Quasi-two-dimensional quantum dots

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 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. 

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. 

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... 

In order to analyze the origin of this difference between the energy spectra of quasi one- and two-dimensional quantum dots in the small confinement regime, the internal space for two electrons is considered as in the quasi-one-dimensional cases. Using a harmonic approximation to the Gaussian confining potential, and neglecting the dependence on the z coordinate, the Hamiltonian of Equation (1) for two electrons takes the form... 

Finite size effects play a central role in dictating the electronic properties of materials at the nanoscale. Due to their unique electronic structure, quasi-zero-dimensional (quantum dots) graphitic structures may exhibit fascinating physical phenomena, which are absent in their quasi-one-dimensional (nanowires, nanotubes, and nanoribbons) coimterparts. Many factors govern the effect of reduced dimensions on the electronic properties of nanoscale materials. Here we focus on two such important factors, which are strongly manifested in the electronic characteristics of graphitic materials, namely, quantum confinement and edge effects ... 

The Hamiltonian (1) for quasi-one-dimensional two-electron quantum dots is simplified by neglecting the x and y degrees of freedom and by approximating the confining Gaussian potential by a harmonic-oscillator potential with >z... 

Quantization refers to the restriction of quasi-freely mobile electrons in a piece of bulk metal and can be accomplished not only by reduction of the volume of a bulk material but also by reducing the dimensionality. A quantum well refers to the situation in which one dimension of the bulk material has been reduced to restrict the free travel of electrons to only two dimensions. Restricting an additional dimension then only allows the electrons to travel freely in one dimension and is called a quantum wire, while restricting all three dimensions results in a quantum dot. 

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