Abstract
Strainmediated magnetism in 2D materials and dilute magnetic semiconductors hold multifunctional applications for future nanoelectronics. Herein, First principles calculations are employed to study the influence of biaxial strain on the magnetic properties of Codoped monolayer \(WS_2\). The nonmagnetic \(WS_2\) shows ferromagnetic signature upon Co doping due to spin polarization, which is further improved at low compressive (2 %) and tensile (+2 %) strains. From the PDOS and spin density analysis, the opposite magnetic ordering is found to be favourable under the application of compressive and tensile strains. The double exchange interaction and pd hybridization mechanisms make Codoped \(WS_2\) a potential host for magnetism. More importantly, the competition between exchange and crystal field splittings, i.e. (\(\Delta _{ex}>\Delta _{cfs}\)), of the Coatom play pivotal roles in deciding the values of the magnetic moments under applied strain. Micromagnetic simulation reveals, the ferromagnetic behavior calculated from DFT exhibits lowfield magnetic reversal (190 Oe). Moreover, the spins of Codoped \(WS_2\) are slightly tilted from the easy axis orientations showing slanted ferromagnetic hysteresis loop. The ferromagnetic nature of Codoped \(WS_2\) suppresses beyond \(\pm 2~\%\) strain, which is reflected in terms of decrease in the coercivity in the micromagnetic simulation. The understanding of lowfield magnetic reversal and spin orientations in Codoped \(WS_2\) may pave the way for nextgeneration spintronics and straintronics applications.
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Introduction
In recent years, to develop new multifunctional materials, tremendous research efforts have been focused on twodimensional (2D) materials due to their potential applications in areas such as electronic, optoelectronic, mechanical and chemical properties^{1,2,3}. Among various 2D materials, transition metal dichalcogenides (TMDCs) such as \(MX_2\) (M: Mo, W, etc.; X: S, Se, etc.) marks significant attention owing to their unique potential applications in fieldeffect transistor (FET), photodetectors, catalysis, Liion batteries etc^{1,2,4,5}. Ever since the discovery of graphene, weak van der Waal systems bring forth a large possibility for hosting magnetism in 2D materials. In the last few years, the focus shifted more on 2D magnetic materials, where the fundamental concept of spin dominate over charge bring new scientific properties and opens the plethora for nanoscale devices and spintronic applications, which have been demonstrated both experimentally and theoretically^{6,7}. Till date several 2D material contributes distinct magnetic properties such as: (i) graphene shows excellent magnetic transport^{8}, (ii) TMDCs (\(MoS_2\), \(WS_2\), \(MoSe_2\), \(WSe_2\), etc.) shows strong spinorbit coupling (SOC) and coupled valley properties^{9,10} and (iii) 2D magnets have the potential candidate for future nonvolatile memory applications^{11,12}. The investigations of new 2D TMDCs or new techniques as an alternative approach have been attracted much attention towards multifunctional applications. In addition, magnetism in 2D materials is creating dilute magnetic semiconductors (DMSs), which have been extensively studied due to their charge carriers making DMSs as the potential of spintronics^{13,14,15,16}.
