Strain Engineering of Intrinsic Ferromagnetism in 2D van der Waals Materials

Since the discovery of the low-temperature, long-range ferromagnetic order in monolayers Cr2Ge2Te6 and CrI3, many efforts have been made to achieve a room temperature (RT) ferromagnet. The outstanding deformation ability of two-dimensional (2D) materials provides an exciting way to mediate their intrinsic ferromagnetism (FM) with strain engineering. Here, we summarize the recent progress of strain engineering of intrinsic FM in 2D van der Waals materials. First, we introduce how to explain the strain-mediated intrinsic FM on Cr-based and Fe-based 2D van der Waals materials through ab initio Density functional theory (DFT), and how to calculate magnetic anisotropy energy (MAE) and Curie temperature (TC) from the interlayer exchange coupling J. Subsequently, we focus on numerous attempts to apply strain to 2D materials in experiments, including wrinkle-induced strain, flexible substrate bending or stretching, lattice mismatch, electrostatic force and field-cooling. Last, we emphasize that this field is still in early stages, and there are many challenges that need to be overcome. More importantly, strengthening the guideline of strain-mediated FM in 2D van der Waals materials will promote the development of spintronics and straintronics.


Theoretical Calculations
In order to understand the essence of 2D ferromagnetism, ab initio Density functional theory, including linear density approach (LDA) [59], local spin density approximation (LSDA) [92], the generalized gradient approach (GGA) [61,93,94], and DFT + U [95,96], was often used to calculate the electronic structure of the system as a starting point. Moreover, the interlayer exchange coupling J was closely related to magnetic anisotropy, and it would also be used to calculate TC [86,[97][98][99][100]. The mean field theory [97,101] would roughly estimate TC but, often, TC overestimated it. Although the random phase approximation (RPA) could more accurately estimate TC of three-dimensional (3D) materials, it may fail in 2D systems with large anisotropy. Notably, classic Monte Carlo (MC) [6,97] simulations can also describe the critical temperature.

Cr-Based 2D van der Waals Materials
Unlike bulk materials, 2D materials can sustain larger strains [33,102]. Similarly, single-layer transition metal trihalides (MX3, I, Cl and Br) can also withstand a strain of about 10% [58]. As a typical example, Yan et al. [59] studied the biaxial strain dependence magnetic anisotropy energy (MAE) of the 2D monolayer CrCl3 (Figure 2a-d). When the compressive strain reached 2.5%, a phase transition from antiferromagnetism (AFM) to FM occurred ( Figure 2e). In addition, when tensile strain was 2.4%, the maximum Curie temperature (TC) reached 39 K. The MAE in the unstrained monolayer was positive, indicating the spins of Cr atoms were off-plane (Figure 2f).

Theoretical Calculations
In order to understand the essence of 2D ferromagnetism, ab initio Density functional theory, including linear density approach (LDA) [59], local spin density approximation (LSDA) [92], the generalized gradient approach (GGA) [61,93,94], and DFT + U [95,96], was often used to calculate the electronic structure of the system as a starting point. Moreover, the interlayer exchange coupling J was closely related to magnetic anisotropy, and it would also be used to calculate T C [86,[97][98][99][100]. The mean field theory [97,101] would roughly estimate T C but, often, T C overestimated it. Although the random phase approximation (RPA) could more accurately estimate T C of three-dimensional (3D) materials, it may fail in 2D systems with large anisotropy. Notably, classic Monte Carlo (MC) [6,97] simulations can also describe the critical temperature.
After applying the strain to the lattice, Cl atoms adjusted their position to minimize the lattice distortion energy at this strain ( Figure 2g). After generating a specific structure, the energy difference between the ferromagnetic and antiferromagnetic states was calculated. Mapping this energy difference to the Hamiltonian (1), Dupont et al. [63] obtained the nearest-neighbor exchange coupling J and magnetic anisotropy K, as shown in Figure 2h,i.
Note that: the spin value of S was 3/2 in the above equation.
As the strain evolved from compression to tension, the system sequentially exhibited the BKT (Berezinskii-Kosterlitz-Thouless) quasi-long-range order (LRO) phase, AFM Ising and FM Ising by QMC (Quantum Monte Carlo) simulations. Although theoretical and experimental studies have been conducted on monolayer and bulk CrCl 3 materials, research on their multilayer structures, including bilayers and trilayers systems, was very limited. Ebrahimian et al. [64] found that biaxial strains could also achieve a phase transition from AFM to FM (Figure 2k-n). In addition, the magnetic anisotropy could be mediated by the strain.

