AB INITIO CALCULATIONS FOR HIGH PRESSURE PROPERTIES OF SOLIDS

HAL Id: jpa-00224300

https://hal.archives-ouvertes.fr/jpa-00224300

Submitted on 1 Jan 1984

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

AB INITIO CALCULATIONS FOR HIGH PRESSURE PROPERTIES OF SOLIDS

M. Cohen

To cite this version:

M. Cohen. AB INITIO CALCULATIONS FOR HIGH PRESSURE PROPERTIES OF SOLIDS.

Journal de Physique Colloques, 1984, 45 (C8), pp.C8-7-C8-11. �10.1051/jphyscol:1984802�. �jpa- 00224300�

JOURNAL DE PHYSIQUE

Colloque C8, supplément au n°11, Tome 45, novembre 1984 page C8-7

AB INITIO CALCULATIONS FOR HIGH PRESSURE PROPERTIES OF SOLIDS

M.L. Cohen

Department of Physics, University of California, Materials and Molecular Research Division, Lawrence Berkeley L a b o r a t o r y , Berkeley, CA 94720, U.S.A.

Résumé - On présente ici une revue des calculs récents, par la méthode du pseudopotentiel, des structures cristallines et des propriétés sous haute pression des solides.

A b s t r a c t - A review of recent pseudopotential c a l c u l a t i o n s of the s t r u c t u r a l and high pressure p r o p e r t i e s of s o l i d s i s presented.

The recent renaissance i n high pressure studies of s o l i d s has been motivated by the development of powerful experimental and t h e o r e t i c a l techniques. On the e x p e r i - mental s i d e , the s i m p l i f i c a t i o n of the high-pressure apparatus, such as developments r e l a t e d t o diamond a n v i l s and ruby pressure sensors, have made i t p o s s i b l e f o r a broad spectrum o f researchers to enter t h i s area. In a d d i t i o n , a v a r i e t y of novel experiments are now p o s s i b l e , and as a r e s u l t , a great deal of important i n f o r m a t i o n on h i g h l y condensed matter has been o b t a i n e d . On the t h e o r e t i c a l s i d e , new models and the a v a i l a b i l i t y of high-speed, large-memory computers allow r e a l i s t i c d e t a i l e d computations o f s t r u c t u r a l and e l e c t r o n i c p r o p e r t i e s at low and high pressures.

This review w i l l concentrate on t h e o r e t i c a l developments, and i n p a r t i c u l a r , the focus w i l l be on the t o t a l - e n e r g y - p s e u d o p o t e n t i a l method / l / . A d e s c r i p t i o n o f the method w i l l be given followed by some recent r e s u l t s .

The pseudopotential model o f a s o l i d considers the system t o be composed o f cores and valence e l e c t r o n s . Using2Si6as an example, the Si core consists o f the nucleus and ten core e l e c t r o n s , 1s22s 2p . The f o u r valence e l e c t r o n s ( h e r e a f t e r r e f e r r e d t o as e l e c t r o n s ) , 3s 3 p , are assumed t o be f r e e t o wander through a p e r i o d i c array o f cores. To compute s t r u c t u r a l p r o p e r t i e s , a Bravais l a t t i c e plus basis is assumed f o r the c o r e s , and the t o t a l energy of the system i s computed. I f the t o t a l energy o f several s t r u c t u r e s are compared, the lowest energy s t r u c t u r e i n t h i s group is considered t o be the appropriate one f o r the system.

To compute the t o t a l energy, the f o l l o w i n g i n t e r a c t i o n s must be determined: core- c o r e , e l e c t r o n - e l e c t r o n , and c o r e - e l e c t r o n . Core-core terms are evaluated using the Madelung approach, and the c o r e - e l e c t r o n c o n t r i b u t i o n can be obtained from p s e u d o p o t e n t i a l s . The 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 v o l v e Coulomb, exchange, and c o r r e l a t i o n e f f e c t s . These l a t t e r terms are approximated by l o c a l d e n s i t y func- t i o n a l s which give an accurate d e s c r i p t i o n of the ground s t a t e / 2 / . Once the core and e l e c t r o n i c k i n e t i c energies are added, the t o t a l energy can be evaluated i n momentum space / 3 / f o r a given c o n f i g u r a t i o n .

Early a p p l i c a t i o n s of t h i s approach focused on Si / 4 , 5 / . By comparing several p l a u - s i b l e s t r u c t u r e s , the diamond s t r u c t u r e was shown t o have the lowest energy f o r volumes near the observed volume. The volume f o r the minimum t o t a l energy gives an e v a l u a t i o n o f the l a t t i c e constant w h i l e the curvature of the energy versus volume c u r v e , E ( V ) , y i e l d s an estimate f o r the bulk modulus ( F i g . 1 ) .

