THE GENERALIZED PSEUDOATOM FORMALISM IN LATTICE DYNAMICS

HAL Id: jpa-00221217

https://hal.archives-ouvertes.fr/jpa-00221217

Submitted on 1 Jan 1981

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.

THE GENERALIZED PSEUDOATOM FORMALISM

IN LATTICE DYNAMICS

M. Ball

To cite this version:

JOURNAL DE PHYSIQUE

CoZZoque C6, suppZ6ment au n o 12, Tome 4 2 , d6cembre 1981 page C6-519

THE GENERALIZED PSEUDOATOM FORMALISM IN LATTICE DYNAMICS

M.A. Ball

D. A. M. T.P., University of Liverpool, Livevpooz L69 3BX, U.K.

Abstract.

The change

1

in potential due to a phonon i s expressed i n terms of the susceptibil and the e f f e c t i v e charge vector. I t i s a n a l y t i c i n metals o r where p i p i s zero S and hence can be uniquely s p l i t i n t o parts which r i g i d l y f6llow the ions and parts which deform as the ions move. This

'generalised pseudoatom' i s neutral. The dynamical matrix i s expressed in terms of I both in the reciprocal l a t t i c e representation and oth6r

representgtions

.

kle show f i r s t t h a t the charge density of a f i n i t e system i s a sum of uniquely- defined pseudoatoms, each consisting of a part moving r i g i d l y with an ion and a part which deforms / l / . Suppose the ion a t R0 moves t o

R?

+ _{The change in}

-

J

charge density t o f i r s t - o r d e r i s f .(r).aR -'NOW f . ( r ) can be written

-J

-

-j'

0 -J

f j ( r ) =

-

p . ( r

-

59)

+ y X

l+(r

-

!j)

J - J (1

l *

Thus to first order i n & R i , the total charge density of the system i s no(r) + / ( p j ( r -*Ej) +

& E j

.

J X B.(r

-

F$))

J J (2).

Translational invariance / l / makes n o ( r ) zero. The term p . ( r

-

R . ) i s the p a r t

J -J

which moves r i g i d l y with the ion and the other term i s the deformation.

In c r y s t a l s the ions move collectively as phonons. For ( 2 ) t o be valid, the charge density m u s t be analytic in q , - the wave-vector of the phonon. Let the equilibrium position vector of the K ion i n the unit c e l l a t Q

-

be R'

-

( Q , K )

_{- -}

S a

_-

t R'

W

( K ) . This i s displaced 6R ( K ) exp tiq

_{- - - -}

.

R O ( a , ~ ) l by the phonon. The change i n the e l e c t r o n i c charge density i s written

lZKz 6 R ( ~ ) . f ~

(9

+ g 9 e x p [;(q 4 - g ) r ] eXp [ig.R(K)]

-

_{- -}

9

- -

where g are reciprocal l a t t i c e vectors and T i s the c e l l volume. The change in the t o t a l charge density i s written f (g +

g , ~ ) .

We write the potential of the nucleus and the core electrons as

W(r-R(&,K);K).

Then

fe(4 + 43") =

x

_{(C! +}

g.

4

_{+ $ ' ) ( C ! +} $')W(! + gt,K)exp[i(g

-

g l ) . R ( ~ ) l ( 4 )

The electron-phonon operator can be expressed in terrns of f and U. 1 ( q + g , ~ ) = i (q + g)W(q + ? , K ) + v(q + g) f e (q + g;)

- . . -

-

-

-

W

-

-

-

( 5 )

where v i s the Coulomb interaction. Then

!

(! + !,K) = i g l Z ~ - l ( q

_{_}

+ 9, 9 +

_-

9' ) ( q + 9' )W(q

_-

+ g' ,r)expIf (9-9' I.R(r)I

_{- .}

_-

(6)

C6-520 JOURNAL DE PHYSIQUE

where E i s the d i e l e c t r i c function matrix. Inverting /2/

(g + g)W(g + % , K ) = i c ~ ( q

-

+ g,

- -

q + g ' ) I ( q

-

-

+ g ' , ~ ) e x p [ i ( g

-

-

g 1 ) . R ( ~ ) 1

-

V ( 7 )

I i s the change in potential f e l t by a classical point charge.

2'

i s the change f e l t by a valence electron: i t includes exchange and correlation.

To investigate the a n a l y t i c i t y of

-

I , we w r i t e I (q

_{- -}

+ g , ~ )

_-

in terms of the ' s u s c e p t i b i l i t y '

2

and the e f f e c t i v e charge vector Z

_-

( q , ~ ) _.. / 3 / :

In i n s u l a t o r s and metals lim (qiK.= ?eff ( K ) - q

-

-

W

where

zeff

( K ) i s the e f f e c t i v e charge tensor. Hence

-

1

I

( 9 , ~ ) =

-

E (q,g) v(g) z ( g , ~ ) (11

and

(12) There can only be non-analytic behaviour in the l i m i t q

- -

-+ 0. In insulators

the second term i n (12) i s non-analytic i f leff ( K ) i s non-zero so

!

(g

+

g , ~ ) i s -1 A

-

a n a l y t i c i f and only i f

kff

( K ) i s zero. In a metal v ( ? ) €

(y,?)

