Abstract
As the density of electrons (N=2, 3, 4, 5, 6, 7, 8,.) increases, complexity arises due to coulomb interactions, inclusion of fermionic exchange symmetry and anisotropy. Consequently, Schrodinger equations of anisotropic quantum dots become non-trivial. Recasting such non-relativistic quantum equations into Whittaker-M basis functions unifies coulomb (exchange) correlation and antisymmetric nature of electrons in two-centered integrals of exact, finite, single-summed, terminated and simplest Lauricella functions (F2) via multi-pole expansion and the subsequent Chu-Vandermonde identity. Coulomb correlations interplay with in-plane and out-of-plane electrical and magnetic confinements in their moderate fields. On the other hand, full-scale fermionic exchange symmetry is incorporated through multiple number of Slater determinants, unlike Hartree-Fock method. In this context, both open-shell and restricted closed-shell configurations are considered for trial Hartree-products which are composed of the basis set of oscillator spin-orbitals. Thus optimized bound states can easily capture large number of mixed term symbols. Consequently, level clustering/accidental degeneracies occur in energy level diagram due to competition among strength of anisotropy in electrical confinements, magnetic field, mass of the carrier and dielectric constant. It brings about orbital induced paramagnetism (T∼(0-1)K), signature of fractional quantum Hall effect (FQHE) in chemical potential cusps (μ) and formation of different ′shell structures′ in capacitive energy respectively, spanning over wide dielectric range of materials (atomic like quantum dot, ZnO, GaAs, CdSe (Cadmium Chalcogenide) and PbSe (Lead Chalcogenide) etc).As the density of electrons (N=2, 3, 4, 5, 6, 7, 8,.) increases, complexity arises due to coulomb interactions, inclusion of fermionic exchange symmetry and anisotropy. Consequently, Schrodinger equations of anisotropic quantum dots become non-trivial. Recasting such non-relativistic quantum equations into Whittaker-M basis functions unifies coulomb (exchange) correlation and antisymmetric nature of electrons in two-centered integrals of exact, finite, single-summed, terminated and simplest Lauricella functions (F2) via multi-pole expansion and the subsequent Chu-Vandermonde identity. Coulomb correlations interplay with in-plane and out-of-plane electrical and magnetic confinements in their moderate fields. On the other hand, full-scale fermionic exchange symmetry is incorporated through multiple number of Slater determinants, unlike Hartree-Fock method. In this context, both open-shell and restricted closed-shell configurations are considered for trial Hartree-products which are composed of the basis set...
Highlights
Combined effect of coulomb interaction, full scale l fermionic exchange symmetry, anisotropy and magnetic field emerges with unusual features in energy spectra, magnetization, chemical potential as level clusterings,[17] paramagnetism and formation of different shell structures respectively, which are studied in detail further
Schrodinger equation of 3-D N − e anisotropic dot is solved exactly in Whittaker-M function eigen-basis that gives a finite summed Lauricella function form to coulomb integral through multipole-expansion, for all bound states
Level clusterings/accidental degeneracies are observed in isotropic system at higher carrier density as a mark of symmetry
Summary
Properties of 2-e quantum dot reveals manifestations like level clustering/dynamical symmetries in response to magnetic field.[16,17] Investigation of wide range of confinement anisotropies and interaction strength on excitation spectra of N=3 and shell structure has been reported.[18,19] As both the single determinant closed and open shell Hartree-Fock method and the single particle averaged potential representation of Kohn-Sham equation in density functional theory are valid for regular and weakly interacting many-body systems, explaining unusual phenomena of highly anisotropic materials in mesoscopic scale is a far reach.[20,21] The crucial attribute to many-electron systems lacking is an exact treatment to coulomb correlations proliferating by a factor of N(N-1)/2 and full-scale implementation of fermionic exchange symmetry. Hazra et al extemporized an exact formalism of Coulomb interactions for both 2-D and 3-D multi-electron systems in multi-pole integrals as finitely single summed, terminating Lauricella function via Chu-Vandermonde theorem.[22,23,24] In section III, we have discussed the electronic structure, magnetization and chemical potential of 3-D N=2, 3, 4, 5, 6, 7, 8,., electron anisotropic quantum dots in detail which reproduces experimental results of Tarucha et al.[25]
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