Energy Spectra of Two Particles in a Parabolic Quantum Dot: Numerical Sweep Method

Abstract

The Schrödinger equation for two particles of the same mass in two‐ and three‐dimensional parabolic quantum dots is analyzed by using the trigonometric sweep method. Novel curves which show quantum‐size effects on the two‐particle energy spectra and take into consideration the repulsive and attractive interaction as particular cases of the same phenomenom are presented. The level‐ordering and crossovers of these curves for 2D and 3D exciton and two‐electron in a spherical quantum dot are analyzed and discussed. Our results for the 2D two‐electron problem are in good agreement with the ones obtained previously by using other methods.

Spatial correlation of two particles in semiconductor quantum ring

“…As it can be shown by expanding expressions (7) and (10) in the series the function q(r) can be approximated for small values of r as qðrÞ % À1 4r 2 þ 2=r for two electrons and as qðrÞ % À1 4r 2 À 2=r for the exciton. Only the second of them provides discrete levels with negative energies similar to two-dimensional hydrogen-like states.…”

“…Particular interest has been related to theoretical study of the effect of the correlation on the spectra of two-particle systems such as two electrons and an exciton confined in QDs. The variational [2][3][4][5], matrix diagonalization [6][7][8][9][10] and finite element [11][12][13][14] methods have been used and the models with rectangular [2,3,11,12], soft-edge-barrier [4], parabolic [6][7][8][9] and charge image [15] confinement potentials have been considered to this end.…”

The Binding Energies of Shallow Donor Impurities in GaAs–(Ga,Al)As Coaxial Quantum-Well Wires

“…To find the eigenvalues k 2 and l 0 we have solved subsequently the equations (3) and (7) by using the numerical trigonometric sweep procedure [9,10]. The nlm j i states binding energies, which are measured with respect to the first energy level of the system without impurity, is defined as…”

Renormalized Schr�dinger Equation for Excitons in Graded Quantum Dot

“…Up to now it has been considered the confinement models with rectangular [1-4], parabolic [5] and charge image [6,7] potentials. The Hamiltonian of an exciton in QD can be separated into center-of-mass and relative-motion terms only in the special case of the parabolic confinement as the electron and the hole effective masses are equal [8]. In other cases, the approximation methods, such as the variational [1-4], matrix diagonalization [5] or finite element [9] have to be used.…”

“…1 for R ¼ 0:1a * 0 ). It should be noted that the definition of the fractal dimension (8), which is valid for the electron-hole separation r eh less than the QD size, becomes ill-conditioned for large values of r eh because the volume measure given by the relation (6) decreases exponentially. For this reason D* in Fig.…”