--l/X(q,qh both and :(cj + g , ? ) tend t o a f i n i t e l i m i t and I(q + g , ~ ) i s always analytic.

X

-

- -

-

T h u s i n metals, and in-those insulators where

zeff

( K ) i s zero

!

i s a n a l y t i c and and so a r e

I '

and f . Each such vector f i e l d c%n be uniquely decomposed i n t o vectors parallel and perpendicu7ar t o q + g, i . e .

f

(9 + %,K) = i ( - (9 + g) P

(4

-

+ ;,l<) V + (q

-

+ g)

-

X B(q

-

+ g , K ) )

-

(13)

In real space, t h i s decomposition gives ( 2 ) f o r f and a similar r e s u l t f o r 1 ' . In insulators where

zeff

( K ) i s non-zero f i s not a n a l y t i c and the charge density can- not completely be described i n terms of pseudoatoms /2/.

One advantage of the formalism i s t h a t the pseudoatom i s neutral e f f

l i m ~ ( q , ~ )

_-

= i

.s

_-

z

_-

( K ) . ~ c ' (q.q)v(?)/4ne2 = 0

Another i s t h a t i t gives the potential which an electron sees when a phonon i s present. The r i g i d p a r t of t h i s potential i s

u(g + g,.) = (g + 4). 1 . ( s + o I K ) / I ~ + 912

V - - -

-

W

The main p a r t of the e f f e c t i v e interaction between the ions i s /3,2/

V(%+%

r

d.1

=

cv+,y,

k )

€-'(v

+S

,9+31)

h&

+$

~ ' ) / v * + ~ )

(17). usin: - (77) and the inversion procedurey7), thcdynamical matrix becomes .

(?Qw

fl[k/>)-''2

7

(v+g,

X)

g

~ ? ~ + g :

k 3

.G?~+~,

9+3

eP

'cqq+93

Q

7

-

9,St

"

-

-

I A & I - . -

-.-

generalised t o other representations by considering E as a matrix and

L

as a

' v e c t o r ' . Then the dynamical.matrix in a general rewesentation i s

wmk)

L

~ f r l ! ' ? - ' 2

--P

c

zle2d

c

p,

(B

(I??,

A .-

Y9&jb1E"&)hp.~P

lia

C Z C ~

-,RC>:#

(191,

~ t , ) ' z

This expression (19) avoids the inversion procedure needed t o d e r i v e x from

X G

.

Using the R.P.A. becomes

A '

t

g

where

Z

e * e l u n t ) r , e h . , t - s (20)

P

however, h i s d e f i n i t i o n o f

8

is d i f f e r e n t from ours.

Equations (20 and (21) a r e useful when we can use the pseudoatom concept. Then we replace by

2'

and incorporate a l l t h e exchange, e t c . , e f f e c t s i n t o

2.

A'

can

U

then be written-in the pseudoatom form (17), i .e.

(22)r the large Coulombic a t t r a c t i v e and repulsive terms has already been accomplished. I f there i s a s u i t a b l e approximation f o r the potential U , a n d 2 can be neglected, i t i s practical to calculate I ' ( ~ , K ) in c e r t a i n representations, e.g. ( ~ , m )

_-."

o r b i t a l s o r the tight-binding representation /4/.

When ?(?,K) i s pure imaginary i n a cubic material, the only direction f o r Z(CJ,K) i s along q , so t h a t I ' ( q , ~ ) _-.

_-

has no deformation part. When there is i n - version symmetry;

2

( g , ! +

9)

i s r e a l , so t h a t L ( ~ , K ) i s pure imaginary provided

~ . R ( K )

_-

_-

i s an integer times n.- This occurs i n cubic-crystals with one ion per unit c e l l o r with the NaCl s t r u c t u r e , but not with the diamond s t r u c t u r e or the A15 s t r u c t u r e . Thus I ' ( q , ~ . )

_-

_-

has no deformation in the a l k a l i metals, nor i n Nb, MO, MoC and TaC, but may have s i g n i f i c a n t deformation i n Ge and Si and the A15 metals. I t would be of i n t e r e s t t o measure the change i n electron-phonon coupling and in

TC i n Nb, Pb and MO caused by s t r a i n s which a l t e r the symnetry and thus allow deformation.

The second term of (12) i s i n the q-direction, so t h a t i f q i s perpendicular t o g i t contributes a' deformation p a r t t o I ( q _-.- + g , ~ ) . If I(q + - g , ~ ) is calculated

_-

from a band-structure calculation, t h i s term i s neglected,-bjt i i metals i t con- contributes t o the e l a s t i c constants.

References

/ l / Ball, M. A . , J . Phys. C: Solid S t a t e Phys. 8 (1975) 3328. Pickett, W . E . , J . Phys. C : Solid S t a t e Phy?. (1979) 1491. /2/ Ball, M. A. J. Phys. C: Solid S t a t e Phys.

10

(1977) 4921. 131 Sham, L . J . , Dynamical Properties of Solids I (1974)

ed. by G. K. Horton and A. A. Maradudin (North-Holland, ~msterdam) /4/ Sinha, S. K . , Dynamical Properties of Solids 111 (1980)