Magnetic Order, Electrical Doping, and Charge-State Coupling at Amphoteric Defect Sites in Mn-Doped 2D Semiconductors

INTRODUCTION

Dilute magnetic semiconductors (DMSs) have attracted much attention because of their potential nanoscale magnetic and electronic applications. [1][2][3][4][5][6][7][8][9][10] A fundamental challenge in nanoscale spin-based electronics or spintronics is finding two-dimensional (2D) magnetic materials with both high Curie temperature (T c ) and large spin polarization, which offer opportunities for the next-generation information technology via faster device operation and high information storage density. [11][12][13][14][15][16][17][18] Recently, 2D-CrI 3 13 and 2D-Cr 2 Ge 2 Te 6 14 have shown promise for low-temperature 2D spintronics. However, the T c of these materials is far below the liquid-nitrogen temperature (77 K) and hence restricts their applications. Therefore, developing 2D magnetic materials possessing robust long-range spin ordering and high T c is a significant outstanding challenge in materials design.

Within all the 2D materials, the transition metal dichalcogenides (TMDs) family, including MoS 2 and WS 2 nanosheets, has been heavily studied for a variety of nanoelectronic and nano-optoelectronic applications. [19][20][21][22][23][24][25][26][27][28][29] Pristine monolayer TMDs are nonmagnetic semiconductors, and the artificial manipulation of the valence d orbitals can induce magnetism, half-metallicity, and electrical control of spin polarization. 22,23,25,[30][31][32] To obtain stable magnetism in 2D-TMDs at room temperature and beyond, various methods have been proposed, and among these, magnetic doping has proven to be a successful strategy. [33][34][35][36][37][38] Due to the many structural and electronic degrees of freedom present at doping concentrations high enough to observe magnetic ordering, the relationship between atomic doping, electronic doping, and magnetic ordering remains underexplored.

Previous studies have shown that isolated Mn doping at the Mo site (i.e., Mn Mo ) in 2D-MoS 2 promotes magnetism, [39][40][41] but experimentally these isolated defects have been observed to cluster together. Apart from isolated Mn Mo , Mn Mo -Mn Mo (i.e., 2Mn 2Mo ) cluster formation is observed in 2D-MoS 2 , and these clusters appear more than the well-dispersed single Mn Mo . 42 Moreover, other experimental reports 43 suggest the presence of many other Mn-defect complexes such as a Mn atom at sulfur site (Mn S ), a Mn atom on top of Mo (Mn-top), and a Mn atom at a Mo site (Mn Mo ). The role of these alternative dopant-defect structures on the magnetic and electronic properties of 2D-TMDs has not yet been considered. Furthermore, in previous studies of isolated Mn Mo , the defect charge state for 3d impurities was not considered, and this is both highly sensitive to the Fermi-level position of the host material and critically important for magnetic ordering. [44][45][46] The charge state can be controlled by manipulating the Fermi level through free-carrier doping [47][48][49] via other structural defects, doping by suitable electron donors and acceptors in 2D-TMDs, or electrostatic gating. 50,51 Therefore, we study the thermodynamics of defect formation and the effect of charge state on the magnetic ordering across experimentally observed Mn defects and their complexes in 2D-MoS 2 and 2D-WS 2 .

