Phase transitions of a few-electron system in a spherical quantum dot

The spin configurations of a spherical quantum dot, defined by a three-dimensional (3D) harmonic confinement potential, containing a few Coulomb Fermi particles (electrons or holes) are studied. Quantum transitions involving a spin transformation and a “cold melting” (from a Wigner crystal-like state, i.e., from regime of strongly correlated electrons, to a Fermi-liquid-like phase) is driven by the dimensionless quantum control parameter q (which is connected with steepness of the confinement potential) is demonstrated. The pair correlation and radial distribution functions which characterize electronic quantum delocalization are analyzed. The calculations using the unrestricted variational Hartree-Fock method (for the ground state at $T=0\mathrm{}\hspace{0.5em}\mathrm{K})$ and the more computer intensive quantum path integral Monte Carlo method (for $T\ne 0\hspace{0.5em}\mathrm{K})$ are performed and compared. For small q, the ground state of the three electron system is crystal-like and has $C_3$ symmetry, i.e., the maxima of electron density are located at the nodes of an equilateral triangle. The preferable spin configuration for small q is “ferromagnetic,” with total spin $S=3/2.\mathrm{}$ As q rises, the widths of the one-electron wave functions grow and become overlapping. At a critical value $q_1$ the ground state changes from $S=3/2\mathrm{}$ to $S=1/2\mathrm{}$ and at the same time, asymmetry appears in the triangle (i.e., spontaneous breaking of the $C_3$ symmetry to $C_2$ symmetry). At a second critical value $q_2$ the electron distribution undergoes a symmetry phase transition, from trianglelike (with $C_2$ symmetry) to axial symmetric (with $C_{\infty }$ symmetry). As q grows further, we obtain a Fermi-liquid-like (non-interacting) electron configuration in the ground state $\mathrm{}(S=1/2\mathrm{}).$ In addition, the $S=3/2\mathrm{}$ state, at a critical q value (which is slightly larger than $q_1)$ undergoes a dramatic charge redistribution.