Computer simulation of electron energy levels for different shape InAs/GaAs semiconductor quantum dots

Abstract

A computational technique for the energy levels calculation of an electron confined by a 3D InAs quantum dot (QD) embedded in GaAs semiconductor matrix is presented. Based on the effective one electronic band Hamiltonian, the energy and position dependent electron effective mass approximation, a finite height hard-wall 3D confinement potential, and the Ben Daniel–Duke boundary conditions, the problem is formulated and solved for the disk, ellipsoid, and conical-shaped InAs/GaAs QDs. To calculate the ground state and first excited state energy levels, the nonlinear 3D Schrödinger is solved with a developed nonlinear iterative algorithm to obtain the final self-consistent solutions. In the iteration loops, the Schrödinger equation is discretized with a nonuniform mesh finite difference method, and the corresponding matrix eigenvalue problem is solved with the balanced and shifted QR method. The proposed computational method has a monotonically convergent property for all simulation cases. The computed results show that for different quantum dot shapes, the parabolic band approximation is applicable only for relatively large dot volume. For the first excited states the non-parabolicity effect also has been found to be stronger than it at ground state. The QD model and numerical method presented here provide a novel way to calculate the energy levels of QD and it is also useful to clarify principal dependencies of QD energy states on material band parameter and QDs size for various QD shapes.