Apart from a high T c , a large magnetic anisotropy energy (MAE), i.e., the energy barrier favoring a particular spatial orientation of the magnetic moments, is a desirable quantity for an ideal dilute magnetic semiconductor. 39,40 Magnetic doping, such as a Mn atom at a Mo site in 2D-MoS 2 or at a W site in 2D-WS 2 , leads to a redistribution of local orbital energies due to the crystal field effect; the dopant orbital magnetic moment distribution and the associated MAE are determined by the crystal field effect. 40,41 In this work, all the calculations have been performed using hybrid density functional theory. We have carried out a comprehensive study of magnetic and electronic properties of experimentally observed Mn defects and complexes such as a Mn atom at a sulfur site (i.e., Mn S ), two Mn atoms at a Mo site (i.e., 2Mn Mo ), a Mn atom on top of a Mo atom (Mn-top), a Mn atom at a Mo site (Mn Mo ) in 2D-MoS 2 , and a Mn atom at a W site (Mn W ) in 2D-WS 2 . Most of them provide deep levels and free electron/hole trapping centers. Interestingly, Mn Mo is stable as both a donor and an acceptor defect within the band gap, which was previously predicted to be an only donor defect. Mn S and 2Mn Mo are stable as an electrically neutral and a negative-U defect, respectively. To simulate the effect of clustering, we consider two Mn dopant atoms at neighboring sites, calculate the formation energy, and compare it to the isolated defects. The defect clusters we considered are Mn Mo -Mn Mo (i.e., 2Mn 2Mo ) and Mn W -Mn W (i.e., 2Mn 2W ) in 2D-MoS 2 and 2D-WS 2 , respectively, and found that they are stable in ferromagnetic (FM) and antiferromagnetic (AFM) coupling in the neutral and +1 charge state, respectively. Unlike isolated Mn Mo , the clustered 2Mn 2Mo provides a shallow donor level in the AFM coupling. Although most of the doping properties in WS 2 are similar to those in MoS 2 , the 2Mn 2W cluster instead provides deep levels in both FM and AFM coupling. We demonstrate the AFM to FM phase transition at a critical electron density n c e = 3.5 × 10 13 cm -2 , in the n-doped 2D-MoS 2 and 2D-WS 2 . Finally, for a reasonable Mn doping concentration of 1.85%, we calculated a Curie temperature of 580 K using the mean-field approximation.

RESULTS AND DISCUSSION

Pristine 2D-MoS 2 and 2D-WS 2 have hexagonal crystal structures, where metal atoms sandwiched between two layers of chalcogen atoms form chalcogen-metal-chalcogen bonds in the unit cell. 25,26 The calculated lattice parameters of 2D-MoS 2 and 2D-WS 2 are essentially identical and found to be 3.18 Å, agreeing with previous reports. 25 The bond length of Mo-S (W-S) is 2.42 Å (2.41 Å), and the distance of two S atomic planes is 3.14 Å (3.13 Å). The HSE06-calculated direct band gaps of 2D-MoS 2 and 2D-WS 2 are 1.9 and 2.1 eV, respectively, at the K-point, agreeing well with previous reports. 52,53 The optimized geometries of all the considered Mn defects are shown in Figure 1. The top views are shown in Figure 1(a to e), and the corresponding side views are shown in Figure 1(f to j). The pristine bond length of Mo-S is 2.41 Å, and after replacing a S with Mn and relaxing the Mn S defect structure, the substituting bond length of Mn-S is 2.62 Å, as shown in Figure 1(a). In 2Mn Mo , the Mn atoms bind with the three upper and lower S atoms with the Mn-S bond length of 2.18 Å, as shown in Figure 1(b). The optimized distance between Mn and S atoms in Mn-top is found to be 2.40 Å Figure 1(c). Figure 1(d) shows that in Mn Mo six S atoms surrounding the Mn site move inward, and the corresponding Mn-S distance is 2.30 Å. In Mn W , the corresponding relaxed W-S bond length is 2.31 Å, as shown in Figure 1(e). Next, Figure 1(k andl) represents the optimized geometry of Mn-Mn clustering or pairs, and the relaxed bond lengths of Mn-S and W-S are 2.30 and 2.31 Å, respectively. The atomic radius of Mn (1.38 Å) is very close to that of Mo (1.40 Å) and W (1.41 Å) atoms, so we expected minimal lattice distortion at the metal sites; however, we found that Mn-Mn clustering leads to strong local lattice distortion as the corresponding Mn-Mn distances are 3.29 and 3.44 Å in 2D-MoS 2 and 2D-WS 2 , vs 3.18 and 3.20 Å in the respective pristine cases.

The spin-polarized projected electronic density of states (PDOS) for all isolated Mn defects in 2D-MoS 2 are analyzed and shown in Figure 2. Removing a S atom as in Figure 2(a) provides two excess electrons, which occupy the singlet d z 2 state near the valence band maxima (VBM) and the two empty doubly degenerate states above the Fermi level (E F ). These defect Kohn-Sham levels originate from Mo-(d xy , d x 2 -y 2 ) orbitals. Upon substituting Mn at the S site (Mn S ), the Mn-3d orbitals break the degeneracy of the spin channels and provide a local magnetic moment of 3 μ B , pushing the defect Kohn-Sham levels [Mo(d xy , d x 2 -y 2 ), Mn(d xy , d x 2 -y 2 ), Mn(d xz , d yz )] near the band edge, as shown in Figure 2(a). The E F is located in the middle of the band gap, and these defect states are present near the band edge; therefore, Mn S is expected to be stable as an electrically inactive defect. The strong lattice distortion in 2Mn Mo leads to several strongly localized defect states near the E F , Figure 2(b). These localized states mainly arise from the Mn-3d, Mo-4d, and S-3p states. A hybridization between Mn-3d, the closest Mo-4d, and S-3p states produces the localized states in the spin-up channel and spin-down channel, providing a local magnetic moment of 7 μ B . 2Mn Mo leads to half-metallicity in 2D-MoS 2 by providing a finite gap in the spin-up and spin-down channel, as shown in Figure 2(b).

