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TOTAL ENERGY CALCULATIONS AND BONDING AT INTERFACES
S. Louie
To cite this version:
S. Louie. TOTAL ENERGY CALCULATIONS AND BONDING AT INTERFACES. Journal de Physique Colloques, 1985, 46 (C4), pp.C4-335-C4-346. �10.1051/jphyscol:1985437�. �jpa-00224688�
JOURNAL DE PHYSIQUE
Colloque C4, suppl6ment au n04, Tome 46, avril 1985 page C4-335
TOTAL ENERGY CALCULATIONS AND BONDING AT INTERFACES S.G. Louie
Department o f Physics, U n i v e r s i t y o f California, and Materials and Molecular Research Division, Lawrence Berkeley Laboratory, Berkeley, CA 94720, U.S.A .
A b s t r a c t - Some o f t h e concepts and t h e o r e t i c a l techniques employed i n r e c e n t ab i n i t i o s t u d i e s o f t h e e l e c t r o n i c and s t r u c t u r a l p r o p e r t i e s o f surfaces and i n t e r f a c e s are discussed. Results o f t o t a l energy c a l c u l a t i o n s f o r t h e 2x1 reconstructed diamond (111) s u r f a c e and f o r s t a c k i n g f a u l t s i n S i a r e reviewed.
I - INTRODUCTION
A f u l l understanding o f a s u r f a c e o r i n t e r f a c e r e q u i r e s knowledge o f both i t s e l e c t r o n i c and geometric s t r u c t u r e s . Hence, a g r e a t deal o f e f f o r t has been devoted i n t h e p a s t decade i n developing t h e o r e t i c a l methods f o r c a l c u l a t i n g t h e p r o p e r t i e s and s t r u c t u r e o f i n t e r f a c e s from f i r s t p r i n c i p l e s . This paper describes some o f the concepts and 1 a t e s t t h e o r e t i c a l techniques used i n addressing the i n t e r f a c e problem. Several r e s u l t s obtained using the pseudopotential d e n s i t y f u n c t i o n a l method a r e presented t o i l l u s t r a t e r e c e n t progress.
Much of the t h e o r e t i c a l work i n t h i s area i n t h e p a s t has been focused on the e l e c t r o n i c p r o p e r t i e s o f t h e i n t e r f a c e s w i t h o u t c o n s i d e r a t i o n o f t h e i r t o t a l
e n e r g e t i c s /I/. Thus, t h e geometric s t r u c t u r e s were necessary i n p u t t o t h e c a l c u l a - t i o n s . The s i t u a t i o n , however, i s changing r a p i d l y . I n the l a s t few years,
i t has become p o s s i b l e t o c a l c u l a t e from f i r s t p r i n c i p l e s the d e t a i l e d e l e c t r o n i c and s t r u c t u r a l p r o p e r t i e s o f m a t e r i a l s and t h e i r surfaces. With b a s i c a l l y t h e atomic number and atomic mass o f t h e c o n s t i t u e n t atoms as i n p u t , many s t a t i c and dynamical p r o p e r t i e s /2/ (e.g., cohesive energies, l a t t i c e constants, b u l k moduli, phonon spectra, c r y s t a l s t r u c t u r e s , e t c . ) have been c a l c u l a t e d t o w i t h i n a few percent o f experiment. This development i s made p o s s i b l e because o f r e c e n t advances i n t h e pseudopotential theory and t o t a l energy c a l c u l a t i onal techniques.
I t has opened up many e x c i t i n g p o s s i b i l i t i e s f o r t h e study o f i n t e r f a c e s s i n c e t h e r e i s l i t t l e t e c h n i c a l d i f f e r e n c e between a s u r f a c e which i s a vacuum-solid i n t e r f a c e and a s o l i d - s o l i d i n t e r f a c e / 3 / . I n t h i s paper, we describe two r e c e n t a p p l i c a t i o n s o f t h e t h e o r y as examples. One i n v o l v e s t h e p r e d i c t i o n o f t h e geometry o f t h e diamond (111) surface /4/, and t h e o t h e r i s a c a l c u l a t i o n of t h e p r o p e r t i e s o f s t a c k i n g f a u l t s i n S i /5/.
The paper i s organized i n t h e f o l l o w i n g manner. I n Sec. 11, a s h o r t d e s c r i p t i o n of t h e t h e o r e t i c a l methods i s given. Section I 1 1 presents a b r i e f review o f p a s t 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 i n which o n l y t h e e l e c t r o n i c p r o p e r t i e s were considered. Results from two r e c e n t t o t a l energy c a l c u l a t i o n s are presented i n Sec. I V . The bonding p r o p e r t i e s and s t r u c t u r e o f t h e diamond (111) surface are discussed, and a minimum energy geometry f o r the 2x1 reconstructed surface i s deter- mined. Also presented i s a study o f t h e i n t r i n s i c and e x t r i n s i c s t a c k i n g f a u l t s
i n Si. The s t a c k i n g f a u l t energies are obtained and found t o be i n good agreement w i t h experimental values. The Hellmann-Feynman f o r c e s a r e c a l c u l a t e d t o study t h e forces on atoms near t h e f a u l t s . L o c a l i z e d d e f e c t ( s u r f a c e and f a u l t ) s t a t e s are determined and t h e i r p r o p e r t i e s analyzed. F i n a l l y , Sec. V presents a sumnary.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985437
JOURNAL DE PHYSIQUE
I 1 - THEORETICAL METHODS
The general approach i n v o l v e s reducing the many-body problem t o t h a t o f a s e t o f s e l f - c o n s i s t e n t f i e l d equations f o r the e l e c t r o n s . For ground-state p r o p e r t i e s , t h i s i s achieved, i n p r i n c i p l e , u s i n g t h e d e n s i t y f u n c t i o n a l formal ism /6,7/.
