Coherent spin ratchets: A spin-orbit based quantum ratchet mechanism for spin-polarized currents in ballistic conductors

We demonstrate that the combined effect of a spatially periodic potential, lateral confinement, and spin-orbit interaction gives rise to a quantum ratchet mechanism for spin-polarized currents in two-dimensional coherent conductors. Upon adiabatic ac driving, in the absence of a net static bias, the system generates a directed spin current while the total charge current is zero. We analyze the underlying mechanism by employing symmetry properties of the scattering matrix and numerically verify the effect for different setups of ballistic conductors. The spin current direction can be changed upon tuning the Fermi energy or the strength of the Rashba spin-orbit coupling.

Figure 1

(Color online) Spin-dependent transmissions as a function of the injection energy $E={\left(kL\right)}^2$ in the presence of Rashba spin-orbit interaction $(k_{\mathrm{SO}}L=1.5)$ for a short periodic chain of five symmetrical potential barriers (see inset, barrier height $U_0=22$) and moderate rocking amplitude $V_0=2$. The dashed (red/gray) and dotted (blue/dark gray) lines indicate $T_S$, Eq. (7), in the two rocking situations. The solid (black) line depicts the ratchet spin transmission, Eq. (5), with the sign indicating the flow direction. For reference, the dashed-dotted (green/light gray) curve shows $T_C$, Eq. (6), and the staircase function $T_C$ for a wire without potential barriers and SO interaction.

Figure 2

(Color online) (a) Ratchet spin transmission as a function of the number of barriers $N$ for $k_{\mathrm{SO}}L=1.5$, $U_0=22$, $V_0=2$, and energies $E={\left(kL\right)}^2=24$ (black symbols, lower line), 33 (red/gray, middle line), and 35.5 (green/light gray, upper line). (b) Ratchet spin conductance $⟨I_S⟩(e∕V_0)$ at zero temperature in units of $e{G}_S$ as a function of applied voltage $V_0$ for $N=20$, $k_{\mathrm{SO}}L=1.5$, $U_0=22$, and $E=24$ (black solid line), 33 (red/gray dashed line), and 35.5 (green/light gray dash-dotted line).

Figure 3

(Color online) Ratchet spin transmission as a function of energy $E={\left(kL\right)}^2$ and SO interaction $k_{\mathrm{SO}}L$ for $N=20$, $V_0=2$, and $U_0=22$. The dashed lines are guide to the eye for the shift of the first Bloch band.

Figure 4

(Color online) Illustration of the spin polarization mechanism for transmission through a strip with a single adiabatic symmetric potential barrier $U\left(x\right)$ (solid line) in the two rocking situations (dashed and dotted line). At points A, B, and C, the position-dependent energy dispersion relation $E_n\left(k_x\right)$ is sketched with respect to the Fermi energy $E_F$ (horizontal line) for two transverse modes and SO-induced spin splitting of each mode.