Finite‐element formulation of quantum wells using transfer matrix

Abstract

AbstractWhen the effective mass approximation is used for analysis of electrons bound in a quantum well, the envelope function generally is not continuous at the heterojunction. The boundary condition at the junction is expressed in terms of the transfer matrix. In this paper, an analysis method based on the finite‐element formulation is proposed for electrons bound in a quantum well made of arbitrary materials. It is shown that by using 3rd‐order Hermitian line elements as the approximation functions, the continuities of the envelope function and its derivative are ensured outside the heterojunction, and that introduction of the transfer matrix becomes possible. It is also shown that the problem becomes an asymmetric generalized eigenvalue problem by means of this formulation. As numerical examples, rectangular quantum wells are analyzed in the cases where the envelope function, and its derivative divided by the effective mass, are continuous at the heterojunction (GaAs/A1GaAs quantum well) where also the function and its derivative are discontinuous (InAs/GaSb quantum well). The validity of the calculations is confirmed by comparison with the analytical solutions.