The eigenvalues o f t h e r e s u l t i n g s i n g l e - p a r t i c l e equations, the Kohn-Sham equations, are a1 so o f t e n i n t e r p r e t e d as e x c i t a t i o n energies w i t h reasonable r e s u l t s when analyzing the spectroscopic p r o p e r t i e s o f m a t e r i a l systems, although t h e r e i s no formal j u s t i f i c a t i o n f o r such a s s o c i a t i o n . The approach i s ab i n i t i o i n t h a t t h e o n l y i n p u t t o t h e c a l c u l a t i o n i s i n f o r m a t i o n about t h e c o n s t i t u e n t atoms and a s e t o f p o s s i b l e s t r u c t u r a l t o p o l o g i e s among which a minimal energy s t r u c t u r e i s derived.
The e f f e c t i v e one-electron eauations i n t h i s formalism a r e o f t h e form ( i n atomic u n i t s ) :
{- +v2 + vkXt(f) +vH(?) + Pxc[n~}*i (f 1 = Eiri
where V e x t i s the e x t e r n a l p o t e n t i a l seen by t h e e l e c t r o n s and VH i s t h e e l e c t r o - s t a t i c o r Hartree p o t e n t i a l . i s the exchange-correlation p a r t o f the e f f e c t i v e p o t e n t i a l which i s given by pGXz 6Exc/6n where Ex, i s t h e exchange-correlation energy o f t h e system. F i n a l l y , t h e d e n s i t y n i s obtained from t h e o n e - p a r t i c l e wavefunctions
N
where N i s the number o f e l e c t r o n s i n the system. What remains i s the s p e c i f i c a t i o n Since t h e exact f u n c t i o n a l i s n o t known, t h e most w i d e l y used scheme
9: :fiz'local d e n s i t y approximation (LDA) /7/:
where c$(n) i s the exchange-correlation energy d e n s i t y of t h e homogeneous e l e c t r o n gas o f d e n s i t y n. Several parameterizations o f e l e c t r o n gas data are i n common use.
I n t h e pseudopotential approach, t h e s o l i d i s considered t o be composed o f r i g i d i o n cores and t h e valence e l e c t r o n s which are i t i n e r a n t . The s e l f - c o n s i s t e n t f i e l d equations are c a r r i e d o u t o n l y f o r the valence e l e c t r o n s . The e x t e r n a l p o t e n t i a l VeXt due t o the cores are modeled by s u p e r p o s i t i o n o f i o n i c pseudopoten- t i a l s which are constructed from a knowledge o f o n l y t h e atomic numbers. The e l e c t r o n wavefunctions are obtained by s o l v i n g t h e Kohn-Sham equations using a basis s e t expansion i n e i t h e r plane waves o r l o c a l i z e d o r b i t a l s .
Once the one-electron wave equation has been solved, the t o t a l energy o f the system may be evaluated by adding t h e core-core and 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 energies t o the c o r e - e l e c t r o n i n t e r a c t i o n energy /8/. The t o t a l energy i s u s u a l l y cast i n t h e form
where the sum i s over a l l occupied s t a t e s and Eion-ion i s t h e e l e c t r o s t a t i c i n t e r - a c t i o n energy among the bare ions. Once t h e lowest energy s t r u c t u r e i s found, the o t h e r s o l i d s t a t e p r o p e r t i e s can be computed.
An i n t e r f a c e being a d e f e c t t o an otherwise p e r f e c t c r y s t a l poses a d d i t i o n a l c o n s t r a i n t s t o t h e c a l c u l a t i o n s . One i s the l a c k o f t r a n s l a t i o n a l symmetry because o f t h e i n t e r f a c e . T h i s can be overcome i n two ways. One approach i s t o match wavefunctions across the i n t e r f a c e allowi'ng both extended and decaying s t a t e s .
Another i s t o use the s o - c a l l e d s u p e r c e l l method i n which t h e i n t e r f a c e i s repeated i n d e f i n i t i v e l y w i t h c e r t a i n s e p a r a t i o n t o prevent i n t e r a c t i o n between i n t e r f a c e s . This method a l l o w s the use o f standard band s t r u c t u r e techniques since p e r i o d i c i t y i s mathematically restored. I t i s t h e most common and i s t h e method employed f o r the s t u d i e s discussed here. Another c o n s t r a i n t i n i n t e r f a c e s t u d i e s i s t h e requirement o f d e t a i l e d s e l f - c o n s i s t e n c y i n t h e c a l c u l a t i o n s because o f the p o s s i b l y i m p o r t a n t charge rearrangement near the i n t e r f a c e . Q u a n t i t i e s such as atomic rearrangements and i n t e r f a c i a l formation energies a r e s e n s i t i v e t o such charge rearrangements.
I 1 1 - SELF-CONSISTENT CALCULATIONS
I n t h e 19701s, many surfaces and i n t e r f a c e s were s t u d i e d u s i n g t h e s e l f - c o n s i s t e n t pseudopotential method /1,9/. I n these studies, t h e e l e c t r o n i c s t r u c t u r e was 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 f o r an assumed s t r u c t u r e . However, no attempt was made t o compute the lowest t o t a l energy s t r u c t u r e because o f t h e o r e t i c a l and computational d i f f i c u l t i e s . Thus, i n the scheme described i n Sec. 11, t h e steps f o r t o t a l energy e v a l u a t i o n were n o t c a r r i e d out. The atomic p o s i t i o n s were e i t h e r i n f e r r e d i n d i r e c t l y from experiments o r taken t o be those o f i d e a l i z e d models. Nevertheless, from these e l e c t r o n i c s t r u c t u r e c a l c u l a t i o n s , a g r e a t deal has been learned about surfaces and i n t e r f a c e s , and the r e s u l t s o f t e n provided t h e needed i n t e r p r e t a t i o n s and explanations o f experimental observations.
