of (16.17) and (16.18) follows that the integral in (16.18) must be equal had. Evaluation of the integral leads to the following sum rule 7r

27r 2

( 5 ( k x ) > F -- ~- +

7r

AF

k--~F(rtad)- -4 -Jr----2-(Ttad)

(16.19)

to be compared with the Sugiyama-Langreth sum rule (7.3). As it is seen, na and the thickness d, which describe the adsorbed slab cannot be chosen independently but in such a way that their product ~ad, should give the number of adsorbed atoms Na in a slab. Equation (16.19) shows that the extra phase-shift due to adsorbed slab is proportional to the number of adatoms. Since the phase-shift of the wave function is related to the electron density of states through the equations (7.5)-(7.10) we can state that the modified sum rule (16.19) manifest the increase of the phase-shift due to the new states generated in the system by the adsorbed slab.

16.5.2

B u d d - V a n n i m e n u s t h e o r e m for a m e t a l - a d l a y e r s y s t e m

The original Budd-Vannimenus theorem (7.20) relates the difference in electrostatic potential at the bare metal surface and that in the bulk, to the energy per electron for the uniform electron gas. The generalization of this theorem for the case of semi-infinite jellium substrate covered with a slab of adsorbate was given by Bigun (1979). More general expression can be derived (Peisert and Wojciechowski, 1994) from Eq. (7.23) for the positive background represented by the multi-step function. For the n+ (x) being the step function of multiplicity s,

n+(x)-

n0 ni ns - O,

x < do, di-1 < x < di, ds-1 < x,

(16.20)

where i = 1, 2, .., s - 1, we have (with 5(x) being the Dirac function) s-1

dn+(x) = Z ( ~ i + l

dx

i=o

_ ~i)5(x - di)

(16.21)

242

C H A P T E R 16. A D S O R P T I O N OF A L K A L I A T O M S

and the formula (7.23) takes the form s-1 1 E(~i+I no i=0

_ ni)~(di) - r

- no

d~T(TtO) --~ C T t ( Z ) dn-------~~o

(16.22)

The second term on the right-hand side accounts for the effect of external potential (which might be introduced to keep the jellium in equilibrium, or to reintroduce the discrete lattice potential). Neglecting this term and taking the double step representation (s = 2) of the positive background with do = 0 and dl = d (Fig. 16.3), expression (16.22) gives the Bigun (1979)result

r

- r

na + -h-[r

- r

- ~ dcT(~) d-----~-

(16.23)

fi = ~0 is the bulk electron density of the substrate and ~ - fil is that of the overlayer. The above generalizations of sum rules for bare metal surfaces are not limited to the adsorption problem but can be employed in studying other surface phenomena as for instance surface segregation, where the positive background may be simulated by the function given by Eq. (16.20).

where,

16.6

Analytical

density

profiles for jellium-alkali

ad-

layer system In the case of metal-metallic adlayer system, in some situations, similarly as in the case of bare metal surface the analytical charge-density profiles are needed. Such profiles allow to estimate qualitatively many properties of the systems considered. The trial functions for the electron density profiles of jellium covered by a submonolayer of alkali adatoms have been proposed by Yamauchi and Kawabe (1976) and by Bigun (1979). However, employed to the calculation of the work function changes, A~, due to alkali atoms adsorption, these density profiles have led to the results which disagree with the experimental data and with self-consistent calculations (Fig. 16.13). The simple analytical model which gives good agreement both with measurements and with self-consistent calculations is the one proposed by Rogowska et al. (1991). The normalized electron density distribution corresponding to the core-charge density, n+(x) of the form (16.9), and is represented by the function /](X)

-

n(x) f fl(x) - F1 (x) _ n ~ fl(x) + f 2 ( x ) - F2(x)

x _~ xo, (16.24) x > x0.

Functions fl(x) and f2(x) describe the electron density profile of the bare metallic substrate and of the adsorbed slab, respectively, x0 is the matching point at which Fl(x) - F2(x). Functions f(x) have the following form (see Eq. 11.17) _ ~ 1 - bexp ['~akTF(X - xo)] f l (~)

[

bl ~ x p [ - ~ k ~ ( ~

- ~o)]

X ~X0, X ~X0,

(16.25)

16.6. A N A L Y T I C A L

243

DENSITY PROFILES

C t - b__x kTF is Thomas-Fermi wave number and b ~ f2(x) - p ( x -

xo)e -q(z-z~

x > xo.

(16.26)

The parameters 7 and q are determined by the condition of minimum of surface energy and p is determined by the charge neutrality condition as

p = sq2d,

(16.27)

where s characterizes the surface concentration of adatoms, s = na/~. The parameter q(N, d) for given d, is a decreasing function of the number of adatoms per unit area, N, and varies from 3.02 to 0.74 for Li, and from 2.99 to 0.49 for Cs, for N = 0.25 x 1 0 1 4 cm -2. The dependence Aq)min versus d, calculated by Rogowska et al. (1991) using tabulated values of q(N, d), is displayed in Fig. 16.13. As is seen from this figure, the presented simple model gives quite a good agreement both with the experimental data and with the self-consistent Lang's calculations.

This Page Intentionally Left Blank

Chapter 17

Adhesion between metal surfaces 17.1

General considerations

It is well known from the experiment that the interaction between two clean metal surfaces brought into close contact leads to their strong bonding. The force acting between surfaces of two metals at small separation is of great interest not only for the understanding of adhesion and cohesion. It is also of great technological importance in such diverse areas as friction and wear, crack propagation, fracture mechanics, interface decohesion, deposition of films, etc. It is also essential for explaining mechanism of scanning tunneling microscopy or atomic force microscopy. In spite of this wide interest, the ability to determine the forces governing the bonding is limited both experimentally and theoretically. Experiments are influenced by different surface imperfections such as impurities, asperities or adsorbates and elastic deformation of the surface layer. The effect of the temperature is also important. Theoretical calculations from first principles are rare and limited to most ideal interfaces only. In this Chapter we will also limit our considerations to the basic or ideal adhesion between a pair of planar metallic surfaces i.e., all defects and imperfections of the real surface will be neglected. Some other important aspects of adhesion not covered here are discussed in the book edited by Lee (1991). The adhesive interaction which develops between two materials brought close together can be sketched in the following way. At larger separations where the electron wave functions do not overlap appreciably the attractive van der Waals or dispersion forces dominate the interaction. The van der Waals forces are of a long-range nature and decrease with the distance to a power law and not as an exponential. As suggested by Lifshitz (1955) the interaction between two condensed bodies treated as a system of atoms acting like isotropic oscillators may be considered as taking place through a fluctuating electromagnetic field in each of the two bodies. 245

C H A P T E R 17. A D H E S I O N B E T W E E N M E T A L S U R F A C E S

246

The resulting force of interaction between two condensed bodies (metals) is

Fvdw-

1

(47r)2x3

du

/0

--1

dvv2

el ( i u ) - 1

c2(iu)- 1

(17.1)

where e(iu) is related to the imaginary part of the complex dielectric functions of the two bodies. The above expression can be simplified without significant loss of accuracy and can be written (Dzyaloshinskii et al., 1961) in the form C 8~2x 3 = x--5

Fvdwwhere

L ~176 (Cl(iU)-I-1)(c2(iu)-d-1)d

(17.2)

u

(17.3)

is the average angular frequency. 1 The force of interaction and the potential energy of adhesive interaction are related by dU Fad d--~" (17.4) -

-

Thus, the van der Waals interaction energy for two metallic plates is of the form

Uvdw --

CV x2

(17.5)

where x is a distance between the plates. For small enough separations (< 10 bohrs) the overlap of wave functions begins to be important and a short-range bonding effects begin to dominate. If the metals in contact have different Fermi energies, charge transfer from one metal to the other occurs and a dipole layer is formed. For alike metals, no net dipole is formed but electrons may tunnel through the barrier from one metal to the other forming a nonvanishing bond charge distribution. A development of adhesive bonding is sketched in Fig.17.1. The maximum in the force curve, which is the force necessary to break the bond, corresponds to the inflection point in the energy curve. The minimum in the energy curve represents the adhesive binding energy or the m a x i m u m work of adhesion. Note that equilibrium occurs at point re where the force is zero. The maximum work of adhesion g a d is the work necessary to increase the separation of the surfaces from the equilibrium distance to infinity. Thus the thermodynamic work of adhesion can be obtained from the Dupr6 (1869) equation

Ead -- (71 + (72 -- (~f2,

(17.6)

where al and a2 are surface energies of metals 1 and 2, respectively, and a/2 is the interfacial surface energy at the contacting surfaces. For the identical metals in contact al = if2, and the adhesive energy becomes

Ead -- 20" -

o "I.

(17.7)

1For two jellium in contact characterized by their electronic p l a s m a frequencies ~Op -- (47rn) 1/2 the c o n s t a n t C takes the form C "-' v/2 Odpl~p2 -- 0.0035169 ~176 1287r Wpl + Wp2 Wpl Jr- Wp2

17.2.

SEMI-INFINITE METALLIC SLABS

247

Energy j r

I

E rnin

I I i i

Force Fmax

re

r

v

Fig. 17.1. Schematic plot of the binding-energy versus separation and force-separation curves. r~ denotes the equilibrium position.

If the surface lattices of two metals are in registry the and the adhesive energy is a double surface energy.

dr I

term in (17.7) is equal zero

Generally, the adhesive interaction energy Earl, is a function of separation, D, of the two metal surfaces. Thus the adhesive energy may be defined as Ead -- E ( D ) - E(oe) 2A

(17.8)

where E is the total energy and A is the cross-sectional area of the contact. For alike metals in contact and for D corresponding to the energy minimum, Ead calculated from (17.8) gives the negative of surface energy or.

