Scattering phase approach for energy spectrum in quantum dots

We present a semiclassical approach to evaluate quantum energy levels in asymmetrical quantum dots and wells, where analytical solutions for Schrödinger equation are not possible. In spatial regions where the potential profile is steep, the wave function is locally solved and gives rise to a momentum-dependent phase correction φs(k) as shown in the figure. For smooth profiles, the usual WKB approximation works. Combining scattering phases accumulated in both steep and smooth regions, we arrive at a generalized EBK quantization rule that can be solved algebraically to obtain the energy levels. We present several examples in which this semiclassical approximation works very well, even for the low-lying excitations.