The \(MoS_2\) and \(WS_2\) TMDCs show greater potential for fabricating magnetic tunnel junctions (MTJs) owing to their unique physical properties such as strong SOC, longranged spin diffusion length, etc^{9,17}. However, these materials do not have their intrinsic magnetism and always depend on external agents. For example, a thin \(MoS_2\) layer can be sandwiched between two ferromagnetic layers as a spacer to achieve MTJ properties^{18,19}. Impurity doping engineering is found to be an appropriate approach to change the electronic and magnetic properties of TMDC semiconductors. Doping marks a significant change in structural stability and magnetism of graphene, which facilitates new multifunctional applications^{20}. Both monolayer (ML) or stacked (\(MoS_2\), \(WS_2\), etc.) TMDCs semiconductors can host the magnetism after doping with either ntype or ptype impurities^{21}. The Mndoped \(MoS_2\) ML depicts potential for a new class of DMSs, which is evident from first principles DFT calculations and MonteCarlo simulations studies^{22}. The ferromagnetic (FM) behavior of Mndoped \(WS_2\) has been verified experimentally and theoretically^{23}. Kang et. al. experimentally demonstrated that the \(MoS_2\) exhibits FM, while \(WS_2\) shows paramagnetic behavior when doping with Fe^{24}. TMrelated defects complexes can induce FM in the \(WS_2\) ML. Eventually, high formation energy makes it difficult for the practical growth of \(WS_2\) ML. Hence, besides the doping of TM elements, vacancies or defects have also been introduced to induce magnetism in TMDC monolayers^{25,26}. Furthermore, lattice strain can be adapted to modulate the physical properties of TMDCs. Tao et. al. predicted singlelayer \(MoS_2\) with single atomic vacancies shows FM under strain for the possible applications in memory switching and logic gates^{27}. However, there is very few limited detailed information available on straininduced FM in \(WS_2\) systems. In addition, \(WS_2\) ML has sufficiently high thermal and oxidative stability compared to \(MoS_2\)^{28,29}. \(WS_2\) ML having \(P6_3/mmc\) space group symmetry, where the W atoms are having trigonal prismatic coordination with the S atoms. The presence of a covalent bond between WS makes it suitable for magnetism in \(WS_2\) after doping, which can be controlled under uniaxial/biaxial strain. Moreover, the straininduced magnetic \(WS_2\) behaves dissimilar for different types of doping. Luo et. al. obtained FM behavior in Aldoped \(WS_2\) under applied compressive strain, while unable to produce any magnetic moment under tensile strain^{30}. Contrary to this, Nadoped \(WS_2\) ML shows weak magnetism at higher compressive strain, while found to be higher at greater tensile strains^{31}. Therefore, these findings motivate us to study the unrevealed straininduced magnetism in Codoped \(WS_2\) ML, which may pave the way for future spintronic applications.
In this work, we studied a possible emergence of FM in Codoped \(WS_2\) ML under strain engineering, which may have applications in TMDC based straintronics^{32}. We employed DFT calculations to understand the mechanism of FM behavior in Codoped \(WS_2\) ML at biaxial compressive and tensile strain. The exact behaviour of the FM nature is understood by using crystalfield and exchangefield splitting. The system is further studied by micromagnetic simulation to address the behavior of reversal magnetization and the magnetic effects under strain. The electronic and magnetic properties are also discussed, which will be significant for future spintronic applications. More importantly, this is the first attempt to understand ferromagnetism in TMDCs of lowdimension DFT calculations with nanoscale micromagnetic simulations.
Computational method
We perform the first principles spinpolarised DFT calculations using the Vienna ab initio simulation package (VASP)^{33} which implements the projector augmented wave (PAW) method to describe electronion interaction. The electronic exchangecorrelation potential is described by the generalized gradient approximation (GGA) in the PerdewBurkeErnzerhof (PBE) parametrization^{27}. The correction of a Hubbard term U (GGA+U) is necessary to describe the strongly correlated 3d orbital of the TM impurities. However, the halfmetallic doped magnetic TMDC systems are found to be independent of U values (U = 2.5, 3). The value of the magnetic moments remain little affected using different U values. Additionally, the optimization of lattice constants have been performed with various U values under unstrained and strained conditions by energy minimization and are observed to be unaltered with GGA+U (see SI). Henceforth, we find no relevant modifications on our conclusions and the further discussions are based on without considering the onsite interaction. We use kinetic energy cutoff of 500\(\sim\)eV for the planewave expansion of the wave functions. All the structures are fully relaxed under conjugate gradient algorithm until a total energy convergence and Hellmann Feynman force up to \(10^{5}\) eV and 0.01 eV/ Å are achieved. The Brillouin zone integration for selfconsistent and projecteddensity calculations is approximated by MonkhorstPack K point mesh of \(5\times 5\times 1\) and \(12\times 12\times 1\) respectively. We use \(4\times 4\times 1\) and \(5\times 5\times 1\) supercells for doping and strainbased calculations. The micromagnetic simulations is carried out by open freeware objectorientedmicromagneticframework (OOMMF)^{34}. By considering the previous experimental reports^{35,36}, the parameters such as magnetic anisotropy, saturation magnetization, exchangelength, etc are employed in Codoped \(WS_2\) by micromagnetic calculations.
