Understanding the Ising zigzag antiferromagnetism of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mtext>FePS</mml:mtext><mml:mn>3</mml:mn></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mtext>FePSe</mml:mtext><mml:mn>3</mml:mn></mml:msub></mml:math> monolayers
Ke Yang, Yueyue Ning, Yuxuan Zhou, Di Lu, Yaozhenghang Ma, Lu Liu, Shengli Pu, Hua Wu
Abstract
Transition metal phosphorous trichalcogenides represent a class of van der Waals magnetic materials ideal for exploring two-dimensional magnetism. This study investigates the spin-orbital states of ${\mathrm{FePS}}_{3}$ and ${\mathrm{FePSe}}_{3}$ monolayers and the origin of their Ising zigzag antiferromagnetism (AFM), using density functional calculations, crystal field level diagrams, superexchange analyses, and parallel tempering Monte Carlo (PTMC) simulations. Our calculations show that under the trigonal elongation of the ${\mathrm{FeS}}_{6}$ (${\mathrm{FeSe}}_{6}$) octahedra, the ${e}_{g}^{\ensuremath{\pi}}$ doublet of the Fe $3d$ crystal field levels lies lower than the ${a}_{1g}$ singlet by about 108 meV (123 meV), which is much larger than the strength of Fe $3d$ spin-orbit coupling (SOC). Then, the half-filled minority-spin ${e}_{g}^{\ensuremath{\pi}}$ doublet of the high-spin ${\mathrm{Fe}}^{2+}$ ions (${d}^{5\ensuremath{\uparrow},1\ensuremath{\downarrow}}$) splits by the SOC into the lower ${L}_{z+}$ and higher ${L}_{z\ensuremath{-}}$ states. The spin-orbital ground state ${d}^{5\ensuremath{\uparrow}}{L}_{z+}^{1\ensuremath{\downarrow}}$ formally with ${S}_{z}=2$ and ${L}_{z}=1$ gives the large $z$-axis spin/orbital moments of $3.51/0.76\phantom{\rule{4pt}{0ex}}{\ensuremath{\mu}}_{\mathrm{B}}$ ($3.41/0.67\phantom{\rule{4pt}{0ex}}{\ensuremath{\mu}}_{\mathrm{B}}$) for ${\mathrm{FePS}}_{3}$ (${\mathrm{FePSe}}_{3}$) monolayer, and both the moments are reduced by the strong (stronger) Fe $3d$ hybridizations with S $3p$ (Se $4p$) states. As a result, ${\mathrm{FePS}}_{3}$ (${\mathrm{FePSe}}_{3}$) monolayer has a huge perpendicular single-ion anisotropy (SIA) energy of 19.4 meV (14.9 meV), giving an Ising-type magnetism. Moreover, via the maximally localized Wannier functions, we find that the first-nearest-neighboring (1NN) Fe-Fe pair has large hopping parameters in-between some specific orbitals, and so does the third-nearest-neighboring (3NN) Fe-Fe pair. In contrast, the second-nearest-neighboring (2NN) Fe-Fe pair has much smaller hopping parameters and the fourth-nearest-neighboring Fe-Fe pair has negligibly small ones. Then, a combination of those hopping parameters and the superexchange picture can readily explain the computed strong 1NN ferromagnetic coupling and the strong 3NN antiferromagnetic one but the relatively much smaller 2NN antiferromagnetic coupling. Furthermore, our PTMC simulations give ${T}_{\mathrm{N}}$ of 119 K for ${\mathrm{FePS}}_{3}$ monolayer and well reproduce its experimental Ising zigzag AFM, and also predict for ${\mathrm{FePSe}}_{3}$ monolayer the same magnetic structure with a close or even higher ${T}_{\mathrm{N}}$.