The Energy Level Structure of Low-dimensional Multi-electron Quantum Dots

Abstract

A unified characterization of the energy-level structure of quasi-one-dimensional quantum dots is presented based on accurate computational results for the eigenenergies and wave functions, as obtained in previous studies for the case of two and three electrons, and in the present study also for four electrons. In each case the quantum chemical full configuration interaction method is adopted employing Cartesian anisotropic Gaussian basis sets. The energy-level structure is shown to be strongly dependent on the confinement strength ω and can be exemplified by the three qualitatively distinct regimes for large, medium, and small confinement strengths. To characterize the energy-level structure in the large or medium ω and the small ω regimes, the polyad quantum number, as well as its extended version, the extended polyad quantum number have been introduced. The degeneracy of energy levels for different spin states in the small ω regime is shown to be caused by the potential walls of the electron-electron interaction potential within the internal space. A systematic way of obtaining the degeneracy pattern of the energy levels in the small ω regime is also presented. Finally, qualitative differences between the energy-level structure of quasi-one-dimensional and quasi-two-dimensional quantum dots in the small ω regime are briefly discussed by referring to the different structure of their internal spaces.

Origin of Hund’s multiplicity rule in quasi-two-dimensional two-electron quantum dots

“…Therefore, the large mixing of these two configurations in the strong confinement regime indicates an improper description of the states of quantum dots that is based on an independent electron model with Hartree-Fock orbitals. In order to properly describe the states in the v p = 2 manifold, appropriate assignments of the normal modes of electrons are required [33,34,61]; these are introduced in the next section.…”

Origin of the first Hund rule and the structure of Fermi holes in two-dimensional He-like atoms and two-electron quantum dots

“…This difference in the shape of the internal wave functions of these two systems can be understood by considering the spatial extent of the relevant one-electron orbitals: In the case of the He-like atomic systems there is a large difference in the size of the (1s) and (2p) orbitals, since they belong to different principal shells of n = 1 and n = 2. On the other hand, in the case of quantum dots, the (0σ g ) and (1π u ) orbitals also reside in different shells of v p = 0 and 1 where the polyad quantum number v p accounts for the shell structure of the two-dimensional harmonic oscillator [40,45,32,46], yet their spatial extent is of the same order of magnitude since the classical turning points of the harmonic oscillator along the x axis, for example, are |x c | = 1 and √ 3 for v = 0 and 1, respectively.…”

A Unitary Transformation and Spectrum of a Three-Electron Quantum Dot

“…The three-electron quantum dots in different potentials including the harmonic, Coulomb and Gaussian terms were studied in Refs. [13][14][15][16][17][18][19][20][21][22] using different methodologies. In the present work, we consider parabolic and Coulomb terms but present a relatively different mathematical way to treat the problem.…”