In Mn-top, Mn-3d hybridizes with S-3p and Mo-4d states and leads to donor states near the conduction band, making the system metallic, as shown in Figure 2(c). After the hybridization, the Mn-3d antibonding states consisting of d xy and d x 2 -y 2 orbitals give states in only one spin channel at the E F , leading to a local magnetic moment of 4 μ B in 2D-MoS 2 ; the other spin states are split into the original valence band. In Mn Mo , Mn-3d orbitals strongly hybridize with the S-3p orbitals and provide midgap states, which further split and give the unoccupied antibonding states Mn(d xy , d x 2 -y 2 ) and Mn(d xz , d yz ) in the spin-down channel and the occupied bonding state Mn(d z 2 ) in the spin-up channel, as shown in Figure 2(d). This occupied spin-up bonding state Mn(d z 2 ) provides 1 μ B magnetic moment per Mn atom. The spin density of Mn Mo is shown in Figure S2 in the Supporting Information. Moreover, we observed no band gap in the spin-up channel and a finite band gap in the spin-down channel, ensuring halfmetallicity when Mn substitutes for Mo in 2D-MoS 2 .

For better insight into hybridization between Mo, W, Mn, and S, we analyzed the molecular orbital (MO) picture of 2D-MoS 2 and 2D-WS 2 as shown in Figure 3(a andb). The MO picture is constructed through the irreducible representation for the D 3h point group symmetry. 54 The lowest a′ 1 , e′, and e″ are all bonding orbitals, where a′ 1 is mostly composed of chalcogen p x and p y orbitals and e′ and e″ lie on both metal and chalcogen atoms. The a″ 2 orbitals which originate through chalcogen p z orbitals do not hybridize with a metal atom. The highest occupied orbital a′ 1 is antibonding (near the Fermi level) and arises through Mo-d [or W-d in Figure 3(b)] orbitals with a small contribution from chalcogen p x and p y orbitals. The e′ and e″ antibonding orbitals are the first unoccupied orbital sets, and these of Mo (or W)-d and S-p orbitals.

Using the band centers method, 55 we analyzed the crystalfield splitting (Δ cf ) and the interatomic Hund's exchange (Δ ex ) of Mn-3d orbitals; the schematic is shown in Figure 3(c to g) for the neutral charge-state configurations. A Mn S defect provides five midgap states, which split into three groups under the trigonal prismatic surroundings of the S atoms, as shown in Figure 3(c). The large overlap of S-3p orbitals with Mn(d xz , d yz ) makes the antibonding orbital (e″) degenerate and raises their energy. The reduced overlapping of the Mn-d z 2 antibonding orbital (a′ 1 ) with the S-3p orbitals does not change the energy of the Mn-3d orbitals. Moreover, the moderate overlapping of Mn(d xy , d x 2 -y 2 ) antibonding orbitals (e′) with S-3p orbitals holds them in between the a′ 1 and e″ orbitals, as shown in the spin-up channel of Figure 3(c). In the case of Mn S , the exchange splitting of the e orbitals, Δ ex = 1.7 eV, is larger than the crystal field splitting, Δ cf = 400 meV. For 2Mn Mo , we obtain Δ ex = 830 meV and Δ cf = 1.22 eV. In Mntop, the interaction of chalcogen p orbitals breaks the symmetry, as shown in Figure 3(e). In Mn Mo and Mn W , Mn substitution leads to occupied Mn-d z 2 states just below the conduction band, as shown in Figures 3(f) and (g). For Mn Mo (Mn W ), we find Δ ex = 381 (431) meV and Δ cf = 690 (552) meV.