These s e l f - c o n s i s t e n t pseudopotential studies i n c l u d e d metal and semiconductor surfaces, metal-semiconductor i n t e r f a c e s (Schottky b a r r i e r s ) , semiconductor- semiconductor i n t e r f a c e s ( h e t e r o j u n c t i o n s ) , and s t a c k i n g f a u l t s . I n t h i s s e c t i o n , we b r i e f l y d e s c r i b e one a p p l i c a t i o n t o i l l u s t r a t e some o f t h e t h e o r e t i c a l concepts and techniques used i n s t u d y i n g i n t e r f a c i a l systems and a l s o t h e l i m i t a t i o n s o f t h e approach w i t h o u t t o t a l energy and f o r c e
0.5 i n f o r m a t i o n .
For a given s t r u c t u r e , the c a l c u l a t i o n s y i e l d a host o f i n f o r m a t i o n on t h e e l e c t r o n i c p r o p e r t i e s i n c l u d i n g t h e i n t e r f a c e energy bands, charge d e n s i t i e s , l o c a l d e n s i t i e s o f s t a t e s , and so f o r t h . The c a l c u l a t e d l o c a l d e n s i t y o f s t a t e s (LDOS) which describes t h e e l e c t r o n energy spectrum as f u n c t i o n o f p o s i t i o n i n r e a l space f o r an A l / S i (111) i n t e r f a c e / l o / .
presented i n F i g . 1. I n t h i s work, t h e E ' s u p e r c e l l c o n s i s t e d o f 12 l a y e r s o f t h e
s i l i c o n c r y s t a l along t h e (111) d i r e c t i o n and an e q u i v a l e n t thickness o f A1 which was approximated by the j e l l i u m model s i n c e the exact geometry a t t h e i n t e r f a c e i s n o t known.
The edge o f t h e j e l l i u m i s taken t o be a t a d i s t a n c e o f one-half t h e S i - S i bond length.
The u n i t c e l l , thus, i s composed o f an A1 h a l f and a S i h a l f , and each i s d i v i d e d i n t o t h r e e regions f o r the purpose o f analyzing the LDOS. As shown i n Fig. 1, t h e LDOS changes from t h a t o f a f r e e - e l e c t r o n model o f a metal w i t h a square r o o t dependence on energy i n r e g i o n I, which i s deep i n the
- 1 4 -12 - 1 0 - 8 -6 - 4 - 2 o 2 4 A1 side, t o t h a t o f b u l k S i i n r e g i o n V I
Energy (eVJ which i s the S i r e g i o n f a r t h e s t from the i n t e r f a c e .
Fig. 1 - Local d e n s i t y o f s t a t e s
near an A1 /Si (111) i n t e r f a c e . Region I V i s t h e t r a n s i t i o n r e g i o n on the ( a r b i t r a r y u n i t s ) S i side and i s the most i n t e r e s t i n g region.
C3-338 JOURNAL DE PHYSIQUE
U n l i k e t h e b u l k LDOS, new metal-induced gap s t a t e s (MIGS) appear i n t h e S i band gap i n t h i s region. These s t a t e s a r e b u l k - l i k e i n t h e metal, have l a r g e amplitude i n t h e d a n g l i n g bond s i t e s o f t h e S i surface, and decay r a p i d l y i n t o t h e semiconductor
/lo/. I t i s these s t a t e s which determine t h e Schottky b a r r i e r p r o p e r t i e s . L o c a l i z e d i n t e r f a c e s t a t e s which decay i n both d i r e c t i o n s away from t h e i n t e r f a c e a l s o e x i s t f o r t h i s system. The peak i n the valence band ( l a b e l e d S ) a t -8.5 eV a r i s e s from such l o c a l i z e d s t a t e s . Because these s t a t e s a r e deep i n [he valence band, they do n o t c o n t r i b u t e t o t h e t r a n s p o r t p r o p e r t i e s o f t h e Schottky b a r r i e r s . From t h e Fermi l e v e l and t h e p o s i t i o n o f t h e conduction band minimum, t h e b a r r i e r h e i g h t can be evaluated as seen i n Fig. 1. The c a l c u l a t e d v a l u e i s 0.6 + 0.1 eV which i s i n v e r y good agreement w i t h t h e measured value o f 0.6 eV /11/.
The MIGS i n t h e band gap are S i dangling bond surface s t a t e s h y b r i d i z e d w i t h the continuum s t a t e s o f t h e metals. They are, t h e r e f o r e , h y b r i d s t a t e s which are i n t e r m e d i a t e between the Bardeen ( s u r f a c e s t a t e ) and t h e Heine ( m e t a l l i c t a i l s t a t e ) p i c t u r e s /12/. A t the i n t e r f a c e , t h e formation o f t h e MIGS and t h e subsequent d i p o l e p o t e n t i a l created due t o the occupation o f them e q u a l i z e the Fermi l e v e l s o f t h e two m a t e r i a l s and determine t h e Schottky b a r r i e r height.
S i m i l a r c a l c u l a t i o n s were done f o r the 111-V and 11-VI compound semiconductor- metal i n t e r f a c e s /13/. The same qua1 i t a t i v e p i c t u r e emerged w i t h t h e t r e n d t h a t
both t h e d e n s i t y o f MIGS and the p e n e t r a t i o n o f t h e MIGS i n t o the semiconductor decrease as the i o n i c i t y o r t h e band gap o f t h e semiconductor increases. This i s i n t u i t i v e s i n c e a b i g g e r band gap imp1 i e s a l a r g e r e f f e c t i v e b a r r i e r f o r these s t a t e s t o p e n e t r a t e i n t o the semiconductor. With the MIGS as a conceptual basis, a microscopic t h e o r y f o r the behavior o f t h e Schottky b a r r i e r has been constructed which, f o r example, e x p l a i n s the change i n b a r r i e r h e i g h t w i t h metal e l e c t r o -
n e g a t i v i t y /13/.
