Abstract
Using firstprinciples calculations, we investigate the magnetic order in twodimensional (2D) transitionmetaldichalcogenide (TMD) monolayers: MoS_{2}, MoSe_{2}, MoTe_{2}, WSe_{2}, and WS_{2} substitutionally doped with period four transitionmetals (Ti, V, Cr, Mn, Fe, Co, Ni). We uncover five distinct magnetically ordered states among the 35 distinct TMDdopant pairs: the nonmagnetic (NM), the ferromagnetic with outofplane spin polarization (Z FM), the outofplane polarized clustered FMs (clustered Z FM), the inplane polarized FMs (X–Y FM), and the antiferromagnetic (AFM) state. Ni and Ti dopants result in an NM state for all considered TMDs, while Cr dopants result in an antiferromagnetically ordered state for all the TMDs. Most remarkably, we find that Fe, Mn, Co, and V result in an FM ordered state for all the TMDs, except for MoTe_{2}. Finally, we show that Vdoped MoSe_{2} and WSe_{2}, and Mndoped MoS_{2}, are the most suitable candidates for realizing a roomtemperature FM at a 16–18% atomic substitution.
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Introduction
The recent experimental realization of twodimensional (2D) magnetic crystals like CrI_{3}^{1,2,3}, CrGeTe_{3}^{4}, and VSe_{2}^{5}, has sparked great interest for possible applications like spintronics^{6,7}, valleytronics^{8}, and skyrmion^{9}based magnetic memories^{10,11}. However, magnetic order in 2D magnetic crystals is hampered because of low magnetic anisotropy and weak exchange interaction strength^{12}, resulting in low Curie temperatures (e.g., 45 K for CrI_{3} and 42 K for CrGeTe_{3}), which limits their use in commercial applications.
Semiconducting 2D materials doped with impurity atoms, e.g., doped graphene^{13} and metaldoped transitionmetal dichalcogenides (TMDs)^{14,15,16,17,18}, have emerged as promising candidates for hightemperature 2D magnetic order. Semiconductors doped with transition metal impurities couple the properties of semiconductors and magnets and are called dilute magnetic semiconductors (DMS). The ability to control magnetic order through charge transfer in a DMS^{19,20,21} opens up the possibility for realizing magnetic devices because their magnetic state can be controlled using an external electric field^{22}.
Among semiconducting 2D materials, which can be used as a base material for 2D DMSs, TMDs are of special interest. The heavy atomic mass of TMDs can lead to larger magnetic anisotropy, which is necessary for the existence of magnetic order in 2D^{23}. The interest in TMDs is further fueled by recent experimental results that have demonstrated the existence of stable magnetic order in TMDs doped with a transition metal impurity^{24,25,26}.
For the technological application of magnetically doped TMDs, it is necessary to find the optimal combination of a TMD and a dopant. However, the number of possible TMDs and dopant combinations is too large for a comprehensive experimental investigation, and theoretical guidance is desired. There have been previous theoretical attempts at modeling doped TMDs and calculating their critical temperature^{14,15,16,27}. However, previous theoretical predictions of the Curie temperature of doped TMDs have predicted unrealistically high Curie temperatures in excess of 1000 K at low doping concentrations (≈5%)^{14,15,16,27}, whereas experimental observations to date suggest a Curie temperature below 350 K at such doping levels^{24,25,28,29}.
The reason for the discrepancy between the experimental and the theoretical work is that previous theoretical works have ignored the effect of magnetic anisotropy^{30} and have used the collinear magnetic approximation^{31}. Most of the previous theoretical works have either calculated the Curie temperature using the Ising model^{15}, or the meanfield approximation^{14}. Unfortunately, both methods (Ising and meanfield) result in an overestimation of the actual Curie temperature^{32}. Moreover, previous theoretical works have taken into account only a few combinations of dopants and TMDs, and a comprehensive study for a vast set of TMDs doped with period four transition metals is still missing.
