Quantum Confinement in Nonadditive Space with a Spatially Dependent Effective Mass for Si and Ge Quantum Wells

We calculate the effect of a spatially dependent effective mass (SPDEM) [adapted from R. N. Costa Filho et al. Phys. Rev. A., \textbf{84} 050102 (2011)] on an electron and hole confined in a quantum well (QW). In the work of Costa Filho et al., the translation operator is modified to include an inverse character length scale, $\gamma$, which defines the SPDEM. The introduction of $\gamma$ means translations are no longer additive. In nonadditive space, we choose a `skewed' Gaussian confinement potential defined by the replacement $x\rightarrow\gamma^{-1}\ln(1+\gamma x)$ in the usual Gaussian potential. Within the parabolic approximation $\gamma$ is inversely related to the QW thickness and we obtain analytic solutions to our confinement Hamiltonian. Our calculation yields a reduced dispersion relation for the gap energy ($E_G$) as a function of QW thickness, $D$: $E_G\sim D^{-1}$, compared to the effective mass approximation: $E_G\sim D^{-2}$. Additionally, nonadditive space contracts the position space metric thus increasing the occupied momentum space and reducing the effective mass, in agreement the relation: $m_o^{*-1}\propto \frac{\partial^2 E}{\partial \v{k}^2}$. The change in the effective mass is shown to be a function of the confinement potential via a point canonical transformation. Our calculation agrees with experimental measurements of $E_G$ for Si and Ge QWs.