Goodenough–Kanamori–Anderson high-temperature ferromagnetism in tetragonal transition-metal xenes

The experimental discovery of the premiere magnetic two-dimensional materials (M2DMs) - CrI₃ [1] and CrGeTe₃ [2, 3]—opened the floodgates to many emergent 2D materials of this class. Numerous theoretical [4–7] and experimental studies [8–11] followed, explaining the origins and possible manipulations of the long-range magnetic order in the monolayer limit. It is needless to emphasize that intrinsically room-temperature M2DMs would be highly beneficial for applications in sensing, spintronics, and otherwise, and bear promise towards high tunability by diverse mechanical, chemical, and electronic means. However, it quickly became clear that critical temperatures (T_c ) for magnetic order to vanish are by rule always smaller in 2D materials compared to their bulk counterparts [1, 12, 13]. Namely, in order to host sizable regions with magnetic order (and circumvent limitations imposed by the Mermin-Wagner theorem [14]), M2DMs require anisotropy in magnetic exchange. That needed anisotropy is known to originate from the spin–orbit coupling (SOC), which is in general much weaker compared to e.g. Coulomb attraction or repulsion of charged particles, hence attains very low magnitudes (of typically 0.01–0.1 meV). Another reason for T_c to decrease as one thins the magnetic material from bulk to a monolayer stems from the correspondingly diminishing magnetic exchange along the third, out-of-plane direction. One may therefore expect that magnetic order in 2D materials is strictly limited to the very low T_c , but that is not necessarily the case. For example, Fe₃GeTe₂ hosts the ferromagnetic (FM) order up to 130 K in the monolayer (ML) limit [15]. In addition, the FM order and T_c of 213 K were measured in few-layer thick 1T-CrTe₂ [16], with the unusual trend that T_c increases as one goes from bulk to few-layers. Furthermore, in a few-layer FePS₃, the antiferromagnetic (AFM) order was observed up to the T_c of 120 K [17]. Finally, FM order was measured even at room temperature in thicker films (${\approx}10$ nm, see e.g. [18, 19]), but also in monolayer MnSe_x [20, 21], VSe₂ [11, 22], and Cr₃Te₄ [23]. The latter samples were deposited epitaxially, which in general involves structural defects [22] and strong interfacial effects with the substrate [22, 23] into the origins of the observed robust magnetic interactions, which complicates theoretical interpretations. Otherwise, the isotropic magnetic interactions in crystalline monolayers are relatively straightforward to extract theoretically in all available first-principles codes. Such studies, on predominantly in silico created 2D materials, have yielded many predictions of high- or even room-temperature intrinsic magnetism [21, 24–28]. However, those predictions typically failed to quantify the underlying microscopic mechanisms for such a large predicted magnetic exchange.

In this paper we therefore take a step back, and explore the route towards room-temperature 2D ferromagnetism starting from the well-established set of Goodenough–Kanamori–Anderson (GKA) empirical theoretical rules from the late 1950s [29–32]. We accordingly aim at monolayers with 90^∘ between the nearest magnetic atoms (A) connected by a ligand (X), thus ideally a Lieb-lattice material of A₂X type. However, a magnetic 2D material of such specific planar structure has not been reported to date, although some Lieb-lattice 2D materials have been considered computationally for other purposes (see e.g. [33]). As a best available choice, for not only geometry but also sizable SOC, we instead focus on the family of tetragonal transition-metal xenes (tTMXs; see figure 1), seeking a square-lattice planar material among them—still with 90^∘ TM-X-TM nearest-neighbor bonds. Such square-lattice planar materials are not yet present in monolayer form, however few nanometers thick (roughly 5–30 nm) layered structures from this family are easily synthesized [34]. Furthermore, in mentioned work, Kang et al grew two distinctive phases of FeTe—low-temperature-grown tetragonal phase that showed AFM order and high-temperature-grown hexagonal phase that showed FM order—by controlling the temperature of crystal formation. Using advanced methodology on top of the standard first-principles approaches based on density functional theory (DFT), we computationally validate the structural stability and strong intrinsic magnetic interactions in these materials, detail the microscopic (orbital-resolved) origin of enhanced magnetic exchange, and identify CrC and MnC as premiere square-lattice monolayer ferromagnets with a particularly high T_c .

