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Cite this:Phys.Chem.Chem.Phys.,

2020, 22, 17255

Magnetism and spin exchange coupling instrained monolayer CrOCl†

Xiaomei Qing,ab Hua Li,a Chonggui Zhong, *ac Pengxia Zhou,*ad

Zhengchao Dongad and Junming Liu d

The magnetism and spin exchange coupling of monolayer CrOCl with different strains are investigated

systematically using first principles. It is found that the magnetic ground state can be changed from

ferromagnetic (FM) to antiferromagnetic (AFM), and the Curie temperature (TC) is enhanced significantly

by applying the uniaxial strain along a- or b-axis direction. The variations of spin exchange coupling are

explained according to the Goodenough–Kanamori–Anderson (GKA) and Bethe–Slater Interaction (BSI)

rules. The strain-dependent magnetic state is mainly attributed to the competition between direct

exchange interactions of cation–cation and indirect superexchange ones of cation–anion–cation in

monolayer CrOCl. The different competitions in a- and b-axis direction determine the different critical

intervals R of magnetic transitions, where R is the distance of the two nearest-neighbor (NN) Cr3+ ions.

The AFM-FM transition occurs at R/r3d = 2.9 and 3.75 in a-axis direction, while it happens at R/r3d = 2.65

along b-axis direction. These results indicate that the sensitive relevancy between the external strain and

magnetic coupling makes monolayer CrOCl a promising candidate for spintronics.

I. Introduction

The emergence of two-dimensional (2D) materials such asgraphene has brought a new opportunity for the developmentof spintronics.1–4 Ferromagnetism is believed to be very usefulfor spintronics, however the Mermin–Wagner theorem statesthat in 2D systems strict long range order at finite temperatureis impossible because of the fluctuations. Therefore it has beena widely held belief that ferromagnetism cannot survive in 2Dsystems until the recent observations of long-range magneticorder in 2D CrI3 and Cr2Ge2Te6.5–7 Moreover, the molecularbeam epitaxial growth has been reported to confirm the existenceof 2D magnetism in Fe3GeTe2, VSe2, MnSex, and Cr2Ge2Te6.8–11

However, the typical Curie temperatures (TC) of 2D magnets aremuch lower than those of their 3D form, such as CrI3 (45 K) andCr2Ge2Te6 (20 K).6,7 The development of new ferromagnetic 2Dcrystals with strong long-range spin order and high TC is a veryimportant and challenging research topic.4,7 Recently a family ofFM semiconducting/metallic monolayers with high TC values has

been theoretically predicted, namely MnNX and CrCX (C = S, Se,Te; X = Cl, Br, I), which enlightens a strategy for searching otherhigh TC FM in polyvalent CrOCl monolayers with two differentvalence states (of anions: O2� and Cl�).12

Mechanical strain is commonly regarded as an effectiveroute to manipulate the physical properties including the spinorder and transition temperature because these propertiescritically depend on the material’s structure.4,13 For the 2Dmaterials, the small Young’s modulus of elasticity make itpossible to withstand large strains in the systems, for example,the monolayer graphene and MoS2 can undergo about 25%strain,14–16 which is much larger than those of perovskite films.The investigation has demonstrated that strain can not onlyresult in the splitting of certain peaks in the Raman spectrumof graphene,17 but also affect the electronic structure of bothmonolayer and bilayer MoS2 and modulate optical properties ofrippled 2D GaSe.18,19 Moreover, the spin–lattice coupling hasbeen investigated experimentally or predicted theoretically inlayered magnets, such as Cr2Si2Te6,20 Cr2Ge2Te6,21 Fe3GeTe2,22

MnPSe3 and CrI3.23,24 Biaxial tensile strain can lead to an AFM-FM transition in 2D MnPSe3,23 and a FM-AFM transition inmonolayer CrI3 by spin–lattice coupling effect.24 These investi-gations indicate that atomic scale in 2D intrinsic magnetic filmprovide a platform for strain to manipulate electronic structuresand spin exchange coupling due to their fascinating highstretchable property.

