Effective computation of complex-shaped quantum-dot structures

Abstract

The possibility of growing complex-shaped nanodot structures of various material composition allows optimization of certain physical parameters. In the present work, we present effective analytical methods for computing conduction-band eigenstates in quantum-dot structures of complex shape. Comparison with detailed finite-element computations is made. The electronic bandstructure model used is a one-band $\vec{k}\cdot\vec {p}$ model assuming infinite barriers. Results based on two semi-analytical models are presented. The first model employs geometrical perturbation theory to obtain the quantitative effect of quantum-dot surface perturbations on electron energy levels. Furthermore, the method output includes the level of degeneracy and variations with geometry to be assessed. The second model allows both energy levels and eigenstates to be easily determined for three-dimensional axisymmetrical GaAs structures of varying radius embedded in an AlGaAs matrix by extending a method originally due to Stevenson on electromagnetic waveguide structures (Stevenson in J. Appl. Phys. 22:1447, 1951) to account for electron states. The latter model simplifies the description of a three-dimensional partial-differential equation problem into a small set of ordinary differential equations. For structures with a large aspect ratio, the small set reduces to a single ordinary differential equation yet maintaining high accuracy. A case study is presented to exemplify the models shown.