METHODS OF CALCULATING THE ELECTRONIC AND ATOMIC STRUCTURES OF INTERFACES

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METHODS OF CALCULATING THE ELECTRONIC AND ATOMIC STRUCTURES OF INTERFACES

A. Sutton

To cite this version:

A. Sutton. METHODS OF CALCULATING THE ELECTRONIC AND ATOMIC STRUC- TURES OF INTERFACES. Journal de Physique Colloques, 1985, 46 (C4), pp.C4-347-C4-359.

�10.1051/jphyscol:1985438�. �jpa-00224689�

JOURNAL DE PHYSIQUE

Colloque Ci, supplément au n°1, Tome 46, avril 1985 page C4-347

METHODS OF CALCULATING THE ELECTRONIC AND ATOMIC STRUCTURES OF INTERFACES

A.P. Sutton

Department of Metallurgy and Science of Materials, Oxford University, Parks Road, Oxford 0X1 SPU, U.K.

Résumé - Des méthodes pour calculer les structures électroniques et atomiques des parois, sont décrites. On donne une introduction aux pseudopotentiels et des méthodes LCAO. Des méthodes pour calculer la structure électronique d'un paroi d'une certaine structure atomique, sont considérées. On y discute la possibilité des calculations d'énergie totale, où les structures atomiques et électroniques sont calculées simultanément.

Abstract - Methods of calculating the electronic and atomic structures of interfaces are described. An introduction to pseudopotentials and LCAO methods is given. Methods of calculating the electronic structure of an interface with a given atomic structure are considered. The feasibility of total energy calculations, in which the atomic and electronic structures are calculated simultaneously, is discussed.

1 - INTRODUCTION

It is well known that the atomic and electronic structures of interfaces are intimately coupled. Coulomb interactions between ion cores in a condensed phase are moderated by the valence electron distribution which responds virtually instantaneously when ions are displaced or a different ionic species is

introduced. In a metal each ion carries with it a neutralizing cloud of valence electron charge forming a neutral 'pseudo-atom'. Pseudo-atoms can be treated as single entities in those metallic systems where linear screening is valid, and their interaction is then described by an effective pair potential. But in other materials the treatment of the valence electron distribution and its response has to be more elaborate. This paper is an introduction to the theoretical methods used to study the electronic and atomic structures of interfaces involving such materials. Much of the quantitative progress in this area is founded on pseudopotentials and, in section 2, we give a brief account of the principles underlying them. A great deal of physical insight into the electronic structure of interfaces (and many other defects) has been provided relatively easily by the empirical tight binding method and other LCAO (linear combination of atomic orbitals) techniques. This method is described and appraised in the second part of section 2. With this background, we proceed, in section 3, to describe methods that have been used to calculate the electronic structure of interfaces of known (or presumed!) atomic structure. Section 4 contains a very brief account of the methods that have been used to calculate the total (internal) energy of

interfaces. These are the methods which lead (at least in principle) to the minimum energy atomic configuration, with the electronic structure of the interface taken fully into account. In section 4 emphasis has been placed on semi^empirical methods because of their greater ability to deal with large numbers of atoms (e.g. for grain boundary calculations).

2 - FORMULATIONS OF THE ELECTRONIC STRUCTURE PROBLEM

The Hamiltonian for a system of interacting electrons and atomic nuclei consists of a sum of three terms. The first, Hj,, is the sum of nuclear kinetic energy Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985438

C4-348 JOURNAL DE PHYSIQUE

o p e r a t o r s and Coulomb i n t e r a c t i o n s . The second, He, i s t h e sum of e l e c t r o n i c k i n e t i c energy o p e r a t o r s and Coulomb i n t e r a c t i o n s , and Gn i s t h e sum of

e l e c t r o n - n u c l e a r Coulomb i n t e r a c t i o n s . S i n c e e l e c t r o n i c motion i s s o much f a s t e r t h a n n u c l e a r motion, e l e c t r o n s a d j u s t t o a ground s t a t e c o n f i g u r a t i o n f o r any p a r t i c u l a r s e t of n u c l e a r p o s i t i o n s . T h i s e n a b l e s u s t o s e p a r a t e t h e e l e c t r o n i c and n u c l e a r motions, and t h e Hamiltonian f o r t h e e l e c t r o n s becomes He + ^Hen

(Born-Oppenheimer approximation). A f u r t h e r s i m p l i f i c a t i o n f o l l o w s from r e c o g n i z i n g t h a t v a l e n c e e l e c t r o n s p l a y t h e dominant r o l e i n chemical bonding.

Core e l e c t r o n s a r e assumed t o be u n a f f e c t e d by s u r r o u n d i n g atoms and t h e y a r e t h e r e f o r e a s s i g n e d t o t h e i r i s o l a t e d atomic s t a t e s . Accordingly, t h e problem i s reduced t o s o l v i n g f o r t h e v a l e n c e e l e c t r o n i c s t a t e s i n t h e f i e l d of o t h e r v a l e n c e e l e c t r o n s and t h e i o n c o r e s .

A s i m p l i f i c a t i o n of t h e many e l e c t r o n problem has been achieved by t h e d e n s i t y f u n c t i o n a l formalism, 111. I n t h i s formalism i t is i s shown t h a t t h e ground s t a t e t o t a l energy of t h e f u l l many e l e c t r o n problem can be e x p r e s s e d a s a unique f u n c t i o n a l of t h e e l e c t r o n d e n s i t y , p ( r ) . The u t i l i t y of t h i s f o r m a l r e s u l t i s l i m i t e d by t h e f a c t t h a t we cannot g i v e an e x a c t f u n c t i o n a l f o r t h e exchange and c o r r e l a t i o n energy of a n i n t e r a c t i n g e l e c t r o n g a s , Ex, [p(f)]. Indeed,

f i n d i n g ~ , ~ [ p ( f ? ] i s e q u i v a l e n t t o s o l v i n g t h e f u l l many e l e c t r o n problem.