The formation energy (E f ) of defects provides important information about their stability, concentration and the charge transition levels within the band gap. The E f of defect [X] in charge state q is given by

where E defect tot and E pristine tot are the relaxed energies of the defective and defect-free supercells, respectively. n i is the number of i-type atoms added (n i > 0) or removed (n i < 0) from the pristine supercell. μ i is the chemical potential of i-type element, referenced to the total energy of a Mo, W, and Mn atom in crystalline bcc phase and a S atom in crystalline orthorhombic α phase. For the stability of 2D-MoS 2 , the μ Mo and μ S satisfy the relation of μ Mo + 2μ S = μ MoS2 , where μ MoS2 is the total energy of 2D-MoS 2 per formula unit. In the Mo-rich limit condition, μ max Mo = E tot Mo , where E tot Mo is the total peratom energy of BCC Mo, and the corresponding S-poor condition is given by μ min S = (μ MoS2 -μ max Mo )/2. In the S-rich limit condition; μ max S = E tot S , in which the E tot S is the total peratom energy of the S orthorhombic crystal, where the S orthorhombic crystal is considered to be the reference source of S. The chemical potential conditions of μ W and μ S in 2D-WS 2 are determined analogously to that of μ Mo and μ S in 2D-MoS 2 . E F is the Fermi-level position, referenced to VBM. Δ q represents the charge-state correction due to the finite size of the supercell.

Defect transition levels (DTLs) are the positions of E F at which the thermodynamically favored charge state of the defect changes from q to q′:

where E f (X q ) and E f (X q′ ) are the formation energies of a defect [X] at the VBM, for charge states q and q′, respectively. For E F positions below ε(q/q′), charge state q is stable, and for the E F positions above ε(q/q′), charge state q′ is stable.

We calculate E f as a function of E F for all the experimentally observed Mn defects in 2D-MoS 2 in the Mo-rich and S-rich condition, as shown in Figure 4(a andb). Only the stable charge states are shown in Figure 4. The E f values as a function of E F of all these Mn defects in the S(Mo)-rich condition for all the possible charge states are shown in Figure S3 in the Supporting Information. Among all the Mn defects, Mn S has the lowest E f , which is 0.65 eV in the Mo-rich condition. However, it has the highest E f (∼3 eV) in the S-rich condition compared to other Mn defects in 2D-MoS 2 , Figure 4(b). Mn S is found to be most stable in the neutral charge state (i.e., Mn 0 S ) within the band gap. Therefore, Mn S is an electrically inactive defect and cannot provide n-type or p-type conductivity to 2D-MoS 2 . Moreover, Mn 0 S induces a local magnetic moment of 3 μ B , which antiferromagnetically couples with the surrounding Mo atoms and leads to an effective ferromagnetism. The E f of Mn 0 Mo is 1.93 and -0.41 eV in the Mo-rich and S-rich condition, respectively, Figure 4. Therefore, Mn Mo is more likely to form in the S-rich condition, suggesting a high concentration under typical S-rich synthesis conditions. As a neutral defect, Mn Mo had been previously predicted to be only a donor defect 39 by density of states analysis; however, interestingly, we find that Mn Mo is most stable in 0, -1 (an electron acceptor), and +1 (single-electron donor) charge state depending on the E F position, suggesting that this is an amphoteric defect. Mn +1 Mo acts as a deep hole trap center (i.e., deep donor defect), and the defect transition level (DTL) ε(+1/0) of this single donor occurs at 1.19 eV below the conduction band minima (CBM). It can also act as a deep electron trap center with the DTL of Mn ε(0/-1)

Mo at 1.41 eV above the VBM. The thermodynamic analysis suggests that Mn Mo is a charge carrier compensating defect that traps free carriers and inhibits electrical conductivity in 2D-MoS 2 . Mn 0 Mo and Mn -1

Mo induce magnetic moments of 1 μ B and 2 μ B , respectively; conversely, Mn +1

Mo is found to be a nonmagnetic defect in 2D-MoS 2 .