The above example serves t o i l l u s t r a t e both t h e power and l i m i t a t i o n o f t h e s e l f - c o n s i s t e n t pseudopotential method. Although we have learned much about the general e l e c t r o n i c nature o f metal-semiconductor i n t e r f a c e s , we have n o t gained i n f o r m a t i o n on t h e d e t a i l geometric s t r u c t u r e o r chemical bonding a t t h e i n t e r f a c e . Since experiments s t i l l cannot r e 1 ia b l y y i e l d i n t e r f a c i a l s t r u c t u r e s , t h i s i s a major d e f i c i e n c y f o r methods which must r e l y on e x p e r i m e n t a l l y determined geometries o r i d e a l i z e d models.
I V - TOTAL ENERGY CALCULATIONS
I n t h e ab i n i t i o t o t a l energy c a l c u l a t i o n s , t h e exact geometry i s no l o n g e r a r e q u i r e d i n p u t . The s t r u c t u r e i s determined by m i n i m i z i n g t h e t o t a l energy w i t h r e s p e c t t o the atomic coordinates near the i n t e r f a c e f o r a given topology. A l t e r n a - t i v e l y , t h e He1 lmann-Feynman f o r c e s on each atom a r e c a l c u l a t e d , and t h e atoms are moved u n t i l a l l f o r c e s are zero. I n e i t h e r approach, the c a l c u l a t i o n must be done i t e r a t i v e l y s i n c e new forces develop Bn t h e i r neighbors when atoms are moved. Several cycles a r e u s u a l l y needed t o achieve a minimum energy, zero f o r c e s t r u c t u r e . I n t h i s approach, both e l e c t r o n i c and s t r u c t u r a l p r o p e r t i e s a r e obtained.
Several important f a c t o r s c o n t r i b u t e d t o the development and use of. ab i n i t i o t o t a l energy c a l c u l a t i o n s i n the 1980's /2/. Among these a r e the refinements i n band s t r u c t u r e c a l c u l a t i o n a l techniques, development o f approximations t o t h e d e n s i t y f u n c t i o n a l formal ism, i n v e n t i o n o f t h e ab i n i t i o pseudopotentials, and development o f techniques f o r c a l c u l a t i n g t o t a l energies and forces. I n t h i s s e c t i o n , we
review two c a l c u l a t i o n s using t h i s approach--the diamond (111) surface as a p r o t o t y p e vacuum-sol i d i n t e r f a c e f o r t e t r a h e d r a l covalent m a t e r i a l s and the i n t r i n s i c and e x t r i n s i c s t a c k i n g f a u l t s along the [Ill] d i r e c t i o n i n S i as -examples o f i n t e r n a l i n t e r f a c e s .
A. Diamond (111) Surface
The determination o f surface s t r u c t u r e i s a major unsolved problem i n s u r f a c e
science. No d e f i n i t i v e experimental probe has been developed y e t f o r g i v i n g d e t a i l e d surface geometry. There are many models proposed t o e x p l a i n the experimental data for t h e diamond (111) s u r f a c e as w e l l as f o r many o t h e r surfaces. Ab i n i t i o t o t a l
Table I. Ground-state properties of energy calculations should be able
diamond and Si. t o distinguish among these models
and suggest a possible low energy structure.
Lattice .Cohesive Bulk
Conztant Energy Modulus Table I summarizes some of the structural ( A ) (eV/atom) (Mbar) properties of bulk diamond calculated
using a l i n e r combination of atomic- l i k e (Gaussian) o r b i t a l s basis s e t 1141.
Di amond This basis s e t i s employed because of
Theory 3.56 7.84 4.37 the localized nature of the carbon
Experiment 3.57 7.37 4.42 electron wavefunctions. Three Gaussian
exponents f o r each of the s , px, p ,
S i l i c o n and pz o r b i t a l s t o t a l i n g 12 b a s ~ s Yunc-
Theory 5.41 4.76 0.93 tions per carbon atom were used. The
Experiment 5.43 4.63 0.99 r e s u l t s i l l u s t r a t e the accuracy of the method and serve as calibrations f o r
the surface study. The computed l a t t i c e constant, cohesive energies, and bulk modulus are a l l i n excellent agreement with experimental values.
The diamond (111) surface i s of i n t e r e s t as the insulating l i m i t f o r the group IV (111) surfaces which show a remarkable variety of surface reconstructions, t h a t i s , atomic rearrangements which r e s u l t in a change in the ideal surface symmetry.
Recent experiments indicate a hydrogen-terminated 1x1 surface a t room temperature, b u t the 2xl/2x2 surface seen by Low Energy Electron Diffraction (LEED) f o r surfaces cleaned by annealing t o above 1000 C i s apparently H-free 115-181. (LEED cannot distinguish between a true 2x2 or disordered domains of 2x1 for t h i s surface;
the s i m i l a r i t y of the angle-resolved photoemission t o that of the 2x1 S i ( l l 1 ) and Ge(ll1) surfaces suggests the l a t t e r . ) Although there a r e many models proposed f o r t h i s surface, the correct s t r u c t u r e remains undetermined. In the study /4/, energy minimization i s carried out f o r a1 1 the topologically d i s t i n c t 2x1 models in the l i t e r a t u r e . These models a r e a l l motivated by experimental data such as those from LEED or angle-resolved photoemission experiments. The structures considered include the ideal relaxed model , the Haneman buck1 ing model 1191,
the Pandey n-bonded chain model /20/,
( a ) ( c ) the Chadi molecule model 1211, and the
Seiwatz single chain model /22/. (See Fig. 2.)
The calculated t o t a l energies for these models a r e summarized i n Table 11. The energy per surface atom f o r the ideal 1x1 model i s used as zero of energy.