N e v e r t h e l e s s , i f p ( r ) i s s u f f i c i e n t l y s l o w l y v a r y i n g we can invoke t h e l o c a l d e n s i t y f u n c t i o n a l approximation and w r i t e

where E~~ ( p ( f ) ) i s t h e exchange and c o r r e l a t i o n energy f u n c t i o n p e r e l e c t r o n of a uniform e l e c t r o n gas of d e n s i t y p ( r ) . Kohn and Sham /1/ showed t h a t e q u a t i o n (1) r e d u c e s t h e many-electron SchrGdinger e q u a t i o n t o a o n e - e l e c t r o n e q u a t i o n :

where

V . . r o ( r ) l i s an e f f e c t i v e one-electron p o t e n t i a l :

pion (L) i s t h e i o n i c charge d e n s i t y . VXc ( p ( r ) ) i s t h e exchange- c o r r e l a t i o n p o t e n t i a l , which i s r e l a t e d t o %, ( p ( L ) ) by

E q u a t i o n s (2-4) a r e s o l v e d i t e r a t i v e l y u n t i l t h e e l e c t r o n d e n s i t y d i s t r i b u t i o n , g i v e n by e q u a t i o n s ( 2 ) and (31, i s c o n s i s t e n t w i t h t h e o n e - e l e c t r o n p o t e n t i a l , g i v e n by e q u a t i o n ( 4 ) . Such c a l c u l a t i o n s a r e d e s c r i b e d a s s e l f - c o n s i s t e n t . One r o l e of Vxc i s t o e l i m i n a t e t h e s e l f - i n t e r a c t i o n of t h e e l e c t r o n s c o n t a i n e d i n t h e f i r s t term of e q u a t i o n ( 4 ) , /2/. A number'of forms f o r Vxc ( p ) have been t e s t e d , / 3 / .

As w i l l be d i s c u s s e d i n s e c t i o n 4, t h e l o c a l d e n s i t y f u n c t i o n a l approximation ( e q u a t i o n (1) ) has been v e r y s u c c e s s f u l a t d e s c r i b i n g ground s t a t e p r o p e r t i e s of v a r i o u s s o l i d s , such a s t h e t o t a l i n t e r n a l energy. However, q u i t e s e v e r e

d i s c r e p a n c i e s w i t h experiments have been found f o r e x c i t e d s t a t e p r o p e r t i e s . For example, band gaps a r e normally u n d e r e s t i m a t e d by -50%. T h i s may i n d i c a t e t h a t t h e r e a r e s i g n i f i c a n t many body e f f e c t s which a r e n o t t r e a t e d a d e q u a t e l y by t h e l o c a l d e n s i t y approximation.

In general the one electron energies, EL, obtained self-consistently from

equations (2-4) are meaningful only if they are occupied states because only then do then contribute to the electron density and hence to the effective one-electron potential used to calculate them self-consistenly. When an electron is promoted to a higher energy state the excitation energy is determined not only by the change in the single particle state but also by the reaction of the system as a whole to the excitation. The latter component is not significant in the case of a spatially extended state, but it can become very important for localised states, such as those bound to defects.

In the all-electron approach the eigenvalues and wave functions of all the

electrons, including the ion core electrons, are calculated. Some of the methods with this approach use muffin tin potentials such as the KKR /4/, APW /5/ and LMTO /6/ techniques. A particularly efficient all-electron method for obtaining one-electron eigenvalues, called the SCF-Xa-SW method, has been developed by Johnson and co-workers /7/ and has appeared prominently in the literature on temper embrittlement in recent years /8/. This method is very similar to muffin- tin approaches since the potential is spherically arveraged within each atomic sphere and spatially averaged between them. The spherical and spatial averaging may be expected to be a good approximation for the potential in close-packed solids, such as pure metals. However, to study the formation of directional bonds and charge densities it is desirable to remove this constraint on the potential. In addition, total energies calculated by the SCF-Xa method are very inaccurate because of the use of the muffin-tin approximation. This has led to some spectacular failures of the method to calculate the shapes of simple molecules such as H20 and NH3. Recent LMTO calculations /6/ drop the muffin tin approximation and give very good results for these two molecules.

Pseudopotential formulations of the Hamiltonian do not suffer from shape constraints on the potential, and they underpin much of the theoretical work on interfaces.

Pseudopotential Formulations

Pseudopotentials were introduced to replace the all electron eigenvalue problem (i.e. core and valence wavefunctions treated on an equal footing) by a simpler eigenvalue problem which applied to the valence subspace only. In the Philips- Kleinman prescription /9/ this was achieved with use of the orthogonalized plane wave (OPW) method of bandstructure /lo/, in which plane waves were combined with Bloch sums of core electron wave functions in such a way that each basis function, x., was orthogonal to the core states. The Hamiltonian, Hps

wjth these basis functions as its eigenfunctions and the valence electron energies, E., as its eigenvalues differed from the all electron Hamiltonian:

J

vH(fl) and Vxc(p(fl)) are the inter-electronic Coulomb and exchange-

correlation potentials for the valence electrons only (V,, is expressed here in the local density approximation). Wps (fl) is the total pseudopotential and represents the residual interaction between the valence electron and the ion cores after orthogonalization to the core states. The effective potential seen by the valence electron is then

C4-350 JOURNAL DE PHYSIQUE

which i s t h e e f f e c t i v e p o t e n t i a l r e f e r r e d t o i n e q u a t i o n ( 4 ) . W (r) i s o f t e n w r i t t e n a s a sum of i o n i c p s e u d o p o t e n t i a l s , which i n generay a r e a n g u l a r momentum dependent, c e n t r e d a t s i t e s &,

A

where P 1 , i s t h e l t h a n g u l a r momentum p r o j e c t i o n o p e r a t o r w i t h r e s p e c t t o t h e c e n t r e a t I&. I o n i c p s e u d o p o t e n t i a l s t h a t are e x p r e s s e d a s a n g u l a r momentum dependent a r e termed ' n o n l o c a l ' , whereas ' l o c a l ' i o n i c p s e u d o p o t e n t i a l s r e p r e s e n t an a v e r a g e 1-independent form, Vps (f):

S e l f - c o n s i s t e n t c a l c u l a t i o n s of t h e v a l e n c e e l e c t r o n i c s t r u c t u r e a r e based on t h e p a r t i t i o n i n g of t h e e f f e c t i v e p o t e n t i a l r e p r e s e n t e d i n e q u a t i o n (8). The s c r e e n i n g , r e p r e s e n t e d by t h e l a s t two terms i n e q u a t i o n ( 8 ) , is c a l c u l a t e d s e l f c o n s i s t e n t l y from t h e v a l e n c e e l e c t r o n d e n s i t y p ( r ) i n t h e system of i n t e r e s t , whereas t h e i o n i c p s e u d o p o t e n t i a l i s f i x e d from t h e o u t s e t f o r a r e f e r e n c e system

( o f t e n t h e p e r f e c t c r y s t a l o r i s o l a t e d i o n ) . Various approaches have been

developed f o r f i n d i n g ' t r a n s f e r a b l e ' i o n i c p s e u d o p o t e n t i a l s , some of which a r e n o t based on c o r e o r t h o g o n a l i s a t i o n . Before we d i s c u s s some of t h e s e we mention t h e e m p i r i c a l p s e u d o p o t e n t i a l approach which is perhaps c l o s e s t t o t h e t i g h t b i n d i n g Hamiltonian.