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We commence our analysis with a computationally crude throughput screening of dynamical stability and magnetic interactions in tTMXs (where TM = V, Cr, Mn, Fe, Co, Ni, Cu, Zn, and X = C, N, O, S, Se, Te). For each stable material we perform total energy mapping between the density functional theory corrected with on-site Coulomb repulsion (DFT+U) and the Heisenberg model Hamiltonian, for six particular magnetic orders—namely FM and AFM orders along three Cartesian directions (see appendix B in supplementary information), in order to extract the governing magnetic interactions in the system. For the sake of screening, Hubbard parameter U in the calculations is taken from the online database, based on bulk oxides of transition metals [35]. In order to decrease computational cost, we consider only the first nearest-neighbor (NN) magnetic interactions and the single-ion anisotropy (SIA) in the Heisenberg Hamiltonian.

The considered primitive unit cell of these materials consists of two TM, and two X atoms—such that each TM atom has four X atoms as the nearest neighbors, and vice versa (see figure 1), where one can apply symmetry rules for the exchange matrix (see appendix A in supplementary information) and SIA (see [36]). In figure 1 each TM atom is sketched with one d orbital, and each X atom with one p orbital, since these atomic orbitals, and their mutual hybridization, are essential for the interactions between magnetic moments on TM atoms. Though the positions of the atoms in the structure are uniquely determined with respect to the in-plane primitive lattice vectors due to the symmetry of tetragonal structures, both atomic species are allowed to relax out-of-plane (along $\vec{a}_{3} \equiv z$ axis).

Our throughput computational screening revealed that out of a 48 materials in total, only the 21 listed in table 1 possess dynamical stability. Out of those 21, we identified five materials with FM order as lowest in energy (out of six possibilities considered), seven materials with AFM order as the lowest-energy one, and nine materials not exhibiting magnetic order, i.e. with magnetic moments on individual atoms below 0.5µ_B in every of the six considered magnetic configurations, and the configuration with magnetic moments exactly 0 as lowest in energy. Out of five FM materials, two of them appear to have completely flat, square lattice when on-site Coulomb repulsion between electrons (U) is properly included—CrC and MnC. Such a structure ensures that the first-nearest-neighbor TM-X-TM bond comprises 90 degrees angle, and second-nearest-neighbor 180 degrees angle, making these structures optimal—according to GKA rules—for 'maximization' of magnetic exchange in the system. This fact served as motivation to thoroughly analyze the structural, electronic, and magnetic properties of these two materials, and discuss the subtle interplay of these properties that optimizes the ferromagnetic order.

Table 1. Magnetic properties of the stable monolayer tTMX structures. FM, AFM, and NM stand for ferromagnetic, antiferromagnetic, and non-magnetic order, respectively. J ${^{xx}}$ and J ${^{yy}}$ mark in-plane, and J ${^{zz}}$ out-of-plane exchange interactions. Out-of-plane exchange anisotropy Δ stands for the difference J ${^{xx}}$-J ${^{zz}}$. A ${_{ii}^{zz}}$ is the single-ion anisotropy (SIA). If signs of SIA and exchange interactions are the same, SIA favors out-of-plane anisotropy, otherwise the in-plane one. J ${^{xx}}$, J ${^{yy}}$, J ${^{zz}}$, Δ, and A ${_{ii}^{zz}}$ are all given in meV. T_c stands for the critical temperature of the magnetic order, Curie for FM and Néel temperature for AFM monolayers.

	Magnetic order	J ${^{xx}}$ = J ${^{yy}}$	J ${^{zz}}$	$\Delta~~$	A ${_{ii}^{zz}}$	Tc (K)
CrC	FM	−52.68	−52.66	0.03	−0.35	515.6
MnC	FM	−105.98	−106.01	−0.03	−0.67	1065.8
VN	FM	−3.73	−10.54	−6.81	−13.94	152.4
CoN	FM	−12.57	−12.94	−0.38	1.28	132.2
NiTe	FM	−7.42	−5.15	2.27	4.53	31.2
FeC	AFM	53.00	53.28	0.28	0.21	5.3
FeO	AFM	5.47	5.46	−0.01	2.48	20.1
MnS	AFM	55.07	55.95	0.88	1.69	15.2
FeS	AFM	16.13	16.38	0.25	−0.56	5.7
MnSe	AFM	50.66	50.65	−0.01	−0.20	10.3
FeSe	AFM	15.30	15.24	−0.06	−1.20	5.8
FeTe	AFM	31.80	−28.89	−60.69	119.63	35.6
CuC	NM	—	—	—	—	—
NiN	NM	—	—	—	—	—
CuN	NM	—	—	—	—	—
NiS	NM	—	—	—	—	—
CuS	NM	—	—	—	—	—
ZnS	NM	—	—	—	—	—
CuSe	NM	—	—	—	—	—
ZnSe	NM	—	—	—	—	—
ZnTe	NM	—	—	—	—	—

