On the problem of the exciton—phonon interaction

Volume 91. number 3

CHEMICAL

PHYSICS

ON THE PROELEM OF THE EXCITON-PHONON

G.S. DAVlDOVI64USTOVSKI Facrdry

of SnuraI

Received 1 October

Sciences.

28 January

LETTERS

1983

INTERACTION

and Lj_M. RISTOVSKI

Univetsiry

of Belgrade.

1982: in fins1 form 4 November

Belgrade.

Yugoslnvia

19S2

It is demonstrated that the theory of the esciton-phonon interaction could bc formulated in such 3 way that the quadratic csciton-phonon coupling terms nrise which lead to the hybridization. Some consequences of tbe tvbridiition

effects arc pointed out.

The hamiltonian of the exciton-phonon interaction is usually derived by espansion of the matrix elements appearing in the exciton hamiltonian in terms of the molecular displacements, i.e. in terms of the phonon creation (C&) and annihilation (C& operators. This procedure gives in the harmonic approximation the exciton-phonon coupling terms of the form Bi,,B&,, and B& B k C+ qs where B’ and B are exciton creation and annihilation it can be seen that in the operators_ However, if one follows the procedure of obtainin, 0 the exciton hamiltonian. general case the hamiltonian of the exciton-phonon interaction must include the quadratic esciton-phonon coupling terms of the formB~Cqs and BqC&. Firstly, we shall justify this statement_ The hamiltonian

of the esciton

tonian of the crystal electron

system in the adiabatic approximation

system by the use of the method

[ I] can be obtained

of approximate

front the hamil-

second quantization

- the ASQ

method [2.3]_ In short, the procedure is the following: The hamiltonian H=nfFf

2

of the crystal electron

%(fifi)oi;fiffnf2

•I- x

system is given as aAflf2f3f4~%zf,

%f2Glfj%f4

*

(1)

fT?f*

where n and m are the lattice vectors (under consideration is the molecular crystal with one molecule per elementary cell); fi are the indices whi& label the molecule energy levels; a+ and (Yare the electron operators; SL, and of the isolated molecule. Qnm are the matrix elements calculated with the wavefunctions Introducing the new electron operators a,,, by the unitary transformation

(2)

(3)

0 009-26 14/83/0000-0000/S

03.00 0 1983 North-Holland

369

CHLXICAL PHYSICS LETTERS

28 January 1983

in the procedure of minimization of the exciton The transformation functions UfS_ which are of importance ground-state energy. will be determined later. III order to obtain the hamiltonian of the rxciton system. the following clumsy calculation steps must be carried out: (1) to set aside all ttrnls with zero values (one or more) of the indices pi from the sums in the hamiltonian (3): = ni,a, o, pnp (2) to introduce the quasi-Ruli operators pi,, and Pnl, which are defined as follows [3] : .pi lnp to 1’ MSS over from the quasi-Pauli operators to the exciton (Bose) operators B,,,, and Bnu in accor=o;,,a,,:(.:) d;tn~c with the formulae given in ref. [3] _In such a way. the hamiltonian of the exciton system, written in the llcitlcr London approsimation. is given as

C’R,,(0000). m

If,, =A’Eo +;

D n,,

=

Lcfl +

HI =

c Q,,JB,+I,i- Bn,> -

JJU

c’ I&,(/WOO) - R,,(OOOO)] ; m

(4)

%~~P~P~) =R,,(/.q~~00).

c,, ;md e,, xc ~hc ground-state cncrgy and p-cscitcd state energy of the molecule_ WC note that in the considered case (adiabatic approximation) the crystal is frozen and the quantities Qnll and 4rII do not depend OII n. However. we shall keep the lattice index n because in the general case, i.e. when the nrok~uhrr Jisplaccmcnts are included. QnU and Dn,, depend on n. TIC han~iltonian (_-I) contains the inconvenient linear part H, which cm be eliminated by the minimization of the ground-state cncrgy. The minimizing procedure which is given in ref. [7] leads to the following equations determining the transfornistion functions Up,:

(5) In or&r to obtain the standard hamiltonian of the esciton-phonon interaction. after the nlinitnization of& (4iulinxtiou of II, 1 D,,,, snd I’f,,,, must be espmded in term of the molecular displacements. We turn our attenII~>IIIO tlw other possibility. Namely. it is possible firstly to espnnd Dnr_Mni,, and Qnr in terms of the molecular Jisplacsrnents :rnd at’tcr 1h3t IO minimize Hu (to eliminate Ht). This approach gives

fYk is the csciton quc~icy.

energy.N

is the

IIXISS

of the molecule.

