On the Frenkel exciton theory of amorphous solids

Physica 46 (1970) 569-576

ON THE

FRENKEL

o North-Holland Publishing Co., Amsterdam

EXCITON

THEORY

I. GENERAL W. CHRISTIAENS

OF AMORPHOUS

SOLIDS

THEORY

and P. PHARISEAU

Laboratorium voor Kristallografie en Studie van de Vaste Stof, Rijksuniversiteit Gent, BelgiB Received 21 July 1969

synopsis In the present paper, the Frenkel exciton is discussed in the case of an amorphous solid, taking into account a three-fold degeneracy of the excited one-electron state. It is shown how this degeneracy gives rise to a supplementary diagonalization of a 3 x 3 matrix, resulting in the longitudinal and transversal p-like exciton eigenvalues. Extending a calculation technique of Born and Bradburn, an explicit expression, useful for numerical calculations, is obtained for the exciton energy. This expression depends on the radial distribution function and the interference function, which are the only available information we have about the geometrical structure of the amorphous material.

1. Introduction. We consider a disordered system of N atoms at rest. Each atom contains one valence electron and the electron spin is neglected. Although the crystal periodicity no longer exists, we assume that a shortrange order is still present in the system. Because the macroscopic density and the radial distribution function are the only experimental information we can obtain about the structure of the amorphous system, these functions should obviously be introduced explicitly into the present calculations. Furthermore we assume that the one-electron ground state is described by an s-state wave function and the one-electron excited state by an orthonormal set of three p-state wave functions. It is shown that for the tight-binding Frenkel exciton state the one-electron functions may be identified with atomic wave functions. Working along similar lines as in the original Frenkel paperr), it is possible to calculate the exciton energy, taking into account an appropriate configurational averaging technique to eliminate the structure of the amorphous substance. The degeneracy of the excited state leads to a supplementary evaluation of a simple characteristic equation. We obtain the longitudinal and transversal p-like exciton eigenvalues, which are similar to what is found in crystal exciton theory. Furthermore we calculate an explicit expression for the exciton energy 569

W. CHRISTIAENS

570

AND P. PHARISEAU

in the form of an integral, having unfortunately a relatively slow convergence. Therefore, the method of Born and Bradburns) for the evaluation of lattice sums, is extended to the case of disordered systems. 2. Theoretical treatment.

A wave function

describing

an excited

state

of

the considered N-electron system, is expressed in the form of a Slater determinant, built up with atomic wave functions v(ri - RI) for the ground state, and yfiLl(‘i- Ri) for each of the three excited states pP (,D = x, y, z). The set of vectors RI, . . . . Rg,....RN are the position vectors of the different constituent atoms. We can write: ~(rr -- RI)

I/", = (N!)-t A three-fold Hamiltonian H = ifI

...

~“(rr -

- RN) Rm,) ... pl(rl ... ... * ... pl"(cv - Rm)

... ...

degeneracy is also present for the wave functions for the N-electron system has the following form: [-(@/2m)

A, + JJ’(c)]

+ 4. Z$ e2/lrt -- rjl. i,$= 1

(1)

#&. The

(2)

The potential function V(rf) is no longer periodic. The accent in the double summation of eq. (2) means that terms i = j must be omitted. The energy matrix corresponding to eqs. (1) and (2) can be written: 8 = (a,)

with

bij = (E;;) )

(3)

where E:; = s #;*Hz,b; dr. Each

element

8u of the matrix

(4) d forms

a 3 x 3 matrix,

given by eq. (4). Since the energy matrix 8 is not diagonal, wave functions I&@by a unitary transformation

we introduce of the set &:

with elements a new set of

(5) (6) Expressing that the energy matrix 6’ corresponding to this set of functions &@ is diagonal and taking into account the conditions (6), we obtain the following equations for the coefficients aB:

(7)

Assuming

571

EXCITON THEORY OF AMORPHOUS SOLIDS. I

THE FRENKEL

a:: can be factorized :

that the coefficients

a;; = aI,*fIF and since the system is homogeneous,

which means that the expression

is only a function of the interatomic distances write eq. (7) into the following simple form: E’0, = z x exp[iK.(Rr 11 Y

-

Rl -

(7)

Rn, we can easily

Rn)] f&EL.

(8)

For the sake of simplicity we dropped the indices j and o. Since eq. (8) still depends on the position vectors RI, . . . . RN, we get, taking an appropriate configurational average : E’O, = x 13,j dR Q(R) exp[iK*

R] Evv(R),

(9)

Y

where

Q(R) = fioJ'(IRI) + d(R).

(10)

function for the considered The function P(IRI) is the radial distribution amorphous material and no is the macroscopic density. The expressions (9) represent a linear and homogeneous set of three equations in OS, Or, and OZ. There exists a non-trivial solution, if the following condition is satisfied: det 1s dR Q(R) exp[iK*R]

Eflv(R) -

E’d,,l = 0.