The cohesive energy o f Si can be obtained from a comparison of the t o t a l energy f o r the observed l a t t i c e constant and f o r a very l a r g e l a t t i c e constant where the atoms are e s s e n t i a l l y n o n i n t e r a c t i n g . Another p r o p e r t y of the system which can

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984802

J O U R N A L DE PHYSIQUE

- 7 92

0.6 0 7 0.8 0 9 1 0 1 1

Volume

Fig. 1 - Energy versus volume (normalized t o the e q u i l i b r i u m volume i n t h e diamond phase) f o r S i i n various assumed c r y s t a l s t r u c t u r e s . The dashed l i n e i s the common tangent between t h e diamond and white t i n curves.

be s t u d i e d using the E(V) curve i s the s o l i d - s o l i d phase t r a n s i t i o n . By c o n s t r u c t i n g a common tangent ( F i g . I ) , the t r a n s i t i o n volumes can be evaluated by examining the endpoints o f t h e tangent l i n e . The slope o f t h i s l i n e gives an estimate o f t h e t r a n s i t i o n pressure. For S i , the r e s u l t s f o r the l a t t i c e constant, cohesive energy, b u l k modulus, diamond t o w h i t e t i n t r a n s i t i o n volume, and t r a n s i t i o n prgssure are (numbers i n parenthesis i n d i c a t e d e v i a t i o n from the measured value): 5.45 A (0.4%); 4.67 eV (1%); 0.98 Mbar (-1%); 0.928 (1.1%); 0.718 (-1.1%); and 99 kbar (10%).

These r e s u l t s are p a r t i c u l a r l y s a t i s f y i n g s i n c e the o n l y i n p u t r e q u i r e d was the atomic number t o generate the pseudopotential and a subset o f c r y s t a l s t r u c t u r e s . If the atomic mass i s i n c l u d e d as i n p u t , the phonon frequencies and Gruneisen para- meters can be evaluated /1,6,7/. The method uses a d i s t o r t e d c r y s t a l t o simulate a frozen phonon. By comparing the c a l c u l a t e d t o t a l energy f o r the i d e a l and the d i s t o r t e d c r y s t a l , t h e energy o f the phonon can be obtained. This technique can be used t o g i v e phonon frequencies a t s e l e c t e d p o i n t s i n the B r i l l o u i n zone and d i s p e r s i o n curves i n symmetry d i r e c t i o n s .

The pseudopotentials used i n the c a l c u l a t i o n s described a r e generated from atomic wavefunctions 18-131. This method i s t h e r e f o r e considered t o be &I i n i t i o since o n l y 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 are used i n computing s o l i d t a t e ef- fects. The e l i m i n a t i o n of t h e core e l e c t r o n s allows a p p l i c a t i o n o f t h i s approach t o heavy atoms w i t h o u t much change i n the requirements f o r computer time. Compari- sons have been made w i t h a l l - e l e c t r o n approaches, and t h e r e s u l t s are c o n s i s t e n t i n the cases I 1 4 1 tested.

Some o f the e a r l y extensions beyond the c a l c u l a t i o n s o f the b u l k p r o p e r t i e s o f the group I V semiconductors were t o surfaces /1,2,14/, and s t r u c t u r a l and h i g h pressure p r o p e r t i e s o f metals /1,15-171, 111-V semiconductors 1181, 11-VI 1191, and I - V I I compounds 1201. These c a l c u l a t i o n s were as successful as those done f o r the group I V m a t e r i a l s and y i e l d e d p r e d i c t i o n s o f the p r o p e r t i e s o f the h i g h pressure phases.

A few p r o t o t y p e examples are discussed below.

Aluminum i s chosen as a p r o t o t y p e metal /15/. The s t r u c t u r a l and v i b r a t i o n a l proper- t i e s were computed and found t o be i n good agreement w i t h the measurements. The c a l c u l a t i o n s for t h e h i g h pressure phases p r e d i c t e d an f c c t o hcp t r a n s i t i o n around 2 Mbar and an hcp t o bcc t r a n s i t i o n a t about 4 Mbar i n reasonable agreement w i t h

e s t i m a t e s based on a n o t h e r c a l c u l a t i o n a l methcd 1211. Experimental d a t a a t these p r e s s u r e s a r e n o t y e t a v a i l a b l e t o o u r knowledge. However, t h e r e appears t o be no r e s t r i c t i o n s i n a p p l y i n g t h i s approach t o m e t a l s a t low and h i g h pressures.

A1 though t h e E(V) curves a r e s i m i l a r f o r S i and Ge, t h e y b o t h d i f f e r from diamond 1221. T h i s d i f f e r e n c e a r i s e s from t h e absence o f p - s t a t e s i n t h e carbon c o r e and t h e absence o f d - s t a t e s f o r t h e p r i n c i p l e quantum number n = 2. The m a j o r f e a t u r e ( F i g . 2) i s t h e l a c k o f a h i g h p r e s s u r e t r a n s i t i o n t o t h e w h i t e t i n phase.

