Ferroelectric heterobilayer with tunable first- and higher-order topological states

As conceptual milestones of nontrivial phenomenon, ${\mathbb{Z}}_2$ topological insulators (TIs) and higher-order TIs (HOTIs) have greatly reshaped the landscape of fundamental physics and materials. However, despite the exciting progress, a tunable topological phase transition between ${\mathbb{Z}}_2$ TIs and HOTIs remains elusive. Here, using a tight-binding model and first-principles calculations, we propose that ferroelectric switching can be a straightforward and efficient way for engineering the ${\mathbb{Z}}_2$ TIs and HOTIs phases with strikingly different bulk-boundary correspondence. Remarkably, based on the Wannier charge centers, edge states, and corner states analysis, we identify the ferroelectric heterobilayer composed of ${\mathrm{MgAl}}_2{\mathrm{Se}}_4$ and ${\mathrm{In}}_2{\mathrm{S}}_3$ as a material candidate of the predicted topological phase transition. Obviously, the ferroelectric switching opens up a technological avenue to bridge the first- and higher-order topologies with high possibility of innovative applications in topotronic and ferroelectric devices.

Figure 1

(a) Top and side views of the TB model. The blue and light green spheres represent flat-layer and buckled-layer atoms, respectively. (b) Lattice of the lower monolayer, where red and black circles represent atoms at different heights. $a_1$ and $a_2$ are lattice vectors. $d_1, d_2$, and $d_3$ are the vectors that connect intralayer nearest-neighbor atoms. (c) Phase diagram of the TB model as energy offset $\mathrm{\Delta }e$ is varied, where the red and blue regions represent the ${\mathbb{Z}}_2$ TI and HOTI, respectively. Band structures of TB model with energy offset (d) $\mathrm{\Delta }e=6.39$ and (g) $\mathrm{\Delta }e=7.21$. The corresponding topological edge spectrum with (e) $\mathrm{\Delta }e=6.39$ and (h) $\mathrm{\Delta }e=7.21$. (f) Evolution of the WCC with energy offset $\mathrm{\Delta }e=6.39$ along $k_2$. (i) Energy spectrum of a triangular nanoflake for the TB model with SOC, where the occupied corner states are marked by red dots. Inset shows the total charge distribution of the six corner states.

Figure 2

(a) Top view of the crystal structure for ${\mathrm{In}}_2{\mathrm{S}}_3$ monolayer, and schematic diagrams of the two polarization states. Red arrows denote the polarization directions. Side views of the ${\mathrm{MgAl}}_2{\mathrm{Se}}_4/{\mathrm{In}}_2{\mathrm{S}}_3$ heterobilayer with (b) outward polarization $P_o$ and (c) inward polarization $P_i$. The inner and exterior In/S layer are labeled as ${\mathrm{In}}_{in}/{\mathrm{S}}_{in}$ and ${\mathrm{In}}_{ex}/{\mathrm{S}}_{ex}$, respectively. Plane-averaged electrostatic potential of (d) ${\mathrm{In}}_2{\mathrm{S}}_3$ monolayer and (e) heterobilayers with the blue and red lines representing inward and outward polarized states, respectively.

Figure 3

Electronic band structures of the ferroelectric ${\mathrm{MgAl}}_2{\mathrm{Se}}_4/{\mathrm{In}}_2{\mathrm{S}}_3$ heterobilayers under outward polarization (a) without SOC and (b) with SOC, and under inward polarization (c) without SOC and (d) with SOC, respectively. Inset shows enlarged view of band around the Fermi level. The color represents the contribution of Sb layer, inner In/S layer (${\mathrm{In}}_{in}/{\mathrm{S}}_{in}$), and exterior In/S layer (${\mathrm{In}}_{ex}/{\mathrm{S}}_{ex}$).

