content_image/419000.jpg content_image/419011.jpg content_image/419008.jpg content_image/419009.jpg content_image/419006.jpg content_image/419007.jpg content_image/419004.jpg content_image/419005.jpg content_image/419010.jpg
<img src='content_image/419000.jpg'>

<img src='content_image/419011.jpg'>

with 

According to this result, the oscillation period of the gap π/b becomes π √ B$_{1}$/M when A$_{1}$ = 0, in accordance with the result obtained by Liu et al $^{32}$. Fig. 2(b) shows the gap oscillation by using the model parameters listed in the second entry of Tab. I. Besides, the sine function implies that Δ may be negative. Later we will see that the sign of Δ can be found by solving E 0 + and E 0 − from Eqs. (30) and (31), respectively. 

<img src='content_image/419008.jpg'>

<img src='content_image/419009.jpg'>

## III. EFFECTIVE CONTINUOUS MODELS 

The solutions of the surface states and thin film of 3DTI can be applied to calculate physical properties ex- plicitly. For instance, we can see whether the ground state of a thin film exhibits QSHE or not by calculating the Chern number or Z$_{2}$ invariant. It is also desirable to establish an effective continuous model to explore the properties of these surface states especially when other interactions have to be taken into account. For this pur- pose, in this section we derive an effective low-energy and continuous models for the surface states and thin film of 3DTI. 

Due to the low-energy long-wavelength nature of the Dirac cone of the surface electrons, we can use the solu- tions of the surface states at the Γ point as a basis to ex- pand the Hamiltonian H ( k ) in Eq.(1), which will be valid when the energy is limited within the band gap between the conduction and valence bands. This is equivalent to a truncation approximation as we exclude the solutions for the bulk states in the basis. In this approach, the Hamiltonian in Eq. (1) can be expressed as 

where 

with 

<img src='content_image/419006.jpg'>

and 

<img src='content_image/419007.jpg'>

The first term can be solved exactly, and the last term describes the behaviors of electrons near the Γ point. 

H$_{0}$ in Eq. (23) is block-diagonal. Its solution can be found by solving each block separately, i.e., h ( A$_{1}$ )Ψ$_{↑}$ = E Ψ$_{↑}$ and h ( − A$_{1}$ )Ψ$_{↓}$ = E Ψ$_{↓}$. Because the lower block is the ”time” reversal of the upper block, the solutions satisfy Ψ$_{↓}$( z ) = ΘΨ$_{↑}$( z ), where Θ = − iσ$_{y}$ K is the time-reversal operator, with σ$_{y}$ the y component of the Pauli matrices and K the complex conjugation operation. Equivalently, we can replace A$_{1}$ by − A$_{1}$ in all the results for the upper block, to obtain those for the lower block. Therefore, we only need to solve h ( A$_{1}$ ). Following the same approach in Sec. II, we put a two-component trial solution 

<img src='content_image/419004.jpg'>

<img src='content_image/419005.jpg'>

## A. Basis states at Γ point 

<img src='content_image/419010.jpg'>

