Electronic Structure and Room Temperature of 2D Dilute Magnetic Semiconductors in Bilayer MoS2-Doped Mn

+e electronic structure and magnetic properties of manganese(Mn-) doped bilayer (BL) molybdenum disulfide (MoS2) are studied using the density function theory (DFT) plus on-site Hubbard potential correction (U). +e results show that the substitution of Mn at the Mo sites of BL MoS2 is energetically favorable under sulfur(S-) rich regime than Mo. +e magnetic interaction between the two manganese (Mn) atoms in BL MoS2 is always ferromagnetic (FM) irrespective of the spatial distance between them, but the strength of ferromagnetic interaction decays with atomic distance. It is also found that two dopants in different layers of BL MoS2 communicate ferromagnetically. In addition to this, the detail investigation of BL MoS2 and its counterpart of monolayer indicates that interlayer interaction in BLMoS2 affects the magnetic interaction in Mn-doped BLMoS2. +e calculated Curie temperature is 324, 418, and 381K for impurity concentration of 4%, 6.25%, and 11.11%, respectively, which is greater than room temperature, and the good dilute limit of dopant concentration is 0–6.25%. Based on the finding, it is proposed that Mn-doped BLMoS2 are promising candidates for two-dimensional (2D) dilute magnetic semiconductor (DMS) for high-temperature spintronics applications.


Introduction
It is known that graphene is the most well-known member in the family of 2D materials. However, its gapless band structure has been deemed as a considerable drawback for realizing switching operation which is essential for digital logic devices [1]. On contrary to zero band gap graphene, other group of 2D materials so called transition metal dichalcogenides (TMDS) with chemical formula MX 2 , where M stands for transition metals like molibidium (Mo), tungeston (W), and so on and X stands for chalcogen atoms like selfur (S), selenium (Se), and telenium (Te), and so on are more recently discovered [2]. e experimental study reveals that the weak van der Waals forces between each 2D monolayer of which they are formed allow thin multilayers to be easily exfoliated from their bulk form [3]. Among those families, monolayer MoS 2 and few layer MoS 2 have attracted interest due to their potential application [2,4]; for instance, the monolayer (ML) MoS 2 has emerged as a semiconductor with a large intrinsic direct band gap of approximately 1.8 eV [3], which makes it suitable for nanoelectronic and optoelectronic applications [5]. In MoS 2 materials, the electronic structure depends on the number of layers which mean that there is a gradual transition from an indirect band gap (1.3 eV) in the bulk material to a direct gap in the ML material with band gap 1.9 eV which reveals that the interlayer van der Waals (VdW) interaction may affect the electronic structure [6]. In addition, it has been reported that the electronic structure of such system depends on stacking patterns [7]. ere are five different stacking configurations for the bilayer MoS 2 system [7], but the most stable one is AA' [8].
On the contrary, during last few decades, the research in dilute magnetic semiconductors (DMSs) have attracted tremendous interest due to their great potential for different spin electronics (spintronics) applications [9]. Spintronics refers to new phenomena of electronic transport for which the electron spin plays a decisive role in contrast to conventional electronics for which the electron spin is practically irrelevant. For a full exploitation of spintronics, one should have materials which show simultaneously semiconducting properties and ferromagnetic ordering at operational temperature [10]. In last two decades, most of DMS research has focused on the transition metal-(TM-) doped III-V and II-VI three-dimensional (3D) semiconductors. However, the ferromagnetic transition temperature in well-studied type of DMSs like Ga 1− x Mn x As is less than 200 K which is far from room temperature [11]. As a result of this, the search for DMS has recently been extended to two-dimensional (2D) transition metal dichalcogenides (TMDs) like MoS 2 -, MoSe 2 -, and WS 2 -doped transition metal [4]. Recent theoretical study reveals that Mn and Fe are the best candidates to generate long-range room temperature in its Ml phase [4]. To date, several groups have reported experimentally [12] and theoretically the feasibility of MoS 2 by substitutional doping with different (TM) : Mn-doped monolayer (ML) MoS 2 [13], iron-(Fe-) doped bilayer MoS 2 [14], Ni-doped ML MoS 2 [15], and Co-doped monolayer MoS 2 [16]. In addition to this, in our previous ab initio study, we have shown that V-doped ML and BL MoS 2 are good candidates for nearly room temperature ferromagnetism [17]. However, electronic structure and magnetic interaction in Mn-doped bilayer (BL) MoS 2 has not yet been studied in detail. In this current study, Mn-doped BL MoS 2 are studied on the basis of spin polarized (DFT + U) formalism. e relative structural stability, electronic structure, and magnetic properties of MoS 2 -doped Mn are studied in detail. To understand magnetic ground state and the magnetic energy ΔE, the energy difference between two dopants in FM and AF cofigurations in the ML and BL phase of MoS 2 was calculated for different impurity concentrations. Furthermore, based on mean field theory together with empirical correction, the Curie temperature (Tc) is also estimated.

