Transport of excitation energy in a three-dimensional doped molecular crystal

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL XXXVI, 141-155 (1989)

Transport of Excitation Energy in a Three-dimensional Doped Molecular Crystal SATYAM PRIYADARSHY AND SAMBHU NATH DATTA Department of Chemistry, Indian Institute of Technology, Powai, Bombay -400 076, India

Abstract We report here a theoretical formulation of the transport of excitation energy in a three-dimensional molecular crystal containing one impurity. The excitation is assumed to be localized in the j t h site at time I , and the expression for the probability of finding the excitation a: some other sjte j ' at a later time t' is derived. The probability is given by the correlation function (P,.(t)P,(O)),where P, represents the site projection operator, Im)(ml. In our derivation we neglect the interaction among excitons of different bands, account for the presence of the impurity by adding a small perturbation term to the pure crystal Hamiltoman, and calculate the exciton solutions through first order. We consider a general impurity; that is, the trap depth is nonvanishing and may even be complex. The exciton-phonon inteJaction is taken to be linear in lattice displacement vectors; we assume that the short time behavior of (X)phonon gives the dominant contribution to the physical property X being studied and solve the dynamical problem by using a timedependent effective potential consisting of fluctuations around the equilibrium average exciton-phonon interaction. Several limiting cases are briefly discussed.

1. Introduction

In an earlier work [l] we reported a model calculation on the transport of a local (site) excitation in a three-dimensional molecular crystal that has one isotopic impurity at site p . In the above work, we assumed that the isotopic impurity has excitation energy equal to that of the host molecule. However, if the characteristics of the impurity excitation differ from those of the excitation of the host molecule, the excitation energy can be trapped at the impurity. In fact, Singh and Thilagam [2] have considered a nonvanishing trap depth and proposed a theory for the trapping of excitons while two phonons are emitted. In this paper we extend our previous formulation of the excitation transport to a general case: we assume that the impurity has a nonvanishing trap depth and that it is even capable of quenching the excitation. The basic theory has been discussed in detail in Ref. 1 and is b$eflyAoutlined here. We are interested in calculating the time correlation function ( j l A (t)B(O)lj),where l j ) represents the state in which thejth moleyle izin an excited state while all other molecules remain unexcited. The operators A and B are chosen as site-excitation projection operators. We have

0 1989 John Wiley & Sons, Inc

CCC 0020-7608/89/020141-15$04.O0

142

PRIYADARSHY AND DATTA

where Ik)s are the eigenfunctions of the exciton Hamiltonian, ,a"d Nks are the corresponding normalization constants. The matrix element (kIA(t)B(0)lk')can be written as [31 (k

JAA (t>S(0)lk') = ( k I exp(iH:xt);i-(t)

exp(-iH:xt)B^ (0)lk')

(2)

where X ( t ) is phonon-averaged form of the operator A^(t).Following Kubo's technique of cumulant expansions [4], x(t)is written as

The cumulants are given by

where V ( t ) is the time-dependent interaction:

v(t>= exp[-i(H:x + H i h ) f ] H e x - p h

exp[i(H:x

+ Hi&].

(5)

For a small V ( t ) we truncate the expansion in (3) as X(r, t ) = exp[K,(t)

+ K2(t)$?(r,0) .

(6)

Equation (6) is utilized in the calculation of the correlation function. Our objective is to calculate the general fomAof the equi1ibri)tm-ave:aged correlati2n function for the following choice of operators: A = Ij ' ) ( j'I = P,,and B = I j ) { j ( = P, . With this choice, the correlation function

r,,,o)= (jtP,.(t)i:(o)tj)

(7)

represents the probability of finding the excitation at the j'th site at time r if at time zero it was created at site j : r,,,.