Nonadiabatic generation of a pure spin current in a one-dimensional quantum ring with spin-orbit interaction

We demonstrate the theoretical possibility of obtaining a pure spin current in a 1D ring with spin-orbit interaction by irradiation with a nonadiabatic, two-component terahertz laser pulse, whose spatial asymmetry is reflected by an internal phase difference $\varphi$. The solutions of the equation of motion for the density operator are obtained for a spin-orbit coupling linear in the electron momentum (Rashba) and they are used to calculate the time-dependent charge and spin currents. We find that there are critical values of $\varphi$ at which the charge current disappears, while the spin current reaches a maximum or a minimum value.

Figure 1

The eigenvalues $E_{l+}$ and $E_{l-}$ of the 1D quantum ring with $N=20$ points vs quantum number $l$. The spectrum is discrete and the eigenvalues are represented by the points. The lines connecting the points are only for guiding the eye. In panel (a), in the absence of the SOI ($V_{\alpha }=0$), the two branches coincide, i.e., $E_{l+}=E_{l-}={\epsilon }_l$. In panel (b), in the presence of the SOI ($V_{\alpha }=0.86$), the two branches split, but $E_{l+}=E_{-l-}$ such that all states are double degenerate. For example, the two horizontal lines show that $E_{3-}=E_{-3+}=1.839$ and $E_{4+}=E_{-4-}=-0.035$. The energy unit is $V=\hslash {}^2/2m^{*}{a}^2$.

Figure 2

Spectrum of the 1D quantum ring with $N=20$ points vs Rashba energy $V_{\alpha }$. The eigenvalues $E_{3\pm }$, $E_{-3\pm }$ are marked with thick lines to illustrate the partial lifting of degeneracy when $V_{\alpha }\ne 0$. The energy unit is $V=\hslash {}^2/2m^{*}{a}^2$.

Figure 3

(a) The time evolution of the azimuthal velocity $v(t)=\langle v_{\theta }\rangle =I^c(t)/e$, (b) the spin current $I^s(t)=\langle I_{2{\theta }_{\alpha }}^s\rangle$ for the selected phase difference $\varphi =0,\pi /2,\pi$. The charge current is zero for both $\varphi =0$ and $\varphi =\pi$, and thus in these cases a pure spin current is created. The inset shows the time dependence of the two components of the radiation pulse described by Eq. (9) with $n=1$ for $\theta =\varphi =0$. The lower peak corresponds to the first term and the higher peak to the second term. The time unit is $\hslash /V=0.0076$ ps.

Figure 4

The stationary values of the velocity $v_{\theta }$ and spin current $I_{2\theta }^s$ after the external pulse vanishes (at $t\gg t_f$) vs the phase difference $\varphi$. The external pulse is $H_1(t)$, Eq. (9) with $n=1$, i.e., a combination of two dipolar fields. The SOI parameter is $V_{\alpha }=0.05$. Pure spin current, corresponding to zero charge current, is obtained when the phase difference $\varphi$ is a multiple of $\pi$.

Figure 5

The stationary values of the velocity $v_{\theta }$ and spin current $I_{2\theta }^s$ after the external pulse vanishes (at $t\gg t_f$) vs the phase difference $\varphi$. The external pulse is $H_2(t)$, Eq. (9) with $n=2$, i.e., a combination of a dipolar and a quadrupolar field. The SOI parameter is $V_{\alpha }=0.05$. Pure spin current is now obtained when the phase difference $\varphi$ is a multiple of $\pi /2$. The results are shown for $0⩽\varphi ⩽\pi$ and they are identical for $\pi ⩽\varphi ⩽2\pi$.