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., 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, γ, which defines the SPDEM.The introduction of γ means translations are no longer additive. In nonadditive space, we choose a 'skewed' Gaussian confinement potential defined by the replacement x → γ −1 ln(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 (E G ) as a function of QW thickness,Additionally, nonadditive space contracts the position space metric thus increasing the occupied momentum space and reducing the effective mass, in agreement the relation: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.

The role of the interface in germanium quantum dots: when not only size matters for quantum confinement effects

“…In order to describe the effect of the interface, we developed a correction to the EMA model through a spatially dependent effective mass (SPDEM) formalism. 26,45 The SPDEM model is directly related to the potential V c (x) for confined carriers and describes the effect of V c on the EM, through the dispersion relationship: . 26,43 The inclusion of the SPDEM into the EMA model gives better agreement between the theory and the experiment, as shown in Fig.…”

A position-dependent mass harmonic oscillator and deformed space

“…The canonical transformation given by Eqs. (16) leads to the new Hamiltonian (see, for instance, [38])…”

“…The energy levels of the quantum harmonic oscillator with PDM given by Eq. (19) are identical to those of a constant mass particle in a constant Morse potential, since these two systems may be mapped into one another by the canonical transform (16). and its associated time independent Schrödinger equation at basis {|x } is…”

Transmission through graded interfaces in the displacement operator method

“…Its classical field theory has also been established [17] . Further more, the formalism has been applied successfully in the study of: the influence of interface potential on the effective mass in Ge nanostructures [18], Quantum confinement in Si and Ge quantum wells [19], the role of quantum confinement in luminescence efficiency of group IV nanostructures [20]. yet its effective application to compositionally graded interfaces has not been addressed.…”