Symbolic-Numeric Algorithms for Computer Analysis of Spheroidal Quantum Dot Models

Abstract

Abstract. A computation scheme for solving elliptic boundary value problems with axially symmetric confining potentials using different sets of one-parameter basis functions is presented. The efficiency of the proposed symbolic-numerical algorithms implemented in Maple is shown by examples of spheroidal quantum dot models, for which energy spectra and eigenfunctions versus the spheroid aspect ratio were calculated within the conventional effective mass approximation. Critical values of the aspect ratio, at which the discrete spectrum of models with finitewall potentials is transformed into a continuous one in strong dimensional quantization regime, were revealed using the exact and adiabatic classifications.Key words: Symbolic-numerical algorithms, boundary value problems, axial confinement potentials, oblate and prolate spheroidal quantum dot models, effective mass approximation, strong dimensional quantization

Algorithm for reduction of boundary-value problems in multistep adiabatic approximation

“…The computational scheme, the SNA, and the complex of programs allow extension for the analysis of spectral characteristics of both electron(hole), impurity and excitonic states in nanoscale quantum-dimensional models like QWs [17], QWrs [18], QDs [12] with different geometry of structure and spatial form of confining potential and external fields.…”

“…The adiabatic approximation is well-known method for effective study of fewbody systems in molecular, atomic and nuclear physics. On the base of pioneering work of Born and Oppenheimer [6] the method was applied in various problems of physics, using the idea of separation of "fast" x f and "slow" x s variables [5] in Hamiltonian composed by fast and slow subsystems H( x f , x s ) = H f ( x f ; x s ) + H s ( x s ) with characterized frequencies ω f > ω s , for example in Hénon-Heiles model [16] or quantum dot (QD) models [12].…”

The role of geometry in tailoring the linear and nonlinear optical properties of semiconductor quantum dots

“…There are two approaches for investigating QDs with complex geometries approximate methods and computational methods. The approximate methods include the geometrical adiabatic approximation methods which can be used to obtain the spectrum and wavefunctions for the oblate and prolate shapes [13][14][15][16][17][18][19][20][21] , and conformal mapping methods which allow to reduce an unsolvable three-dimensional problem into lower dimensions causing a possible loss of information [22][23][24][25] . The next is numerical modeling, which by far provides the best flexibility in terms of the shapes of QDs.…”