Theoretical description of electronic properties of vertical gated quantum dots

A computational method for studying the electronic properties of vertical gated quantum dots is presented. This method is based on the self-consistent procedure of the solution of the three-dimensional Poisson-Schrodinger problem for few-electron systems confined in the quantum dots. In the present paper, we have applied this method to a quantitative description of transport spectroscopy [S. Tarucha et al., Phys Rev. Lett. 77, 3613 (1996) and L.P. Kouwenhoven et al., Science 278, 1788 (1997)] in vertical gated quantum dots of the cylindrical symmetry. For the entire nanodevice we have obtained the realistic profile of the confinement potential from the Poisson equation. This potential takes into account all the voltages applied to the leads, the spatial distribution of the ionized donors, and number N of electrons confined in the quantum dot. For small N the calculated lateral confinement potential is approximately parabolic, which supports the previous conjectures that the two-dimensional harmonic-oscillator model can be used for a qualitative description of the gated quantum dots. The present study shows that the approximate parabolicity of the lateral confinement potential is a nontrivial property, since it results from a summation of nonparabolic contributions. The nonparabolic corrections should be included in order to obtain an accurate quantitative description of the transport-spectroscopy data. We have solved the N-electron Schrodinger equation by the unrestricted Hartree-Fock method and calculated the chemical potential for the electrons confined in the gated quantum dot. The chemical potential is found to be a nonlinear function of the gate voltage. We have determined the conversion factor, relating the gate voltage with the energy scale, which enabled us to perform a direct quantitative comparison of the computational results with the experimental data. The present results very well reproduce the measured positions of the current peaks for small source-drain voltage. In particular, we have quantitatively described the shell filling and Hund's rule for artificial atoms. We have also determined the conditions of the single-electron tunneling as functions of the source-drain voltage and the gate voltage and obtained the boundaries of the Coulomb diamonds on the stability diagram. The calculated positions, sizes, and shapes of the Coulomb diamonds are in a very good agreement with experiment. We have also evaluated the distribution of the ionized donors and the surface charge induced on the gate and discussed the problem of screening of interelectron interactions in the quantum dot by the electrodes.