Distinctive magnetic properties of CrI3 and CrBr3 monolayers caused by spin-orbit coupling
DDistinctive magnetic properties of CrI and CrBr monolayerscaused by spin-orbit coupling C. Bacaksiz,
1, 2, 3, 4
D. ˇSabani,
1, 2
R. M. Menezes,
1, 2, 5 and M. V. Miloˇsevi´c
1, 2, ∗ Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium NANOlab Center of Excellence, University of Antwerp, Belgium Bremen Center for Computational Material Science (BCCMS), Bremen D-28359, Germany Computational Science Research Center, Beijing and ComputationalScience and Applied Research Institute Shenzhen, Shenzhen, China Departamento de F´ısica, Universidade Federal de Pernambuco,Cidade Universit´aria, 50670-901, Recife-PE, Brazil (Dated: March 1, 2021)After the discovery of magnetism in monolayer CrI , the magnetic properties of different 2D ma-terials from the chromium-trihalide family are intuitively assumed to be similar, yielding magneticanisotropy from the spin-orbit coupling on halide ligands. Here we reveal significant differencesbetween the CrI and CrBr magnetic monolayers in their magnetic anisotropy, resulting Curietemperature, hysteresis in external magnetic field, and evolution of magnetism with strain, all pre-dominantly attributed to distinctly different interplay of atomic contributions to spin-orbit couplingin two materials. PACS numbers: Valid PACS appear here
I. INTRODUCTION
In the family of two-dimensional (2D) materials, ex-hibiting a range of exciting and advanced properties, theintrinsic magnetism was long evasive. The first pathwayto realize the magnetic 2D crystal has been exfoliationfrom layered bulk magnets. Nearly a decade after real-ization of graphene, exfoliated monolayer FePSe andCrSiTe were reported to have long-range magnetic or-der, indirectly demonstrated via Raman and conductiv-ity measurements, respectively. However, the field of 2Dmagnetism has truly boomed only after the premiere di-rect evidence of two-dimensional (2D) ferromagnetism, inmonolayer CrI and bilayer Cr Ge Te , that attractedmuch attention of the scientific community. Since then,a number of new 2D magnets were synthesized and uti-lized in different heterostructures, including monolayerCrBr and CrCl , other members of the Cr-trihalidefamily next to CrI .Although the number of 2D magnets exfoliated eitherfrom magnetic or nonmagnetic bulk counterparts isincreasingly large, the physics behind the magnetism inthese materials is common. Namely, the magnetic mo-ment originates from unpaired d -electron of the transi-tion metal atoms, but the stability of magnetization atfinite temperature comes as the consequence of magneticanisotropy, lifting the restrictions stipulated by Mermin-Wagner theorem. There are two possible sources ofthe magnetic anisotropy in 2D materials: one is theanisotropy of the magnetic ion due to the character andsymmetry of bond coordination with the non-magneticatoms, known as single-ion anisotropy (SIA); the other isthe anisotropy in magnetic exchange interaction betweenthe magnetic atoms within the crystal. In each case, thespin-orbit coupling (SOC) is a direct responsible for thearising magnetic anisotropy. Before the experimental realization of monolayer CrI ,the chromium-halides were predicted to be ferromagneticin the monolayer form, where by including SOC inthe consideration, magnetic anisotropy energies (MAEs)were calculated. After the actual exfoliation of CrI , amore detailed study on the chromium-halides reportedthe tunability of MAE by strain. Other studies dis-cussed accurate calculation of the critical temperature ofCrI and other 2D magnetic materials using spin-wavetheory and Monte-Carlo (MC) simulations on top of abinitio results, and were mostly focused on the properdescription of the magnetic interactions. Very recentworks then considered adsorption and substitution of for-eign atoms, such as hydrogen or oxygen, to manip-ulate the exchange interaction in order to increase thecritical temperature.It is now well established that the origin of the mag-netic anisotropy in monolayer CrI , for both SIA andanisotropy of the exchange interactions, is associatedwith the SOC of iodine atoms rather than chromiumones. As a consequence one can intuitively predictthat the magnetic anisotropy of monolayer CrBr shouldbe lower as compared to that in CrI due to differencein SOC between I and Br. As a corroboration to thesepredictions, recent experiment established the Curie tem-perature of CrBr as 21 K, significantly lower than oneof monolayer CrI ( T C = 45 K). However, the extentand