Unique electronic state in ferromagnetic semiconductor FeCl₂ monolayer

Reduced dimensionality and interlayer couplings of bulk vdW materials trigger intriguing electronic, optical and other quantum properties Novoselov et al. (2004, 2005); Zhang et al. (2005); Cao et al. (2018); Sun et al. (2019). According to the Mermin-Wagner theorem Mermin and Wagner (1966), long-range magnetic order at finite temperature is prohibited in 2D isotropic Heisenberg spin-rotational-invariant systems. However, magnetic anisotropies can break spin-rotation symmetry, thus bringing about possible long-range magnetism. Recently, 2D vdW magnets with tunable magnetic anisotropy have attracted a large volume of attention as 2D FM was observed in atomically thin CrI₃ Huang et al. (2017) and Cr₂Ge₂Te₆ Gong et al. (2017). These 2D vdW magnets exhibit appealing properties, such as the magnetic anisotropy Kim et al. (2019); Yang et al. (2020); Liu et al. (2020a), the exotic quantum spin liquid states Xu et al. (2020) and aniferromagnetic (AF) topological insulators Gong et al. (2019); Deng et al. (2020); Otrokov et al. (2019). The abundant properties open a new avenue to spintronic applications, such as spin valves Cardoso et al. (2018), spin filters Klein et al. (2018); Song et al. (2018), and data storage Soumyanarayanan et al. (2016). Moreover, owing to the layered structure, one could be able to control the 2D magnetic properties by strain Yang et al. (2020); Liu et al. (2020a), doping Deng et al. (2018); Huang et al. (2018a), heterostructure Gibertini (2019); Liu et al. (2020b) or applying magnetic/electric field Huang et al. (2018b); Jiang et al. (2018). For example, the $T_{\rm C}$ of monolayer Fe₃GeTe₂ can exceed room temperature via ionic gating.

The binary transition metal dihalides MX₂ (M = transition metal; X = halogen: Cl, Br, I) are vdW layered materials containing a triangular lattice of M²⁺ cations McGuire (2017). The bulk magnetism was analyzed several decades ago Wilkinson et al. (1959); Lines (1963); Birgeneau et al. (1972); Jacobs and Lawrence (1967). Very recently, FeCl₂ monolayer has been synthesized by molecular-beam epitaxy Zhou et al. (2020); Cai et al. (2020), and it is a new 2D vdW magnet with a semiconducting gap of about 1.2 eV Cai et al. (2020). However, several theoretical studies suggest that it is a high-spin ($S$ = 2) half-metallic (HM) ferromagnet with a conducting down-spin $t_{2g}$ channel in the local octahedral coordination Torun et al. (2015); Ashton et al. (2017); Feng et al. (2018); Kulish and Huang (2017); Ghosh et al. (2021). Another two studies present the FM insulating solution Botana and Norman (2019); Yao et al. (2021), but the ground state was not yet firmly determined through total energy calculations. Moreover, the orbital singlet solution $d$^5↑$a$${}^{\downarrow}_{1g}$ was obtained when the reasonable $U$ and SOC parameters were used Yao et al. (2021). Then, only the spin moment is expected, and it should be smaller than 4 $\mu_{\rm B}$ per Fe²⁺ ($S$ = 2) reduced by the Fe-Cl covalence; and a small orbital moment, if any, would be in the plane (see more below, e.g. Table 1). Note that FeCl₂ bulk was reported to have interlayer AF but intralayer FM couplings, with the Néel temperature $T_{\rm N}$ = 24 K and a perpendicular magnetic moment of 4.3 $\mu_{\rm B}$ exceeding the spin-only value Wilkinson et al. (1959); Birgeneau et al. (1972); Jacobs and Lawrence (1967). So far, the intralayer FM coupling of most concern for FeCl₂ monolayer has been discussed in the FM HM solution in the previous studies Ashton et al. (2017); Kulish and Huang (2017); Ghosh et al. (2021), in which the $T_{\rm C}$ was too much overestimated by the fictitious itinerant magnetism, instead of the true superexchange in this FM semiconducting monolayer. Therefore, the FM semiconducting behavior of FeCl₂ monolayer and the perpendicular magnetic moment remain largely unresolved.

