Two-dimensional (2D) structures that exhibit intriguing magnetic phenomena such as perpendicular magnetocrystalline anisotropy (PMA) have become a focus of spintronic research due to their potentials in maximizing the information storage density. Herein we perform density-functional theory plus U (DFT+U) calculations to investigate the binding affinity and intrinsic magnetic properties of an individual rare-earth (RE) Sm atom on WSe2 monolayer. Our calculations show that Sm adatom energetically prefers to adsorb at the W-top site in WSe2 rather than the Se-top and hollow sites. We predict extremely large PMA values of ∼7–33 meV per Sm at the most stable W-top site, depending on U parameter in DFT+U calculations, while it is negligibly small for the Se-top and hollow sites. The underlying mechanism for large PMA is elucidated in terms of the strong spin–orbit coupled Sm 4f – W 5d orbital states and large 4f orbital magnetic moment in the high-spin crystal field. These results provide a viable route to achieving an atomic scale f-electron PMA in 2D structures, opening interesting prospects in two-dimensional semiconducting spintronics.

Over the years after the discovery of graphene,1 the exploration of two-dimensional (2D) transition metal dichalcogenides (TMDs) has been active in materials science, owing to their unique atomic and electronic structures. In particular, researchers focus on engineering the physical and chemical properties of TMD MX2, where M is a transition metal element (M = Mo or W) and X is a sulfur element (X = S or Se), by decorating with functional species or/and metal dopant atoms.2–5 Nevertheless, rare-earth (RE) elements have not been seriously considered as dopants, leaving unanswered questions what would be the effect of RE dopant atoms on the physical properties of MX2 monolayers. Recently, MoS2 monolayer with Sm dopant atoms has been successfully grown by chemical vapor deposition method.6 The authors reported that the presence of Sm substantially alters the semiconducting electronic properties of MoS2 monolayer. Furthermore, there have been few more experimental and theoretical studies on transition metal (TM) and RE doped MX2 monolayers.7–9 For instance, the possible magnetic nature of Sm-doped MoS2 monolayer has been predicted by the density-functional theory (DFT) calculations, which is attributed to the hybridization between the Sm-4f and S-3p orbital states.8 It has also been experimentally reported that the V-doped WSe2 structures exhibit magnetic ordering at room temperature.10,11 In addition to these remarkable findings, the promising alternative seemingly resides in the adoption of the strong spin–orbit coupling (SOC) features of f and d electrons for exploring the magnetocrystalline anisotropy (MA). If ever realized, large perpendicular MA (PMA) is of crucial significance for stable long-term spintronic applications.

In this paper, we perform the DFT plus U (DFT+U) calculations to investigate the binding affinity and intrinsic magnetic properties of an individual Sm atom on WSe2 monolayer. Our calculations show that Sm adatom energetically prefers to adsorb at the W-top site rather than the Se-top and hollow sites. We predict extremely large MA energy (MAE) values of 7–33 meV per Sm adsorbed at the most stable W-top site in WSe2 monolayer, depending on U parameter in DFT+U calculations, in contrast to the negligible MAE of Sm at the Se-top and hollow sites. Through the single-particle energy spectra analyses, we elucidate the underlying mechanism for large PMA with the strong spin–orbit coupled Sm 4f – W 5d orbital states and large orbital magnetic moment of Sm in the high-spin crystal field.

The DFT calculations were performed using the Vienna ab initio simulation package (VASP).12 Generalized gradient approximation, parameterized by Perdew, Burke, and Ernzerhof, was employed to describe the exchange-correlation functional.13 The strongly correlated Sm 4f electrons were treated with the Hubbard method.14,15 The onsite U parameter was chosen as 6 eV,16,17 which is sufficient to split the f orbital bands into lower and upper Hubbard bands, as discussed in the following paragraphs. Our model structure consists of a single Sm deposited on a 4 × 4 supercell lattice of WSe2 monolayer, as illustrated in Figs. 1(a) and 1(b). In order to avoid spurious interactions between periodic images of WSe2 monolayers, a vacuum spacing perpendicular to the plane was employed to be no less than 15 Å. An energy cutoff of 550 eV, a 3 × 3 × 1 k-point mesh, and relaxation force criterion of 10−2 eV/Å were adopted for the structure optimization. MAE is determined by the total energy difference when the magnetization directions are in the ab plane (E) and along the c axis (E): MAE = EE. We imposed a denser k-point mesh of 5 × 5 × 1 in noncollinear calculations, which is sufficient to obtain well-converged MAE values. The SOC term was included in a second-variational way employing scalar relativistic calculations of the valence state.18 

FIG. 1.

