Reformulated Hamiltonian for nonparabolic bands in semiconductor quantum wells

Abstract

A new form of the Hamiltonian and a boundary condition on the derivatives of the wave function are presented for use in the Schrödinger wave equation to calculate the energy eigenvalues in semiconductor quantum wells. This fourth-order Hamiltonian allows the use of a constant effective mass for the electron, and accounts for the effect of band nonparabolicity on the energy eigenvalues in a simple way. Calculations show that nonparabolicity contributes only a small effect for quantum wells grown in the material systems ${\mathrm{Al}}_{0.35}$ ${\mathrm{Ga}}_{0.65}$As/GaAs and InP/${\mathrm{Ga}}_{0.47}$ ${\mathrm{In}}_{0.53}$As. Larger effects are predicted in the [111] crystal-growth direction than the [100] direction.