First-principles design of a half-filled flat band of the kagome lattice in two-dimensional metal-organic frameworks

We design from first principles a type of two-dimensional metal-organic framework (MOF) using phenalenyl-based ligands to exhibit a half-filled flat band of the kagome lattice, which is one of a family of lattices that show Lieb-Mielke-Tasaki's flat-band ferromagnetism. Among various MOFs, we find that trans-Au-THTAP (THTAP=trihydroxytriaminophenalenyl) has such an ideal band structure, where the Fermi energy is adjusted right at the flat band due to unpaired electrons of radical phenalenyl. The spin-orbit coupling opens a band gap giving a nonzero Chern number to the nearly flat band, as confirmed by the presence of the edge states in first-principles calculations and by fitting to the tight-binding model. This is a novel and realistic example of a system in which a nearly flat band is both ferromagnetic and topologically nontrivial.

Figure 1

(a) Molecular structure of the phenalenyl-based ligand ($Z={\mathrm{C}}^{•},\mathrm{N}$; $X^{\prime },Y^{\prime }=\mathrm{OH},\mathrm{SH},{\mathrm{NH}}_2$). (b) Structure of the unit cell (solid line) of the proposed MOFs ($M=\mathrm{Ni},\mathrm{Cu},\mathrm{Pt},\mathrm{Au}$; $X,Y=\mathrm{O},\mathrm{S},\mathrm{NH}$). (c) Band structure of the single-orbital tight-binding model for the kagome lattice with a nearest-neighbor hopping parameter $t.$ The brown dashed line shows the Fermi energy when there is one electron per each orbital, while the arrow indicates that the flat band becomes half-filled when two unpaired electrons are transferred from radical phenalenyl. (d) Structure of trans-Au-THTAP, where $M=\text{Au},X=\text{NH},Y=\text{O},Z={\text{C}}^{•}$, and L stands for a neighboring ligand. (e) Top view of trans-Au-THTAP. The solid line indicates a unit cell.

Figure 2

(a) Band structure of trans-Au-THTAP calculated without SOC. The black solid (blue dashed) lines show the spin-up (spin-down) bands. (b) PDOS of the spin-down bands for each element in trans-Au-THTAP calculated without SOC. (c) Zoom-in of (a) around the Fermi energy. (d) Zoom-in of the band structure calculated with SOC. (e) Comparison between the band structures calculated with SOC by DFT (solid lines) and by TB (circles) with the calculated Chern numbers (Ch). In each panel, the Fermi energy is taken to be zero.

Figure 3

Band structure for a ribbon of trans-Au-THTAP. The black solid lines represent the bulk states, and the red (blue) dashed lines represent topological edge states along the bottom (top) edge.