Strain-Modulated Magnetism in MoS2

Since the experiments found that two-dimensional (2D) materials such as single-layer MoS2 can withstand up to 20% strain, strain-modulated magnetism has gradually become an emerging research field. However, applying strain alone is difficult to modulate the magnetism of single-layer pristine MoS2, but applying strain combined with other tuning techniques such as introducing defects makes it easier to produce and alter the magnetism in MoS2. Here, we summarize the recent progress of strain-dependent magnetism in MoS2. First, we review the progress in theoretical study. Then, we compare the experimental methods of applying strain and their effects on magnetism. Specifically, we emphasize the roles played by web buckles, which induce biaxial tensile strain conveniently. Despite some progress, the study of strain-dependent MoS2 magnetism is still in its infancy, and a few potential directions for future research are discussed at the end. Overall, a broad and in-depth understanding of strain-tunable magnetism is very necessary, which will further drive the development of spintronics, straintronics, and flexible electronics.

Notably, strain engineering [3][4][5][6] is also an effective way to mediate the magnetism of 2D materials. However, most of the previous work mainly focused on theoretical calculations. We have first introduced biaxial strain into the MoS 2 film through spontaneous buckling and found that biaxial strain can enhance its room-temperature FM (RTFM) [72][73][74]. As a whole, an extensive and in-depth understanding of strain-mediated magnetism in MoS 2 is needed, which would provide new avenues for spintronics and straintronics.

Nanoribbons
Similar to ZnO [80,137,138] and graphene [4,10] NRs with zigzag-terminated edges, zigzag MoS2 NRs also exhibit FM [17,80] independent of NRs width and thickness due to the edge atoms. In contrast, armchair NRs show non-magnetism (NM). Interestingly, introducing adatoms can enhance the net magnetic moment of armchair NRs, but the FM of zigzag NRs is inhibited by the defects caused by adatoms [75]. Because the edge atoms are passivated, their spin polarization at the Fermi level is suppressed. Furthermore, an external static electric field can also reduce the energy gap of armchair NRs [76]. In detail, this electric field will drive metal-insulator phase transformation, which modulates or even suppresses FM.
In addition, monolayer and bilayer MoS2 are also sensitive to tensile strain but cannot produce the long-range magnetic order in Figure 2. However, the magnetic moment in zigzag NRs is nearly doubled by 10% strain [51], as shown in Figure 2B, which may be related to the magnetic coupling from different edge atoms. As shown in Figure 2B-E, the variation is generally not monotonous [48,49,51].
Interestingly, applying tensile strain and an electric field in the zigzag direction can cause the reversible modulation of FM [48]. The applied strain is within the elastic limit of the material, which achieves the reversibility of regulation. Even for zigzag Janus MoSSe NRs, the magnetism shows a multi-stage change with the increase in strain, which is closely related to the electronic phase transition. After the electric field is applied again, the magnetism can be regulated more effectively [81].
However, this modulation is obviously different from that of the zigzag MoS2 NRs shown in Figure 2E-G. Indeed, the difference in local spin density distribution determines the different modulation results of zigzag MoS2 NRs and zigzag Janus MoSSe NRs.

