Efficient Calculation of Electron States in Self-Assembled Quantum Dots: Application to Auger Relaxation

An efficient method for calculation of self-assembled dot states within the effective mass approximation is described and its application to the calculation of Auger relaxation rates is detailed. The method is based on expansion of the dot states in a harmonic oscillator basis whose parameters are optimised to improve the convergence rate. This results in at least an order of magnitude reduction in the number of basis states required to represent electron states accurately compared to the conventional plane wave approach. Auger relaxation rates are calculated for harmonic oscillator model states and exact states for various pyramidal models. The dipole approximation, previously used to calculate Auger rates, is found to be inaccurate by a factor of around 2–3. The harmonic oscillator states do not reproduce the rates for the more realistic pyramidal models very well and even within the set of pyramidal models variations in the dot shape and size can change the rates by up to an order of magnitude. Typical Auger relaxation rates are on a picosecond timescale but the actual value is strongly dependent on the density of electrons outside the dot.