(a) Top view and (b) side view of a 4 × 4 supercell structure of WSe2 monolayer with a Sm dopant atom. Bigger gray and red spheres represent W and Sm atoms, respectively, while smaller green is Se. In (a) the letters of W, Se, and H superimposed on the red spheres denote the three possible adsorption sites of Sm adatom: W-top, Se-top, and hollow site. The letter a at the bottom denotes the lattice constant of the primitive unit cell of WSe2. In (b) the letter h denotes the height of Sm adatom with respect to the Se-plane. (c) Electronic density of states DOS of W (red solid) and Se atoms (green dotted), and formula unit (f.u.) of WSe2 monolayer (blue solid line). (d) The same for the Sm 4f orbital states in Sm/WSe2 structure from DFT+U calculations for different U parameters. The Fermi level is set to zero in energy.

FIG. 1.

(a) Top view and (b) side view of a 4 × 4 supercell structure of WSe2 monolayer with a Sm dopant atom. Bigger gray and red spheres represent W and Sm atoms, respectively, while smaller green is Se. In (a) the letters of W, Se, and H superimposed on the red spheres denote the three possible adsorption sites of Sm adatom: W-top, Se-top, and hollow site. The letter a at the bottom denotes the lattice constant of the primitive unit cell of WSe2. In (b) the letter h denotes the height of Sm adatom with respect to the Se-plane. (c) Electronic density of states DOS of W (red solid) and Se atoms (green dotted), and formula unit (f.u.) of WSe2 monolayer (blue solid line). (d) The same for the Sm 4f orbital states in Sm/WSe2 structure from DFT+U calculations for different U parameters. The Fermi level is set to zero in energy.

Close modal

We first inspected the atomic and electronic structures of WSe2 monolayer. The optimized lattice constant of WSe2 monolayer is 3.31 Å, which is in reasonable agreement with the experimental (3.28–3.29 Å)19,20 and previously reported theoretical values (3.316 Å).21 From the electronic density of states (DOS) in Fig. 1(c), our calculated semiconductor bandgap of WSe2 monolayer is 1.54 eV, which in an experiment is 1.64–1.65 eV.22,23 The majority and minority spin subbands of both W and Se atoms in WSe2 structure are degenerate, which indicates a nonmagnetic nature of WSe2. There are three possible adsorption sites of Sm on WSe2 monolayer: W-top, Se-top, and hollow site [See Fig. 1(a)]. In Table I we show the optimized in-plane lattice a and height h of Sm with respect to the Se-plane. Upon the adsorption of Sm, the calculated lattice constant of WSe2 slightly increases from 3.31 Å to 3.32–3.33 Å. While Sm is above the Se-top site with h = 3.14 Å, Sm is placed rather toward the WSe2 plane for the W-top and hollow sites (h = 2.22–2.54 Å). Accordingly, the latter two sites (W-top and hollow) exhibit higher binding affinities than the former Se-top site. Here, the binding energy Eb is defined as Eb = E(hequilibrium) − E(hisolated), where E(hequilibrium) and E(hisolated) are the total energies of WSe2 with Sm at the equilibrium coordinate and with Sm isolated from the WSe2 surface, respectively. The calculated values of Eb are −1.03, −0.43, and −0.72 eV for the W-top, Se-top, and hollow sites, respectively. This indicates that the W-top site is more favored than the Se-top and hollow sites.

TABLE I.

Optimized in-plane lattice a (Å) and height h (Å) of Sm adatom with respect to the Se-plane, binding energy Eb (eV), and spin magnetic moment ms (μB) and charge transfer Δρ (e) of Sm, W, and Se atoms in Sm/WSe2 structure, from DFT+U calculations with U = 6 eV, for different adsorption configurations.