Computational Details
DFT + U calculations were performed using the plane-wave pseudopotential method with the aid of QUANTUM ESPRESSO code. On-site Hubbard parameter, U � 4 eV, was assigned for dopants to take into account the strong correlation in the Mn 3d state [18]. Ultrasoft pseudopotentials (UPPs) were used to deal with the interaction between valence electrons and the ion core. e plane wave basis set is given a cutoff energy of 60 Ry used after performing the convergence test with respect to total energy. A unit cell with the periodic boundary condition was adopted to simulate the infinite x-y plane. For the BL crystal, Grimmes DFT-D2 dispersion correction [19] was applied to account for the long-range van der Waals interactions between layers. e equilibrium interlayer distance was obtained by careful minimization of total energy with respect to distance between layers. To investigate the doping effects of Mn impurities on bilayer MoS 2 , the BL MoS 2 was modeled by supercell of (3 × 3 × 1), (4 × 4 × 1), and (5 × 5 × 1) which contains 36 S and 18 Mo, 64 S and 32 Mo, and 50 Mo and 100 S atoms, respectively. e vacuum space 20Å thickness along the z axis was used to avoid any self-interaction of the slabs for ML (to make sure there is no interaction along the z-axis). Integrations over the Brillouin zone (BZ) were sampled based on a Monkhorst pack 2D grid [20] based on the size of supercells.

Defect Formation Energy and Structural Stability.
To understand relative stability of dopant atom (Mn) in BL MoS 2 , dopant formation energy (E form ) was carried out employing the following equation [17,21]: where E (Mn, MoS 2 ) and E (MoS 2 ) are total energy of doped and pure BL MoS 2 , respectively, n i is the corresponding number of species that has been added to or removed from the supercell, and μ Mn and μ Mo are chemical potentials of Mn and Mo, respectively. Both Mo-and S-rich conditions were considered for chemical potential calculation of Mo. In the Mo-rich condition, the chemical potential for μ Mo is obtained from its bulk body-centered cubic (BCC) structure of Mo [21], whereas, under the S-rich condition, μ Mo is obtained from energy difference between formula unit of MoS 2 and ring form of sulfur molecules. e dopant formation energy calculated using equation (1) for different impurity configurations of impurity atoms is summarized in Table 1.
All calculated values under the S-rich condition is negative which reveal that doping Mn under the S-rich condition of BL MoS 2 is favorable in comparison with the Mo-rich condition in agreement with previous theoretical report [21] and experimental result [22]. Besides to this, the least formation energy − 2.0144 eV is obtained in doping single Mn on 4 × 4 × 1 BL MoS 2 (6.25%) compared to 3 × 3 × 1 (11.11% Mn doping) which indicates that Mn dopants are more energetically favorable to occupy the substitutional lattice site (Mo) at low impurity concentration (dilute magnetic limit) than at high concentration.