The E f of Mn-top 0 is 2.85 eV in the Mo-rich and S-rich conditions and found to be most stable in +2, +1, 0, and -1 charge state within the E F range, Figure 4. The DTLs of Mntop ε(+2/+1) and Mn-top ε(+1/0) occur at 1.59 and 0.7 eV, respectively, below the CBM and ensure a deep donor defect. It is also stable as a deep acceptor defect, and the associated DTL ε(0/-1) is located 1.58 eV above the VBM. Therefore, although expected to be a shallow donor, we find that Mn-top is both a acceptor and deep donor defect. It also induces the magnetic moment of 5 μ B , 4 μ B , and 4 μ B in +2, +1, and 0 charge states, respectively. Figure 4 shows that 2Mn Mo is most stable as a negative-U defect, where +1 and -1 charge states (higher charge state) are more stable than the neutral charge state (lower charge state). The negative-U behavior of 2Mn Mo also suggests a strong local lattice distortion, leading to a high formation energy in Mo(S)-rich conditions. The DTL of 2Mn Mo ε(+1/-1) occurs 0.93 eV above the VBM (or 0.97 eV below the CBM) and indicates a two-electron transfer phenomenon in the system at that particular value of E F . 2Mn Mo will pin the Fermi level at E F = E VBM + 0.93 eV, where the E f of 2Mn +1

Mo and 2Mn -1 Mo are equal. Therefore, 2Mn Mo is always a free charge carrier compensating defect for the entire range of E F within the gap and reduces the n-type (ptype) conductivity of 2D-MoS 2 . Furthermore, 2Mn +1

Mo and 2Mn -1

Mo induce the magnetic moment of 1.3 μ B and 1 μ B , respectively. Therefore, our thermodynamic analysis of all Mn defects in 2D-MoS 2 shows that they are generally free charge compensating centers with high formation energies (except for Mn Mo in S-rich conditions and Mn S in Mo-rich conditions) and inhibit electrical conductivity for 2D-MoS 2 . On the other hand, fortunately, all these defects provide local magnetism, and hence their formation is ideal for the application in semiconducting spintronics.

Next, we test the impact of dopant clustering on the magnetic property by comparing the spin-polarized PDOS of Mn Mo -Mn Mo clustering (or 2Mn 2Mo ) with the isolated Mn Mo from Figure 2(d), as shown in Figure 5(a,c). In Mn Mo , the hybridization of Mn-3d and S-4p states provides AFM coupling. On the other hand, when the S spins interact with a Mn Mo defect, the AFM coupling between Mn and S provides an effective ferromagnetic coupling of all the Mn Mo spins. Isolated Mn Mo induces a local magnetic moment of 1 μ B , Figure 5(a). We considered the parallel (FM coupling) and antiparallel (AFM coupling) spin configurations for the adjacent two Mn atoms in 2Mn 2Mo . We found that FM coupling is slightly more stable than AFM coupling. The spin density of Mn Mo has been calculated and is shown in Figure S2 in the Supporting Information. The PDOS of 2Mn 2Mo suggests that the hybridization of Mn-3d, S-3p, and Mo-4d orbitals induces only localized spin-down states near the Fermi level and are identical for FM and AFM coupling, resulting in a magnetic moment of 2 μ B for both couplings, Figure 5(c). Therefore, clustering of Mn Mo defects with low formation energies preserves magnetic ordering.

Since the electronic structure of MoS 2 is very similar to WS 2 , we expect that all the Mn defects discussed above will show the same behavior in 2D-WS 2 . Hence, here we directly consider the Mn W -Mn W (or 2Mn 2W ) clustering effect in 2D-WS 2 . The spin-polarized PDOS of 2Mn 2W is calculated and compared with the isolated Mn W , which is shown in Figure 5(b,d). Unlike 2Mn 2Mo , the identical spin-up and spin-down states of 2Mn 2W indicate no magnetic ordering at the defect site, Figure 5(d). 2Mn 2W induces stronger structural distortion in WS 2 than in MoS 2 , pulling the Mn atoms apart and delocalizing the spin-polarized states, leading to zero magnetic moment. Isolated Mn W shows a similar behavior to Mn Mo and depicts that there are more spin-up than spin-down states, signifying that Mn atoms couple with neighboring S atoms in the FM order. Isolated Mn W induces a local magnetic moment of 1 μ B , Figure 5(b). The spin density of Mn Mo has been calculated and is shown in Figure S2 in the Supporting Information. We attribute this change in magnetic defect clustering behavior to the greater difference in atomic radius between Mn and W than Mn and Mo, which introduces larger lattice distortion at the substitution site.

