Spin-orbit coupling effects in two-dimensional circular quantum rings: Elliptical deformation of confined electron density

We study electron states confined in two-dimensional circular quantum rings in the presence of spin-orbit coupling due to both structure and crystal inversion asymmetry in the external magnetic field. It is demonstrated that the confined electron density loses the circular symmetry of the confinement potential provided that both Rashba and Dresselhaus coupling constants are nonzero, with the exception of a special case of equal coupling constants and absence of the spin Zeeman interaction. An elliptical deviation from the circular symmetry—present already for a single confined electron—is for two electrons strengthened by the Coulomb repulsion. We discuss signatures of the charge-density deformation in the experimentally accessible quantities: magnetization and charging properties of the ring. Relevance of the results of one-dimensional ring models for description of spin-orbit coupling effects is also discussed.

Figure 1

(Color online) Single-electron energy spectrum for the ring defined by Eq. (4) without spin-orbit coupling. The red line near the bottom of the plot shows the ground-state angular momentum.

Figure 2

(Color online) (a) Blue solid curves show the energy spectrum for a single type of the spin-orbit coupling [$\alpha =10.8\text{meV}\text{nm}$, $\beta =0$ or equivalently $\alpha =0$, $\beta =10.8\text{meV}\text{nm}$] for $g=0$ as obtained by the present approach. The black lines show the results given by the analytical formula Eq. (8) for a strictly one-dimensional quantum ring shifted down on the energy scale by $-40.94\text{meV}$. The dashed red curves indicate the results obtained in the lowest-radial-state approximation (see text) referred to the right axis. (b) Solid blue and red dashed curves at the top of the plot show the value of the average spin as obtained for the two-dimensional quantum ring with the exact diagonalization approach and with the basis restricted to the lowest radial state, respectively. The black dotted line and the solid green line indicate the ground-state total and orbital angular momentum, respectively. The black solid line near the bottom of the plot shows the $s$ parity of the ground state. Pure Rashba coupling was applied for this figure ($\alpha =10.8\text{meV}\text{nm}$, $\beta =0$).

Figure 3

(Color online) Spin-up (red curves) and spin-down densities (blue curves) as obtained in the ground-state for $B=0.6\text{T}$ (a) and $B=0.4\text{T}$ (b) for the pure Rashba coupling and $g=0$ (parameters of Fig. 2). The dashed vertical lines show the positions of the maxima of the majority and minority-spin distributions. The black solid lines show the confinement potential.

Figure 4

(Color online) (a) Blue solid curves show the spectrum for $\alpha =\beta =10.8\text{meV}\text{nm}$ and $g=0$ as obtained with the unrestricted basis. Red dotted curves indicate the results of the lowest-radial-state approximation shifted down on the energy scale by 0.088 meV. The black dotted curves show the results for the spin-orbit coupling excluded. The dashed lines show the parameter $\rho$ characterizing deviation of the charge density from the circular symmetry [Eq. (9)], as obtained by the exact diagonalization (blue line) and with the lowest-radial-state approximation (red line). (b) Charge density obtained for $B=0.75\text{T}$ plotted along the center of the ring $r=50\text{nm}$ as obtained by the exact diagonalization (blue curve) and with the basis restricted to the lowest radial state (red line) for the ground state of odd $s$ parity.

Figure 5

(Color online) Charge and spin densities as obtained for $g=0$, $\alpha =\beta =10.8\text{meV}\text{nm}$ and $B=0.75\text{T}$ in the basis restricted to the lowest-radial-state (a) and in the unrestricted basis (b) for the ground state of odd $s$ parity (for the ground state of even $s$ parity the spin-up and spin-down densities are inverted).

Figure 6

(Color online) (a) The blue solid and the red dotted curves show the spectrum as calculated for $\alpha =\beta /2=5.4\text{meV}\text{nm}$ with the unrestricted basis and in the lowest-radial-state approximation, respectively. The dashed curves indicate the value of the $\rho$ parameter as calculated by the exact (blue) and restricted (red) bases. The inset shows the ground-state charge density for $B=3.77\text{T}$ along the circumference of the ring as obtained by the basis restricted to the lowest radial state (red curve) and for unrestricted basis (blue curve). (b) The black line at the top of the plot show the average value of the $z$ component of the spin, the blue and red lines the average values of the $J_{-}$ and $J_{+}$ total angular-momentum operators, and the green line near the bottom of the plot the ground-state s-parity obtained in the exact calculation.

