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 Costa Filho et al. (2011)] on an electron and a 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, γ, which defines the SPDEM. The introduction of γ   means that translations are no longer additive. In nonadditive space, we choose a ‘skewed’ Gaussian confinement potential defined by the replacement x→γ−1ln(1+γx)x→γ−1ln(1+γx) in the usual Gaussian potential. Within the parabolic approximation γ 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 (EG) as a function of QW thickness, D:EG~D−1, compared to the effective mass approximation: EG~D−2EG~D−2. Additionally, nonadditive space contracts the position space metric thus increasing the occupied momentum space and reducing the effective mass, in agreement with the relation: mo⁎−1∝∂2E/∂k2. 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 EG for Si and Ge QWs.