Magnetic anisotropy of 4f atoms on a WSe₂ monolayer: a DFT + U study

Abstract

Inspired by recent advancements in the field of single-atom magnets, particularly those involving rare-earth (RE) elements, we present a theoretical exploration employing DFT+U calculations to investigate the magnetic properties of selected 4f atoms, specifically Eu, Gd, and Ho, on a monolayer of the transition-metal dichalcogenide WSe₂ in the 1H-phase. This study comparatively examines RE with diverse 4f orbital fillings and valence chemistry, aiming to understand how different coverage densities atop WSe₂ affect magnetocrystalline anisotropy. We observe that RE lacking 5d occupation exhibit larger magnetic anisotropy energies at high densities, while those with outer 5d electrons show larger anisotropies in dilute configurations. Additionally, even half-filled 4f shell atoms with small orbital magnetic moments can generate substantial energy barriers for magnetization rotation due to prominent orbital hybridizations with WSe₂. Open 4f shell atoms further enhance anisotropy barriers through spin-orbit coupling effects. These aspects are crucial for realizing stable magnetic information units experimentally.

Introduction

The continuous demand for computational power and the continuous generation of increasing amounts of data motivates novel computing paradigms and improved memory capacity in electronic devices. A promising approach for both is the use of single magnetic atoms with large magnetic anisotropies deposited on 2D materials or surfaces^1,2,3,4,5. These magnetic atoms can act as information carriers, with their magnetization aligned in specific preferred directions, energetically separated from other orientations. This property can also give rise to two-level quantum systems^6,7.

In this context, rare-earth (RE) atoms on 2D materials offer intriguing possibilities for stable magnetic units in classical and quantum applications, serving as bits and qubits^8,9,10,11,12. This is attributed to their significant spin and orbital magnetic moments arising from the strongly localized 4f electrons. The selection of RE atoms as a magnetic source is indeed justified by the limited hybridization effects of the 4f electrons, ensuring the generated magnetization is protected and rigid. In fact, the essential requirement for utilizing magnetic atoms as information carriers is a stable magnetization, both in magnitude and orientation, safeguarded against reversal due to thermal or quantum fluctuations as well as environmental factors like scattering with substrate conduction electrons and phonons^{13,14,15,16,17}. Achieving this stability demands substantial energy barriers between different spin states, meaning significant magnetocrystalline anisotropies.

Foundational work has been laid through experimental studies investigating rare-earth atoms on graphene or metallic substrates^{18,19,20,21,22,23,24,25}, employing techniques such as x-ray adsorption spectroscopy, x-ray magnetic circular dichroism, and scanning tunneling microscopy to discern magnetic ground states, excitation processes, and relaxation times^{26,27,28,29,30}, and showing that achieving stable magnetic units in the single-atom limit is possible in some instances, as demonstrated for single Ho atoms adsorbed on MgO³¹. However, predicting which rare-earth atom might yield a substantial magnetic anisotropy on a specific substrate is challenging due to variations in the lanthanide series’ chemical nature and its interactions with the underlying material. To address this issue, we propose a theoretical study based on DFT + U calculations, as implemented within the the full-potential linearized augmented planewave code FLEUR³², following the rotationally invariant formulation introduced by Liechtenstein et al.³³. The objective is to compare the magnetic anisotropy of two different rare-earth representatives from a chemical perspective on a selected 2D material, a transition-metal dichalcogenide (TMDC) monolayer, and to examine variations in magnetic properties with the concentration of the rare-earth atoms adsorbed on the 2D material, providing insights into combinations of rare-earth atoms and coverage density that might be experimentally appealing for applications relying on magnetic stability.

In detail, we conduct a comparative study on the electronic and magnetic properties of three distinct RE atoms, Eu, Gd, and Ho, deposited at two different coverages on a monolayer of 1H-WSe₂, a valleytronic semiconductor, focusing on the ferromagnetic order. The two different adsorption density scenarios include one with a full monolayer coverage of RE atoms in a 1 × 1 unit cell, equivalent to one magnetic atom per substrate unit cell, and the other with a diluted arrangement in a $\sqrt{3}\times \sqrt{3}$ supercell configuration, representing a 1/3 coverage, as depicted in Fig. 1a and b, respectively.

Fig. 1: Structural models of the two simulation cells and differential charge densities.

Rare-earth atoms (depicted as red spheres) adsorbed on the T-W site on the W atoms (grey spheres) of a 1H-WSe₂ monolayer: (a) 1 × 1 unit cell, representing a high coverage scenario, and (b) $\sqrt{3}\times \sqrt{3}$ supercell, illustrating a low coverage scenario of rare-earth atoms. Se atoms are shown in green. The view is from the top. c Differential charge density along the (110) plane of 1 × 1 Eu/WSe₂. d Differential charge density along the (010) and ($1\bar{1}0$) crystallographic planes of $\sqrt{3}\times \sqrt{3}$ Eu/WSe₂. Blue regions denote a loss of charge, and red regions signify a gain in charge. Units in e/bohr³.

The choice of RE atoms includes representatives with different electron configurations: On the one hand, Eu and Gd with half-filled 4f shells (4f⁷), characterized by a small orbital angular momentum, and on the other hand, Ho with an open 4f shell (4f¹¹), which exhibits strong spin-orbit coupling (SOC) effects due to its non-zero orbital angular momentum. This variation in electronic structure affects the shape of the 4f charge cloud³⁴, leading to different interactions with the C_3v crystal field of the substrate, depending on the direction of magnetization. Another aspect of the choice of RE atoms are their valencies. Gd is considered as a special case due to its unique atomic valence d¹ configuration in 4f⁷5d¹ having an additional outer 5d electron, which results in distinct chemical behaviors compared to Eu and Ho, which typically have an atomic configuration of 4fⁿ5d⁰ with n = 7, 11, respectively, lacking the outer 5d electron. Consequently, this investigation sheds light on how to manipulate magnetocrystalline anisotropy by exploiting the chemical properties of the RE atom and its coverage on the 2D material.