(11)

As mentioned in the introduction, a diagonalization of a 3 x 3 matrix results from the fact that the excited one-elecmstate is 3-fold degenerate. Carrying out these calculations, we obtain the well-known longitudinal and transversal p-like exciton energy, in a quite srmilar way as introduced in crystal exciton theory by Heller and Marcuss). 3. Explicit expressions of the elements E@v(R). are given by eq. (4). After some calculations pression for a “diagonal” element : EL

= (N -

1) b,v j v*,(rd

N&v

C [s V*,(Q)

~vn(n)

The energy matrix elements we obtain the explicit ex-

dn + j p);*(n)

Z&(Q)

dXf.2)

fJ12~&1)

in

d712

d@2)

fh2~4~2)

R&I)

dd

dv

-

-

m -

+ 2s d*Pd -

j d*(u)

h*,Pl) dP2)

H12~~n(n)

~32)

dTl2

vW2)

fh2dr2)

FJ;(~)

dTl21,

+

(12)

where Hrs = e2/2 jrr -

rs].

(13)

The operator X is the self-consistent Hartree-Fock operator. Taking into account the expression for the Hartree-Fock ground-state energy and the

572

W. CHRISTIAENS

AND P. PHARISEAU

assumption that there is no mixing the expression (12) :

of p-states,

we can formally

write for

EMU lLlz zz Ep@ nnSPV. A non-diagonal

(14)

element

is explicitly

Cl!?& = 2[J p?Z(rl) ~34 -

HisMri)

j p);*(ri) @$(rs) Risqn(rs)

given by: p?“,(rs) dTis pj”,(ri)

dwl.

(15)

Since we are working with atomic wave functions p(r), the second integral occurring in eq. (15) will be small with respect to the first one. Although it has been pointed out by some authors4) that this overlap integral cannot always be neglected, we restrain our further calculations to the non-overlap integral. The technique to calculate such integrals is usually to expand jr1 - r$1 in Taylor series. If we break off after the dipolar terms and take into account a set of orthonormality functions 91 and @ we get for eq. (15) : E@v(R) =

(Dv”.DO~) R-3 -

conditions

for the one-electron

3(ZP’.R)(D’~.l?)R-5,

(16)

where Dvo = e j &*(ri)(ri Doll = e J

yZ(r2)(

r2

Inserting the expression element of eq. (11):

-

Rn) v,(ri) &)

ph(r2)

dTi,

(17)

dT2.

(16) into eq. (1 I), we can write for a non-diagonal

no j dR P(R) exp[iK.R][(D’a.DOfi)

R-3 -

(ZW.R)(lW.R) R-51.

(18)

In the case of crystal exciton theory, the expression (18) comes out to be a lattice sum. Making use of the theta-transformation technique of Ewalds), Born and Bradburns) succeeded in transforming such a lattice sum into several much more rapidly converging series. In the next section it is shown how we can simply generalize that method to the case of amorphous materials.

4. Calculation integral

of the exciton of the form :

j dR P(R) R-n exp[iK. Taking

into account

energy.

R],

the standard

~R~-~=~(~)~'~u

The main problem

n = 1,2, . . .

is to calculate

an

(19)

result :

exp[-R%]ti(n/2)-1,

(20)

a double integral,

in which we split the inte-

0

the expression

(19) becomes

THE

FRENKEL

EXCITON

THEORY

gration over u into an integration

F(M,

U) = J dR P(R) exp[-_lR

-

Assuming that the Fourier transform F(M,

U) = J dg expPg.Ml

expression

F(g, 4 = @W-1

I

parameter

573

T and

U), defined by

Ml2 u + iKa(R

-

M)].

(21)

of F exists, we have the basic relations:

F(g, 4;

F(g, a) = (2x)-3 j dM exp[-ig* The explicit We get

SOLIDS.

from 0 to some arbitrary

from 7 to +oo. We consider a priori the function F(M,

OF AMORPHOUS

(22)

M]F(M,

U).

(23)

of F(g, u) can be calculated

a(g)lK

from eqs. (21) and (23).

+ g12/4ul,

(24)

where a(g)

=

(2743 %?Lqg)

A(g)- 1 =

(24-3

+

no J

now!!)

11;

-

dR [P(R)

-

(25)

l] exp[ig.R].

(26)

The function A(g) is the so called interference function and is connected to the radial distribution function by a Fourier transform. Introducing eq. (24) into (22), taking into account the definition (21) and putting afterwards M = 0 we can deduce the following relation: jdRP(R)

exp[-R2u+iK.R]

=2-s(~)-*Jdga(g)

exp[-jK+g]s/4u]. (27)

This equation enables convenient form :

us to bring

dR P(R) R-n exp[iK.

the expression

(19) into a much more

R] =

s

=P(gll’[

2-3x-*

J

dg u(g) +m

+

s

dR P(R) exp[iK.

The latter equation

R]

i 0

du u(+s)@exp[-

s 7

du u.(“/s)-1

can be simplified,

/K + g12/4u] +

exp[-_R2u 1. (28)

introducing

a function

cumdefined by:

m

a&)

=

j v” exp[-vx]

dv = x-m-rZ’(1

-

m, x),

1

where r represents

the incomplete

gamma function.

