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

Abstract

Linearity of the topological insulator edge state spectrum plays the crucial role for various transport phenomena. The previous studies found that this linearity exists near the spectrum crossing point, but did not determine how perfect the linearity is. The purpose of the present study is to answer this question in various edge states models. We examine Volkov and Pankratov (VP) model [1] for the Dirac Hamiltonian and the model of [2, 3] (BHZ1) for the Bernevig, Hughes and Zhang (BHZ) Hamiltonian [4] with zero boundary conditions. It is found that both models yield ideally linear edge states. In the BHZ1 model the linearity is conserved up to the spectrum ending points corresponding to the tangency of the edge spectrum with the boundary of 2D states. In contrast, the model of [5] (BHZ2) with mixed boundary conditions for BHZ Hamiltonian and the 2D tight-binding (TB) model from [4] yield weak non-linearity.

Conductivity of a two-dimensional HgTe layer near the critical width: The role of developed edge states network and random mixture of p - and n -domains

“…Often, the edge states in a 2D TI are considered based on the 6-band Bernevig-Hughes-Zhang 15 model with zero boundary conditions on the external border. Less known is the minimal Dirac-like 2-band model by Volkov-Pankratov (VP) 16 first invented for the 3D case and used for the 2D case later [19][20][21][22][23] .…”

Effects of Electron-Hole Asymmetry on Electronic Structure of Helical Edge States in HgTe/HgCdTe Quantum Wells

“…First, let us consider the case γ = 0, which corresponds to the isotropic model. Detailed study of the edge states structure within isotropic approximation is presented in [25]. In this case the Hamiltonian (1) is composed of two independent 2×2 blocks.…”

“…In this case the Hamiltonian (1) is composed of two independent 2×2 blocks. At γ = 0 the Schrödinger equation (2) has an analytical solution, valid for a wide range of k y [20,25] ε (0)…”

“…It can be seen that the spectrum of edge electrons deviates from the linear one with a velocity (11), and this deviation is well described by the dependence ε kys = ε (0) kys +ε (2) kys . Corrections to the linear dispersion of edge electrons were also studied in [25] in the framework of the isotropic model, but with boundary conditions of a more complex form. The nonlinearity of the spectrum leads, for example, to the generation of edge photocurrent in the edge channels due to indirect optical transitions [16].…”

Exact solution for many-body Hamiltonian of interacting particles with linear spectrum

“…The linear spectrum of the 2D TI edge states is the main reason of disappearance of the electron correlation energy 13 . In 11 we have studied the linearity of the energy spectrum in the edge states of the 2D topological insulator. We concluded that in two models of the edge states 12 and 1 -3 the linearity is absolute, while in other cases the non-linear corrections are extremely weak.…”