The choice of the TMDC 1H-WSe₂ as 2D material finds justifications in its inherent characteristics. This includes its robust SOC induced by the tungsten W atoms and its chemical composition, which is anticipated to result in strong orbital hybridizations with the adsorbed RE atoms. This contrasts with previous theoretical studies^{35,36,37,38,39,40} that focused on the magnetic properties of RE elements on a graphene sheet, where the interaction process involved only the π orbitals of the 2D material, and spin-orbit coupling would arise solely from the RE element. Moreover, the semiconducting 1H phase of WSe₂ is expected to exhibit a low density of states around the Fermi energy, potentially reducing scattering events between the magnetization of the RE atoms and conduction electrons, which is advantageous for the investigation of the quantum spin behavior of RE atoms at low temperatures.

Limited information is available regarding the magnetic characteristics of RE atoms on TMDCs, as previous studies have primarily concentrated on introducing RE defects into TMDC monolayers to modify their electrical properties^41,42. In contrast, our investigation focuses on the adsorption of magnetic RE atoms onto these 2D materials, exploring the interactions within this heterostructure and their impact on magnetocrystalline anisotropy.

To guide the reader through the manuscript more smoothly, we want to emphasize that we have intentionally structured the comparison of rare-earth atoms on a property-by-property basis, as this is the central focus of our work. Accordingly, the manuscript is organized to first discuss the chemical properties related to adsorption, followed by electronic properties, and finally, magnetism. Tables 1 and 2 in the first and second sections are based on the same set of calculations.

Table 1 Adsorption properties for Eu, Gd and Ho on a 1H-WSe₂ monolayer located at the T-W, T-Se and H sites for the 1 × 1 and $\sqrt{3}\times \sqrt{3}$ coverages: adsorption energy in eV, adsorption distance in Å (taken as the perpendicular distance from the first Se layer), and d occupation of the magnetic RE atom measured in the muffin-tin sphere with radius of 2.8 a₀

Results

Adsorption properties

We begin the discussion by examining the adsorption features of the three rare-earth atoms on the 1H-WSe₂ monolayer. A concise discussion on the choice of coverage density in the Eu/WSe₂ system is available in ref. ⁴³, exploring the potential for engineering anomalous Hall conductivity. On the WSe₂ substrate, we explore three distinct adsorption sites for the rare-earth atoms: atop a W atom (T-W), atop a Se atom (T-Se), or within the hexagonal “Hollow” site (H). To determine the adsorption energy, we subtract the total energy of the heterostructure, denoted as ${E}_{{{\rm{RE}}}/{{{\rm{WSe}}}}_{2}}$, from the total energies of the isolated components, represented by ${E}_{{{\rm{RE}}}}$ and ${E}_{{{{\rm{WSe}}}}_{2}}$,

$${E}_{{{\rm{ads}}}}={E}_{{{\rm{RE}}}/{{{\rm{WSe}}}}_{2}}-{E}_{{{\rm{RE}}}}-{E}_{{{{\rm{WSe}}}}_{2}}.$$

(1)

Table 1 presents the calculated adsorption properties in the 1 × 1 and $\sqrt{3}\times \sqrt{3}$ simulation cells, for all three positions. Comparing the adsorption energies of different adsorption sites, it is evident that the preferred position for the RE atoms is atop the W atom in both instances. The adsorption of the RE element on WSe₂ induces a C_3v symmetry at the local position. In addition, the RE atom at the T-W site is closer to the TMDC substrate compared to the T-Se and H sites, likely resulting in a more pronounced crystal field effect and charge transfer.

Starting with the discussion on Eu, in the diluted $\sqrt{3}\times \sqrt{3}$ configuration, the observed larger distances d₀ at the T-Se and H-sites, compared to the T-W site, correlate with a systematic decrease in the 5d occupation of the Eu atom from T-W to H to T-Se. This reduction in d occupation primarily results from decreased interactions with the underlying substrate, and the smaller the interaction, the closer one comes to the atomic divalent limit with the d⁰ configuration. Conversely, in the 1 × 1 unit cell, the d_occ values remain similar across adsorption sites, indicating that they predominantly originate from hybridization with neighboring Eu atoms in the magnetic monolayer and are not influenced by the distance from the substrate. Moreover, by comparison of the equilibrium distances between the two coverages, it is seen that the magnetic dilution only affects the perpendicular distance at the hexagonal H-site, which is reduced by 0.25 Å. Focusing on the T-W site, the most notable distinctions between the high coverage and low coverage scenarios are evident in the d occupation. In the diluted limit, the occupation numbers approach the atomic configuration d⁰ with a reduced d occupation to 0.158 electrons. This mirrors the contributions from neighboring Eu atoms and the WSe₂ monolayer discussed earlier.

In Gd/WSe₂, the 1 × 1 configuration exhibits a larger 5d occupation as compared to the $\sqrt{3}\times \sqrt{3}$ scenario across all adsorption sites. Specifically, at the favored T-W position, it is larger by 0.19 electrons, implying charge transfer from Gd 5d valence shell to the substrate in the dilute limit. This is further reflected by the increased adsorption energy observed when Gd atoms are more diluted, indicating notable hybridization effects involving these electrons. Intriguingly, this enhanced interaction with WSe₂ results in closer proximity, manifested in the considerably reduced equilibrium adsorption distance d₀. Precisely, d₀ measures 2.377 Å in the 1 × 1 coverage, lowered to 2.049 Å in the diluted T-W configuration. These data reveal a clear difference in chemical behavior compared to Eu, where dilution did not significantly affect the adsorption distance.

