# Spin-crossover and high magnetic anisotropy in organometallic structures: W3C6O6 and Re3C6O6

Adam Hassan Denawi, Holger Vach, Kausik Das, Vincent Oison, Roland Hayn

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Adam Hassan Denawi, Holger Vach, Kausik Das, Vincent Oison, Roland Hayn. Spin-crossover and high magnetic anisotropy in organometallic structures: W3C6O6 and Re3C6O6. Materials Today Communications, 2025, 44, pp.112165. 10.1016/j.mtcomm.2025.112165. hal-05018681

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# Spin-crossover and high magnetic anisotropy in organometallic structures: $W_3C_6O_6$ and $Re_3C_6O_6$

Adam Hassan Denawi¹*, Holger Vach¹, Kausik Das², Vincent Oison³ and Roland Hayn³

¹CNRS, LPICM, École Polytechnique, IP Paris, Palaiseau 91128, France.

²Institute of Physics, Polish Academy of Sciences, aleja Lotników 32/46, PL 02-668 Warszawa, Poland.

³Aix Marseille Université, CNRS, IM2NP UMR 7334, 13397, Marseille, France.

## ABSTRACT

Organometallic monolayers of the form $TM_3C_6O_6$ where $TM$ is any possible transition metal are a very promising material class. Some members of this family with 3d transition metals had been synthesized on noble metal surfaces and we extend here the study to two metals, Tungsten (W) and Rhenium (Re), of the 5d series by the use of ab-initio calculations. One of the key findings is the discovery of spin-crossover behavior in both monolayers. Especially interesting is the spin-crossover behavior of $Re_3C_6O_6$ which exhibits a transition from intermediate-spin (semiconducting) to high-spin (metallic) states, leading to a significant change in the bandgap, affecting its electrical behavior. In $W_3C_6O_6$, a small gap in the electronic band structure has been observed. Furthermore, a remarkable high magnetic anisotropy energy has been found in these organometallics with 5d transition metals. Besides the ab-initio energy differences, also the parameters of a Heisenberg model with local anisotropy energies have been determined. The presence of high magnetic anisotropy energies in these organometallics makes them beneficial for potential technological applications, particularly in magnetic storage, spintronics, and other magnetic-based technologies. The high magnetic anisotropy also holds promise for quantum computing, where stable and robust magnetic qubits are essential for information storage and manipulation. Additionally, these materials with high magnetic anisotropy energies are of great interest in fundamental research as they can provide valuable insights into the underlying magnetic interactions and electronic structure of the compounds. In summary, our
study highlights the fascinating properties of TM₃C₆O₆ organometallics, with implications for future technologies, quantum computing, and fundamental research in the field of magnetic materials. Furthermore, the discovery of spin-crossover behavior of these materials opens up exciting possibilities for their use in future technologies and epitaxy on different substrates.

## INTRODUCTION

The kagome Heisenberg antiferromagnet¹⁻⁸ and spin-crossover materials are two captivating research areas in condensed matter physics.⁹⁻¹⁴ So, spin-crossover materials demonstrate reversible electronic spin-state switching under external stimuli, holding both fundamental and practical significance. The kagome Heisenberg antiferromagnet reveals the complexities of frustrated magnetism and fractionalized excitations, with potential applications in topological quantum computation.¹⁵,¹⁶ The kagome lattice's peculiarities, including frustration, the existence of Dirac points¹⁷,¹⁸ and flat excitation branches, make it a fascinating system to study exotic electronic behavior and investigate topological phases, particularly when considering spin-orbit coupling.¹⁷ Furthermore, the kagome lattice's unique structure allows for local excitations of magnetic or electronic nature, resulting in dispersionless (flat) excitation branches in the electronic band structure. Flat-band systems have drawn significant attention in various areas of physics due to the rich and unusual phenomena associated with their unique electronic properties.¹⁹,²⁰ The electronic and transport properties of kagome lattices, especially in the presence of spin-orbit coupling and magnetism, have opened up exciting possibilities for exploring topological quantum materials and exotic electronic states.²¹⁻²⁴ Spin-crossover materials provide insights into spin-state switching mechanisms and have promising applications in molecular devices and technologies.²⁵⁻³¹

Metal-organic frameworks (MOFs) are indeed a fascinating class of porous materials with highly ordered structures, consisting of metal ions or clusters coordinated with organic ligands.
Combining kagome lattices with MOFs is an exciting research direction that leverages the unique properties of kagome lattices, which lead to frustration and exotic magnetic behavior in magnetic systems. By incorporating the kagome lattice structure into MOFs, one can explore the interplay between structural and magnetic properties, opening up new avenues for understanding and harnessing emergent quantum phenomena. This area of research holds promise for applications in quantum information technologies and the development of novel materials with tailored electronic and magnetic properties.