The first important finding concerns the structure. Namely, as indicated above, the relative atomic arrangement along the z axis strongly depends on the on-site Coulomb repulsion (U) between electrons on TM atoms. In figures 2(a), (b), (e) and (f), we present the relaxed structures for CrC and MnC with no U included, and for the U value calculated self-consistently—using the linear response theory as introduced by Timorov et al [37]. The latter U value is found as 3.29 eV for CrC, and 3.81 eV for MnC. In case of U = 0, the structure buckles, and two X atoms relax above and below the plane of TM atoms. Furthermore, the X atoms create the tetrahedral structure around each TM atom. On the contrary, when using a realistic value for U, both CrC and MnC exhibit an ideally planar structure, i.e. a 2D square lattice as desired in the GKA argumentation towards the enhanced ferromagnetic order. For certainty, we tested the dynamical stability of both these planar structures—and the phonon dispersions shown in figures 2(c), (d), (g) and (h) exhibited no imaginary phonon frequencies.

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After determining the on-site Coulomb repulsion and the planar structural stability, we move on to the magnetic properties of CrC and MnC. The necessary, yet insufficient condition for long-range magnetism is the non-zero magnetic moment per unit space. The TM atoms are expected to provide the latter, due to the localized unpaired electrons in their 3d shell, each electron carrying spin $\frac{1}{2}$, and spin magnetic moment of $1~\mu_{B}$. In most of the structures based on TMs, the contribution of the orbital magnetic moment is negligible compared to the spin magnetic moment, due to the quenching of orbital momentum, hence the magnetic moment can be assumed to originate purely from the spin of the electron. Consequently, the total magnetic moment on each TM atom is $N\times1~\mu_{B}$, where N is the total number of unpaired electrons in the 3d shell of each TM atom.

The basic ionic theory suggests 4+ oxidation state of Cr and Mn cations in our monolayers, since C atom receives 4 electrons to reach stable octet configuration. Furthermore, Cr⁴⁺ ion has 20 electrons and the $1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}3d^{2}$ electronic configuration, while Mn⁴⁺ ion 21 electrons and the $1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}3d^{3}$ electronic configuration. This means that one expects $2~\mu_{B}$ per Cr atom and $4~\mu_{B}$ per primitive unit cell in CrC, and $3~\mu_{B}$ per Mn atom and $6~\mu_{B}$ per primitive unit cell in MnC. Ionic theory predicts no moment per C atom in either cases, due to the mentioned stable octet configuration.

For a more precise account of magnetic moments per atom we resort to DFT calculations, and find that: (1) in case of CrC, the magnetization per Cr atom is 2.85 µ_B and per C atom $-0.72~\mu_{B}$, resulting in 4.27 µ_B per primitive unit cell; (2) in case of MnC, the magnetization per Mn atom is 3.88 µ_B and per C atom $-0.77~\mu_{B}$, resulting in 6.22 µ_B per primitive unit cell. The DFT results do corroborate the crude ionic theory regarding the total magnetization of the unit cell, but also reveal the significant hybridization between (d) orbitals of TM atoms and (p) orbitals of C atoms—causing a rather significant magnetization on otherwise non-magnetic C atoms.

The origin of the magnetic moments on all atoms involved is discussed in detail in the chapter 'MAGNETIC MOMENTS' in the supplementary information. Next to it, we comment on rather large discrepancy between ionic and DFT predictions of the magnetic moments on both TM and C atoms.

However, in order for a system to host measurable magnetic order, next to the non-zero magnetic moments on TM ions, it must also host significant interaction between them. Unlike the initial estimates using the method based on mapping between total energies of the DFT and the Heisenberg Hamiltonians, we now calculate the magnetic exchange by mapping the energy variations due to the infinitesimal rotation of the magnetic moment on TM atoms from the reference FM state, between the DFT Hamiltonian in the localized-orbital basis set, and the Heisenberg Hamiltonian $H = \sum_{\lt i,j\gt} \boldsymbol{S}{_i}\boldsymbol{J}{_{i,j}}\boldsymbol{S}{_j}$, as implemented in the TB2J code [38]. In the latter Hamiltonian, S _i denotes the unit 3D vector of the magnetic moment on the ith TM atom; $3\times3$ matrix $\boldsymbol{J}_{i,j}$ stands for interaction between magnetic moments on ith and jth TM atoms; and $\lt i,j\gt$ denotes i ≠ j, with avoided double counting. The SIA matrix cannot be calculated using this formalism (for details see [38]), therefore it is not explicitly written in the used Heisenberg Hamiltonian. However, SIA is not negligible, and correspondingly must be taken into account during e.g. the calculation of T_c for the FM order. Therefore, we combine SIA reported in table 1, together with the $\boldsymbol{J}_{i,j}$ calculated using TB2J to construct the total model Hamiltonian for eventual 2nd-principles calculations:

$$\begin{equation} H = \sum_{\lt i,j \gt} \boldsymbol{S}{_i}\boldsymbol{J}{_{i,j}}\boldsymbol{S}{_j} + \sum_{i} \boldsymbol{S}{_i}\boldsymbol{A}{_{i,i}}\boldsymbol{S}{_i} .\end{equation} \tag{ 1 }$$

The main advantage of TB2J and the Green's-function-based methodology over the total energy mapping is the ability to orbitally disentangle the origins of magnetic interactions [39], and also the ability to calculate interactions between all different neighbors within a large supercell upon a single DFT calculation on the primitive unit cell. In particular, we have calculated the matrices $\boldsymbol{J}_{i,j}$ up to the 284th NN for both materials.

Our results obtained using TB2J obey the symmetry-imposed constraints—i.e. all off-diagonal elements of all $\boldsymbol{J}_{i,j}$ matrices are exactly 0. The diagonal part of each matrix can further be split—as presented in table 1—into the isotropic (J $_{i,j} =$ J $^{xx}_{i,j} =$ J $^{yy}_{i,j}$) and the anisotropic part ($\Delta =$ J $^{zz}_{i,j}-$ J $^{xx}_{i,j}$). The anisotropic part of the exchange, Δ, in either system does not exceed few (1-6) µeV, and is comparable with the rounding error in our calculations. Therefore, we consider Δ as effectively 0, and we subscribe the stabilization of the magnetic order in these 2D materials to just J $_{i,j}$ and SIA ⁶ .

In figure 3 we present our TB2J results for J $_{i,j}$ up to the 6th NN. One notices in figure 3 that 1st and 2nd NN interactions are strong (few tens of meV), an order of magnitude larger compared to usually encountered isotropic exchange values (few meV) in 2D materials. Another observation is that even pairs over 1 nm distance have small but non-zero exchange interaction, of few hundreds of µeV. However, since these are much smaller than the 1st and 2nd NN interactions, in what follows, we focus on the first two NN pairs with giant J $_{i,j}$, being essential for the high-T_c GKA ferromagnetism. We use exchange interaction parameters calculated with TB2J, and SIA from table 1 to build the Heisenberg Hamiltonian as explained above, to then employ Monte Carlo simulations to evaluate the stability of the FM order with respect to temperature. In figure 4, we present the thereby obtained evolution of (normalized) magnetization (M$_{st}$/M_s ), magnetic susceptibility (χ), and specific heat (C_v ), as a function of temperature. One clearly sees that estimated critical temperatures of the FM order in both CrC and MnC exceed the room temperature, being 307 K and 428 K respectively. Even though anisotropy is in general required to stabilize magnetic order in 2D above 0 K—in our case that is SIA—the main reason for such a large T_c lies in the particularly large isotropic exchange between the 1st as well as the 2nd NN pairs of TM atoms. Table 1 provides Curie and Neel temperature values for t-TMX structures. Notably, VN, CoN, and NiTe structures exhibit ferromagnetic properties; however, their respective Curie temperatures are significantly lower compared to those of CrC and MnC. This phenomenon can be primarily attributed to two key factors: weakened exchange interactions and strong Single Ion Anisotropy (SIA). This weakened exchange interaction and strong SIA inherently limits their ability to maintain ferromagnetic order at elevated temperatures, thus resulting in lower Curie temperatures when contrasted with CrC and MnC. Conversely, antiferromagnetic materials exhibit Neel temperatures in proximity to absolute zero (0 K) due to the significant impact of SIA. This low Neel temperature signifies the point at which antiferromagnetic materials undergo a transition to a non-magnetic state. The details about calculations of Curie and Neel temperatures can be found in supplementary information.

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Having presented the core results, we next detail the origin of the magnetic interactions behind the observed high Curie temperature in monolayer tTMXs, bearing in mind the original assumptions following from the GKA rules. To shed the light on the source of the large exchange J $_{i,j}$, we look into the orbitally-resolved contributions.

