Electron-electron interactions in a one-dimensional quantum-wire spin filter

The combined presence of a Rashba and a Zeeman effect in a ballistic one-dimensional conductor generates a spin pseudogap and the possibility to propagate a beam with well-defined spin orientation. Without interactions, transmission through a barrier gives a relatively well-polarized beam. Using renormalization group arguments, we examine how electron-electron interactions may affect the transmission coefficient and the polarization of the outgoing beam.

Figure 1

(a) The thin lines denote the energies $E+E_{\alpha }$ of the two Rashba bands as a function of wave-vector ratio $k_x∕k_{\alpha }$ both on the left of the potential step and on the right, for $\eta =0.3$ and $\Gamma =1.5$. The potential step of height $V_1$ is indicated as a thick line. The dashed line is the Fermi energy $E_F$. (b) One of the three scattering states. An incident wave $e^{i{k}_{+}x}$ is injected on the left-hand side of the step and generates two transmitted waves and one reflected wave.

Figure 2

Solid line: location of the initial points. Upper dashed line: renormalization group flow for $\Gamma =1$. The starting point is at $t_{1,1}=1$. Lower dashed line: renormalization group trajectory for $\Gamma =10$. The dashed-dotted line is simply the first diagonal, corresponding to totally unpolarized beams and reaches $t_{1,1}^2=1$ at $t_{1,2}^2=1$.

Figure 3

Total transmission and polarization as a function of the logarithm of the temperature. The two curves marked with left arrows represent the total transmission for $\Gamma =1.01$ (solid line) and $\Gamma =10$ (dashed line). The two other curves represent the spin polarization for $\Gamma =1.01$ (dash-dotted line) and $\Gamma =10$ (dashed line), respectively.