Exotic Topological Bands and Quantum States in Metal–Organic and Covalent–Organic Frameworks

Conspectus Metal–organic and covalent–organic frameworks (MOFs/COFs) have been extensively studied for fundamental interests and their promising applications, taking advantage of their unique structural properties, i.e., high porosity and large surface-to-volume ratio. However, their electronic and magnetic properties have been somewhat overlooked because of their relatively poor performance as conductive and/or magnetic materials. Recent experimental breakthroughs in synthesizing two-dimensional (2D) π-conjugated MOFs/COFs with high conductivity and robust magnetism through doping have generated renewed and increasing interest in their electronic properties. Meanwhile, comprehensive theoretical studies of the underlying physical principles have led to discovery of many exotic quantum states, such as topological insulating states, which were only observed in inorganic systems. Especially, the diversity and high tunability of MOFs/COFs have provided a playground to explore novel quantum physics and quantum chemistry as well as promising applications. The band theory has empowered us to understand the most fundamental electronic properties of inorganic crystalline materials, which can also be used to better understand MOFs/COFs. The first obvious difference between the two is that instead of atomic orbitals residing at lattice sites of inorganic crystals, molecular orbitals of organic ligands are predominant in MOFs/COFs. The second key difference is that usually all atomic orbitals in an inorganic crystal are subject to one common group of lattice symmetry, while atomic orbitals of metal ion and molecular orbitals of different organic ligands in MOFs/COFs belong to different subgroups of lattice symmetries. Both these differences will impact the band structure of MOFs/COFs, in particular making it more complex. Consequently, which subset of bands are of most importance depends strongly on the location of Fermi level, i.e., electron counting and charge doping. Furthermore, there are usually two types of characteristic electrons coupled in MOFs, i.e., strongly correlated localized d and f electrons and diffusive s and p electrons, which interplay with lattice, orbital, and spin degrees of freedom, leading to more exotic topological and magnetic band structures. In this Account, we present an up-to-date review of recent theoretical developments to better understand the exotic band structures of MOFs/COFs. Starting from three fundamental 2D lattice models, i.e.