| Abstract: | This paper investigates the effects of mass dicontinuity on the numerical solutions to quantum wells using the effective mass equation. The numerical methods utilized are the finite element method with first-order elements, and the finite difference method with the entire truncated solution domain discretized by equally spaced nodes. The three Hamiltonians explored are the convention Hamiltonian, the BenDaniel and Duke Hamiltonian, and the Bastard Hamiltonian. It is shown that the proper discretization patterns for both numerical schemes may drastically improve the solution accuracy. The finite difference representation of the BenDaniel and Duke Hamiltonian using the direct mass average is found more accurate than the one using the harmonic mass average. It is further pointed out that, at the mass profile discontinuities, the commonly accepted interface conditions for the Bastard Hamiltonian are not natural conditions. This observation is critical if the Bastard Hamiltonian is to be solved numerically. |
