Two-electron anharmonic quantum dot: Effects of uniform magnetic field

The confinement of two electrons by an external potential in a semiconductor and hence a two-electron quantum dot (TEQD) is a well-investigated problem. These external potentials vary from a square well to a harmonic one. The harmonic/parabolic confinement dominates over the other confining potentials. The real physical situations demand the augmentation of the anharmonic term. Using the anharmonic contribution, we explore the effects of the magnetic field on the said TEQD. The Hamiltonian of TEQD under magnetic fields is separated in center of mass (CM) and in relative coordinates (RCs). The Z-component of the CM reduces to the one-dimensional quartic anharmonic oscillator and is unaffected by the magnetic field. The remaining two-coordinates of the CM give rise to the two-dimensional anharmonic oscillator Hamiltonian plus the Larmor term. The equilibrium distance between the two electrons and hence the minimum potential is found to obey the polynomial of order 12. The twelfth-order polynomial is solved analytically. The appearance of Landau levels is indicated. Under small anharmonicity, the analytical solutions are exhibited through some interesting figures. The radial part of the Hamiltonian and hence the Schrödinger equation is separated from the angular part. The potential involving RC also exhibits a stable minimum, and a Taylor series expansion is being made. The minimum position is found by solving the sixth-order polynomial equation analytically. The renormalized frequency and the anharmonic constant are expressed as functions of equilibrium distance. Effects of the magnetic field and anharmonic constants on the energy eigenvalues of the TEQD are indicated.