Finite Element Solutions to GaAs-AlAs Quantum Wells with Connection Matrices at Heterojunctions

Abstract

In this paper, the irreducible finite element formulations for the solutions to the single-band effective-mass equation are developed to incorporate the connection conditions at the heterojunctions. The exact interface formulation utilizes the exact form of the connection matrix. The finite-difference interface formulation uses the finite difference approximation to the connection matrix. For the investigation of the GaAs-AlAs quantum well, the finite-difference interface formulation has to be employed, and the relative errors are shown to be less than 103. Then, this finite element formulation is utilized to compare the effects of the electric field on the GaAs-AlAs quantum well with the heterojunctions modeled by the conventional, the effective-mass-dependent, and the energy-dependent connection matrices. It is found that the eigenenergies predicted by the models with non-conventional connection matrices are larger in magnitude and less dependent on the electric field than the conventional model.