In this work, we calculate the critical temperature (Curie temperature or KosterlitzThouless (K–T) transition temperature) for a set of 2H TMDs: MoS_{2}, MoSe_{2}, WSe_{2}, WS_{2}, and MoTe_{2}, substitutionally doped with all the period four transition metals starting from Ti to Ni (Ti, V, Cr, Mn, Fe, Co, Ni). To model the magnetic structure of doped TMDs, we use our method, developed in ref. ^{12}. We model the magnetic exchange interaction (J) using a parametrized functional form (J(r)) and fit its parameters to firstprinciples calculations. Finally, we calculate the concentrationdependent critical temperature using the MonteCarlo method. First, we apply our method to one of the TMDs, MoSe_{2}, doped with all period four transition metals, and show all the possible magnetic ordered states originating from different dopants. Next, we present the magnetically ordered states and the critical temperature for all the TMDs doped with period four transition metals. We find that out of the thirtyfive material combinations investigated, ten are nonmagnetic (NM), nine are antiferromagnetic (AFM), and sixteen are ferromagnetic (FM). Out of the 16 FMs, 6 are FMs with an inplane magnetic easyaxis, and the other 10 have an outofplane easyaxis. We find that the most promising FMs can be realized by doping MoSe_{2} and WSe_{2} with V along with doping MoS_{2} with Mn, which have a Curie temperature of approximately 200 K at an atomic substitution of 15%.
Results
Computational model
Figure 1 illustrates our computational model, which has two parts: DFT and SpinModel. To reduce the computational cost and weed out the candidate ferromagnets, in the first part (DFT), we determine the magnetic ground state of the doped TMD by substituting the transition metal with two dopants in a supercell of size 3 × 3 × 1. If out of all possible magnetic configurations, the ferromagnetic state has the lowest energy, we make bigger supercells (4 × 4 × 1, 5 × 5 × 1, and 7 × 7 × 1) of the corresponding TMD and dopant. We substitute two transitionmetal atoms (W/Mo) in the TMD supercells with dopant atoms separated at distances ranging from the nearest neighbor to the fifth neighbor. We calculate the total energy for both the FM and the AFM magnetic orders with both the inplane and the outofplane magnetic easy axis for the bigger supercells of TMDs^{12}.
In the second part (Spinmodel), we model the magnetic structure of doped TMDs using a classical Heisenberg Hamiltonian, which features parameterized exchange tensor J_{i,j}, and onsite anisotropy D determined from DFT^{12}. Specifically, we approximate the elements of J_{i,j} tensor as a continuous function of distance J(r_{i} − r_{j}) (see Eq. (2) in the Methodology section). We go beyond the nearestneighbor interaction because longrange interaction plays a decisive role in determining the magnetically ordered state of doped materials^{33,34}. We take into account the exchange interactions up to the 5th neighbor (N = 5), beyond the 5th nearest neighbor the exchange interaction (J(r)) is numerically truncated.
To obtain the parameters for J(r), we fit the parameterized Heisenberg Hamiltonian to the total energy obtained in the DFT step. The details of our fitting procedure are outlined in ref. ^{12}. We study the phase change of the parameterized Heisenberg Hamiltonian for large (40 × 40) supercells with an atomic substitution ranging from 6% to 18%, using the MonteCarlo (MC) algorithm. We obtain the median critical temperature (Curie/K–T) from the peak of the average susceptibility for each percent atomic substitution, obtained from the MC simulations. For each percent substitution, we average over 20 different substitutional configurations to account for configurational entropy. We provide further computational details and parameters at the end of this article.
Magnetic order in MoSe_{2}
We first apply our computational method to MoSe_{2} doped with all period four transition metals. We present all possible magnetically ordered states in doped MoSe_{2}: the nonmagnetic state (NM), the ferromagnetic state with outofplane spin polarization (Z FM), the outofplane polarized clustered FMs (clustered Z FM), the inplane polarized FMs (X–Y FM), and the antiferromagnetic state (AFM). We then apply our method to all the TMDs doped with period four transition metals and calculate their critical temperature and uncover their magnetically ordered phase at the atomic substitution of 15%.
We perform DFT calculations on a 3 × 3 × 1 cell of MoSe_{2} doped with two dopant atoms (Ti, V, Mn, Cr, Fe, Co, Ni), which amounts to 22.22% atomic substitution. We define the atomic substitution using, \(\frac{{N}_{{\rm{dopant}}}}{{N}_{{\rm{transitionmetal}}}}\), where N_{dopant} is the number of dopant atoms and N_{transitionmetal} is the total number of transition metal atoms in the TMD supercell. We find that Ni and Ti as dopants result in a negligible magnetic moment of 0.09 μ_{B} per dopant. Hence, Ni and Ti as dopants result in a weaklymagnetic/nonmagnetic (NM) state in MoSe_{2}. On the other hand, V, Cr, Mn, Fe, and Co have a magnetic moment of 1.7 μ_{B}, 3.0 μ_{B}, 3.7 μ_{B}, 3.1 μ_{B}, and 1.6 μ_{B} per dopant, respectively. We find that the magnetic easyaxis is inplane for Fe and Mn, whereas it is outofplane for V and Codoped MoSe_{2}.