Different from the bulk AFM state,25 monolayer CrOCl ispredicted to be a new type of potential 2D intrinsic ferromagnetic

a School of Sciences, Nantong University, Nantong 226019, China.

E-mail: chgzhong@ntu.edu.cn, ntzhoupx@ntu.edu.cnb School of General Education, Nantong Institute of Technology, Nantong, 226602,

Chinac School of Physical Science and Technology, Soochow University, Suzhou, 215006,

Chinad Laboratory of Solid State Microstructures and Collaborative Innovation Center of

Advanced Microstructures, Nanjing University, Nanjing 210093, China

† PACS: 75.70.Ak; 74.62.-c; 75.30.Et; 77.80.bn.

Received 29th February 2020,Accepted 13th July 2020

DOI: 10.1039/d0cp01160f

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materials for its small exfoliation energy, strain-dependent TC,great dynamical and thermal stabilities.26 Similarly to thepreparation of graphene, the monolayer CrOCl can also besuccessfully mechanically exfoliated,27 however, the magneticproperties of the monolayer CrOCl were theoretically predictednot to follow the GKA rules28,29 by which the graphene abide,especially magnetic exchange coupling, even though the GKArules have been successful in explaining the magnetic propertiesinduced by superexchange interactions on a lot of 2D and bulkmaterials, e.g. recently-highlighted 2D CrX3 (X = F, Cl, Br, I)materials and bulk ABX3 (A = Mn, Cr; B = P, Si; X = S, Se, Te) etc.Besides the GKA rules, an extended superexchange theory (ESET) isalso proposed to explain the magnetic properties in monolayerCrOCl.30 However, it remains not clear whether the ESET theorycan be applied to the strained monolayer CrOCl, and many relevantproperties of monolayer CrOCl are still under investigation,especially effects of strain on electrical, magnetic and opticalproperties.

In the present paper, using first-principles calculations, weinvestigated the magnetic phase transition, the spin exchangeinteraction and mechanism of monolayer CrOCl materials understrain engineering. By applying the 0–16% uniaxial strain, includingcompression and tensile ones along a- and b-axis, we find that theuniaxial compressive strain along a-axis (15%) or b-axis (8%), andtensile strain along a-axis (16%) both can lead to FM-AFM transi-tion. The strain-dependent magnetic properties of monolayer CrOClis mainly attributed to the competition of the direct interaction andindirect superexchange interaction between the two NN Cr3+ atoms.

II. Computation detailII.1. Crystal structure

The bulk transition-metal oxyhalide CrOCl is an AFM semi-conductor, which has orthorhombic structure (space groupPmmn) with 2D rectangle sublattice in the a–b plane, seeingFig. 1(a–c). The Cr ions bounded by O atoms form alternatedpyramids while the planar layers sandwiched by Cl atomiclayers are stacked along the c-axis with a large van der Waalsgap, as shown in Fig. 1(b) and (c). Every Cr ion is coordinated bysix covalent bonds including neighboring four O and two Clions, and these bonds generate a strong twisted octahedron ofCrO4Cl2, as plotted in Fig. 1(d).

II.2 Computation method

All calculations were performed based on density functionaltheory by using the projector-augmented wave (PAW) method asimplemented in Vienna ab initio Simulation Package (VASP).31,32

The generalized gradient approximation (GGA) with Perdew–Burke–Ernzerhof parametrization (PBE) functional is used forthe exchange–correlation functional calculations.33 The valenceelectron configuration is set by: Cr 3d54s1, O 2s22p4 and Cl3s23p5 in our calculations. The strong on-site Coulomb interactionfor the transition-metal Cr-3d orbital are considered by the GGA+Umethod following the Dudarev scheme to compensate for the self-interaction errors of GGA.34,35 We tested several U values, and the

test results showed that when U = 7.5 eV and J = 0.5 eV, theoptimization results were more consistent with the experimentalparameters.25 For all calculations, we set the plane-wave cutoffenergy to 450 eV, the convergence energy to 1.0 � 10�6 eV and theresidual forces on atoms to 0.001 eV Å�1. A 2� 2� 2 supercell andone 2 � 2 � 1 supercell are adopted in optimizing the bulk andmonolayer structures, respectively. The Brillouin zone integral wasgenerated automatically by the Monkhorst–Pack method, andthe k-point meshes for bulk and monolayer were 5 � 6 � 5 and5 � 6 � 1 respectively. For monolayer CrOCl, the vacuum spacethickness of 20 Å was set along the c-direction, and for bulk, theDFT-D3 method was used to consider van der Waals (vdW)correction.36