17.2

A d h e s i o n of s e m i - i n f i n i t e m e t a l l i c slabs

In order to calculate adhesive energy, Ead, from (17.8) we have to determine the total energy E. In this purpose we will again make use of the density functional formalism of Chapter 5 (Bennett and Duke, 1967). The total energy of the ground state can be expressed as a functional of the electron density n(r)

E[n(r)] - Ts[n(r)] + Ees[n(r)] + E=~[n(r)]

(17.9)

CHAPTER 17. ADHESION B E T W E E N METAL SURFACES

248

where Ts is the kinetic energy, Ees is the electrostatic energy, and Exc is the exchangecorrelation energy contribution. In a more explicit form this can be written as E[n(r)]

-

T~[n(r)] +

1~

/

1//n(r)n(r')drdr' v(r)n(r) dr + ~ i r _ r' I

ZiZ j -~- Exc[n(r)]

+2

(17.10)

n~j

where v(r) is the ionic potential. The second and fourth term in Eq. (17.10) represent the electron-ion and ion-ion interaction energies respectively. For simple metals represented by the jellium model, v(r) represents the potential produced by jellium and the ion-ion interaction energy term (classical cleavage) is absent. Looking for an improvement on the jellium model we may introduce the interaction of electrons with discrete ions using the first order perturbation theory. In order to account for this, similarly like in the calculation of the lattice corrections to the surface energy in Chap. 8, we may represent the electron-ion interaction by a weaker pseudopotentials. Treating the difference potential 5v(r) between the lattice of pseudopotentials and the potential due to jellium as a small one, the total energy E, is to a first order perturbation approximation (Ferrante and Smith, 1973) given by

1/

E[n] - T~[n] + E~[n] - ~

r

dx + Wint -~- A

/

5v(x)n(x) dx

(17.11)

where the third and the Win t term result from the combination of the electrostatic terms appearing in (17.10). r represents the electrostatic potential of the system and can be obtained from the Poisson equation

d2r dx 2

= 47r[~10(-x) + ~20(x - D) - n(x)]

(17.12)

n l and ~2 are the positive background charge densities of metals 1 and 2 respectively and n(x) is the electron density profile determined for a given separation (Fig. 17.2). Wint is the exact difference between the point ions and the jellium-jellium interaction. In order to determine electron density at the interface, in the first approximation, we may expect that when two metals are brought into proximity or contact the charge rearrangement at each of the surfaces, at the metal side, will not be significant. This is supported by a simple argument resulting from the stationary property of the energy functional E[n]. Let the supperposed electron density n(x), differ from the exact one by 5n, i.e.

where

5n -

n-

n(x)

where n is the exact electron density. Then we may write

E[n(x)]

-

E[n] + 1

6n(r) 5n(r)dr

ff 62E[n] 5 n ( r ) h n ( r ' ) d r d r ' + . . . . + 2 y y ~n(r)hn(r')

(17.13)

249

17.2. SEMI-INFINITE METALLIC SLABS

D=oo

J

0

Fig. 17.2. Schematic plot of the electron charge density profile for two metals: (a) at infinite separation, (b) separated by D. Making use of the stationary property of E[n], we have

/ 6E[] 6n(r)

6n(r)dr

(17.14)

-- 0

what means that E[n(x)] is accurate to second order in 5n. Therefore we may assume that the electron distribution at the interface can be approximated by the superposition of metal-vacuum distributions of each metal. This approximation breaks, however, at very small separations. This is evident since in reality, at vanishing separations, the amplitude of Friedel oscillations appearing on the each of the metal halves should decrease to zero. The electron density distribution for the metal-metal or metal-vacuum-metal system may be represented either by the variational density profile or by the self-consistent density profile resulting from the jellium surface calculations of Chapter 8. The advantage of the variational density profiles is the possibility of performing most of the calculation analytically. A disadvantage is a necessity of using density gradient expansion in evaluation of the kinetic energy. Assuming Smoluchowski's form (see Chap. 11) of the metal-vacuum electron distributions, the electron distribution for two metals, at separation D, is (Ferrante and Smith, 1973) -

1

e3~

1

e ( - x ) + E~le-~l~e(x)

CHAPTER 17. ADHESION BETWEEN METAL SURFACES

250

1 + n 2 - ~n2e -z2(x-D) O ( x - D) +~n21 eZ2(x_D)o( D -- x),

(17.15)

where the parameters 31 and 32 are determined variationally for the metal-vacuum interface and O(x) is the step function. Evaluation of the Wint term describing the exact difference between the ion-ion and the jellium-jellium interaction requires quite tedious lattice summation. For the like metals in contact this term in equal to the classical cleavage energy introduced in Chap. 8. Generally Wint can be written as

Wint(D) - / f

n+(r)n+(r') drdr '

(17.16)

Ir-r'I

where n + (r)

- ZE m

~(x - Xm) E E l

~(Y - yt)~(z - Zh) --p E

h

~(X -- Xm)

(17.17)

m

p is the ionic charge per unit area in a given lattice plane, 1 and h are summation indices in the plane and m enumerates the planes. Thus the second term represents the jellium in a plane-wise expansion. The expansion of n + (r) in a double Fourier series for each lattice plane parallel to the surface shows (Ferrante and Smith, 1973) that the resulting series for Wint is rapidly converging. It appears also that Wint term is negligible unless the facing planes are in registry what means that it is of importance only for the like faces of the same metal. For two most densely packed surfaces of the fcc or hcp lattices in perfect registry i.e., for fcc(lll)-fcc(lll) or hcp(0001)-hcp(0001) contact the Wint can, to a very good approximation, be expressed as

W~nt(D)=A -2v/3Z------~2 e X P c 3 [ v/347r(d+c D)]

(17.18)

where d is the interplanar spacing in the metal and c is the interionic distance in a given lattice plane. The similar expression for the perfect registry of bcc(110)-bcc(110) fragments is

-~(d + D)) v~ exp

---c

(d

(17.19)

The calculations of adhesive energies for several simple metals, based on the simple overlap density distribution (17.15), have shown that all systems were bound (Ferrante

17.2. SEMI-INFINITE METALLIC SLABS

251

(a} Commensurate adhesion -200

-200 f

_,oo

-~176176~"'"'-~'""'

-

8

0

0

1 -~~176 n,OOO~)-Zn,O00~)

~

-800[

~

0

I

l

r

E

,

-200

-200

(0001)

-400 cn (i,)

i

i

~

0

0.2

124

-600

-400

0

-600

0.6

( 110) i

0

0.2

'!

0.4

0.6

(b) Incommensurate adhesion

or tll

Z LU -100 -200 -300 I:~-400 /AI(111)-Zn(0001[ < -500 - 600

0 -100 2~ 3~ 4~ 500 6~

i

,

,

,

,

,

" I

I

I

I

Mg(0001)- No(1101

I

I

I

I

I

i

'VZnI0001)-Mg(0001i 0.2

0.4

0.2

0.4

I

I

0.2

I

I

0.4

I

SEPARATION D(nm) Fig. 17.3. Adhesive binding energy curves versus separation. (a) Commensurate adhesion (W~nt ~= 0), (b) incommensurate adhesion (W~nt = 0). Redrawn with permission from Ferrante and Smith (1985).

252

C H A P T E R 17. ADHESION B E T W E E N M E T A L SURFACES

3000 2500 2000 E

o L_

>O

rr

uJ

izl l

1500

c Energy

1000 500 0 9 9

.

..

:ft.- "-~'~"

Pseudopotentiol Energy -500 . 9f ~ . : ~ " / ~ ~ Electrostatic Energy ./ -1000 - /,'~-...._ Total Energy _/ - ' Exchange Corretation Energy -1500 / -2000

l

I

I

0.2

I

I

0.4

I

0.6

SEPARATION D (nm) Fig. 17.4. Components of the binding energy for an A I ( l l l ) - M g ( 0 0 0 1 ) adhesion. Redrawn with permission from Ferrante and Smith (1985).

and Smith, 1973; Kiejna and Zi~ba, 1985). The adhesive energy curves showed a minimum close to a zero separation. These results were confirmed by the self-consistent calculation (Ferrante and Smith, 1979; 1985). Fig. 17.3 exhibits the binding energy curves calculated self-consistently 2 for several metals in contact assuming commensurate (Wint ~ 0) and incommensurate (Wint = 0) adhesion. The curves show that the range of strong bonding is about 2 /~ for bimetallic contact. For most of the contacts the minimum in the adhesive energy curves does not occur at zero separation but is slightly shifted. The reason for this is that the bulk lattice constants used to determine the average electron density in the bulk (the electron density parameter r~) are not exactly the same that minimize the bulk cohesive energy. It means that they are also not completely consistent with the core radius of the Ashcroft pseudopotential. The equilibrium lattice constants differ slightly from those which were employed in calculating the energy curves. The curves computed for the equilibrium lattice constants have a minimum shifted to zero separation (Ferrante and Smith, 1979). In Fig. 17.4 different components of the adhesive binding energy are displayed for Al(111)-Mg(0001) contact. At large separation the attractive kinetic energy initiates 2For the details of self-consistency procedure see Ferrante and Smith (1985) and McCann and Brown (1988).

253

17.3. E X A C T RELATIONS FOR BIMETALLIC INTERFACES Table 17.1

Adhesive binding energies (in erg/cm 2) for different metal combinations. From Ferrante and Smith (1985).