CrBr 3
More interestingly, the Curie temperature of monolayer CrBr 3 could be increased to 314 K by doping [103], which was between CrCl 3 (323 K) and CrI 3 (293 K). Although both hole doping and electron doping could enhance ferromagnetic coupling, the effect of hole doping was better at the same doping concentration. After applying biaxial strain to a unit cell, its magnetic moment remained unchanged, which indicated that the biaxial strain could not effectively enhance the ferromagnetic coupling of monolayer CrX 3 . In addition, Webster et al. [59] found that applying a tensile strain of 2.1% could increase the T C to 44 K, which was about 5 K higher than when no strain was applied. However, at a compressive strain of −4.1%, a FM to AFM phase transition, similar to CrCl 3 [59,63,64] and CrI 3 [59,[70][71][72], also occurred.

CrI 3
Unlike monolayer CrCl 3 , the electronic bandgap of monolayer CrI 3 remained almost unchanged after applying biaxial tensile strain; after applying biaxial compressive strain, the electronic bandgap decreased significantly and MAE increased significantly. When compressive strain reached 5%, MAE increased by 47% [59]. Continuing to increase the strain (−5.7%) resulted in a phase transition from FM to AFM. Similar to CrCl 3 , Wu et al. [69] also found that the CrI 3 monolayer underwent a complex phase transition from magnetic metals, half-metal, half-semiconductor to magnetic semiconductor as the strain evolved from compression (−15%) to tension (10%).
As a typical example, Vishkayi et al. [71] investigated the effects of biaxial and uniaxial strain on the magnetism of monolayer CrI 3 (Figure 3a-e). A similar phase transition from FM to AFM [59] was also observed when a compressive strain greater than 7% was applied ( Figure 3d). As the strain increased, the electronic bandgap showed an opposite trend when applying compressive or tensile strain ( Figure 3d). Interestingly, uniaxial strain enhanced the nearest neighbor, Dzialoshinskii-Moriya (DM) interaction, by breaking the inversion symmetry, and its effect was stronger than biaxial strain.

CrTe2
Guo et al. [104]. found that applying uniaxial or biaxial te CrTe2 in the T phase did not cause a phase transition from di Under biaxial strain, the CrTe2 monolayer [105] underwent pha compressive strain in Figure 4a  The phase transition from FM to AFM under compressive strain (−3% or −5%) had also been discovered in the CrI 3 bilayer [70], similar to other previously reported systems [59,71]. In addition, Safi et al. [72] found that the phase transition occurred at −6% compressive strain. More importantly, they also discovered a second phase transition point from FM to AFM, which occurred near −2.5% compressive strain.

CrTe 2
Guo et al. [104]. found that applying uniaxial or biaxial tensile strain to monolayer CrTe 2 in the T phase did not cause a phase transition from direct to indirect bandgap. Under biaxial strain, the CrTe 2 monolayer [105] underwent phase transformation at −1% compressive strain in Figure 4a  sion from [71]. Copyright 2020, American Physical Society).

CrTe2
Guo et al. [104]. found that applying uniaxial or biaxial tensile strain to monolayer CrTe2 in the T phase did not cause a phase transition from direct to indirect bandgap. Under biaxial strain, the CrTe2 monolayer [105] underwent phase transformation at −1% compressive strain in Figure 4a-k.  Interestingly, the CrTe 2 monolayer with strain-free was a FM state. After the tensile strain was applied, its T C could rise to 1022.8 K [106], and the magnetic moment of Cr atom increased linearly, which may be caused by the increase of the density of states at Fermi energy N EF . Magnetic anisotropy exhibited a different sensitivity to uniaxial and biaxial strain, as shown in Figure 4l. A monolayer was more sensitive to tensile strain, while a bilayer was more sensitive to compressive strain, and bulk was insensitive to the applied strain. Furthermore, band filling [108] of the monolayer also underwent a transition from out-of-plane to in-plane, while the bilayer and bulk did not undergo this transition ( Figure 4m). More interestingly, the strain had the greatest impact on the dihedral angle θ D , followed by the effect of Cr-Te-Cr bond length ( Figure 4n). Notably, the charge density wave (CDW) phase [107] promoted greater stability of the long-range FM order.

