Boundary conditions and surface state spectra in topological insulators

Abstract

We study spectra of surface states in 2D topological insulators (TIs) based on HgTe/(Hg,Cd)Te quantum wells and 3D Bi2Se3-type compounds by constructing a class of feasible time-reversal invariant boundary conditions (BCs) for an effective kp-Hamiltonian and a tight-binding model of the topological insulators. The BCs contain some phenomenological parameters which implicitly depend on both bulk Hamiltonian parameters and crystal potential behavior near the crystal surface. Space symmetry reduces the number of the boundary parameters to four real parameters in the 2D case and three in the 3D case. We found that the boundary parameters may strongly affect not only an energy spectrum but even the very existence of these states inside the bulk gap near the Brillouin zone center. Nevertheless, we reveal in frames of the tight-binding model that when surface states do not exist in the bulk gap in the Brillouin zone center they cross the gap in other points of the Brillouin zone in agreement with the bulk-boundary correspondence.

Linearity of the edge states energy spectrum in the 2D topological insulator

“…The general form of mixed boundary conditions was derived in [5]. We shall consider the case when boundary potential does not mix the spin of electrons,…”

“…We also conclude that the linearity of two edge states is conserved in this model. Besides, we will deal with the BHZ2 model [5] which is the generalization of BHZ1 model. In the BHZ2 model the mixed boundary condition is used instead of the zero boundary condition in the BHZ1 model.…”

“…It seems that the k-p method applied to the 1D Hamiltonian of edge states would yield the corrections to the spectrum of higher orders in the 1D momentum of the edge state. Hence, then we will study the spectrum linearity in other different known models: BHZ1 [2,3], BHZ2 [5] and TB [4].…”

Band bending driven evolution of the bound electron states at the interface between a three-dimensional topological insulator and a three-dimensional normal insulator

“…Following Ref. [33], one can formally write the boundary conditions in the matrix form with any model parameters. Unfortunately, it is not clear what set of parameters is more appropriate to describe the system under consideration, since the bulk wave functions of the TI and NI materials are classified according to different irreducible representations of different crystalline symmetry groups.…”

Edge absorption and pure spin current in a 2D topological insulator in the Volkov–Pankratov model

“…The edge states in TI were subject of numerous investigations [1][2][3][4][5][6][7][8][9][10][11][12][13]. Most of them rest on the so called Bernevig, Hughes and Zhang (BHZ) Hamiltonian [14] with zero boundary conditions and its generalizations.…”