Ab initio statistical mechanics of the Néel transition in monolayer FePS 3

We present analysis of temperature- and field-dependent magnetic order and N\'eel transition in monolayered ${\mathrm{FePS}}₃$, exhibiting a zigzag antiferromagnetic (zAFM) order below a N\'eel temperature (${T}_{\mathrm{N}}$) of 120 K, using a combination of first-principles density functional theoretical calculations, effective spin Hamiltonian, and Monte Carlo simulations. As a consequence of spin-phonon coupling, the honeycomb lattice of Fe ions distorts upon zAFM ordering, with ferromagnetic chains exhibiting long intrachain Fe-Fe bonds and short interchain Fe-Fe bonds. We construct an effective spin Hamiltonian to capture low-energy physics determining exchange interactions from first-principles using the four-state method. Analyzing this spin Hamiltonian with Monte Carlo (MC) simulations, we estimate ${T}_{\mathrm{N}}$ (85 K), which changes upon quantum correction to 127 K, closer to the experimental ${T}_{\mathrm{N}}$ of 120 K. Our low temperature- and field-dependent MC simulations capture the experimentally observed plateau in $M$ versus $H$ at the half-magnetization and reveal its magnetic structure: the half-plateau phase (HPP) exhibits a structure cell-doubled along the chain direction ($a$ axis) of zAFM state, with ferromagnetic ($\ensuremath{\uparrow}\ensuremath{\uparrow}\ensuremath{\uparrow}\ensuremath{\uparrow}$) and antiferromagnetic ($\ensuremath{\uparrow}\ensuremath{\uparrow}\ensuremath{\downarrow}\ensuremath{\downarrow}$) chains alternating along $b$ axis. Density functional theoretical calculations reveal that the HPP exhibits a gap of 0.6 (1.27) eV in electronic bands of down (up)-spin channel. Our MC simulations reveal domains with distinct orientations of HPP at higher fields and transformation to ferromagnetic (FM) state at $H>60$ T, which is overestimated relative to the experiments at ${H}_c\ensuremath{\sim}40$ T. The FM state of monolayered ${\mathrm{FePS}}₃$ is half-metallic with a gap of 2.2 eV in the majority-spin channel. We find that the coupling of phonons with orbital moments is weak, while that with spins is moderate, and predict phonon anomalies at ${T}_{\mathrm{N}}$.