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. 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 7 where H SIA eff = H$_{eff}$ $_{+}$   0 0 ˜ V 0 0 0 0 ˜ V ∗ ˜ V ∗ 0 0 0 0 ˜ V 0 0   $^{,}$ (51) ˜ V = ∫ L/ 2 − L/ 2 dz 〈 ϕ ( A$_{1}$ ) | V$_{a}$ ( z ) | χ ( A$_{1}$ ) 〉 . (52) H SIA eff = [ h$_{+}$ ( k ) ˜ V σ$_{0}$ ˜ V σ$_{0}$ h$_{−}$ ( k ) ] . (53) H SIA eff = [ h$_{+}$ ( k ) ˜ V σ$_{z}$ − ˜ V σ$_{z}$ h$_{−}$ ( k ) ] . (54) E$_{±}$ = E$_{0}$ − Dk 2 ± √ ( Δ 2 − Bk $^{2}$)2 + ( ℏ v$_{F}$ ) $^{2}$k$^{2}$, (55) E$_{1}$$_{,}$$_{±}$ = E$_{0}$ − Dk 2 ± √ ( Δ 2 − Bk $^{2}$)2 + ( | ˜ V | + ℏ v$_{F}$k ) $^{2}$, E$_{2}$$_{,}$$_{±}$ = E$_{0}$ − Dk 2 ± ( Δ 2 − Bk $^{2}$)2 + ( | ˜ V | − ℏ v$_{F}$k ) $^{2}$, (56) 〈 z 〉 α = ∫ L 2 − L 2 z | ψ$_{α}$ | $^{2}$dz. (57) B. Location of the surface states −0.1 −0.05 0 0.05 0.1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 k$_{x}$ ( ˚ A − $^{1}$) E (eV) 4 QL Top surface Bottom surface 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 FIG. 3: (Color online) Energy spectra of surface states (four branches in the middle) and several branches of bulk states (those at the top and the bottom) for a film with the thickness of 4 QL in the presence of the structure inversion asymmetry. The color of lines corresponds to the spatial distribution of the wavefunctions in z direction. Dark blue (light green) rep- resents that the wavefunctions mainly distribute on the side of the top (substrate) surface. The model parameters are listed in the first row of Table I. 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 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 〉 , By this definition, 〈 z 〉 α ∈ [ − L $_{2}$, L $_{2}$] and 〈 z 〉 α becomes 0 for a symmetric spatial distribution. 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