0l6 017 0 8 i 3 1'0 1'1 1'2 ?3 VOLUME

F i g . 2 - T o t a l - e n e r g y versus volume ( n o r m a l i z e d ) curves f o r s i x phases o f carbon.

For t h e phases considered, t h e r e i s a c a l c u l a t e d t r a n s i t i o n from t h e diamond t o t h e s i m p l e c u b i c phase a t 23 Mbar. More r e c e n t c a l c u l a t i o n s by Biswas g

c.

by Y i n have c o n s i d e r e d t h e BC-8 phase and f i n d a l o w e r p r e s s u r e t r a n s i t i o n around 12 Mbar. The g r a p h i t i c phases o f C and S i have a l s o been s t u d i e d /23/. For g r a p h i t e t h e c a l c u l a t e d v a l u e s f o r t h e i n - p l a n e and i n t e r l a y e r spacings a r e i n good agreement w i t h experiment. S i m i l a r l y , r e s u l t s a r e good f o r t h e computed i s o t r o p i c b u l k modulus and t h e graphite-diamond s t r u c t u r a l energy d i f f e r e n c e .

Recent c a l c u l a t i o n s on S i p r e d i c t t h a t g r a p h i t i c S i i s weakly bound w i t h an energy 0.71 eYlatom h i g h e r t h a n t h e diamond phase. A l a r g e n e g a t i v e p r e s s u r e o f -69 k b a r i s r e q u i r e d f o r i t s f o r m a t i o n . Hence, t h i s phase o f S i i s n o t l i k e l y t o be formed.

A r e c e n t c a l c u l a t i o n by Chang on s i m p l e hexagonal ( s h ) and hcp S i e x p l o r e s

t h e t r a n s i t i o n from w h i t e t i n t o sh t o hcp ( F i g . 3 ) . The s h phase i s m e t a l l i c w i t h a r e a s o n a b l y l a r g e d e n s i t y o f s t a t e s a t t h e Fermi energy. E s t i m a t e s o f t h e e l e c t r o n - phonon c o u p l i n g suggest t h a t t h i s m a t e r i a l i s p r o b a b l y a semiconductor.

The e x t e n s i o n s o f t h e t h e o r y t o i o n i c m a t e r i a l s l i k e NaCl /20/ and MgO 1241 a r e of i n t e r e s t because o f t h e p o s s i b l e s t r u c t u r a l t r a n s i t i o n s from t h e NaCl t o CsCl (B1 t o B2) s t r u c t u r e s . For NaC1, t h e t r a n s i t i o n p r e s s u r e o f 27 GPa i s i n good agree- ment w i t h t h e measured v a l u e /25/, b u t some d i f f e r e n c e s a r e found between t h e c a l c u -

l a t e d 1201 and measured 1251 pressure-volume curves. The most r e c e n t e x p e r i m e n t a l r e s u l t s by Heinz and Jeanloz a r e i n c l o s e r agreement w i t h t h e t h e o r y .

For ~ 9 ' 0 , th e p r e d i c t i o n 1211 o f t h e t r a n s i t i o n p r e s s u r e f o r B1 t o 82 i s a t 10 Mbar.

T h i s r e s u l t suggests t h a t i t i s u n l i k e l y t h a t t h e 82 phase e x i s t s even i n t h e l o w e r m a n t l e o f t h e e a r t h .

A l t h o u g h t h e total-energy-pseudopotential approach has been a p p l i e d s u c c e s s f u l l y t o a broad spectrum o f s o l i d s w i t h a minimum o f i n p u t 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, i t i s s t i l l l i m i t e d by t h e c h o i c e o f c r y s t a l s t r u c t u r e s t e s t e d . Hence,

JOURNAL DE PHYSIQUE

-7.83 -

S i

-

-

\

%

- 7 . 8 7

% PD L

a, C -7.89

- 7 . 9 1

0.6 0.8 1 . O

V o l u m e

F i g . 3 - T o t a l s t r u c t u r a l energy o f S i v e r s u s volume n o r m a l i z e d t o t h e e q u i l i b r i u m volume i n t h e diamond phase.

i t i s d i f f i c u l t t o know whether a computed h i g h p r e s s u r e phase i s t h e l o w e s t p o s s i - b l e . One method which can be used as an a i d i n choosing good c a n d i d a t e s f o r s t r u c - t u r e s i n v o l v e s t h e computation o f Hellmann-Feynman f o r c e s . T h i s approach has been a p p l i e d t o s u r f a c e s , and some e x t e n s i o n s t o b u l k s o l i d s a t h i g h p r e s s u r e s a r e i n p r o g r e s s .