Figure 4

Evolution of WCC for ${\mathrm{MgAl}}_2{\mathrm{Se}}_4/{\mathrm{In}}_2{\mathrm{S}}_3$ heterobilayers with (a) outward polarization and (d) inward polarization. The corresponding topological edge spectrum with (b) outward polarization and (e) inward polarization. (c) Illustration of the 2D ${\mathbb{Z}}_2$ TI and the helical edge states on the boundary. (f) Energy spectrum and charge distributions of $C_3$-symmetric triangle-shaped nanoflake for the ${\mathrm{In}}_2{\mathrm{S}}_3$/Sb heterobilayer with inward polarization, where the occupied corner states are marked by red dots.

Figure 5

The band structures of TB model with energy offset $\mathrm{\Delta }e=6.39$ for difference magnitude of onsite SOC hopping term ${\lambda }_{\mathrm{soc}}^u$, from 0.0 to 0.25.

Figure 6

[(a)–(e)] The band structure evolution of ${\mathrm{MgAl}}_2{\mathrm{Se}}_4/{\mathrm{In}}_2{\mathrm{S}}_3$ heterobilayer during the transition from $P_o$ phase to $P_i$ phase with SOC. The band crossing and reopening process is clearly shown. [(f)–(j)] The band structures of TB model with SOC for different magnitude of energy offset $\mathrm{\Delta }e$, from 6.39 to 7.21.

Figure 7

Side views of the ${\mathrm{Cu}}_2{\mathrm{Br}}_2/{\mathrm{In}}_2{\mathrm{S}}_3$ heterobilayer with (a) outward polarization $P_o$ and (c) inward polarization $P_i$. Electronic band structures of the ${\mathrm{Cu}}_2{\mathrm{Br}}_2/{\mathrm{In}}_2{\mathrm{S}}_3$ heterobilayer (b) under outward polarization with SOC, and (f) under inward polarization with SOC, respectively. The corresponding topological edge spectrum with (c) outward polarization and (g) inward polarization. (d) Evolution of WCC for ${\mathrm{Cu}}_2{\mathrm{Br}}_2/{\mathrm{In}}_2{\mathrm{S}}_3$ heterobilayer with outward polarization. (h) Energy spectrum and charge distributions of triangle-shaped nanoflake for the heterobilayer with inward polarization, where the occupied corner states are marked by red dots.

Figure 8

Side views of the $\mathrm{Sb}/{\mathrm{In}}_2{\mathrm{Se}}_3$ heterobilayer with (a) outward polarization $P_o$ and (c) inward polarization $P_i$. Electronic band structures of the $\mathrm{Sb}/{\mathrm{In}}_2{\mathrm{Se}}_3$ heterobilayer (b) under outward polarization with SOC, and (f) under inward polarization with SOC, respectively. The corresponding topological edge spectrum with (c) outward polarization and (g) inward polarization. (d) Evolution of WCC for $\mathrm{Sb}/{\mathrm{In}}_2{\mathrm{Se}}_3$ heterobilayer with outward polarization. (h) Energy spectrum and charge distributions of triangle-shaped nanoflake for the heterobilayer with inward polarization, where the occupied corner states are marked by red dots.

Figure 9

Side views of the ${\mathrm{Cu}}_2{\mathrm{I}}_2/{\mathrm{In}}_2{\mathrm{Se}}_3$ heterobilayer with (a) outward polarization $P_o$ and (c) inward polarization $P_i$. Electronic band structures of the ${\mathrm{Cu}}_2{\mathrm{I}}_2/{\mathrm{In}}_2{\mathrm{Se}}_3$ heterobilayer (b) under outward polarization with SOC, and (f) under inward polarization with SOC, respectively. The corresponding topological edge spectrum with (c) outward polarization and (g) inward polarization. (d) Evolution of WCC for ${\mathrm{Cu}}_2{\mathrm{I}}_2/{\mathrm{In}}_2{\mathrm{Se}}_3$ heterobilayer with outward polarization. (h) Energy spectrum and charge distributions of triangle-shaped nanoflake for the heterobilayer with inward polarization, where the occupied corner states are marked by red dots.