Massive Dirac fermions and spin physics in an ultrathin film of topological insulator

We study transport and optical properties of the surface states, which lie in the bulk energy gap of a thin-film topological insulator. When the film thickness is comparable with the surface-state decay length into the bulk, the tunneling between the top and bottom surfaces opens an energy gap and form two degenerate massive Dirac hyperbolas. Spin-dependent physics emerges in the surface bands, which are vastly different from the bulk behavior. These include the surface spin Hall effects, spin-dependent orbital magnetic moment, and spin-dependent optical transition selection rule, which allows optical spin injection. We show a topological quantum phase transition where the Chern number of the surface bands changes when varying the thickness of the thin film.

Figure 1

(Color online) Schematic comparison between (a) the gapped ${\text{K-K}}^{\prime }$ valleys in the staggered graphene and (b) the twofold degenerate hyperbolas described by our effective Hamiltonian in Eq. (10).

Figure 2

(Color online) [(a)–(c)] Twofold degenerate $\left({\tau }_z=\pm 1\right)$ energy spectra of surface states for thicknesses $L=20,25,32\text{Å}$ (solid lines), and $L\to \infty$ (dash lines). The gray area corresponds to the bulk states. The energy spectra are obtained by solving $H\left(k,-i{\partial }_z\right)\Psi \left(z\right)=E\Psi \left(z\right)$ under the boundary conditions $\Psi \left(z=\pm L/2\right)=0$. Please note that the scales of energy axis in (a)–(c) are different. The model parameters are adopted from Ref. 9: $M=0.28\text{eV}$, $A_1=2.2\text{eV}\text{Å}$, $A_2=4.1\text{eV}\text{Å}$, $B_1=10\text{eV}{\text{Å}}^2$, $B_2=56.6\text{eV}{\text{Å}}^2$, $C=-0.0068\text{eV}$, $D_1=1.3\text{eV}{\text{Å}}^2$, and $D_2=19.6\text{eV}{\text{Å}}^2$. [(d)–(g)] The parameters for the new effective model $H_{\text{eff}}$: $D$, $B$, the energy gap $\Delta$, and the Fermi velocity $v_F$ vs $L$.

Figure 3

(Color online) Schematic illustration of the $\mathbf{k}$-dependent spin configurations in the conduction band (top) and valence band (bottom) in Dirac hyperbola ${\tau }_z=+1$ when (a) $\Delta /B<0$ and (b) $\Delta /B>0$. The center and corners of each panel correspond to that $k=0$ and $k$ is large enough, respectively.

Figure 4

(Color online) [(a) and (b)] Energy spectra of surface states (dash lines) and orbital angular moment $m$ of the conduction bands (solid lines) as functions of $k$ for $L=20$ and $32\text{Å}$. [(c) and (d)] Berry curvature $\Omega \left(k\right)$ of the conduction band. (e) Orbital angular moment $m$ of the conduction band as a function of thickness $L$ for $k=0.01$, 0.001, and $0{\text{Å}}^{-1}$. The singularities occur at where $\Delta$ changes sign. Only results for the hyperbola ${\tau }_z=+1$ are shown. The results for ${\tau }_z=-1$ are right opposite to those for ${\tau }_z=+1$.

Figure 5

(Color online) The hyperbola-dependent Hall conductance vs Fermi level $\mu$ for (a) $L=20\text{Å}$ and (b) $32\text{Å}$, respectively.