Two-Dimensional $\mathbf{\pi }$-Conjugated Covalent-Organic Frameworks as Quantum Anomalous Hall Topological Insulators

The quantum anomalous Hall (QAH) insulator is a novel topological state of matter characterized by a nonzero quantized Hall conductivity without an external magnetic field. Using first-principles calculations, we predict the QAH state in monolayers of covalent-organic frameworks based on the newly synthesized $X_3{({\mathrm{C}}_{18}{\mathrm{H}}_{12}{\mathrm{N}}_6)}_2$ structure where $X$ represents $5d$ transition metal elements Ta, Re, and Ir. The $\pi$ conjugation between $X$ $d_{xz}$ and $d_{yz}$ orbitals, mediated by N $p_z$ and C $p_z$ orbitals, gives rise to a massive Dirac spectrum in momentum space with a band gap of up to 24 meV due to strong spin-orbit coupling. We show that the QAH state can appear by chemically engineering the exchange field and the Fermi level in the monolayer structure, resulting in nonzero Chern numbers. Our results suggest a reliable pathway toward the realization of a QAH phase at temperatures between 100 K and room temperature in covalent-organic frameworks.

Figure 1

(a) Top view of a unit cell of $X_3{(\mathrm{HITP})}_2$ monolayer with lattice vectors ${\mathit{a}}_{\mathbf{1}}$ and ${\mathit{a}}_{\mathbf{2}}$ in the $xy$ plane; (b) a schematic showing that $X$ atoms form a kagome lattice with three atomic sites $A$, $B$, and $C$ in a unit cell; and (c) the first Brillouin zone of the structure, with reciprocal lattice vectors $b_{\mathbf{1}}$ and ${\mathit{b}}_{\mathbf{2}}$ and high-symmetry points $\mathrm{\Gamma }$, $M$, and $K$.

Figure 2

A schematic of the electronic band structure around the $E_f$ that explains the origin of ferromagnetism in certain $X_3{(\mathrm{HITP})}_2$ monolayers with magnetic exchange field $\mathrm{\Delta }$.

Figure 3

Schematic of the changes in crystal structure and electron filling configuration of $X_3{(\mathrm{HITP})}_2$ when (a) all N in ${\mathrm{Ta}}_3{({\mathrm{C}}_{18}{\mathrm{H}}_{12}{\mathrm{N}}_6)}_2$ are replaced by O, resulting in the ${\mathrm{Ta}}_3{({\mathrm{C}}_{18}{\mathrm{H}}_{12}{\mathrm{O}}_6)}_2$ structure, and (b) half of N atoms in ${\mathrm{Ir}}_3{({\mathrm{C}}_{18}{\mathrm{H}}_{12}{\mathrm{N}}_6)}_2$ are replaced by O, resulting in the ${\mathrm{Ir}}_3{({\mathrm{C}}_{18}{\mathrm{H}}_{12}{\mathrm{N}}_3{\mathrm{O}}_3)}_2$ structure.

Figure 4

Band structure and ${\sigma }_{xy}$ of ${\mathrm{Ta}}_3{({\mathrm{C}}_{18}{\mathrm{H}}_{12}{\mathrm{O}}_6)}_2$: (a) the band structure without considering SOC; (b) and (c) the band structure after turning on SOC for the $|d_{yz}^{1,2,3},\uparrow ⟩$ and the $|d_{xz}^{1,2,3},\uparrow ⟩$ bands, respectively; and (d) and (e) ${\sigma }_{xy}$ of ${\mathrm{Ta}}_3{({\mathrm{C}}_{18}{\mathrm{H}}_{12}{\mathrm{O}}_6)}_2$ as a function of the Fermi energy tuning corresponding to (b) and (c), respectively.

Figure 5

Band structure and ${\sigma }_{xy}$ of ${\mathrm{Ir}}_3{({\mathrm{C}}_{18}{\mathrm{H}}_{12}{\mathrm{N}}_3{\mathrm{O}}_6)}_2$: (a) the band structure without considering SOC; (b) and (c) the band structure after turning on SOC for the $|d_{yz}^{1,2,3},\downarrow ⟩$ and the $|d_{yz}^{1,2,3},\uparrow ⟩$ bands, respectively; and (d) and (e) ${\sigma }_{xy}$ of ${\mathrm{Ir}}_3{({\mathrm{C}}_{18}{\mathrm{H}}_{12}{\mathrm{N}}_3{\mathrm{O}}_6)}_2$ as a function of the Fermi energy tuning corresponding to (b) and (c), respectively.