Effective-mass Hamiltonians for abrupt heterojunctions in three dimensions

Abstract

We find the matching conditions on the wave function and on its normal derivative across an abrupt heterojunction in three dimensions between two otherwise uniform crystals for an effective-mass Hamiltonian with kinetic energy operator of the form (1/2$A_{\mathrm{ij}}$p${^}_i$ $B_{\mathrm{jk}}$p${^}_l$ $C_{\mathrm{kl}}$ (summed on repeated indices i,j,k,l=1,2,3) where $A_{\mathrm{ij}}$ $B_{\mathrm{jk}}$ $C_{\mathrm{kl}}$=$M^{\mathrm{-}1}$ ${\mathrm{}}_{\mathrm{il}}$, the inverse effective-mass tensor. For a simple parametrization of A, B, and C our results rule out such Hamiltonians for most applications unless A=C=I and ${\mathrm{B}=\mathrm{M}}^{\mathrm{-}1}$, in which case the wave function and the normal component of the current are continuous across the heterojunction.