Current-induced switching of magnetic molecules on topological insulator surfaces

Electrical currents at the surface or edge of a topological insulator are intrinsically spin polarized. We show that such surface or edge currents can be used to switch the orientation of a molecular magnet weakly coupled to the surface or edge of a topological insulator. For the edge of a two-dimensional topological insulator as well as for the surface of a three-dimensional topological insulator the application of a well-chosen surface or edge current can lead to a complete polarization of the molecule if the molecule's magnetic anisotropy axis is appropriately aligned with the current direction. For a generic orientation of the molecule a nonzero but incomplete polarization is obtained. We calculate the probability distribution of the magnetic states and the switching rates as a function of the applied current.

Figure 1

Current-induced switching of a localized spin $1/2$ weakly coupled to the helical edge of a two-dimensional topological insulator. Backscattering of a right-moving electron is accompanied by a flip to the spin-“up” state of the localized spin, leaving it in a fully polarized state after a single backscattering event. For a sufficiently large applied current the reverse scattering process, which would return the localized spin to the “down” state, is strongly suppressed.

Figure 2

Schematic drawing of energy levels and transition rates ${\mathrm{\Gamma }}_{s\pm 1,s}$ (a) for a spin-1 molecule and (b) for a spin-$3/2$ molecule.

Figure 3

Steady-state probability distribution (25) for $S=5$ and for angles $\theta =0,\theta =\pi /6,\theta =\pi /3,\theta =5\pi /12,\theta =\pi /2,\theta =7\pi /12,\theta =2\pi /3,\theta =5\pi /6$, and $\theta =\pi$.

Figure 4

Mean spin component $\langle \mathbf{S}·\mathbf{e}\rangle$ along the direction of the anisotropy axis (solid curves) and the mean spin component $\langle S_z\rangle$ in the current direction (dashed curves) as a function of the angle $\theta$ for $S=1$ (left panel) and $S=10$ (right panel).

Figure 5

Molecular magnet on the surface of a three-dimensional topological insulator. The TI surface is the $xz$ plane. A current is applied in the positive $z$ direction.

Figure 6

The probabilities $P_1,P_0$, and $P_{-1}$ for a spin-1 molecule on the surface of a three-dimensional topological insulator to be in the corresponding spin state as a function of $2\pi \hslash I/k_{\mathrm{B}}T$. We have set $2\pi \hslash D/k_{\mathrm{B}}T=5$. The inset shows the maximum value of the probability $P_1$ for a molecule with $S=1$ as a function of $2\pi \hslash D/k_{B}T$.

Figure 7

Mean spin component $\langle \mathbf{S}·\mathbf{e}\rangle$ along the direction of the anisotropy axis (solid curve) and the mean spin component $\langle S_z\rangle$ in the current direction (dashed curve) as a function of the angle $\theta$ for total spin $S=5$. Left panel: intermediate-current regime $k_{\mathrm{B}}T\ll \hslash I\ll \hslash D(2S-1)$; right panel: high-current regime $I\gg D(2S-1)$.

Figure 8

Mean spin component in the current direction $\langle S_z\rangle$ per molecule as a function of the total spin $S$ for an ensemble of randomly oriented molecular magnets on the surface of a three-dimensional topological insulator. Open circles are for the intermediate-current regime $k_{\mathrm{B}}T\ll \hslash j\ll \hslash D(2S-1)$; solid circles are for the high-current regime $j\gg D(2S-1)$.