Figure 2a shows the concentrationdependent critical temperature (Curie temperature for outofplane FM and KosterlitzThouless transition temperature for inplane FM) of MoSe_{2} doped with V, Mn, Fe, and Co. We observe that Vdoped MoSe_{2} exhibits roomtemperature outofplane FM at an atomic substitution of about 16.5%, and Fedoped MoSe_{2} exhibits roomtemperature inplane FM at an atomic substitution of 16%. We also observe that the variance in the obtained critical temperatures is very low across different substitutional configurations (<30 K) except for Co doping. Low variance implies that the critical temperature is robust to the random position of dopants in V, Fe, and Mndoped MoSe_{2}.
Figure 2b shows the saturation magnetization (M = √(S_{x}^{2} + S_{y}^{2} + S_{z}^{2})) per dopant atom in V, Mn, Fe, and Codoped MoSe_{2}, obtained from the MC simulations at an atomic substitution ranging from 6% to 18%. The dotted lines show the starting magnetization of each magnetic dopant at the start of the MC simulation (also shown in Table 1). We observe that the saturation magnetization in Vdoped MoSe_{2} remains almost flat, and decreases slightly at a percent substitution below 7%. For Fe, the saturation magnetization increases with increasing atomic substitution and saturates to its maximum magnetization at around 18% atomic substitution. However, for both Mn and Codoped MoSe_{2}, the saturation magnetization remains far below their respective maximum magnetization.
Figure 3a shows the magnetically ordered state in a sample of V, Cr, and Codoped MoSe_{2} at an atomic substitution of 15%, at a temperature of 5 K. We plot the magnetization in the outofplane direction \(({\hat{S}}_{{\rm{z}}}={S}_{{\rm{z}}}/ S )\). The magnetically ordered state in V and Codoped MoSe_{2} is FM with an outofplane easy axis, whereas, for Cr substitution, the magnetically ordered state is AFM. We observe that the magnetically ordered state of Vdoped MoSe_{2} saturates to a perfect FM state with an outofplane easyaxis. For Co dopants, we find that there are clusters of FMoriented Co ions, but the longrange order is missing. In the case of Cr, we observe that the magnetically ordered state has a randomized magnetic order with spins orienting randomly. The reason for such a randomized magnetically ordered state is that the AFM order leads to a magnetic frustration for some clusters, which leads to a randomized orientation for the magnetic moments.
Figure 3b shows the magnetically ordered state in a sample of Fe and Mndoped MoSe_{2} at an atomic substitution of 15% at a temperature of 5 K. Because the magnetic easyaxis is inplane, we plot the inplane angle (ϕ) of each dopant atom: \(\phi ={\cos }^{1}({S}_{{\rm{x}}}/ {S}_{\parallel } )\), where, S_{∥} is the inplane magnetization. We observe that for both Mn and Fe, the orientation of the magnetic moment remains randomized with some shortrange (<8 Å) order. For both Fe and Mn, we observe two effects. First, we observe domain formation with FM clusters. Second, the FM clusters are not perfectly ferromagnetic because their magnetic order breaks at slightly longer distances. We will discuss domain formation separately later in this section. The slight breaking of magnetic order at a longer distance appears due to the KosterlitzThouless transition behavior^{35}, where, even at temperatures below the KT transition temperature, longrange magnetic order does not exist.
Interestingly, we see some level of quasi vortex formation both for Fe and Mndoped MoSe_{2}. The observation is in line with the K–T physics for magnets with inplane anisotropy^{35}. However, due to broken translational symmetry because of the random placement of defects, as well as finite inplane anisotropy, vortex formation is imperfect. Nevertheless, observation of quasi vortices in doped MoSe_{2} implies that the topological vortices in the K–T transition are quite robust to lattice imperfections. However, an indepth analysis of this phenomenon is beyond the scope of this article.
Figure 4 shows the exchange function J(r) for doped MoSe_{2}. The solid line shows the average between the exchange between electrons with moments along the x and zdirection (J(r) = (J^{x}(r) + J^{z}(r))/2), and the dots show the calculated discrete J parameters obtained from the DFT calculations using the J_{1} − J_{2} model^{31}. We define a distance >7 Å as longrange, and <7 Å as shortrange (up to the thirdnearest neighbor).