Nanoribbons
Similar to ZnO [80,137,138] and graphene [4,10] NRs with zigzag-terminated edges, zigzag MoS 2 NRs also exhibit FM [17,80] independent of NRs width and thickness due to the edge atoms. In contrast, armchair NRs show non-magnetism (NM). Interestingly, introducing adatoms can enhance the net magnetic moment of armchair NRs, but the FM of zigzag NRs is inhibited by the defects caused by adatoms [75]. Because the edge atoms are passivated, their spin polarization at the Fermi level is suppressed. Furthermore, an external static electric field can also reduce the energy gap of armchair NRs [76]. In detail, this electric field will drive metal-insulator phase transformation, which modulates or even suppresses FM.
In addition, monolayer and bilayer MoS 2 are also sensitive to tensile strain but cannot produce the long-range magnetic order in Figure 2. However, the magnetic moment in zigzag NRs is nearly doubled by 10% strain [51], as shown in Figure 2B, which may be related to the magnetic coupling from different edge atoms. As shown in Figure 2B-E, the variation is generally not monotonous [48,49,51].
Interestingly, applying tensile strain and an electric field in the zigzag direction can cause the reversible modulation of FM [48]. The applied strain is within the elastic limit of the material, which achieves the reversibility of regulation. Even for zigzag Janus MoSSe NRs, the magnetism shows a multi-stage change with the increase in strain, which is closely related to the electronic phase transition. After the electric field is applied again, the magnetism can be regulated more effectively [81].
However, this modulation is obviously different from that of the zigzag MoS 2 NRs shown in Figure 2E-G. Indeed, the difference in local spin density distribution determines the different modulation results of zigzag MoS 2 NRs and zigzag Janus MoSSe NRs.
In addition, the contribution of V atoms to magnetism is much greater than that of S or Se atoms [50]. In contrast, unstrained NbS 2 and NbSe 2 monolayers [52] are NM but can produce between 0.50 and 0.61 µ B per unit cell after applying 5% biaxial tensile strain. This novel magnetic behavior of NbS 2 and NbSe 2 monolayers is related not only to the bond length increased by strain but also to the metallic properties.
In fact, the self-exchange of populations between 4d orbitals of Nb atoms can lead to spin splitting [52,56]. Overall, V or Nb 4d states contribute mainly to the metallic state near the Fermi energy level [50,52,56]. By applying strain, the Curie temperature of the materials may be raised above room temperature [52], which will accelerate the spintronic application of 2D magnetic materials. However, MoS 2 , WS 2 , MoSe 2 , and WSe 2 have no intrinsic magnetism [52] due to their characteristic band structures.
Hydrogen atoms [53,67] can modify the electronic structure of pristine 2H-MoS 2 , but cannot produce spontaneous magnetism under <3% tensile strain [53], as shown in Figure 3A. With the increase in biaxial tensile strain, the magnetic moment and stability will be enhanced, as shown in Figure 3B. When the strain reaches 6.6%, the supercell obtains the most stable FM state, and the magnetic moment reaches 0.57 µ B per unit cell. In addition, its Curie temperature (Tc,~232 K) is much higher than that of the transition metal (TM)-doped system (Tc,~40 K) [84]. VSe2, exhibit FM [50], and the FM will increase rapidly when the strain increases from −5% to 5%. Metallic materials such as pristine 1T-MoS2 [33,37,41,67,139,140], VS2 [50,141,142] and VSe2 [50,[141][142][143][144][145] monolayer are more likely to form spontaneous magnetization. In addition, the contribution of V atoms to magnetism is much greater than that of S or Se atoms [50]. In contrast, unstrained NbS2 and NbSe2 monolayers [52] are NM but can produce between 0.50 and 0.61 µB per unit cell after applying 5% biaxial tensile strain. This novel magnetic behavior of NbS2 and NbSe2 monolayers is related not only to the bond length increased by strain but also to the metallic properties.
In fact, the self-exchange of populations between 4d orbitals of Nb atoms can lead to spin splitting [52,56]. Overall, V or Nb 4d states contribute mainly to the metallic state near the Fermi energy level [50,52,56]. By applying strain, the Curie temperature of the materials may be raised above room temperature [52], which will accelerate the spintronic application of 2D magnetic materials. However, MoS2, WS2, MoSe2, and WSe2 have no intrinsic magnetism [52] due to their characteristic band structures.
Hydrogen atoms [53,67] can modify the electronic structure of pristine 2H-MoS2, but cannot produce spontaneous magnetism under <3% tensile strain [53], as shown in Figure  3A. With the increase in biaxial tensile strain, the magnetic moment and stability will be enhanced, as shown in Figure 3B. When the strain reaches 6.6%, the supercell obtains the most stable FM state, and the magnetic moment reaches 0.57 µB per unit cell. In addition, its Curie temperature (Tc, ~232 K) is much higher than that of the transition metal (TM)doped system (Tc, ~40 K) [84].  Mo atom for 2H-MoS 2 H and the magnetic moment of Mo d orbitals per Mo as a function of strain (Reprinted/adapted with permission from Ref. [53]. Copyright 2013, American Physical Society). (C) Monolayer 1T-MoS 2 model without and with hydrogen adsorption (Reprinted/adapted from Ref. [67] with permission from the Royal Society of Chemistry). (D) The function of magnetic moments of Mo atom in 1T-MoS 2 as tensile strain (Reprinted/adapted from Ref. [67] with permission from the Royal Society of Chemistry). Note that: The 2H phase structure with space group Pm2 has hexagonal symmetry and the primitive unit cell of the single-layer has three atoms. The S atom is with trigonal prismatic coordination around Mo atoms; The 1T phase is also with hexagonal symmetry and the primitive unit cell of the single-layer has three atoms. In the 1T phase with space group Pm1, the S atom is with octahedral coordination around Mo atoms.
However, 1T-MoS 2 and 1T-MoS 2 H show FM behaviors, as shown in Figure 3C. Unlike 2H-MoS 2 , the relationship between magnetic moments and strain is linear, as shown in Figure 3D [67]. The crystal field makes a great contribution to the magnetism of the system.
Similarly, the biaxial tensile strain can also modulate the magnetism of nitrogen-doped 2H-MoS 2 [70]. When the strain gradually increases to 17.09%, a single nitrogen doping structure (NMo 16 S 31 ) shows different magnetic phases. However, the magnetic moment of a dense nitrogen doping structure (NMo 4 S 7 ) steps from 0 up to 1 µ B under 14% strain. In detail, unpaired electrons doped with nitrogen atoms will induce magnetic order. When the doped nitrogen atoms are too dense, the magnetic order will be weakened. However, the biaxial tensile strain has a good modulation effect on these two structures.

