Effective Model for Massless Dirac Electrons on a Surface of Weak Topological Insulators

Abstract

In a typical situation, gapless surface states of a three-dimensional (3D) weak topological insulator (WTI) appear only on the sides, leaving the top and bottom surfaces gapped. To describe massless Dirac electrons emergent on such side surfaces of a WTI, a two-dimensional (2D) model consisting of a series of onedimensional helical channels is usually employed. However, an explicit derivation of such a model from a 3D bulk Hamiltonian has been lacking. Here, we explicitly derive an effective 2D model for the WTI surface states starting from the Wilson-Dirac Hamiltonian for the bulk WTI and establish a firm basis for the hitherto hypothesized 2D model. We show that the resulting 2D model accurately reproduces the excitation spectrum of surface Dirac electrons determined by the 3D model. We also show that the 2D model is applicable to a side surface with atomic steps.

Engineering Dirac electrons emergent on the surface of a topological insulator

“…Similarly, for N z even, the vanishing of equation ( 16 ) at implies where and is an arbitrary odd integer. Since the spectrum of the surface state is given as [ 31 ] equations ( 17 ) and ( 18 ) signify that the surface spectrum is gapless when N z is odd, while it is gapped when N z is even [ 30 , 31 ]. Also, replacing N z with N h , one can equally apply equations ( 17 ) and ( 18 ) for characterizing the 1D helical modes that appear along a step formed on the surface of a WTI [ 18 ] (cf figure 3 and discussion given in the next section).…”

“…( 17), (18) signify that the surface spectrum is gapless for N z odd, while it is gapped when N z is even. 30,31 Also, note that replacing N z with N h , one can equally apply Eqs. ( 17) and ( 18) for characterizing the 1D helical modes that appear along a step formed on the surface of a WTI 18 ; c.f.…”

Finite-Size Scaling Analysis of the Conductivity of Dirac Electrons on a Surface of Disordered Topological Insulators

“…Electron transport in the presence of disorder has also been examined in the WTI case. [38][39][40][41] Dirac electrons on a side surface of WTIs are described by a 2D chain model [40][41][42][43] consisting of one-dimensional helical channels, each of which is coupled with its nearest neighbors. The number of helical channels plays an important role; if it is odd, the system has gapless excitations, while it acquires a finite-size gap if it is even.…”

Weak Topological Insulators with Step Edges: Subband Engineering and Its Effect on Electron Transport

“…Although the case with no step edge is implicitly assumed above, H 2D can be used even in the presence of straight step edges if we appropriately reduce B that connects two neighboring helical modes across each step edge. 17 Let η(w) be the factor of reduction for a step edge of width w. This is determined by the overlap of the basis functions for neighboring helical modes across a step edge as 33…”

“…where ↑, ↓ and 1, 2 respectively represent the spin and orbital degrees of freedom. The tight-binding Hamiltonian for WTIs is expressed as H 3D = H x + H y + H z with11,17 …”