N is the number

of molecules

and ok

is the phonon

fre-

The espression for x4 and Fks cm be found in ref. [I]. In the calculation the two-level approximation

Volume 94, number 4

CHEMICAL PHYSICS LETTERS

28 Januruy 1983

is used and only one phonon branch is taken into account_ The expansion in terms of the molecular displacements

is carried out according

to the relations

R ~,=~~1~R~exp[ik-(n-m)+ik-(x~-~xm)]~N-1~R~exp[ik-(n-m)][1+i(~~-~x,)], k xn -xm

(?i&?7Vwq)1~Z[exp(in

= F

k

-4) + exp(im-q)](Cf4

+

Cq) ,

B,=_J'V-~/~CB~ exp(ik-n). k

where xn is the molecular displacement of the molecule n, CG and C4 are the phonon operators. The minimization procedure does not affect the hamihoniansHif,r and H$_ That means that it can be carried out in the same way as in the case of the frozen crystal_ Comparing to the standard theory of exciton-phonon interaction, the proposed approach gives the additional interaction terms which are included in Ht. So,the change of the procedures of the minitnization and the expatlsion in terms of the molecular displacements gives different results. We shall restrict the analysis to the Hint because the analysis of If!*,nl can be found in many places (see, for example, ref. [I]). Including the quadratic operator terms only, the hamiltonian of the interacting exciton-phonon system cw be expressed as follows:

It is easy to conclude that this hamiltonian leads to the hybridization of the elementary excitations in the cmsidered system. The energies of the hybrid excitations can be determined by the diagonalization of the hamiltonian. According to the known procedure of the diagonalization, we introduce the new Bose operators .$

(9)

(10) ePk are the energies of the hybrid excitations_ As can be seen, the first consequence of the hybridization. concerning the optical properties of the considered system. is the renormalization of the esciton energy and consequently the renorn~alization of the exciton mass. More about the consequences of the esciton-pbonon hybridization can be found in ref. [I]. where tlie mechanism of the metastable exciton-phonon hybridization is considered_ The authors of this paper consider a unitary transformation of the standard hami!tonian of the interacting esciton-phonon system which gives an equivalent quadratic hamihonian of the exciton-phonon interaction. Although the physical meanings of that hamiltonian and of the liamiltonian (8) are quite different, the method of analysis of these hamiltonians is the same. That means that rlie results which are obtained in ref. [4] (escluding the results obtained in part 1) could be applied straightforwardly to our case. We give a short description of the method which is used in ref. [4] and after that the main results which are also valid in our case. The hybridization affects the dielectric permeability of the considered system (crystal)+ The tensor of the dielectric permeability is defined by the material equation Q,(/c, 6~) = e&c, w) && w) where eep(k. o) is the 371

PHYSICS

28 January 1983

LETTERS

cO,(k. w) are the components of the electric induction vector and C&, w) are the components of the field vcmor. Using rhc calibration r?,(k_ w) = (iw/c)d’,(k_ a). where $(I& o) are the components of

tensor. &xtric he

CHEMICAL

94. number 4

Voluurr

vector

potential.

om

obtain

can

5-d& w) = -(4x/c)K,&

w)j#c.

w) .

w) =

K&.

(wqc7-)q&.

0)

+x-,x-,- A,,

y

(11)

where j,(k. w) is the density of the external curreuts which perturb the system. i\csording

squalized

IO

rh

approach

givsn

with he non-equilibrium relation is obtzhxi:

l;~lknving

K;j(k_

wj = (c2/w2)[?juG

whcrc I‘+(k.

in ref. [S] the phenomenological vahe

mcm

- i(27i3/fi)rag(k.

o) is Ihe Fourier component

I;&

@i,(r.

-- r’. I - I’) = U(t - r’)(E,(r.

I) =

The cocfiicicnrs The relaIions

I

d’k

I-,(k)[Bk(r)

+

ir tUllvws

be the

(12)

t’) - kp(‘.. t’) &Jr_ I)) _

r) i,(/.

operators

(13) WJy

in the following

[s]:

- r) _

Pk(f)]exp(ik

(14)

ciln be found in ref. [S] _

I,,(k)

8 = ,]!iyo (i12ii)Gp,./(w

GP&‘k’

could

this intO 3CCoul&

of the Green’s function

(9) and (14) enable us to espress I‘uncriorls ((cPpR[ tz.k.))_ Since

Green’s

potentialAA,

Taking

t).?(t)).

a)] .

The elcc1ric field vector cm be expanded over the es&on C,(r.

value of the vector

t)> = (S-l(f)~a(r.

the Green’s

- wPk + i7j) :

funcrion

r,,(/c.

upk = c&l

o)

through

the hybrid-excitation

_

(19

tha1

In1 A-&l

(k.

w)

=

-- g;

I_,(k)L,(k)

c

[(Ob’,‘)’

+

(#‘I

[6(w

-

Wpk)

P

TIIC I‘unctions O,,

md

upk

are introduced

in eq_ (9)_ It

is easy

to

show

+ 6(0

+

Wpk)l _

(16)

thar

(17) of the csciton-phonon hybridization. 311 additional peak of of this result is the peak which is identified 3s 311optical ~11~1og~ ul‘ IIK hliissbauer peak (see ref. 161. p_ 166). II‘ one includa tile retarded inreraction, the esciton-phonon-photon hybridization appears which leads to the dipole ~displmmr~lt coupling. The opto-mechrtnic~l tensor cm also be defined (for details see ref. [4])From

q.

111~diclecrric

( 16) it follows permeribility

rlrar 3s a consequence appears. The possible

proof