In this work, we critically examine the electronic and magnetic structures of FeCl₂ monolayer using a set of delicate first-principles calculations, crystal field level analyses, and Monte Carlo simulations. Our calculations find a moderate $t_{2g}$-$e_{g}$ crystal field splitting for the Fe $3d$ states, and thus a high-spin ($S$ = 2) Fe²⁺ state is achieved by the Hund exchange. Moreover, under the global trigonal crystal field, Fe $3d$ $t_{2g}$ triplet splits and the $e^{\prime}_{g}$ doublet is lower than $a_{1g}$ singlet by 30 meV. Owing to the strong correlation effect of the narrow Fe $3d$ bands in the triangular lattice, the band gap is determined by the Hubbard $U$. But note that here the orbital states must be delicately handled, and only when the SOC is properly included, the correct electronic ground state $d$^5↑$l$${}^{\downarrow}_{z+}$ would be achieved. Then, an out-of-plane orbital moment, strong perpendicular magnetic anisotropy of single ion type, and intralayer FM coupling, all well account for the FM semiconducting behavior of FeCl₂ monolayer, with $T_{\rm C}$ = 25 K being estimated by our Monte Carlo simulations. Furthermore, we predict that the $d$^5↑$l$${}^{\downarrow}_{z+}$ ground state can be further stabilized by a compressive strain, and –3% (–5%) strain would raise the $T_{\rm C}$ to 69 K (102 K). Therefore, FeCl₂ monolayer is indeed an appealing 2D FM semiconductor.

Density functional theory (DFT) calculations were carried out using the full-potential augmented plane wave plus local orbital code (Wien2k) Blaha et al. (2001). The optimized lattice parameter of FeCl₂ monolayer is $a$ = $b$ = 6.708 Å, which is close to (within 1.5%) the experimental value of 6.809 Å for the bulk Wilkinson et al. (1959). A vacuum slab of 15 Å was set along the $c$-axis. The muffin-tin sphere radii were chosen to be 2.2 bohr for Fe atoms and 1.8 bohr for Cl. The plane-wave cut-off energy of 12 Ry was set for the interstitial wave functions, and a $12\times 12\times 1$ k-mesh was sampled for integration over the Brillouin zone. To describe the electron correlation effect, several methods may be used, e.g., the local spin density approximation plus Hubbard $U$ (LSDA+$U$) method Anisimov et al. (1993), the self interaction correction Perdew and Zunger (1981); Shinde et al. (2021), the hybrid functional Becke (1993); Andriyevsky and Doll (2009), and the GW theory van Schilfgaarde et al. (2006). Here the economic and practical LSDA+$U$ method was employed, with the common values of Hubbard $U$ = 4.0 eV and Hund exchange $J_{\rm H}$ = 0.9 eV for Fe $3d$ electrons. As seen below, our LSDA+SOC+$U$ calculations well reproduce the experimental band gap, which also justifies the choice of the $U$ value. The SOC is included for both Fe 3$d$ and Cl 3$p$ orbitals by the second-variational method with scalar relativistic wave functions. We critically examine the orbital states and hence determine the correct electronic ground state by calculating the crystal field level splittings and by comparing orbital multiplets using total energy calculations. For this purpose, we perform DFT calculations using spin-restricted LDA, spin-polarized LSDA, LSAD+$U$, and LSDA+SOC+$U$, as detailed below. Moreover, we perform Monte Carlo simulations on a $8\times 8\times 1$ spin matrix to estimate the $T_{\rm C}$ of FeCl₂ monolayer, using the Metropolis method Metropolis and Ulam (1949) and the obtained exchange parameter and magnetic anisotropy from the LSDA+SOC+$U$ calculations.

