Two-orbital spin-fermion model study of ferromagnetism in the honeycomb lattice

The spin-fermion model was previously successful to describe the complex phase diagrams of colossal magnetoresistive manganites and iron-based superconductors. In recent years, two-dimensional magnets have rapidly risen up as a new attractive branch of quantum materials, which are theoretically described based on classical spin models in most studies. Alternatively, here the two-orbital spin-fermion model is established as a uniform scenario to describe the ferromagnetism in a two-dimensional honeycomb lattice. This model connects the magnetic interactions with the electronic structures. Then the continuous tuning of magnetism in these honeycomb lattices can be predicted, based on a general phase diagram. The electron/hole doping, from the empty $e_{\mathrm{g}}$ to half-filled $e_{\mathrm{g}}$ limit, is studied as a benchmark. Our Monte Carlo result finds that the ferromagnetic $T_{\mathrm{C}}$ reaches the maximum at the quarter-filled case. In other regions, the linear relationship between $T_{\mathrm{C}}$ and doping concentration provides a theoretical guideline for the experimental modulations of two-dimensional ferromagnetism tuned by ionic liquid or electrical gating.

Figure 1

Comparison of low-dimensional honeycomb magnets and [111]-bilayer of ${AB\mathrm{O}}_3$ perovskites. (a) The local octahedral structure. The red and yellow circles represent the magnetic cations and ligand anions, respectively. The $d$ orbitals split into the $e_{\mathrm{g}}$ and $t_{2\mathrm{g}}$ orbitals in an undistorted octahedral crystal field in the absence of spin-orbit coupling (SOC). The neighboring octahedra are connected in an edge-sharing manner ($M-X-M$ bond angle $\sim {90}^{\circ }$), which is the common case for $M{X}_3$, $MA{X}_3$, and $M_2{A}_2{X}_6$ type van der Waals magnets. (c) The neighboring ${M\mathrm{O}}_6$ octahedra share a common corner ($M$-O-$M$ bond angle $\sim {180}^{\circ }$) in perovskites. The bold solid lines denote the [111]-oriented bilayer, which forms a buckled honeycomb lattice. (d) Top view of the [111] bilayer. The hopping integrals of $e_{\mathrm{g}}$ electrons are schematically shown by double arrows.

Figure 2

Electronic band structures of two-orbital model on the honeycomb lattice. Here the plain ferromagnetic background is used. (a) A comparison between DFT band structure and model band structure for ${\mathrm{MnF}}_3$ monolayer. Here the values of $t^{bb1}$ and $t^{bb2}$ are listed in Table 1, obtained from the MLWFs fitting. In the DFT calculation, the electron-hole symmetry is broken, leading to wider conducting bands than valence bands. (b) The 3D dispersion curve near a single Weyl point at $K$ for ${\mathrm{MnF}}_3$. (c) The 3D dispersion curve for a Weyl nodal line when using $t^{bb1}=139$ meV and $t^{bb2}=227$ meV.

Figure 3

Model simulated ferromagnetic transitions in undoped/doped ${\mathrm{CrGeTe}}_3$ monolayers. The static structure factor of $t_{2\mathrm{g}}$ spins is defined as $S\left(q\right)={\sum }_{ij}\langle S_i·S_j\rangle exp[-iq·(R_i-R_j)]$, where $\langle S_i·S_j\rangle$ is the spin correlation function in real space. (a) The $S(q=0)$ curves for different doping levels as a function of temperature. The $T_C$ can be significantly enhanced upon electron doping. (b) The Monte Carlo $T_C$ as a function of doping level, in comparison with the values obtained in the organic intercalation experiments. (c), (d) The Monte Carlo snapshots at (c) 42 K and (d) 300 K, respectively, for the $0.5$ electron/Cr doping case.

Figure 4

(a) The evolution of magnetic transition temperatures ($T_C$ or $T_{\mathrm{N}}$) as a function of chemical potential $\mu$. Here two types of $J_{ex}$ are considered: $J_{ex}=-3.2$ meV and $J_{ex}=10$ meV. The former leads to paramagnetic-ferromagnetic transitions in the whole range, while the latter can lead to paramagnetic-antiferromagnetic transitions in two ends. Inset: The corresponding $e_{\mathrm{g}}$ electron density ($n$) as a function of $\mu$ at $4.1$ K. (b-c) Typical spin structure factors at $36.7$ K. (b) The ferromagnetic state with Bragg peak centered at the $\mathrm{\Gamma }$ point and (c) the Néel antiferromagnetic state with Bragg peaks resided on the corner of 1st Brilliouin Zone.