We analyzed the DTLs of clustered 2Mn 2Mo and 2Mn 2W and compared them with the DTLs of isolated Mn Mo and Mn W as a function of the Fermi level (E F ) in Mo-rich, W-rich, and S-rich conditions, as shown Figure 6. The E f of 2Mn Mo is 3.30 eV under Mo-rich conditions and -1.4 eV under S-rich conditions. Therefore, 2Mn 2Mo is more likely to form in the S-rich growing sample of 2D-MoS 2 . On the other hand, isolated Mn Mo is thermodynamically more stable than the 2Mn 2Mo clusters in the Mo-rich condition. Noticeably, 2Mn 0 2Mo favors FM coupling over AFM coupling by 150 meV, Figure 6(a,b). 2Mn +1 2Mo provides a shallow donor level in AFM coupling, and the DTL of 2Mn ε(+1/0) 2Mo occurs at E CBM = 0.22 eV, which means that when E Fermi = E CBM , 2Mn 0 2Mo is the most stable defect and the electron excitation of 2Mn 0 2Mo → 2Mn +1 + e -requires 0.22 eV, where e -is a conduction electron, Figure 6(a,b). However, in the FM coupling, 2Mn 2Mo provides a deep donor level and the associated DTL occurs at E CBM = 0.51 eV. Therefore, 2Mn 2Mo is a useful magnetic defect in AFM coupling, which can enhance the n-type conductivity of 2D-MoS 2 . Interestingly, when the E F is more than 0.22 eV below the E CBM , 2Mn +1 2Mo is thermodynamically more stable in AFM coupling than FM coupling, as shown in Figure 6(a,b). Moreover, 2Mn +1 2Mo induces a local magnetic moment of 3 μ B in the AFM coupling. On the other hand, the isolated Mn 0 Mo coupled with the nearest six S atoms is most stable in an AFM configuration, but Mn -1

Mo prefers FM coupling. The added electron occupies the previously unoccupied defect state just above the E F and gives a local magnetic moment of 2 μ B . Mn +1 Mo exhibits nonmagnetic behavior, where the donor state just below the E F (Figure 4(a)) transfers its electron to the conduction band and the net magnetic moment goes to zero.

As-grown 2D-MoS 2 is an n-type semiconductor, and it is found that the equilibrium E F position is located at 0.6 eV below the CBM; this does not change significantly with the growth conditions. 56 Therefore, we have considered E F 300 K to be the same for Mo-rich and S-rich conditions, as shown in Figure 6(a,b). At E F 300 K , 2Mn 2Mo and Mn Mo are most stable in +1 and neutral charge states, respectively. The high E f of 2Mn +1 2Mo at E F 300 K the Mo-rich condition requires high temperature to be activated and supports the intrinsic n-type conductivity of 2D-MoS 2 , Figure 6(a). Fortunately, the low E f of 2Mn +1

2Mo at E F 300 K in the S-rich condition provides excess free electron carriers and increases the n-type conductivity of 2D-MoS 2 even at low temperatures, Figure 5(b). As Mn Mo is an electrically inactive defect at the E F 300 K , it does not play any role in electrical conduction in the thermodynamic equilibrium condition.

The E f of Mn 0 W is 1.93 and -0.3 eV in the W-rich and S-rich condition, respectively, which suggests its high concentration in the S-rich growth condition, Figure 6(c,d). Similar to Mn Mo , isolated Mn W is found as an amphoteric defect in 2D-WS 2 . The DTLs of Mn ε(+1/0) W and Mn ε(0/-1) W occur at 0.95 eV below the CBM and 1.25 eV above the VBM, respectively, leading to deep defect behavior, Figure 6(c,d). In Mn W , the Mn atoms coupled with the nearest six S atoms give an AFM configuration in the neutral charge state and FM configuration in the -1 charge state. Mn +1 W is found to be a nonmagnetic defect. The induced magnetic moment by Mn W is 1 μ B and 2 μ B in the neutral and -1 charge state, respectively. The clustered 2Mn 2W is more stable in AFM configuration than the FM configuration by 150 meV. The E f of 2Mn 0 2W in AFM configuration is 3 and -1.37 eV in the W-rich and S-rich condition, resulting in its high concentration in S-rich grown samples of 2D-WS 2 . Unlike 2Mn 2Mo , 2Mn 2W is found to be an amphoteric defect, and the thermodynamic properties of FM and AFM configuration are nearly identical, as shown in Figure 6(c,d). The DTLs of 2Mn ε(+1/0) 2W and 2Mn ε(0/-1) 2W are located at 1.4 eV below the CBM and 1.22 eV above the VBM in the AFM configuration, and in the FM configuration, DTLs are located at 1.48 below the CBM and 1.27 eV above the VBM. Therefore, 2Mn 2W is a charge-compensating defect and cannot lead to electrical conductivity to 2D-WS 2 . Moreover, 2Mn 2W provides a magnetic moment of 1 μ B in the -1 charge state of FM coupling, but does not induce the magnetic moment in any other charge state. As-grown 2D-WS 2 is an ntype semiconductor with E F 300 K positioned at 0.9 eV below the CBM, as shown in Figure 6(c,d). Interestingly, isolated Mn W and cluster 2Mn 2W are stable in their neutral charge state at E F 300 K . Therefore, they are electrically inactive defects in the thermodynamically equilibrium condition and will not lead to any electrical conduction in 2D-WS 2 .