I n t h e e m p i r i c a l p s e u d o p o t e n t i a l approach /11/ t h e e f f e c t i v e p o t e n t i a l , e q u a t i o n ( 8 ) , is w r i t t e n a s a sum of atomic p s e u d o p o t e n t i a l s which a r e r e p r e s e n t e d by j u s t a few terms i n a F o u r i e r expansion. The c o e f f i c i e n t s a r e a d j u s t e d t o a g r e e w i t h some e x p e r i m e n t a l l y determined f e a t u r e s of t h e energy bands. S i n c e one f i t s t h e s c r e e n e d i o n i c p s e u d o p o t e n t i a l s i n t h e r e f e r e n c e system t h e s c r e e n i n g i s f i x e d and hence s e l f - c o n s i s t e n c y i s o u t s i d e t h e scope of t h i s approach. Only a few terms a r e needed i n t h e F o u r i e r expansion of t h e s e p s e u d o p o t e n t i a l s , t o o b t a i n good f i t s t o t h e i n t e r b a n d t r a n s i t i o n e n e r g i e s . However, n e a r a d e f e c t t h e p o t e n t i a l cannot be r e p r e s e n t e d by j u s t a few terms i n a F o u r i e r expansion, and a l l f i x e d - s c r e e n i n g t h e o r i e s a r e of d o u b t f u l v a l i d i t y where t h e r e i s a d r a m a t i c change i n t h e e l e c t r o n d e n s i t y s u c h as a t s u r f a c e s and v a c a n c i e s .

A l a r g e number of s e l f - c o n s i s t e n t s u r f a c e and i n t e r f a c e c a l c u l a t i o n s have been c a r r i e d o u t w i t h u s e of semi-empirical i o n i c p s e u d o p o t e n t i a l s (12). The approach i s t o assume a f u n c t i o n a l form f o r t h e i o n i c p s e u d o p o t e n t i a l , e i t h e r i n r e a l s p a c e o r wave-vector s p a c e , w i t h a number of f r e e parameters which a r e a d j u s t e d t o reproduce e i t h e r t h e observed f r e e i o n term v a l u e s o r t h e p e r f e c t c r y s t a l band s t r u c t u r e . The f i t t i n g procedure can be performed w i t h a r b i t r a r y forms f o r t h e p s e u d o p o t e n t i a l i n t h e c o r e and t h e u s u a l c h o i c e is t o make t h e p o t e n t i a l ' s o f t '

( i . e . f i n i t e and smooth) i n t h e c o r e region. The r e a s o n f o r t h i s c h o i c e i s t h a t a r e l a t i v e l y s m a l l p l a n e wave b a s i s s e t i s needed f o r convergence of t h e charge d e n s i t y w i t h s o f t p o t e n t i a l s . The v a l e n c e e l e c t r o n c h a r g e d e n s i t y is c a l c u l a t e d from e q u a t i o n (3) w i t h t h e assumption t h a t t h e pseudo-wave f u n c t i o n s (Xj i n e q u a t i o n ( 7 ) ) a r e t h e r e a l v a l e n c e wave f u n c t i o n s . But, a l t h o u g h t h e v a l e n c e e l e c t r o n e i g e n v a l u e spectrum may be f i t t e d a c c u r a t e l y by a semi-empirical i o n i c p s e u d o p o t e n t i a l t h e r e i s no g u a r a n t e e t h a t t h e valence e l e c f r o n wave-functions a r e a c c u r a t e l y reproduced f o r t h e r e f e r e n c e system. I n a d d i t i o n , i n a t t a i n i n g s e l f - c o n s i s t e n c y s o f t - c o r e p s e u d o p o t e n t i a l s can s u f f e r from o v e r p e n e t r a t i o n of t h e c o r e r e g i o n by t h e v a l e n c e e l e c t r o n d e n s i t y when t h e i o n i s p l a c e d i n a n environment d i f f e r e n t from i t s r e f e r e n c e system. Hard-core p s e u d o p o t e n t i a l s ( s t r o n g l y r e p u l s i v e a t s m a l l L ) , on t h e o t h e r hand, e n s u r e minimal c o r e p e n e t r a t i o n

f o l l o w i n g changes of environment and t h e r e f o r e have a g r e a t e r t r a n s f e r a b i l i t y . These f a c t o r s have l e d t o t h e r e c e n t development of 'norm-conserving' i o n i c p s e u d o p o t e n t i a l s /13/ which g i v e t h e t r u e v a l e n c e e l e c t r o n wave f u n c t i o n s o u t s i d e a core r a d i u s , re, a s w e l l a s t h e c o r r e c t valence e l e c t r o n e i g e n v a l u e s . T h e i r d e f i n i t i o n makes no r e f e r e n c e t o c o r e e l e c t r o n s t a t e s . The method of

c o n s t r u c t i o n p e r m i t s a continuous range from s o f t t o hard c o r e p o t e n t i a l s w i t h a trade-off between p o t e n t i a l s o f t n e s s and pseudo-wave f u n c t i o n accuracy away from t h e core. These p o t e n t i a l s a r e non-local and have optimum t r a n s f e r a b i l i t y . They a r e f i t t e d by a s m a l l s e t of a n a l y t i c f u n c t i o n s which f a c i l i t a t e s t h e i r u s e e i t h e r w i t h a p l a n e wave b a s i s s e t f o r &-space c a l c u l a t i o n s o r a l o c a l Gaussian b a s i s s e t f o r r e a l s p a c e c a l c u l a t i o n s .