We observe that J(r) is the strongest for V, Fe, and Codoped MoSe_{2} in the short range. For Mn, J(r) is weaker in the short range. Whereas, for Cr, J(r < 7 Å) is negative, signifying an AFM interaction. Looking at the longrange interaction beyond 8 Å (inset of Fig. 4), we find that the longrange interaction is strongest in Vdoped MoSe_{2} and significantly weaker in Co and Mndoped MoSe_{2}. In Fedoped MoSe_{2}, the longrange interaction is the second highest.
Analyzing the ordered magnetic states for each dopant shown in Fig. 3a and b, their saturation magnetization in Fig. 2b, and their J(r), we find that a strong shortrange interaction results in FM cluster formation, e.g., in Co, Fe, and Mndoped MoSe_{2}, and the clusters then orient randomly due to weak longrange intercluster interaction. At higher percent atomic substitution in Fedoped MoSe_{2}, we find that bigger clusters start forming as seen by the increase in saturation magnetization for Fedoped MoSe_{2} at higher atomic substitution, as shown in Fig. 2b. However, the X–Y nature of magnetism prohibits the longrange order in Fedoped MoSe_{2}. It should also be noted that the appearance of a saturation magnetization at these concentrations in the X–Y magnets is due to the finite size of the lattice, the random position of the magnetic ions, and the longrange behavior of J(r).
To summarize this section, we find that five magnetically ordered states are possible, depending on the J(r) in a doped TMD including,

1.
The nonmagnetic state (Ti and Nidoped MoSe_{2}).

2.
The frustrated AFM ordered state (Crdoped MoSe_{2}).

3.
The Z FM ordered state with strong longrange interactions, where we observe full FM orientation with their magnetic easyaxis in the outofplane direction (Vdoped MoSe_{2}).

4.
The clustered Z FM ordered state with strong shortrange and weak longrange interactions, where we observe weakly interacting clusters of FMs (Codoped MoSe_{2}).

5.
The X–Y ordered state, where we observe weakly aligned inplane FM clusters, but the longrange order remains significantly randomized at any finite temperature (Mn and Fedoped MoSe_{2}).
Critical temperature of MoS_{2}, WS_{2}, MoSe_{2}, WSe_{2}, and MoTe_{2}
Figure 5 shows the magnetically ordered state, as well as the critical temperature at an atomic substitution of 15% for the selected combination of the TMD and the dopant. Material combinations indicated with "Z" are ferromagnetic (FM) with their magnetic easyaxis in the outofplane direction, while "X–Y" indicates the X–Y magnets, with an inplane easy axis. The TMD dopant combinations, which are shaded, are clustered Z FM. Materials indicated as AFM have an antiferromagnetic (AFM) ordered state, while (NM) represents a nonmagnetic (paramagnetic) state. As discussed above, FMs with an outofplane magnetic easy axis have a Curie temperature, while, FMs with an inplane magnetic axis have a KosterlitzThouless phase transition, which results in a quasiordered magnetic state, but the longrange order remains randomized. We provide the full atomic substitutiondependent (6% to 18%) critical temperature for all the Z and the X–Y ferromagnets in the supplementary documentation (Supplementary Table. 2).
Some general trends can be extracted from Fig. 5. We observe that Cr dopants result in an AFMordered state for all the TMDs. Ti and Ni as dopants result in a nonmagnetic state. Interestingly, V, Fe, and Mn always result in an FM ordered state for all the TMDs except for MoTe_{2}, for which only Fe and Co result in an FM state. Co dopants result in an AFM ordered state for disulfides (MoS_{2}, WS_{2}), but they result in an FM ordered state for diselenides (MoSe_{2}, WSe_{2}), and MoTe_{2}. V dopants always yield an outofplane FM. Whereas, Mn and Fe substitution result in an X–Y magnetic order for WS_{2}, MoSe_{2}, and WSe_{2}.
The main highlights of Fig. 5 are the material combinations that result in an outofplane FM with strong longrange interaction. We find five combinations with V as a dopant for all the TMDs except MoTe_{2} and Mndoped MoS_{2}. Mn substitution in MoS_{2} results in an FMordered state, with high median Curie temperature of 190 K at 15% atomic substitution. Also, V as a dopant in MoSe_{2} and WSe_{2} results in an outofplane FM with a high Curie temperature measuring ≈200 K.