msΔρ
ahEbSmWSeSmWSe
W-top 3.33 2.22 −1.03 6.01 0.07 0.00 −0.97 −0.02 0.05 
Se-top 3.32 3.14 −0.43 5.94 0.01 0.00 −0.54 −0.03 0.06 
Hollow 3.33 2.54 −0.72 5.94 0.02 0.00 −0.84 −0.02 0.05 
msΔρ
ahEbSmWSeSmWSe
W-top 3.33 2.22 −1.03 6.01 0.07 0.00 −0.97 −0.02 0.05 
Se-top 3.32 3.14 −0.43 5.94 0.01 0.00 −0.54 −0.03 0.06 
Hollow 3.33 2.54 −0.72 5.94 0.02 0.00 −0.84 −0.02 0.05 

It is known that the standard DFT fails to accurately describe the electronic structure and magnetic properties of the strongly localized 4f orbital states due to the oversimplified treatment of electron correlations, where the 4f states are pinned right at the Fermi level (EF).14,15 This is also the case for the present Sm atom in Sm/WSe2 structure, as seen in DOS of Sm 4f orbital states in Fig. 1(d). On the other hand, the DFT+U approach provides a more realistic treatment for the Sm 4f states, as this splits the f bands into lower and upper Hubbard bands.16,17 It is obvious from the Sm DOS in Fig. 1(d) that the 4f bands at EF are split into the occupied and unoccupied states in DFT+U calculations. In DFT+U, it is ambiguous as to the proper choice of U value. One practical way to determine U is direct comparison with experimental data. To the best of our knowledge, no experimental results have been reported thus far to compare with the present calculations. Hence we analyze the magnetic properties of Sm/WSe2 structure with different U parameters. The spin magnetic moment (ms) of Sm adatom at the W-top site increases linearly from 5.83 μB at U = 0 (DFT) to 6.01 μB at U = 6 eV. We ascribe this peculiar behavior to the high-spin state of the Sm 4f6 electrons. Furthermore, the W atoms underneath and surrounding Sm have induced magnetic moments of 0.04–0.07 μB (Table I), which is prominent beyond those (negligible) of the other W sites away from Sm.

The induced magnetism in Sm/WSe2 structure can be understood in terms of the spin-polarized charge transfer (Δρ) between Sm and W atoms. We find that for the pristine WSe2 monolayer, the electronic charges of W and Se atoms are 4.34 and 6.83 e, respectively. Further analysis from Bader charge (Table I) indicates that upon the Sm adsorption, the Se 4p orbitals accumulate even charges of 0.05–0.06 e from its surrounding Sm 4f6 (0.54–0.97 e) and W 5d orbitals (0.02–0.03 e) since the Se has a larger electronegativity (2.55) than Sm (1.17) and W (2.36). For the less energetical Se-top and hollow sites, the magnetic moments of Sm and W are ∼5.94 and 0.01–0.02 μB for U = 6 eV in DFT+U, respectively.

Figure 2(a) displays the calculated MAE of Sm/WSe2 for different U. In DFT (U = 0), the calculated MAE values of Sm/WSe2 are 7.2, 10.9, and 10.1 meV for the W-top, Se-top, and hollow site, respectively. For the W-top site, MAE increases with U, reaching ∼33.2 meV at U = 6 eV. On the contrary, MAE becomes negligibly small with U ≥ 2 eV for the Se-top and hollow sites; 0.7 meV for the Se-top and −0.4 meV for the hollow site at U = 6 eV. These MAE values of the W-top site Sm in positive stand for the preferable direction of magnetization perpendicular to the WSe2 plane. From a practical viewpoint, the atomic-scale MAE values up to 60 meV/adatom have been achieved in individual Co, Fe, and Mn atoms on Pt, CuN, and MgO surfaces.24–27 Our prediction in support with these experimental studies suggest a possible achievement of the 4f-orbital-driven large PMA at the atomic length scale. This prevents a stable magnetization axis from thermal fluctuations, which is one of the major problems in memory applications. The thermal stability factor Δ is maintained by the large PMA through Δ = (KV)/(kBT), where K, V, kB, and T are the magnetic anisotropy constant, volume, Boltzmann constant, and temperature, respectively.

FIG. 2.

(a) Magnetocrystalline anisotropy energy MAE and (b) orbital magnetic anisotropy OMA of Sm/WSe2 structure from DFT+U calculations for different U parameters. Difference of spin–orbit coupling energies, ΔEsoc, between in- and out-of-plane magnetization orientation for the (c) W 5d and (d) Sm 4f orbital states of Sm/WSe2 structure. The positive and negative contributions to ΔEsoc are denoted by the orange and blue cones, respectively.

FIG. 2.