Surfaces in contact

AI(lll)-AI(lll) Mg(0001)-Mg(0001) Na(110)-Na(ll0) AI(111)-Mg(0001) AI(lll)-Na(110) Mg(0001)-ia(110)

Incommensurate

Commensurate

Win t - 0

Win t ~ 0

490 460 195 505 345 310

715 550 230

the bonding. In analogy with the molecular bond, the negative kinetic energy results from the smoothing of the wave functions when the orbitals of two metals begin to overlap. For a discussion of the paradoxical role of the kinetic energy in the binding energy we refer the reader to Feinberg and Ruedenberg (1970). At shorter distance the kinetic energy becomes repulsive term and the dominant attractive term is the exchange-correlation energy. The van der Waals dispersion forces (not shown in the picture) become dominant only at approximately 6 A separations for two A1 slabs in contact (Inglesfield, 1976; Vannimenus and Budd, 1975) i.e., far beyond the region of strong bonding. A comparison of the adhesive binding energies at the minimum (Table 17.1) allows to predict a possibility of transfer of atoms of one metal onto the other metal surface. It is seen that binding energy for Na-Na contact is weaker than the strength of Al(111)Na(ll0) or Mg(0001)-Na(ll0) contact. It means that atoms at the Na surface are weaker bounded to the atoms lying underneath than to A1 or Mg atoms and they may be transferred to the A1 or Mg surface, respectively.

17.3

E x a c t r e l a t i o n s for b i m e t a l l i c i n t e r f a c e s

As we will see, important exact relations between the electrostatic potential and the bulk properties of interacting semi-infinite jellium can be derived. In analogy to the metal-vacuum interface these relations may serve as a test of the internal consistency of microscopic calculations for the bimetallic contacts. Let as consider two slabs of different metals, 1 and 2, separated by a vacuum gap of width D. Let the thickness of the first metallic slab be L1 - D/2 and that of the second slab L2 - D/2 (Fig. 17.5). The force per unit area that the slab 1 exerts on the slab 2 is obtained by summing the force acting on each layer of charge (Budd and

254

C H A P T E R 17. A D H E S I O N B E T W E E N M E T A L S U R F A C E S

S/

//

•

-t-1

L2

o

S/ Fig. 17.5. Schematic representation of two interacting macroscopic jellium films separated by D.

Vannimenus, 1973) F12(D) - n2

(17.20)

dx E2(x, D) /2

where E2(x,D) - -

dr

(17.21)

dx

is the electric field at point x, in the film 2, when the separation is D. Integrating (17.20) we get (neinrichs and Kumar, 1976)" F12(D) = - ~ 2 [r

D) - r

D)].

(17.22)

For the macroscopic samples the electrostatic potential energy r D) = r i.e., it is independent of D. On the other hand, sufficiently deep inside slab 2, the electrostatic potential takes the bulk value r Taking this as a reference potential we may write Ar Ar

= r = r

- r D) - r

(17.23)

(17.24)

and (17.22) can be written in the form F12(D) - - 5 2 [Ar

- Ar

(17.25)

Similar expression can be derived for the force F21 (D) exerted by the metal 2 on the metal 1. Now, recalling the exact relation (7.20) for a half-space jellium system, which may be written in the form ~Ar = -pi, i = 1, 2 (17.26)

17.3. E X A C T R E L A T I O N S FOR B I M E T A L L I C I N T E R F A C E S

255

OE and E denotes the total

where p denotes the bulk electronic pressure, p energy, the force F12(D) is given by P2 + ~2Ar

(17.27)

F21(D) = Pl + n l A r

(17.28)

F12(D)

-

Similarly,

We stop here for a while by noting that the second term in the expressions (17.27) and (17.28) vanishes for D --+ 0, and in the case of identical metals F12 = F21 =- F, reduces to the bulk electronic pressure p = -~t2deT/d~t. Thus, the bulk pressure may be interpreted as the force exerted on the one of two parallel jellium half-spaces by the second half-space in the limit of vanishing separation. The existence of the electronic pressure implies an intrinsic instability of jellium as already discussed in Sec. 4.7. In the general case, by the Newton law, we must have F12(D) = F21(D) and consequently equating expressions (17.27)and (17.28) we have (17.29)

Pl + nl Ar (D) = p2 + ~2Ar

which for separation D ~ 0 reduces to the equation derived by Heinrichs and Kumar (1975). Using (17.23) and (17.24)equation (17.29)can be written as

r (D)pl-P2--~-rt2r -

n2

- n1r

_t_ __ ( _ nrl n 2

-2-'DD ) .

(17.30)

Denoting the first term on the rhs of (17.30) as a21, we get the linear dependence between the two surface potentials r

,D

---~-,D

= a21 + - - r

n2

.

(17.31)

This is the exact relation between the surface potentials and bulk properties of two jellium slabs (Raykov, 1978). For the two like metals in contact a21 = 0, and we get the equality of the potentials i.e., (~2(D/2, D) -- r D), which follows from the requirement of continuity of the potential. In order to determine potentials r (i = 1, 2) appearing in a21, we make use of the requirement of equality of the bulk electro-chemical potentials of two slabs, i.e., #1

-

r

= #2

-

r

(17.32)

where the bulk electrostatic potential of one of the slabs may be chosen arbitrarily. Let us put r = 0. Then we have #1 - #2 - r and a21 can be written in the form

a21 --

Pl - P2

_ n2

+ EF2 -- EEl + # x c ( n 2 ) - #xc(nl)

(17.33)

256

C H A P T E R 17. A D H E S I O N B E T W E E N M E T A L S U R F A C E S

there we have made use of the relation, # - EF + #xc, where EF is the Fermi energy and #xc is the exchange-correlation part of the chemical potential (see Eq. (9.11)). On the other hand, as we remember, the electronic pressure is given by

p_

dcv(n) d~

(17.34)

where, s is the total average energy per particle in the uniform electron gas of density ~. Thus for given metals in contact a21 is a definite constant and equation (17.31) may serve, and serves, as a check of self-consistency of the calculations (Ferrante and Smith, 1985).

17.4

The force between

metal

surfaces at small sep-

arations Let us discuss now the force acting between surfaces of two metals at small separation. The maximum cohesive/adhesive force determines the ideal fracture strength which is an important parameter for theories of fracture (Thomson, 1986). In general, the calculation of this force for a real metal is a complicated problem. In a simplest case one can consider force acting between the two like metal surfaces. Denoting by F ( x ) the force per unit area at the separation x, the work performed in increasing separation of two semi-infinite segments of a metal from D to infinity, defines the adhesive energy (per unit area), 1 Ead(D) - -~

E

F(x)dx,

(17.35)

where the factor of 1 takes into account the existence of two surfaces. In the limit of D --+ 0, Eq. (17.35) gives the surface energy a, i.e. a = -~

F ( x ) dx.

(17.36)

At small separations, the expression for the force between two metal fragments versus separation, can be generally written as F ( x ) - Fo + A x + O(x2).

( 7.37)

For real metals F(0) = 0, whereas for two jellium fragments brought into contact the repulsive force F0, does not disappear because of the instability of this model (compare Eqs (17.27-17.28)) and is given (Budd and Vannimenus, 1976) by the electronic pressure term (17.34) Fo =- p. (17.38) The existence of this term is illustrated in Fig. 17.6 which shows the force between two similar slabs of jellium (Nieminen, 1977). The appearance of the linear attractive term for jellium fragments at small separation is the result of a readjustment of the

17. 4. A D H E S I V E FORCE A T S M A L L S E P A R A T I O N S

~

257

rs-3 rs-4

d

U_.

0

I

I

I

I

I

I

I

I

,

-1 -2 1

3

I

5

D(~) Fig. 17.6. The adhesive force between two similar jellium slabs. Redrawn with permission from Nieminen (1977).

electron density distribution, towards lower densities, what leads to a lowering of the pressure exerted by electron gas in the vicinity of the cleavage plane (Heinrichs, 1985). As we have seen in Sec. 15.2, in order to get realistic description of surface interactions it is essential to incorporate the ionic lattice effects. Fig. 17.7 displays the adhesive force acting between two pieces of A1 oriented with (111) face, which results from the differentiating of the adhesive energy curve with respect to the separation D. As is seen the force is vanishing at zero separation, and it takes the maximum value at about 1 A separation. The maximum gives the force (per unit area) or breaking stress necessary for brittle fracture. In the case illustrated in the Fig. 17.7, we have F m a x --~

2.2 x 10 -4 a.u. = 6.5 x 109Nm -2,

whereas the Orowan formula,

Fmax- v/Eo/d,

(17.39)

which is an usual expression used to estimate the breaking stress (Inglesfield, 1976) gives the value of Fmax which is one order of magnitude larger than that given by (17.39). Here, E is Young's modulus, a is surface energy and d is the interplanar spacing. In order to discuss the nature of adhesive force for real metals it is reasonable to assume that at small separations this force can be treated as elastic one with the proportionality constant A, which can be expressed by the uniaxial elastic modulus

258

C H A P T E R 17. ADHESION B E T W E E N M E T A L SURFACES

I

xlO-4

I

I

I

I

I

I

.Cteavaae Pseudopotential

.,.--,,,.

d

I

0I..

\'~

\'\

,,o

"'...

van der Waais "'..

.j ,~"

-1 1

3

D(A)

5

7

Fig. 17.7. Components of the force between two AI(lll) surface in perfect registry. The curve labeled 'cleavage' corresponds to Wint term. The van der Waals force is also shown for comparison. Redrawn with permission from Nieminen (1977).

Cll associated with the direction perpendicular to the interface (Kittel, 1967). Thus, one can write F(x) = C l l x / d = Ax (17.40) where d is the distance between lattice planes. At large separat!ons the two halves of the crystal interact via the long-range polarization (van der Waals) force of the form (17.2). As it is seen from Fig. 17.7 the van der Waals forces begin to dominate at separations greater than 5 A (Inglesfield and Wikborg, 1975). Basing on these facts, and on the picture emerging from Fig. 17.7, one can look for the interpolation between the small and large x behavior of the force, namely

F(x)

=

l

Ax + O(x 2)

for x --+ 0,

C - j + O(1/x 4)

for x -+ co.

