content_image/411225.jpg content_image/411226.jpg content_image/411221.jpg content_image/411222.jpg content_image/411223.jpg content_image/411224.jpg content_image/411229.jpg content_image/411227.jpg
it in details in this paper. Here we focus on the case of the antisymmetric term, V ( z ) = V$_{a}$ ( z ), which breaks the top-bottom inversion symmetry in the Hamiltonian. A detailed analysis demonstrates that V$_{a}$ ( z ) couples Φ$_{1}$ (Φ$_{3}$) to Φ$_{2}$ (Φ$_{4}$), which can be readily seen according to their spin and parity natures. The modified effective Hamiltonian in the presence of V ( z ) becomes 

where 

Comparing this definition with that of ˜ A$_{2}$ in Eq. (42), we find that ˜ V also can be either real or purely imaginary. In the case of a purely imaginary (case II) ˜ A$_{2}$ , ˜ V must be real (see Appendix VII), and the effective Hamiltonian with SIA can be written as 

In the case of a real ˜ A$_{2}$ , ˜ V must be purely imaginary, and the effective Hamiltonian with SIA then has the form 

Without the SIA term, the effective Hamiltonian (44) gives the energy spectra of the gapped surface states as 

<img src='content_image/411225.jpg'>

where + ( − ) sign stands for the conduction (valence) band, each of which has double spin degeneracy due to time-reversal symmetry. When the SIA term is included, the Hamiltonian (51) gives 

<img src='content_image/411226.jpg'>

where the extra index 1 (2) stands for the inner (outer) branches of the conduction or valence bands. The en- ergy spectra in the presence of ˜ V is shown in Fig. 3. Each spin-degenerate dispersion in Eq. (55) shifts away from each other along k axis. Both the conduction and valence bands show Rashba-like splitting. An intuitive understanding of the energy spectra in Fig. 3 can be 

<img src='content_image/411221.jpg'>

<img src='content_image/411222.jpg'>

<img src='content_image/411223.jpg'>

<img src='content_image/411224.jpg'>

given with the help of Fig. 4. On the left is for a thicker freestanding symmetric TI film, and it has a single gap- less Dirac cone on each of its two surfaces, with the solid and dash lines for the top and bottom surface, respec- tively. The two Dirac cones are degenerate. The top of Fig. 4 indicates that the inter-surface coupling across an ultrathin film will turn the Dirac cones into gapped Dirac hyperbolas. On the bottom of Fig. 4, SIA lifts the Dirac cone at the top surface while lowers the Dirac cone at the bottom surface. The potential difference at the top and bottom surfaces removes the degeneracy of the Dirac cones. On the right of Fig. 4, the coexistence of both the inter-surface coupling and SIA leads to two gapped Dirac hyperbolas which also split in k -direction, as shown in Fig. 3. 

<img src='content_image/411229.jpg'>

Location of the surface states can be revealed by eval- uating the expectation of position z of these states. The spatial distributions along the z direction of a state ψ$_{α}$ can be evaluated by the expectation of position in z di- rection 〈 z 〉 , L 

By this definition, 〈 z 〉 α ∈ [ − L $_{2}$, L $_{2}$] and 〈 z 〉 α becomes 0 for a symmetric spatial distribution. 

## B. Location of the surface states 

<img src='content_image/411227.jpg'>

