Quasi-one-dimensional quantum dots

In the present contribution the interpretation of the energy-level structure of quasi-one-dimensional quantum dots of two and three electrons is reviewed in detail by examining the polyad structure of the energy levels and the symmetry of the spatial part of the Cl wave functions due to the Pauli principle. The interpretation based on the polyad quantum number is applied to the four electron case and is shown to be applicable to general multi-electron cases. The qualitative differences in the energy-level structure between quasi-one-dimensional and quasi-ta>o-dimensional quantum dots are briefly discussed by referring to differences in the structure of their internal space. 

The total energies and wavefunctions of the Hamiltonian (1) have been calculated as the eigenvalues and eigenvectors of a Cl matrix. Full Cl has been used for all calculations of quasi-one-dimensional quantum dots and for quasi-two-dimensional quantum dots with N = 2, while multi-reference Cl has been used for quasi-two-dimensional quantum dots with N = 3 and 4. The results are presented in atomic units. They can be scaled by the effective Bohr radius of 9.79 nm and the effective Hartree energy of 11.9 meV for GaAs semiconductor quantum dots [25,26]. 

The polyad quantum number is defined as the sum of the number of nodes of the one-electron orbitals in the leading configuration of the Cl wave function [19]. The name polyad originates from molecular vibrational spectroscopy, where such a quantum number is used to characterize a group of vibrational states for which the individual states cannot be assigned by a set of normal-mode quantum numbers due to a mixing of different vibrational modes [19]. In the present case of quasi-one-dimensional quantum dots, the polyad quantum number can be defined as the sum of the one-dimensional harmonic-oscillator quantum numbers for all electrons. 

In order to explain the band structure for the small confinement regime the nature of the potential energy function in the Hamiltonian has been examined in the internal space. Since, for quasi-one-dimensional quantum dots, the electrons can only move along the z coordinate, their x and y dependence is neglected in the analysis. The internal space is defined by a unitary transformation from the independent electron coordinates (z, Z2, , zn) into the correlated electron coordinates (za, zp,...). The coordinate za represents the totally symmetric center-of-mass coordinate za = 7=(zi + Z2 + + zn), and the remaining correlated electron coordinates zp,..., zn represent the internal degrees of freedom of the N electrons [20,21]. In the case of two electrons the correlated coordinates are defined by... 

The energy spectrum of two electrons confined in a quasi-fwo-dimensional Gaussian potential has also been studied for the same set of the strengths of confinement as the corresponding quasi-one-dimensional cases, and are compared to them. The energy spectrum of the quasi-two-dimensional quantum dot is qualitatively different from that of the quasi-one-dimensional quantum dot in the small confinement regime. The origin of the differences is due to the difference in the structure of the internal space. 

---