17.4. A D H E S I V E F O R C E A T S M A L L S E P A R A T I O N S

259

Making use of the force-potential-energy relation, such interpolation formula can be written (Zaremba, 1977; Kohn and Yaniv, 1979) in the form: F=

dU dx'

(17.41)

where 1( C ) U - - ~ (72+x2 ,

(17.42)

and G 2 -(C/A)

1/2.

(17.43)

Inserting this form into (17.36) yields

O'--~1 (AC)I/2

(17.44)

Since the constant C is known from the Lifshitz formula (17.2) it remains to determine A. In the calculations of Zaremba (1977) and Kohn and Yaniv (1979), the constant A was expressed by the lattice-phonon dispersion relation. Instead of choosing the particular form (17.41)-(17.44) for F(x), Kohn and Yaniv (1979) postulated to express F(x) in terms of the universal function f*(x). Let us scale the distance x in units of 1 - (C/A) 1/4 and the force (per unit area) in units of (AaC) 1/4. Then F - ( A 3 C ) l / 4 f * ( x *) (17.45) where x* - x/1 = x ( A / C ) 1/a.

(17.46)

The force versus distance dependence is depicted schematically in Fig. 17.8. Thus the surface energy is given by a - a*(AC) 1/2 (17.47) where oe* = -1 2

f* ( x *) dx *

(lr.4s)

is a dimensionless universal constant of the order of unity. This constant can be calculated by integrating the scaled force function X ~

-

(1 +

(17.49)

corresponding for instance to (17.41)-(17.44). This gives a* = 1/4. On the other hand, c~* can be estimated from (17.47) provided we know the experimental values of surface energy a, and the constants A and C are determined by measured phonon dispersion curves and by (17.3) through the measured optical spectra, respectively. Plotting a versus (AC) 1/2 one finds the best fit line which determines empirical c~* = 0.476. In order to see the influence of the electronic structure on the force-distance curves it would be interesting to determine the force constant A in terms of the electronic response function. The idea of linking the elastic force constant, A, with the electronic

260

C H A P T E R 17. ADHESION B E T W E E N M E T A L SURFACES

properties of jellium was proposed by March and Paranjape (1984) who calculated constant A at x = 0, using the Thomas-Fermi approximation and the virial theorem to relate it to the curvature of the kinetic energy at zero separation. For a cleaved jellium the force constant can be expressed rigorously by an integral over the static dielectric function, c(k), of the homogeneous electron gas (Budd and Vannimenus, 1973; Heinrichs, 1985) A- -2~ 2

~ k2e(k).

(17.50)

In the Thomas-Fermi approximation, which is valid at high densities, we have (Ashcroft and Mermin, 1976) q2 e(k)- 1 + ~ (17.51) where q-1 = ATF is the Thomas-Fermi screening length (see Appendix A). Inserting (17.51) into (17.50) one obtains the Thomas-Fermi expression for the force constant: A - 2 7 r ~ 2 - 32 ( 3/ ) 3 - q

8

~

11/2"1

(17.52)

rs

The explicit expression for A, valid at arbitrary density, should be calculated from (17.50) using a better approximation for c(k). The above approach can be extended to calculate the force constant for two different metals in contact. The general expression is (Streitenberger, 1986) A = - n l n 2 ~ ( x , x')l~=~,=0.

(17.53)

where ~ and ~2 represent the uniform background densities of metal 1 and 2, and ~(x, x') is the density-potential response function defined by ~v(x, x')

-

5r

(17.54)

~n+(x')

F(x]

c ~3 X

Fig. 17.8. Schematic plot of the force-distance dependence for adhesive interaction.

17.4. ADHESIVE FORCE A T SMALL SEPARATIONS

261

where r is the total electrostatic potential of the system and 5n+(x) describes small perturbation of the positive charge. According to (17.54), ~o(x,x') can be interpreted as the electrostatic potential of a charged sheet linearly screened in the uniform electron gas of the two half-planes. Again, if we consider the high density limit of metals 1 and 2, at small separation we may make use of the Thomas-Fermi approximation for ~o(x, x'). Within this approximation the force constant, A12, for the bimetallic interface is given by A12 =

47r~1n2

(17.55)

ql +q2

where q,

-

\ d~i J

'

i - 1, 2

(17.56)

and Pi is the internal chemical potential of i-th metal. In the high density limit one may neglect the exchange and correlation contributions to pi and A12 will take the form A12 =

7 3 5/2 3 5/2 rslrs2 + rs2rsl )

(17.57)

where 7 is the numerical constant, 3 -

= 0.4582,

(17.5s)

and r sl, r s2 are the Wigner-Seitz density parameters of metals 1 and 2. For the like metals rsl = r~2 = r~ and (17.57) turns into (17.52). Now me may use A12, given by (17.57), to extend the described above idea of scaling surface energies (Kohn and Yaniv, 1979) to the adhesive energies. This will be discussed in the following Chapter.

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Chapter 18

Universal scaling of binding energies 18.1

Scaling of adhesive binding energies

As we have seen in Chapter 17, different metals in contact have binding energy curves of similar shape which differ, however, one from the other in the binding energy depth and the position of equilibrium distance. Rose, Ferrante and Smith (1981) have proposed a simple universal relation to scale the energy distance dependence for flat metal interfaces. The energy can be scaled by dividing by the equilibrium (or minimum) binding energy AE, to yield

E*(g) = E ( a ) / A E ,

(18.1)

where E* (g) is a universal function describing the binding energy curve. The argument is a scaled length determined by

gt- a-a,~ 1 '

(18.2)

where a is the separation distance, am is the equilibrium separation and 1 is a length scale describing the width of the binding energy curves. It is found that l is reasonably fixed by the Thomas-Fermi screening length l ( r r ) 1/6 1//6 ATE -- -~ -~ n(am)

(18.3)

where n(a,~) is the electron density at the minimum of the curve. Such a choice of the scaling length is not unique and it can be chosen in a number of different ways, for example as a fitting parameter. Generally, it should be a measure of the distance over which electronic forces can act. This point is discussed further on. Now we may ask about the functional form of the universal function E*(g). It appears that it is well represented by the Rydberg function E* (g) = - ( 1 + fig) exp(-flg) 263

(18.4)

C H A P T E R 18. SCALING OF BINDING E N E R G I E S

264

m

I

A

_

"1 ,.,%

-0.2

+

/" >(.D I:E hi Z hi

o

Al-Zn

a

AI--Mg

s

-0.4

hi

Zn-t.,Cg

9 AI-AI

m kiJ "1-1:3

hi

9 Zn-Zn

IL

-0.6

(.J tm

f

1

9 Mg4~g

13

I -0.8 _

| O

? I

-1.0 0

o

AI-Na

v

Zn-Na

o

Mg-Na 9 Na-Na

1 2

.1

4

I

I

6

8

10

SCALED SEPARATION, a"

Fig. 18.1. Adhesive energy curves for different metallic contacts from Fig. 17.3 scaled according to Eq.(18.1)-(18.3). Reprinted with permission from Rose et al. (1981). @1981 The American Physical Society.

where ~ is the fitting parameter. In the original work (Rose et al., 1981) the best fit was achieved with ~ = 0.9. The calculated adhesive energies for ten different bimetallic contacts, scaled according to (18.1)-(18.4)using )~TF = (ATF1 +)~TF2)/2 are displayed in Fig. 18.1. Here, /~TF1 and )~TF2, represent the screening lengths of the two metals in contact. The closeness of the scaled results to the universal curve is remarkable. The plausibility argument for the existence of universal binding energy curves in adhesion may be provided based on the jellium model. The discussion in Chapter 6 have shown that at the metal-vacuum interface the electronic density distribution, n(x), scales with the Thomas-Fermi screening length ATE. Following (6.9) we can write -

(is.5)

18.1.

265

S C A L I N G OF A D H E S I V E E N E R G I E S

and -

(18.6)

X/1TF,

where ~ is the electron density in the bulk and ~(~) is the universal form for the electron density. Application of this scaling to the electron density profiles at different metal surfaces results in the curves shown in Fig. 18.2 (Smith et al., 1982). It is seen that all scaled curves merge into one universal curve quite accurately except of the Friedel oscillations on the metal side. This is understood as the electron density outside a jellium surface varies exponentially with distance for all jellium bulk densities. A reasonable charge-conserving fit to the scaled densities is given by the Perdew function (11.17), with "7 = 1.02, which neglects Friedel oscillations. For two identical metals separated by a, and located symmetrically with respect to the origin, the electron density at the surface of one of the metals can be written as n (x-

-

2)

#'~

[(x-

2)/ATE].

(18.7)

The similar scaling should apply to the Kohn-Sham effective potential which decays outside the metal surface exponentially (see discussion in Chap. 6), so we have

1.2

I

W rn

D Z D IJJ ._I

<

0.8

I

I

I

oooO~176176

1.0

Z U.I a

I

_

dg A

% #

o Ai, r s=2.07

0.6

B

o~

o Zn, r s = 2 . 3 0 ,, Mg, r s = 2.65

0.4

~

O

o Na, r s = 3 . 9 9

(,9 Or;

8

\

0.2

1 -6

I -4

1 .-2

1 0

-

"%._ ! 2

4

S C A L E D DISTANCE a*

Fig. 18.2. Electron density profile at the surface of jellium of different bulk density versus distance scaled with the Thomas-Fermi screening length )~TF. Reprinted with permission from Smith et al. (1982). 9 The American Physical Society.

