Spin-orbit-based device for electron spin polarization

We propose quantum devices having spin-orbit coupling (but no magnetic fields or magnetic materials) that, when attached to leads, yield a high degree of transmitted electron polarization. An example of such a simple device is treated within a tight binding model composed of two one-dimensional chains coupled by several consecutive rungs (i.e., a ladder) and subject to a gate voltage. The ensuing scattering problem (with Rashba spin-orbit coupling) is solved, and a sizable polarization is predicted. When the ladder is twisted into a helix (as in DNA), the curvature energy augments the polarization. For a system with random spin-orbit coupling, the distribution of polarization is broad; hence a high degree of polarization can be obtained in a measurement of a given disorder realization. When disorder occurs in a double helix structure then, depending on scattering energy, the variance of the polarization distribution can increase even further due to helix curvature.

Figure 1

Geometry of the device and the tight-binding model. Electrons hopping on two parallel chains (lying on the $x-z$ plane) can also hop between chains along $N$ adjacent consecutive rungs. Free electrons moving in the leads (tight-binding sites marked by red points) are scattered off the sample in which SO is active (tight-binding sites marked by blue points). Two cases are considered. (1) When the sample is acted upon by a homogeneous electric field $E\widehat{\mathbf{y}}$, Rashba SO coupling generates SU(2) hopping matrices $U\left(z\right)=e^{\pm i\lambda {\sigma }_z}$ along horizontal links and $U\left(x\right)=e^{\pm i\lambda {\sigma }_x}$ along vertical links, i.e., the rungs (this case is shown in the figure, and can be naturally extended to include other SO scattering mechanisms, e.g., Dresselhaus). (2) Any arbitrary specified inhomogeneous field configuration or Rashba/Dresselhaus coupling case can also be solved. When the field configuration is too complicated for accurate description (e.g., in the case of DNA), the corresponding hopping can be encoded by random SU(2) matrices (in the same spirit as the treatment of energy levels of complex nuclei with random matrices).

Figure 2

Results for nine links. (a) Conductance as function of $\lambda$ with $k=1$. (b) Conductance as function of $k$ with $\lambda =1$. (c) $P_{T,z}$ as function of $\lambda$ with $k=1$. The sharp variation, accompanied by sign change, around $\lambda =\pi /2$ is discussed in the text. (d) $P_{T,z}$ as function of $k$ with $\lambda =1$. The results shown in (c) and (d) imply that the sign of polarization can be controlled either by tuning the Fermi energy or by tuning the strength of the SO coupling.

Figure 3

(a) Conductance $g$ and (b) polarization $P_{T,z}$ as function of number of links $N$ with $k=1.57$ and $\lambda =1$ for the non-twisted ladder (as shown in Fig. 1, with zero curvature ${\varepsilon }_c=0$) and for the twisted ladder with finite curvature ${\varepsilon }_c=-1/4$.

Figure 4

Polarization distributions $D\left(P_{T,z}\right)$ in a disordered ladder with 15 links, for fixed wave number $k=1.5$, based on ensembles of 6000 samples. A comparison is made between the cases of ${\varepsilon }_c=0$ (orange) and ${\varepsilon }_c=-1/4$ (blue); see Eq. (5) (corresponding to nontwisted and twisted ladder, respectively). The first moment vanishes for both distributions but the variances (and also the average conductance, not shown here) are very different; for ${\varepsilon }_c=0$ we find $\langle g\rangle =0.211, v\left(P_{T,z}\right)=0.0689$, while for ${\varepsilon }_c=-1/4$ we find $\langle g\rangle =0.629, v\left(P_{T,z}\right)=0.101$. Thus the effect of curvature is to augment the conductance by a factor $\simeq 3$ and the variance of the polarization distribution by a factor $\simeq 1.5$.

Figure 5

Electron hopping on the corners of a square in the $x-z$ plane subject to a homogeneous electric field $E\widehat{\mathbf{y}}$. Rashba (actually Pauli) SO coupling generates SU(2) hopping matrix elements $e^{\pm i\beta {\sigma }_z}$ along the horizontal links and $e^{\mp i\beta {\sigma }_x}$ along the vertical links.

Figure 6

$cos{\lambda }_{\mathrm{AC}}$ vs the SO coupling strength $\lambda$ for the square system in Fig. 5 [see Eq. (B3)].

Figure 7

1D ring interferometer (schematic). Electrons approaching the sample from left at polar angle $\theta =0$ are partially reflected and partially transmitted at the second lead at polar angle $\theta =\alpha$ (reflection matrix $r$ and transmission matrix $t$). The ring is subject to a homogeneous perpendicular electric field, so that the Rashba SO mechanism generates an effective magnetic field along the radial direction $\widehat{\mathbf{n}}\left(\theta \right)=(cos\theta ,sin\theta ,0)$. The local phase factor for polar angle $\theta$ is $e^{i\lambda \widehat{\mathbf{n}}\left(\theta \right)·\mathbf{\sigma }}$.