Adiabatic description of nonspherical quantum dot models

Abstract

Within the effective mass approximation an adiabatic description of spheroidal and dumbbell quantum dot models in the regime of strong dimensional quantization is presented using the expansion of the wave function in appropriate sets of single-parameter basis functions. The comparison is given and the peculiarities are considered for spectral and optical characteristics of the models with axially symmetric confining potentials depending on their geometric size making use of the total sets of exact and adiabatic quantum numbers in appropriate analytic approximations

Analytical and numerical calculations of spectral and optical characteristics of spheroidal quantum dots

“…For OSQD and PSQD the crossings of the energy levels that occur with increasing γ F are similar to the exact crossings of the energy levels with decreasing c semiaxis in OSQD and PSQD without electric field (γ F = 0), i.e., we observe the accidental degeneracy, which is known to be generally associated with the existence of an additional integral of motion [27] and with the separability of variables in oblate and prolate spheroidal coordinate systems.…”

“…Below we present the analysis of the spectrum under the variation of parameters, which opens the questions about the additional symmetry of the problem, connected with the existence of exact and approximate integrals of motion [27,38]. In Fig.…”

“…At γ F = 0 for IPBM we have the interband overlapĨ ν,ν = δ nρo,n ρo δ nzo,n zo δ m,−m for OSQD, orĨ ν,ν = (J 1+|m| (α nρp+1,|m| )/J 1−|m| (α nρp+1,|m| )) 2 δ nzp,n zp δ nρp,n ρp δ m,−m for PSQD, where α nρp+1,|m| is the positive root of the Bessel function, and the selection rules m = −m , n zo = n zo , n ρo = n ρo , or n ρp = n ρp , n zp = n zp [27], while at γ F = 0 one should calculate the interband overlap (53) in accordance with the selection rules m = −m , n ρo = n ρo , or n ρp = n ρp , respectively. Note,that in the adiabatic limit and at small γ F the contributions of non-diagonal matrix elements to the energy values are about 1% for IPBM of OSQD and PSQD; then in the Born-Oppenheimer approximation of the order b max for the AC we get…”

“…The 3D BVP for SQDs at fixed values of m is reduced to 2D BVP with respect to fast x f and slow x s variables: oblate x f = z (minor axis), x s = ρ (major axis) and prolate x f = ρ (minor axis), x s = z (major axis) [27]:…”

“…In our ear-lier work we elaborated the calculation schemes, symbolic-numerical algorithms (SNAs) and programs, based on the generalized Kantorovich method (KM) for numerical solving with required accuracy the boundary-value problems (BVPs) of discrete and continuous spectra describing the axial-symmetric models of quantum wells(QWs), quantum wires(QWrs) and quantum dots(QDs) in external fields within the framework of the effective mass approximation [23][24][25][26][27][28][29][30][31][32][33][34][35]. Meanwhile, for the analysis and estimations of the appropriate range of material parameters, spectral and optical characteristic of quantum dots at the first stage of investigation, conventionally, approximate eigenvalues and eigenfunctions evaluated in the analytical form were applied [6-8, 14, 22].…”