Tuning three-dimensional higher-order topological insulators by surface state hybridization

Higher-order topological insulators (HOTIs) are a novel class of materials that exhibit exotic boundary states. The finite size effect induced hybridization between the boundary state HOTIs, however, remains largely unexplored. In this work, we analytically and numerically study the hybridization in films of three-dimensional chiral second-order topological insulators (SOTIs). We show that the gaps of two gapped surfaces (Chern insulators) are increased upon reducing the film thickness to a finite value. Meanwhile, in the presence of the hybridization of chiral edge states, we find that the gapless chiral hinge states (i.e., the boundary states of the surface Chern insulators) can be gapped, destroying the SOTIs. Moreover, we show that the gapless surfaces would in general be gapped due to the hybridization of surface states for a finite-sized system. Our work demonstrates that HOTIs exhibit a rich variety of physics under the influence of finite-size effects.

Figure 1

Probability distributions of a chiral SOTI films of different thicknesses, illustrating the crossover from a 3D crystal to a 2D thin film. (a) The 3D geometry: For sufficiently thick slabs, probability is peaked at the boundaries while it exponentially decays into the bulk, indicating surface states localized at $x=\pm L/2$. (b) The quasi-2D geometry: Reducing the thickness, the surface states begin to overlap across the bulk but nonzero real part of $\lambda$. (c) The 2D geometry: For very small thickness, the surface states of the 3D crystal hybridize and eventually turn into the bulk states of the 2D thin film marked by $\mathrm{Re}\left\{\lambda \right\}=0$.

Figure 2

The thickness $L$ dependence of the surface energy gap ${\mathrm{\Delta }}_t$. (a) The surface energy gap of a TI in the parameter regime: ${\lambda }_{1,2}=a_{\lambda }\pm i{b}_{\lambda }, v=0.8ta, B=2.0t{a}^2, m=0.2t$, and $D=0$. Here, $t$ is the hopping strength, and $a$ is the lattice constant. (c) The surface energy gap of the SOTI for the same parameters of TI case except for $D=0.2t{a}^2$. (b), (d) are the same as (a), (c), respectively, except that we set ${\lambda }_2\gg {\lambda }_1$ and $B=0.6t{a}^2$. In (c), (d), the dashed lines mark the surface gaps of infinitely sized SOTI, $\gamma =2mD/\sqrt{B^2+D^2}$, and the inset diagrams show the finite-sized gaps defined as ${\mathrm{\Delta }}_f={\mathrm{\Delta }}_t-\gamma$. For all diagrams, the blue dots are obtained by numerically solving Eq. (18), and the red lines are approximate results for the same parameters.

Figure 3

(a)–(e) The dependence of the parameters $v_1, B_1, B_2, D_1$, and $D_2$ in Eq. (22) on the film thickness $L$. (f) The energy bands of the film in the large-thickness limit. The dashed black lines in (b) and (e) are $\pm BD/\sqrt{B^2+D^2}$, respectively. For all diagrams, we set $m=0.2t, B=0.6t{a}^2, v=1.0ta$, and $D=0.2t{a}^2$.

Figure 4

The dashed lines represent the bulk and surface bands, while the colored bands represent the hinge modes. The hinge modes gap out as the length decreases. The parameters used in this picture are $v=1.0ta, B=0.5t{a}^2, m=1.0t$, and $D=0.1t{a}^2$.

Figure 5

(a) Dependence of the (001) surface gap on system length $L$. (b)–(d) The band structures for blue, red, and black points in (a), respectively. The parameters chosen are $D=0.2t{a}^2,B=2.0t{a}^2, v=0.6ta$, and $m=0.2t$.