III. Result and discussionsStructure optimization

In order to investigate the magnetic coupling in monolayerCrOCl, firstly, four different magnetic configurations, includingFM, AFM-Neel, AFM-stripy-1, and AFM-stripy-2 state, are constructedin a 2 � 2 � 1 supercell of monolayer CrOCl, as shown in Fig. 2.When the four magnetic state structures are optimized inunstrained monolayer CrOCl, we find that the energy differencesbetween the three AFM and FM state are 109.11, 175.48, and101.35 meV per unit cell respectively, as listed in Table 1, whichindicates FM order is the ground state (G.S.) in monolayer CrOCl.Moreover, the obtained lattice constants, band gap, and magneticmoment of each metal ion of unstrained monolayer CrOCl atFM ground state is a = 3.941 Å, b = 3.272 Å, Eg = 2.35 eV andM = 3.235 mB respectively, which are consistent with the previousresults.26 Of course, it should be noted that the non-magnetic (NM)state has not been considered owing to the great energy disparitybetween the NM state and magnetic states.

To confirm the magnetic order in bulk CrOCl without strain,we combine two FM state layers with the interlayer FM and AFM

Fig. 1 Crystal structures of bulk and monolayer CrOCl. (a) Top view of2 � 2 planer from c-axis. (b) Side view of the supercell of 2 � 2 � 2 bulkalong a-axis. (c) Side view of the supercell of 2 � 2 � 1 monolayer alongb axis. (d) The twisted octahedron of CrO4Cl2.

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coupling for magnetic calculations, respectively. It shows thatthe energy of interlayer AFM coupling is favored by 1.68 meVper u.c. lower than the FM state, which indicates that the bulkCrOCl is an intrinsic antiferromagnet.25 While for the AFM bulkCrOCl, the obtained lattice constants a, b and c are respectively2%, 2.7%, and 0.8% larger, and electronic band gap andmagnetic moment are approximately 3.5% less than the experi-mental values,25,26 as illustrated in Table 2.

In the meantime, allowing for the effect of spin–orbit coupling(SOC) on the electronic structure and magnetic properties, we takeit into account in the self-consistent calculations to clarify thenature of magnetic anisotropy in CrOCl. Then, we calculate theunit cell total energy E100, E010 and E001 with the magnetic momentalong the [100], [010] and [001] direction respectively, as shown inTable 3. It is found that the E001 is 0.74 and 0.72 meV less than E010

and E100, respectively. In other words, monolayer CrOCl has theout-of-plane magnetic anisotropy, but the magnetic anisotropyenergy is small. Then, the energies of four magnetic structuresare calculated with SOC, respectively. It is found that the unit celltotal energy EFM of the FM state is 116.42, 178.37 and 100.85 meVless than those of the AFM-Neel, AFMstripy-1 and AFMstripy-2 state

when the easy axis lies along the [001] direction, respectively. Itfollows that the FM order is still G.S.

Further, to examine the influence of SOC on electronicproperties, we also compared the band structure with SOCand without SOC, as shown in Fig. 3. The results show thatthe inclusion of the SOC in the FM state hardly changes thecorrespondingly band dispersions near the band edges, and theindirect energy gap with the SOC is just 0.01 eV less than thatwithout SOC. Compared with the band without SOC, The effectof SOC is only to shift the whole bands slightly towards thelower energy region, while the location of the CBM and VBMremain the same. Apparently, distinct from the chromiumtriiodide CrI3,37,38 the SOC effect is weak in monolayer CrOCl.This findings agrees well with a very recent theoretical resultsby Wang et al.12 Therefore, based on this weak influence of theSOC on the energy band, G.S. and magnetism of monolayer CrOCl,we neglect the effect of SOC in the following investigations.

