Gate-tunable imbalanced Kane-Mele model in encapsulated bilayer jacutingaite

We study free, capped, and encapsulated bilayer jacutingaite $\left({\mathrm{Pt}}_2{\mathrm{HgSe}}_3\right)$ from first principles. While the freestanding bilayer is a large-gap trivial insulator, we find that the encapsulated structure has a small trivial gap due to the competition between sublattice symmetry breaking and sublattice-dependent next-nearest-neighbor hopping. Upon the application of a small perpendicular electric field, the encapsulated bilayer undergoes a topological transition towards a quantum spin Hall insulator. We find that this topological transition can be qualitatively understood by modeling the two layers as uncoupled and can be described by an imbalanced Kane-Mele model that takes into account the sublattice imbalance and the corresponding inversion-symmetry breaking in each layer. Within this picture, bilayer jacutingaite undergoes a transition from a $0+0$ state, where each layer is trivial, to a $0+1$ state, where an unusual topological state relying on Rashba-like spin orbit coupling emerges in only one of the layers.

Figure 1

Band structure and lateral view of the crystal structure for (a) free, (b) h-BN capped, and (c) h-BN encapsulated bilayer jacutingaite. In the top panels, the full band structure along the path $\mathit{\Gamma }\text{−}\mathbf{M}\text{−}\mathbf{K}\text{−}\mathit{\Gamma }$ is shown with dots, while lines represent the Wannierized band structure of the eight bands closest to the Fermi level. The red rectangle in (c) highlights the region around the gap magnified in Fig. 2. In the bottom panels, we note that in each ${\mathrm{Pt}}_2{\mathrm{HgSe}}_3$ layer Hg atoms form a buckled honeycomb lattice [see also Fig. 3 for a top view in the case of (c)], with Hg atoms in the two sublattices alternating above and below a ${\mathrm{Pt}}_2{\mathrm{Se}}_3$ layer. Dashed lines highlight the vertical position of Hg planes, showing the inequivalence of inner and outer planes, the latter tending to extend farther away from the ${\mathrm{Pt}}_2{\mathrm{Se}}_3$ layer. This tendency, most apparent in the free bilayer, is suppressed by the presence of h-BN, with the encapsulated bilayer recovering a more symmetric structure of each layer.

Figure 2

Top: Band structure of h-BN encapsulated bilayer jacutingaite around the K point in a small energy window close to the Fermi energy for three different values of the applied external electric field. Note that at zero field all bands are doubly degenerate due to inversion symmetry, whereas at nonzero fields this degeneracy is lifted. Bottom: The direct gap at the $\mathbf{K}$ point as a function of perpendicular external electric field. Around $E_{\mathrm{ext}}=0.3\mathrm{V}/\AA$ the gap closes. For larger fields, a band inversion occurs, and the system is a quantum spin Hall insulator with $\nu =1$. The indirect band gap is shown in Appendix pp4, while the direct gap computed with different functionals is reported in Appendix pp5.

Figure 3

Top and lateral views of the crystal structure of h-BN encapsulated bilayer jacutingaite. In the top view, the green shaded area highlights the hexagonal Wigner-Seitz unit cell and the (buckled) honeycomb lattice formed by Hg atoms. Two Wannier functions associated with the two sublattices of the upper layer are also reported as isosurfaces for both positive (blue) and negative (red) values. The Wannier functions associated with the bottom layer can be simply obtained by inversion symmetry.

Figure 4

Bilayer jacutingaite can be qualitatively understood as two decoupled layers. (a) To verify this, we calculated the gap at $\mathbf{K}$ as a function of external field for the individual layers using the tight-binding model with interlayer couplings set to zero. Though the gap is quantitatively underestimated (compare with Fig. 2), we still find a topological transition in the top layer. (b) The dispersion close to $\mathbf{K}$ changes only subtly when we have interlayer coupling (black solid lines) or not (red and blue dashed lines) at zero field. (c) At a finite external field of $E_{\mathrm{ext}}=0.46\mathrm{V}/\AA$, the dispersion for just the top layer (red) and bottom layer (blue) still has a large overlap with the full bilayer band structure. (d) The WCCs (in units of the lattice parameter $a$) computed for the full tight-binding model (black) are the same as the WCCs computed per layer (red and blue). (e) Same calculation as in (d), but now at a finite field $E_{\mathrm{ext}}=0.46\mathrm{V}/\AA$. We clearly see the topological nature of the bands in the top layer (red).

Figure 5

Terms in the imbalanced Kane-Mele model. We include nearest-neighbor hopping $t$ and next-nearest-neighbor hopping $t_2$. Sublattice symmetry breaking is included by the term $m{\sigma }^z$. The spin-orbit terms include the regular Kane-Mele term ${\lambda }_{\mathrm{KM}}{\sigma }^z{s}^z$ and a sublattice-symmetric Kane-Mele term ${\lambda }_{\mathrm{KM}}^{\prime }{s}^z$. The arrow directions indicate the sign of the imaginary hopping. Finally, we include a nearest-neighbor Rashba term ${\lambda }_R$ that stabilizes an unusual topological insulator phase when ${\lambda }_{\mathrm{KM}}^{\prime }>{\lambda }_{\mathrm{KM}}$.

Figure 6

(a) Lateral view of the crystal structure of free trilayer jacutingaite. The central layer is symmetric, while in the other layers the outermost Hg atoms are farther away from the central plane of Pt atoms with respect to the inner ones. (b) Electronic band structure of trilayer jacutingaite, where symbols represent direct calculations, while lines are the result of a minimal tight-binding model based on Wannier functions. The zero of energy is set at the Fermi level.

Figure 7

Evolution of Wannier charge centers (WCCs) for (a) free and (b) h-BN capped bilayer ${\mathrm{Pt}}_2{\mathrm{HgSe}}_3$. Both systems are topologically trivial. Note that the WCCs (in units of the lattice parameter $a$) computed for the full tight-binding model (black) are the same as the WCCs computed per layer (red and blue).

Figure 8

The indirect band gap (measured as the minimum of the conduction band minus the maximum of the valence band) of the encapsulated bilayer as a function of external electric field. In the topological phase the gap is smaller than the direct gap at K in Fig. 2 but still positive, suggesting the presence of a fully developed band gap.

Figure 9

Direct band gap at the K point for h-BN encapsulated bilayer jacutingaite as a function of the external electric field computed using either the PBE [56] or the HSE [57] functional. Dots represent actual calculation results, while lines are linear extrapolations.