While a considerable number of topological quantum materials have been discovered and examined within the realm of inorganic layered crystals, progress in the exploration of organometallic lattices has been comparatively limited, despite existing proposals ³². Nevertheless, the on-surface coordination reaction on noble-metal surfaces has led to the recent synthesis of kagome systems such as Mn₃C₆O₆, Cu₃C₆O₆, and Fe₃C₆O₆³³⁻³⁵. These metal-organic coordination networks are comprised of C₆O₆ rings and form a dense kagome lattice featuring 3d transition metal (TM) ions. In a previous theoretical study, we extended the search of exotic systems to all 3d TM ions going from Sc to Cu.³⁶ In this study, we focus on investigating the properties of freestanding monolayers for W₃C₆O₆ and Re₃C₆O₆, providing valuable insights into their geometric, electronic, and magnetic characteristics. Our primary focus is on exploring the geometric structures of the kagome 2D c-MOF, aiming to comprehend the bonding between metal centers and organic ligands. Additionally, we investigate spin-crossover phenomena and magnetic properties to uncover potential magnetic ordering or spin-liquid behavior. Employing Density Functional Theory with a Hubbard term correction (SGGA+U) in our calculations, with and without spin-orbit coupling, allows for precise examination of the electronic and magnetic properties of W₃C₆O₆ and Re₃C₆O₆, offering valuable insights into their specific characteristics and potential applications. Our research uncovers essential information regarding the bonding between metal centers and organic
ligands in the kagome 2D c-MOF, shedding light on its distinctive electronic properties. The analysis of the electronic band structure enhances our understanding of its electronic behavior. Additionally, the study of spin-crossover behavior provides valuable information on reversible transitions between different spin states, holding significance for potential applications in electronics and spintronics. The investigation into magnetic properties using DFT provides insights into the presence of magnetic ordering or possible spin-liquid behavior, characteristic of frustrated magnetic systems. Understanding these magnetic properties is crucial for harnessing the material's potential in magnetic-based technologies. The comprehensive DFT study of the kagome 2D c-MOF has furnished valuable information on its structural, electronic, and magnetic properties, positioning it as a versatile candidate for various electronic applications. The exploration of spin-crossover phenomena and magnetic properties in these structures deepens our understanding of the material's behavior, opening new avenues in surface science, nanotechnology, and potential magnetic applications. Our research significantly contributes to the understanding of kagome 2D c-MOF and its potential as an electronic material. The insights gained can pave the way for further research and analysis of kagome MOFs, leading to advancements in materials science and potential applications across various technological fields.
METHOD

First-principles calculations have been performed in the framework of the density functional
theory (DFT) using the projector augmented wave (PAW) method³⁷⁻⁴⁰ as implemented within
VASP⁴¹ with an energy cutoff of 480 eV for the plane-wave expansion. The GW pseudo-
potentials were used. The structures were relaxed until the convergence criteria for the energy
deviations was 10⁻⁷ eV. The Gaussian smearing method was used in these calculations and a
width of 0.01 eV was adopted. The Brillouin zone was determined by a set of 6×6×1 k-points
in the unit cell using the Monkhorst-Pack points.⁴² Exchange and correlation are described
within the generalized gradient approximation using the Perdew-Burke-Ernzerhof (PBE)
parametrization. It is well known that the standard Density Functional Theory (DFT) has
difficulties to describe correctly the electronic structure of transition metal compounds. For that
reason, we apply here the Spin polarized Generalized-Gradient Approximation with Hubbard
term U (SGGA+U). The SGGA+U corrections were introduced by Liechtenstein et al. ⁴³ We
take in the following U = 5 eV and an exchange energy of J = 0.90 eV. The necessity of the
SGGA + U method for metal-organic compounds with transition metal ions is proven by many
examples such as TM-TCNB⁴⁴,⁴⁵ or TM-TCNQ⁴⁶, TM-ZQ²⁵,⁴⁷⁻⁵⁰ and TM₃C₆O₆³⁶,⁵¹ where TM
is a transition metal. We also perform calculations without the Hubbard U correction, i.e. we
apply the SGGA method, to compensate the uncertain knowledge of the U parameter.
# RESULTS AND DISCUSSION