Magnetic phase transition

To investigate the variations of magnetism and spin exchangecoupling of the strained monolayer CrOCl, different tensile andcompressive strains along a- and b-axis are applied respectively

Fig. 2 Four magnetic configurations and magnetic constants of mono-layer CrOCl films. (a) FM; (b) AFM-Neel; (c) AFM-stripy-1; (d) AFM-stripy-2.J1, J2 and J1 are the magnetic constants of the first, second and thirdnearest neighbor in (a).

Table 1 The total energy of 2 � 2 � 1 supercell in the FM and threedifferent AFM states of monolayer CrOCl, here DE = Eother � EFM

Magnetic state FM AFM-Neel AFM-stripy-1 AFM-stripy-2

E (eV) �131.65795 �131.54884 �131.48247 �131.55660DE (meV) 0 109.11 175.48 101.35

Table 2 The lattice constants, band gap energy, magnetic moment of unstrained monolayer and bulk CrOCl

Type

Monolayer

G.S.

Bulk

G.S.a (Å) b (Å) Eg (eV) M (mB/Cr) a (Å) b (Å) c (Å) Eg (eV) M (mB/Cr)

This paper 3.941 3.272 2.35 3.235 FM 3.943 3.270 7.754 2.70 3.241 AFMExperiment25 — — — — — 3.863 3.182 7.694 2.8 3.36 AFMOther paper26 3.942 3.256 2.38 3.24 FM 3.947 3.264 7.711 2.71 3.22 AFM

Table 3 The G.S., band gap, magnetic moment, unit cell total energy E100,E010 and E001, easy axis of unstrained monolayer CrOCl with SOCcalculations

G.S. Eg (eV) M (mB/Cr) E001 (eV) E010 (eV) E100 (eV) Easy axis

FM 2.34 3.234 �131.6768 �131.67608 �131.6761 c

Fig. 3 The calculated band structures in FM state (a) without SOC (b) withSOC.

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for the above four magnetic structures, while the c-direction iscompletely relaxed. The strain is defined as e = (a � a0)/a0,where a0 and a are the lattice constants of unstrained andstrained monolayer CrOCl, e 4 0 and e o 0 denote the tensilestrain and compressive strain, respectively. In our calculation,the strain e is varied from �16% to 16%. Then, the calculatedtotal energies of supercell with different magnetic states underuniaxial strain along a- and b-axis are obtained, as shown inFig. 4(a) and (b). The parabolic curves indicate that the mono-layer CrOCl is in the range of elastic variation.

From Fig. 4(a), it is found that monolayer CrOCl do notchange the FM state until a rather large uniaxial strain e (�15%and 16%) along a-axis, where the magnetic transitions between FM

and AFMstripy-1 occur obviously, seeing Fig. 4(c). However, there is alittle difference under strain along b-axis, the FM-AFMstripy-2

magnetic transition will be observed when the strain reach toe = �8%, as shown in Fig. 4(b) and (d), but the tensile strain inthis direction can’t realize a magnetic transition. Of course, thestrain-dependent magnetic transition has also been revealed inmany other 2D magnetic materials, such as MnPSe3, CrI3,CrSiTe3 and CrPS4.23,24,39,40

As seen from Fig. 4(c) and (d), the magnetic order ofmonolayer CrOCl is sensitive to strain and exhibits the obviousanisotropy, however, the magnetic exchange interaction do notconform entirely to the GKA rules, which has revealed that theAFM (FM) order is along the bond angle of 1801 (901) in cation–anion–cation interaction path, for example MnO.30 In 2Dsemiconductor MnPSe3,23 FeOCl and CrX3 (X = F, Cl, Br, I),41,42

it is predicted theoretically that the strain-dependent magneticstability is mainly attributed to the competition of the directinteraction and indirect superexchange interaction between thetwo NN magnetic atoms. Then, as for whether the similarcompetition also exists in monolayer CrOCl, it is something thatneeds to be discussed in depth.