LCAO Methods

The hallmark of LCAO methods is t h e expansion of t h e one-electron wave f u n c t i o n , a s a l i n e a r combination of a t o m i c o r b i t a l s , @a:

where a i s a composite i n d e x d e n o t i n g o r b i t a l type (s,p,d e t c . ) and t h e a t o m i c s i t e on which i t i s c e n t r e d ( f o r a review s e e r e f . / 1 4 / ) . Each v a l e n c e e l e c t r o n i s assumed t o move i n an e f f e c t i v e p o t e n t i a l ( e q u a t i o n ( 8 ) ) t h a t is r e p r e s e n t e d i n t h e one e l e c t r o n Hamiltonian ( e q u a t i o n ( 2 ) ) as a sum of atom-centred p o t e n t i a l s .

As i n t h e e m p i r i c a l p s e u d o p o t e n t i a l method t h e s e atom-centred p o t e n t i a l s can be i d e n t i f i e d a s s c r e e n e d i o n i c p s e u d o p o t e n t i a l s . The Schr8dinger e q u a t i o n reduces t o a set of l i n e a r e q u a t i o n s .

which r e q u i r e s

f o r n o n - t r i v i a l s o l u t i o n s . and S a r e t h e i n t e g r a l s

(0.1~

1%)

e l e m e n t s , Ha

,

I n s o - c a l l e d a b i n i t i o c a l c u l a t i o n s t h e i n t e g r a l s Hap and Sap a r e

e v a l u a t e d e x p l i c i t l y f o r an assumed s e t of atomic o r b i t a l s . Semi-empirical methods, such a s Extended Hiickel Theory, assume s i m p l e p r e s c r i p t i o n s f o r H

a 8

and Sap. The m a j o r i t y of s u r f a c e and i n t e r f a c e LCAO c a l c u l a t i o n s have used t h e e m p i r i c a l t i g h t b i n d i n g (ETB) approach which has i t s o r i g i n s i n t h e work of S l a t e r and K o s t e r /16/. T h i s i s an i n t e r p o l a t i o n scheme whereby t h e m a t r i x e l e m e n t s Hap a r e t r e a t e d a s parameters t h a t a r e f i t t e d t o t h e energy bands

( o b t a i n e d e x p e r i m e n t a l l y o r from a c c u r a t e c a l c u l a t i o n s ) a t p o i n t s of h i g h symmetry i n t h e B r i l l o u i n zone /17/. The atomic b a s i s f u n c t i o n s 0, a r e assumed t o form an orthonormal s e t s o t h a t t h e o v e r l a p m a t r i x Sag i s t h e i d e n t i t y matrix.

JOURNAL DE PHYSIQUE

Ldwdin / l a / s u p p l i e d a j u s t i f i c a t i o n f o r t h i s assumption by showing t h a t one can always c o n s t r u c t an orthonormal s e t of atomic b a s i s f u n c t i o n s , Qa from a

nonorthogonal s e t , ga, by t h e t r a n s f o r m a t i o n

S l a t e r and Koster / 1 6 / showed t h a t t h e Qa have t h e same t r a n s f o r m a t i o n p r o p e r t i e s a s t h e ga and one i s t h e r e f o r e j u s t i f i e d i n assuming an o r t h o g o n a l s e t . However, i t i s c l e a r from e q u a t i o n ( 1 4 ) t h a t t h e o r t h o g o n a l s e t is more s p a t i a l l y d e l o c a l i s e d t h a n t h e s e t Oa, which i s f a m i l i a r from t h e t h e o r y of Wannier f u n c t i o n s .

I n i t s s i m p l e s t and most widely used form only one and s e l e c t e d two-centred i n t e r a c t i o n s a r e r e t a i n e d i n t h e t i g h t b i n d i n g approximation (e.g. /17/). Often t h e two-centred i n t e r a c t i o n s a r e c o n f i n e d t o n e a r e s t neighbours only. Having f i t t e d t h e non-zero Hamiltonian m a t r i x e l e m e n t s t o s e l e c t e d p o i n t s of an a c c u r a t e band s t r u c t u r e one s t u d i e s t h e e l e c t r o n i c s t r u c t u r e of an unrelaxed d e f e c t by assuming t h e same m a t r i x elements a r e t r a n s f e r a b l e t o t h e d e f e c t environment / 1 9 / . K r i e g e r and Laufer / 2 0 / c r i t i c i s e d t h i s l a s t assumption, p a r t i c u l a r l y i n i t s a p p l i c a t i o n t o a c o o r d i n a t i o n d e f e c t such a s a vacancy o r a f r e e s u r f a c e , because i t i s tantamount t o i g n o r i n g t h e change i n t h e e f f e c t i v e p o t e n t i a l i n t h e v i c i n i t y of t h e d e f e c t . The v i n d i c a t i o n of t h e u s e of a l o c a l i s e d and approximately t r a n s f e r a b l e a t o m i c - l i k e b a s i s s e t came w i t h t h e work on "chemical

p s e u d o p o t e n t i a l s " / 2 1 / . A d e f i n i n g e q u a t i o n f o r t h e l o c a l i z e d atomic l i k e o r b i t a l s was developed / 2 1 / i n which t h e i n f l u e n c e of t h e atomic environment o c c u r s o n l y a s a weak p e r t u r b a t i o n . T h i s i s achieved by u s i n g t h e v a l e n c e o r b i t a l s on an a d j a c e n t atom t o c a n c e l off most of t h a t atom's p o t e n t i a l . These o r b i t a l s r e p r e s e n t t h e

exact

s e c u l a r m a t r i x a r e n o t m a t r i x e l e m e n t s of t h e t r u e Hamiltonian, but t h o s e of a

non-Hermitian pseudo-Hamiltonian. However t h e pseudo-Hamiltonian becomes

Hermitian when t h e b a s i s f u n c t i o n s a r e g e o m e t r i c a l l y e q u i v a l e n t o r b i t a l s on d i f f e r e n t s i t e s o r s e t s of symmetry r e l a t e d o r b i t a l s .