Results and discussion
To study the effects of Codoping for the magnetic properties of \(WS_2\) mono layer (ML), we replace one host Watom by foreign Coatom in \(4\times 4\times 1\) [\((Co, W)S_2\)] and extended \(5\times 5\times 1\) supercell with doping concentrations 6.25% and 4.16% respectively. The graphical representation of \((Co, W)S_2\) ML with its hexagonal unit cell (black solid line) is shown in Fig. 1a, where the optimized lattice constant is found to be 3.16 Å. The structural and magnetic parameters extracted from the calculated results are listed in Table 1. The obtained results show that the bond lengths of WS shrink around the doping sites locally after Codoping when compared to pristine \(WS_2\). However, the CoS bond length remains same with WS bond length in pristine \(WS_2\) at various doping concentration. Such behavior may be attributed to various factors such as electron affinities, ionic radii differences between the Co and W ions after the geometrical relaxation and the changed electrostatic forces that arise due to the occurrence of multivalent Co ions at the host W site^{37}. It is apparent to notice that \(WS_2\) exhibits effective magnetic ordering with net magnetic moments \(2.62~ \mu _B\) and \(2.58~\mu _B\) at different Codoping concentration of 4.16% and 6.25%, respectively. The obtained results offer similar/higher magnetic moments than the previously studied transition metaldoped \(WS_2\) ML system^{21}. The observation of such high magnetic moments in Codoped \(WS_2\) ML suggests its potential candidature for 2D DMSs. In order to investigate the effects of external perturbation such as strain on the magnetic properties, both compressive (\(5\%~to~ 0\%\)) and tensile (\(0\%~to~5\%\)) isotropic biaxial strain (along x and yaxis simultaneously) have been applied to \((Co, W)S_2\) ML. From Fig. 1a, strained unit cells (dotted orange and violet cells for tensile and compressive strain, respectively) have been shown schematically along with the unstrained unit cell. It has been previously observed that the biaxial strain attributes isotropic changes to the magnitude of the lattice vectors of various TMDCs ML^{38}. The relationship between the reciprocal lattice in Brillouin zone and the primitive vectors in real space is expressed as:
where \(a_i\) and \(b_i\) are the primitive vectors and reciprocal lattice, respectively. Accordingly, under an isotropic/uniform biaxial tensile (compressive) strain, the lattice constant will be increased (decreased), and the reciprocal lattice will be shrinked (enlarged), which in turn reduces (expands) the Brillouin zone as shown in Fig. 1b^{38}. During our calculations, the periodicity is well preserved under the both biaxial strains.
The structural stability of \((Co, W)S_2\) ML under isotropic biaxial strain can be estimated by formation energies, which can be expressed as: \(E_{FE}= E_{doped}E_{pure}+n(\mu _W\mu _{Co})\), where \(E_{pure}\) and \(E_{doped}\) represent the total energy of the pristine \(WS_2\) and Codoped \(WS_2\) ML, respectively. \(\mu _W\) and \(\mu _{Co}\) are the chemical potential for host W and foreign Coatom, respectively. n is the number of dopants in the studied supercell. The formation energies at various biaxial strains for W and Srich conditions are listed in Table 2. The formation energies of the strained systems are seen to increase monotonically with the increase in both compressive and tensile strains, as displayed in Fig. 2a. Previous experimental results evident that the Srich condition is more suitable for the practical growth of pristine \(WS_2\) ML than the Wrich condition^{39}. For our system, it can be noticed that \(E_{form}\) for Srich is lower when compared to Wrich conditions at each studied strain system, as supported by the experimental condition. From Fig. 2b, it can be worth noting that the magnetic moments are further modified under various applied strains, which explore possible directions for TMDCs based spintronic and straintronic application^{9,32}. Among all the applied strains, the maximum magnetic moment of \(3.25~\mu _B\) and \(2.69~\mu _B\) are achieved for +2% tensile and 2% compressive strain, respectively. The obtained results ensure a higher magnetic moment at low strain as compared to other TM doped TMDCs^{30,31,40}. According to Luo et. al., the enhancement in the magnetic moments was only observed under compressive strain in Aldoped \(WS_2\) ML^{30}; however, the present study reveals the improvement of the magnetic moment under both compressive and tensile strain by Codoping. The enhanced magnetic properties under strained conditions may be attributed to the overlapping of spinpolarized electronic band alignments originating from both magnetic cations and nearest neighbour anions.
