Diffusion and recombination of optically-injected carriers in a semiconductor wafer in 3-dimensions

Abstract

In this work, we derive a general equation describing the transmission of a probe beam in a pump/probe experiment upon optical injection of carriers into a semiconductor. The pump/probe radial overlap equation generalizes previous pump/probe approaches by considering the pump and probe beam sizes relative to each other and to the diffusion length. The pump/probe equation leverages a powerful solution to the free-carrier density under optical injection that is also derived in this work. The free-carrier density solution extends the work of Luke and Cheng to 3-dimensions, incorporating the effects of radial diffusion in a plane parallel to the semiconductor surface. The pump/probe equation quantifies the magnitude of free-carrier absorption of a probe beam induced by free-carriers optically injected via a pump beam. We show that when the pump/probe beams are much smaller than the carrier diffusion length, radial diffusion effects dominate. Measurements in this regime can be used to uniquely and simultaneously determine both the effective carrier lifetime and the diffusion coefficient. The equation agrees well with experimental measurements using a recently developed single-beam pump/probe technique, which ensures a perfect overlap of the pump and probe beams. Based on this equation, measurement criteria are developed for accurate determination of carrier lifetime and to correct for the effects of radial diffusion.

Optimization of the s-SNOM near-field nanoscopy for the investigation of transport phenomena on the nanometer length scale

“…(1.13)) are smaller than the typical diffusion length scale (e.g. L ∼ 400 µm [184]), therefore the excited carriers have enough time to diffuse and dilute from their generation volume fig. 4.15.…”

“…For a deeper understanding of the charge carrier excitation and diffusion in Si, there are several excellent works dealing with those questions in the scope of solar cells. The photo-excitation in Si has been analysed analytically for the 1d case for CW [185] and pulsed [186] illumination; the latter case is even treated in 3d [184]. Further analytical models can be found in the general literature about (charge carriers) diffusion [187,188].…”

“…7 At the end, not the absolute n value is important, but its dependences because later a fit parameter scales the entire data set. The decisive differential diffusion equation (diffusion constant D) [184][185][186] is given by The constant A can be solved by inserting eq. (4.15) into Fick's first law of diffusion eq.…”