(a) Magnetocrystalline anisotropy energy MAE and (b) orbital magnetic anisotropy OMA of Sm/WSe2 structure from DFT+U calculations for different U parameters. Difference of spin–orbit coupling energies, ΔEsoc, between in- and out-of-plane magnetization orientation for the (c) W 5d and (d) Sm 4f orbital states of Sm/WSe2 structure. The positive and negative contributions to ΔEsoc are denoted by the orange and blue cones, respectively.

Close modal

We further inspect the relationship between the orbital magnetic anisotropy (OMA) and MAE according to Bruno’s model:28  MAE=ξ4μBΔmo, where ξ is the strength of SOC and Δmo is the orbital moment difference defined as Δmo=momo. For the W-top Sm, the absolute values of mo and mo are 0.38 (3.64) and 0.99 (3.80) μB at U = 6 (0) eV, respectively. The calculated Δmo values of Sm adatom are shown in Fig. 2(b) for different U in DFT+U. The main trend of MAE is well preserved for OMA in inverse. These results thus adequately obey the Bruno relation – the easy magnetization axis coincides with the direction that has the largest orbital moment. Naively, one can note that the Bruno model is illustrative in the analysis of an individual RE magnet’s anisotropy.

In order to understand the physical origin of large PMA of the W-top Sm on WSe2 monolayer, we analyze the difference of SOC energies between in-plane and out-of-plane magnetization, ΔEsoc=EsocEsoc, in Figs. 2(c) and 2(d) for the W 5d and Sm 4f orbital states, respectively. The expectation value of Esoc is twice the actual value of the total energy correction to the second order in SOC, i.e., MAE ≈ 1/2ΔEsoc.29,30 Our calculations indicate that MAE (33.2 meV) from the total energy calculations is in agreement within a few percent accuracy with that (31.9 meV) obtained from the atom projected SOC energy calculations. The other 50% of the SOC energy translates into the crystal-field energy and the formation of the unquenched orbital moment.31,32 The MAE values of Sm and W atoms obtained from the atom decomposed SOC energy calculations are 22.8 and 1.5 meV, respectively. Note that there are six neighboring atoms of W to Sm, which all provide similar contributions (1–1.5 meV) to MAE. As shown in Figs. 2(c) and 2(d), the Sm 4f orbitals dominate MAE whereas the contributions from the W 5d orbitals are an order of magnitude smaller. For the W 5d orbitals, we find that the positive MAE arises primarily from the SOC ⟨±2|∓2⟩ pairs. For the Sm 4f, the SOC pairs involving the m = ±2 states provide the dominant contribution to the large MAE.

For more feasibility and insights, it would be instructive to explore the vacancy defects in WSe2 and Sm/WSe2 structures. In the 6 × 6 supercell calculations of Sm-free WSe2 monolayer, we have considered various vacancy defects from a single atom vacancy to pairs of W and Se vacancies (not shown). Our calculations show that the presence of 2–6 Se vacancies next to a W vacancy can induce magnetism to their neighboring W and Se sites (about 2–4 μB) due to the lattice distortion and charge redistribution effects. The similar results for some configurations have been reported in previous first-principles calculations.33,34 These findings should be clarified and suggest the consideration of possible vacancy defects incorporation to the RE dopant atoms on TMD structures in the further calculations.

In summary, we performed the DFT+U calculations to investigate the binding affinity and intrinsic magnetic properties of an individual Sm on WSe2 monolayer. Our calculations demonstrate the favorable formation of Sm and WSe2 pairs, where Sm adatom prefers to adsorb at the W-top site rather than the Se-top and hollow sites in WSe2 structure. We predicted large MAE values of ∼7–33 meV per Sm adatom, depending on U parameter in DFT+U calculations, adsorbed at the most stable W-top site in WSe2 monolayer. This large perpendicular anisotropy with low magnetization and small volume is the key factor to realizing materials with a low switching current and high thermal stability for spintronics applications.

This work is supported by the US DoD Office of Naval Research Global under award No. N62909-23-1-2035 and National Research Foundation of Korea (NRF) grant No. RS-2023-00257666 funded by the Korea government (MSIT).

The authors have no conflicts to disclose.

N. Batnyam: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). T. Ochirkhuyag: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). D. Odkhuu: Conceptualization (equal); Funding acquisition (equal); Supervision (equal); Writing – original draft (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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