For s - p bonded semiconductors, such a s S i , Ge, GaAs e t c . , a "minimal b a s i s s e t " i s o f t e n used c o n s i s t i n g of one s and t h r e e p atomic-like o r b i t a l s on each of t h e two atoms i n a p r i m i t i v e c e l l (e.g. / 1 7 / ) . Very good f i t s t o t h e v a l e n c e energy bands have been o b t a i n e d i n t h i s way b u t t h e lower cohduction bands a r e l e s s

s a t i s f a c t o r i l y d e s c r i b e d . But i t is c r u c i a l t o have an a c c u r a t e d e s c r i p t i o n of both t h e v a l e n c e and lower conduction bands because t h e y both c o n t r i b u t e t o d e f e c t s t a t e s i n t h e p r i n c i p a l band gap. One way of improving t h e f i t t o t h e conduction bands i s t o i n c l u d e d - s t a t e s i n t h e b a s i s s e t , but t h i s i n c r e a s e s s e v e r e l y t h e s i z e of a d e f e c t computation. Louie / 2 2 / has developed a s u c c e s s f u l method of i n c l u d i n g t h e e f f e c t of t h e d - s t a t e s i n c o v a l e n t systems w i t h o u t i n c r e a s i n g t h e s i z e of t h e s e c u l a r d e t e r m i n a n t . This method was used i n r e f . / 2 3 / t o s t u d y t h e e l e c t r o n i c s t r u c t u r e of t h e u n r e c o n s t r u c t e d 30" p a r t i a l d i s l o c a t i o n c o r e i n S i , and i t is expected t o be used i n c r e a s i n g l y i n t h e f u t u r e .

ETB c a l c u l a t i o n s a r e , s t r i c t l y speaking, n o t s e l f - c o n s i s t e n t a s t h e method does n o t make u s e of e x p l i c i t b a s i s o r b i t a l s , s o t h a t i t cannot y i e l d v a l e n c e charge d e n s i t i e s . The s q u a r e s of t h e c o e f f i c i e n t s , c i a , of t h e o f b i t a l s a r e of t e n t r e a t e d a s e f f e c t i v e atomic c h a r g e s . T h i s i s a q u e s t i o n a b l e p r a c t i c e because, i f one assumes o r t h o g o n a l o r b i t a l s , e q u a t i o n ( 1 4 ) r e v e a l s t h a t t h e c o e f f i c i e n t s correspond t o charge d i s t r i b u t e d over s e v e r a l atoms, whereas w i t h nonorthogonal o r b i t a l s t h e sum of t h e s q u a r e s of t h e c o e f f i c i e n t s i s n o t u n i t y . N e v e r t h e l e s s t h i s d e f i n i t i o n of t h e l o c a l charge d e n s i t y can be used t o mimic s e l f - c o n s i s t e n c y i n s i m p l e models (e.g. / 2 4 / ) where d i a g o n a l Hamiltonian m a t r i x e l e m e n t s a r e c o n s i d e r e d t o be charge dependent.

F i n a l l y we mention t h e semi-empirical s e l f - c o n s i s t e n t f i e l d LCAO methods of quantum c h e m i s t r y such a s t h e n e g l e c t of d i f f e r e n t i a l o v e r l a p methods /25/. As w e l l a s being s e l f - c o n s i s t e n t (without invoking t h e l o c a l d e n s i t y approximation) t h e s e methods have well-defined t o t a l energy a l g o r i t h m s 1261. However, I know of no a p p l i c a t i o n of t h e s e methods t o s t u d y i n t e r f a c e s .

3

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Given t h e atomic s t r u c t u r e of an i n t e r f a c e one can c a l c u l a t e t h e c o r r e s p o n d i n g e l e c t r o n i c s t r u c t u r e by a v a r i e t y of methods. I n a l l such c a l c u l a t i o n s , however, one must f i r s t d e c i d e how t o d e a l w i t h t h e l o s s of t r a n s l a t i o n a l symmetry normal t o t h e i n t e r f a c e , and p o s s i b l y p a r a l l e l t o i t a s w e l l . I n g e n e r a l , p e r i o d i c i n t e r f a c e s ( i . e . t h o s e which d i s p l a y two-dimensional symmetry p a r a l l e l t o t h e i n t e r f a c e ) a r e r a r e between c r y s t a l s of d i f f e r e n t m a t e r i a l s . A p e r i o d i c i n t e r f a c e s can be s t u d i e d only by c l u s t e r methods, whereby a r e g i o n of t h e i n t e r f a c e i s modelled by a c l u s t e r of atoms w i t h a p p r o p r i a t e boundary c o n d i t i o n s on t h e s u r f a c e of t h e c l u s t e r (e.g. / 2 7 / ) . Various methods of 'embedding' t h e s e c l u s t e r s have emerged i n r e c e n t y e a r s /28/. F o r t u n a t e l y , t h e semiconductor h e t e r o j u n c t i o n s of t e c h n o l o g i c a l s i g n i f i c a n c e a r e c h a r a c t e r i s e d by very s m a l l l a t t i c e mismatches 1291. I n t h o s e c a s e s t h e mismatch i s i g n o r e d and i n ' i d e a l ' i n t e r f a c e s atoms a r e assumed t o be i n t h e i r i d e a l c r y s t a l p o s i t i o n s r i g h t up t o t h e i n t e r f a c e .

F r e q u e n t l y , however, t h e mismatch a t metal-semiconductor and m e t a l s i l i c i d e - semiconductor i n t e r f a c e s is n o t n e g l i g i b l e . Louie and Cohen 1301 took t h e view t h a t t h e e x i s t e n c e of a continuum of f r e e - e l e c t r o n - l i k e s t a t e s is t h e e s s e n t i a l p r o p e r t y of t h e m e t a l a t a metal-semiconductor i n t e r f a c e . Accordingly, t h e y modelled t h e A1 c r y s t a l i n t h e i r s t u d y of Al-Si i n t e r f a c e s by a j e l l i u m of a p p r o p r i a t e d e n s i t y , which e l i m i n a t e d t h e d i f f i c u l t y of l a t t i c e mismatch. The main c o n c l u s i o n s of t h i s work have r e c e i v e d s u p p o r t from c a l c u l a t i o n s on t h e Al-Ge (001) i n t e r f a c e (31) i n which t h e j e l l i u m approximation was n o t made. For t h e r e s t of t h i s s e c t i o n we w i l l c o n s i d e r only p e r i o d i c i n t e r f a c e s .

