Pseudo-magnetic-field and effective spin-orbit interaction for a spin-1/2 particle confined to a curved surface

By considering the spin connection, we deduce the effective equation for a spin-1/2 particle confined to a curved surface with the nonrelativistic limit and in the thin-layer quantization formalism. We obtain a pseudo-magnetic-field and an effective spin-orbit interaction generated by the spin connection. Geometrically, the pseudo-magnetic field is proportional to the Gaussian curvature and the effective spin-orbit interaction is determined by the Weingarten curvature tensor. In particular, we find that the pseudo-magnetic-field and the effective spin-orbit interaction can be employed to separate the electrons with different spin orientations. All these results are demonstrated in two examples: a straight cylindrical surface and a bent one.

Figure 1

(a) Illustration of a cylinder. (b) Energy dispersion diagram without considering the spin connection. (c) Energy dispersion diagram with the consideration of the spin connection. (d) Conductance of a cylinder with (solid line) and without (dash line) the consideration of the spin connection at zero temperature.

Figure 2

(a) A bent cylindrical surface. The Gaussian curvature $K>0$ at the outer points, and $K<0$ at inner points. (b) The outer part of the bent cylindrical surface, $\theta \in [-{\theta }_0,{\theta }_0]$, where ${\theta }_0$ is small. The yellow arrow denotes the incident charge current, and the red and blue arrows denote the currents with spin orientation pointing inwards and outwards, respectively. (c) The inner part of the bent cylindrical surface, $\theta \in [\pi -{\theta }_0,\pi +{\theta }_0]$.