The f o r c e c a l c u l a t i o n s f o r s u r f a c e s proceed i n t h e f o l l o w i n g way. An i d e a l geometry, t h a t i s , ( l x l ) , i s chosen f o r t h e s u r f a c e , and t h e Hellmann-Feynman f o r c e s a r e com- p u t e d on t h e s u r f a c e atoms. I f t h e s u r f a c e i s s t a b l e , moving t h e atoms r e s u l t s i n r e s t o r i n g f o r c e s . I f a s u r f a c e r e c o n s t r u c t i o n s h o u l d r e s u l t , t h e r e a r e r e s i d u a l f o r c e s . By moving t h e a t m s i n t h e d i r e c t i o n s suggested by t h e r e s i d u a l f o r c e s , a new s t r u c t u r e i s o b t a i n e d . Each new c o n f i g u r a t i o n r e q u i r e s a new s e l f - c o n s i s t e n t f o r c e c a l c u l a t i o n . Once t h e f o r c e s a r e near zero, a l o w energy s t r u c t u r e has been obtained. I t i s s t i l l n o t p o s s i b l e t o c l a i m t h a t t h e s t r u c t u r e i s t h e minimum ener- gy s t r u c t u r e f o r t h e assumed volume because t h e c a l c u l a t i o n may y i e l d o n l y a l o c a l minimum.

O t h e r approaches i n c l u d i n g Monte C a r l o samplings o f c o n f i g u r a t i o n space a r e p o s s i b l e , b u t a t t h i s p o i n t , t h e r e i s no c e r t a i n p r e s c r i p t i o n f o r o b t a i n i n g t h e p r o p e r s t r u c - t u r a l s u b s e t t o t e s t . Most choices a r e made by t r a i l and e r r o r .

Acknowledgement - T h i s work was supported by N a t i o n a l Science Foundation G r a n t No.

DMR8319024 and by t h e D i r e c t o r , O f f i c e o f Energy Research, O f f i c e o f B a s i c Energy Sciences, M a t e r i a l s Sciences D i v i s i o n o f t h e U.S. Department o f Energy under Con- t r a c t No. DE-AC03-76SF00098.

References /1/ COHEN M. L., F h y s i c a S c r i p t a T 1 (1982) 5.

/2/ KOHN W. and SHAM L. J., ~ h y s . R e v . , 140 (1965) A1333.

/3/ IHM J., ZUNGER A. and COHEN M. L., J. Phys. C

12

/4/ IHM J. and COHEN M. L.. , Phys. Rev. B 1 (1980) 1527; e r r a t u m Phys. Rev. B 22 (1980) 2135.

/5/ V N M. T. and COHEN M. L., Phys. Rev. L e t t . 45 (1980) 1004; Phys. Rev. B 6,

(1982) 5668.

/6/ YIN M: T. and COHEN M. L., Phys. Rev. B 26 (1982) 3259.

/7/ KUNC K. and MARTIN R. M., Phys. Rev. L e t t , 48 (1982) 406.

/8[ STARKLOFF T. and JOANNOPOULOS J. D., Phys. E v . B 16 (1977) 5212

ZUNGER A. and COHEN M. L., Phys. Rev. B 18 (1978) 5449.

HAMAHN D. R., SCHLUTER M., and CHIANG C.,~hys. Rev. L e t t . 43 (1979) 1494.

KERKER G., J. Phys. C 13 (1980) L189.

LOUIE S. G., FROYEN and COHEN M. L., Phys. Rev. B 26 (1982) 1738.

YIN M. T. and COHEN M. L., Phys. Rev. B 25 (1982) 7 4 0 x

McMAHAN A. K., YIN M. T., and COHEN M. L z Phys. Rev. B 4 (1981) 7210.

LAM P. K. and COHEN M. L. , Phys. Rev. B 3 (1983) 5986.

CHOU M. Y., LAM P. K., and COHEN M. L., Phys. Rev. B 28 (1983) 4179.

LAM P. K., CHOU M. Y., and COHEN M. L., J. Phys. C 1 7 7 1 9 8 4 ) 2065.

FROYEN S. and COHEN M. L., Phys. Rev. B 28 (1983) 3258.

CHANG K. J. and COHEN M. L., S o l i d S t a t e T o m . 50 (1984) 487.

FROYEN S. and COHEN M. L., Phys. Rev. B 9 ( 1 9 8 q 3770.

MORIARTY J. A. and McMAHAN A. K., Phys. Rev. L e t t . 3 (1982) 809.

CHANG K. J., FROYEN S., and COHEN M. L., Phys. Rev. L e t t . 50 (1983) 2006.

YIN M. T. and COHEN M. L., Phys. Rev. B 29 (1984) 6996.

CHANG K. J. and COHEN M. L., Phys. Rev. n o be p u b l i s h e d ) . SATO-SORENSEN Y., J. Geophys. Res. 88 (1983) 3543.