Calculation of the Deformation Potentials by the Green Function Method

Revista Brasileira de Física, Vol. 8, Nº 1, 1978

Calculation of the Deformation Potentials by the Green Function Method: Niobium* E. Z. SILVA*" and L. G. FERREIRA *, Universidade de São Paulo, São Paulo SP

Instituto de Física

Recebido em 23 de Setembro de 1977

I n t h i s paper we show how t o introduce, i n the band s t r u c t u r e c a l c u l a t i on by the KKR method, a simultaneous e v a l u a t i o n o f the i s o t r o p i c mation p o t e n t i a l s , wi t h o u t al t e r i n g the s t r u c t u r e f a c t o r s .

An

deforappl i c a -

t i o n t o niobium i s presented.

Neste a r t i g o , mostramos como i n t r o d u z i r o c á l c u l o dos p o t e n c i a i s de deformação em um c á l c u l o de e s t r u t u r a de f a i x a s p e l o &todo Kohn e Ros toker. estrutura.

A v i r t u d e de nosso método

é

de

Korringa,

de preservar os f a t o r e s de

Como aplicação, estudamos o caso do metal Niõbio.

1. INTRODUCTION The Green f u n c t i o n (KKR, m u l t i p l e s c a t t e r i n g ) c r y s t a l and molecules,

method (Refs.l,2

though much f a s t e r than o t h e r methods,

3,4) !;

o u s l y handicapped whenever one i s faced w i t h the c a l c u l a t i o n o f the trix elements

of

a Hamiltonian p e r t u r b a t i o n .

These

for

seri ma-

c a l c u l a ions are

u s u a l l y compl i c a t e d due t o the p e c u l i a r expansion o f the wave f u n c t i o n i n the r e g i o n o u t s i d e the spheres c i r c u m s c r i b i n g the atoms.

*

Work supported by the

Paulo (FAPESP)

** Present

.

address: H.H.

TOL BS8 ITL, U.K.

***

fundaçãode Amparo à Pesquisa do Estado de W i l l s Physics Laboratory, Tyndal

.

Postal address: C.P.

I n that re-

20516, 01000

-

São Paulo SP.

são

Avenue, BRIS-

gion, t h e expansion i s rnade i n terms o f s c a t t e r e d f u n c t i o n s o f many cent e r s thus c o m p l i c a t i n g the c a l c u l a t i o n o f the volume i n t e g r a l s .

l n what

follows, we consider a s p e c i a l p e r t u r b a t i o n which can be handled by the Green f u n c t i o n method ir1 a s u r p r i s i n g l y simple way: the p e r t u r b a t i o n

of

an i s o t r o p i c l a t t i c e expansion i n cubic c r y s t a l s . A s s t r a i n deformation i f the l a t t i ce a f f e c t s the band s t r u c t u r e by means o f coef f i c i e n t s known as

deformation potentiats. For a given s t r a i n ten-

the energy o f a c e r t a i n s t a t e i n the B r i l l o u i n zone i s s h i f t e d by

where D..

23

i s the deformé~tionp o t e n t i a l tensor.

I n t h e special

case

of

an i sotrop~ i sc t r a i n i n e cubic c r y s t a l , Eq.(l .l) i s e q u i v a l e n t t o

where 6a i s the change i n the l a t t i c e parameter. note i s t o show how t o c:alculate C, i n Eq.(l .2),

The purpose

of

this

simultaneously w i t h the

band s t r u c t u r e c a l c u l a t i o n .

Z. THEORY I n order t o f i x our notation, we review the equations o f the KKR rnethod:

i s the secular equation, where X denotes the pai r (L,m) def i n i n g an g u l a r momentum;

sX

=

where jk and nR are the spheri c a l Bessel and Neumann functions; whi l e

an-

are t h e l o g a r í t h m i c d e r i v a t i v e s a t the sphere radius Fi and energy E;

,

K2 = € - V , , where

(2.4)

V , , i s t h e constant p o t e n t i a1 i n t h e region outs de the spheres ; G(X,X')

+

= 4a C IA(X,X1) r A ( K , 2)

,

(2.5)

A the I A * s denoting Gaunt i n t e g r a l s f o r A

r A ( s D = cot

(KR)

m

( L , M ) , and

4a

wL(m'

are the s t r u c t u r e f a c t o r s expanded as s e r i e s i n the reciproca1 la t t i c e

+

+

vectors g, f o r a given p o i n t k i n the B r i l l o u i n zone. When s t r a i n e d i s o t r o p i c a l l y ,

the l a t t i c e parameter changes from a

and a given energy s t a t e from

E,

i n the secular equation (2.1)

i s a f u n c t i o n o f a and E, and i f E

are t o be s o l u t i o n s t o (2.1)

to

E.