Examining the adsorption characteristics of Ho atoms on WSe₂, we note a chemical behavior similar to that of Eu. Particularly, in the preferred T-W and T-Se sites, the equilibrium distance from the 2D material remains relatively similar upon dilution of the magnetic atom, while it undergoes changes in the H-site. Regarding the favored T-W adsorption site, the d_occ is larger in the 1 × 1 cell compared to the $\sqrt{3}\times \sqrt{3}$ supercell, suggesting again an almost atomic configuration in the diluted scenario.

Electronic properties

After exploring the adsorption geometries, our focus shifts to the electronic properties of the magnetic 2D heterostructures. Table 2 presents the f occupation, spin (m_s), and orbital (m_l) magnetic moments of the RE atoms in the system, measured in Bohr magnetons. The m_l value is specifically assessed for the favored adsorption site T-W in the case of half-filled 4f shells (Eu and Gd). In the case of Ho, we computed the orbital moment for all three positions for a comprehensive comparison.

Table 2 Ground state properties for Eu, Gd and Ho on a WSe₂ monolayer in the T-W, T-Se and H sites in the 1 × 1 and $\sqrt{3}\times \sqrt{3}$ coverages: f occupation of the magnetic RE atom, spin magnetic moment of the RE atom in μ_B, and orbital magnetic moment in μ_B, all measured in the muffin-tin sphere with radius of 2.8 a₀

Examining Eu/WSe₂, we observe a few general trends: Firstly, irrespective of the Eu density, the spin moments of the Eu atom is larger than the occupancy of the localized 4f electrons. Assuming full spin-polarization of the 4f electrons, the intra-atomic exchange polarizes the primarily 5d valence electrons ferromagnetically. Secondly, reducing the concentration of the Eu atoms, we notice that Eu becomes more atomic like, the f occupancy increases and the d occupancy decreases, the spin moment results more from the 4f electrons and less from the d electrons. Thirdly, We find a small induced orbital moment. The negative sign indicates that it couples antiparallel to the spin moment. We will see below that Gd/WSe₂ exhibits similar behavior, whereas Ho shows different behavior due to its open 4f shell.

Focusing on Eu in the T-W site, we observe that, akin to the earlier-discussed d occupation, the f occupation increasingly mirrors the atomic configuration 4f⁷ when the atom becomes more diluted, increasing the f occupancy from ~6.87 to ~6.92. The distinct atomic behavior is also evident by the fact that in case of the $\sqrt{3}\times \sqrt{3}$ supercell, the total spin magnetic moment comes practically only from the 4f electrons, while the 6.865 unpaired 4f electrons in the 1 × 1 unit cell contribute to a value of ~7.130 μ_B. The additional 0.265 μ_B results from the intra-atomic spin-polarization of the acquired 5d electrons. Conversely, in the $\sqrt{3}\times \sqrt{3}$ system, where the acquired d occupation is only 0.158 electrons, the spin magnetic moment of the RE is larger by merely 0.07 μ_B compared to the pure 4f magnetic moment. Additionally, the orbital magnetic moment undergoes a reduction by one order of magnitude upon dilution of the Eu atom.

As demonstrated in ref. ⁴³, diluting the magnetic Eu atom results in a reduction of magnetic proximity, manifested by a reduced exchange splitting of the energy states. This is qualitatively evident in the differential charge densities depicted in Fig. 1c and d, calculated as $\Delta \rho (r)={\rho }_{{{\rm{Eu}}}/{{\rm{WSe}}}_2}(r)-{\rho }_{{{\rm{Eu}}}}(r)-{\rho }_{{{{\rm{WSe}}}}_{2}}(r)$. Red regions indicate a gain of charge, while blue regions signify a loss of charge. The more intense color spots suggest greater charge involvement in the formation of the heterostructure in the magnetic monolayer case, indicating stronger orbital hybridizations and magnetic proximity in the system.

Gd, on the other hand, exhibits a significantly larger overall spin magnetic moment, with values of 7.509 μ_B in the 1 × 1 unit cell and 7.407 μ_B in the $\sqrt{3}\times \sqrt{3}$ supercell, compared to 7.130 μ_B and 6.994 μ_B for the Eu case at the favored T-W site. These increased moments are attributed to the higher 5d occupation, which is intra-atomically spin-polarized by the 4f magnetic moment. Another difference between the two coverages is also evident in the change of orbital magnetic moment, which increases in magnitude from −0.023 μ_B to −0.166 μ_B when diluting the Gd atom, in contrast to the decreasing trend observed for Eu from −0.032 μ_B to −0.005 μ_B.

Regarding the open 4f-shell system of Ho, in the 1 × 1 unit cell the f_occ, along with the spin and orbital magnetic moments, reveals that Ho essentially retains its atomic 4f occupation with 11 electrons, three of which are unpaired. However, in the T-W site, the orbital magnetic moment deviates from Hund’s rules, which predict an orbital moment of 6 μ_B, resulting in a reduction of 1 μ_B and a final value of ~5 μ_B for ${m}_{l}^{{{\rm{Ho}}}}$. Consequently, the adsorption on top of WSe₂ induces quenching of the magnetic orbital moment driven by the competition between the crystal field and the intra-orbital exchange interaction. This phenomenon is also observed in the other adsorption sites, leading to values of m_l ~ 4.9 μ_B. The observed quenching can be attributed to both crystal field effects from the substrate and the closely lying Ho nearest neighbors in the magnetic monolayer.