Defective Strained Systems
Inspired by the magnetism caused by conductive electrons in defective graphene. Many research groups tried to introduce single vacancies into the MoS 2 monolayer [57,94]. Experimentally, atomic single vacancies [91] (V Mo : mono-molybdenum vacancy; V S : monosulfur vacancy; V 2S: disulfur vacancy), vacancy complexes (V MoS3 : vacancy complex of Mo and nearby three sulfur; V MoS6 : vacancy complex of Mo nearby three disulfur pairs) and antisite defects [61,63] (S 2Mo : an S2 column substituting a Mo atom; Mo S : a Mo atom substituting an S column; Mo 2S : a Mo atom substituting an S2 column) have been observed in CVD (chemical vapor deposition)-grown MoS 2 monolayer by atomic-resolution annular dark field (ADF) imaging on an aberration-corrected scanning transmission electron microscope (STEM) [86]. Through first-principles calculations shown in Table 1, it is found that pristine [60,94] and single vacancy [57,60,94]-MoS 2 monolayer are NM. Notably, when 19% biaxial tensile strain is applied to the pristine MoS 2 monolayer, 4 × 4 supercells produce a magnetic moment of 5 µ B . However, the uniaxial strain cannot cause a magnetic phase transition regardless of the applied direction.
Interestingly, unstrained MoS 2 monolayers with V Mo [60,94], V S [57,60], V 2S [57,60,94], V MoS [60], V MoS3 [57], S 2Mo [61] and Mo S2 [61] are NM, as shown in Figure 4, while unstrained MoS 2 monolayers with V MoS2 [46,60,146], V MoS6 [57] and Mo S [63] are magnetic. In detail, the charge transfer and Mo atoms around the defects contribute mainly to magnetism. Furthermore, spin reorientation and the largest magnetic moment occur in the V 2S -MoS 2 monolayer [60], as shown in Figure 4A,B, which is related to magneto-crystalline anisotropy. With the increase in the tensile strain, FM-NM-FM phase transformation has been observed in V S -MoS 2 . Li et al. [91] have also drawn the magnetic phase diagram caused by strain and external electric field, as shown in Figure 4C,D. After applying strain, the charge sulfur vacancy defect shows rich magnetic responses.  Since Zhou et al. [86] and Jin et al. [147] found the antisite defects in the MoS 2 monolayer by STEM imaging in 2013, researchers have been trying to understand their magnetic characteristics in Figure 4E-H. In detail, the defect is an intrinsic structural defect. After applying 8% biaxial tensile strain, the system will produce long-range magnetic order [61]. Overall, the spin density is mainly distributed in the sulfur atom and its nearest or second neighbor, the Mo atom. However, the antisite-doped monolayer exhibits a high spin state under the biaxial strain from −7% to 4%. With the further increase in tensile strain, magnetism will vanish. The position of the antisite atom is related to the magnetism of the system. In addition, it is found that strained V S can greatly improve the hydrogen evolution activity of MoS 2 basal planes [148]. The sulfur vacancy will become a new active site and tune the adsorption-free energy of the hydrogen atom.

