Screening two-dimensional materials with topological flat bands

The topological flat band (TFB) has been proposed theoretically in various lattice models, to exhibit a rich spectrum of intriguing physical behaviors. However, the experimental demonstration of flat band (FB) properties has been severely hindered by the lack of materials realization. Here, by screening materials from a first-principles materials database, we identify a group of two-dimensional materials with TFBs near the Fermi level, covering some simple line-graph and generalized line-graph FB lattice models. These include the kagome sublattice of O in $\mathrm{Ti}{\mathrm{O}}_2$ yielding a spin-unpolarized TFB, and that of V in ferromagnetic ${\mathrm{V}}_3{\mathrm{F}}_8$ yielding a spin-polarized TFB. The monolayer ${\mathrm{Nb}}_3\mathrm{Te}{\mathrm{Cl}}_7$ and its counterparts from element substitution are found to be breathing-kagome-lattice crystals. The family of monolayer $\mathrm{II}{\mathrm{I}}_2\mathrm{V}{\mathrm{I}}_3$ compounds exhibit a TFB representing the coloring-triangle lattice model. $\mathrm{Re}{\mathrm{F}}_3$, $\mathrm{Mn}{\mathrm{F}}_3$, and $\mathrm{Mn}{\mathrm{Br}}_3$ are all predicted to be diatomic-kagome-lattice crystals, with TFB transitions induced by atomic substitution. Finally, $\mathrm{Hg}{\mathrm{F}}_2$, $\mathrm{Cd}{\mathrm{F}}_2$, and $\mathrm{Zn}{\mathrm{F}}_2$ are discovered to host dual TFBs in the diamond-octagon lattice. Our findings pave the way to further experimental exploration of eluding FB materials and properties.

Figure 1

Flowchart of computational screening for 2D crystals with ideal TFBs.

Figure 2

Lattice structure and tight-binding bands of (a) kagome lattice, and its derivatives including (b) breathing-kagome, (c) coloring-triangle, (d) diatomic-kagome lattices, and (e) diamond-octagon lattice. $t_{\mathrm{NN}}$ and $t_{\mathrm{NNN}}$ represent the NN and NNN hopping integral, respectively. Gray and blue dots/lines in upper panels show the original lattices and their (generalized) line-graph lattices, respectively. For standard construction, (a) and (e), a line graph takes edge centers of the original graph as its vertices, while for generalized construction, a line graph takes off-edge-center positions, along the edge (b) or off the edge (c), as its vertices. Panel (d) can be viewed as two copies of the generalized line graph in (b), leading to dual FBs.

Figure 3

Ideal kagome bands in monolayer kagome atomic crystals. (a) Oxygen atoms in monolayer $\mathrm{Ti}{\mathrm{O}}_2$ form upper and lower kagome layers connected by middle sites, with titanium atoms locating at the center of oxygen tetrahedrons. The electronic structure exhibits obvious kagome bands (bold lines in left panel), and the charge of TFB near the Fermi level is contributed by $p$ orbitals of oxygen atoms (right panel). (b) Vanadium atoms in monolayer ${\mathrm{V}}_3{\mathrm{F}}_8$ form the perfect kagome lattice, resulting in the typical kagome bands (bold lines in left panel) with a TFB exactly locating at Fermi level. The $d$ orbitals of vanadium atoms contribute to the electron charge of the spin-up kagome bands (right panel).

Figure 4

Left: The electronic structure of monolayer ${\mathrm{Nb}}_3\mathrm{Te}{\mathrm{Cl}}_7$ with niobium atoms sitting on a breathing-kagome lattice. Right: Electron charge distribution contributing to the breathing-kagome bands with a gapped Dirac cone in the left.

Figure 5

(a) Electronic structure and orbital composition of monolayer ${\mathrm{B}}_2{\mathrm{S}}_3$ with +6% tensile strain (left panel). Δ marks the Γ-point energy interval between $p_{\mathrm{z}}$ and $p_{\mathrm{x},\mathrm{y}}$ bands just below the Fermi level. The $p_{\mathrm{z}}$ orbitals of sulphur atoms, occupying the sites of a coloring-triangle lattice (right panel), contribute to the TFB of kagome bands near the Fermi level. (b) The variation of Δ with external tensile strain.

Figure 6

Atomic and electronic structure for diatomic-kagome crystals of (a) $\mathrm{Re}{\mathrm{F}}_3$, (b) $\mathrm{Mn}{\mathrm{F}}_3$, and (c) $\mathrm{Mn}{\mathrm{Br}}_3$. Left: Band structure with the diatomic-kagome bands highlighted by thick lines. Right: The distribution of electron charge density for the diatomic-kagome bands highlighted in the left.

Figure 7

Dual TFBs in monolayer $\mathrm{Hg}{\mathrm{F}}_2$ with a diamond-octagon lattice. Left: Electron band structure. Right: Charge density plot showing the bands in the left are mainly contributed from $s$ orbitals of mercury atoms.

Figure 8

Honeycomb TFB crystal of monolayer $\mathrm{V}{\mathrm{F}}_3$. Left: Band structure. Right: Electron charge density distribution contributing to the highlighted bands in the left.