(29)

W. CHRISTIAENS

574 For expression

(28), we get

AND P. PHARISEAU

:

dR P(R) R--a exp[iK. R]= s = [I.(+)-j-l [2t-l.i(“-s)/s~dg dR P(R) exp[iK*

+ +I2

a(g) cYo_&

IK + g12/4r) +

R] OI~~,.+~(R~T),

(30)

s

To calculate the first integral of the non-diagonal element (18), we simply put 12 = 3 in the result (30). The second integral in (18) can be found, using the equation J dR P(R) exp[iK*

R](DyOR)(DOpR)R-5 =

= - C D;‘Df” 8/8Kc

a/aK, [jdR P(R) R-5

exp[iK*

R]].

hi After to

some straightforward 120j dR P(R) exp[iK*

= no[r($)]-1 + 2-47+7-1

calculations

the matrix

R][(DvO.DO@) R-3 -

C D;“D,O”[~*&jj dR i,i

element

(31)

(18) reduces

R)(Doa* R)R-51 = P(R) exp[iK* R]a*(R%) + 3(ZF.

j dg 4 g) ao( IK + gi2/4T) (Ki + gd(Kj + a) -

- 2~~ j dR RiRjP(R) exp[iK.

(32)

R]CX@(R~T)].

It will be interesting to investigate the expression (32) for small values of the exciton wave vector K. In this particular case, the expression (32) reduces to +cno z D;“DiOp[3K-2KrKj -

Su].

(33)

i,i Furthermore

it can be verified that the dipole moments

0: = e s P’*(P)

lP*V(P)

satisfy the relations

(34)

do;

l&q = 101/l = /lIZI. That means that the dipole moment from zero. Taking into account

the properties

(35)

DVhas

only one component

0: different

(34) and (35), we get for the expression

(33) : $wzo )D’j2 (--6,w + 3K2K,Kv),

where Kfl and Kv are the projections the considered

system

is invariant

(36) of the vector K on Dfland D*.Since for the choice of the orthogonal set of

THE

FRENKEL

p-functions,

EXCITON

THEORY

OF AMORPHOUS

SOLIDS.

it can be chosen in such a way that the z axis coincides

This will immediately diagonalize We obtain for the eigenvalues

expression

I

575

with K.

(36).

-$Cn()D2

(37)

+cnoD2.

(38)

and

Expressions (37) and (38) correspond, respectively, with a so called transversal and a longitudinal p-like Frenkel exciton. We remark that these results are quite similar to those found in crystal exciton theory, because they are independent of the structure of the considered material. The only difference is that the explicit value of the macroscopic density no will be slightly smaller in an amorphous material in comparison to the same substance in the crystal phase. The expression (32) can also be diagonalized for any value of K, in a similar way as done for expression (36). After some calculations we obtain for a transversal exciton state, i.e. for p = x or p = y : Q[Z’(#)]-1

D2{4rc+K-1

YdRR P(R) a&(R%) sin KR + 0

+

8~1~1K-V%ci exp[-K2/47].

rdg.g.

[A(g) -

l] x

0 x

exp[-g2/4T] -

7 dg [A(g) -

&;1K-3d

X [Ei(-(K -

[+gKc1 ch(2gK) +

+

;)‘/4T)

-

(K2 + g2--- 1) - l] sh(2gK)] -

[(h-l

l](K2 - g2)2x

Ei(-(K

-

g)2/4T)]

dR R4P(R) CQ(R~T)[(KR)-3

8dT

-

sin KR - (KR)-2 cosKR]),

0 (39)

where -Ei(--x)

= rdzl u exp[-~1, Z

cq(~) = x exp[--x] a&)

+ &TCX-~[~-

= (1 t- Qx) x-l exp[--x]

Erf(x*)],

+ +~-~[l

(40) -

Erf(xa)].

The expression (39) will be very interesting for numerical calculations, since the functions defined by the expressions (40) converge rapidly to zero, so that the integrals in (39) can be taken over a relatively small interval. The macroscopic density, the radial distribution function and the interference function appear explicitly in (39). These functions are the only information

THE

576

FRENKEL

EXCITON

THEORY

about the structure of the amorphous perimentally by diffraction methods.

OF AMORPHOUS

SOLIDS.

I

system which can be determined

ex-

This work is part of a research program sponsorAcknowledgements. ed by I.R.S.I.A. (Brussels). One of us (W.C.) thanks this organisation for a grant.

We wish also to thank

interest

Professor

Dr. W. Dekeyser

for his continuous

and discussions.

REFERENCES 1) Frenkel, 2)

Born,

3)

Heller,

J., Phys.

Rev.

37 (1931)

M. and Bradburn,

M., Proc.

W. R. and Marcus,

4)

Demidenko,

5)

Ewald,

A. A., Soviet

P. P., Ann.

Phys.

17, 1276. Cambridge

A., Phys. Physics 64 (1921)

Rev.

Solid State 253.

Phil.

84 (1951)

Sot. 39 (1943) 809.

3 (1961)

869.

104.