3d Transition Metal (TM) Ion-Doped Systems
Doping engineering [149][150][151][152][153][154][155][156][157] is a traditional way to control the properties of materials, especially for 2D materials. Recently, it has been confirmed experimentally [149,150,[158][159][160][161][162] Nanomaterials 2022, 12,1929 8 of 20 that 3d TM doping can induce ferromagnetism in nonmagnetic MoS 2 , which can be combined with strain engineering to tune the magnetism, as shown in Table 2. Interestingly, TM-doped systems show different magnetic responses. Except for V, Cr, Ti, and Sc atoms [88], the TM-doped systems without strain are nonmagnetic, and no matter how much biaxial strain is applied, there will be no long-range magnetic order. Arguably, Ma et al. [100] reported that V-doped monolayer MoS 2 exhibits magnetic half-metal at zero strain. After 2% compressive strain or 3% tensile strain is applied, the system will change from an FM state to an antiferromagnetic state.
Notably, the magnetic properties of Co/Ni/Cu/Zn-doped molybdenum disulfide show nonlinear changes with strain. After applying 20% compressive strain, the system is nonmagnetic. When the applied tensile strain reaches a specific value, the system will obtain a high spin state (5 µ B for the Co-doped; 5 µ B for the Ni-doped; 5 µ B for the Cudoped; 3 µ B for the Zn-doped). However, the magnetic moment will reduce to 0 under 20% tensile strain, except for the Co-doped system (3 µ B ).
The linear monotonicity of magnetism with strain has also been found in Mn-doped and Fe-doped MoS 2 systems, which is similar to those of 1T-MoS 2 and 1T-MoS 2 H. In detail, the systems are NM under a 20% compressive strain. Appling 20% tensile strain, the systems have obtained high spin states (2.8 µ B for Mn-doped; 4.3 µ B for Fe-doped).
In general, strain engineering is an effective method to control the magnetism of the TM-doped molybdenum disulfide system.

Methods of Appling Strain
Since the experiments revealed that 2D materials can withstand up to 20% strain, strain-modulated magnetism has gradually become an emerging research field. However, it is difficult to apply strain directly in suspended 2D materials in Table 3. In 2013, Andres et al. [104] created wrinkles in few-layer MoS 2 by pre-stretching the gel-film substrate, resulting in uniaxial tensile strain up to 2.5%. In the same year, uniaxial tensile strain (0-2.2%) was also applied in the MoS 2 /polycarbonate system by using fourpoint bending equipment [106]. Since then, many research groups have tried to apply strain through a variety of flexible substrates, including polymers [107], polyethyleneterephthalate (PET) [108,109], polyvinyl alcohol (PVA) [110], polyimide (PI) [111,112] and polydimethylsiloxane (PDMS) [113,114].
Recently, it has also been experimentally found that the strain can be introduced into the materials through patterned substrates such as holey Si 3 N 4 [125], rippled Si/SiO 2 [126], SiO 2 nanocones [127], SiO 2 nanopillars [128], pyramid/cones Al 2 O 3 [129], ZnO nanorods arrays [130], nanodots arrays, and so on. During the transfer of MoS 2 samples, bubbles [131][132][133] are often formed to introduce large strains into the samples. Notably, most of the methods required additional equipment to provide external stimulation, such as an AFM tip [134,135], an electromechanical device [74], or a focused laser beam [136]. Because scanning superconducting quantum interference device (SQUID) needs to be conducted in a cryogenic temperature and vibration environment, it is difficult to detect the strained material system. So far, material systems that can spontaneously form buckles [72][73][74] are more suitable for magnetic study.

