Two-Electron Spherical Quantum Dot in a Magnetic Field

Abstract

We investigate three-dimensional, two-electron quantum dots in an external magnetic field B. Due to mixed spherical and cylindrical symmetry the Schrödinger equation is not completely separable. Highly accurate numerical solutions, for a wide range of B, have been obtained by the expansion of wavefunctions in double-power series and by imposing on the radial functions appropriate boundary conditions. The asymptotic limit of a very strong magnetic field and the 2D approach have been considered. Ground state properties of the two-electron semiconductor quantum dots are investigated using both the 3D and 2D models. Theoretical calculations have been compared with recent experimental results. IntroductionThe influence of spatial confinement on properties of quantum systems have been widely studied in the literature and remains subject of continuous interest in both theoretical and experimental fields [1][2][3][4][5]. One of the most interesting confined quantum systems is the two-electron Hooke's-law atom (HA) also known as hookium or harmonium [6]. The HA refers to a model system composed of two electrons interacting by the Coulombic potential and confined in an external harmonic potential. Many applications for this system ensues from some unique properties of the HA. For the parabolic confinement the two-electron Hamiltonian separates. This property significantly simplifies the two-particle Schrödinger equation leading, in fact, to the one-dimensional radial problem. We note that the separability of the Schrödinger equation describing many interacting particles is rather exceptional. In particular, it is possible when interactions between disjoint pairs of particles, interacting by arbitrary two-particle potentials, are harmonic [7,8]. For the HA the Schrödinger equation not only separates exactly but also, for a set of the coupling constants, closed-form analytical solutions exist [9,10]. This is of particular importance for the understanding of the role of electron-electron interaction and correlation effects.When the electron mass is replaced by the relevant effective mass and the e-e interaction coupling constant is supplemented by the dielectric constant of semiconductor, the HA atom may be regarded to as a twoelectron quantum dot (QD). During last few decades many theoretical and experimental studies on QDs have been performed within parabolic confinement models as well as beyond this approximation. Both the 2D and 3D QD's have been investigated in a framework of strict quantum-mechanical and semiclassical approaches, including also effects of an external magnetic field [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. Semiclassical solutions for a two-electron QD in a magnetic field have been investigated in terms of action-angle variables using the classical adiabatic approximation [19]. Spin-singlet and spin-triplet transitions have been studied and the magnetic moment and susceptibility have been obtained as functions of the magnetic field [20].A.

Oscillating properties of a two-electron quantum dot in the presence of a magnetic field

“…Other interesting observation is that the J norm oscillations occur for larger b values in the 3D case (w z = 0.000 111) than in the quasi-2D one (w z = 1.11). This behavior is also observed in the study of the chemical potential and the addition energy at reference [34].…”

The weakly bound states in Gaussian wells: From the binding energy of deuteron to the electronic structure of quantum dots

“…Furthermore, in two and three dimensions, Gaussian wells and barriers have proven to be effective in the examination of diverse quantum systems, such as quantum rings and polarons [4–6]. Specifically, quantum wells are often employed to simulate the artificial confinement of electrons within quantum dots [7–11], as opposed to harmonic confinement [12–14], which does not allow for dissociation. Additionally, the Gaussian potential offers a straightforward approach to modeling short‐range interactions among nucleons [15].…”

Quantum correlations in one-dimensional Wigner molecules

“…In particular, the study of the manybody properties of various quantum composite systems is nowadays one of the most active areas of theoretical physics [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]. Investigating quantum correlations in systems of interacting particles trapped in external potentials is not only important in view of the context of quantum information technology [24] but also a key to improving our understanding of quantum matter.…”