The determinant D o f the

O

t o a, matrix and E,

f o r the l a t t i c e parameters a and ao,we must

have

Expanding D(E,~) t o the f i r s t order i n (E at

-

E

o

) and ( a

- a o ) , We

avive

-

I n a band s t r u c t u r e c a l c u l a t i o n , one has simple access t o the denominator

I

i n Eq. (2.8) because, i n o r d e r t o f i n d the eigenvalue E one has o' an t o c a l c u l a t e the determinant f o r a n e t o f E values. A numeriral d e r i v a -

t i v e o f the determinant i n t h i s n e t produces

The c a l c u l a t i o n o f the numerator

1

ao

can be s i m p l i f i e d i f we consider E

O

the s p e c i a l form o f the s t r u c t u r e f a c t o r s

rA, Eq.(2.6),

now w r i t t e n as

-+

I n a l a t t i c e expansion, the r a t i o s W a 3 , R/a, and the vectors are kept constant. Thus, TA depends on

Ka,

E

-+

ak and ag

and a o n l y through the

product

that i s

Thus, i n s t e a d o f c o n s i d e r i n g the determinant D as a f u n c t i o n o f we may a l s o consider i t as f u n c t i o n of a and Ka. between

E

and K, we o b t a i n

E

and a,

Using t h e r e l a t i o n (2.4)

,

I n (2.101, performed.

the d e r i v a t i v e s appearing i n the second term can

be

readily

I n the f i r s t term, the numerator becomes

where MXX i s the minor o f the element ()i,X)

Denoting by Hhh,

,

i n the secular m a t r i x , and

the s e c u l a r m a t r i x element, we have

and o b t a i n

Thus one sees t h a t i n o r d e r t o make a c a l c u l a t i o n o f the c o e f f i c i e n t s

I

C

along w i t h the band s t r u c t u r e c a l c u l a t i o n , we a l s o need, besides the 10g a r i t h m i c d e r i v a t i v e , the values o f i n a n e t o f energies

E.

a~ a~ .1a 2 IE, 2 a

n d s

tabulated E

These values depend on the rnodel assumed f o r the

p o t e n t i a l and i t s change i n a l a t t i ce expansion.

3. APPLICATION TO NIOBIUM

The procedure described above was a p p l i e d t o the band s t r u c t u r e o f bium.

nio-

The p o t e n t i a l was assumed equal t o a sum o f atomic p o t e n t i a l s .

a l a t t i c e expansion,

In

the atomic p o t e n t i a l s were supposed t o move r i g i d l y .

S l a t e r ' s average o f the exchange i n t h e f ree- electron gas

approximation

was used5. The r e s u l t s f o r a l a t t i c e parameter a = 6.2377 a.u. b l e and i n Figs. 1 and 2. being numbered from below.

I n Fig.1, I n Fig.2,

are shown i n t h e Ta-

we p l o t the energy bands, the bands we show t h e behaviour o f t h e coef-

f i c i e n t C i n the B r i l l o u i n zone f o r the bands numbered i n Fig.1.

It

is

q t i t e remarkable t h a t C v a r i e s much w i t h i n a band, and any model, f o r the e l e c t r o n a c o u s t i c phonon i n t e r a c t i o n f o r instance, based on a constant C, i s very unreal i s t i c .

TABLE

-

Energy eigenva!ues and isotropic deformtion poten-

t i a l s i n ths B r i l l o u l n zone of nlobium.

REFERENCES

1. Korringa, J.,

Physica 13, 392 (1947).

2. Kohn, W. and Rostoker, N.,

3. Ham, F.S.

and Segall, B.,

4. Johnson, K.H., 5. S l a t e r , J.C.,

Phys. Rev. 94,

111 1 (1954).

Phys. Rev. 124,

1786 (1961).

J. Chern. Phys. 45, 3085 (1966). Phys. Rev. 81, 385 (1951).