The pristine \(WS_2\) ML exhibits a direct bandgap of \(E_g=1.87~eV\) (see SI: Fig S2) at the Kpoint having very close approximation with both experimental (\(E_g=1.88~eV\))^{41} and other theoretical (\(E_g=1.68~eV\)) results^{42}. Figure 3 elucidates the spinpolarized band structures of unstrained and strained \((Co,W)S_2\) ML at their equilibrium lattice constant. However, it can be noticed from Fig. 3b that some impurity states appear within the bandgap when a single Watom is replaced by Co in both spinup and spindown channels. These impurity states are mainly contributed by Coatom, while the effects from the nearest neighboring W and S atoms can be neglected, which has been explained in other TM doped TMDCs^{21}. The asymmetric behavior of spinup and spindown components of Codoped \(WS_2\) ML indicates \(WS_2\) behaves magnetically active after Codoping. Moreover, the doped ML systems show halfmetallic characteristics due to the suppression of bandgap. The spinup component in the doped band structure shows dispersion behavior due to crossing of band lines across the Fermi level, as shown in Fig. 3b. Similarly, the spindown component of the doped system has a bandgap of 1.168 eV with a halfmetallic spacing of 0.294 eV for \((Co, W)S_2\) ML. Furthermore, the tuning of bandgap at various isotropic biaxial strains brings out effective variations in the magnetic properties of the doped system. The increase in the bondlengths and bondangles under various isotropic tensile strains are listed in Table 2. This indicates the decrease in the bondenergy between TM elements and S atoms leads to increase the number of spinpolarized electrons at the vicinity of fermilevel. As a result of which, the maximum obtained magnetic moment (\(3.25~\mu _B\)) is observed at +2% tensile strain. Figure 3a and c shows the spinpolarized band structure under the application of −2% compressive and +2% tensile strain, respectively. Moreover, one can infer that in the case of −2%, spinup impurity states are localized near the Fermi level occupying Cod states, whereas, for +2%, the spindown Cod states are filled to produce the magnetism. This behavior can be correlated to the opposite nature of the magnetic ordering in compressive and tensile strains and may show good performance in transferring one particular spin oriented electrons such as in a spin filtering devices. The highly dispersive impurity dband lines around the fermi level in case of +2% tensile strain renders more electrons on the edge of Co atom spinpolarised when compared to the less dispersive dband lines in case of −2% compressive strain. Interestingly, the bandgap is minimum at strain \(\pm 2\%\), which ensures the bandgap tunability plays a significant role in TMDC based DMSs. The variation of halfmetallic gaps with two different U values (U = 2.5, 3) under unstrained and strained (both compressive and tensile) conditions have also been incorporated which are listed in supplementary information (see SI: Table S3). However, the bandgap is further increased at higher applied strains (see SI: Fig. S2) resulting in the decrease of the magnetic moments in both strain direction.