266

C H A P T E R 18. S C A L I N G OF B I N D I N G E N E R G I E S

where VB is the value of effective potential in the bulk. For some calculations the electron density at the bimetallic interface, to a reasonable accuracy, might be represented by a simple overlap of metal vacuum density and potential distributions (compare Sec. 17.2). Thus, applying first-order perturbation theory we may write

i ? (o)

Ead(a) ~-- ~

n x-

(o)

-~ Ve// x + -~ dx,

(18.9)

O0

where A is the area of the surface. Using (18.7) and (18.8) we have

(x

Ead(a) r'~ A ~'I'VB/c~

,1 10,

It is seen that the quantities in front of the integral are constants independent of a. On the other hand the integrand of (18.10) is independent of rs. Thus equation (18.10) is expressed in the form similar to Eq. (18.1) which gives the energy curve of Fig. 18.1. A similar scaling applies to diverse metal-alloy or alloy-alloy binding-energy curves which can be fitted to the form (18.4) using simple, analytical form (17.15) for the the electron density profile (Kiejna, 1987).

18.2

Universal binding energy curves

The universality of the binding-energy relation is not limited to the simple metals and to adhesion. The theoretical total cohesive-energy curves calculated for bulk metals as a function of interatomic separation can be also scaled into a universal function according to (18.1)-(18.3). The separation between atoms is expressed in terms of the Wigner-Seitz radius r w s = (3/(47rnA)) 1/3, where nA is the electron density in the atomic sphere. Thus, the scaled distance is 5

--

r w s

-

rwsm

ATF

(18.11)

where rWSm is the Wigner-Seitz radius corresponding to equilibrium spacing. The Thomas-Fermi screening length, ~TF, w a s determined using the equilibrium interstitial electron density (Moruzzi et al., 1978). A E is the cohesive energy at the equilibrium Wigner-Seitz radius r w s m . The scaled binding energies (Rose et al., 1981) are shown in Fig. 18.3. The smooth curve is of the form (18.4) with/3 = 1.16 which differs slightly from that used for adhesive-energy curves. Universal binding-energy-distance relationship can be found also for the atoms chemisorbed on a metallic surface. Fig. 18.4 shows the dependence of the energy of adion-substrate system on the adion separation from the jellium surface. The variety of shapes ranging from slowly varying function for low density alkalis (K, Rb, Cs) to a relatively strong-varying function for hydrogen, again, can be scaled (Smith et al., 1982) onto a single universal curve (Fig. 18.5).

18.2. B I N D I N G E N E R G Y CURVES

267

i

w

-0.2

>0 nLU Z uJ - 0 . 4 uJ :> o3 uJ -tO o r uJ ..J <:

o

I~

_

--

9 9

. -0.6

-

Mo

9

_

K Srn

0

-0.8

pa')exp(-pa')

41 +

{

_

Srn -

4fS(5d,6s 4f6(Sd,6s)

)3 2-

Cu

| -1.0

SCALED

I

I

I

I

0

2

4

6

CHANGE

IN W I G N E R

SEITZ

RADIUS,

a"

Fig. 18.3. Binding energy of bulk metals scaled into a universal curve. The distance is scaled with (18.11). Reprinted with permission from Rose et al. (1981). @1981 The American Physical Society.

W h a t is more important, it occurs that the resultant universal relations E*(5) for energy-distance dependence in adhesion, cohesion and adsorption as well as for diatomic molecules (Ferrante et al., 1983) fall to a high degree of accuracy into a one universal curve. Despite its intuitive appeal a screening length (18.3) cannot be used as a scaling length in the systems like diatomic molecules where it is ill defined. To avoid this difficulty, and to make the results more general, we may take the scaling length, l, as a free parameter, being measure of the distance over which the interatomic forces act. Now, if we know the universal relation E*(g), say relation (18.4) with/3 = 1, A E and 1 can be found from any two independent points determined (experimentally or theoretically) on the binding energy relation E(a). If, as before, A E is the equilibrium binding energy, then in order to account for the curvature of the binding energy the screening length l may be determined by the second derivative of binding energy at

268

CHAPTER

18.

SCALING

OF BINDING ENERGIES

2.0 Cs

0 >.. (.9

t

'

'

0.2

0.3

/

n-" - 2 . 0 I..U Z Lit 0

z

s Z r,n

-4.0

c) -6.0 Z 0

-8.0 J

0

i

0.1

I

0.4

SEPARATION (nm) Fig. 18.4. Total energy of the adion-substrate system versus adion distance from the surface of jellium (rs = 1.5). Redrawn with permission from Smith et al. (1982). @1982 The American Physical Society. the minimum:

1/2

d E*(a)

AE d2E(a)

l -

d5 2

(18.12) ~=0

da 2 a~a

m

The value of the second derivative of the scaled energy-distance curve is arbitrary and might be set equal unity at equilibrium. That is, we get d2E(a)]_ 1 l-

{AE

da2

a=a~

}1/2 (18.13)

The use of this form for l, allowed to maintain or to improve the accuracy of universal curves for adhesion, chemisorption and cohesion. Moreover, the resulting universal relations E*(g) for adhesion, chemisorption, cohesion and diatomic molecules were all the same showing that there is a single universal relation for all these diverse systems. This indicate the existing correspondence between metallic and molecular bond. The occurrence of universality of the binding-energy curves appears to have many implications on relationships between observables. Perhaps the most important one,

18.2. BINDING ENERG Y CURVES

269

-0.1 >,(3 Q:

-0.2

uu -0.3 z uJ

(3

z

1:3

z

u

-0.4

O

Si on rs = 2.0

El

H on r s = 2 . 0 7 (AI)

o

0 on r s = 2.0

H on r s = 2 . 6 5 (Mg)

-0.5

m

)(-9 Rb

0 -0.6 I.-<:

~7 Li

c~ -0.7

0

Na

<: u -0.8 v)

0

K

UJ ..J

onrs=l.5

C) Cs (5

-0.9

H

-1.0 -1

0

i

I

i

1

I

I

1

2

3

4

5

6

SCALED SEPARATION a"

Fig. 18.5. Scaled binding-energy curves for atoms chemisorbed on jellium surfaces. Reprinted with permission from Smith et al. (1982). @1982 The American Physical Society.

from the point of view of the material covered by this book, is the relation of surface and cohesive energies. From Eqs (18.1)-(18.3) we have

d2E(a)

ZXE 12

da 2 a=a

(18.14)

d52 5=0

m

For adhesion of two identical metal halves, the equilibrium binding energy AE, is the double surface energy (7. The above expression gives a slope of the force at the interface and can be expressed via the elastic stiffness constant Cl1, associated with the direction normal to the surface (Kittel, 1967). Thus,

2(7 d2Ead(5) ] 12

dg 2

Cll -

d

(18.15)

5=0

where d is the interplanar spacing. Similarly, for the cohesive energy, putting A E -

270

C H A P T E R 18. S C A L I N G OF B I N D I N G E N E R G I E S

Ecoh, and assuming a uniform dilatation of a bulk metal we have

E oh d: Ead( ) 12

- 127rBrws

d~ 2

(18.16)

5=0

where B is the bulk modulus. Dividing (18.15) by (18.16) we get

ZE~d(5)]

d2Ead(5) Ecoh

-1

da2

d~t2

Vii

1

a:0 2d 127rBrws

(18.17)

5=0

The ratio of the second derivatives is independent of the metal considered and can be evaluated from Figs 18.1 and 18.3 to get

47rr~sa- 1.7 Cll 2d ?'ws 3---B--Ecoh"

(18.18)

The term standing in front of Ecoh on the rhs of (18.18) is constant, within 20%, for

(eV/atom)

5.0 I

2.5 I

7.5

7.5

RU//x

10.0 V

E

0

Cr

0 If)

5.0

O4

Zr

MoX/x ~f

To

.Lr

5.0

,-e ,~;t

E O

Ti)~ Ni

O

,x'Au

m

>

AcJ~ ' x C.,u

"7 O

Ma . / "xTt /

b

/'

ID

N3

0

xpb

2.5

XAL

xBe

5.O

I0.0

0.82 Eco h(lO -12 .e rgs /o tom )

Fig. 18.6. Plot of the relation (18.19) between surface and cohesive energy of metals (straight line). The experimental data are denoted by the x 's. Reprinted with permission from Rose et al. (1983). @1983 The American Physical Society.

18.2. B I N D I N G E N E R G Y CURVES

271

a wide range of metals (Rose et al., 1983). Replacing the ratio of the elastic constants C l l / B , by the value averaged over the various crystal faces, one gets the relation between metal's surface and cohesive energies per atom (18.19)

47rrwsa ~- 0.82Ecoh

which resembles the empirical relation for the surface energy. Equation (18.19) is plotted in Fig. 18.6 as a straight line and compared with the experimental values for metals. As is seen the agreement of the experimental data with theory is rather good. By closing, we only would like to mention a possibility of simple prediction of the equation of state (i.e., p = p(9, T)) of metals and alloys from a knowledge of universal binding energy curves. As it was shown (Rose et al., 1984) the equation of state is given by

OE(a)

P-

0f~

=

-

(367rf~2)1/3/

d5

(18.20)

where p is the pressure, [~ the volume per atom. From (18.13) we have,

l-

127rrwsB

with A E , ryes and B, being metal's zero-pressure values of cohesive energy, WignerSeitz radius and bulk modulus, respectively. The values of p(f~) predicted from (18.20) compare very well with the measured numbers. Note, however, a slightly different form of E*(5) employed in the derivation of (18.20).

