Surface State Magnetization and Chiral Edge States on Topological Insulators

We study the interaction between a ferromagnetically ordered medium and the surface states of a topological insulator with a general surface termination that were identified recently [F. Zhang et al. Phys. Rev. B 86, 081303(R) (2012)]. This interaction is strongly crystal face dependent and can generate chiral states along edges between crystal facets even for a uniform magnetization. While magnetization parallel to quintuple layers shifts the momentum of the Dirac point, perpendicular magnetization lifts the Kramers degeneracy at any Dirac points except on the side face, where the spectrum remains gapless and the Hall conductivity switches sign. Chiral states can be found at any edge that reverses the projection of the surface normal to the stacking direction of quintuple layers. Magnetization also weakly hybridizes noncleavage surfaces.

Figure 1

The definition of crystal and local frames and the sketch of surface state conduction band spin textures. ${\widehat{k}}_z$ is perpendicular to QLs. ${\widehat{k}}_3$ is outwardly normal to the face $\Sigma (\theta ,\varphi )$, and ${\widehat{k}}_1$ (${\widehat{k}}_2$) is in-plane tangent to the longitude (latitude) circle, with ${\widehat{k}}_2={\widehat{k}}_3\times {\widehat{k}}_1$. The spin textures described in the text are shown in local frames, with ${\widehat{k}}_1$ toward the right and ${\widehat{k}}_2$ toward the top.

Figure 2

Constant energy contour plot of the lowest conduction band of the hybridized surface states on two opposite side faces. (a) ${\Delta }_x=0$, ${\Delta }_z=0.2$; (b) ${\Delta }_x=0.2$, ${\Delta }_z=0$. We use arbitrary units, with $v_{\parallel }=1$ and $v_z=0.7$.

Figure 3

Three TI shapes that support chiral edge states in the presence of a uniform exchange field ${\Delta }_z{\sigma }_z$ on the surface. (a) A spherical TI, (b) a TI slab, and (c) a TI with zigzag side faces. In all panels, the faces with $\theta <\pi /2$ ($\theta >\pi /2$) are colored in light green (cyan), and the chiral edge states are denoted by bold lines with arrows.

Figure 4

(a) A schematic TI crystal that has a top face with $0<{\theta }_t<\pi /2$, a bottom face with ${\theta }_b=\pi$, and four surrounding faces with the same ${\theta }_s$ and $\pi /2<{\theta }_s<\pi$. (b) The top view of (a). When perpendicular magnetization is present, there is a chiral state along the edges of the top face but none on the bottom edges. The four surrounding faces have $\Delta \varphi ={\varphi }_s-{\varphi }_t=0$, $\pi /2$, $\pi$, and $3\pi /2$, respectively [47].