In the diluted $\sqrt{3}\times \sqrt{3}$ case, simulations on the T-W site lead to two different energy minima: one following Hund’s rules with ${m}_{l}^{{{\rm{Ho}}}} \sim 6\ {\mu }_{{{\rm{B}}}}$, and another with a quenched value of ${m}_{l}^{{{\rm{Ho}}}} \sim 5\ {\mu }_{{{\rm{B}}}}$. In the following, we will designate the Ho atom case with an orbital magnetic moment of approximately m_l = 6 μ_B as T-W_I, and the case with approximately m_l = 5 μ_B as T-W_II. Similarly to the 1 × 1 case, in the T-Se site, the orbital moment is again quenched, while it follows Hund’s rules in the H-site. These behaviors can be understood by considering the hybridizations with the surrounding crystal field, which appear stronger when the adatom adsorbs directly on top of an atom of the WSe₂ monolayer. Conversely, the effect is reduced when adsorbing in the middle of the hexagon formed by Se and underlying W atoms. It is important to note that these calculations of Ho atoms have been performed in the presence of self-consistently included SOC effects by choosing a perpendicular spin-quantization axis (along z) to the substrate. Additionally, in the T-W site, the T-W_II Ho atom has not been further relaxed compared to the T-W_I case. All the calculations discussed were carried out without initializing the occupation matrix of the 4f shell for the Ho atom, allowing the 4f configuration to relax naturally to the preferred occupation. In every case, across all adsorption sites and densities, this led to a 4f¹¹ orbital occupation for the Ho atom.

To examine the distinctions of the electronic structure among the investigated rare-earth systems, we refer to Fig. 2, which illustrates the spin-resolved density of states (DOS) for each chosen rare-earth atom under the two concentration scenarios. Specifically, Fig. 2a–c correspond to the 1 × 1 magnetic monolayer case, while Fig. 2d–f pertain to the diluted situation in the $\sqrt{3}\times \sqrt{3}$ supercell, with the T-W_I configuration shown for Ho. A comparison between the electronic structures of the T-W_I and T-W_II orbital occupations for Ho/WSe₂ in the $\sqrt{3}\times \sqrt{3}$ supercell can be found in the Supplementary Material.

Fig. 2: Spin-resolved density of states of rare-earth atoms on WSe₂ at the two adsorption densities.

a Spin-polarized density of states of Eu adsorbed on WSe₂ in the 1 × 1 unit cell. b Spin-polarized density of states of Gd adsorbed on WSe₂ in the 1 × 1 unit cell. c Spin-polarized density of states of Ho adsorbed on WSe₂ in the 1 × 1 unit cell. d Spin-polarized density of states of Eu adsorbed on WSe₂ in the $\sqrt{3}\times \sqrt{3}$ supercell. e Spin-polarized density of states of Gd adsorbed on WSe₂ in the $\sqrt{3}\times \sqrt{3}$ supercell. f Spin-polarized density of states of Ho adsorbed on WSe₂ in the $\sqrt{3}\times \sqrt{3}$ supercell in the T-W_I configuration. The shaded curves represent the total density of states (TDOS) of the heterostructures, while orange and green denote the 4f and 5d states, respectively, of the rare-earth atoms. The upper half of the plots corresponds to the spin-up channel, and the lower half to the spin-down channel.

Notably, both Eu and Ho exhibit a similar behavior, with the occupied 4f states (orange peaks) positioned closer to the Fermi energy compared to Gd. This holds true for both coverages, especially in the 1 × 1 unit cell, where the 4f peaks tend to come in close energetic proximity with other states of the heterostructure, such as the d states of the RE atom itself. Particularly, Ho exhibits a splitting of the 4f manifold attributable to the influence of SOC and electron correlation, thus covering a broader energy range. This proximity facilitates orbital hybridization, as qualitatively shown in Fig. 3, which illustrates the product of the weights of different orbitals projected to various combinations of chemical species in 1 × 1 Ho/WSe₂. The remarkably flat 4f bands are observable within an energy span of ~ −4 eV to ~ −0.8 eV. Additionally, the distinctive K-valleys of the TMDC manifest around ~ −1.8 eV at the high-symmetry points K and K’, with an energy splitting induced by spin-orbit coupling. Similar to the findings in ref. ⁴³ for Eu on WSe₂, we observe a direct interaction between the 4f electrons and the delocalized electrons constituting the crystal field, e.g. f electrons of Ho with d electrons of W, along with interactions between the spin-polarized delocalized electrons of the system, such as between d of Ho and d of W.

Fig. 3: Spin-orbit coupling included band structure calculations of orbital interactions in 1 × 1 Ho/WSe₂ along the Γ-K-M-K’-Γ path.

The interactions include: a f states of Ho with d states of W, b d states of Ho with d states of W, c s states of Ho with p states of Se, and d s states of Ho with d states of W. In the color map, yellow stands for strong interactions, purple defines weak interactions.

Examining the DOS for Gd atoms on WSe₂ (Fig. 2b and e), the scenario is notably different. The occupied 4f peaks are deep in energy (~ −8 eV for 1 × 1 and ~ −10 eV for $\sqrt{3}\times \sqrt{3}$). This considerable energy separation prevents orbital hybridization since these peaks are energetically far removed from all other states in the system. However, the spin-polarized d states exhibit a significant enhancement at the Fermi energy compared to Eu and Ho, particularly in the $\sqrt{3}\times \sqrt{3}$ diluted case.