To further elucidate the electronic and magnetic properties in strained \((Co, W)S_2\) ML, the total density of states (TDOS) and partial density of states (PDOS) are plotted in Fig. 4. The pristine \(WS_2\) behaves as a nonmagnetic semiconductor, which can be inferred from the symmetric nature of spinup and spindown states (see SI: Fig. S3), which is well consistent with previously studied results^{42}. Unlike pristine \(WS_2\), in \((Co, W)S_2\) ML, the splitting of spin states is observed, giving rise to net magnetic moment (\(2.58~\mu _B\)), as depicted in Fig. 4b. The origin of magnetization in \((Co, W)S_2\) ML is mainly contributed from the additional three unpaired electrons of \(Co3d^74S^2\) than \(W5d^{4}4s^2\) with \(1.67~\mu _B\) per Coatom giving rise to ntype doping. However, the contribution from the nearest neighbor W (\(0.03~\mu _B\) per W atom) and S (\(0.11~\mu _B\) per S atom) atoms to the net magnetic moment is less than the contribution arising from Co. Similarly, the contribution from the Coatom and the nearest neighbour W, S atoms at various studied compressive and tensile strains are listed (see SI: Table S1). The TDOS and PDOS for extended \(5\times 5\) supercell (see SI: Fig. S3) show no significant modification in magnetization after Codoping. To understand the magnetic exchange behavior in \((Co, W)S_2\) ML, the interaction between foreign Cod with neighbouring Wd and Sp are need to be considered. The values of magnetic moments depend on the hybridization among the Cod, Wd and Sp. The occupied states near the Fermi level mainly arise from Co3d and W5d orbitals in the majority spin channel, as evident from Fig. 4b. As the Fermi level lies within the partially occupied majority band of the impurity states, expecting a double exchange coupling between Co and W^{43,44}. Similarly, from Fig. 4a, it can be inferred that the broadening of the majority bands along with the Fermi level passing through them indicates a stronger double exchange mechanism under compressive strain^{40}. However, in tensile strain, the broadening of the minority charge carriers can be noticed across the Fermi level forming highly delocalized minority spin channels which allows maximum magnetic moment under strained \((Co, W)S_2\) ML. The interaction of Cod with nearest adjacent Sp can be explained in terms of pd hybridization mechanism. The Co atoms are strongly coupled with their neighboring S atoms around the doping sites due to the hybridization of outofplane Co3d and S3p orbitals creating considerable unbalanced spin populations in spinsplit impurity bands near the fermi level. It is worth noting from Table 2 that the CoS bond length is lower than the WS bond under compressive strain, which is found to be reversed in the case of applied tensile strain. This change in bond lengths leads to different hybridization mechanisms for varied strains. From the above analysis, it can be believed that the competition between the CoS and WS bond lengths may be considered as the prime factor in modulating the magnetic properties under strain engineering.
Next, we consider the spin density distribution to support our understanding of the exchange coupling from the PDOS near the Fermi level, shown in Fig. 5. Figure 5b elucidates the spin polarization between the Co atom and its nearest W/S in unstrained \((Co, W)S_2\) ML. The coupling between the foreign Coatom and host Watom results in parallel spin alignment indicating a double exchange interaction as evident from PDOS in Fig. 4b. However, the interaction between Co and nearest neighbor S results in pd hybridization from the outofplane orbitals as depicted from Fig. 5b. The magnetic coupling behavior in a TM doped TMDCs systems can also be tuned by the introduction of cation and anion vacancies^{45}. In such cases, the trapped electrons in the vacancies couple with the magnetic moments of the TM impurity ions form bound magnetic polarons (BMP) within the Bohr radius of the impurity site. The recently studied experimental results confirms the strong ferromagnetism induced by BMP driven by Svacancies in \(MoS_2\) nanosheet^{46}. However, in our DFT calculations, the magnetic properties of Codoped \(WS_2\) monolayer have been established by considering the Srich environment due to their lower formation energies. Additionally, The contribution from BMP and Sdeficiencyinduced spin delocalization in Codoped \(WS_2\) monolayer under both unstrained and strained condition has weak effect to the ferromagnetism. From the spin density distribution of \((Co, W)S_2\) ML, it can be perceived that Codoping induces a longrange magnetic interaction with nearest neighboring W/S atoms. Moreover, the spin distribution is more localized around its magnetic centres for strained systems when compared the spin distribution around the magnetic centres of unstrained system. In case of unstrained \((Co, W)S_2\) ML, a continuous network of the magnetically coupled TM impurities over local clusters establishes a longrange magnetic interaction leading to lower percolation threshold. However, under the application of compressive and tensile strains, the local spin clusters around the impurity atoms establishes a shortrange magnetic interaction, which limits to the nearest neighbour S atoms. In such a scenario, a higher doping concentration is required to reach the percolation threshold^{47}. Additionally, it can be concluded that the induced spin density at the dopant site is maximum under +2% tensile strain, which reflects the ultimate magnetic moment in this case. For the extended \(5\times 5\times 1\) Codoped \(WS_2\) supercell, the magnetic coupling between dopant and host atoms shows similar behavior (see SI: Fig. S4). However, with the increase in either compressive or tensile strain, the magnetic moment decreases due to lower spin polarization.