## Structural properties

Ab-initio calculations are used to determine the structural properties and the electronic structure of the TM metals which is analyzed by calculating the band structure and density of states (DOS). The magnetic couplings between the metals were also analyzed. Our initial calculation involves the structural optimization of the $TM_3C_6O_6$ networks. This is achieved by computing the total energy as a function of the lattice constant to determine the fundamental state. The SGGA functional is employed for this purpose, with the Hubbard U correction having a negligible impact on the structural properties. The iteration process continues until convergence is achieved in the total energy calculation. As the Hubbard U term does not influence a nonmagnetic state, the SGGA method (without U) is utilized to compare the lattice constants between the magnetic and nonmagnetic states in Figure 1. For $W_3C_6O_6$, the equilibrium structure exhibits a W–O average distance that varies from 2.05 Å in the low-spin (LS) state ($S=0$) to 2.14 Å in the high-spin (HS) state ($S=2$), along with a 0.21 Å elongation of the W–W distance, ranging from 3.82 Å to 4.03 Å. In the case of $Re_3C_6O_6$, the calculated average lengths of the equatorial and axial Re–O bonds are 2.10 Å in the intermediate-spin (IS) state, and 2.03 Å and 2.66 Å, respectively, in the high-spin (HS) state (See Figure 1 and Table 1). We predict the $W_3C_6O_6$ and $Re_3C_6O_6$ to be two-dimensional spin crossover compounds (with a spin transition). For the $W_3C_6O_6$ free-standing structures we find two local minima with different spin states: a low-spin (LS) state $S=0$ per W and the equilibrium lattice constant of 7.64 Å, and a high-spin state $S=2$ per W by variation of the lattice constant. The high-spin state has an energy minimum for a lattice constant of 8.05 Å. For the $Re_3C_6O_6$ structure, we find two possible spin states, namely an intermediate-spin (IS) state ($S=3/2$) with a lattice constant of 7.96 Å and a high-spin state ($S=5/2$ per Re) with a lattice constant of 8.57 Å. Spin crossover complexes can be switched between low-spin (LS) (or IS) and high-spin (HS) states.
magnetic states. This work also provides more insight into the potential coordination preferences of Rhenium ions ($S = 3/2$ vs. $S = 5/2$) based on their spin states and the structural flexibility of the $C_6O_6$ rings within the crystal lattice. The crystal structure of $Re_3C_6O_6$ consists of an isolated layer of the compound and this layer has different configurations depending on the lattice constant, which affect the rotation of the carbon-oxygen ($C_6O_6$) rings. Importantly, the kagome lattice for the metal ions (Re) remains fixed, and the oxygen ions are tightly bound to the $C_6$ rings, so they will rotate along with the carbon ring. The rotation of the carbon-oxygen rings can enable the adjustment of the magnetic moment environment around the Re ions and this suggests that the spin states of Re ions ($S = 3/2$ vs. $S = 5/2$) may influence their preferred coordination geometry. Based on the calculations we performed for $Re_3C_6O_6$ ($S = 3/2$) and $Re_3C_6O_6$ ($S = 5/2$), we found that $Re_3C_6O_6$ ($S = 5/2$) prefers the coordination with just two oxygen neighbors over a configuration with four ($S = 3/2$). These proposed coordination preferences are influenced by the spin states of Re ions, which influences the magnetic behavior and electronic structure of the compound. It is important to note that these results are highly important and likely rooted in materials for Nano Technology-Spintronics. The behavior of $Re_3C_6O_6$ can indeed be complex and influenced by various factors, including crystal structure, spin states, and the interactions between different elements within the material. Experimental investigations would be valuable in confirming these theoretical calculations and understanding the relationship between spin states, coordination environments, and magnetic properties in $Re_3C_6O_6$.
<figure><img src="image_3.png" /><figcaption>**Figure 1:** Total energy versus lattice constant for the freestanding $TM_3C_6O_6$ structures calculated with the SGGA and without (GGA) magnetic state *S* of TM atom.</figcaption></figure>

## Electronic and Magnetic properties

The charge density represents the distribution of electric charge in a material. In the context of $TM_3C_6O_6$ monolayers, the charge density shown in Figure 2 is likely obtained by summing the electron densities for spin up and spin down states. It provides information about the spatial distribution of the electrons in the *x* - *y* plane. The spin density, shown in Figure 3, is defined as the difference between the electron densities for spin-up and spin-down states. It provides insights into the magnetic properties of the material. Positive spin density indicates an excess of spin-up electrons, while negative spin density indicates an excess of spin-down electrons.

The study of the spin density is important in understanding the magnetic behavior of $TM_3C_6O_6$ monolayers. The analysis of charge and spin densities in $TM_3C_6O_6$ monolayers is a significant step in characterizing the electronic and magnetic properties of the material. The magnetization is predominantly concentrated on the metal sites within the structures of $W_3C_6O_6$ and $Re_3C_6O_6$. In addition to the magnetization on the metal sites, which build a kagome lattice, some magnetic moments also appear at the oxygen sites that bridge the metal atoms. These moments on the
oxygen sites contribute to the overall magnetic behavior of the materials. The observation of a kagome lattice generated by the magnetic moments at the metal sites is significant, as it can lead to the exploration of unique electronic and magnetic phenomena in these materials. Understanding the magnetization distribution is crucial for characterizing the magnetic properties and potential applications of W<sub>3</sub>C<sub>6</sub>O<sub>6</sub> and Re<sub>3</sub>C<sub>6</sub>O<sub>6</sub> in various technological fields, including quantum computing, spintronics, and magnetic fields.

<figure><img src="image_4.png" /><figcaption>**Figure 2:** (a, b) Geometrical structure and (c and d) charge density of a free-standing structure of W and Re atoms and C<sub>6</sub>O<sub>6</sub> molecule (C: grey, O: red, and TM: bleu).</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 3: Cut plane of the spin density contribution with isovalue of ±0.02 of W₃C₆O₆ and Re₃C₆O₆. The spin surface is shown in the x-y plane like the atomic structure.</figcaption></figure>