Spontaneous Formation of Web Buckles
Spontaneous buckling [163,164] is frequently observed in the film system of traditional materials. When the residual strain in the film reaches its critical value, it will drive the film to delamination from the substrate and from spontaneous wrinkles [72][73][74]. Interfacial adhesion [73,165] is one of the key factors in determining whether buckling is formed or not. Relatively low adhesion is conducive to the formation and propagation of buckles. Because there is no hanging bond on the surface of 2D materials such as MoS 2 , the van der Waals (vdW) force is the interaction between the material and the substrate, and its interface adhesion is relatively low. Since then, MoS 2 films are very likely to become the perfect platform for understanding the phenomena of spontaneous buckling [73].
Recently, our group prepared ultra-smooth MoS 2 films [72,73] by polymer-assisted deposition (PAD), as shown in Figure 5. When the thickness of the film is about 400 nm, its roughness is about 1 nm. In the laboratory environment, MoS 2 films will also spontaneously form buckles due to external disturbance. Inspired by this experimental observation, we have used a tungsten probe close to the touch film to apply a point load. Once the probe touches the film, web buckles will be formed and further spread to the whole film surface. The formed large-area film with web buckles is very suitable for the SQUID test. Surprisingly, there is no obvious damage to the web buckle's structure after the magnetic test.
Since Ferrari et al. [167] successfully measured the uniaxial and biaxial strain in graphene samples in 2009, Raman spectroscopy has become a powerful tool to characterize the strain deformation of two-dimensional materials. Soon after 2013, the strain-tunable energy gap was studied in mono-, bi-, and tri-layer MoS 2 [104,106,136,168,169]. Notably, Yagmurcukardes et al. [170] studied how the strain modulated the Raman characteristics of single-layer materials by first-principle calculation. Therefore, we used Raman spectroscopy to quantify the strain in web buckles ( Figure 6) [72][73][74]. In detail, it is estimated by Raman mapping that about 68% of the region in the flat film has strain variations.
Since Ferrari et al. [167] successfully measured the uniaxial and biaxial strain in graphene samples in 2009, Raman spectroscopy has become a powerful tool to characterize the strain deformation of two-dimensional materials. Soon after 2013, the strain-tunable energy gap was studied in mono-, bi-, and tri-layer MoS2 [104,106,136,168,169]. Notably, Yagmurcukardes et al. [170] studied how the strain modulated the Raman characteristics of single-layer materials by first-principle calculation. Therefore, we used Raman spectroscopy to quantify the strain in web buckles So far, we cannot distinguish the magnetism from different buckled areas. Although traditional magnetic force microscopy can be obtained, we believe that there are too many impurity signals to identify the information in the samples. Hopefully, the newly emerging magnetic imaging technologies will provide technical support for further research. In order to clarify the strain-dependent ferromagnetism, we selected flat films and buckled films to test, as shown in Figure 7. After buckling, the saturation magnetization at 300 K increases to 7.5 times that before buckling. This is because the biaxial tensile strain induced by web buckles produces the generation of more defects such as VS. The enhancement of magnetism may be related to the decrease in compressive strain and the increase in defects.
So far, we cannot distinguish the magnetism from different buckled areas. Although traditional magnetic force microscopy can be obtained, we believe that there are too many impurity signals to identify the information in the samples. Hopefully, the newly emerging magnetic imaging technologies will provide technical support for further research.

Conclusions and Outlook
In this review, we have summarized the recent developments in strain-dependent magnetism in MoS2. First, we reviewed the progress of the theoretical study. Then, we compared the experimental methods of introducing strain and their effects on the ferromagnetism. We emphasized the roles played by web buckles since they could induce biaxial tensile strain conveniently for further tests, including magnetic measurements. Obviously, despite some progress, the study of strain-dependent MoS2 magnetism is still

Conclusions and Outlook
In this review, we have summarized the recent developments in strain-dependent magnetism in MoS 2 . First, we reviewed the progress of the theoretical study. Then, we compared the experimental methods of introducing strain and their effects on the ferromagnetism. We emphasized the roles played by web buckles since they could induce biaxial tensile strain conveniently for further tests, including magnetic measurements. Obviously, despite some progress, the study of strain-dependent MoS 2 magnetism is still in its infancy.
Although RTFM has been enhanced experimentally by biaxial strain [72] induced by web buckles, the magnetism contributions from different zones cannot be distinguished experimentally. Since most conventional magnetic probes [171] require the sample area to be at the millimeter level, magnetic testing of the micron wrinkled area is a great challenge. Very recently, magnetic imaging techniques have emerged as important tools for investigating 2D materials, such as magnetic force microscopy (MFM) [172][173][174][175][176][177][178], SQUID [179,180], magneto-optical Kerr effect (MOKE) [181,182] and scanning nitrogen-vacancy center microscopy (SNVM) [183][184][185][186][187]. These techniques make it possible to detect the magnetism of the wrinkled area.
Since the modulation effect of uniaxial strain on the properties of materials is weaker than that of biaxial strain, whether the RTFM of molybdenum disulfide can be regulated by uniaxial strain has always been a mystery, which is worthy of further exploration. In addition, the substrates commonly used in experiments are isotropic, so it is relatively easy to introduce isotropic strain (such as biaxial strain) into 2D materials. Recently, anisotropic substrates such as m-quartz [121,188] have been used in experiments, which provides a new idea for introducing uniaxial strain into MoS 2 . We believe that the regulation of uniaxial strain on FM can be explained clearly by combining nanoscale magnetic detection instruments.

Conflicts of Interest:
The authors declare no conflict of interest.