Decay length of edge/surface states in topological insulators

Edge states of 2D quantum spin Hall (QSH) systems or surface states of 3D QSH systems are localized near the boundary of the systems, whereas their decay length $\ell$ may vary. We study their generic behaviors in the presentation. We first note that $\ell^{-1}$ plays the role of the imaginary part of the wavenumber perpendicular to the edge/surface, and $\ell=\infty$ when the edge/surface states are absorbed into the bulk bands. We calculate $\ell$ and discuss their behaviors by using effective models. In accordance to our expectations, we can show that the $\ell$ is shorter when the edge/surface states are far away from the points where the edge/surface states are absorbed into the bulk bands. The edge states in HgTe quantum well or the surface states in Bi$_2$Se$_3 $ have Dirac-like dispersion and longer $\ell$, while in Bi ultrathin films or Bi$_{1-x}$Sb$_x$, $\ell$ is as short as the lattice constant because the edge/surface states spread almost over the Brilloin zone. In particular, in the Dirac-like bands, the minimum $\ell$ corresponds to the inverse of the $k$-space size of the Dirac cone of the edge/surface states. The QSH systems with shorter $\ell$ will be more favorable for real-space observation of edge/surface states.