Spin texture in weak topological insulators: the role of bulk states and band bending

TL;DR

This paper derives the spin textures in weak topological insulators considering bulk states and band bending, revealing four possible textures and emphasizing the importance of band bending in determining spin polarization.

Contribution

It introduces a supersymmetric approach to analyze spin textures in weak topological insulators, incorporating bulk states and band bending effects for the first time.

Findings

Four distinct spin textures identified: helical, hyperbolic, hedgehog, hyperbolic hedgehog.

Band bending influences the winding number and spin polarization.

In SnTe, band bending is essential for correct spin texture characterization.

Abstract

We derive the spin texture of a weak topological insulator via a supersymmetric approach that includes the roles of the bulk gap edge states and surface band bending. We find the spin texture can take one of four forms: (i) helical, (ii) hyperbolic, (iii) hedgehog, with spins normal to the Dirac-Weyl cone of the surface state, and (iv) hyperbolic hedgehog. Band bending determines the winding number in the case of a helical texture, and for all textures can be used to tune the spin texture polarization to zero. For the weak topological insulator SnTe, we show that inclusion of band bending is crucial to obtain the correct texture winding number for the (111) surface facet $\Gamma$-point Dirac-Weyl cone. We argue that hedgehogs will be found only in low symmetry situations.