Valley splitting in finite barrier quantum wells

Abstract

The valley splitting (VS) in a silicon quantum well is calculated as a function of barrier height with both the multiband $s{p}^{3}{d}^{5}{s}^{\ensuremath{\ast}}$ model and a simple two-band model. Both models show a strong dependence of the VS on barrier height. For example, in both models some quantum wells exhibit a sharp minimum in the valley-splitting amplitude as the barrier height is changed. From the simple two-band model we obtain analytic approximations for the phases of the bulk states involved in the valley-split doublet, and from these we show that such sharp minima correspond to parity changes in the ground state as the barrier height is increased. The two-band analytic results show a complicated dependence of the valley splitting on barrier height, with the phases essentially being determined by a competition among effective quantum wells of differing length. These analytic results help explain the VS in realistic structures where different finite barrier heights are possible depending on the confining heterojunctions used.