Motional narrowing effect in one-dimensional Frenkel chains with configurational disorder
Introduction
The concept of motional narrowing, earlier raised by Knapp [1] for the on-site energy (or diagonal) disorder, has been very fruitful in explaining many optical phenomena in quasi-one-dimensional systems like J aggregates of polymethine dyes and conjugated polymers (for a review, see Refs. 2, 3, 4 and references therein). This effect is manifested as a decrease of the magnitude of the diagonal disorder as soon as the states of individual molecules are collectivized due to the intermolecular interaction and form excitonic states. The decrease depends on whether or not the disorder is small enough to be regarded as a perturbation. In the perturbative case, the suppression factor is determined by the square root of the whole number of molecules in an aggregate, while in the nonperturbative case this number should be substituted by the number of coherently bound molecules [1]. A similar narrowing effect, but different suppression factor, was recently reported for the case of dynamic disorder 5, 6, 7.
It was found in Ref. [8] that the numerically simulated absorption spectrum of polysilane with an uncorrelated Gaussian distribution of nearest-neighbor couplings are similar to those for an uncorrelated diagonal disorder (see also [9]). In contrast, numerical simulations of off-diagonal disorder given by Gaussian randomness in the molecular positions 2, 10 found that the behavior of the optical observables was very different from that expected from standard motional narrowing arguments [1].
The main goal of the present Letter is to uncover the origin of such a difference. It will be shown that, in the case of configurational disorder, certain correlations appear in the distribution of hopping integrals, in spite of the fact that the distribution of the molecular positions is uncorrelated. This finally results in a scaling law for the typical fluctuation of the Frenkel Hamiltonian matrix with respect to the magnitude of positional disorder and the number of molecules different from that for uncorrelated diagonal disorder. As a consequence, the main features of the exciton optical response are largely affected.
Section snippets
Description of the model
Consider a collection of N two-level molecules forming a one-dimensional chain. For our present purposes, we will neglect the static inhomogeneous offset energy of the molecules imposed by the surrounding host medium (diagonal disorder). Under this assumption, the effective Frenkel Hamiltonian describing the system can be written in the formHere, the state vector |n〉 denotes the nth molecule being excited and the site index n lies within the symmetric domain −(N−1)/2≤n≤(N−1)/2 (N
Motional narrowing effect
To gain insight into the magnitude of the typical fluctuation of the scattering matrix ΔKK′, one should calculate either its distribution function or its moments, using the distribution (2) of the positional fluctuations. We chose the second procedure, so that the magnitude of interest will be the mean square deviation, defined aswhere brackets denote the average over the joint probability distribution , with P(ξn) of the form (2).
In what follows, we assume that the
Discussion of the numerical results
As it was already mentioned in the Introduction, numerical simulations of optical properties of linear molecular aggregates with off-diagonal disorder generated by Gaussian uncorrelated fluctuations in the molecular positions yielded different behaviors of the optical observables as compared to those for diagonal disorder 2, 10. In this section, it will be shown that the peculiarities found in Refs. 2, 10 can be qualitatively explained from the viewpoint of the modified motional narrowing
Conclusion
The motional narrowing effect in one-dimensional Frenkel chains with off-diagonal disorder arising from Gaussian fluctuations in the molecular positions is found to be different from that for diagonal disorder and for off-diagonal disorder with uncorrelated randomness in the nearest-neighbor couplings. Such distinction is due to the fact that the fluctuations of the dipolar coupling of a given molecule to the adjacent ones are correlated, even if the fluctuations of the molecular positions are
Acknowledgements
The authors thank A. Sánchez for critical reading of the manuscript. This work was supported by CAM under Project 07N/0034/98. V.A.M. thanks UCM for the support under Sabáticos Complutense as well as partial support from the Russian Foundation for Basic Research under Project 97-03-09221.
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On leave from All-Russian Research Center “Vavilov State Optical Institute”, Saint-Petersburg, Russia.