The l o s s of t r a n s l a t i o n a l symmetry normal t o t h e i n t e r f a c e i n a b i c r y s t a l s i g n i f i e s t h a t c r y s t a l band s t r u c t u r e methods a r e i n a p p l i c a b l e . There a r e e s s e n t i a l l y two ways of coping w i t h t h i s . The f i r s t is t o c o n s i d e r a s u p e r l a t t i c e of i n t e r f a c e s i n which a l t e r n a t e s l a b s of 5 t o 12 l a y e r s of each m a t e r i a l , p a r a l l e l t o t h e i n t e r f a c e , a r e s t a c k e d p e r i o d i c a l l y on top of e a c h o t h e r . T h i s r e s t o r e s t r a n s l a t i o n a l symmetry normal t o t h e i n t e r f a c e and t h e c a l c u l a t i o n is e s s e n t i a l l y a band s t r u c t u r e c a l c u l a t i o n w i t h a l a r g e u n i t c e l l . S i m i l a r l y , t h i s geometry has been used t o s t u d y f r e e s u r f a c e s , chemisorption and t h e e a r l y s t a g e s of i n t e r f a c e f o r m a t i o n (i.e. up t o a few monolayers coverage) by p e r i o d i c a l l y i n s e r t i n g vacua between ' s l a b s ' of c r y s t a l , i n c l u d i n g chemisorbed atoms, o v e r l a y e r s e t c . The second approach i s t o c o n s i d e r a t r u e b i c r y s t a l geometry, and t r e a t t h e i n t e r f a c e a s a l o c a l i s e d p e r t u r b a t i o n (i.e. l o c a l i s e d i n t h e d i r e c t i o n normal t o t h e i n t e r f a c e ) of t h e bulk p r o p e r t i e s of t h e a d j o i n i n g c r y s t a l s . The Green's f u n c t i o n formalism i s i d e a l l y s u i t e d t o t h i s problem and has been developed s u c c e s s f u l l y f o r s u r f a c e s and i n t e r f a c e s 1321. We s h a l l b r i e f l y d i s c u s s t h e s e two approaches i n t u r n .

Both p l a n e wave and LCAO b a s i s sets have been used i n s u p e r l a t t i c e c a l c u l a t i o n s . The s e l f - c o n s i s t e n t p s e u d o p o t e n t i a l s u p e r l a t t i c e c a l c u l a t i o n s , i n t r o d u c e d by Cohen, S c h l i i t e r and coworkers 1121, employed semi-empirical i o n i c

p s e u d o p o t e n t i a l s and a p l a n e wave b a s i s s e t f o r t h e v a l e n c e wavefunctions. Even w i t h s o f t c o r e p s e u d o p o t e n t i a l s very l a r g e b a s i s s e t s were needed t o a c h i e v e

convergence of t h e charge d e n s i t y ( t y p i c a l l y 300-600 p l a n e waves f o r c e l l s c o n t a i n i n g o n l y 20-40 atoms). T h i s r e s u l t s i n enormous m a t r i c e s t o be

d i a g o n a l i s e d and i t i s computing power which l i m i t s t h e a p p l i c a t i o n of t h i s method t o c e l l s c o n t a i n i n g g r e a t e r numbers of atoms. The r e s u l t s of such a c a l c u l a t i o n c o n s i s t of t h e e n e r g i e s and wavefunctions of a l l s u p e r l a t t i c e s t a t e s , i n c l u d i n g any i n t e r f a c e s t a t e s , t h e t o t a l s e l f - c o n s i s t e n t charge d e n s i t y and charge

d e n s i t i e s of i n d i v i d u a l s t a t e s and atom-resolved l o c a l d e n s i t i e s of s t a t e s . S e l f -

JOURNAL DE PHYSIQUE

consistent superlattice calculations have been carried out with LCAO bases in refs. /33,34,35,36/. In general hard-core pseudopotentials require a

prohibitively large plane wave basis set for convergence and this is the reason for using an atomic basis (e.g. 1351). Tight binding superlattice calculations have also been carried out (e.g. /37,38,39/).

The very large matrices one has to diagonalise with the superlattice method are a consequence of the fact that bulk states of the adjoining crystals are produced together with any interface induced states. In the second approach 1321 this shortcoming is avoided by first calculating the band structures of the adjoining crystals, and then focussing on the interface induced changes. These changes are readily found by a Green's function tight binding formalism once a matrix has been written down which represents the procedure of cutting two infinite solids and joining them to form a bicrystal (plus two semi-infinite free crystals which can be discarded). The Green's function formalism distinguishes clearly between interface induced and bulk states. It provides a powerful method of studying bound interface states, resonances and antiresonances within the bands, and wave vector-, atom- or orbital- resolved local densities of states. The sizes of the matrices involved in this method are determined by the range of the interface potential, which is usually much more localized than the wave functions of interface states. For example, Pollmam and Pantelides 1321 used an empirical tight binding Hamiltonian with a minimal basis set (sp3) and only nearest neighbour interactions to study ideal Ge-GaAs and Ge-ZnSe (100) interfaces. In that case the matrices involved were only 64 x 64, although it should be stressed that the calculations were not self-consistent. Although the matrices involved in the Green's function method are relatively small a large number of Green's

functions for the perfect crystal (evaluated at different crystal wavevectors) have to be stored. This is not a serious problem with the large storage capacities of modern computers.

It is interesting that the results obtained for the Ge-GaAs (110) interface by the self-consistent pseudo-potential method I401 and the empirical tight binding Green's function method 1411 are in qualitative agreement. The results differ slightly in the number and wave-vector dependences of the low-lying interface states. It is not clear whether these differences are due to self-consistence effects or the relatively poor representation of the conduction bands in the empirical tight binding Hamiltonian.

Pollmann 1421 has shown how overlayers, chemisorbed species, superlattices and defects at surfaces and interfaces can be treated by the Green's function formalism.

4

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Ideally we would be able to calculate the minimum free energy structure of an interface, taking into account self-consistently its electronic structure.