Relaxing the f i r s t two surface bonds (Fig. 3 ( a ) ) lowers the energy by 0.37 eV. Buckling of the 1x1 surface by r a i s - ing and lowering a l t e r n a t e rows of surface atoms, on the other hand, i s found t o r a i s e the energy. Of the other three 2x1 models, the ideal Pandey chain model 1231 has the lowest energy. The Seiwatz single chain model i s c l e a r l y unfavorable, and the Chadi molecule model, which has the second lowest energy, has not been relaxed f u r t h e r because the calculated surface s t a t e dispersion i s inconsis- t e n t with angle-resolved photoemission Fig. 2 - Geometries of 1x1 and 2x1 data /15/.
diamond (111) surfaces: ( a ) ideal
s t r u c t u r e , ( b ) Pandey chain model, The energy of the Pandey model i s ( c ) Seiwatz single chain model, and further minimized by adjusting the four ( d ) Chadi molecule model. surface-most bond 1 engths to give the
C4-340 JOURNAL DE PHYSIQUE
Table 11. Calculated t o t a l e n e r g i e s of C(111J 1 x 1 and 2x1 s u r f a c e r e c o n s t r u c t i o n model s.
Surface Energy
model eV/(surface atom)
Ideal 1x1 0.00
Relaxed 1 x 1 -0.37
Buckled (Az = k0.26 A) 0.35
Chadi a-bonded molecule 0.28
Sei watz s i ngl e chain 1.30
Ideal Pandey a-bonded chain -0.05 Relaxed Pandey a-bonded chain -0.47
Sane with k2% dimerization -0.46
Same with 24% dimerization -0.43
Same with +6% dimerization -0.38
Full y re1 axed Pandey chai n -0.68
"relaxed" s t r u c t u r e of Fig. 3 ( b ) lowering t h e Fig. 3 - Bond length changes energy t o -0.47 eV. A r a t h e r unexpected f e a t u r e (with r e s p e c t t o bulk) of of t h e r e s u l t i n g geometry i s t h e 8% lengthening ( a ) relaxed 1x1 and ( b ) re- of t h e subsurface i n t e r l a y e r bond. In c o n t r a s t , laxed 2x1 Pandey chain models t h e s u r f a c e chain bond was only shortened by 4% f o r t h e diamond (111) s u r f a c e . t o a length which i s approximately midway between
t h a t o f g r a p h i t e and diamond. Another important
r e s u l t i s t h a t , c o n t r a r y t o some s p e c u l a t i o n s , 8 dimerization of t h e chain i s found t o r a i s e t h e
s u r f a c e energy. The s t r u c t u r e can be f u r t h e r
relaxed by allowing t h e atoms below t h e f i r s t 6 two l a y e r s t o move. This movement r e l i e v e s some
of t h e bond angle s t r a i n s on the t h i r d l a y e r
atoms. The f i n a l f u l l y relaxed s t r u c t u r e has - 4
an energy lowered by an additional 0.21 eV per
s u r f a c e atom. 2 -
A 2 Once a minimum energy s t r u c t u r e i s determined, L 0
t h e c a l c u l a t e d s u r f a c e s t a t e d i s p e r s i o n can be Q, used t o compare with experimental r e s u l t s f o r 15 O confirmation. Figure 4 shows t h e c a l c u l a t e d
s u r f a c e band s t r u c t u r e f o r t h e f u l l y relaxed
Pandey chain model with t h e Fermi l e v e l a t near - 2 2 eV. Experimental angle-resolved photoemission
d a t a /15/ f o r t h e occupied s u r f a c e s t a t e s a r e
shown f o r comparison. There i s good agreement - 4 - - -
between theory and experiment f o r t h e band r J K
d i s p e r s i o n . However, t h e c a l c u l a t e d band i s t o o
high by a r i g i d s h i f t of -1 eV. This kind of Fig. 4 - Calculated s u r f a c e discrepancy i s most l i k e l y caused by using t h e bands ( s o l i d l i n e s ) and reson- l o c a l d e n s i t y approximation which i s well-known ances (dashed l i n e s ) f o r t h e t o g i v e excel l e n t s t r u c t u r a l (ground-state) 2x1 diamond (111) s u r f a c e f o r p r o p e r t i e s but t o o small e x c i t a t i o n e n e r g i e s f u l l y relaxed Pandey chain S i m i l a r c a l c u l a t i o n s /24/ have been c a r r i e d o u t model. Black d o t s a r e experi -
f o r t h e 2x1 phases of t h e (111) s u r f a c e s of Si mental d a t a of Ref. 15.
and Ge. The r e s u l t s obtained a r e q u a l i t a t i v e l y
t h e same. Namely, t h e relaxed a-bonded chain geometry y i e l d s the lowest t o t a l energy among the s t r u c t u r e s t e s t e d , and t h e s u r f a c e band d i s p e r s i o n s a r e i n good agreement with experiment but not t h e band p o s i t i o n s .
From t h e s e r e s u l t s , t h e s t r u c t u r e and bonding a t t h i s s u r f a c e can be described i n
the f o l l o w i n g terms. The d r i v i n g mechanism f o r
<,,,, t h e r e c o n s t r u c t i o n i s the existence o f half-occu-
7
8
p i e d dangling bonds o f t h e i d e a l (111) surface1 which are h i g h l y unfavorable e n e r g e t i c a l l y . I n t h e n-bonded chain geometry, t h e surface i s
O y - s t a b i l i z e d by a l l o w i n g t h e dangling bonds t o move i n t o near-neighbor p o s i t i o n s where they can
Y p a r t i c i p a t e i n v bonding. However, t h i s geometry r e s u l t s i n l a r g e bond angle d i s t o r t i o n s which g i v e r i s e t o a s t r a i n energy. The second and t h i r d l a y e r atoms are, thus, f o r c e d t o move t o re1 ie v e the s t r a i n s r e s u l t i n g i n a f i n a l r e l a x e d geometry.
VL7
The same general mechanism appears t o be o p e r a t i v e f o r the (111) surfaces o f a l l t h r e e o f the group I V elements.
