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 $xrightarrowgamma^{-1}ln(1+gamma x)$ 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, $D$: $E_Gsim D^{-1}$, compared to the effective mass approximation: $E_Gsim D^{-2}$. 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.