The role of the interface potential on the effective mass of charge carriers is elucidated in this work. We develop a new theoretical formalism using a spatially dependent effective mass that is related to the magnitude of the interface potential. Us
ing this formalism we studied Ge quantum dots (QDs) formed by plasma enhanced chemical vapour deposition (PECVD) and co-sputtering (sputter). These samples allowed us to isolate important consequences arising from differences in the interface potential. We found that for a higher interface potential, as in the case of PECVD QDs, there is a larger reduction in the effective mass, which increases the confinement energy with respect to the sputter sample. We further understood the action of O interface states by comparing our results with Ge QDs grown by molecular beam epitaxy. It is found that the O states can suppress the influence of the interface potential. From our theoretical formalism we determine the length scale over which the interface potential influences the effective mass.
Experimental results obtained previously for the photoluminescence efficiency (PL$_{eff}$) of Ge quantum dots (QDs) are theoretically studied. A $log$-$log$ plot of PL$_{eff}$ versus QD diameter ($D$) resulted in an identical slope for each Ge QD sam
ple only when $E_{G}sim (D^2+D)^{-1}$. We identified that above $Dapprox$ 6.2 nm: $E_{G}sim D^{-1}$ due to a changing effective mass (EM), while below $Dapprox$ 4.6 nm: $E_{G}sim D^{-2}$ due to electron/ hole confinement. We propose that as the QD size is initially reduced, the EM is reduced, which increases the Bohr radius and interface scattering until eventually pure quantum confinement effects dominate at small $D$.
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 tr
anslation 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.