The origin of FM behavior in \(WS_2\) ML after doping and strain engineering can be further explained based on orbital decomposed PDOS analysis of the Coatom, as shown in Fig. 6. According to ligand field theory, the 3d states of Co atom can be split into single [\(a_1\) (\(d_{Z^2}\))] and two twofold degenerate [\(e_1~(d_{xy, x^2y^2})\), \(e_2~(d_{xz,yz}\)] states. Intraatomic Hund’s exchange splitting (\(\Delta _{ex}\)) is determined by the energy difference of \(e_1\) orbital between the spinup and spindown states, whereas the energy difference between \(e_1\) and \(e_2\) orbitals is referred to as crystal field splitting (\(\Delta _{cf}\))^{48,49}. The spinsplitting in Codoped and strain engineered \(WS_2\) near the Fermi level mainly results from the exchange splitting and crystal field splitting. As suggested by Pan et. al., the FM behavior in TMDCs due to TM doping arises from the competition between the exchange splitting and crystal field splitting^{50}. Figure 6d represents the schematic for exchange and crystal field splitting under − 2% compressive and +2% tensile strain compared with the unstrained condition. Moreover, the exchange splitting dominates over crystal field splitting in our studied system due to ntype Codopant. From Fig. 6d, it can be observed that the difference between \(\Delta _{cf}\) and \(\Delta _{ex}\) increases under the application of strains, which in turn reflects the increased magnetic moments, as listed from Table 2. This \(\Delta _{ex}\) and \(\Delta _{cf}\) at each studied strain is plotted in Fig. 6e. The detailed analysis confirms that the larger the separation between \(\Delta _{cf}\) and \(\Delta _{ex}\), the greater will be magnetic moments, as evident from Fig. 6e^{50}.
Our detailed observation from the DFT calculation reveals that the Codoped \(WS_2\) ML behaves as FM in nature. In order to better implement an application point of view, the magnetization reversal of \(WS_2\) under Codoping engineering needs to be understood. After substituting Co at Wsite, there are two possibilities of formation of anisotropy, one is uniaxial, and another one is biaxial anisotropy. As suggested from the previous reports, the uniaxial anisotropy value is high due to the formation of a larger coercive field (\(H_c\)) with the easy axis measurement ^{51}. In contrast, the biaxial anisotropy strength is lower, resulting in low \(H_c\) values. Experimental results show that the Codoped \(WS_2\) bulk, nanosheets, ML exhibit hysteresis at very low \(H_c\) (few hundreds of Oe) at room temperature^{35,36,52}. A similar FM signature is also reported in the Codoped \(WSe_2\) system^{36}. However, the nature of the hysteresis is found to be slanted in all these cases. This intrigues us to understand the behavior of magnetization in our system. However, the FM behavior in Codoped in \(WS_2\) has not been thoroughly understood in terms of magnetization reversals. The quest for slanted hysteresis in all the reported cases is still elusive in the scientific community. Here we address the behavior of magnetization reversal and the magnetic effects of the system under strain. We approach micromagnetic modeling by using an open freeware objectorientedmicromagneticframework (OOMMF) package to perform qualitative analysis for ferromagnetism^{34}. This is a to understand the intrinsic magnetic properties of the FM systems, which has very limited reports available on TMDCs materials. For practical applications, we need a sample dimension of nminrange. For micromagnetic modeling, we used a sample dimension of \(50\times 50\times 1~nm^3\), and cell size is \(1\times 1\times 0.5~nm^3\) to compute the simulation. This micromagnetic simulation governs by LandauLifshitzGilbert (LLG) equation, can be written as^{53}:
\(\frac{d\mathbf{M} }{dt}\) provides the information of the M over time, first term represents precision of moments, while the second term is responsible for damping. \(\gamma\) denotes the gyromagnetic ratio, \(\alpha\) stands for damping factor, these values are kept constant throughout the process of simulation. \(M_s\) represents saturation magnetization, and \(\mathbf{H} _{eff}\) is the effective field of demagnetization and external magnetic field. \(M_s\) is correlated with the total energy and \(\mathbf{H} _{eff}\) of the system, \(\mathbf{H} _{eff}= (E_{total}/\mathbf{M} )/(\mu _0 M_{s}\)), where \(E_{total}\) is the total energy density of the system. \(E_{total}\) would be the sum of all micromagnetics energies, which can be written as:
\(E_{exch}\) is the exchange energy, \(E_{anis}\) is the magnetocrystalline anisotropy energy, \(E_{demag}\) is the demagnetizing or stray field energy, \(E_{Zeeman}\) is due to an external field, and \(E_{me}\) is the magnetoelastic energy.
In the micromagnetic simulation, magnetic parameters are set the same throughout the cells. A stable state is achieved from a random energetic state. The simulation is performed implementing finite difference methods. The parameters are optimized by looking at various Codoped systems and the magnetic properties associated with atomic layers of Co films^{51}. The optimized parameters like exchange length: \(2.1\times 10^{6}~erg/cm\), saturation magnetization: \(30~emu.cm^{3}\), anisotropy values: \((46)\times 10^{3}~ erg/cm^3\) are considered for micromagnetics simulation. When Co is doped initially at Wsite, the magnetic moments are randomly dispersed in the cell matrix. On applying a sufficient magnetic field, the Co moments get oriented in the direction of external field. The snapshots of the various simulated states for the orientation of the spin moments while achieving from a random state to uniform ground state with systematic increase in applied magnetic field is shown in the supplementary information (see SI: Fig. S5). A single magnetic domain state is observed throughout the hysteresis. Figure 7a represents two kinds of hysteresis taken at two different easy axes. The hysteresis for (1, 0, 0) is quite squareinnature, where the nucleation of domains occurred near to the remanence. However, this signature of the loop contradicts the loop obtained for Codoped in \(WSe_2\), and \(WS_2\) ML systems^{35,36}. To achieve the experimental signature of the loop, which is slightly slanted and nucleation occurs at a distance from the remanence, we consider magnetization orientation tilted by 20% off from the original orientation, i.e, (1, 0.2, 0) direction. As a result, the obtained hysteresis is slightly slanted from the previous orientation (1, 0, 0), which is nearly similar to the hysteresis obtained in the \(WS_2\) system^{35}. From the previously studied experimental results, it has been observed that \(WSe_2\) shows relatively strong ferromagnetism after doping with TM Co^{36}. The experimental observation shows tilting magnetization hysteresis curve at all studied temperatures ranging from very low 5 K to 300 K^{36}. A similar FM response was reported in a wide range of temperature including room temperature of \(WS_2\)^{35}. Although the magnetic parameters of \(WS_2\) system are tuned for various factors such as size dependent (power and nanoribbons), thermal effect, doping, defect engineering, etc.; but, the tilting behavior is persist with all conditions^{25,35,36,52,54}. Hence, the behavior of tilting MH behavior has unaffected under these conditions, which ensures the anisotropic energy of the system attributes the tilting behavior in magnetization hysteresis curve of TMDCs. The coercive field for (1, 0, 0) orientation is around 345 Oe, whereas the same has been reduced to 190 Oe for (1, 0.2, 0). This coercive field is quite agreeing with the result reported in the \(WS_2\) system^{35}. Therefore, after Co doping into the system, it is worthy of mentioning that the effective magnetization orientation is slightly off from the easy anisotropic axis.