To describe the magnetic moments at the TM sites which build a kagome lattice, we calculated the exchange couplings $J_1$ and $J_2$ to nearest and second nearest neighbors on the kagome lattice in the Heisenberg Hamiltonian

$$H_H = \sum_{\langle i,j \rangle} J_{ij} \vec{S}_i \cdot \vec{S}_j = \frac{1}{2} \sum_{i,j} J_{ij} \vec{S}_i \cdot \vec{S}_j = \frac{1}{2} \sum_{r,g} J_{rg} \vec{S}_r \cdot \vec{S}_{r+g} \quad (1)$$

where $\langle i,j \rangle$ means that we sum over each bond only once. We defined the exchange energy by the energy difference of the 2D structures between the ferromagnetic (FM) arrangement and the antiferromagnetic (AFM) ones after relaxation of all atomic positions to calculate $E_{ex} = E_{AF} - E_{FM}$ which we define per magnetic ion. If the exchange energy is positive (negative), it indicates that the ground state of the system is ferromagnetic (antiferromagnetic). For a
ferromagnetic arrangement of the spins $S$ on the kagome lattice we obtain $E_{FM} = 2(J_1 + J_2)S^2$
whereas the two antiferromagnetic arrangements³⁶ give $E_{AF1} = \frac{2}{3}(J_1 - J_2)S^2$, and $E_{AF2} =
-\frac{2}{3}(J_1 + J_2)S^2$. Taking the two exchange energies $E_{ex,1} = E_{AF1} - E_{FM}$ and $E_{ex,2} = E_{AF2} - E_{FM}$
(see Table 1) in our study of $W_3C_6O_6$ and $Re_3C_6O_6$, we found antiferromagnetic exchange
couplings with the SGGA+U method. The exchange coupling $J_1$ represents the interaction
strength between neighboring magnetic moments and is a key parameter in describing the
magnetic ordering and stability of the material. Using the SGGA+U method, we found a large
value of $J_1 = 3.7$ meV for $W_3C_6O_6$ with a high-spin (HS) state. This indicates a strong
antiferromagnetic interaction between the magnetic moments of the material. For $Re_3C_6O_6$, the
SGGA+U method predicted two different $J_1$ values, one for the intermediate-spin (IS) state and
another for the high-spin (HS) state. The calculated $J_1$ values were 2.4 meV for $S = 3/2$ (IS) and
1.3 meV for $S = 5/2$ (HS). The fact that the SGGA+U values of exchange couplings are
confirmed by the SGGA calculations is remarkable. This suggests that the Hubbard U
correction used in the SGGA+U method is appropriate for capturing the electron correlation
effects in these magnetic materials. The agreement between the two methods reinforces the
reliability of the SGGA+U approach in describing the magnetic properties of $W_3C_6O_6$ and
$Re_3C_6O_6$. It also indicates that the strong antiferromagnetic interactions between the magnetic
moments are well-captured by the SGGA+U method and are consistent with the intrinsic
electronic and magnetic behaviors of these materials. This information is valuable in
understanding the magnetic behavior and stability of $W_3C_6O_6$ and $Re_3C_6O_6$, and it can provide
guidance for further studies and potential applications of these materials in magnetic field and
technologies.
<table><caption>**Table 1.** Distance between the TM atoms and the O atoms ($d_{TM-O}$), lattice constant (a), exchange energies ($E_{ex-1/2}=E_{AFM (1/2)}-E_{FM}$ per TM atom), exchange coupling constants $J_1$ and $J_2$ as explained in the text, total magnetic moments (M per TM atom), local magnetic moments of the d orbital at the TM atoms ($M_d$ per TM atom), local magnetic moments at the TM atoms ($M_m$ per TM atom), energy band gaps (spin up ($E_a$) and spin down ($E_b$)), and overall energy gaps ($E_g$) for two-dimensional (2D) $TM_3C_6O_6$.</caption><thead><tr><th rowspan="2"></th><th colspan="2">W<sub>3</sub>C<sub>6</sub>O<sub>6</sub><br>High-spin state (HS)</th><th colspan="2">Re<sub>3</sub>C<sub>6</sub>O<sub>6</sub><br>Intermediate-spin(IS)</th><th colspan="2">Re<sub>3</sub>C<sub>6</sub>O<sub>6</sub><br>High-spin state (HS)</th></tr><tr><th>Without<br>U</th><th>U= 5eV</th><th>Without U</th><th>U= 5eV</th><th>Without U</th><th>U= 5eV</th></tr></thead><tbody><tr><td>d<sub>TM-O</sub> [Å]</td><td>2.14</td><td>2.16</td><td>2.10</td><td>2.12</td><td>2.03 ; 2.66</td><td>2.04 ; 2.60</td></tr><tr><td>a [Å]</td><td>8.05</td><td>8.13</td><td>7.96</td><td>8.00</td><td>8.57</td><td>8.54</td></tr><tr><td>E<sub>ex-1</sub> [meV]</td><td>-11.55</td><td>-22.11</td><td>-63.87</td><td>-21.63</td><td>-1.21</td><td>-18.37</td></tr><tr><td>E<sub>ex-2</sub> [meV]</td><td>-55.23</td><td>-41.91</td><td>-116.17</td><td>-41.38</td><td>-46.83</td><td>-29.22</td></tr><tr><td>J<sub>1</sub> [meV]</td><td>8.19</td><td>3.713</td><td>6.276</td><td>2.370</td><td>5.474</td><td>1.302</td></tr><tr><td>J<sub>2</sub> [meV]</td><td>-3.012</td><td>0.216</td><td>0.694</td><td>0.113</td><td>-2.665</td><td>0.451</td></tr><tr><td>M [μ<sub>B</sub>]</td><td>3.65</td><td>4</td><td>3</td><td>3</td><td>4.19</td><td>5</td></tr><tr><td>M<sub>d</sub> [μ<sub>B</sub>]</td><td>2.42</td><td>2.67</td><td>2.24</td><td>2.27</td><td>3.02</td><td>3.48</td></tr><tr><td>M<sub>m</sub> [μ<sub>B</sub>]</td><td>2.72</td><td>3.01</td><td>2.29</td><td>2.39</td><td>3.50</td><td>4.04</td></tr><tr><td>E<sub>a</sub> [eV]</td><td>0</td><td>1.45</td><td>0</td><td>1.89</td><td>0</td><td>0</td></tr><tr><td>E<sub>b</sub> [eV]</td><td>0</td><td>1.71</td><td>0</td><td>1.24</td><td>0</td><td>0</td></tr><tr><td>E<sub>g</sub> [eV]</td><td>0</td><td>0.37</td><td>0</td><td>1.24</td><td>0</td><td>0</td></tr></tbody></table>