Cr 2 Ge 2 Te 6
As early as 2014, Li et al. [73] predicted that Cr 2 X 2 Te 6 (X = Si, Ge), a layered crystal with intrinsic FM, could be obtained experimentally by exfoliation. Furthermore, the T C of Cr 2 Ge 2 Te 6 (Cr 2 Si 2 Te 6 ) was increased by 51.7(56) K under an isotropic tensile strain. Similarly, Dong et al. [75] also found that Cr 2 Ge 2 Se 6 and Cr 2 Ge 2 Te 6 were a stable FM semiconductor, as shown in Figure 5a-e.

Cr2Ge2Te6
As early as 2014, Li et al. [73] predicted that Cr2X2Te6 (X = Si, Ge), a layered crystal with intrinsic FM, could be obtained experimentally by exfoliation. Furthermore, the TC of Cr2Ge2Te6 (Cr2Si2Te6) was increased by 51.7(56) K under an isotropic tensile strain. Similarly, Dong et al. [75] also found that Cr2Ge2Se6 and Cr2Ge2Te6 were a stable FM semiconductor, as shown in Figure 5a-e.  Through Monte Carlo simulation, it was found that the T C of Cr 2 Ge 2 Se 6 (144 K) was five times that of Cr 2 Ge 2 Te 6 (30 K) (Figure 5d). Interestingly, when compressive strain was applied, the T C decreased, and even at 2% strain, the phase became AFM. After applying tensile strain, the T C would be increased, even up to 500 K. Notably, the T C of Cr 2 Ge 2 Se 6 with 5% strain was always higher than RT (Figure 5e). To our knowledge, 2D Cr 2 Ge 2 Se 6 has not yet been prepared experimentally.
As a member of Cr-based materials, 2D CrPbTe 3 (CPT) has a stable monolayer structure and a higher T C , as shown in Figure 5f. Similar to Cr 2 Ge 2 Se 6 , its T C (110 K) gradually increased from compressive strain (61 K) to tensile strain (150 K) (Figure 5g-j). In addition, strain can also induce spin reorientation from the in-plane to the out-of-plane.

CrPS 4
AgVP 2 Se 6 , as a typical quaternary FM semiconductor, was synthesized by heating the elements in evacuated silica tubes as early as 1988 [88]. Ouvrard et al. found that its polycrystalline powder showed low-temperature FM, high-temperature PM and T C of about 29 K. Recently, single crystal AgVP 2 Se 6 samples [90] exhibited better stability than MX 3 materials. As another typical example, single crystal CrPS 4 [91] was obtained by the chemical vapor transport method and mechanical exfoliation. Interestingly, odd layers were ferromagnetic at lower temperatures, while even layers were antiferromagnetic, as shown in Figure 6. To the best our knowledge, there were currently no reports on regulating the magnetic properties of AgVP 2 Se 6 and CrPS 4 through strain engineering. crystalline powder showed low-temperature FM, high-temperature PM and TC of about 29 K. Recently, single crystal AgVP2Se6 samples [90] exhibited better stability than MX3 materials. As another typical example, single crystal CrPS4 [91] was obtained by the chemical vapor transport method and mechanical exfoliation. Interestingly, odd layers were ferromagnetic at lower temperatures, while even layers were antiferromagnetic, as shown in Figure 6. To the best our knowledge, there were currently no reports on regulating the magnetic properties of AgVP2Se6 and CrPS4 through strain engineering.

Fe-Based 2D van der Waals Materials
The bulk Fe3GeTe2 (FGT) with the hexagonal platelets was first synthesized by a direct solid-state reaction (SSR) [81]. Zhuang et al. [82] found that applying biaxial strain on single-layer metallic Fe3GeTe2 could enhance the MAE and total magnetic moment ( Figure  7a-d). When 2% tensile strain was applied, its MAE increased by 50%; After the applied strain changed from compression to tension, the total magnetic moment also increased with the increase of strain. Through DFT calculations, Hu et al. [85] discovered that monolayer FGT was a metallic type ferromagnet, with its magnetic moment mainly localized on Fe atoms and its metallicity mainly derived from the Fe d orbitals in Figure 7e-g.

