Electronic states in a quantum lens
Abstract
We present a model to find analytically the electronic states in selfassembled quantum dots with a truncated spherical cap (``lens'') geometry. A conformal analytical image is designed to map the quantum dot boundary into a dot with semispherical shape. The Hamiltonian for a carrier confined in the quantum lens is correspondingly mapped into an equivalent operator and its eigenvalues and eigenfunctions for the corresponding Dirichlet problem are analyzed. A modified RayleighSchrödinger perturbation theory is presented to obtain analytical expressions for the energy levels and wave functions as a function of the spherical cap height b and radius a of the circular cross section. Calculations for a hard wall confinement potential are presented, and the effect of decreasing symmetry on the energy values and eigenfunctions of the lensshape quantum dot is studied. As the degeneracies of a semicircular geometry are broken for b≠a, our perturbation approach allows tracking of the split states. Energy states and electronic wave functions with m=0 present the most pronounced influence on the reduction of the lens height. The method and expressions presented here can be straightforwardly extended to deal with more general Hamiltonians, including strains and valenceband coupling effects in Group IIIV and Group IIVI selfassembled quantum dots.
 Publication:

Physical Review B
 Pub Date:
 March 2001
 DOI:
 10.1103/PhysRevB.63.125319
 arXiv:
 arXiv:condmat/0008404
 Bibcode:
 2001PhRvB..63l5319R
 Keywords:

 73.61.r;
 73.21.b;
 03.65.Ge;
 78.30.Fs;
 Electrical properties of specific thin films;
 Electron states and collective excitations in multilayers quantum wells mesoscopic and nanoscale systems;
 Solutions of wave equations: bound states;
 IIIV and IIVI semiconductors;
 Condensed Matter  Mesoscale and Nanoscale Physics;
 Condensed Matter  Materials Science
 EPrint:
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