Electronic Structures and Magnetism of Pure and Single
Mn-Doped BL MoS 2 . e calculated equilibrium lattice constant after optimization is 3.18Å, which is closer to the experimental value [23] and in good agreement with theoretical value 3.18Å [24]. In addition to this, the calculated interlayer distance (the distance between two ML) of BL MoS 2 is found to be 6.543Å and the band gap calculated at this interlayer distance is 1.3 eV ( Figure 1) closer with the previous reported value 1.29Å [25]. Furthermore, the band gap increases with increasing interlayer distance, as shown in Figure 1(a). To investigate the effects of single Mn doping on the electronic and magnetic properties of pure BL MoS 2 , 4 × 4 × 1 and 3 × 3 × 1 BL MoS 2 model supercells which result in a magnetic impurity concentration of 3.125% and 5.55%, respectively, after doping a single Mn atom were considered (see Figure 2). As seen from Table 2, the total magnetic moment of the system is 1 and 1.02 μ B after introducing single Mn in supercell of 3 × 3 × 1 and 4 × 4 × 1, respectively.
To understand how the states are distributed, the total density of state (DOS) for pure and single Mn-doped BL MoS 2 is plotted. As shown from Figure 3(a), for the pure system, the Fermi level is located at the middle and spin up and spin down states are symmetric indicating the pure BL MoS 2 are nonmagnetic semiconductors. However, after introducing single Mn, the spin degeneracy of the band structure is broken and the minority of states remain semiconductors, whereas the majority (spin up state) impurity state is formed in the vicinity of the Fermi level; as a result, it behaves as metallic, leading to magnetic and half metallic behavior of the total DOS. On the contrary, with increasing concentration of Mn dopant (3.125% to 11.11%), the impurity state is broaden (moves closer to conduction band minimum (CBM)) Figures 3(b)-3(e) which reveals that the doped system behaves as the n-type of semiconductor. Furthermore, to understand the nature of band structure and defect state, the band structure of pure and single Mn doped in one of its layers is plotted. As shown from Figures 4(a) and 4(b), the pure system is nonmagnetic semiconductor with direct band gap measured to be 1.3 eV. However, in single Mn-doped system, the majority band structure and the impurity states are formed above the Fermi level (within the gap) (Figure 4(c)). But, the minority state remains semiconductor, but the band gap is suppressed by 0.1 eV compared to the pure system (Figure 4(d)), which also gives further confirmation half metallic and magnetic behavior of the doped system. On the contrary, the impurity state closer to CBM reveals that the system is more likely the n-type of semiconductor. Based on those observation, it is suggested that the origin of magnetism as an isolated Mn atom with electron configuration [Ar]3d 5 4S 2 has one more d-orbital electron than Mo (with electron configuration [Ar]3d 4 4S 2 ); thus, the extra one electron is responsible for the observed defect state in the gap.

