Form of the quantum kinetic-energy operator with spatially varying effective mass

Abstract

Through a nonrelativistic limit of the Dirac Hamiltonian and considering the spatial dependence of the particle mass in the Foldy-Wouthuysen transformation, it is shown that the quantum kinetic-energy operator with spatially varying effective mass has the form |P[p-hat[1/$\sqrt{\mathrm{m}(\mathbf{r})}$]p-hat[1/$\sqrt{\mathrm{m}(\mathbf{r})}$]+[1/$\sqrt{\mathrm{m}(\mathbf{r})}$]p-hat[1/$\sqrt{\mathrm{m}(\mathbf{r})}$]p-hat|P]. However, the transmission properties of an electron through a GaAs/${\mathrm{Al}}_{\mathrm{x}}$ ${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$ As heterojunction calculated both with this form and the BenDaniel and Duke operator p-hat[m(r)${]}^{\mathrm{\ensuremath{-}}1}$p-hat/2, the most used kinetic-energy operator, differ at most by 1% if the existence of interface regions as thin as two GaAs lattice units is taken into account.