We first analyze the crystal field levels of the Fe²⁺ ion in FeCl₂ monolayer, which is crucial for understanding of the electronic ground state. As seen in Fig. 1, FeCl₂ monolayer adopts the 1-$T$ structure with P$\overline{3}$m1 space group Zhou et al. (2020). Each Fe²⁺ ion is surrounded by six Cl^- ions, thus forming an FeCl₆ octahedron, and the edge-sharing octahedra comprise a triangular lattice. The local octahedral crystal field splits the Fe 3$d$ orbitals into the lower-lying $t$_2g triplet and higher $e$_g doublet. The $t$_2g triplet further splits into the $e^{\prime}_{g}$ doublet and the $a$_1g singlet in the global trigonal crystal field. The global coordinate system was used in the following calculations, with the $z$ axis along the $\left[111\right]$ direction of the local FeCl₆ octahedron and $y$ along the $\left[1\overline{1}0\right]$ direction Wu et al. (2005); Ou and Wu (2014).

To see the crystal field effect, we first perform the spin-restricted LDA calculation. Our results show that the $t$_2g-$e$_g octahedral crystal field splitting is about 1 eV, see Fig. 2(a). Among the $t_{2g}$, the $a$_1g singlet and $e^{\prime}_{g}$ doublet are almost degenerate at a first look. In reality, by a close look the $e^{\prime}_{g}$ is found to be slightly lower than the $a_{1g}$ by 30 meV, by calculating the center of gravity of their respective partial density of states (DOS). This result is crucial for the following discussion of the correct electronic ground state. The Cl 3$p$ state mainly lies in the range of 3-6 eV below the Fermi level, and its strong $pd\sigma$ hybridization with Fe $3d$-$e_{g}$ and weak $pd\pi$ hybridization with Fe $3d$-$t_{2g}$ are both evident.

Apparently, the Fe $3d$ orbitals form narrow bands in the triangular lattice, with the bandwidth less than 1 eV. Therefore, these localized $3d$ orbitals would be strongly spin-polarized by the local Hund exchange. Through the spin-polarized LSDA calculation, we find that indeed the Fe²⁺ ion is in the high-spin $S$ = 2 state, see Fig. 2(b). The up-spin $3d$ orbitals are fully occupied, and the down-spin $t_{2g}$ ($a_{1g}$ and $e^{\prime}_{g}$) is 1/3 partially occupied, seemingly giving a HM solution. Note that this HM solution is fictitious, as the strong correlation effect of the narrow bands is absent in the LSDA calculations and in reality FeCl₂ monolayer has a semiconducting gap of about 1.2 eV Cai et al. (2020).

Now we carry out LSDA+$U$ calculations to elucidate the electron correlation effect. As seen in Fig. 3, we obtain two contrasting high-spin solutions, one with the down-spin $a_{1g}$ occupation, and the other with the down-spin $e^{\prime}_{g}$ half filling. Apparently, the former solution has a semiconducting gap of 1.42 eV (well comparable with the experimental one of about 1.2 eV Cai et al. (2020)), while the latter one is again half metallic. Note that although the $e^{\prime}_{g}$ doublet is lower than the $a_{1g}$ singlet in the crystal field level diagram as discussed above, the down-spin $e^{\prime}_{g}$ doublet has to be half filled for the high-spin Fe²⁺ ion due to the present symmetry constriction. Thus, this solution has to be computationally half metallic, even with strong correlation effect which is sufficient to open a band gap. Owing to the half filling of the narrow $e^{\prime}_{g}$ band which strides over the Fermi level, this half-metallic solution is less stable, by about 162 meV/fu in our LSDA+$U$ calculations (see Table 1), than the semiconducting solution with the down-spin $a_{1g}$ occupation. The present LSDA+$U$ results show that the experimental band gap is determined by the Hubbard $U$. Then one may assume that the semiconducting solution with the down-spin $a_{1g}$ occupation is the correct one. However, this solution is an orbital singlet, and in principle it has no orbital moment but just a spin moment (3.55 $\mu_{\rm B}$/Fe, see Table 1) for the high-spin $S$ = 2 state. Even considering the SOC mixing between the nearly degenerate $a_{1g}$ and $e^{\prime}_{g}$ (see below), a small in-plane orbital moment is expected for this solution. Then this semiconducting solution with the down-spin $a_{1g}$ occupation cannot account for the experimental perpendicular magnetic moment of 4.3 $\mu_{\rm B}$ Wilkinson et al. (1959); Birgeneau et al. (1972); Jacobs and Lawrence (1967). In this sense, this semiconducting $d^{5\uparrow}a^{\downarrow}_{1g}$ solution is not the correct ground state.

