Spin-dependent transport through a chiral molecule in the presence of spin-orbit interaction and nonunitary effects

Recent experiments have demonstrated the efficacy of chiral helically shaped molecules in polarizing the scattered electron spin, an effect termed chiral-induced spin selectivity. Here we solve a simple tight-binding model for electron transport through a single helical molecule, with spin-orbit interactions on the bonds along the helix. Quantum interference is introduced via additional electron hopping between neighboring sites in the direction of the helix axis. When the helix is connected to two one-dimensional single-mode leads, time-reversal symmetry prevents spin polarization of the outgoing electrons. One possible way to retrieve such a polarization is to allow leakage of electrons from the helix to the environment, via additional outgoing leads. Technically, the leakage generates complex site self-energies, which break unitarity. As a result, the electron waves in the helix become evanescent, with different decay lengths for different spin polarizations, yielding a net spin polarization of the outgoing electrons, which increases with the length of the helix (as observed experimentally). A maximal polarization can be measured at a finite angle away from the helix axis.

Figure 1

Tight-binding model of a single helical molecule with radius $R$, pitch $h$, and twist angle $\mathrm{\Delta }\phi$. Electrons can hop between adjacent sites along the helix with hopping amplitude $J$ or vertically to the $N\mathrm{th}$ neighbor with hopping amplitude $\tilde{J}$. Spin-orbit interaction is assumed to act only between NN sites. (a) Schematic view of the helical molecule. (b) Mapping of the model onto a one-dimensional chain of $M$ unit cells, each containing $N$ sites.

Figure 2

Band structure of the infinite helix for $N=3$ sites in each unit cell. The site energy is ${\varepsilon }_0^{}=0$ and the values of $\tilde{J}/J$ and $\theta$ are specified in the legend of each panel. Solid (black) and dashed (red) bands correspond to spin up and down along the vector $\widehat{\mathbf{n}}$, respectively.

Figure 3

Spin polarization (solid blue) in a chain of $M=6$ unit cells (other parameters are $N=2, J=1.5J_0^{}, \tilde{J}=0.6J_0^{}$, and $J_x^{}=0.2J_0^{}$) (a) as a function of energy (in units of $J_0^{}$) with $\theta =0.4\pi$ and (b) as a function of $\theta$ with $E=0$ (center of the band). The spin polarization vanishes (dashed magenta) if either $J_x^{}=0$ (unitary chain), $\tilde{J}=0$ (NN chain), or $\theta =0$ (no SOI). Inset: Reflection (black) and transmission (green) coefficients for spin up (solid) and spin down (dashed). The red line is the sum these coefficients.

Figure 4

Decay length of a pair of spin up (solid black) and spin down (dashed red) as function of $J_x^{}/J_0^{}$ for $N=2, J=1.5J_0^{}, \tilde{J}=0.6J_0^{}, \theta =0.4\pi$, and $E=0$. The inset shows the band structure corresponding to the chosen parameters, and the pair of opposite spins propagating upward inside the chain ($v=dE/dk>0$) at energy $E=0$ is indicated by blue points.

Figure 5

Spin polarization along ${\widehat{\mathbf{n}}}^{\prime }$ as a function of the number of unit cells $M$ for various values of $\theta$, with $E=0, J=1.5J_0^{}, \tilde{J}=0.6J_0^{}$, and $J_x^{}=0.2J_0^{}$.