B . Stacking F a u l t s i n Si
0 Stacking f a u l t s a r e probably t h e s i m p l e s t o f the p l a n a r defects o r i n t e r n a l i n t e r f a c e s . For diamond s t r u c t u r e semiconductors, s t a c k i n g f a u l t s along
t h e [Ill) d i r e c t i o n correspond t o Fig. 5 - Geometry o f t h e diamond s t r u c t u r e . a misplacement o f t h e t h i r d nearest-
neighbor arrangement, and the systems are o n l y s l i g h t l y d i s t u r b e d compared
( a 1 I S F t o those due t o o t h e r bond-breaking
[ I I I I d e f e c t s . There are, however, r a t h e r
few t h e o r e t i c a l i n v e s t i g a t i o n s o f t h e s t a c k i n g f a u l t s i n semiconductors.
I n p a r t i c u l a r , t h e r e has n o t been a complete study f o r the t o t a l energy o r t h e s t r u c t u r e o f Si s t a c k i n g f a u l t s i n the 1 i t e r a t u r e .
Experimentally, t h e i n f e r r e d s t a c k i n g f a u l t energies f o r (about 50-60 erg/cm 3 i ) a r e very small /25,26/. A recent charge c o l l e c t i o n scanning e l e c t r o n microscopy experiment on an e x t r i n s i c s t a c k i n g f a u l t i n n- type s i l i c o n suggested t h e existence o f s t a c k i n g f a u l t s t a t e s w i t h ener- gies a t about 0.1 eV below t h e con- d u c t i o n band minimum /27/. Photo- 1 umi nescence s p e c t r a o f p l a s t i c a l l y deformed samples showed a d e f e c t s t a t e near 0.15 eV above the valence Fig. 6 - Atomic p o s i t i o n s i n the (110) plane band maximum /28/.
of S i f o r ( a ) ISF and ( b ) ESF. The dashed
1 in e i n d i c a t e s a s t a c k i n g f a u l t plane. The The s t a c k i n g sequence along t h e i l l 1 1
n e t force on each atom i n t h i s i d e a l geome- d i r e c t i o n i n t h e diamond s t r u c t u r e t r y i s marked i n u n i t s o f 10-2 Ry/a.u. i s AA'BB'CC. (See Fig. 5.) An
i n t r i n s i c s t a c k i n g f a u l t (ISF) i s a planar d e f e c t corresponding t o a p a i r o f atomic planes m i s s i n g from the i d e a l s t a c k i n g sequence, and an e x t r i n s i c s t a c k i n g f a u l t i s a d e f e c t r e s u l t i n g from adding a p a i r o f atomic planes (e.g., AA' i n s e r t e d between BB' and CC'). The atomic p o s i t - ions near the f a u l t s i n the (110) plane are shown i n Fig. 6. For both types o f f a u l t s , the o r i e n t a t i o n o f t h e Si-Si bond i s r o t a t e d 120" from i t s normal d i r e c t i o n when passing through a f a u l t plane (dashed l i n e i n Fig. 6). As a consequence, for t h e atoms near a f a u l t (e.g., atom 2 i n Fig. 6(a) and atoms a and 4 i n Fig. 6 ( b ) ) , t h e numbers o f f i r s t and second nearest neighbors a r e n o t changed, b u t the number of t h i r d nearest neighbors i s reduced from 12 t o 9 w i t h one a d d i t i o n a l neighboring atom a t a d i s t a n c e s l i g h t l y l a r g e r than the second nearest-neighbor distance.
C4-342 JOURNAL DE PHYSIQUE
I n t h e c a l c u l a t i o n /5/, the geometries shown i n Fig. 6 are used t o c a l c u l a t e
the t o t a l energies o f t h e ISF and the ESF. Supercells o f 10 (14) atoms a r e construc- t e d f o r the ISF (ESF). The t o t a l energies o f t h e p e r f e c t c r y s t a l and t h a t o f t h e c r y s t a l w i t h s t a c k i n g f a u l t s are compared t o o b t a i n t h e s t a c k i n g f a u l t energies.
Since we a r e c a l c u l a t i n g extremely small energy d i f f e r e n c e s , g r e a t care i s needed i n t r e a t i n g the p e r f e c t c r y s t a l and the s u p e r c e l l s w i t h the f a u l t s i n equal f o o t i n g n u m e r i c a l l y so t h a t the r e q u i r e d p r e c i s i o n s o tained. The c a l c u l a t i o n s are c a r r i e d o u t using a plane wave b a s i s w i t h (i+~)! up t o 10 Ry which corresponds t o about 70 plane waves per atom.
Table I d i s p l a y s some o f t h e c a l c u l a t e d s t a t i c s t r u c t u r a l p r o p e r t i e s o f t h e S i p e r f e c t c r y s t a l u s i n g a six-atom s u p e r c e l l o f t h e same symmetry as the s u p e r c e l l s c o n t a i n i n g t h e s t a c k i n g f a u l t s . The c a l c u l a t e d l a t t i c e constant i s 5.41 A, which i s i n e x c e l l e n t agreement w i t h t h e experimental value o f 5.43 A. A l l subsequent r e s u l t s are obtained a t t h i s c a l c u l a t e d e q u i l i b r i u m l a t t i c e constant. The c a l c u l a t e d s t a c k i n g f a u l t energy i s 40 erg/cmZ and 26 erg/cm2 f o r the ISF and ESF r e s p e c t i v e l y . The u n c e r t a i n t y i n t h e t h e o r e t i c a l values r e s u l t i n g from the f i n i t e number o f planes waves and % p o i n t s i s estimated t o be 20%. The experimental values /25,26/
which are e x t r a c t e d u s i n g e l s t i c i t y models vary from 50 t o 70 erg/cmZ f o r the ISF and from 50 t o 60 erg/cml f o r the ESF. Thus, the theory i s i n reasonable agreement w i t h experiment. Although t h e r e has been previous t h e o r e t i c a l work /29-31/, t h i s i s t h e f i i - s t time t h a t s t a c k i n g f a u l t energies a r e obtained from an ab i n i t i o 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 .