Exchange coupling

Both magnetic transition and variation of Curie temperaturecan be tuned, which may be ascribed to the strain-tuningmagnetic exchange coupling in monolayer CrOCl. To clarifythe relations among them, it is necessary to estimate the spinexchange coupling of the neighboring level. We considered thefirst, second and third nearest-neighbor magnetic exchangecoupling as: J1, J2, and J3, which are shown in Fig. 2(a). Usingthe spin Hamiltonian based on the 2D Heisenberg model:

H ¼ �Ji;jP

~Si � ~Sj ,26,42 where Ji,j is the adjacent spin exchange

coupling constants, and mapping the DFT energies of differentmagnetic states to the Heisenberg model, the four spin con-figurations energies satisfy the following equations:12,38,42

EFM = E0 � (4J1 + 2J2 + 2J3)|-

S|2, (1)

EAFM-Neel = E0 � (�4J1 + 2J2 + 2J3)|-

S|2, (2)

EAFM-stripy-1 = E0 � (2J2 � 2J3)|-

S|2, (3)

EAFM-stripy-2 = E0 � (�2J2 + 2J3)|-

S|2, (4)

where E0 is the nonmagnetic total energy of one magnetic atom,and |

-

S| is the magnetic moment of Cr ions that equals to 3m0.Firstly, for the unstrained monolayer CrOCl, the calculated

results are J1 = 0.189 meV, J2 = 0.162 meV and J3 = 0.420 meV. Allof them are positive, indicating their FM interactions. TheJ3 4 J1 4 J2 relation suggests the largest spin exchangecoupling existed along a-axis, which is in agreement with theresult of previous calculations.30 According to GKA rules,28–30 acrossover angle of 127 � 0.51 is proposed to be related to theFM-AFM transition.42,43 The two Cr–O–Cr angles are 951, veryclose to 901, so J1 is a FM superexchange coupling through O2�

ions, as shown in Fig. 5(a). Similarly, as the Cr–Cl–Cr angle isvery close to 901, seeing Fig. 5(b), J2 is also FM superexchangethrough Cl� ion. The Cr–O–Cr angle 1541 along a-axis is much

Fig. 4 The total energy E of supercell with uniaxial strains in (a) a-axis,(b) b-axis directions for different magnetic states, and schematic plot ofthe effect of uniaxial strain in (c) a-axis, (d) b-axis directions on themagnetic coupling.

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larger than 1301, suggesting that J3 should be AFM couplingthrough O2� ion, as shown in Fig. 5(c). However, the actual calcula-tion shows that J3 is a strong FM coupling between two Cr3+ ionsalong a-axis, which is obviously incompatible with the GKA rules.

Further, we investigated the variations of adjacent exchangecoupling constant J1, J2 and J3 with different strains alonga- and b-axis, as shown in Fig. 6(a) and (b). From Fig. 6(a), itis found that the effects of the strain along a-axis on exchangecoupling J1 and J2 are weak and insensitive to the strain,while the variation of J3 is very obvious throughout the wholeconsidered strain range. The slight variations and positivevalues of J1 and J2 indicate that the system in the c- andb-directions always remain in the FM state. The J3 will becomenegative when the �13% compression or 12% tensile strain isapplied, and reach the maximum under �8% compressivestrain. However, the strain along b-axis acts significantly onthe exchange coupling J2, but have negligible influence on thepositive J1 and J3, as seen from Fig. 6(b).