To understand the electronic and magnetic properties of the $Re_3C_6O_6$ and $W_3C_6O_6$ MOF structures, we studied the DOS and PDOS in detail (See Figures 4 and 5). We first performed DFT SGGA calculations, which usually can provide very fundamental information about the electronic properties of the system. But the missing of a localized spin state is a known failure of the GGA due to insufficient treatment of the strong correlation effect. A common remedy is to apply the SGGA+U method to properly treat the localized d electrons. Our SGGA+U (U=5 eV, J=0.9 eV) calculations show clearly isolated spin-up and spin-down bands separated by the Fermi level, consistent with the physical picture of localized S = 2 spins for the $W_3C_6O_6$ structure.

The electronic band structure of $W_3C_6O_6$ in the low-spin state (S = 0) shows a small bandgap of approximately 0.10 eV. The band gap increases when $W_3C_6O_6$ is in the high-spin state (S =
2), the electronic band structure shows a slight increase of the bandgap by approximately 0.30 eV. This means that the energy difference between the valence and conduction bands is larger compared to the low-spin state. The increased bandgap in the high-spin state suggests that $W_3C_6O_6$ may exhibit improved semiconductor behavior. A larger bandgap means higher electron energies to be required for excitation, making the material potentially more suitable for applications in high-performance electronics and optoelectronic. The presence of a small bandgap in both the low-spin and high-spin states indicates that $W_3C_6O_6$ possesses semiconductor-like behavior. This property opens up exciting possibilities for its use in electronics, where the control of electron flow and bandgap engineering are crucial for device performance. The spin-dependent bandgap variations in $W_3C_6O_6$ suggest that the electronic properties of the material can be modulated by tuning its spin states. Such tunability may enable the design of spintronics where the magnetic and electronic properties can be controlled simultaneously. Overall, the electronic band structure of $W_3C_6O_6$ with its small and tunable bandgap, along with the influence of spin states, makes it a promising candidate for future electronic and spintronic applications. Since the research in the field of material science advances, the understanding of the properties and potential applications of $W_3C_6O_6$, with its small gap of almost 0.40 eV, may continue to expand, opening up new avenues for its use in cutting-edge electronic technologies.

Certain transistors, especially those in high-frequency and power amplifiers, employ narrow-bandgap semiconductors. The reduced bandgap characteristics of these materials facilitates higher electron mobility, resulting in faster switching speeds and improved overall performance in electronic systems. LEDs designed to emit light in the near-infrared range utilize specific narrow-bandgap semiconductors. In the realm of spintronics, which harnesses both electron spin and charge for information processing, these materials play a crucial role due to their distinctive electronic properties. Moreover, they prove valuable in quantum technology,
especially in contexts requiring precise control of electronic states, such as quantum computing and other technologies.

For Re₃C₆O₆ with intermediate-spin state (S = 3/2), the electronic band structure displays a bandgap of approximately 1.24 eV. This bandgap indicates that the material can act as a semiconductor, as it has a range of energies where no electronic states are available. Semiconductors can have applications in electronics, as their conductivity can be controlled by doping or applying an external electric field. On the other hand, when Re₃C₆O₆ is in a high-spin value (S = 5/2), the electronic band structure shows the absence of a bandgap at the Fermi level. In other words, the material exhibits metallic behavior due to the availability of electronic states at the Fermi level. This indicates that Re₃C₆O₆ behaves more like a metal, with good electrical conductivity. Especially remarkable, the results for the Re₃C₆O₆ structure for two possible spin states, namely an intermediate-spin state (S = 3/2) and a high-spin state (S = 5/2) which are summarized in Figures 4 and 5, show that biaxial tensile strain leads to a transition from intermediate-spin to high-spin state. At the critical tensile strain, the gap closes, leading to a degeneracy at the gamma point. We further calculated the strained system by turning on the SOC interaction (see Figure 7) and the discovery of two Dirac cones close to the K point in Re₃C₆O₆ is a significant finding. Dirac cones are a unique feature in the electronic band structure of certain materials and represent a fascinating aspect of condensed matter physics and materials science, offering a rich playground for exploring novel physical phenomena.
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>
<figure><img src="image_12.png" /><figcaption>**Figure 4:** Band-structures and total DOS of the 2D systems built of transition-metal (W and Re) atoms and the C<sub>6</sub>O<sub>6</sub> molecule with the PAW-SGGA+U method.</figcaption></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>
<figure><img src="image_17.png" /><figcaption>**Figure 5:** Projected DOS of the atoms (TM, C and O) and d orbitals on the TM atoms in the 2D $TM_3C_6O_6$ materials with the PAW-SGGA+U method.</figcaption></figure>

