Spin filtering in all-electrical three-terminal interferometers

Unlike the two-terminal device, in which the time-reversal invariant spin-orbit interaction alone cannot polarize the spins, such a polarization can be generated when electrons from one source reservoir flow into two (or more) separate drain reservoirs. We present analytical solutions for two examples. First, we demonstrate that the electrons transmitted through a “diamond” interferometer into two drains can be simultaneously fully spin polarized along different tunable directions, even when the two arms of the interferometer are not identical. Second, we show that a single helical molecule attached to more than one drain can induce a significant spin polarization in electrons passing through it. The average polarization remains nonzero even when the electrons outgoing into separate leads are eventually mixed incoherently into one absorbing reservoir. This may explain recent experiments on spin selectivity of certain helical-chiral molecules.

Figure 1

Illustration of a three-terminal diamond interferometer. The electrons moving on the diamond edges (of length $L)$ are subjected to spin-orbit interactions.

Figure 2

Spin polarization in the first (solid) and second (dashed) output leads, along the directions ${\widehat{\mathbf{n}}}_1$ and ${\widehat{\mathbf{n}}}_2$, respectively, as a function of $\omega$ at the center of the energy band in the leads, $E=0$. For the black (thick) curves $y_b=y_c=y=2J_0,J_{0b}=J_{0c}$, and $J=\sqrt{y^2+J_0^2}=\sqrt{5}{J}_0$. Full spin polarization is obtained for $\omega =arctan(J_0/y)\approx 0.15\pi$. For the red (thin) curves $J_{0b}=0.7J_{0c}$ and only a partial spin polarization is obtained.

Figure 3

Illustration of the model of a single helical molecule, of radius $R$ and pitch $h$. In the tight-binding picture, electrons hop between adjacent sites along the helix (hopping amplitude $J)$ and along the vertical direction $\widehat{\mathbf{z}}$, to the $N\mathrm{th}$ neighbor (hopping amplitude $\tilde{J})$. Spin-orbit interaction is assumed to act only between nearest neighbors along the helix. The molecule is connected to a single lead at one edge and to $N$ leads at the sites of the last $\left(M\mathrm{th}\right)$ unit cell.

Figure 4

Spin polarization in the first (solid black) and second (dashed blue) output leads, along the directions ${\widehat{\mathbf{n}}}_1$ and ${\widehat{\mathbf{n}}}_2$, respectively, in a molecule of $M=6$ unit cells (the other parameters are $J=1.2J_0^{}$ and $\tilde{J}=0.5J_0^{})$. (a) As a function of energy (in units of $J_0^{})$ with $\tilde{\lambda }=0.1\pi$. (b) As a function of $\tilde{\lambda }$ at the center of the energy band in the leads, $E=0$.

Figure 5

Total spin polarization along the $z$ axis in a right-handed (solid) and left-handed (dashed) molecule of $M=6$ unit cells with $h/\ell =0.8$ (the other parameters are as in Fig. 4). (a) As a function of energy (in units of $J_0^{})$ with $\tilde{\lambda }=0.1\pi$. (b) As a function of $\tilde{\lambda }$ at the center of the energy band in the leads, $E=0$.