This Page Intentionally Left Blank

Appendix A

A.1

Fundamental constants

T h e f u n d a m e n t a l physical constants which are used to express different physical quantities are given in the Table A.1. Table A. 1

Quantity Electron mass Electron charge Planck's constant Speed of light Boltzmann constant

A.2

CGS m = 9.109534 e = 1.602189 4.803242 h = 1.054589 c = 2.997925 kB = 1.380662

xlO -28 g

SI • 10 -31 kg • 10 -19 C

• 10 -10 ESu •

-27 erg.s

x 101~ c m / s x 10 -16 e r g / K

• 10 -34 J.s x l0 s m / s x 10 .23 J / K

Atomic units

In describing different microscopic quantities we have widely employed the a t o m i c system of units t h r o u g h o u t . In this system the electronic mass m, the electronic charge e a n d the Planck constant h are taken to have unit value: m = h = e = 1. T h u s the unit of distance is the Bohr radius of the h y d r o g e n a t o m a0 = h 2 / r n e 2 = 1. T h e unit of energy is the hartree, which corresponds to the energy of an electron in the first Bohr orbit, Eh = e2/a0 9 T h e unit of t i m e is defined as t i m e required for an electron in the first Bohr orbit of a h y d r o g e n a t o m to describe one radian. These definitions are based on the electrostatic Gaussian system of units which is convenient for theoretical consideration of the microscopic properties of the elect r o m a g n e t i c systems. On the other hand, due to the close connection of the surface p h e n o m e n a with technical applications it is useful to know how to convert a t o m i c units into SI-units. 273

274

APPENDIX

A.

We may define atomic units starting from the SI units introducing the constant of the Coulomb interaction ~2 Z

e2 =

47re0

2.30712

x

10-28j.m

where eo - 107/(47rc 2) m - 3 k g - l s 4 A 2 . Then the Bohr radius is ao - h 2 / m e 2 and the unit of energy is Eh -- e2/ao. Thus to convert the SI units into atomic units one has to put m - h - c - 1. The physical quantities encountered in this text and their equivalents in electrostatic and SI systems are summarized below in the Table A.2. Table A.2

Quantity

Symbol

Length Energy

ao e2/ao

Surface energy Potential Electric field Surface charge Time

e2 / a 3o e/ao

liao / e 2

Electrostatic

SI

0.529177 • 10 - s cm 4.359815 • 10-11 erg 27.2116 eV 1.55692 • 106 e r g / c m 2 9.07682 • 10 -2 stVolt 1.71527 • 107 s t V o l t / c m 1.71527 • 107 s t C o u l / c m 2 2.41888 • 10 -17 s

0.529177 • 10 -1~ m 4.359815 • 1 0 - 1 8 J 27.2116 eV 1.55692 x 103 J / m 2 27.21158 V 5.14224 • 1011 V / m 57.21530 C / m 2 2.41888 • 10 -17 S

A.3.

A.3

CHARACTERISTICS

OF T H E E L E C T R O N

GAS

275

The quantities characteristic for the electron gas and screening

In this appendix, for a quick reference, we recapitulate some quantities for the homogeneous electron gas and screening. For the average electron concentration ~ - N/f~, the Wigner-Seitz density parameter in atomic units is: rs -

(

~

ao.

The Fermi wave number: kF -- (37r2~) 1/3 ao 1,

( 9 7 r ) 1/3 1 _ kF =

where a-

4)1/3 -

~

rs

1

1

ars ao

1_(9)1/3

0.52106,

a -

~

- 1.91916.

Fermi wave-length:

271 )~F = ~ F Fermi energy: EF--

h2k~

1

2m

2a2r 2

hartree.

Thomas-Fermi wave number: 4 F kTF ~-

3a 1 ~.1/2 ao t8

EF ]

Thomas-Fermi screening length: 1 ) ~ T F - kTF

1/2 rs a0. 3a

The frequency of plasma oscillations:

a3p

hartree/h.

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Appendix B

Planar average of the potential difference 5v(r) Let w ( r - Ri) be the pseudopotential at position r due to the ion at Ri and r be the potential at x due to a semi-infinite jellium occupying the half-space x < O. The potential difference ~v(r) is 5v(r) - ~

w(] r - Ri 1)- r

(B.1)

i

The planar average of (%(r) is

~v(x)-

(~v(r))-(

~

w( r - R i

]))- r

(B.2)

i

(~
where the brackets denote an average over the yz-plane. Following Lang and Kohn (1970) it is convenient to separate ~v(x) into two parts ~(x) - ~1(~)+ ~(x).

(B.3)

The first term

~Vl(X)-(

-Ze 2 ~ (,r z(x<0)

Ri 1 ) ) - r

(B.4)

due to the averaging over yz-plane can be regarded as a sum of contribution from the ionic charges, smeared out uniformly over the lattice planes and the contribution from the uniform electron gas. Note that now summation runs over the lattice planes. The second term in (B.3) comes from the repulsive region of the ion cores represented by the repulsive part WR of the pseudopotential

(~v2(x)- ~ (WR(Ir z(x<0) 277

Rz I)}.

(B.5)

A P P E N D I X B. T H E P O T E N T I A L D I F F E R E N C E

278

By application of the Gauss law, the field within the surface layer due to the uniform electron distribution and the positive ion sheet is determined and the line integral of this field gives for ~vl (x) the following periodic function

(~Vl(X) -- / 0,

[

(X > 0)

Ul(X -+-/d),

(B.6)

[-(/-4- 1)d < x < -ld]

where I - 0, 1, 2, 3 , . . . labels the lattice planes, and ul(x)=-2rr~e 2 x+dO

-x-

(B.7)

To evaluate 5v2(x) we employ the Ashcroft model potential for which

Ze 2 wn(I r - Rz l) - [ r -

Rz119(rc- { r - Rz l)

(B.8)

where rc is the cut off radius of the pseudopotential. Proceeding in a similar way like by evaluation of (B.7) the general form for ~v2(x) can be written as oo

5~(x) - ~

~ ( ~ + ld)

(B.9)

/=0 where

d

(B.10)

For nonoverlapping ion cores i.e., in the case when r~ <_ d/2

0,

(x > 0)

uz(x + ld),

[ - ( / + 1)d < x < - l d ]

(B.11) 5v2(x) -

For overlapping ion cores, when [ d < rc < 3d, we have

~(~)

O,

(rc - d/2) < x

~(~) + ~:(x + d),

[-d < 9 < ( , ~ - d/Z)] (B.12)

u2(x + ld) + u2[x + ( 1 - 1)d] +u2[x + (1 + 1)d],

[-(l + 1)d <__x <_ - l d ],

where 1 - 1, 2, 3, .... The surface-ion-layer relaxation process shifts the position of first plane of ions from x - - d / 2 to xx - (- 89+ A)d. This leads also to a modification of the expressions for ~Vl and ~v2 given above. For nonoverlapping ion cores the corresponding expressions

are: ~v),(x) - 5Vl,X(x) + ~v2,x(x)

(8.13)

279 where (~Vl,),(X) -- --27l'?~e2 [(X -~- d) 2 - x 2 0 ( x )

- 2d(x

-

x~)O(x

-

x~)],

(B.14)

and 5v2,~(x) - 21rfie2d (rr

I x - x~ [ ) O ( r ~ - I x - x~ I).

(B.15)

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Appendix C

Surface correlation energy for the Ceperley-Alder parameterizat ion The parameterized Ceperley-Alder correlation energy can be written as

CA

~C

0.1423 - - --

__

1 + 1.0529r~/2 + 0.3334r~

a

~

a

1+ p + q

=

~"

s

(C 1) "

W i t h this notation for the Smoluchowski trial density profile (Eq. (11.16)) the correlation part of surface energy yields:

~2 ~

=

_6_~ A

ly~l(y~+l) dy_

r s2

r8

1

y4 + y2 + 1 d y -

a[~l S

r

1 y4 § y2 § 1

/01

y5(y2 _

yll

1)dy

]

(P+q)Y+qdy

1 (Y -~- 1 ) ( y 4 -~- y 2 + 1) y 2 -iV-p y + q

jfo~y5(y

1 ) (y2P ++qpy ) Y ++ qqd y ] }

(c.2)

where/3 is a variational parameter and

d = 0.45 (322) 1/3 , ~

B-

1.125

(2)1/3 ,

~1 -- 2 - 1 / 6 .

The integrals in the first two square brackets yield:

~ 1 yll(y2 _~_1) 1

y4 + y2 + 1

dy -

y5 (y4 fo ~1 281

_ 1)dy

-

7.532134,

(c.3)

282

APPENDIX C. SURFACE CORRELATION E N E R G Y

1 y4 + y2 § 1 d y -

y5(y2 _ 1 ) d y - 3.646312.

(C.4)

The remaining integrals have to be evaluated numerically for a specific electronic density (rs).

Appendix D

Linear p o t e n t i a l a p p r o x i m a t i o n for a m e t a l surface In the linear potential model, the effective potential to be of the form

Yeff(x ) at the surface is assumed

Veff(x) - FxO(x)

(D.1)

where the field strength F is defined in terms of the slope parameter x F as F =

h2k~/xF (see Fig. D 1) The solutions of the SchrSdinger equation for this potential 2rn

"

"

/~Veff FERMI

~es

LEVEL

.----z- . . . . . . . .

~'"/ ,,," /I,

[

n

0

0

XF

~

89F

I{

= Fx O(x)

T

I . . . . . . . . . . I--

Aoo

/~lc,x X

Fig. D.1. Energy diagram for the linear potential at the metallic surface. The hatched region represents the jellium.

