Levy Flight of Photoexcited Minority Carriers in Moderately Doped Semiconductors: Theory and Observation

Abstract

Spatial spread of minority carriers produced by optical excitation in semiconductors is usually well described by a diffusion equation. The classical diffusion process can be viewed as a result of a random walk of particles in which every step has the same probability distribution with a finite second moment. This allows applying the central limit theorem to the calculation of the particle distribution after many steps. However, in moderately doped direct-gap semiconductors the photon recycling process can radically modify the spatial spread. For this process, the steps in the random walk are defined by the reabsorption length of photons produced in radiative recombination. The step distribution has an asymptotic power-law decline. Moments of this distribution diverge and the displacement is governed by rare but large steps. Random walk of this kind is called the Levy flight. It corresponds to an anomalously large spread in space and a modified ("super-diffusive") temporal evolution. Here we discuss the first direct observation of the hole profile in n-doped InP samples over distances of the order of a centimeter and more than two orders of magnitude in hole concentration. Luminescence spectra and intensity were studied as a function of distance from the photo-excitation in a rather unusual geometry (homogeneous excitation of the wafer edge and observation of the luminescence spectra from the broadside). The intensity is proportional to the minority-carrier concentration and exhibits a slow power-law drop-off with no changes in the spectral shape. This power law gives a direct evidence of Levy-flight transport. It has enabled us to evaluate the index of the distribution, the characteristic distance of the minority-carrier spread and the photon recycling factor. The results are in good agreement with the theoretical analysis.