The Hubbard $U$ determines the band gap of FeCl₂ monolayer and produces a strong orbital polarization for the down-spin $t_{2g}$ ($a_{1g}$ and $e^{\prime}_{g}$) states. In order to find the correct electronic ground state, one needs to delicately handle the orbital degrees of freedom. When the SOC is included, the $e^{\prime}_{g}$ doublet splits into $l_{z+}$ and $l_{z-}$ states, with the respective orbital moments of +1 $\mu_{\rm B}$ and –1 $\mu_{\rm B}$ along the $z$ axis (i.e., the crystallographic $c$ axis). Then the (near) degeneracy of the $t_{2g}$ states is completely lifted upon the SOC effect, and each of them can now be subject to an orbital polarization by the Hubbard $U$. We now perform the LSDA+SOC+$U$ calculations and carefully handle the orbital multiplets, see Table 1. In particular, we now find the $d^{5\uparrow}l^{\downarrow}_{z+}$ ground state, and it is semiconducting with a band gap of 1.17 eV (very close to the experimental one of about 1.2 eV Cai et al. (2020)), see Fig. 4(a). By a simple comparison between this semiconducting solution and the LSDA+$U$ HM solution [Fig. 3(b)], one might infer that it is the SOC which opens the band gap. However, this is a wrong statement. How can the SOC (a few tens of meV in strength) open the gap more than 1 eV? Actually, here the SOC offers new orbital degrees of freedom, and the Hubbard $U$ determines the band gap. The semiconducting $d^{5\uparrow}a^{\downarrow}_{1g}$ solution remains almost unchanged by a comparison between Figs. 3(a) and 4(b), and this solution is less stable than the $d^{5\uparrow}l^{\downarrow}_{z+}$ ground state by about 85 meV/fu, see Table 1.

The $d^{5\uparrow}l^{\downarrow}_{z+}$ ground state has the orbital moment of 0.59 $\mu_{\rm B}$ along the $z$ axis, in addition to the spin moment of 3.56 $\mu_{\rm B}$. It has the easy perpendicular magnetization and is more stable than the planar magnetization by 13.43 meV/Fe, see Table 1. Therefore, the total magnetic moment of 4.15 $\mu_{\rm B}$/Fe along the $z$ axis well accounts for the experimental perpendicular moment of 4.3 $\mu_{\rm B}$ Wilkinson et al. (1959); Birgeneau et al. (1972); Jacobs and Lawrence (1967). Moreover, the $d^{5\uparrow}l^{\downarrow}_{z+}$ ground state prefers a FM coupling in FeCl₂ monolayer, and it is more stable than the stripe AF state by 2.83 meV/Fe. Using the magnetic exchange expressions $-JS^{2}$ for each FM pair with $S$ = 2, $-3JS^{2}$ per fu for the FM state, and $JS^{2}$ per fu for the stripe AF state, we derive the FM exchange parameter $J$ = 2.83/4$S^{2}$$\approx$ 0.18 meV. In addition, for the metastable $d^{5\uparrow}a^{\downarrow}_{1g}$ state, the SOC mixes the nearly degenerate $a_{1g}$ and $e^{\prime}_{g}$ states and produces a planar orbital moment of 0.23 $\mu_{\rm B}$. Note that this easy planar magnetization would fail to explain the experimental perpendicular magnetization. Therefore, all the above LSDA+SOC+$U$ results lead us to a conclusion that $d^{5\uparrow}l^{\downarrow}_{z+}$ (but not $d^{5\uparrow}a^{\downarrow}_{1g}$) is the correct ground state and it consistently explains the FM semiconducting behavior of FeCl₂ monolayer with the perpendicular magnetization.

