Abstract

Exciton superradiance from systems with restricted geometries, such as molecular aggregates, semiconductor microcrystallites, and quantum wells has recently attracted consider able attention in the literature.^[1] These systems exhibit ultrafast spontaneous emission rates, because all oscillator strength is contained in one (or a small fraction) of all the extended excited states (excitons). By contrast, for an in finite crystal each exciton mode couples to exactly one photon mode to form stable (non-decaying) polaritons. It is an interesting question how the transition between the superradiant exciton and the stable-polariton regimes occurs when the system size is continuously increased. In this paper, I investigate this problem by studying the radiative exciton dynamics in molecular crystal slabs as a function of the slab thickness L. My approach exists of deriving a general dispersion relation for the excitons that accounts for both the instantaneous and the retarded dipole-dipole interactions within the slab. The retarded (radiation-mediated) interactions are responsible for the cooperative spontaneous decay in the slab and are contained in a frequency- and wavenumber-(k-) dependent self-energy of the excitons, which is obtained in analytic form for all values of the slab thickness. The non-Markovian nature of the self-energy turns out to be essential for a smooth transition to bulk polaritons. By solving the dispersion relation perturbatively around the static excitons, one recovers the Fermi golden rule for the spontaneous emission rate. In this approximation, the k=0 exciton is found to be superradiant for L≪λ (λ is the exciton transition wavelength) with a rate proportional to L. The superradiant nature of this exciton breaks down at L≈λ/2, and for L≫λ its decay rate drops off as 1/L, giving a smooth transition to the stability of the bulk polariton (Fig. 1).^[2] For wavenumbers in the exciton-photon resonance region (k=2π/λ), however, the Fermi golden rule yields a decay rate that grows linearly with L for all values of L, predicting superradiance at all length sclaes and contradicting a smooth approach to stable polaritons in the infinite limit (Fig. 2). This problem is common to all approximations that equate the frequency and the wavenumber in the self-energy.^[4] I show that in a higher-level approximation, existing of expanding the dispersion relation around the polariton frequency, the spontaneous emission rate behaves like 1/L (L large) for all values of the wavenumber (Fig. 2). This “polariton-pole approximation” reduces to the Fermi golden rule far from the exciton-photon resonance and (or) for slabs thin compared to λ. I finally note that in practice the spontaneous decay rate of polaritons in finite crystals is often modelled using the time-of-flight and internal reflection coefficient of eletromagnetic wave packets.^[3] It turns out that the microscopic decay rates obtained in the polariton-pole approximation can be related to this macroscopic picture over a large range of wavenumbers.

© 1992 IQEC