Fe-Based 2D van der Waals Materials
The bulk Fe 3 GeTe 2 (FGT) with the hexagonal platelets was first synthesized by a direct solid-state reaction (SSR) [81]. Zhuang et al. [82] found that applying biaxial strain on singlelayer metallic Fe 3 GeTe 2 could enhance the MAE and total magnetic moment (Figure 7a-d). When 2% tensile strain was applied, its MAE increased by 50%; After the applied strain changed from compression to tension, the total magnetic moment also increased with the increase of strain. Through DFT calculations, Hu et al. [85] discovered that monolayer FGT was a metallic type ferromagnet, with its magnetic moment mainly localized on Fe atoms and its metallicity mainly derived from the Fe d orbitals in Figure 7e-g.
After applying the biaxial strain shown in Figure 7h, there was a significant change in the spin-density distributions of the FGT monolayer. The magnetism of this system mainly came from Fe atoms (Figure 7i), especially Fe3 atoms at the inequivalent site (Figure 7f), with little contribution from Ge and Te atoms (Figure 7j). Due to the Poisson effect, when stretched in the in-plane direction, it contracted in the out of plane direction. Therefore, as the bond lengths of Fe1-Te and Fe3-Te were reduced in the plane, the bond lengths of Fe3-Te in the vertical direction were increased (Figure 7k). Biaxial strain caused a change in bond length, thereby promoting charge transfer within the monolayer (Figure 7l). When the biaxial strain increased from −5% to 5%, the spin splitting of the Fe1 3d orbital near the Fermi level became larger, as shown in Figure 7m; after the strain was applied, the spin polarization of Fe3 atoms would increase in Figure 7n, which would cause the magnetic moment to increase. In conclusion, biaxial strain-mediated FM in the FGT monolayer was closely related to the charge transfer between Fe1 and Te atoms.
Previous studies have mainly focused on biaxial strain-mediated FM in the FGT monolayers [82,85], without studying the modulation mechanism of uniaxial strain on FM. As a typical example, Zhu et el. [86] investigated the uniaxial (ε a and ε c ) and isotropic (ε iso. ) strain modulation of the exchange coupling constant J ij and T C in FGT. As shown in Figure 8, three different magnetic configurations (FM, AFM1, and AFM2) were considered. The total energy of AFM2 was much higher than those of FM and AFM1.
fore, as the bond lengths of Fe1-Te and Fe3-Te were reduced in the plane, the bond lengths of Fe3-Te in the vertical direction were increased (Figure 7k). Biaxial strain caused a change in bond length, thereby promoting charge transfer within the monolayer ( Figure  7l). When the biaxial strain increased from −5% to 5%, the spin splitting of the Fe1 3d orbital near the Fermi level became larger, as shown in Figure 7m; after the strain was applied, the spin polarization of Fe3 atoms would increase in Figure 7n, which would cause the magnetic moment to increase. In conclusion, biaxial strain-mediated FM in the FGT monolayer was closely related to the charge transfer between Fe1 and Te atoms.   The system exhibited a phase transition from AFM to FM, similar to the modulation result of biaxial strain [59]. Interestingly, when the strain (ε a ) along the a-axis was applied, the phase transition occurred at a tensile of~4% or compressive of~2%; when the strain (ε c ) along the c-axis strain was applied, it occurred at a compressive of~8-10%. Furthermore, the strain-mediated variation of J ij total is shown in Figure 8g-i. T C could be estimated by the following equation [86,98,99]: Note that J max is the maximal eigenvalue of the matrix consisting of the exchange coupling between different atoms [86,109]. After applying the uniaxial strain, the T C could be raised to room temperature (Figure 8k-m). When isotropic strain is applied, the T C exhibited complex nonlinear changes and failed to reach room temperature (Figure 8m). Their results demonstrated that applying uniaxial strain was an effective way to elevate the T C .
Regarding another interesting compound, Fe 5 GeTe 2 [110,111], recent experiments have found that it has a higher T C . However, little is known about the electronic and magnetic properties of its monolayer. Joe et al. [83] found that the magnetism of bulk and monolayer metallic Fe 5 GeTe 2 originated from Fe d orbital. Moreover, biaxial strain could also enhance the Fe magnetic moment from 1.65 µ B to 2.66 µ B .