Magnetic anisotropy energy (MAE) is a crucial property in magnetic materials, particularly in ferromagnetic and ferrimagnetic systems. It refers to the energy required to change the orientation of the magnetization vector from one direction to another within the material. In simpler terms, it measures the material's preference for aligning its magnetic moments in a particular direction. The concept of magnetic anisotropy arises from the crystal symmetry of a material. Crystals can exhibit different degrees of symmetry along different crystallographic directions. This anisotropy in the structure influences the magnetic behavior of the material. High magnetic anisotropy energy (MAE) is essential for maintaining a stable magnetization in
materials, even when exposed to external magnetic fields or varying temperatures. This stability
is critical for ensuring reliable performance in applications like magnetic storage. Particularly,
materials with high MAE are sought after in data storage such as hard disk drives and magnetic
memory, enhancing data storage robustness and retention.

In magnetic sensors, the use of high MAE materials becomes valuable for precise detection
of magnetic fields. The stable magnetization direction ensures accurate and consistent responses
from the sensors. In the realm of spintronics, where electron spin is harnessed for information
processing and storage, high MAE materials serve as crucial components. They act as "spin
filters" and "spin valves," exerting control over the flow of spin-polarized electrons. For
nanoscale magnetic, the importance of high MAE materials cannot be overstated. These
materials contribute to the creation of stable magnetic bits, minimizing energy dissipation in
miniaturized devices. Furthermore, high MAE materials play a pivotal role in achieving
increased magnetic recording density in storage media, thereby enabling higher information
storage capacity. In our research on W<sub>3</sub>C<sub>6</sub>O<sub>6</sub> and Re<sub>3</sub>C<sub>6</sub>O<sub>6</sub> (see Table 2), understanding the
magnetic anisotropy energy can provide insights into their magnetic properties, stability of
magnetic order, and potential applications in magnetic-based technologies.

The orbital magnetic moments, as well as the magnetic anisotropy energy (MAE), of
TM<sub>3</sub>C<sub>6</sub>O<sub>6</sub> are systematically studied here by using the spin-orbit coupling (SOC) and the effect
of SOC was invoked into the self-consistent implementation of the density-functional theory
together with an explicit inclusion of the Hubbard U term (DFT+U). The MAEs, i.e. the energy
differences E<sub>x</sub> - E<sub>z</sub> and E<sub>x</sub> - E<sub>y</sub> are presented in Table 2. The MAE and the orbital moment
(<span style="text-decoration: overline">M</span><sub>L</sub>) are intimately connected <sup>52-54</sup>. However, the values of orbital moments for W and Re,
which are given in Table 2, remain much smaller than the spin moments due to the crystal field
influence being more important than spin-orbit coupling for these 5d ions in the kagome lattice.
That leads to the quenching of the orbital moments.
The MAE can also be cast into an anisotropy term $H_{aniso}$ to be added to the Heisenberg Hamiltonian (Eqn. (1)) to give the complete Hamiltonian in the form $H = H_H + H_{aniso}$. Due to the quenching of the orbital moments, we are justified to formulate the anisotropy part of the Hamiltonian in terms of local spins $\vec{S}_i$, which are connected with the local spin moments by $\vec{M}_{S,i} = 2\mu_B \vec{S}_i$. Taking into account the symmetries of the planar kagome lattice we can write

$$H_{aniso} = -\frac{K_z}{4} \sum_i (S_{iz}^2 - (S_{ix}^2 + S_{iy}^2)) - \frac{K_{hexa}}{2} \sum_i \sum_{j=A,B,C} (\vec{S}_i \cdot \vec{e}_j)^6, \quad (2)$$

where $K_z$ and $K_{hexa}$ are two anisotropy parameters corresponding to the planar and hexagonal symmetry elements, and the three unit vectors $\vec{e}_A = \vec{e}_x$, $\vec{e}_B = -(1/2)\vec{e}_x + (\sqrt{3}/2)\vec{e}_y$, and $\vec{e}_C = -(1/2)\vec{e}_x - (\sqrt{3}/2)\vec{e}_y$ give the three symmetry axes of the lattice. The Hamiltonian (2) describes the simplest form of MAE to be compatible with the lattice symmetry and we are going to check how well the ab-initio results follow that analytical form. The two anisotropy parameters can be determined with the ab-initio results for a ferromagnetic arrangement of spins along the three Cartesian axes $x$, $y$, and $z$ (see Table 2). More generally, the spin can also point into an arbitrary direction which is given in spherical coordinates: $S_{ix} = S \sin \theta \cos \varphi$, $S_{iy} = S \sin \theta \sin \varphi$, $S_{iz} = S \cos \theta$. With that replacement, the anisotropy energy per TM ion becomes

$$E_{aniso} = E_z + \left(\frac{S^2 K_z}{2}\right) \sin^2 \theta - \left(\frac{32 S^6 K_{hexa}}{3}\right) \sin^6 \theta \left(\frac{9}{2} + \cos^2 (3\varphi)\right). \quad (3)$$

