Quantum Dots with Rashba Spin-Orbit Coupling

We present results on the effects of spin-orbit coupling on the electronic structure of few-electron interacting quantum dots. The ground-state properties as a function of the number of electrons in the dot $N$ are calculated by means of spin-density functional theory. We find a suppression of Hund’s rule due to the competition of the Rashba effect and exchange interaction. Introducing an in-plane Zeeman field leads to a paramagnetic behavior of the dot in a closed-shell configuration and to spin texture in space.

Figure 1

(a) Low-energy part of the single-particle spectrum for the dot, calculated both by perturbation theory (dots) and by numerical diagonalization (squares) for $k_{\mathrm{s}\mathrm{o}}{l}_{\omega }=0.2633$ [this value corresponds to the data for $k_{\mathrm{s}\mathrm{o}}=0.01{\mathrm{n}\mathrm{m}}^{-1}$ for the addition energies of the realistic dot shown in panel (b)]. The label $p$ is an index that enumerates the eigenstates in order of ascending energy. Inset: Low-energy part of the single-particle spectrum calculated numerically for $k_{\mathrm{s}\mathrm{o}}{l}_{\omega }=0.7896$ [this value corresponds to $k_{\mathrm{s}\mathrm{o}}=0.03{\mathrm{n}\mathrm{m}}^{-1}$ in panel (b)]. In this case the SO coupling dominates the single-particle spectrum. (b) Addition energy vs number of electrons in the dot for different values of the SO coupling strength. In this figure and in the following ones, the dot is defined by a confining potential of strength $\hslash \omega =5\mathrm{m}\mathrm{e}\mathrm{V}$.

Figure 2

(a) Average values of $S_x$ vs the strength of the in-plane magnetic field, computed by means of SDFT, for two closed-shell configurations ($N=2$ and $N=6$) and for different strength of the spin-orbit interaction. Without the Rashba term, $⟨S_x⟩$ would be zero. (b) Variation of the ground-state energy $\Delta E=E(B)-E(0)$ vs the strength of the in-plane magnetic field, computed by means of SDFT. The ground-state energy shows a quadratic dependence on the in-plane magnetic field: $E(B)=E(0)-A{B}^2$, with $A>0$. The susceptibility is positive and the system is paramagnetic.

Figure 3

Spin density projected along $x$ (a), $y$ (b), and $z$ (c), computed by means of SDFT, for a dot with $N=6$ electrons, $k_{\mathrm{s}\mathrm{o}}=0.02{\mathrm{n}\mathrm{m}}^{-1}$, and $\hslash {\omega }_z=1\mathrm{m}\mathrm{e}\mathrm{V}$. In (a), for ease of visualization, we plot $-n_x$ instead of $n_x$. Spatial integration of $n_y$ and $n_z$ gives zero, yielding a zero average for the corresponding total-spin components.