Considerable progress has been made towards this goal in recent years, principally because of the use of density functional theory /I/ and greater computing power. For example, it is now possible to predict accurately not only the correct lattice parameter and bulk modulus of elemental semiconductors 1431, compound semiconductors 1441 and metals 1451 but also crystal stability and pressure induced phase transformations 1431. These calculations require only the atomic numbers of constituent elements and a subset of crystal structures as input information. Norm-conserving pseudopotentials, combined with the local density functional approximation are employed in the total Hamiltonian and the problem is solved self-consistently either in momentum space 1461 or real space 1471. Having achieved such an accurate description of the ideal crystal its application to interfaces is currently limited by computing power. The principal problem is convergence of the basis set. In order to predict, for example, the various reconstructed configurations of the Si (111) surface the total energy calculation would require an accuracy of hetter than O.OleV/surface atom 1481. At present

t h i s could o n l y be achieved w i t h a p r o h i b i t i v e l y l a r g e b a s i s s e t of e i t h e r p l a n e waves o r l o c a l Gaussian o r b i t a l s . N e v e r t h e l e s s , a number of t o t a l energy c a l c u l a t i o n s have been performed, w i t h l i m i t e d r e l a x a t i o n s i n c l u d e d . Many of t h e s e c a l c u l a t i o n s have probed t h e geometry and l o c a l bonding of chemisorbed atoms on semiconductor 136,491 and m e t a l 1331 s u r f a c e s , w i t h p h o t o e l e c t r o n s p e c t r o s c o p y experiments p r o v i d i n g a v a l u a b l e check.

A more t r a c t a b l e ( b u t l e s s a c c u r a t e ) approach t o minimizing t h e t o t a l i n t e r n a l energy of l a r g e systems i n semiconductors has been developed by Chadi /50,51,52/, i n an ETB framework. The t o t a l energy i s w r i t t e n a s

E = .E E,(!3) + .E (U,E; +YE;)

^U.N*

7-

occupied bkdc (15)

The f i r s t term i s t h e band s t r u c t u r e energy and i s t h e sum of e n e r g i e s of a l l occupied o n e - e l e c t r o n s t a t e s . The second and t h i r d terms d e n o t e t h e c o r r e c t i o n due t o o v e r c o u n t i n g of t h e e l e c t r o n - e l e c t r o n i n t e r a c t i o n s i n t h e band s t r u c t u r e energy and i t a l s o i n c l u d e s t h e ion-ion Coulomb i n t e r a c t i o n s . The summation i n t h e second term is o v e r a l l bonds and ci d e n o t e s t h e f r a c t i o n a l change i n bond l e n g t h from t h e r e f e r e n c e v a l u e i n t h e p e r f e c t c y r s t a l . The e m p i r i c a l c o n s t a n t s U1 and U 2 a r e o b t a i n e d from t h e c o n d i t i o n t h a t t h e t o t a l energy of t h e p e r f e c t c r y s t a l i s a minimum a t t h e e x p e r i m e n t a l l y observed l a t t i c e parameter and by f i t t i n g t h e bulk modulus. The t i g h t b i n d i n g i n t e r a c t i o n parameters a r e assumed t o have an i n v e r s e s q u a r e dependence on bond l e n g t h 1531. Nb i n t h e l a s t term i s t h e number of bonds i n t h e system and Uo i s determined from t h e cohesive energy of t h e p e r f e c t c r y s t a l . The l a s t term i n e q u a t i o n (15) is v i t a l i n t h o s e c a s e s where bonds a r e broken o r formed; i n i t s absence t h e t o t a l energy would d e c r e a s e m o n o t o n i c a l l y w i t h i n c r e a s i n g atomic c o o r d i n a t i o n . The energy of a d e f e c t i n a system c o n t a i n i n g N atoms i s g i v e n by AET = ET (N)

-

Eo i s t h e t o t a l energy p e r atom i n t h e p e r f e c t c y r s t a l . As i n o t h e r e m p i r i c a l approaches, t h e r e i s t h e q u e s t i o n of t r a n s f e r a b i l i t y of t h e parameters Uo, U1, U p and t h e t i g h t b i n d i n g i n t e r a c t i o n p a r a m e t e r s , from t h e p e r f e c t c r y s t a l , where t h e y a r e f i t t e d , t o t h e d e f e c t environment. T h i s can be t e s t e d only by comparison w i t h experiment and indeed e q u a t i o n (15) has been s u r p r i s i n g l y s u c c e s s f u l a t p r e d i c t i n g some r e c o n s t r u c t e d s u r f a c e g e o m e t r i e s 1501.

Bond d i s t o r t i o n s and r e c o n s t r u c t i o n s can g i v e r i s e t o charge t r a n s f e r which i n t r o d u c e s Coulomb i n t e r a c t i o n s t h a t a r e n o t i n c l u d e d i n e q u a t i o n (15). Although t h e s e i n t e r a c t i o n s would be s c r e e n e d by v a l e n c e e l e c t r o n s t h e i r e f f e c t i s t o reduce t h e degree of charge t r a n s f e r which i n t u r n a f f e c t s t h e r e s u l t a n t atomic c o n f i g u r a t i o n . I n p r i n c i p l e , t h e s e e f f e c t s could be t a k e n account of by making t h e i n t r a - a t o m i c t i g h t b i n d i n g m a t r i x elements charge dependent, a s i n r e f 1241.

An e x p r e s s i o n f o r t h e f o r c e on an atom i s o b t a i n e d by d i f f e r e n t i a t i n g e q u a t i o n (13) w i t h r e s p e c t t o a n atomic c o o r d i n a t e . Chadi 1511 h a s shown how t h e Hellmann Feynman theorem can be used t o o b t a i n a simple e x p r e s s i o n f o r t h e d e r i v a t i v e of t h e band s t r u c t u r e energy. Using t h i s method he c a l c u l a t e d a r e l a x e d s t r u c t u r e of t h e C=9 (221), 3 8 . 9 4 ° / [ 1 ~ 0 ] symmetrical t i l t boundary i n S i 1521.