--
I n a d d i t i o n t o t h e f a u l t energies, f o r c e s a r e c a l c u l a t e d u s i n g t h e Hellmann- Feynman theorem. This provides i n f o r m a t i o n on t h e tendency o f t h e atoms t o r e l a x near the f a u l t . The f o r c e on each atom near the f a u l t s i n the i d e a l geometry are shown i n Fig. 6. The t h r e e - f o l d r o t a t i o n a l symmetry around the [111] a x i s d i c t a t e s t h a t the o n l y forces e x i s t i n g are along the [lll] d i r e c t i o n . As seen from the f i g u r e , atoms belonging t o the same double l a y e r immediately n e x t t o a f a u l t plane (e.g., atoms b and 5) tend t o lengthen t h e c o v a l e n t bond between them. Also, t h e atoms belonging t o d i f f e r e n t double l a y e r s (e.g., atoms 3 and d i n Fig. 6 ( b ) ) a l s o tend t o r e p e l each other. The n e t e f f e c t i s t h a t t h e mis- - o r i e n t e d bonds along t h e chain want t o move away from t h e bond d i r e c t l y below o r above it. Thus, these r e s u l t s p r e d i c t t h a t t h e f u l l y r e l a x e d s t r u c t u r e f o r both ISF and ESF would be s l i g h t l y d i l a t e d along t h e [lll] d i r e c t i o n .
S i I S F
The c a l c u l a t e d energy bands i n t h e two-dimensional B r i l l o u i n zone a r e presented i n Fig. 7 f o r the ISF. The allow-
4 ed b u l k s t a t e s ( t h e p r o j e c t e d
band s t r u c t u r e ) are given by the
3 shaded regions. Stacking f a u l t
s t a t e s (dashed 1 in e s ) aye found i n t h e gap r e g i o n near r and
2 R. There are two f a u l t s t a t e s
- a t f o r the ISF. One i s a two-
3 f o l d degenerate s t a t e a t about
E 1 0.1 eV above t h e valence band
> mhximum; t h e o t h e r i s s i n g l y
w degenerate a t about 0.3 eV below
rr 0
w t h e conduction band minimum. The
L
W charge d e n s i t y o f these s t a t e s
- 1 i s p l o t t e d i n Fig. 8. The s t a t e
near the valence band maximum i s a bonding s t a t e w i t h charge
-2 d e n s i t y m a i n l y concentrated be-
tween f a u l t atoms ( F i g . 8 ( a ) ) . The l e v e l p o s i t i o n o f t h i s s t a t e
-3 - - - i s c o n s i s t e n t w i t h t h e photolumin-
W r K escence spectrum f i n d i n g /28/.
The s t a t e below t h e conduction F i g . 7 - Calculated s t a c k i n g f a u l t s t a t e s band minimum a t i s , on the o t h e r (dashed l i n e s ) f o r the ISF. hand, an anti-bonding s t a t e
( F i g . 8 ( b ) ) . S i m i l a r s t a c k i n g f a u l t states are found f o r t h e ESF.
V - SUMMARY
We have presented a review o f t h e ab i n i t i o pseudopotenti a1 d e n s i t y f u n c t i o n a l method f o r studying surfaces and i n t e r f a c e s . Results from c a l c u l a t i o n s on t h e diamond (111) surface and t h e s t a c k i n g f a u l t s i n S i are discussed. For t h e case o f t h e diamond surface, t h e bonding p r o p e r t i e s and s t r u c t u r e o f t h e 2x1 r e c o n s t r u c t i o n a r e examined. A f u l l y r e l a x e d n-bonded geometry i s p r e d i c t e d from t o t a l energy m i n i m i z a t i o n . For t h e case o f s t a c k i n g f a u l t s , c a l c u l a t i o n s are c a r r i e d o u t f o r e x t r i n s i c and i n t r i n s i c f a u l t s along t h e [lll] d i r e c t i o n . The s t a c k i n g f a u l t ener- g i e s a r e calculated, and forces on t h e atoms a r e determined f o r t h e i d e a l geometries.
The existence and the p r o p e r t i e s o f l o c a l i - zed d e f e c t s t a t e s are a l s o examined and Fig. 8 - Charge d e n s i t y of s t a c k i n g compared w i t h experiment. The o n l y i n p u t f a u l t s t a t e s of t h e ISF a t w i t h en- t o the c a l c u l a t i o n s a r e t h e atomic numbers e r g i e s (a) 0.1 eV above the valence and a s e t o f p o s s i b l e s t r u c t u r a l topologies.
band maximum and (b) 0.3 eV below t h e
conduction band minimum. The charge Acknowledgement - T h i s work was supported d e n s i t y i s i n u n i t s of e l e c t r o n per by N a t i o n a l Science Foundation Grant No.
c e l l volume and i s normalized t o .one DMR8319024 and by a program development e l e c t r o n p e r c e l l . The contour I n t e r - fund from t h e D i r e c t o r o f t h e Lawrence v a l s are (a) 1.5 and (b) 3.0. Berkeley Laboratory.
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[I] For a r e c e n t review, see M. L. Cohen, Advances i n E l e c t r o n i c s and E l e c t r o n Physics, Vol. 51, eds. L. Marton and C. Marton, Academic Press, New York, pp. 1-62 (1980).
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DISCUSSION
D. A s t ; Two comments. ( i ) If I r e c a l l it c o r r e c t l y , it was l a t e r shown t h a t t h e r e s u l t s o f Kimmerling e t a l . were obtained on a decorated s t a c k i n g f a u l t . ( i i ) I n high r e s o l u t i o n TEM o f twins t h e r e a r e some i r r e g u l a r i t i e s i n c o n t r a s t which may be compatible with a d i l a t a t i o n occurring a t t h e boundary.