Magnetic Interaction between Dopants in Monolayer (ML) and Bilayer (BL) MoS 2 Doped with a Pair of Mn Atoms.
e magnetic interaction between dopants in doped ML and BL MoS 2 systems is studied by calculating the total energy difference between FM and AF configurations at the same impurity separation [17]. e magnetic energy (ΔE) is given by where E FM and E AF are the total energies of the supercell in FM and AF states, respectively. Employing model supercells,  Tables 2 and 3. As can be seen from Tables 2 and 3, the calculated ΔE for all impurity configurations are negative which show that FM interaction is favorable for all concentrations of impurity in agreement with previous study in ML MoS 2 -doped Mn [26]. In addition to that, the strength of FM interaction decays from N to NNN for all concentrations of impurity. Furthermore, in order to get insights into FM interaction between the two nearest neighbor (N) Mn dopants, the total density of states (DOS) are plotted as shown in Figures 3(e) and 3(e); the dopants are ferromagnetically interacting, and the impurity states push towards the CBM with increasing impurity concentration (6.25-11.11%), which is in broad agreement with dilute magnetic semiconductor properties [11]. We further extend our investigation by doping two Mn in different layers BL MoS 2 , as seen in the input structure in Figure 2(d) and Table 4, and ΔE � − 0.006 and − 0.0025 eV when two dopants are separated by d 0 = 6.53Å (at equilibrium interlayer distance) and d � 7.8Å, respectively. Interestingly, the result reveals that two dopants in different layers can interact ferromagnetically, and the strength of ferromagnetism suppresses with atomic distance and impurity concentration. is is also seen in the DOS plot in Figure 3(e), where spin degeneracy of spin up and spin down is breaking and system becomes ferromagnetic. For further information, the electronic band structure is drawn, as shown from the band plot for those system seen in Figures 4(e) and 4(f ), the system becomes magnetic even when two dopants are in different layers, and this makes the physics of system under investigation more interesting that how two dopants in different layers are communicating. On the contrary, understanding mechanism of exchange interaction in magnetic system is another issue. Goodenough-Kanamori-Anderson rules which state that the magnetic ion-ligand-magnetic ion angle is 180°of two magnetic ions with partially filled d shells is strongly antiferromagnetic, whereas the magnetic ion-ligand-magnetic ion angle is 90°and is ferromagnetic super exchange. us,    erefore, we report that interlayer interaction in Mn-doped BL MoS 2 system can affect its magnetic interaction, and similar finding was recently reported in irondoped BL MoS 2 [14].

Ferromagnetic Transition Temperature (T c ). By mapping
Heisenberg Hamiltonian together with mean field approximation, the Curie temperature (T c ) below which the system develops a long-range ferromagnetic ordering can be found as follows [27]: where ΔE � E FM − E AF is the magnetic energy obtained from the first principle spin polarized DFT calculation and N is the number of dopants in supercell. Using the value ΔE for first nearest neighbor impurity configurations (N) in Table 3 and N = 2, we have calculated ferromagnetic transition (T c ) for Mn-doped BL MoS 2 . However, it is well known that the magnetic ordering in the doped system is strongly influenced by percolation, and thus mean-field approximation cannot capture this behavior and tends to systematically overestimate T c in these systems [13]. To overcome this, we make use of some empirical relation which connects the mean field value critical temperature T MFA c with corrected critical temperature (T corr c ) as T corr c � (0.506)T MFA c [28], where T corr c is the exact (corrected) critical temperature for 2D hexagonal lattice calculated using Ising model and T MFA c is the predicted critical temperature using mean field theory.
e calculated result is presented in Table 4. As shown in Table 4, T c in Mn-doped BL MoS 2 is found to be above the room temperature (RT), and a nonmonotonic behavior of T c is observed; T c increases with Mn concentrations in a range of dilute limit (0 to 6.25%) and then decreases with further increasing Mn concentration above this value (6.25 to 11.11%), which confirm that, at high concentrations, the system exhibits FM instability due to some sort of direct interaction. us, the result indicates that ferromagnetism in this system is tunable by controlling the concentration of magnetic dopants (Mn), and such kind of properties are seen in well-known III-V DMS [10].

Conclusion
In conclusion, Mn dopant in BL MoS 2 are energetically favorable to occupy the substitutional lattice site (Mo) under sulfur-(S-) rich regime than Mo. e magnetic interaction between dopants in Mn-doped BL MoS 2 is always ferromagnetic. Moreover, the strength of ferromagnetism decays with atomic distance. Interlayer interaction in Mn-doped BL MoS 2 affects its magnetic properties. Super exchange mechanism is primarily responsible for ferromagnetic interaction between a pair of dopants. e calculated T c shows that good dilute limit of dopant concentration is 0 − 6.25%, and further increasing dopant concentration results in FM instability. Based on the result, it is suggested that Mn-doped BL MoS 2 are promising candidates for 2D dilute magnetic superconductors for high-temperature spintronics applications.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest.  Advances in Condensed Matter Physics 7