Magnetic anisotropy

The impact of magnetic dilution on the magnetic anisotropy energy (MAE) is shown in Fig. 4, depicting the total energy variation as a function of the magnetization orientation for Gd and Eu on WSe₂ in the two coverage scenarios. Each plot illustrates the MAE for the entire magnetic heterostructure using solid lines, along with the corresponding contributions from the individual components, RE+Se and the 2D material W+Se, represented by dotted lines and obtained by switching off sequentially the SOC contribution at the W atoms and at the RE atom. Figure 4a and b correspond to 1 × 1 and $\sqrt{3}\times \sqrt{3}$ Eu/WSe₂, while Fig. 4c and d present the analogous data for Gd/WSe₂. Specifically, the magnetization rotates from perpendicular to the WSe₂ plane (along the z-axis, θ = 0°) to parallel to the WSe₂ plane (along the x-axis, θ = 90°). All energy values are shifted with respect to the lowest energy point, set at 0 eV.

Fig. 4: Magnetic anisotropy energy curves of Eu/WSe₂ and Gd/WSe₂ at the two adsorption densities.

a Magnetic anisotropy energy (MAE) curve of the 1 × 1 Eu/WSe₂ total system (solid purple line), additionally showcasing the distinct contributions from Eu+Se (dashed orange line) and W+Se (dashed yellow line) components. In (b) the MAE of the diluted $\sqrt{3}\times \sqrt{3}$ configuration of the overall Eu/WSe₂ system is presented (solid dark green line), along with the individual contributions from Eu+Se (dashed blue) and W+Se (dashed turquoise) components. c MAE curve of the 1 × 1 Gd/WSe₂ total system (solid purple line) along with the Gd+Se (dashed orange line) and W+Se (dashed yellow line) contributions. d MAE of the diluted $\sqrt{3}\times \sqrt{3}$ configuration of the overall Gd/WSe₂ system is shown (solid dark green line), along with the individual contributions from Gd+Se (dashed blue) and W+Se (dashed turquoise) components. The total energy of the system is plotted against the polar angle of the magnetization, measured from the z-axis.

The overall anisotropic behavior for both RE atoms exhibits a low-order nature, primarily governed by the first magnetic anisotropy constant, denoted as K₁ in ${E}_{an}={K}_{1}{\sin }^{2}\theta$, where θ represents the polar angle associated with out-of-plane anisotropy. For both magnetic systems and coverages, the system favors an out-of-plane magnetization direction, as evidenced by the lowest energy at θ = 0° in the solid lines in purple and dark green.

Specifically for Eu/WSe₂, in comparing the magnetic anisotropy between the 1 × 1 and $\sqrt{3}\times \sqrt{3}$ cells (Fig. 4a and b), we observe that the energy barrier required for magnetization rotation from perpendicular to parallel to the 2D material is approximately 1.75 meV for the 1 × 1 cell and roughly 0.20 meV for the $\sqrt{3}\times \sqrt{3}$ cell. This suggests the need for a high coverage of Eu atoms to achieve stable magnetic states. The difference can be explained in terms of orbital hybridizations. In the 1 × 1 high coverage scenario, the close proximity of the spin-up occupied 4f states to Eu 5d states and other delocalized substrate states (e.g. d states of W) leads to orbital hybridization (see Fig. 2). Thus, the pronounced anisotropy can be fundamentally attributed to two main factors:

(1) Direct interaction: an interaction between the 4f charge and other delocalized electrons of WSe₂, such as the d states of W. This interaction induces orbital angular momentum in the 4f shell, resulting in an anisotropic interaction with the surrounding crystal field upon rotation of the magnetization. (2) Indirect contribution: this is driven by intra-atomic spin-polarization from the 4f states to the d states of Eu. The latter, being more spatially extended, interact anisotropically with the environment based on the rotation of the 4f magnetic moment through intra-atomic exchange interaction. However, given the small 5d occupation, this contribution is expected to have a lesser influence on the overall energy scale.

In the diluted $\sqrt{3}\times \sqrt{3}$ case, the factors contributing to magnetic anisotropy are expected to be analogous but less pronounced due to the lower density of magnetic atoms per W atom. The W atoms act also as the origin of spin-orbit coupling, essentially “inducing” SOC in the magnetic source. Consequently, the interplay between magnetism and SOC results in a less prominent outcome compared to the high coverage scenario.

Upon examining the contributions from Eu+Se and W+Se in the two coverage scenarios by switching on SOC only in the MT spheres of specific atoms, it becomes apparent that in the 1 × 1 case, the predominant contribution arises from Eu atoms. Their spin-orbit contribution favors an out-of-plane easy axis, closely resembling the anisotropy energy barrier observed in the total system. In contrast, the TMDC alone favors an in-plane magnetization orientation (indicated by the yellow dotted line). With dilution, the Eu+Se contribution vanishes, revealing that all MAE contribution is attributed to the slightly spin-polarized WSe₂. Intriguingly, magnetic dilution induces a shift in the easy-axis of the magnetic TMDC.

Shifting our attention to Gd/WSe₂ (refer to Fig. 4c and d), which trend is reversed, suggesting that the diluted situation yields larger magnitudes compared to the high-coverage scenario. Specifically, for the $\sqrt{3}\times \sqrt{3}$ scenario, an energy difference of approximately ~5 meV is observed from the perpendicular to the in-plane direction, while in the 1 × 1 unit cell, this is reduced to about ~0.4 meV.