That formula allows to calculate the anisotropy parameter $K_z$ and $K_{hexa}$ from the energy differences $E_x - E_z$ and $E_x - E_y$, and they are given in Table 2. However, as can be seen in Figure 6, the energy variation with the angles $\varphi$ and $\theta$ is in reality more complicated than the simple form (3). As an example, we consider the energy variation with $\varphi$ in the $x-y$ plane for $W_3C_6O_6$ without U correction.
A giant easy-axis behavior parallel to the z axis can be obtained for W₃C₆O₆, meaning that it has a large positive MAE of 48.61 meV for Eₓ - E_z. Correspondingly, the orbital magnetic moments are M_L(x) = M_L(y) = -0.144 µ_B and M_L(z) = -0.210 µ_B. This substantial MAE, characterized by a large positive value of K_z and small K_{hexa}, suggests that the material has a highly stable magnetic easy axis (z direction), making it very resistant to flipping its magnetization direction even under external influences. Different, but very instructive, is the case of Re₃C₆O₆ with a remarkably high easy-plane (x-y) anisotropy of -16.7 meV/atom (Eₓ - E_z) for S = 5/2 (HS) in SGGA+U+SOC. Without the U correction, the MAE values increase to -133.37 meV/atom. For the HS state of Re₃C₆O₆, the energy varies a little when the magnetic moment rotates in the x - y plane indicated by a small K_{hexa} value. As a rule, the amplitudes of M_L(x) and M_L(y) are larger than those of M_L(z) for cases with easy-plane behavior where the magnetization prefers to lie in the plane perpendicular to the z axis, i.e., negative MAE. Also, the d shell is only partially filled which allows an orbital current in the x - y plane and we find orbital moments of M_L(x) = M_L(y) = 0.095 µ_B in the plane, which are larger in amplitude than M_L(z) = -0.020 µ_B. Even more impressive are the orbital moments for the IS state of Re₃C₆O₆. The large K_{hexa} value in that case indicates that the energy variation in the plane is much larger than between in-plane and out-of-plane magnetization directions.
<table><caption>**Table 2:** Magnetic anisotropy energy (MAE, per TM atom in meV), total magnetic moments $M_i$, local magnetic moments of the d orbitals at the TM site $M_d(i)$ and orbital moments ($M_L(i)$) per TM atom in $\mu_B$) for $TM_3C_6O_6$ (without substrate). The data result from SGGA-PBE and SGGA+U calculations including SO coupling with the magnetic moments directed towards the $i=\{x,y,z\}$ direction.</caption><thead><tr><th rowspan="2"></th><th colspan="2">W<sub>3</sub>C<sub>6</sub>O<sub>6</sub><br>High-spin state (HS)</th><th colspan="2">Re<sub>3</sub>C<sub>6</sub>O<sub>6</sub><br>Intermediate-spin(IS)</th><th colspan="2">Re<sub>3</sub>C<sub>6</sub>O<sub>6</sub><br>High-spin state (HS)</th></tr><tr><th>Without<br>U</th><th>U= 5eV</th><th>Without<br>U</th><th>U= 5eV</th><th>Without<br>U</th><th>U= 5eV</th></tr></thead><tbody><tr><td>(E<sub>x</sub>-E<sub>z</sub>) [meV]</td><td>45.13</td><td>48.61</td><td>-85.23</td><td>-42.17</td><td>-133.37</td><td>-16.67</td></tr><tr><td>(E<sub>x</sub>-E<sub>y</sub>) [meV]</td><td>1.91</td><td>2.59</td><td>-74.79</td><td>-32.09</td><td>19.907</td><td>0.66</td></tr><tr><td>S</td><td>2</td><td>2</td><td>3/2</td><td>3/2</td><td>5/2</td><td>5/2</td></tr><tr><td>M<sub>x</sub> [μ<sub>B</sub>]</td><td>3.50</td><td>3.87</td><td>1.30</td><td>2.17</td><td>3.35</td><td>3.70</td></tr><tr><td>M<sub>y</sub> [μ<sub>B</sub>]</td><td>3.51</td><td>3.81</td><td>2.47</td><td>2.32</td><td>3.34</td><td>3.69</td></tr><tr><td>M<sub>z</sub> [μ<sub>B</sub>]</td><td>3.50</td><td>3.74</td><td>2.61</td><td>2.54</td><td>3.47</td><td>3.73</td></tr><tr><td>M<sub>d</sub>(x) [μ<sub>B</sub>]</td><td>2.30</td><td>2.56</td><td>1.89</td><td>1.59</td><td>2.86</td><td>3.14</td></tr><tr><td>M<sub>d</sub>(y) [μ<sub>B</sub>]</td><td>2.30</td><td>2.51</td><td>1.60</td><td>1.68</td><td>2.88</td><td>3.13</td></tr><tr><td>M<sub>d</sub>(z) [μ<sub>B</sub>]</td><td>2.31</td><td>2.49</td><td>1.91</td><td>1.96</td><td>2.95</td><td>3.18</td></tr><tr><td>M<sub>L</sub>(x) [μ<sub>B</sub>]</td><td>-0.102</td><td>-0.144</td><td>0.250</td><td>0.310</td><td>0.124</td><td>0.095</td></tr><tr><td>M<sub>L</sub>(y) [μ<sub>B</sub>]</td><td>-0.098</td><td>-0.144</td><td>0.393</td><td>0.314</td><td>0.133</td><td>0.097</td></tr><tr><td>M<sub>L</sub>(z) [μ<sub>B</sub>]</td><td>-0.222</td><td>-0.210</td><td>-0.064</td><td>-0.068</td><td>-0.025</td><td>-0.020</td></tr><tr><td>K<sub>z</sub> [meV]</td><td>17.31</td><td>17.21</td><td>289.89</td><td>134.30</td><td>-77.74</td><td>-6.50</td></tr><tr><td>K<sub>hexa</sub> [meV]</td><td>-0.32</td><td>-0.43</td><td>70.04</td><td>30.03</td><td>-0.87</td><td>-0.03</td></tr></tbody></table>