Figure 7

(Color online) Same as Fig. 6 but for a ring of four times smaller width: $R_i=55\text{nm}$ and $R_o=60\text{nm}$. The energy spectrum obtained by the restricted basis in (a) was shifted down by 0.067 meV.

Figure 8

(Color online) (a) Black curves show the energy spectrum as calculated by the exact diagonalization for $\alpha =\beta =10.8\text{meV}\text{nm}$ and $g=-2.15$. The red symbols show the deformation parameter $\rho$, which in wider $B$ range is presented also in panel (b). (c) The black curve at the top of the plot shows the ground-state average value of the spin component, the red curve presents the average orbital angular momentum and the plot at the bottom of the plot indicates the $s$ parity.

Figure 9

(Color online) Charge and spin densities for the parameters considered in Fig. 8 at $B=2\text{T}$.

Figure 10

(Color online) (a) Black curves show the energy spectrum as calculated by the exact diagonalization for $\alpha =\beta =0$ and $g=-2.15$. (b) The black curve at the top of the plot shows the ground-state value of the spin component, the red curve presents the orbital angular momentum and the plot at the bottom of the plot indicates the s-parity.

Figure 11

(Color online) Same as Fig. 8 but for $\alpha =\beta /2=5.4\text{meV}\text{nm}$, $g=-2.15$.

Figure 12

(Color online) Same as Fig. 11 but for $\alpha =10.8\text{meV}\text{nm}$, $\beta =0$ and $g=-2.15$.

Figure 13

Two-electron spectrum for $g=0$ in the absence of the spin-orbit coupling (a), and for $\alpha =\beta =10.8\text{meV}\text{nm}$ (b).

Figure 14

(Color online) Two-electron energy spectrum for $g=-2.15$ and equal coupling constants $\alpha =\beta$. Blue (red) curves show the energy levels of odd (even) s-parity. Plots (a), (b) and (c) correspond to $\alpha =0$, $\alpha =10.8\text{meV}\text{nm}$, and $\alpha =5.4\text{meV}\text{nm}$, respectively.

Figure 15

(Color online) Two-electron charge and spin ground-state densities for $B=4\text{T}$ for $\alpha =\beta =10.8\text{meV}\text{nm}$. $g=0$ is assumed in (a) and (c), and $g=-2.15$ in (b) and (d). Electron-electron interaction is neglected in (a), (b) and accounted for in (c), (d).

Figure 16

(Color online) Parameter $\rho$ quantifying the deviation of the electron density from the circular symmetry as obtained for two electrons for $\alpha =\beta =10.8\text{meV}\text{nm}$ and $g=-2.15$ with (blue solid line) and without electron-electron interaction (blue dashed line). The black solid line corresponds to $\alpha =\beta /2=5.4\text{meV}\text{nm}$ (electron-electron interaction included).

Figure 17

(Color online) Magnetization for a single (upper plot) or two confined electrons (lower plot) without the spin-orbit coupling and for a single type of the spin-orbit coupling present. $g=-2.15$ is assumed for the Landé factor.

Figure 18

(Color online) Magnetization for a single (upper plot) or two confined electrons (lower plot) without the spin-orbit coupling and for both types of spin-orbit coupling present. $g=-2.15$ is assumed for the Landé factor.

Figure 19

(Color online) Chemical potentials for one (three lower curves) and two (three upper curves) confined electrons without the spin-orbit coupling and for a single type of spin-orbit coupling present $\left(g=-2.15\right)$. The chemical potential for two electrons is shifted down by 1 meV.

Figure 20

(Color online) Chemical potentials for one (three lower curves) and two (three upper curves) confined electrons without the spin-orbit coupling and for both types of spin-orbit coupling present $\left(g=-2.15\right)$. The chemical potential for two electrons is shifted down by 1 meV.