Singlet-triplet instability for multi-electron system confined by a harmonic potential in a two-dimensional quantum dot―generalized unrestricted Hartree–Fock approach with the real-space finite difference method―

An electron confined by the harmonic potential in a two-dimensional quantum dot (2DQD) provides the QD orbitals characterized by the radial n and angular l quantum number. Rashba electric field (Ξ) breaks the structure symmetry, and then resolves spin degeneracy in each QD state via spin-orbit interaction (SOI). However, the applied Ξ direction (zenith angle θ) varies the resolved separation D from zero (in-plane direction,(θ=π/2) to maximum (θ=0). Focusing on this directional feature, we study the singlet-triplet (ST) instability for the multi-electron system driven by the Rashba field.

Since the Rashba SOI couples opposite spin states, we use the generalized unrestricted Hartree–Fock (GUHF) approach. We solve the GUHF equation using the real-space finite-difference method. The single-particle first-excited states (n, l) = (0,±1), e.g., conserve spin degeneracy when X is applied in the in-plane direction. Thus, a four-electron system is expected to produce a ground-state triplet. As θ decreases, the Rashba SOI increases the energy separation Δ. However, until the exchange energy overcomes the single-particle excitation energy Δ, the ground-state triplet will remain. We explore the ST instability by varying θ and the harmonic confinement potential ω₀  for different numbers of electrons and reveal the ground-state spin-multiple phases of multi-electron systems.