Then why don’t the variations of exchange coupling J2 and J3

under different strains completely meet the GKA rules? Firstly,there is no doubt when the tensile strain is beyond 12% ina-axis direction of the monolayer CrOCl, the J3 can be ascribedto the Cr–O–Cr superexchange coupling interaction, and meetthe GKA rules. However, while the strain is less than about12%, we note that the variation of J3 between two NN Cr3+ ionsalong a-axis direction is highly consistent with the Bethe–SlaterInteraction (BSI) curve.44,45 According to the BSI theory, thedirect exchange function J can be positive or negative, dependingon the ratio of the interatomic spacing R to the orbital radius r3d

of 3d shell. As shown in Fig. 6(c), the curves of J3 and J2 vs. R/r3d

are also plotted. It follows that the J3 can be ascribable to thedirect exchange coupling, obviously, it cannot be explainedideally by the GKA rules and the extended superexchange theory(ESET).30 Especially, the transition point of FM-AFM coupling forJ3 under compression strain is at R/r3d = 2.9 in J3 � R/r3d curve,which is also good in line with the value in BSI rules and thetiny difference is due to the effect of Cr–O–Cr superexchange

coupling. This phenomenon has also been revealed in themonolayer CrPS4 AFM-FM transition, which is driven by thechange of distance between Cr–Cr.40

In detail, the AFM-FM transition in the monolayer CrOClwith stain can be understood by the intuitive physical picture ofFig. 5(d) and (e), where the bond angles of Cr–O–Cr are givenfor the two cases of �15% and 16% strain. When the �15%compression strain along a-axis is applied in the system, theCr–O–Cr angle decreases from the nostrained 1541 to 1261 dueto the decrease of lattice constant a, and the distance R of twoNN Cr3+ ions in a direction decreases correspondingly from3.942 Å (R/r3d = 3.34 4 2.9) to 3.31 Å (R/r3d = 2.8 o 2.9). Then,according to the BSI curve, the J3 exhibits the direct AFM exchangecoupling due to the Pauli exclusion principle.46 Furthermore, this

Fig. 5 Schematic diagrams of magnetic exchange interactions. Theexchange coupling for (a) J1, (b) J2, (c) J3 in unstrained monolayer CrOCl,respectively; (d) J3 under �15% compressive and (e) J3 under 16% tensilestrain along a-axis direction, the former is the AFM direct exchange and thelatter indirect superexchange coupling; (f) J2 under �8% compressivestrain along b-axis direction, which is AFM direct exchange coupling.

Fig. 6 Variations of magnetic exchange coupling J1, J2 and J3 withuniaxial strain e. Here, the applied e is along (a) a-axis and (b) b-axis.(c) J2 and J3 are plotted as a function of the R/r3d.

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type of direct exchange coupling between the NN cations throughthe short distance cation–anion–cation path have been reported inbulk TiOCl and TiOBr.47,48 However, under the 16% tensile strainalong a-axis, the Cr–O–Cr bond angle increases from 1541 to 1621,and the increase of R makes the direct exchange interactionbecome weak, while the superexchange effect become dominant,as a result, the AFM coupling appears according to the GKA rules.

In addition, it is obvious to deduce that the variation ofexchange coupling J2 along b-axis may be ascribable to thecompetition between the superexchange coupling of Cr–Cl/O–Cr and direct exchange coupling of two NN Cr3+ ions. When thetensile or small compression strain is applied, the super-exchange interactions between two Cr3+ ions by Cl� and O2�

ions play the major roles. According to the GKA rules, twosuperexchange couplings should be FM due to about 901 bondangle of Cr–Cl/O–Cr, thus, the J2 shows FM coupling, as shown inFig. 6(b). Likewise, this type of the FM superexchange couplingshas also been reported to dominate in many 2D materials.42 Onthe contrary, if a relatively large compression such as �8% strainis applied, the J2 shows AFM coupling, which is due to the smallspacing R (R/r3d = 2.55) between two NN Cr3+ ions under thisstrain (see Fig. 5(f)). Since the AFM direct exchange couplingoverwhelms gradually the two FM superexchange ones withdecreasing the spacing R, the AFM interaction plays a major role,just like the phenomenon in MnPSe3.23