To explain the reduced MAE in the high-coverage scenario compared to Eu, it must be noted that the direct contribution is absent, and only indirect contributions are observed, as the 4f states are energetically too distant to interact with other states. While this holds true for the diluted scenario of Gd on WSe₂, the computed magnetic anisotropy remains remarkably large. In this case, the effect likely originates from the extensive spin-polarized d DOS observed at the Fermi energy, coupled with the adsorption characteristics that result, as observed previously, in a very short equilibrium distance compared to all other cases. Therefore, the magnetic anisotropy in this case arises from the robust interaction of the spin-polarized d states with the surrounding C_3v crystal field.

Analyzing the individual contributions to the MAE, we observe that in the 1 × 1 scenario, both Gd and the spin-polarized substrate contribute similarly in magnitude and preferring an out-of-plane magnetization direction. In the diluted case, the primary contribution comes from the Gd atoms, although the effect of WSe₂ increases significantly by one order of magnitude due to its closer proximity to the Gd atom and, consequently, its stronger spin-polarization.

In both Eu/WSe₂ and Gd/WSe2, the variation in the magnitude of MAE with magnetic dilution can also be explained by considering the change in the magnetic orbital moment of the RE atoms. The larger MAE of Eu/WSe₂ in the 1 × 1 unit cell corresponds to a larger value of m_l by one order of magnitude. Similarly, the increased MAE in Gd/WSe2 from 0.4 meV to approximately 5 meV, as coverage is reduced, reflects a change in m_l from −0.023μ_B to −0.166μ_B. More details on the orbital moment anisotropy of the two RE/WSe₂ systems in the two coverage scenarios are available in the Supplementary Material. This includes the variation of the magnetic orbital moment relative to the magnetization direction.

Magnetic anisotropy of Ho atoms on a WSe₂ monolayer

In examining the magnetic anisotropy of Ho/WSe₂, we observe that, in contrast to Gd, Ho follows a pattern similar to Eu: larger MAEs are evident in the 1 × 1 unit cell compared to the $\sqrt{3}\times \sqrt{3}$ coverage. This is depicted in Fig. 5, where blue points indicate DFT + U calculations for the 1 × 1 unit cell of Ho/WSe₂, while red points are the DFT + U calculations in the $\sqrt{3}\times \sqrt{3}$ supercell considering the T-W_I occupation. It can be noticed that energy values peak at around 16 meV in the 1 × 1 cell, covering a range from the canted easy-axis direction at approximately θ = 70° to the hard-axis found at ~θ = 30°. In the $\sqrt{3}\times \sqrt{3}$ supercell, there is an energy difference of about 6 meV between the easy (~θ = 50°) and hard (θ = 0°) axes.

Fig. 5: Magnetic anisotropy energy curves of Ho/WSe₂.

Magnetic anisotropy energy curves of the 1 × 1 magnetic monolayer (blue curve) and the diluted $\sqrt{3}\times \sqrt{3}$ scenario (red curve) of Ho/WSe₂. The points indicate the DFT + U calculations, while the solid lines illustrate the fitting according to Eq. (2). The total energy of the system is plotted versus the polar angle of the magnetization, measured from the z-axis.

Additionally, in this case, the open 4f-shell induces a highly anisotropic behavior in space, resulting in energy curves that deviate from the first contribution that is proportional to ${K}_{1}{\sin }^{2}\theta$. Specifically, in a trigonal C_3v crystal field, the DFT+U calculations can be fitted using the equation³⁴:

$$\begin{array}{l}{E}_{an}\,=\,{K}_{1}{\sin }^{2}\theta +{K}_{2}{\sin }^{4}\theta \\\qquad \,+\,{K}_{2}^{{\prime} }{\sin }^{3}\theta \cos \theta \cos \,(3\varphi)\\\qquad \,+\,{K}_{3}{\sin }^{6}\theta +{K}_{4}{\sin }^{6}\theta \cos\, (6\varphi)\\\qquad \,+\,{K}_{5}{\sin }^{3}\theta\, {\cos }^{3}\theta \cos \,(3\varphi),\end{array}$$

(2)

where the K_i are the magnetic anisotropy constants, and (θ, φ) are the polar and azimuthal angle describing the magnetization direction. Focusing solely on the out-of-plane contribution (φ = 0), we need to consider the magnetic anisotropy constants ${K}_{1},{K}_{2},{K}_{2}^{{\prime} },{K}_{3}^{{\prime} },$ and K₅, where we define ${K}_{3}^{{\prime} }={K}_{3}+{K}_{4}$. The values obtained from the fitting procedure are presented in Table 3. It is noteworthy that substantial contributions come from all K_i constants, highlighting the significantly anisotropic behavior compared to the Eu and Gd systems.

Table 3 Magnetic anisotropy constants obtained via fitting of DFT + U data depicted in Fig. 5 b for 1 × 1 and $\sqrt{3}\times \sqrt{3}$ Ho/WSe₂

We can attribute the highly anisotropic behavior of the Ho atoms to the direct and indirect contributions observed earlier for Eu, along with the influence of the open 4f-shell that inherently induces substantial spin-orbit coupling effects. This effect is evident, for instance, in ref. ⁴⁰, where large magnetic anisotropy values of Ho are found on a bare graphene monolayer that lacks SOC. In this scenario, the Ho atom resides within a high-symmetry hexagonal crystal field created by the C atoms, resulting in relatively weak orbital interactions. This allows for a straightforward application of a point-charge model to theoretically describe the magnetic states. The heightened complexity of magnetic anisotropy in the WSe₂ heterostructure is evident not only in the reduced symmetry leading to a larger number of magnetic anisotropy constants (for a C_6v field, only K₁, K₂, K₃, and K₄ are relevant³⁴) but also in the discussed orbital interactions due to the chemical variety in the chosen 2D material. These interactions, while complicating the adoption of a point-charge model, can still be harnessed to achieve larger magnitudes in the MAE.