Wrinkle-Induced Strain
The previously discovered strain-mediated FM phenomena were all in 2D materials without intrinsic long-range magnetic order [27,28,41]. Very recently, Seidel et al. [42] found a clear dependence of T C on the strain state and the thickness of Cr 2 Ge 2 T 6 (CGT). The layered CGT lattice with intrinsic magnetism was shown in Figure 9a. A stripe domain structure was observed with MFM ( Figure 9b). After buckling, the wrinkled area had a higher T C than the flat area on (Figure 9c). DFT calculations indicated that the strain could elevate the T C in monolayer and bilayer Cr 2 Ge 2 T 6 (Figure 9d-g). In addition, the strain of three different wrinkles in Figure 9h was confirmed by COMSOL simulations. To the best of our knowledge, this was the first time that the strain distribution in CGT wrinkles was mapped with the COMSOL simulation. Furthermore, the magnetic signals could be observed at the wrinkled regions at RT through MFM.

Bending or Pre-Stretching Flexible Substrates
Similar to pre-stretched flexible substrates, including Gel-Film [30], polydimethylsiloxane (PDMS) [30,[112][113][114][115][116][117], polyethyleneterephthalate (PET) [118], polyimide (PI) [47,119,120], and polyvinyl alcohol (PVA) [121,122], strain can also be introduced into 2D materials by bending flexible substrates. As a typical example, Yan et al. [121] designed a novel polymer-buried strategy to apply tensile strain on Fe 3 GeTe 2 (FGT) nanoflakes ( Figure 10a). Firstly, PVA was spin-coated onto pre-stretched mechanical exfoliated FTG nanosheets. Then, the PET sheet was attached to the surface of the PVA film and peeled off with tweezers. The complex film (FGT/PVA/PET) was placed into a non-magnetic plastic tube and then the tensile strain was applied on the FGT nanosheets through a three-point fixing device. As the applied strain gradually increased, the sample transitioned from its original PM state to FM states, as shown in Figure 10b,c. When the strain reached 4.7%, the hysteresis loop evolved from soft magnet to hard magnet; however, when the strain reached 7.0%, the hysteresis loop displayed an opposite evolution trend. Moreover, the M s and H c were very close to those found for the case of tensile strain. Unlike zero-strain samples, the sample with a 3.4% tensile strain always exhibited centrosymmetry and no exchange bias (Figure 10d,e). The hidden AFM state and interface-exchange interaction could be revealed by controlling the strain. The frequency shift was inverted after relaxing the strain, indicating the transition from FM state to PM state (Figure 10f,g). As the strain increased, the T C was elevated above RT in Figure 10h,i. In short, strain engineering is an efficient way to increase T C compared to other methods (Figure 10j). main structure was observed with MFM ( Figure 9b). After buckling, the wrinkled area had a higher TC than the flat area on (Figure 9c). DFT calculations indicated that the strain could elevate the TC in monolayer and bilayer Cr2Ge2T6 (Figure 9d-g). In addition, the strain of three different wrinkles in Figure 9h was confirmed by COMSOL simulations. To the best of our knowledge, this was the first time that the strain distribution in CGT wrinkles was mapped with the COMSOL simulation. Furthermore, the magnetic signals could be observed at the wrinkled regions at RT through MFM.  the Ms and Hc were very close to those found for the case of tensile strain. Unlike zerostrain samples, the sample with a 3.4% tensile strain always exhibited centrosymmetry and no exchange bias (Figure 10d,e). The hidden AFM state and interface-exchange interaction could be revealed by controlling the strain. The frequency shift was inverted after relaxing the strain, indicating the transition from FM state to PM state (Figure 10f,g). As the strain increased, the TC was elevated above RT in Figure 10h,i. In short, strain engineering is an efficient way to increase TC compared to other methods (Figure 10j).  [111] and Co [123] represent pure and Co-doped Fe 5 GeTe 2 , respectively; Fe 4 refers to Fe 4 GeTe 2 [124]; Au [125], FePS 3 [37] and Bi 2 Te 3 [34] denote the corresponding heterojunctions with Fe 3 GeTe 2 . Gating means the electrostatically gated Fe 3 GeTe 2 [126], FIB [127] and Ga [128] are focused ion beam-treated and Ga ion-implanted Fe 3 GeTe 2 , respectively. (Reproduced with permission from [121]. Copyright 2021, The Royal Society of Chemistry).
As another typical example, Miao et al. [47] found that when 0.32% uniaxial tensile strain was applied to the FGT nano sheet, its H c increased by more than 150%. In order to apply strain, they directly exfoliated the FGT nanosheets onto the PI film. By controlling the distance of the needle tip pushed at the center of the substrate, the PI film was bent to apply different uniaxial tensile strains to the FGT sample. The difference of magnetic anisotropy energy was attributed to the strain-mediated FM. More importantly, they realized a magnetization reversal with the limited strain. Similarly, Xu et al. [48] also observed a reversible phase transition from AFM to FM in a CrSBr nanoflake at cryogenic temperature. Their strain equipment consisted of three piezoelectric actuators glued to a titanium flexure element. By cleaving a silicon substrate to form a micrometer-scale gap, the sample was suspended at the gaps. Furthermore, a piezo voltage was used to continuously apply strain reversibly to the CrSBr flake.