In order to estimate the $T_{\rm C}$ of FeCl₂ monolayer, we assume the spin Hamiltonian and carry out Monte Carlo simulations

where the first term represents the isotropic Heisenberg exchange with $J=0.18$ meV, and the $D$ parameter in the second term stands for the magnetic anisotropy of the single ion type. Our LSDA+SOC+$U$ results listed in Table 1 suggest the easy $z$-axis magnetization with the anisotropy energy of 13.43 meV/Fe and $S_{z}=2$, and thus $D=3.36$ meV can be derived. Using these $J$ and $D$ parameters, our Monte Carlo simulations give $T_{\rm C}$ = 25 K for the pristine FeCl₂ monolayer, see Fig. 5.

As a lattice strain is an effective way to tune the electronic state and magnetism of 2D materials, here we also study a biaxial strain effect on the FM of FeCl₂ monolayer. A compressive strain would raise the crystal field level of the $a$_1g singlet, and thus further stabilizes the $d$^5↑$l$${}^{\downarrow}_{z+}$ ground state. As shown in Table 1 and Fig. 6, the 3% or 5% compressive strain gives a larger orbital moment, a stronger perpendicular anisotropy, and a stronger FM coupling, according to the LSDA+SOC+$U$ calculations. Therefore, the $T_{\rm C}$ is naturally expected to be enhanced, and this is indeed confirmed by our Monte Carlo simulations (see Fig. 5): $T_{\rm C}$ is 69 K for the –3% strain and 102 K for –5% strain. In addition, we study a tensile strain, which would gradually destabilize the $d$^5↑$l$${}^{\downarrow}_{z+}$ ground state. Our LSDA+SOC+$U$ calculations show that the orbital moment gets smaller, the perpendicular magnetic anisotropy shrinks, and strikingly, the intralayer magnetic coupling changes its sign: the stripe AF state gets more stable than the FM state upon 3% and 5% strain. All these results show that a compressive strain would significantly enhance the $T_{\rm C}$ of FeCl₂ monolayer, but that a tensile strain could trigger an interesting FM-AF transition. Therefore, FeCl₂ monolayer could be an appealing 2D magnetic semiconductor potentially suitable for spintronic applications.

In summary, using a set of delicate DFT calculations including the SOC and Hubbard $U$, aided with the crystal field level analyses, we achieve the correct $d^{5\uparrow}l^{\downarrow}_{z+}$ ground state for FeCl₂ monolayer. This ground state well explains, in a consistent way, the experimental FM semiconducting behavior with the perpendicular magnetization, and our Monte Carlo simulation gives $T_{\rm C}$ = 25 K for the pristine FeCl₂ monolayer. Moreover, we find that upon the compressive strain, the $d^{5\uparrow}l^{\downarrow}_{z+}$ ground state gets more stable, and the enhanced FM coupling and the perpendicular magnetic anisotropy raise the $T_{\rm C}$ a lot, up to 69 K (102 K) for –3% (–5%) strain. We also predict an interesting FM-AF transition for FeCl₂ monolayer under a tensile strain. All these results suggest that FeCl₂ monolayer is an appealing 2D magnetic semiconductor.

This work was supported by National Natural Science Foundation of China (Grants No. 12104307 and No. 12174062).