Model effective-mass Hamiltonians for abrupt heterojunctions and the associated wave-function-matching conditions

Abstract

We consider a class of Hermitian effective-mass Hamiltonians whose kinetic energy term is $\frac{({m}^{\ensuremath{\alpha}}\stackrel{^}{p}{m}^{\ensuremath{\beta}}\stackrel{^}{p}{m}^{\ensuremath{\gamma}}+{m}^{\ensuremath{\gamma}}\stackrel{^}{p}{m}^{\ensuremath{\beta}}\stackrel{^}{p}{m}^{\ensuremath{\alpha}})}{4}$ with $\ensuremath{\alpha}+\ensuremath{\beta}+\ensuremath{\gamma}=\ensuremath{-}1$. We apply these Hamiltonians to an abrupt heterojunction between two crystals and seek the matching conditions across the junction on the effective-mass wave function ($\ensuremath{\psi}$) and its spatial derivative ($\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\psi}}$). For $\ensuremath{\alpha}\ensuremath{\ne}\ensuremath{\gamma}$ we find that the wave function must vanish at the junction thus implying that the junction acts as an impenetrable barrier. Consequently, the only viable cases are for $\ensuremath{\alpha}=\ensuremath{\gamma}$ where we show that ${m}^{\ensuremath{\alpha}}\ensuremath{\psi}$ and ${m}^{\ensuremath{\alpha}+\ensuremath{\beta}}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\psi}}$ must be continuous across the junction.