Doping a semiconducting material with electron and hole carriers changes the interaction between defects and, by extension, the magnetic coupling of that material. Therefore, we calculate the energy difference between FM and AFM coupling (Δ FM-AFM ) of 2Mn 2Mo and 2Mn 2W as a function of electron and hole doping, as shown in Figure 7(a,b). Negative values of Δ FM-AFM indicate a preference for FM coupling. In the neutral system of 2Mn 2Mo and 2Mn 2W , FM coupling is more favorable than AFM coupling by Δ FM-AFM = -150 meV. We have previously discussed that in n-doped 2D-MoS 2 and 2D-WS 2 , 2Mn 2Mo and 2Mn 2W are stable in the neutral charge state and -1 charge state, respectively (Figure 6). Here, we observe that when 2D-MoS 2 and 2D-WS 2 are n-doped, 2Mn 2Mo and 2Mn 2W are stable in FM coupling until the minimum value reaches 0.25e/Mn (n e = 1.37 × 10 13 cm -2 ); beyond that, AFM coupling is stable, as shown in Figure 7(a). AFM stability reaches its maximum value at 0.5e/Mn (n e = 2.75 × 10 13 cm -2 ) and then starts decreasing until the AFM coupling is no longer stable after n e = 3.5 × 10 13 cm -2 . Therefore, there is a phase transition from AFM to FM coupling at a critical electron density n c e = 3.5 × 10 13 cm -2 , as shown in Figure 7(a). In the p-doped 2D-MoS 2 and 2D-WS 2 , 2Mn 2Mo and 2Mn 2W are stable in the +1 charge state, Figure 6. We found that for p-doped 2D-WS 2 , the difference Δ FM-AFM of 2Mn 2W increases first until n p = 0.68 × 13 cm -2 and then decreases monotonically until a minimum value at 0.5e/Mn (n p = 2.75 × 10 13 cm -2 ). The Δ FM-AFM of 2Mn 2Mo decreases monotonically until a minimum value at 0.5e/Mn (n p = 2.75 × 10 13 cm -2 ). After 0.5e/Mn, the Δ FM-AFM of both 2Mn 2Mo and 2Mn 2W increases until the FM coupling is no longer stable at 0.87e/Mn (n p = 4.81 × 10 13 cm -2 ) for p-doped 2D-MoS 2 and 2D-WS 2 , Figure 7(b).

Next, we calculate the Curie temperature (T c ) from Δ FM-AFM using the following equation, which is based on the mean-field approximation (MFA);

where k B is the Boltzmann constant and n = 2 is the number of Mn atoms as clustered in the supercell. The calculated T c as a function of carrier density is shown in Figure 8. At a 1.85% concentration of Mn, we calculate a Curie temperature of 580 K in the MFA. Furthermore, we estimate that T c can reach up to 1800 and 1500 K at 1e/Mn (n e = 5.5 × 10 13 cm -2 ) for the ndoped 2D-MoS 2 and WS 2 , respectively. Moreover, for the pdoped condition, the calculated T c is 700 K at 0.5e/Mn (n p = 2.75 × 10 13 cm -2 ). A recent report 57 suggests that isolated Mn W (Mn Mo ) in 2D-WS 2 (2D-MoS 2 ) shows a huge perpendicular magnetic anisotropy of 35 meV (8 meV). However, 2Mn 2w and 2Mn 2Mo show a relatively weak MAE compared to isolated Mn W and Mn Mo in 2D-WS 2 and 2D-MoS 2 , respectively. This means that although clustering of defects can preserve magnetic ordering, the easy axis may be less well determined in these clustered arrangements; in future work, we expect to investigate the effect of electrical doping on the MAE in these systems.