Then, we return back to the J1, J2 of the non-strainedmonolayer CrOCl, it is found that even if the R/r3d is alsosmaller than 2.9, the J1 and J2 still show the FM coupling, whichindicate that although the AFM direct exchange coupling isstrong at the R/r3d = 2.63 for J1 and 2.77 for J2 according to theBSI curve, as shown in Fig. 5(a) and (b), the combined interactionsof two FM superexchange couplings still prevail over the singleAFM coupling. For J2 in b-axis direction, we can obtain that theequilibrium point of competition between AFM direct exchangeand two FM superexchange couplings by Cr–O–Cr and Cr–Cl–Croccurs at R/r3d = 2.65 from the Fig. 6(c). Moreover, it can beinferred that the transition point of J1 has a smaller value than2.63. Obviously, the transition point at R/r3d = 2.90 along a-axisdirection is largest in three different directions, which may beascribable to the different competitions in different directions. Ina-axis direction, the competition exists between a direct exchangeand a superexchange coupling, while in the other two directions,the competition exists between a direct exchange and two super-exchange couplings.

Transition temperature

In the mean-field approximations, the transition temperatureTC(TN) of monolayer CrOCl can be evaluated by the followingrelationship:38,49

T Si!D E

z¼ 2

3kB

Xiaj

Ji;j Sj!D E

z; (5)

where kB is the Boltzmann constant. We consider the exchangecoupling Ji,j to the third nearest neighbor. With this method,the calculated TC for unstrained monolayer CrOCl is 133 K,

which is dramatically larger than the values of 2D CrI3 (45 K)and Cr2Ge2Te6 (20 K),6,7 and just a little smaller than that(160 K) calculated by Monte Carlo simulations for monolayerCrOCl,26 where only the NN exchange coupling is considered.However, as the same neighboring exchange interactions areconsidered, it is found the obtained TC is higher than the valuespredicted using Ising model, anisotropic Heisenberg model byMetropolis Monte Carlo simulations.12

To understand the effect of strain on the magnetic transition,we plotted TC of the strained monolayer CrOCl in Fig. 7, wherethe uniaxial strains ranging from �16% to 16% along a- andb-axis are considered, respectively. It’s found that the TC can beeffectively enhanced from 133 K to 190 K by applying thecompressive strain along a-axis or the tensile strain alongb-axis. The similar strain effects have also been reported experi-mentally in many 2D systems, such as CrX3 and CrSiTe3,42,50,51

and the peak value (190 K) is comparable with the previousrecord in the diluted magnetic semiconductors GaMnAs.52

Moreover, the variation trend of TC under small uniaxial strainse (from �5% to 5%) are perfectly consistent with that of MonteCarlo simulations.26 It’s worth noting that TC is not alwaysincreased under the compressive strain along a-axis direction,and when e is about�8%, there is a turning point of rapid declinein the TC curve, which also indicates that strain engineering is aeffective means of improving TC of FM monolayer CrOCl to acertain extent.

At the same time, it is not difficult to see that the variation ofCurie temperatures TC tuned by uniaxial strains at FM state arehighly consistent with those of exchange coupling in corres-ponding direction of monolayer CrOCl by comparing the datain Fig. 6 and 7, which confirmed directly that the magnitude ofTC is determined by the corresponding exchange couplinginteraction ( J3 in a-axis direction, and J2 in b-axis direction).

IV. Conclusion

In summary, by using first principles calculations, we find thatthe magnetic transition FM-AFM in the monolayer CrOCl underuniaxial strain can be manipulated by the strain along either

Fig. 7 The TC(N) in monolayer CrOCl film under uniaxial strains alongdifferent directions.

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a- or b-axis direction, and the transition temperature TC can beeffectively increased, which indicate it has potential applicationprospect in the field of 2D spintronics. To clarify the AFM-FMtransition and transition temperature, and thus the magneticexchange coupling, we consider the possible competition betweenthe significant direct exchange and the indirect superexchangecoupling in monolayer CrOCl according to the GKA and BSI rules,which depends on the Cr–O/Cl–Cr bond angle, the spatial distanceR of NN Cr3+ ions, as well as the number of superexchange paths.These conclusions will enable us to better explore and apply 2Dmagnetic materials.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the National Natural ScienceFoundation of China (Grants No. 11504093 and No. 11604164),the Natural Science Foundation of the Jiangsu Province of China(Grant No. BK2012655).

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