Electronic spectra and chemical potential of 2-D multi-electron quantum dots in magnetic field: exact multi-pole expansion of coulomb correlation

Abstract

Two-dimensional superlattices, TMDCs and graphene family exhibit strong coulomb correlations among e–e, e–h, electron-phonon and etc. As the number of electron increases, confining gate voltage and transverse magnetic field are superseded by increasing coulomb interactions (N(N−1)/2 factors) that composes non-trivial Schrödinger equations. Representing such equations in Whittaker-M functions yields a modest alternate formalism, that accommodates integrals of coulomb (exchange) correlations in single-summed, finite and exact Lauricella functions via Chu-Vandermonde identity. For higher carrier density (N = 3, 4, 5, 6, ..), the multipole expansion is incurred as exact and finite-summed coulomb, coulomb-type and dipole-type integrals. Although, fermionic exchange symmetry of many electron systems could be included in terms of various two-electron integrals, for the sake of brevity we have aimed to reproduce experimental results without spin. Signature of interplay among gate voltage, magnetic field, dielectric constant, mass and density of carriers is examined in electronic spectra, magnetization, chemical potential for the systems spanning over wide range of materials (He, BN, GaAs and etc.). Interestingly, chemical potential and addition energy as a function of magnetic field and number of carriers monitor the statistics between strongly degenerate to weakly degenerate composite fermions, coulomb blockade and shell structure of 2-D superlattices. At the most, quadrapole and octapole suffice the convergence.