Lattice Mismatch
High-quality, single-crystalline Fe 4 GeTe 2 thin films [43] were grown on sapphire substrate by molecular beam epitaxy (MBE). As shown in Figure 11a, the thin films exhibited a rhombohedral structure. During the MBE epitaxy preparation, due to a lattice mismatch of about 20% between the Fe 4 GeTe 2 sample and the sapphire substrate, the lattice rotated 30 • to form a perfect single crystal.  [77,124,[129][130][131]. (Reproduced with permission from [43]. Copyright 2023, Springer Nature).
However, there was a tensile strain of about 2% inside the Fe4GeTe2 film. The in-plane M-T curve of the 16 nm FGT film in Figure 11b indicated that its TC was close to RT. The results of XRD ( Figure 11c) and HRTEM (Figure 11d) indicated that the FGT film had a perfect single crystal, and the atomic ratio of the Fe:Ge:Te element was about 4:1:2. At 300 K, a 4 nm film exhibited robust in-plane magnetic anisotropy, as shown in Figure 11e. By fitting the Mr-T curve in Figure 11f, the TC could be deduced as high as 530 K. Furthermore, in Figure 11, it was confirmed through M-H loops that the few layered FGT sample exhibited high-temperature FM. In addition, the thickness dependence of TC showed that when the thickness was thinner, its TC was higher, as shown in Figure 11h,i. Notably, MBE-prepared FGT had the higher TC and maintains a relatively high-magnetic anisotropy (Figure 11j). Through DFT calculations, it was found that a 2% tensile strain was not the reason for the enhanced TC.
However, there was a tensile strain of about 2% inside the Fe 4 GeTe 2 film. The in-plane M-T curve of the 16 nm FGT film in Figure 11b indicated that its T C was close to RT. The results of XRD ( Figure 11c) and HRTEM (Figure 11d) indicated that the FGT film had a perfect single crystal, and the atomic ratio of the Fe:Ge:Te element was about 4:1:2. At 300 K, a 4 nm film exhibited robust in-plane magnetic anisotropy, as shown in Figure 11e. By fitting the M r -T curve in Figure 11f, the T C could be deduced as high as 530 K. Furthermore, in Figure 11, it was confirmed through M-H loops that the few layered FGT sample exhibited high-temperature FM. In addition, the thickness dependence of T C showed that when the thickness was thinner, its T C was higher, as shown in Figure 11h,i. Notably, MBE-prepared FGT had the higher T C and maintains a relatively high-magnetic anisotropy (Figure 11j).
Through DFT calculations, it was found that a 2% tensile strain was not the reason for the enhanced T C .