S.G. Louie: There is a paper by Y. Hanoka (Mobil Tyco Co., Waltham, Mass) i n Am.
Rev. Mat. S c i . which, I b e l i e v e , g i v e s numbers f o r energy l e v e l s o f twins i n S i a s derived from temperature dependent EBIC d a t a .
R. Havdock: How s e n s i t i v e a r e your r e s u l t s t o t h e pseudopotential you d e r i v e from t h e f r e e atom? How good a r e t h e atomic wavefunctions t o which you f i t t h e pseudo wavefunctions?
S.G. Louie: The r e s u l t s a r e i n s e n s i t i v e t o t h e d e t a i l s o f form o f t h e &initio i o n i c pseudopotentials near t h e nucleus provided they a r e derived using t h e norm-conserving c r i t e r i a such a s t h o s e described i n t h e Hamann-SchluterChiang o r t h e Kerkev scheme. The a l l - e l e c t r o n atomic wavefunctions a r e 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 w i t h i n t h e l o c a l d e n s i t y approximation. The a l l - e l e c t r o n wavefunctions and t h e pseudo-wavefunctions a r e i d e n t i c a l beyond a chosen c o r e r a d i u s from t h e nucleus i n t h e Kerker scheme. T h i s c o r e r a d i u s h a s t o be chosen small enough t o ensure high degree o f t r a n s f e r a b i l i t y o f t h e i o n i c p o t e n t i a l .
D.P. DiVincenzo: Are your r e s u l t s f o r t h e i n t r i n s i c and e x t r i n s i c s t a c k i n g f a u l t s ( i n p a r t i c u l a r t h e boundary d i l a t a t i o n s and t h e valence-derived gap s t a t e ) q u a l i t a t i v e l y o r q u a n t i t a t i v e l y a p p l i c a b l e t o t h e coherent twin boundary i n Si?
S.G. Louie: We have some preliminary r e s u l t s on t h e S i twin boundary. I t seems t h a t t h e q u a l i t a t i v e f e a t u r e s found f o r t h e i n s t r i n s i c and e x t r i n s i c s t a c k i n g f a u l t s a l s o apply f o r t h e twin boundary.
R.C. Pond: There is some experimental evidence supporting your suggestion t h a t t h e r e is some d i l a t i o n a t s t a c k i n g f a u l t s i n S i . I f t h e r e is an expansion, it follows t h a t t h e d i s l o c a t i o n s which bound s t a c k i n g f a u l t s do n o t have t h e Burgers v e c t o r s u s u a l l y assumed. Does t h i s imply t h a t t h e values o f s t a c k i n g f a u l t e n e r g i e s , obtained by experimental o b s e r v a t i o n s , should be r e v i s e d , and would t h e r e v i s e d v a l u e s correspond more o r l e s s c l o s e l y with your c a l c u l a t e d values.
S.G. Louie: Yes. However, I b e l i e v e t h e measured s t a c k i n g f a u l t e n e r g i e s a r e only approximate s i n c e they a r e derived from experimental d a t a using e l a s t i c i t y models.
The r e v i s i o n s discussed h e r e a r e probably w i t h i n t h e u n c e r t a i n t i e s o f both t h e experimental and t h e o r e t i c a l values.
M. Schluter: ( i ) Did you check hex diamond? Could it be t h a t a t high p r e s s u r e t h e hex phase is lower i n energy than BC8? ( i i ) Is t h e s t a c k i n g f a u l t s t a t e near t h e VBM s t a b l e a g a i n s t t h e r e l a x a t i o n s o f t h e s t r u c t u r e ?
S.G. Louie: ( i ) A t p r e s e n t we have n o t c a l c u l a t e d carbon i n t h e hexagonal diamond s t r u c t u r e . R e s u l t s from S i work show t h a t t h e hexagonal diamond s t r u c t u r e has s l i g h t l y h i g h e r energy than t h e cubic diamond s t r u c t u r e up t o very high p r e s s u r e . ( i i ) Preliminary r e s u l t s i n d i c a t e t h a t they a r e s t a b l e .
M. Riihle: ( i ) You determined t h e f o r c e s on t h e atoms i n S i i n ( o r n e a r ) a twin.
The f o r c e s w i l l r e s u l t i n a r e l a x a t i o n o f t h e atomic s t r u c t u r e (observed by HREM).
W i l l t h e r e l a x a t i o n i n f l u e n c e a l l your r e s u l t s ? ( i i ) Is it p o s s i b l e t o do s i m i l a r c a l c u l a t i o n s f o r metals?
S.G. Louie: ( i ) The c a l c u l a t e d value o f t h e f o r c e s i n d i c a t e s t h a t t h e r e l a x a t i o n w i l l be r a t h e r small. I expect only small changes i n t h e c a l c u l a t e d q u a n t i t i e r
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such a s t h e f a u l t e n e r g i e s and s t a c k i n g f a u l t s t a t e p o s i t i o n s . The e x i s t e n c e o f t h e s e d e f e c t s t a t e s is n o t expected t o change. ( i i ) Yes. We can do s i m i l a r c a l c u l a t i o n s f o r metals.
E. Molinari: Could you comment about t h e comparison between your r e s u l t s and t h e experiments, a s f a r a s t h e presence o f s t a c k i n g f a u l t s t a t e s i n t h e s i l i c o n bond gap is concerned?
S.G. Louie: The s t a c k i n g f a u l t s t a t e s t h a t we f i n d near t h e t o p o f t h e valence band maximum a g r e e w e l l with a s t a c k i n g f a u l t l e v e l observed i n photoluminescence measurements by Weber and Alexander (Ref. 28). The o t h e r p r e d i c t e d d e f e c t s t a t e s w i l l be more d i f f i c u l t t o observe because they a r e not i n s i d e t h e minimum gap o f t h e S i p r o j e c t e d band s t r u c t u r e .