From a quantum perspective, it is crucial to note that reducing the local symmetry around the magnetic source introduces additional quantum operators capable of mixing magnetic states. This potential mixing can favor quantum tunneling of magnetization^14,44 and, consequently, destabilize the magnetization. For example, in a hexagonal C_6v crystal field, the Stevens operator^45,46 ${\hat{O}}_{6}^{6}$ mixes states labeled by the quantum number J_z in a total angular momentum manifold J with a difference of ΔJ_z = ±6. On the other hand, a trigonal C_3v crystal field includes extra operators that can generate quantum superpositions of states differing by ΔJ_z = ±3 in addition to the ${\hat{O}}_{6}^{6}$ operator. In principle, adsorbing the magnetic atom onto a high-symmetry site is preferred. However, if the mixed states do not represent the magnetic ground state, choosing a 2D material that creates substantial energy gaps between magnetic states (e.g., through orbital hybridizations) could lead to a favorable outcome for generating stable magnetic units.

Heading back to the comparison with Eu, in Ho/WSe₂ it is crucial to consider that in the high-density 1 × 1 magnetic monolayer the compact 4f charge clouds themselves contribute to the crystal field, leading to a 4f-4f repulsion—a factor absent in Eu due to its spherical 4f charge cloud. Collectively, these factors contribute to an enhanced magnetic anisotropy compared to 1 × 1 Eu/WSe₂. Nonetheless, the pattern seems analogous, as the magnetic anisotropy energy decreases with the dilution of Ho atoms. In both Eu and Ho cases, this reduction can be attributed to a decreased magnetic proximity and interaction with the overall crystal field—both from the substrate and neighboring RE atoms. Another noticeable factor is the diminishing spin-polarized d occupation near the Fermi energy, signifying a reduction in the indirect contribution.

Table 2 for Ho indicates that for the $\sqrt{3}\times \sqrt{3}$ diluted coverage in the preferred adsorption site, T-W, the energy minimization occurs with two distinct 4f orbital occupations. One adheres to Hund’s rules, exhibiting a magnetic orbital moment of ${m}_{l}^{{{\rm{Ho}}}} \sim 6\ {\mu }_{{{\rm{B}}}}$, signif ying 7 electrons occupying orbitals with magnetic quantum numbers from −3 to +3 with aligned spins to maximize the total spin moment, while the remaining 4 electrons then fill the orbitals with magnetic quantum numbers +3, +2, +1, and 0, maximizing the total angular moment. In contrast, the configuration with ${m}_{l}^{{{\rm{Ho}}}} \sim 5\ {\mu }_{{{\rm{B}}}}$ involves displacing one spin-down electron from the orbital with magnetic quantum number 0 to −1, thereby quenching the orbital magnetic moment by 1 μ_B. In the case of Hund’s rules orbital occupation (T-W_I), the total angular momentum is J = 15/2, whereas for the deviation (T-W_II), it is J = 13/2. As previously explained, this reduction in the orbital moment stems from a competition between the crystal field interaction and the intra-atomic exchange, indicating that the crystal field effect surpasses the latter in energy. This behavior contrasts with what is observed for Ho atoms on a graphene monolayer, as discussed in ref. ⁴⁰, where the 4f orbital occupation adheres to Hund’s rules.

To assess the impact of the quenching of the orbital magnetic moment on magnetic anisotropy, DFT + U calculations were employed to compute the total energy for different out-of-plane magnetization directions, ranging from θ = 0° to θ = 90° for the magnetic state J = 13/2. For angles between 0° and 20°, the J = 13/2 state is energetically favored, being 0.33 eV lower in energy compared to J = 15/2. However, as the magnetization rotates further, an energy inversion occurs, and the J = 15/2 state becomes the new ground state. Due to difficulties in achieving convergence for the J = 13/2 state at larger angles, the self-consistent procedure was performed by fixing the 4f occupation matrix to the desired value of J. Consequently, we refrain from discussing energy differences between the two magnetic states.

Nevertheless, the consistent symmetry of the crystal field in both scenarios emphasizes that eventual differences in the MAE, both in shape and in magnitude, arise from the spatial geometry of the 4f charge cloud, determined by the varying 4f orbital occupation. This can be qualitatively observed in Fig. 6, where we show the computed magnetization densities of Ho within the two analyzed 4f orbital occupations in presence of a perpendicular and a parallel spin quantization axis. These calculations underline the necessity of an accurate description of the 4f states, as the magnetic properties can drastically change for different energy minima.

Fig. 6: Magnetization densities of the Ho atom on top of WSe₂ with J = 15/2 and J = 13/2 orbital occupations.

The red clouds correspond to the magnetization densities in the (a) J = 15/2 configuration with ${m}_{l}^{{{\rm{Ho}}}}=6 \ {\mu }_{{{\rm{B}}}}$ and (b) J = 13/2 configuration with ${m}_{l}^{{{\rm{Ho}}}}=5 \ {\mu }_{{{\rm{B}}}}$. In both cases the magnetization density is shown for a perpendicular (θ = 0°) and parallel (θ = 90°) magnetization to the 2D material. The isosurface level is 0.01 e/bohr³. The red arrow symbolizes the spin quantization axis.

Discussion

In this manuscript, we have demonstrated the tunability of magnetocrystalline anisotropy in rare-earth atoms on the valleytronic semiconducting 1H-WSe₂ monolayer based on their adsorption density. Specifically, the magnetic anisotropy is closely tied to the specific rare-earth atom chosen, and, in general, open 4f-shells result in more significant energy variations as the magnetization is rotated in space. This is attributed to the non-spherical nature of the 4f charge density and the pronounced spin-orbit coupling arising from a non-zero orbital angular momentum.