Electrostatic Force
Considering the abnormally large mechanical response of Cr 2 Ge 2 Te 6 (CGT) thin films caused by strong magnetostriction at T C , it was expected that this reverse effect could achieve strain-controlled T C . As shown in Figure 12a-c, the electrostatic strain-tuning T C was realized in a suspended Cr 2 Ge 2 Te 6 /WSe 2 heterostructure [44]. After applying a gate voltage V g , there was electrostatic force (F el ) in Figure 12c between the heterostructure and the bottom silicon substrate, resulting in strain in Cr 2 Ge 2 Te 6 thin films. The strain of the CGT layer can be estimated using the following formula: Nanomaterials 2023, 13, 2378 15 of 23 the bottom silicon substrate, resulting in strain in Cr2Ge2Te6 thin films. The strain of the CGT layer can be estimated using the following formula: Note that Δ is the total strain in the CGT layer; is the electrostatic pulling strain; is bending strain; is the dielectric constant of vacuum; is the gate voltage; is the membrane radius; is the Poisson's ratio; is total tension in suspended heterostructures at = 60 K; and is the separation between the heterostructure and the bottom Si substrate. Interestingly, when the electrostatic force induced a strain of 0.026%, the TC of the suspended Cr2Ge2Te6 /WSe2 heterostructure was increased by about 2.6 K, as shown in Figure 12d-g.

Field-Cooling
Due to the fact that the lattice parameters of CGT materials at 5 K are greater than those at 270 K, it would cause in-plane expansion during cooling [133,134]. Phatak et al. [45] found that field-cooling could cause the material expansion in CGT flakes, leading to line-the estimate of electrostatically induced strain ∆ as a function of V g . The shaded magenta region shows the uncertainty in ∆ . The inset shows as a function of added voltage-induced strain ∆ in addition to calculations from Li and Yang [73] (solid green line) and from Dong et al. [132] (solid orange line). The vertical error bars in T C were estimated from determining the peak position in (e) within 2% accuracy in the measured maximum. (Reproduced with permission from [44]. Copyright 2022, American Chemical Society).
Note that ∆ is the total strain in the CGT layer; el is the electrostatic pulling strain; b is bending strain; ε 0 is the dielectric constant of vacuum; V g is the gate voltage; r is the membrane radius; v is the Poisson's ratio; n(T 0 ) is total tension in suspended heterostructures at T 0 = 60 K; and g 0 is the separation between the heterostructure and the bottom Si substrate. Interestingly, when the electrostatic force induced a strain of 0.026%, the T C of the suspended Cr 2 Ge 2 Te 6 /WSe 2 heterostructure was increased by about 2.6 K, as shown in Figure 12d-g.

Field-Cooling
Due to the fact that the lattice parameters of CGT materials at 5 K are greater than those at 270 K, it would cause in-plane expansion during cooling [133,134]. Phatak et al. [45] found that field-cooling could cause the material expansion in CGT flakes, leading to inplane strain. Furthermore, they directly observed strain-induced evolution of the magnetic domain structure by cryogenic Lorentz transmission electron microscopy (LTEM), which was closely related to the magnetoelastic coupling between strain and magnetization. This work revealed how to directly measure magnetic domain structures at the nanoscale.

Conclusions and Outlook
In this review, we have summarized the recent progress of strain-mediated intrinsic FM in 2D van der Waals materials with long-range order. First, we introduce how to explain the strain-mediated intrinsic FM on Cr-based and Fe-based 2D van der Waals materials with long-range FM order through ab initio Density functional theory (DFT), and how to calculate magnetic anisotropy energy (MAE) and T C from the interlayer exchange coupling. Subsequently, we focus on numerous attempts to apply strain to 2D materials in experiments, including wrinkle-induced strain, flexible substrate bending or stretching, lattice mismatch, electrostatic force and field-cooling. However, research in this field is still in its early stages and there are many challenges that must be overcome.
By selecting substrates with different thermal expansion coefficients, including xquartz [135][136][137][138], sapphire [27,28,139], and SiO 2 [114]), uniaxial [135][136][137][138] or biaxial strain [27,28,114,139,140] could be introduced into 2D materials. The introduction of biaxial strain in this way was limited to the study of 2D materials without intrinsic long-range magnetic order, such as MoS 2 [28] and ReS 2 [27]. So far, this strategy has not been applied to the research on 2D intrinsic van der Waals materials. Very few equipment [47,48,121] could apply strain to 2D materials, but less could be coupled with magnetic testing equipment [41,141]. Especially, it is very challenging that one could accurately determine the strain while measuring the magnetic properties and T C in different regions.