From DFT calculations it is evident that the FM behavior can be tuned under the application of strain. Here, the micromagnetic simulation provides the effect of strain on the larger scale in Codoped \(WS_2\) ML. In this case, if strain is applied to the system, we expect an alteration in the anisotropic values. In the literature also, it is reported that strain can certainly control the anisotropy in thin films^{55,56}. Here, we vary anisotropy values to understand the effect of strain in the Codoped \(WS_2\) system. In this case, we tune the anisotropy values from \(4\times 10^{3}~erg/cm^3\) to \(7.5\times 10^{3}~erg/cm^3\) to observe the changes in the magnetic properties. The coercive field is gradually decreased with the increase in the anisotropy value, which is represented in Fig. 7b. On the other way, FM properties are getting affected by enhancing anisotropy values. This behavior is quite analogous with the results obtained from DFT calculations, where the increase in the percentages in strain values leads to a decrease in FM nature. Our results reinvigorate the FM coupling behavior in Codoped \(WS_2\) ML under various strain conditions; a further understanding of magnetic reversal in this system may pave the way for nextgeneration spintronics and straintronics applications.
Conclusion
We explore the straininduced ferromagnetism in transition metal Codoped \(WS_2\) monolayer by using firstprinciples DFT calculations and micromagnetic simulation. Codoping marks a significant change in magnetic properties with an impressive magnetic moment of \(2.58~\mu _B\). The magnetic exchange interaction is found to be double exchange coupling between Co and W and strong pd hybridization between Co and nearest S, which is further verified from spin density distribution. We find that the resultant impurity bands of the Codoped \(WS_2\) plays a role of seed to drive novel electronic and magnetic properties under applied strain. Among several biaxial strains, the magnetic moment is found to be a maximum of \(3.25~\mu _B\) at 2% tensile and \(2.69~\mu _B\) at −2% compressive strain due to strong double exchange coupling and pd hybridization among foreign Co and W/S. Further magnetic moments at higher applied strain is decreased due to reduced spin polarization. In addition, the competition between exchange splitting and crystal field splitting of Co dorbital plays a significant role to determine these values of magnetic moments under the application of strain. From the micromagnetic simulation, it is confirmed that the Codoped \(WS_2\) monolayer shows slanted ferromagnetic hysteresis with a low coercive field. The effect of higher strain suppresses the ferromagnetic nature, which has a good agreement with the results obtained from DFT calculations. Our findings indicate that induced magnetism in \(WS_2\) monolayer under Codoping promotes the application of 2D TMDCs for the nanoscale spintronics, and especially, the strainmediated magnetism can be a promising candidate for future straintronics applications.
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Acknowledgements
S.K.G. thanks the Science and Engineering Research Board (SERB), India, for financial support (Grant YSS/2015/ 001269). R.A. thanks the Swedish Research Council (VR 2016 06014 & VR202004410) and J. Gust. Richert stiftelse, Sweden (202100655) for financial support. Authors gratefully acknowledge computational resources from the Swedish National Infrastructure for computing SNIC (2021/142) and HPC2N.
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The idea was proposed by A.J., S. K. M., and S.P.S. The simulations and data analyses were performed by A.J., S. K. M., and A.K.S. A.J., S.K.M., S.S., M.C.S., A.K.S., N.K.S., J.M., S.G., R.A., S.S. discussed the results and reviewed the manuscript.
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Jena, A.K., Mallik, S.K., Sahu, M.C. et al. Strainmediated ferromagnetism and lowfield magnetic reversal in Co doped monolayer \(WS_2\). Sci Rep 12, 2593 (2022). https://doi.org/10.1038/s4159802206346w
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DOI: https://doi.org/10.1038/s4159802206346w
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