Edge states and integer quantum Hall effect in topological insulator thin films

The integer quantum Hall effect is a topological state of quantum matter in two dimensions, and has recently been observed in three-dimensional topological insulator thin films. Here we study the Landau levels and edge states of surface Dirac fermions in topological insulators under strong magnetic field. We examine the formation of the quantum plateaux of the Hall conductance and find two different patterns, in one pattern the filling number covers all integers while only odd integers in the other. We focus on the quantum plateau closest to zero energy and demonstrate the breakdown of the quantum spin Hall effect resulting from structure inversion asymmetry. The phase diagrams of the quantum Hall states are presented as functions of magnetic field, gate voltage and chemical potential. This work establishes an intuitive picture of the edge states to understand the integer quantum Hall effect for Dirac electrons in topological insulator thin films.

SUPPLEMENTARY NOTE 3 | EDGE STATES AT OPEN BOUNDARY.

Without SIA
In the absence of SIA, i.e., V = 0, the Hamiltonian (1) is block-diagonalized and the Hamiltonian for one block is given by and the Hamiltonian h 2 (∆, B) for the other block is related to h 1 by h 2 (∆, B) = h 1 (−∆, −B). Using the trial wave function, we find two λ's for each given eigenenergy E where Each λ corresponds to two linearly independent eigenstates of the system without boundary and A general wave function is then a linear combination of four eigenstates Consider a semi-infinite geometry y ∈ [0, +∞) with open boundary conditions. Since V λ (ξ) are exponentially divergent while U λ (ξ) are vanishing as ξ approaches +∞, the wave function Ψ(ξ, E) can only contain the U λ (ξ) components as required by the normalizablity. The boundary condition Ψ(ξ 0 , E) = 0 with ξ 0 ≡ − √ 2k x B then gives an equation to determine the eigenenergies E for the semi-infinite system Solving this equation of E for each given k x , we obtain the energy dispersions of LLs with an open boundary.
In the limit ∆, B → 0, the two λ are approximated by As ξ 2 0 λ + , we can use [1] where Γ(t) is the Gamma function. Using the asymptotic expression for the Γ(t) function when t → ∞, where e is Euler's number which can be expressed as we find that On the other hand, we have Therefore, we can finally reduce Eq. (25) as We can use this equation to determine the energy dispersions of LLs in the limit ∆, B → 0, while use Eq. (25) to determine the edge dispersions for a finite B case. Supplementary Figure 1 shows the edge dispersion of the LL of n = 0 for several values of B near the edge. The LL of n = 0 suddenly changes from electron-like to hole-like when B changes from a positive infinitesimal to a negative infinitesimal.

With SIA
In the presence of SIA, we shall put the trial wave functions in terms of U λ (ξ) and V λ (ξ) functions Eq. (10) in the main text into the eigen equation, and then find four λ's for each given E. The general wave function Ψ(E, ξ) is then constructed as a linear combination of eight eigenstates.
The allowed eigenenergies E and the superposition coefficients are found by applying the open boundary condition at the boundary. For the semi-infinite geometry y ∈ [0, +∞) with open boundary conditions, the wave function Ψ(ξ, E) can only contain the U λ (ξ) components. The boundary condition Ψ(ξ 0 =, E) = 0 can be rewritten as a determinant of a 4 × 4 matrix The solutions of E for each given k x form the edge dispersions of LLs. Generally speaking, in the four λ's, λ 1,2 are real while the other two λ 3,4 are complex and λ 3 = λ * 4 . The explicit expressions for the four λ's and the four corresponding eigenstates ϕ u (λ i , ξ 0 ) (i = 1, 2, 3, 4) can be found, but are too complex. However, in the small coupling limit ∆, B → 0, we have the two real λ's as λ 1,2 = 1/2 − (V ± E) 2 /η 2 and the corresponding eigenstates The two complex λ's can be approximated by λ 3,4 ∼ η 2 /ω 2 ± iδ with |δ| |λ 3,4 | and the corresponding eigenstates Putting Eqs. (35-36) into the boundary condition Eq. (34) and using the following approximation at ω → 0 where | | |η/ω| and |δ/(2 )| ∼ |η/ω| 1, the equation of E is finally simplified as This is the equation that we can use to find the energy dispersion for the massless surface electrons near the edge. When V = 0, λ 1 = λ 2 = λ ≡ 1/2 − E 2 /η 2 and thus Eq. (38) is further reduced to We can see that one of these two equation must resemble to Eq. (32).