Finally, we would like to mention that the electronic origin of magnetism in doped TMDs is a result of the superexchange interaction in the shortrange^{14,16} and carriermediated interaction in the long range. For example, the electronic origin of magnetism and the doping stability in MoSe_{2} are briefly discussed in the supplementary document (Supplementary Fig. 4).
Discussion
We have presented the magnetic order in TMDs doped with period four transition metals. We have determined the nature of the magnetically ordered states, as well as their critical temperature as a function of percent atomic substitution. We showed that there are five possible magnetically ordered states for doped TMDs, depending on the nature of their exchange interaction J(r), the magnetic anisotropy, and the atomic substitution. The possible magnetically ordered states are nonmagnetic (NM), perfectly ferromagnetic (FM Z), clustered ferromagnetic (clustered FM Z), X–Y ferromagnetic (X–Y FM), and randomized antiferromagnetic (AFM).
We have shown that Ti and Ni dopants always result in a nonmagnetic state. Moreover, Cr dopants result in an AFM configuration for all the TMDs. Both Mn and Fe dopants result in an X–Y magnet for MoSe_{2}, WS_{2}, and WSe_{2}. From this study, we conclude that the best chance of realizing a 2D DMS using TMDs with roomtemperature Curie temperature is found in Mndoped MoS_{2} and Vdoped MoSe_{2} and WSe_{2} at an atomic substitution in excess of 16.5%.
We have provided a generalized method of modeling the magnetic interaction in doped 2D materials. For further usability of our method, the parameters of the functional form for all the TMD and dopant combinations are provided in a supplementary document (Supplementary Table 1).
There have been recent experimental reports regarding magnetic order in TMDs^{24,25,26}, and FM clusters have been detected in Vdoped WSe_{2}, using magnetic scanning tunnel microscopy (MTM)^{26}. The magnetically ordered states presented in this work for transitionmetal doped TMDs, and their critical temperature can be verified experimentally using a similar procedure as used in^{26}. Moreover, recent experimental reports have shown that the transitionmetal substitution is often accompanied by vacancies^{36,37}, and a possible future extension of our work will be to include the impact of structural defects on the magnetic order of TMDs.
Methods
Magnetic structure and the exchange interactions
We model the magnetic structure of doped TMDs using a parameterized Heisenberg Hamiltonian assuming a localized nature of the magnetic interaction^{38},
The first term is the exchange term between the i^{th} and the j^{th} magnetic atom (dopant) with S = S_{x}x + S_{y}y + S_{z}z, as the magnetic moment vector. J_{i,j} is the strength of the exchange interaction between the i^{th} and the j^{th} magnetic atoms and is a tensor as described in ref. ^{12}. Because anisotropy plays an important role in determining the magnetic ground state of doped magnetic systems^{39}, we take into account the J_{i,j} tensor instead of an effective isotropic exchange. The second term is the onsiteanisotropy with strength D. We only use the diagonal elements of J_{i,j} which are, J^{xx}, J^{yy}, and J^{zz} for the magnetic axis in the x, y, and z direction, respectively. Because of the inplane isotropy in TMDs, we modify the J_{i,j} tensor by choosing, J^{xx} = J^{yy} = J^{∥} and J^{zz} = J^{⊥}, with J^{∥} being the inplane exchange interaction, and J^{⊥} being the outofplane exchange interaction.
We approximate J_{i,j} as a function of distance J(r) because we go beyond the nearestneighbor interaction. The functional form is,
Here, h(r) is the Heaviside step function. r_{c} is the cutoff radius within which we approximate the J parameters using Bsplines B_{i}(r)^{40} with order 3, and outside r_{c} we approximate them using an exponential decay \({c}^{\perp /\parallel }\ \exp (r/\lambda )\)^{33,34}. Parameters A^{⊥/∥} and c_{i} are the free parameters. We choose r_{c} to be within the third nearestneighbor range, which is 7 Å for all the TMDs. Because of the continuity at the boundary r = r_{c}, the parameter c^{⊥/∥} and λ have an analytical form in terms of the spline functions,
Note that, traditionally, going beyond the nearestneighbor interaction increases the number of parameters as 2N, where N is the interaction range. For example, for nextneighbor interaction, N = 2, and the total number of J_{i,j} parameters required to model the magnetic structure is 4. However, in our method, we take into account the exchange interactions up to the 5th neighbor (N = 5). Thanks to the functional form (J(r)) we use, the number of free parameters remains fixed to five. The parameters of the functional form for all the TMDs and the dopant combinations are provided in a separate supplementary document (Supplementary Table 1).
