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

Abstract

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 $x\to {\gamma }^{-1}\ln (1+\gamma x)$ 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: $m_o^{\ast -1}\propto \frac{{\partial }^{2}E}{\partial {\check{k}}^2}$. 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.

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Cited by in corpus (8)

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• Supersymmetric quantum mechanics and coherent states for a deformed oscillator with position-dependent effective mass
• Influence of interface potential on the effective mass in Ge nanostructures
• Two dimensional electron gas in a non-Euclidean space
• Quantum confinement in Si and Ge nanostructures: Effect of crystallinity