Our initial assumption was that the gross of the exchange interactions originates from the exchange between the d orbitals of the interacting TM atoms (d-d exchange). In order to properly describe the physics behind our observations, we treat 3s, 3p, 3d, 4s, and 4p as valence orbitals on TM atoms. Further, we quantify the contributions of each orbital-to-orbital interaction between the neighboring TM atoms (e.g. 4s on TM1 and 3d on TM2), to the total exchange between them. For facilitated interpretation, we consider the contributed interactions between each type of orbital on TM1 (a = 3s, 4s, 3p, 3d, and 4p) and only 3d orbitals on TM2 (b = 3d), for the first three NN magnetic interactions $\textit{J}^{a,b}_{\#NN}$, as shown in figure 5. It is rather obvious from figure 5 that TM1(3d)-TM2(3d) interactions dominate (being several tens of meV strong), and determine the magnetic order in the system—in this case the FM one. Interactions between other types of orbitals on one TM and 3d orbitals on the other are generally at least an order of magnitude smaller. Therefore, after d-d exchange is proven to be crucial for the large exchange and room-temperature magnetism in tTMXs, we next decompose it into the exchanges between individual d-orbitals of the interacting pair of TM atoms, to disentangle the key contributors. The results per d-orbital and per material are shown in figure 6.

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3.1. The nearest-neighbor interaction

3.2. The next-nearest and further-neighbor interactions

In summary, after a computational throughput screening of the whole family of tTMX materials, we have identified two dynamically stable and ideally 2D flat Lieb-like magnetic crystals, CrC and MnC. According to the seminal GKA rules, materials of such symmetry are prone to host pronouncedly high ferromagnetic exchange interaction. Our detailed analysis of the magnetic properties has shown that both Cr and Mn ions indeed have sizable magnetic moments, that materials host large isotropic magnetic interactions (order of 10 meV), that exchange anisotropy is negligible in these systems (order of 1 µeV), and that stabilization of the long-range magnetic order in these materials should be prescribed to single-ion anisotropy (order of 0.1 meV). As a result of this large isotropic exchange and non-zero single-ion anisotropy, the Curie temperature for ferromagnetic transition of these materials exceeds the room temperature—307 K in CrC and 428 K in MnC.

In detailed analysis presented, we pinpoint the sole physical mechanism that lies in the origin of room temperature magnetism in these systems—geometry-selected single-p-mediated superexchange. We indisputably show that this mechanism leads to the large FM contributions for both the first- and the second-nearest-neighbor magnetic exchange between TM atoms, in both CrC and MnC.

In case of the 3rd NN, which was of secondary importance in our systems, we could not establish a criterion for single dominant contribution based on geometry, however, even there two contributions were identified as leading ones. Finally we have shown that interactions beyond the 3rd NN are far smaller and may be neglected in any prediction of experimentally measurable physical properties of these systems.

In conclusion, we want to outline few general rules for room temperature magnetism in this material family:

• (i)  
  Matrix elements in TB-like Hamiltonian associated with direct d-d orbital interaction must be negligible—at least ∼5–10 times smaller than matrix elements connected with the indirect interaction.
• (ii)  
  The superexchange process should be mediated only with the single p orbital of single non-magnetic atom - 'L' superexchange in case of the first-nearest-neighbor and '—' superexchange in case of the second-nearest-neighbor. This is guaranteed in the flat square-lattice.
• (iii)  
  Magnetic atoms should be chosen from the first half of the first row of transition metals (V, Cr, Mn, Fe) in order to guarantee 'small' number of d-electrons (${\unicode{x2A7D}} 5$), which will ensure the high-spin state. Other typical magnetic elements (Co, Ni, Cu) may host low-spin state and therefore not comply with our conclusions.

Consequently, in any such system with sufficient anisotropy in magnetic interactions, the latter features will foster the large isotropic exchange and high critical temperature of the long-range magnetic order in two dimensions.

This work was supported by the Research Foundation-Flanders (FWO) and the Technological Research Council of Turkey (TUBITAK) under Contract No. 118F512. D Š is a doctoral fellow of FWO under Contract No. 11J4322N. The computational resources and services for this work were provided by the VSC (Flemish Supercomputer Center), funded by the FWO and the Flemish Government—department EWI.

The data cannot be made publicly available upon publication because no suitable repository exists for hosting data in this field of study. The data that support the findings of this study are available upon reasonable request from the authors.