<figure><img src="image_18.png" /><figcaption>**Figure 6:** Comparing the magnetic anisotropy energy (MAE) of $W_3C_6O_6$ with respect to $\phi$ in the x-y plane between the analytical formula and the ab-initio results.</figcaption></figure>
Having calculated the exchange couplings and the MAE does not mean that we have already determined the magnetic state. That can be done by minimizing the energy of the total Hamiltonian. In a first step we can treat each spin as a classical vector which is especially justified for large spins like $S = 5/2$ or 2, but should also be an acceptable approximation for $S = 3/2$. In a second step one could try to calculate the influence of quantum corrections. Since the magnetic anisotropy energy prefers the three directions of the kagome lattice, it is highly probable that the magnetic state will be collinear. That is especially true for high values of $K_{hexa}$. It should be remarked that the antiferromagnetic Heisenberg model on the kagome lattice allows in principle also a spin liquid state, but only for $S = 1/2$ or $S = 1$ and without MAE.

<figure><img src="image_19.png" /><figcaption>**Figure 7:** Band-structure including spin orbit coupling of the Re$_3$C$_6$O$_6$ network with the PAW-SGGA+U method.</figcaption></figure>
CONCLUSIONS

This research findings on W₃C₆O₆ and Re₃C₆O₆ organometallics are highly intriguing and hold great potential for various technological and scientific applications. Ab-initio calculations were employed to determine the structural properties and electronic structure of the TM₃C₆O₆ systems. Notably, both W₃C₆O₆ and Re₃C₆O₆ exhibited intriguing two-dimensional spin crossover behavior, a phenomenon wherein they transition between low-spin or intermediate-spin states to high-spin states per transition metal element.

In the case of W₃C₆O₆, it was identified as a spin-2 kagome-lattice antiferromagnetic semiconductor with a distinctive easy axis anisotropy perpendicular to the x-y plane. This property makes it an ideal candidate for constructing two-dimensional spin-valves, crucial components in the field of spintronics. On the other hand, Re₃C₆O₆ displayed a transition from an intermediate-spin state (S = 3/2) to a high-spin state (S = 5/2), accompanied by a structural change involving the rotation of C₆O₆ hexagons while maintaining the kagome lattice structure. This transition significantly impacted the bandgap of Re₃C₆O₆, rendering it either metallic or a gapless semiconductor, thus offering versatility for electronic applications.

Furthermore, both W₃C₆O₆ and Re₃C₆O₆ exhibited remarkably high magnetic anisotropy energies (MAE), a characteristic that holds immense promise for various technological applications. In particular, materials with high MAE are essential for magnetic storage devices such as hard drives, as they ensure stable and reliable data storage by maintaining robust magnetic orientations. Moreover, in the realm of spintronics, these materials could be instrumental in developing efficient and low-power electronic devices. Additionally, their high MAE makes them valuable candidates for magnetic qubits in quantum computing, facilitating information storage and manipulation in quantum systems.

In conclusion, the exploration of the unique properties of W₃C₆O₆ and Re₃C₆O₆ organometallics unveils exciting opportunities for advancements in technology, particularly in the fields of
spintronics, magnetic storage, and quantum computing. Continued research in this area holds
the promise of unlocking the full potential of these materials and their magnetic properties,
paving the way for transformative developments in various scientific and technological
domains.

## AUTHOR INFORMATION

### Corresponding Author

*E-mail: hassan.denawi@polytechnique.edu (A.H.D.)

## ACKNOWLEDGEMENTS

This work was supported by the Computer resources of the Centre Informatique National de l'Enseignement Supérieur (CINES), Project No. A0020906873, the High Performance Computing (HPC) resources of Aix-Marseille University financed by the project Equip@Meso (ANR-10-EQPX-29-01) and the HPC centers of IDRIS and TGCC (Grant 90642) are gratefully acknowledged for computational resources and the dedicated and efficient help of the members of their assistance teams. We thank M. Abel, and M. Koudia for helpful discussions.

## Conflict of interest

The authors declare that they have no conflict of interest.

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**Declaration of interests**

[x] The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

[ ] The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
**Authorship contribution statement**

**Adam Hassan Denawi:** Writing - original draft, Writing - review & editing, Conceptualization, Methodology, Investigation.
**Holger Vach, Kausik Das, Vincent Oison and Roland Hayn:** Review & editing.