We adopted density functional theory calculations to reveal that rare-earth atoms exhibiting chemical similarities to Eu, such as Ho without an external 5d valence electron, demonstrate more substantial magnetic anisotropies in high concentrations on the 2D material. Conversely, rare-earth atoms like Gd, possessing 5d valence electrons, exhibit the opposite behavior.

These contributions involve a direct interaction between localized 4f electrons and the environment, as well as an indirect contribution arising from intra-atomic spin-polarization and ferromagnetic exchange interaction between 4f electrons and delocalized electrons like d electrons. These, in turn, interact with the crystal field. The strength of these interactions depends on coverage density, the proximity to the substrate, position of electronic states in the energy spectrum, and the geometry of the 4f charge density. These effects are intrinsic to the specific valence configuration of the rare-earth atom.

Compared to graphene, a WSe₂ monolayer as a 2D material for 4f atom adsorption can result in larger energy barriers during magnetization rotation. This is due to the stronger orbital hybridization with the 4f atom and the prominent spin-orbit coupling present in WSe₂. The largest magnetic anisotropies are achieved by adsorbing open 4f shells with dense coverage on WSe₂, reaching approximately 16 meV from the easy to hard-axis for Ho, where the anisotropic 4f charge clouds contribute significantly to the spin-orbit coupling and crystal field, especially when compared to the half-filled cases Eu and Gd. However, one of the most surprising findings is that even for half-filled 4f shells, which are typically less interesting in this context due to their relatively isotropic charge clouds, we observed that large magnetic anisotropy energy can still be attained. This is possible by adjusting the coverage density in relation to the valence shell configuration, particularly the presence of 5d electrons. This leads to energy barriers as high as 1.75 meV for Eu and around 5 meV for Gd, for the rotation of magnetization from the out-of-plane to in-plane direction. As a result, these rare-earth atoms also become promising candidates for memory devices. This emphasizes the need to account for chemical properties, such as the valence configuration that affects adsorption behavior—for example, the distance from the substrate, coverage density, and direct contributions to anisotropy energy through the interaction of valence electrons with the environment—when investigating magnetic anisotropy effects, as these factors play a critical role in shaping the overall outcome.

These findings bear experimental significance for applications requiring stable magnetizations, such as hard magnets, and they also encourage additional investigation into how the chemical properties of rare-earth atoms influence magnetic stability. Indeed, future analysis is required, considering various lanthanide species, both heavy and light (with more and fewer than half-filled 4f shells). Additionally, exploring the quantum-level impact on magnetic anisotropy, including determining the quantum multiplet splittings of magnetic states, would provide a more profound understanding of magnetization stability against quantum fluctuations. This analysis would also enable the eventual determination of two-level quantum systems. Moreover, such assessments should cover multiple options of transition metal dichalcogenide materials, involving variations in both metallic and chalcogen atoms within the substrate.

Methods

General DFT setup

The results presented were obtained using the full-potential linearized augmented planewave (FLAPW) method as implemented in the FLEUR code^32,47. Reliable self-consistent results have been obtained adopting the PBE prescription of the generalized gradient approximation (GGA)⁴⁸ exchange-correlation functional with a 10 × 10 k-point mesh, while the magnetic anisotropies were computed using a 21 × 21 k-point mesh. The 1 × 1 unit cell, which included one RE atom, two Se atoms, and one W atom, was simulated with the equilibrium GGA lattice constant of a = 3.327 Å, as illustrated in Fig. 1a. Concerning the LAPW basis functions, the plane-wave basis cut-off was set to ${K}_{\max }=4.0\ {a}_{0}^{-1}$, and the maximum angular momentum inside the MT spheres was set to ${l}_{\max }=10$ for the RE atom and ${l}_{\max }=8$ for W and Se. For simulations of the dilute case, we utilize a supercell (Fig. 1b) with a lattice constant of $a=\sqrt{3}\times 3.327$ Å and maintain the same self-consistent field parameters as in the 1 × 1 cell, except for employing a k-point mesh of 20 × 20 for the spin-orbit coupling calculations.

Treatment of the 4f electrons

To account for the highly localized 4f electrons of the magnetic atoms, the DFT + U method was employed, considering the fully-localized limit for the double-counting term⁴⁹. The DFT + U parameters were set to U = 6.7 eV and J = 0.7 eV for Eu and Gd, as per^49,50, and U = 7.03 eV and J = 0.83 eV for Ho, based on reported values in⁵¹. It is noteworthy that for RE atoms, U values around 7 eV are widely accepted as they have been shown to reproduce experimental results for various properties of bulk RE systems⁵². Calculations related to Ho atoms as well as those conducted for the magnetic anisotropy curves included the self-consistent treatment of the spin-orbit coupling by means of the second variation formulation⁵³. The Hubbard U was added in the (scalar-relativistic, collinear) first-variation step. Specifically, the calculations for the Ho/WSe₂ systems were conducted without initializing the occupation matrices and incorporating SOC and the Hubbard U correction from the start. These calculations converged to a 4f¹¹ configuration with the different orbital occupations described in the manuscript: approximately 5 μ_B for the 1 × 1 high coverage, and both ~5 μ_B and ~6 μ_B for the $\sqrt{3}\times \sqrt{3}$ low coverage. For the calculation of the magnetic anisotropy curves of the Ho systems, each computation at a given angle used the charge density from the previous angle as input, along with the density matrix appropriately rotated with the spin quantization axis. From a methodological perspective, future developments could involve improving the treatment of 4f electron correlations by moving beyond DFT + U, for instance, by adopting the Hubbard-I approximation^54,55.