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 $\frac{(m^{\alpha }\hat{p}{m}^{\beta }\hat{p}{m}^{\gamma }+m^{\gamma }\hat{p}{m}^{\beta }\hat{p}{m}^{\alpha })}{4}$ with $\alpha +\beta +\gamma =-1\mathrm{}$. 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 ($\psi$) and its spatial derivative ($\dot{\psi }$). For $\alpha \ne \gamma$ 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 $\alpha =\gamma$ where we show that $m^{\alpha }\psi$ and $m^{\alpha +\beta }\dot{\psi }$ must be continuous across the junction.