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 00ej=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~ = 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

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~ 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 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) , 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 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

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 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

,._,..



"-" ,.~ <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 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


.. (.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.-