On the Ferrimagnetic State of CrCl$_2$(pyz)$_2$

Van der Waals layered ferromagnetic compounds with high two-dimensional electronic conductivity holds strong potential for quantum computing, future unconventional superconductors, catalysts, batteries, and fuel cells. We suggest a minimal theoretical model to understand the magnetic properties of the metal-organic framework CrCl$_2$(pyz)$_2$ (pyz=pyrazine). Using a Hubbard model we show that the groundstate is dominated by a specific configuration of delocalized electrons on the pyz sites with a ferrimagnetic coupling to the localized spins on the Cr sites. This model suggests a magnetic moment of $2μ_B$ which is remarkably close to the experimental value of $1.8 μ_B$ [K. S. Pedersen et al., Nat. Chem. 10, 1056-1061 (2018)]. From Weiss mean-field theory we predict a weak ferromagnetic Cr-Cr coupling of $\approx 0.9$ meV. This is consolidated by second order perturbation theory of the RKKY interaction yielding a Cr-Cr coupling of $\approx 5$ meV. Understanding the interactions in these types of compounds can facilitate designs of metal-organic compounds with tailored magnetic properties.

💡 Research Summary

The manuscript presents a theoretical investigation of the ferrimagnetic ground state observed in the two‑dimensional metal‑organic framework CrCl₂(pyz)₂, where “pyz” denotes the pyrazine ligand. The authors aim to rationalize the experimentally measured saturation magnetization of ~1.8 μ_B per formula unit and the modest Curie temperature of 55 K by constructing a minimal model that captures the essential physics of localized Cr³⁺ spins interacting with itinerant electrons on the pyrazine ligands.

First, a comprehensive tight‑binding description of a monolayer of CrCl₂(pyz)₂ is derived using Slater–Koster parametrization. The basis set comprises five Cr d‑orbitals, three Cl p‑orbitals, and two tilted pyrazine p′‑orbitals, leading to a 15 × 15 Hamiltonian. The resulting band structure reproduces the semi‑metallic character reported in density‑functional theory (DFT) calculations: flat Cr‑derived d‑bands lie near the Fermi level, indicating strong localization, while the pyrazine‑derived p‑bands are highly dispersive, signifying delocalized carriers. The Cl states are far below the Fermi level and are omitted from the low‑energy model.

Motivated by this electronic landscape, the authors introduce a reduced Hamiltonian that contains only the two Cr³⁺ ions (each with spin S = 3/2) and four pyrazine sites, each capable of hosting a single spin‑½ electron. The lattice topology is that of a Lieb lattice, where each Cr couples antiferromagnetically (exchange constant J < 0) to three of the four pyrazine sites, while the pyrazine electrons can hop between neighboring sites with amplitude t and experience an on‑site Hubbard repulsion U. The Hamiltonian reads

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