Prediction of coexistence of anomalous valley Hall and quantum anomalous Hall effects in breathing kagome-honeycomb lattices

Abstract

To explore the behaviors of valley degrees of freedom, we built a breathing kagome-honeycomb (BKH) lattice composed of a breathing kagome sublattice (BKL) and a breathing honeycomb sublattice (BHL). Many metal-organic frameworks (MOFs) without space-inversion symmetry (SIS) can be modeled as BKH lattices. The electronic states and the evolution of Dirac cones of the constructed BKH lattice under magnetic exchange and spin-orbit coupling interactions were investigated based on a tight-binding model and first-principles calculations. Coexistence states of anomalous valley Hall (AVH) and quantum anomalous Hall (QAH) effects, namely, the multiple Hall effects, were achieved in the BKH lattice with ${p}_{z}$ orbitals located at the BKL and ${p}_{x}$, ${p}_{y}$ (and also ${d}_{xy}$, ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$) orbitals located at the BHL. The properties of the coexisting state, including the valley polarized status (at $K$ or ${K}^{\ensuremath{'}}$) and the chirality of the QAH effect, were determined by the overlapping styles of the bands from the two sublattices. An MOF of ${\mathrm{C}}_{6}{\mathrm{N}}_{3}{\mathrm{H}}_{3}\mathrm{Au}$ with a BKH lattice is proposed. Given the coexistence of the AVH and QAH effects in this built MOF, the metallic edge state protected by the topology of the MOF is 100% valley polarized, different from the ordinary QAH effect. Our results provide insights into the electronic states of the MOFs without SIS and a material platform with potential applications in not only valleytronics but also electronics and spintronics.