It should be noted that the generalized functional form of the Eq. (2) is useful for materials with inplane rotational invariance. Materials with broken rotational symmetry require an additional parameter to account for the angular dependence of J(r).
The magnitude of the magnetic moment for the MonteCarlo simulations are obtained from DFT using
Here, \({M}_{{\rm{DFT}}}^{l,j}\) is the average of the magnetic moment of the jth magnetic atom of lth magnetic configuration obtained from DFT. N_{c} and N are the total number of the magnetic configurations simulated and the magnetic atoms, respectively^{12}.
DFT calculations
All the abinitio DFT calculations reported in this work were performed using the Vienna abinitio simulation package (VASP)^{41,42}. The ground state selfconsistent field (SCF) calculations were performed using a projectoraugmented wave (PAW) potential^{41} with a generalizedgradient approximation as proposed by PerdewBurkeErnzerhof (PBE)^{43}. We have used a kinetic energy cutoff of 450 eV for our DFT calculations. The Brillouin zones were sampled using a Γcentered kpoint mesh of size 5 × 5 × 1 points for 4 × 4 × 1 supercells, and 3 × 3 × 1 points for 5 × 5 × 1 and 7 × 7 × 1 supercells. The TMD supercells doped with transition metals were relaxed until the force on each of the ions was below 10 meV/Å. The energy convergence criterion for the subsequent SCF calculations was set to 10^{−4 }eV.
We have used the Hubbard U model within DFT + U^{44} to take into account the electronelectron interaction in the d orbital of the magnetic transitionmetal atoms. We use the linear response method^{45} to determine the Hubbard U parameter for the d orbitals of the dopant atoms. The U values we obtain from the linear response calculation are in the range 4 − 6 eV for Ti, V, Cr, Mn, Fe, Co, and Nidoped TMDs. For the transitionmetal and the chalcogen atoms of the TMDs, we use a U = 0 for all their orbitals. We have verified our results by applying a U on the dorbital of the base transition metal atoms for the TMDs, and our result does not change qualitatively and quantitative changes were small. For example, for Crdoped MoS_{2} the nearneighbor antiferromagnetic interaction increased by a mere 4% when we applied a U = 4 eV on the Mo atoms.
To obtain Eq. (2) parameters, we place two dopant atoms in a supercell of the base TMD at various positions (near, next, and nextnext neighbor) and calculate the total energy of various magnetic configurations. We use supercells of sizes 4 × 4 × 1, 5 × 5 × 1, and 7 × 7 × 1 to calculate the total energies of various magnetic configurations.
MonteCarlo simulations
We simulate the phase change of the Heisenberg Hamiltonian using MonteCarlo (MC) simulations. We obtain the critical temperature from the peak of the susceptibility obtained from the MC simulations. The critical temperature was calculated only for the samples with M_{sat }≥ 0.33 × M_{start}, where M_{sat} is the saturation magnetization per dopant atom obtained from MC, and M_{start} is the starting magnetization per dopant atom obtained from DFT. To ensure that we capture the effect of configurational entropy, we run 20 separate MC calculations for each material combination and percent atomic substitution, each starting from randomly doped configurations of a 40 × 40 supercell of a TMD. We use a pseudorandom number generator to generate random positions in a pristine TMD lattice, and place the dopant atoms at those positions. We investigate TMDs with an atomic substitution ranging from 6% to 18%. For each randomly doped configuration, we use 1000 equilibration steps, and 1000 MC steps for averaging observables, at each temperature step. For each equilibration and MC step, we perform N_{atom} spinflip steps, where N_{atom} is the number of dopant atoms in the unit cell.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The codes that are necessary to reproduce the findings of this study are available from the corresponding author upon reasonable request. All DFT calculations were performed by using the Vienna abinitio simulation package (VASP).
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S.T., W.G.V. and B.S. conceived the project. S.T. developed the code and performed the simulations. S.T., M.L.V.de.P. and W.G.V. analyzed the obtained results. S.T. wrote the paper with all the authors contributing to the discussion and preparation of the manuscript.
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Tiwari, S., Van de Put, M.L., Sorée, B. et al. Magnetic order and critical temperature of substitutionally doped transition metal dichalcogenide monolayers. npj 2D Mater Appl 5, 54 (2021). https://doi.org/10.1038/s41699021002330
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DOI: https://doi.org/10.1038/s41699021002330
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