Low-energy effective Hamiltonian for giant-gap quantum spin Hall insulators in honeycomb $X$-hydride/halide $(X=\mathrm{N}–\mathrm{Bi})$ monolayers

Using the tight-binding method in combination with first-principles calculations, we systematically derive a low-energy effective Hilbert subspace and Hamiltonian with spin-orbit coupling for two-dimensional hydrogenated and halogenated group-V monolayers. These materials are proposed to be giant-gap quantum spin Hall insulators with record huge bulk band gaps opened by the spin-orbit coupling at the Dirac points, e.g., from 0.74 to 1.08 eV in $\mathrm{Bi}X$ ($X=\mathrm{H}$, F, Cl, and Br) monolayers. We find that the low-energy Hilbert subspace mainly consists of $p_x$ and $p_y$ orbitals from the group-V elements, and the giant first-order effective intrinsic spin-orbit coupling is from the on-site spin-orbit interaction. These features are quite distinct from those of group-IV monolayers such as graphene and silicene. There, the relevant orbital is $p_z$ and the effective intrinsic spin-orbit coupling is from the next-nearest-neighbor spin-orbit interaction processes. These systems represent the first real 2D honeycomb lattice materials in which the low-energy physics is associated with $p_x$ and $p_y$ orbitals. A spinful lattice Hamiltonian with an on-site spin-orbit coupling term is also derived, which could facilitate further investigations of these intriguing topological materials.

Figure 1

(a) The lattice geometry for 2D $X$-hydride/halide $\left(\mathrm{X}=\mathrm{N}–\mathrm{Bi}\right)$ monolayer from the side view (top) and top view (bottom). Note that two sets of sublattice in the honeycomb group V element $X$ are not coplanar (a buckled structure). The monolayer is alternatively hydrogenated or halogenated from both sides. (b) The first Brillouin zone of 2D $X$-hydride/halide monolayer and the points of high symmetry.

Figure 2

(a) and (b) The partial band-structure projection for NH and NF without SOC, respectively. The size of the symbols is proportional to the weight of the band eigenfunctions on different atomic orbitals (color coded). The Fermi level is indicated by the dotted line. (c) and (d) Band structures for BiH and BiF without (black dash lines) and with (red solid lines) SOC. The four band structures are obtained from the first-principles methods implemented in the vasp package [32] using projector augmented wave pseudopotential, and the exchange-correlation is treated by PAW-GGA. The Fermi level is indicated by the solid line.

Figure 3

A comparison of the band structures for monolayer SbH calculated using FP and TB methods with SOC. The dashed green curve is the FP result. The solid red and blue curves are the TB model results. The red curve is with the NN hopping only, while the blue curve also includes the NNN hopping terms. For the NN case, the parameters are taken as $V_{pp\sigma }=1.68\mathrm{eV},V_{pp\pi }=-0.60\mathrm{eV}$. For the NNN case, the parameters are taken as $V_{pp\sigma }=1.69\mathrm{eV},V_{pp\pi }=-0.62\mathrm{eV},V_{pp\sigma }^{\mathrm{NNN}}=0\mathrm{eV},V_{pp\pi }^{\mathrm{NNN}}=-0.23\mathrm{eV}$. For both cases, ${\lambda }_{\mathrm{so}}=0.21\mathrm{eV}$. The superscript NNN means the next-nearest-neighbor hoping. The Fermi level is set to zero.

Figure 4

The leading-order effective SOC processes in $X$-hydride or $X$-halide $\left(X=\mathrm{N}–\mathrm{Bi}\right)$, silicene and graphene. (a) and (b) Sketches of the huge effective on-site SOC in $X$-hydride systems and $X$-halide systems. (c) Sketch of the effective SOC from NNN hopping processes caused by the buckling in silicene. (d) Sketch of the second-order effective SOC from NNN hopping processes in graphene.