283

A P P E N D I X D. L I N E A R P O T E N T I A L

284

APPROXIMATION

are given by

_ ~ Asin[kx + 5(k)],

Ck(x)

[

forx<0

CkAi((),

(D.2)

forx>0

where A = - ( 2 / L ) 1/2 is the normalization constant, 5(k) is the phase shift. CkAi(~) is the solution of the following Airy differential equation d2r - ~r d~ 2

= 0

(D.3)

where ~ = ( x - E / F ) ( 2 F ) 1/3, E is energy and Ck is the normalization factor. The requirement of the continuity of the wave function and its logarithmic derivative at x = 0 allows to determine the phase shift factor 5(k) and the coefficient Ck to yield

1

5(k, xF) -- arc ctg

Ai'(-~0) ]

(D.4)

(~0)1/2 A i ( - ~ o )

and

Ck = - ( 2 / L ) 1/2 sin 5(k, XF)[Ai(-~o)] -1

(D.5)

where ~0 - (k2/k2F)(kFXF) 2/3 and Ai' (~) is the derivative of the Airy function (abramowit2 and Stegun, 1972). The limit x F = 0 corresponds to the infinite barrier model. For the linear potential model it is convenient to introduce the dimensionless coordinates y = kFx and q = k/kF. Then the slope parameter is YF = kFXF and the variable ~ is

-- (y -- q2yF)yF1/3 ,

and~o-q

2 2/3 YF

9

(D.6)

Denoting 1/3 A i ( - q 2~yF2/3) ec(q, yF) -- UF Ai' ( -'~2~~F )

such that ~(0)

_ 1/3 -- YF

(D.7)

Ai(O)

(D.8)

Ai'(O)

the phase shift may be written as

5(q, YF) -- arc ctg [

(D.9)

1

The electronic density can be expressed semi analytically by the integrals of the Airy functions:

n(y) = 1

1/01 dq(1 - q2) cos 2[qy + (~(q, YF)] 2 ' 1

- 3

~0

Ai2(~) dq(1 - q2)sin 2 5(q, YF) A i 2 ( ~ o ) ,

Y< 0 -

y>_o.

(D.10)

(D.11)

285 The other surface properties of interest: jellium surface position, electrostatic potential, surface dipole barrier and surface kinetic energy can be presented in terms of universal functions of the slope parameter YF by using known integral expressions involving the Airy functions (Abramowitz and Stegun, 1972; Sahni et al., 1977). From the charge neutrality condition (4.48) one gets for metal surface position Ya = k F a ,

371"

2

3 ~1

Y a -- -- --~- + -~ Y F -- -2 , u d q (1 - q2)

sin 25(q, YF) 2q "

(D.12)

Alternatively, applying the Sugiyama sum rule (7.2) we get Ya --

8

3

/o 1d q q a ( q ,

(D.13)

yf).

The surface dipole barrier can be written in the form

A r - ( 4-~) where

(

3

(D.14)

2 2

7r ~(0) +

/o1dq (11 +- q2)n2'~ n2q 2 ; '

(D.15)

-- ~ -

]i 1d q ( 1 -

q2)qsin25(q, yF).

(D.16)

J(YF) -- -4 1 - sy a +-~ and

K(yF) The electrostatic potential: For y < 0:

4

37c

2

- ~ ( J ( Y F ) + --~-Y + ~ O ( y a

kF

-- y ) +

1 2

-~y( Ya -- y)O(y - Ya)

3/ol

cos 25(q, YF) sin 2 qY q2

3 fo 1

sin 25 (q, YF) sin 2qy

+-4

+ -~

dq (1 _ q2) dq (1 - q2)

q2

)

(D.17)

For y > 0: 2 ( y - ya ) 2 (F)( ya -- y) -~- -~--~ kF

]~F

YF2/3 jr01 dq (1 - q2) sin 2 5(q, YF) [2(2Ai2(~) -2~Ai'2(() - Ai(()Ai'(()].

(D.I8)

286

APPENDIX D. LINEAR POTENTIAL APPROXIMATION

The surface kinetic energy

0"~1) 1 [1 + -80(3/01 dq q~(q, YF) ~01dqq3~(q, YF))] k~ 160~ -

-

,

(D.19)

and a~2)

1 K(yF)

k~ - 67r2

YF

(D.20)

Appendix E

Finite linear potential model The linear potential model of Appendix D is a special case of the following effective potential at the surface

V~ff(x ) = Fx[O(x) - O(x - b)] + W O ( x - b)

(S.~)

where F is the field strength and W is the barrier height (see Fig.E.1). Note also that this potential generalizes also the step-potential model of Section 4.2 for which V~ff (x) = W O ( x - b ) . Thus the barrier height can be written in terms of the parameter /3 where

82 -- ( h2k2F ) / w -

(E.2)

The electronic wave functions generated by this potential are

g,k(x)-

A sin[kx + 5(k)],

for x _< 0

BkAi(~) + CkBi(~),

for 0 < x < b

Dk e x p ( - a x ) ,

for x > b,

(E.3)

where the constant A = - ( 2 / L ) 1/2 is determined by the normalization condition, k - ~ v/2mE, ~ - 1 v / 2 m ( W _ E), ~ - (x - E / F ) ( 2 F ) 1/3 and E is the energy. Ai(~) and Bi(~) are the linearly independent solutions of the Airy differential equation. The phase-shift factor 5(k) and the coefficients Bk, Ck and Dk are determined from the continuity condition for the wave function and its logarithmic derivative at x = 0 and x = b, to yield: ~(k) - ~rc ctg

Bk--A

Ck - A

1

N(-~0) ]

(~0)~/2 M ( - ~ 0 )

~o ) 1/2 A(-(o)

~0 )l/2X(~b) Y((b)'

A(-(o)

287

'

(E.4)

(E.5)

(E.6)

A P P E N D I X E. F I N I T E L I N E A R P O T E N T I A L M O D E L

288

V~. _--~-A. . . . . . . . . .

FERMI LEVEL

#/*

..... .

.

.

.

/"

,..."~/ I I

I

I{

I

', ',

.

0

a

1 ~

_

l--

xF b

x

Fig. E.1. Energy diagram for the finite-linear potential at the surface.

~0 ) 1/2 Dk--A

A(-~0)

M(~b) exp[(~b + ~0)~/2],

(E.7)

where

X(~b) - Ai'(~b) + ~/2Ai(~b),

(E.S) (E.9)

M(~) = Ai(~)- Bi(~)X(~b)/Y(~b),

(E.10)

N(~) = Ai'(~) - Bi'(~)X(~b)/Y(~b),

(E.11)

A ( - ( o ) - (oM2(-~o) + N e ( - ( o )

(E.12)

where ~0 - (k2/k2F)(kFXF) 2/3, ~b - ~Fb(kFXF) - 1 / 3 - ~0 and the primes denote the derivatives of the Airy functions. The remaining parameter, the jellium edge position, is determined either from the total charge neutrality condition or by the Sugiyama phase-shift sum rule (Section 7.1). Introducing the dimensionless variables, y = kFX and q = k / k F , all surface properties of interest can now be written in terms of universal functions of barrier height and slope parameters (Sahni et al., 1978).

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Index Dirichlet, 204 Neumann, 204, 206 boundary conditions, 86 boundary conditions Born-von-Karman, 58 periodic, 61, 73 Bravais lattice, 3, 5, 27, 65, 71 Budd-Vannimenus theorem, 99,127, 142, 143, 145, 174, 241

adatom, 224-228, 231-234, 242, 243 adatom energy level shift, 227 ionization of, 227, 230 adhesion, 199, 245, 246, 264, 266-269 adhesion commensurate, 252 ideal, 245 incommensurate, 252 adsorption, 11, 15, 16, 20, 27, 199,223, 224, 226, 227, 229, 231, 233, 234, 237, 242, 267 alkali atoms, 223, 227, 231, 232, 237, 242 alkali metals, 136, 137, 188, 191, 193, 194, 199, 223, 224, 228, 237 alloy surface, 198, 201,202 alloys, 147, 187-194, 196, 199 alloys alkali-metal, 194, 202 binary, 191,194 disordered, 192, 198 ordered, 194, 198 Ashcroft's pseudopotential, 111, 117, 133, 134, 151, 188, 193, 201, 252, 278 atomic units, 55, 75, 77, 100, 168, 180, 228

cell

Bravais elementary, 4 elementary, 4 symmetric, 6

unit, 4, 5, 11, 17 charge neutrality condition, 63-65, 74, 105, 127, 134, 142, 145, 165, 172, 177, 228, 240, 243, 285, 288 chemical potential, 34, 56, 79, 117, 123, 124, 157, 158, 188, 190, 256, 261 chemisorption, 223, 233, 234, 268 clusters, 213-215,217-219 cohesion, 245, 267, 268 cohesive energy, 252, 266, 269, 271 contact potential difference, 190 correlation (Coulomb) hole, 70, 162,165, 168 correlation energy, 68, 72, 80-82, 100, 105, 125, 131, 146, 150, 151, 189, 197, 281 correlation energy surface, 148 Coulomb interaction, 62, 65, 80, 81, 113, 227, 274 crystal structure, 4, 6, 10, 110,115, 119

binding energy, 49, 219, 223, 246, 253, 263, 267 binding energy curves, 252,263, 266, 268 curves universal, 266, 271 boundary condition, 55, 86, 102, 172 boundary condition 299

300 crystallographic plane, 7, 8, 11, 14, 19, 21,160, 198, 228 Debye temperature, 27-31 Debye temperature surface, 29, 30 Debye-Waller factor, 28 density functional theory (formalism), 77, 80, 81, 85, 103, 113, 119, 125, 142, 157, 178, 187, 192, 199, 207, 227, 247 density of states, 56, 91, 94, 96, 206, 207, 233, 241 depolarization, 225, 226, 231 dipole moment, 112,210,211,224-227, 233, 237 Dirichlet boundary condition, 53 discrete-lattice corrections, 73, 90, 107, 110, 112 dispersion forces, 245, 253 electron affinity, 213-215,217, 226 electron density parameter, 55, 72,120, 138, 146, 150, 151, 187-189, 196, 201,213, 232, 252 electron density profile, 59, 64, 87, 89, 114, 126, 129, 131, 141, 143, 151, 172, 183, 193, 199, 228, 242, 248, 265, 266 electron density profile exact, 114, 143 self-consistent, 141,177, 249 Smoluchowski's, 146,147, 200,281 Thomas-Fermi, 85 trial, 87, 144, 197 variational, 141,249 electronegativity, 190, 191 electrostatic potential, 53, 71, 78, 79, 81, 85, 88, 97, 99, 100, 104, 107, 115, 123, 124, 126-128, 141, 143, 147, 151, 157, 159, 167, 172, 197, 214, 241, 248, 253-255, 261,285 electrostatic theorem, 97 entropy, 33, 47 equilibrium shape of crystal, 36, 38-41

