ELECTROSTATIC ENERGY AND LATTICE VIBRATIONS IN THIN IONIC SLABS

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ELECTROSTATIC ENERGY AND LATTICE

VIBRATIONS IN THIN IONIC SLABS

G. Kanellis, J. Morhange, M. Balkanski

To cite this version:

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JOURNAL DE PHYSIQUE

Colloque C6, suppldment au n o 12, Tome 42, de'cembre 1981 page C6-341

ELECTROSTATIC ENERGY AND LATTICE VIBRATIONS IN THIN IONIC SLABS

G. Kanellis", J.F. Morhange and M. Balkanski

Laboratoire de Physique des Solides de Z1Uniuersit6 Pierre e t Marie Curie, associd au CNRS, 4 Place Jussieu, 75230 Paris Cedex 05, France.

A b s t r a c t . - We have c a l c u l a t e d t h e E l e c t r o s t a t i c energy p e r c e l l f o r t h i n i o n i c s l a b s of any s t r u c t u r e and any o r i e n t a t i o n , a s a f u n c t i o n of t h e p o s i t i o n of t h e c e l l and t h e t h i c k n e s s of t h e s l a b . I t i s shown t h a t a macroscopic d e p o l a r i z i n g f i e l d i s c r e a t e d i n s i d e t h e s l a b s of c e r t a i n o r i e n t a t i o n s . The Coulomb i n t e r a c t i o n betweenpla- ne l a t t i c e s i s a l s o c a l c u l a t e d f o r t h e g e n e r a l c a s e and t h e r e s u l t s a r e a p p l i e d t o G a s .

1 . The E l e c t r o s t a t i c energy.- Considering a s l a b p a r a l l e l t o t h e ( h k l )

p l a n e of t h e c r y s t a l , one can choose such a u n i t c e l l t h a t t h e p r i m i t i v e t r a n s l a t i o n v e c t o r s

Z l

and

a2

l i e on t h e p l a n e ( h k l l and

a3

o u t of i t . The e x p r e s s i o n f o r t h e e l e c t r o s t a t i c energy p e r c e l l of t h e c e l l Z = l l l , Z2, Z 3 1 , analogous t o t h e Madelung c o n s t a n t i s N ( 1 )

t

K

z

j = Z

2' 2' K' rink- blende 1 2 -0.90

-

0Qb40Q-0r0 9 where r o i s t h e n e a r e s t neighbor d i s t a n c e , K 01 ( 1 0 0 )

-1 .oo

-

and

run

over a l l atoms i n t h e above d e f i n e d

u n i t c e l l , Z 1 , Z 2 r u n from -a t o +m, EK i s t h e -1.30 - _{charge f r a c t i o n a t r i b u t e d t o i o n} K and N mea-

$

s u r e s t h e t h i c k n e s s of t h e s l a b i n u n i t c e l l s . By s e t t i n g f o r each p o s i t i o n v e c t o r

1slZ

=

I G , , ~ ~

+

[Z1I

2 (2) where II and L d e s i g n a t e t h e p a r a l l e l a n d p e r - p e n d i c u l a r t o t h e s l a b components of t h e v e c t o r s r e s p e c t i v e l y , and using a v a r i a n t of Ewald's method we e v a l u a t e t h e two -dimensional i n f i -

-1 -07 n i t e sum i n r e l a t i o n (1). For 2 = ( 0 0 1 3 ) we t h i c k n e s s (number o f c e l l s ) o b t a i n , F i g . 1. E l e c t r o s t a t i c energy p e r c e l l for c e n t r a l c e l l s i n a s l a b as a f u n c t i o n o f i t s t h i c k n e s s .

*Perm. a d r e s s : F i r s t Laboratory of P h y s i c s , U n i v e r s i t y of T h e s s a l o n i k i

-

Greece.

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JOURNAL DE PHYSIQUE

Ti-

"'*

₁

'K'K''

1

H ( T / ~ ~ / R ) E ~ { - ~ ~ ? . s ( K K ~ ) - ~

1

~ ~ ~

( K ~ I

~ r

,

(3) l s

f -

2Rsa K P

M'

' a KK'

where

r runs

over all atoms in the unit cell lying on the same plane I h k Z ) ,

g

and are vectors of the two -dimensional direct and recipro- cal lattices respectively, Sa is the area of the two-dimensional unit cell and

It is expected that for very thick slabs ( N + - l expression ( 3 ) gives the Nadelung constant of the structure. This is the case indeed only if the plane ( h k 2 ) is neutral. I f not, the limit of a ( Z 3 , N ) for N + m is different by the amount

where is the dipole monent of a unit cell, v _a its volume and

Go

is the unit vector perpendicular to the plane ( h k Z ) . Aa(m) expresses the energy of a unit cell in the macroscopic depolarizing field created by the charged planes of the slab. In fig. 1 we give some results for slabs of different orientations in the zink -blende structure.

11. The Coulomb Interaction.- The field at a lattice site ~ ( Z K ) due to all dipoles F ( Z k 3 at lattice sites

G f k ?

is given by,

where

q

is the wave vector.

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+

i h { ( ~ , +qa)cosB <- ( Tf - q ~ e q { - 2 ~ 1 ? +GI

1;

( Z 3 ~ ; Z r ~ 7 3

1 - ~ n \ i ? ~ j E f Z ~ ~ ; Z k ~ 1

,

(gal

c o s a been the direction cosines of

7

0'

As it can be seen, all terms in the above expressions are regular functions of

q

for

q

+O. Hence there is no macroscopic field lying on the plane of the slab.

By differentiating the total electrostatic energy of the slab in afield equal in magnitute and opposite in direction to the depolarizing field,

the nacroscopic field due to vibrations of the plane lattices along

c0

is obtained. For a slab parallel to the plane f 1 1 i ) of the zink -blende structure it is,

Emac _{(ZK) =}

_-

4 l ~

cosa

x

(6Z3N6~R + 6 Z 3 1 6 ~ 1 +6ZjN6~2

+

a ' a L

12,

where L is the thickness of the slab and d the shortest distance between two successive planes.

Expression ( 9 ) gives the well known nacroscopic field for infinite thickness, while for thinner slabs implies interaction between each surface plane and the rest of the slab.