2D metal-organic framework: Altermagnetic high-order topological insulator and tunable spin-polarized corner states

Altermagnetic high-order topological insulators (HOTIs) represent an emerging class of magnetic topological phases. However, their realization in metal-organic framework (MOF) materials remains rare. Here, we show that monolayer MOF Cr(diz)2 (diz = 1,3-diazete) hosts such a state. The material exhibits the altermagnetic ground state, with the magnetic moments mainly from Cr and C atoms. In this magnetic ground state, both the bulk and edge states are gapped, and spin-up and spin-down corner states reside degenerately within this gap. On the square nanodisk, these corner states exhibit opposite spin directions at adjacent corners. In addition, these corner states can also be tuned by multiple approaches. Specifically, the biaxial strain can tune the corner states toward or away from the Fermi level. The electric field can induce energy splitting between the spin-up and spin-down corner states, and the energy difference is proportional to the electric field strength. When stacked into a bilayer, the system transitions from a d-wave to a g-wave altermagnet, with the corner states displaying opposite spin directions both between adjacent layers and between neighboring corners within each layer. Our work not only extends the altermagnetic HOTI state to MOF materials but also provides a platform for manipulating spin-polarized corner states toward spintronic applications.