Establishment of an effective-mass Hamiltonian for abrupt heterojunctions

Abstract

For an abrupt heterojunction between two otherwise uniform semiconductors in one dimension we suggest using the effective-mass Hamiltonian H=-(1/2)${ħ}^2$[m(z)${]}^{\alpha }$∇[m(z)${]}^{\beta }$∇[m(z)${]}^{\alpha }$+V(z) with 2α+β=-1 and where m(z) is the position-dependent effective mass and V(z) is the position-dependent conduction band edge. The wave-function matching conditions across the heterojunction are continuity of $m^{\alpha }$φ and $m^{\alpha +\beta }$∇φ. By imposing a simple physical criterion on the solution, φ, of the eigenvalue problem for H we find an experssion for β involving the underlying Bloch functions appropriate to the two-component semiconductors, evaluated at the heterojunction. In a model calculation we estimate β≊0 for GaAs-${\mathrm{Al}}_{\mathrm{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$As, independent of x.