Chadi's f o r m u l a t i o n f o r s-p c o v a l e n t s o l i d s has some s i m i l a r i t i e s w i t h t h a t used by Masuda and S a t o 1541 t o c a l c u l a t e d i s l o c a t i o n c o r e s t r u c t u r e s i n b.c.c.

t r a n s i t i o n m e t a l s . These a u t h o r s a l s o e x p r e s s t h e t o t a l energy a s a sum of a band s t r u c t u r e term ( a r i s i n g from t h e d band o n l y ) and a s h o r t range r e p u l s i v e

i n t e r a c t i o n . The band s t r u c t u r e energy c o n t r i b u t i o n i s f o r m u l a t e d i n r e a l s p a c e u s i n g t h e l o c a l d e n s i t y of s t a t e s on each atom, and i n t e g r a t i n g t h e one-electron e n e r g i e s t o t h e Fermi l e v e l . The l o c a l d e n s i t y of s t a t e s is approximated by a Gaussian f i t t e d t o t h e second moment 1551 and t h e ddo, ddx and dd6 t i g h t b i n d i n g i n t e r a c t i o n s a r e assumed t o vary e x p o n e n t i a l l y w i t h t h e i n t e r a t o m i c d i s t a n c e . The assumption of a Gaussian l o c a l d e n s i t y of s t a t e s o b l i t e r a t e s a l l t h e s t r u c t u r e which t h i s i m p o r t a n t f u n c t i o n c o n t a i n s . The assumption could be avoided by u s i n g

C4-356 JOURNAL DE PHYSIQUE

t h e ' r e c u r s i o n method' /56/ t o c a l c u l a t e t h e l o c a l d e n s i t y of s t a t e s /57/.

A l t e r n a t i v e l y , one could u s e t h e Hellmann-Feynman f o r c e s /51/ t o g i v e an e x a c t e x p r e s s i o n i n t h e framework of t h e d-band t i g h t b i n d i n g Hamiltonian, without making any assumption about t h e l o c a l d e n s i t y of s t a t e s . W e a l s o n o t e t h a t H a r r i s o n /53/ has p r e s e n t e d arguments, based on muffin t i n o r b i t a l t h e o r y , f o r an i n v e r s e f i f t h power dependence of t h e d-d t i g h t b i n d i n g i n t e r a c t i o n s on

i n t e r a t o m i c d i s t a n c e .

F i n a l l y we mention t h e use of v a l e n c e f o r c e f i e l d s (e.g. / 5 8 / ) t o s t u d y g r a i n boundary s t r u c t u r e s i n c o v a l e n t l y bonded s o l i d s /59,60/. I n t h i s approach t h e t o t a l energy i s e x p r e s s e d a s a sum of s t r a i n energy terms (bond s t r e t c h i n g , bending and p o s s i b l y t o r s i o n ) and bond b r e a k i n g e n e r g i e s w i t h t h e p e r f e c t c r y s t a l a s t h e r e f e r e n c e c o n f i g u r a t i o n . Valence f o r c e f i e l d s have enjoyed c o n s i d e r a b l e s u c c e s s i n d e s c r i b i n g phonon d i s p e r s i o n r e l a t i o n s f o r t h e p e r f e c t c r y s t a l /58/.

However, t h e i r a p p l i c a t i o n t o s i t u a t i o n s i n v o l v i n g l a r g e atomic d i s p l a c e m e n t s i s l i m i t e d by t h e f a c t t h a t they do n o t t a k e i n t o account e l e c t r o n i c

r e h y b r i d i z a t i o n . For example, H a r r i s o n /61/ has shown t h a t t h e 2x1

r e c o n s t r u c t i o n of t h e S i (111) s u r f a c e i s due t o a b a l a n c e between a n e t energy g a i n from r e h y b r i d i z a t i o n of t h e d a n g l i n g h y b r i d s , accompanying t h e b u c k l i n g of t h e s u r f a c e , and an energy p e n a l t y r e s u l t i n g from t h e d i s t o r t i o n . N e v e r t h e l e s s , t h e v a l e n c e f o r c e f i e l d approach i s expected t o be r e l i a b l e when t e t r a h e d r a l bonding i s p r e s e r v e d , a s i n r e f . /59/.

5

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Norm-conserving p s e u d o p o t e n t i a l combined w i t h t h e l o c a l d e n s i t y f u n c t i o n a l formalism a r e c u r r e n t l y t h e most a c c u r a t e method of d e s c r i b i n g t h e ground state e l e c t r o n i c s t r u c t u r e of s u r f a c e s and i n t e r f a c e s . I n p r i n c i p l e , t o t a l energy c a l c u l a t i o n s f o r a d i v e r s e range of h e t e r o p h a s e and homophase i n t e r f a c e s can be performed w i t h t h i s formalism, but i n many c a s e s t h e s i z e of t h e c a l c u l a t i o n i s p r o h i b i t i v e l y l a r g e f o r t h e c u r r e n t g e n e r a t i o n of computers. However, i t has been argued /62/ t h a t u n d e r e s t i m a t i o n of t h e band gap by -50%, by d e s c r i p t i o n s u s i n g t h e l o c a l d e n s i t y f u n c t i o n a l approximation, i m p l i e s s i m i l a r e r r o r s f o r t h e p r e d i c t e d p o s i t i o n s of d e f e c t energy l e v e l s i n t h e fundamental gap. I n s i l i c o n , f o r example, t h i s would imply a maximum e r r o r of 9.3eV f o r t h e d e f e c t energy l e v e l s p r e d i c t e d by t h e most a c c u r a t e c a l c u l a t i o n s . Semi-empirical LCAO methods, s u c h a s t h e e m p i r i c a l t i g h t b i n d i n g method, can p r o v i d e a good d e a l of q u a l i t a t i v e i n s i g h t , w i t h a r e l a t i v e l y s m a l l c a l c u l a t i o n . Recent improvements on t h e u s u a l t i g h t b i n d i n g scheme were mentioned i n s e c t i o n 2 and an e f f i c i e n t method of

c a l c u l a t i n g f o r c e s on atoms i n t h e t i g h t b i n d i n g scheme was r e f e r r e d t o i n s e c t i o n 4.

ACKNOWLEDGEMENTS

I am g r a t e f u l t o Drs.K.W.Lodge and J.C.H.Spence and P r o f e s s o r J.W.Christian,F.R.S.

f o r t h e i r comments on t h e manuscript. T h i s r e s e a r c h was funded by t h e Royal S o c i e t y through a 1983 U n i v e r s i t y Research Fellowship.

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