Three-dimensional analysis of modulated photoreflectance in a silicon wafer

Abstract

The effects of probe and pumping beam size and modulation frequency on photoreflectance were investigated for a silicon wafer by considering one- and three-dimensional generation and propagation of thermal and plasma waves,PR _1D andPR _3D. The magnitude ofPR _1D decreased as the inverse square of the effective beam radius and that ofPR _3D was 100 times smaller thanPR _1D at 0.1 μm effective beam radius and decreased with the effective beam radius. The phase shift ofPR _1D was nearly constant at 225°, whereas that ofPR _3D increased with the effective beam radius from 0° to 225°. The magnitude and phase ofPR _3D become the same as those ofPR _1D by satisfying the equivalence conditions, where the probe and pumping beam radii are larger than the thermal and plasma wavelengths, when the effective beam radius was larger than 112 μm.PR _1D decreased with modulation frequency as ω^−1/2, whereas the magnitude ofPR _3D was nearly constant and 100 times smaller than that ofPR _1D at 1 kHz modulation frequency. ThePR _1D phase varied from 180° to 225°, but that of thePR _3D increased from 0° to that ofPR _1D with increase of the modulation frequency. As the modulation frequency increased, the magnitude and phase ofPR _3D approached to those ofPR _1D by approaching the equivalence conditions, owing to the decrease of the thermal and plasma wavelengths. The good agreements in the modulation frequency dependence of the magnitude and phase ofPR _3D with those measured, justified the three-dimensional analysis of the photoreflectance.