INDEX

exchange (Fermi) hole, 69, 70,161-163, 165, 167, 168 exchange charge, 67, 69, 165 exchange charge density, 67, 69, 162, 163, 165 exchange energy, 67, 68, 72, 79, 81, 82, 88, 105, 125, 131 exchange potential, 167, 169 exchange-correlation energy, 103, 105, 113, 124, 142, 147, 149, 155, 167, 184, 248, 253 exchange-correlation hole, 90, 142,156, 162, 167, 168 exchange-correlation potential, 83, 8890, 120, 131, 132, 165, 168, 207, 214 facet, 37-39, 41,136 Fermi energy, 54, 57, 61, 62, 65, 75, 112, 114, 125, 204, 206, 207, 219, 246, 256, 275 Fermi level, 61, 74, 210, 226,227, 233 Fermi momentum, 54, 205 Fermi sphere, 54 field emission, 135 field evaporation, 181,182, 185 finite potential well, 58, 60, 61 fracture, 245, 256, 257 free electrons, 53, 54, 58, 60, 68, 70, 204, 206 free energy, 33, 34, 36, 37, 47 Friedel oscillations, 59, 87, 89, 93, 94, 134, 142-144, 172, 174, 209, 210, 249, 265 Friedel sum rule, 96 geometric surface, 23, 63, 65, 74, 75, 154, 160, 171,179 gradient expansion, 81, 82, 104, 114, 142, 151,249 ground state energy, 54, 62, 68, 81, 83, 112, 123, 128, 142, 199, 247 Gurney effect, 226 Hartree energy, 147 Hartree potential, 81, 83, 88

INDEX

Hartree-Fock approximation, 65, 68,161 Hartree-Fock equations, 66, 67, 162 Hellmann-Feynman theorem, 95, 97 ideal metal, 115 image charge, 70, 153, 162, 165 image force, 69, 70, 226 image plane, 153-155, 159, 178, 179, 181 image plane position, 154, 156, 159-161, 175, 180 image potential, 90, 153-156,165,167169, 208 infinite barrier model, 58, 64, 75, 76, 154, 163, 165, 207, 284 infinite potential barrier, 53, 58, 65, 107, 162, 163 internal energy, 33, 46, 47 ionic radius, 237 ionization energy, 125, 127, 226 ionization potential, 213-215, 217 jellium edge, 63, 65, 96, 126, 143, 144, 155, 159, 160, 163, 165, 175, 288 jellium model, 62, 65, 88, 93, 97, 104, 107, 112, 114, 115, 118, 120, 122, 124, 142, 151, 159, 199, 202, 207, 211,217, 248, 264 jellium model spherical, 217 jellium-on-jelliummodel, 227, 230,237, 238, 240 Kohn-Sham equations, 82, 83, 90, 131, 143, 155, 169, 172, 183, 214, 232 Koopmans theorem, 125 Koopmans work function, 125, 127 lattice bcc, 3, 6, 9, 10, 110 Bravais, 3, 5, 13, 14, 27 fcc, 3, 4, 9-11,110, 250 hcp, 9, 10, 110, 250 hexagonal, 3, 13, 137

301 two-dimensional, 13, 14, 21, 24, 26, 27, 224 two-dimensional reciprocal, 15 lattice constant, 5, 6, 10, 20, 110, 196, 252 lattice parameter, 5, 7, 9, 10, 41, 42, 187 lattice plane, 9, 10, 14, 20, 250, 258, 277, 278 lattice relaxation, 21,147, 183, 211 lattice relaxation field induced, 181 Laue condition, 27 linear potential model, 283, 284, 287 linear potential model finite, 287 linear response, 100, 156,157, 159, 172, 174 local density approximation (LDA), 84, 88-90,105,131,142,148,155, 159, 160, 207, 208 local density of states, 91, 93, 94, 226, 227, 233 low energy electron diffraction (LEED), 21, 27, 184 Madelung energy, 109, 116, 188 magic numbers, 213, 218 melting, 31 melting temperature, 43, 44 Miller indices, 7-10, 14, 16, 29, 37, 38, 40, 135 Miller-Bravais notation, 8 monolayer, 16, 19, 137, 223, 224, 228, 230, 239 multilayer relaxation, 24, 25 packing density, 11, 12, 21,137 perturbation theory, 107, 109,112,128, 178, 188, 248, 266 phase shift, 60, 89, 95, 96, 107, 172, 240, 241,284, 287 phase shift sum rule, 96, 172, 240, 288 phase transition, 27, 44 photoemission, 60, 231

302 physical surface, 63, 65, 74, 154 plasma frequency, 246 pseudopotential, 73, 90, 108, 112, 113, 115, 116, 128, 133, 134, 139, 140, 151, 183, 188, 192, 197, 211,248, 277, 278 pseudopotential core radius, 111,115,134, 139, 188, 192, 196, 201,252 quantum size effect (QSE), 30, 61,203, 205, 209, 211,213 quantum size effect (QSE) non-oscillatory, 203, 204 oscillatory, 203, 206, 207, 211 reciprocal lattice, 6, 9, 15, 26, 27 reciprocal lattice vector, 10, 26 response function, 177, 259, 260 saturated image barrier, 155 screening charge, 153, 171,179-181 second harmonic generation, 178 self-consistent method, 83, 87, 88, 183 single crystal, 3, 19-21, 27, 36-38, 40, 41,136, 137, 185, 237 size effect, 30, 42, 43, 203 Slater potential, 165, 167, 169 slope parameter, 165, 283-285, 288 Smoluchowski rule, 137 solid solution, 189, 193, 194, 200 Sommerfeld's model, 53, 58, 73, 103, 125 square potential barrier, 63, 65, 207 stability condition, 117, 134, 139, 188 stabilized-jellium model, 73, 115, 118, 134, 138, 139, 145, 159, 161, 162, 175, 179 strain, 48 structure bcc, 7, 21,135 close-packed, 3, 7 fcc, 7, 11, 13, 20, 21,135 hcp, 3, 8, 11, 12, 135 structureless pseudopotential, 115, 118

INDEX

substrate, 137, 194, 199, 203, 211 sum rules, 95-97, 100, 240, 242 surface barrier, 75, 155, 170 surface charge density, 127, 172 surface density of states, 96 surface dipole barrier, 123-125,131,134, 224, 229, 285 surface double layer, 73 surface energy, 36, 38, 41, 42, 44, 45, 49, 73, 75, 100-103, 107, 109, 111, 114, 118, 120, 122, 128, 136, 142, 147, 150, 151, 183, 184, 187, 190-192, 194, 199, 201, 202, 211, 243, 246-248, 256,257, 259, 261,269, 271 surface energy classical cleavage, 109 correlation, 281 electrostatic, 109, 121 exchange-correlation, 104, 105,114, 122 kinetic, 73, 104, 114, 285, 286 pseudopotential, 110,111,151,183, 197 surface excess, 46, 47 surface free energy, 36, 37, 40, 46-48 surface reconstruction, 16, 20, 21, 27, 30, 119, 122 surface relaxation, 20, 21, 24, 27, 30, 211 surface segregation, 20, 187, 198, 199, 201,242 surface segregation criterion, 199, 200 surface stress, 27, 42, 45, 48, 49, 103, 119-122 surface tension, 45, 48 surface virial theorem, 101 thermionic emission, 60, 135 Thomas-Fermi method (model), 78, 79, 82, 85, 87, 89, 143, 144 Thomas-Fermi screening length, 86,143145, 260, 263, 264, 266, 275 Thomas-Fermi-Dirac approximation, 80, 87

INDEX

303

translation vectors, 3, 4, 7, 13, 15-17, 26

virtual crystal approximation, 188, 192, 196

ultra-high vacuum, 136, 223 uniform electron gas, 71, 78, 81, 82, 88, 101, 105, 163, 241, 256, 261, 277 uniform positive background, 62, 71, 72, 107, 109, 128, 133, 151, 160, 213, 227, 230 unit cell, 4, 5, 11, 17, 211 unit cell area of, 16 conventional, 4, 5, 11 primitive, 4-6 primitive volume of, 5 Wigner-Seitz, 6 two-dimensional, 20 volume of, 5, 12

Weyl theorem, 204 Wigner-Seitz cell, 6, 23, 56, 71, 115, 116, 134, 190 Wigner-Seitz polyhedron, 71 Wigner-Seitz radius, 55, 56, 187-189, 194, 201,202,266, 271 work function, 61, 100, 103, 107, 123125, 127, 128, 131, 134-138, 140, 147, 151, 187, 190, 191, 193, 194, 198, 208, 211, 215, 217, 219, 224, 226, 227, 229, 231,237, 239 work function change due to adsorption, 227, 229, 242 face-dependent, 128, 129,134-137, 140 mean, 135-137, 140 minimum, 229, 231,237, 238 polycrystalline, 133, 135 Wulff's constant, 38, 39 Wulff's theorem, 38, 40

van der Waals forces, 245, 253, 258 Vannimenus-Budd sum rule, 100 Vegard's law, 187-189 vibrations of atoms, 28-31 vicinal plane,40, 41 virial theorem, 95, 100-102, 122, 260