manner of how individual atomic contributions toSOC affect the magnetic anisotropy in different mono-layer magnets remained unaddressed to date. Therefore,in this article we perform a thorough comparison betweenseemingly similar monolayer magnets, CrI and CrBr , inorder to provide a comprehensive (qualitative and quan-titative) understanding of the atomically-resolved effectsof spin-orbit coupling on magnetism. Specifically, we ex-plore the magnetic anisotropy, variation of magnetism a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b FIG. 1: Schematic representation of the structure of monolayer CrI (a) and CrBr (d). Panels (b) and (e) show the differencebetween the charge distribution after crystallization and the total charge distribution of bare atoms, which then indicatesthe bonding and anti-bonding charges in the two materials. (c) and (f) panels show the density of states of two materials,decomposed according to the atomic orbitals. Subscripts x (cid:48) , y (cid:48) and z (cid:48) in the orbitals denote the local coordinates of thecorresponding atoms. under biaxial strain, critical temperature and responseto the magnetic field, and link the differences in SOC intwo similar materials to their rather dissimilar emergentmagnetic properties.The paper is organized as follows. In Sec. II we providedetails of our computational methodology. The struc-tural and electronic properties, the magnetic properties,and the effect of strain are presented and discussed inSec. III A, III B, and III C, respectively. In Sec. III D,the temperature-dependent magnetism and magnetiza-tion reversal of the two materials in external magneticfield are discussed. Sec. IV summarizes our findings. II. COMPUTATIONAL METHODOLOGY
In order to investigate the structural, electronicand magnetic properties we use calculations based ondensity functional theory (DFT). To obtain magneticparameters, we employed four-state energy mappingmethodology.
When different magnetic configura-tions are examined, the magnetic moments are con-strained in desired directions, in order to prevent theirrelaxation into the ground state configuration (or anystable configuration other than desired one) during theself-consistent procedure. Heisenberg spin Hamiltonianis considered in the form: H = 12 (cid:88) i,j S i J ij S j + (cid:88) i S i A ii S i , (1)where S i = ( S xi , S yi , S zi ) is a vector. J ij and A ii are 3 × ab initio simulation packageVASP which solves the Kohn-Sham equations itera-tively using a plane-wave basis set. To describe electron exchange and correlation, the Perdew-Burke-Ernzerhof(PBE) form of the generalized gradient approximation(GGA) was adopted. The spin-orbit coupling (SOC)was included in all calculations, as VASP includes SOCvia following added term to DFT Hamiltonian: H SO = ¯ h (2 m e c ) (cid:18) − V ( r )2 m e c (cid:19) − dV ( r ) dr (cid:126)σ · (cid:126)L, (2)where (cid:126)σ = ( σ x , σ y , σ z ) stands for 2 × (cid:126)L = (cid:126)r × (cid:126)p is the angular momentum, and V ( r )is the spherical part of the electron potential. The vander Waals (vdW) forces were taken into account usingthe DFT-D2 method of Grimme. In order to calcu-late charge transfer between the atoms we employed theBader charge technique. The kinetic energy cut-off of the plane-wave basis setwas 600 eV and energy convergence criterion was 10 − eV in the ground-state calculations. Gaussian smearingof 0.01 eV was used and the pressures on the unit cellwere decreased to a value lower than 1.0 kbar in all threedirections. On-site Coulomb repulsion parameter, U ,was taken as 4 eV for magnetic Cr atom. To avoid in-teractions between periodically repeating monolayers invertical direction, our calculations were performed withsufficiently large vacuum space of ∼
13 ˚A.For subsequent considerations of the temperature-dependent magnetization and the critical temperature( T C ), we performed spin dynamics simulations basedon stochastic Landau-Lifshitz-Gilbert (LLG) equation,using simulation package Spirit , adapted to accom-modate the anisotropic interactions of our Hamiltonian[Eq. (1)]. 50 ×
50 supercell of the spin lattice was con-sidered. The spin system is initialized in random con-figuration at high temperature and then cooled downwith the temperature steps of 0.125 K. To obtain equilib-rium magnetization took relaxation over 10 time steps(with time step ∆ t = 1 fs) for each temperature. Forthe field-dependent calculations, the Zeeman term H B = gµ B (cid:80) i S i · B has been included in Eq. (1), where B is theapplied magnetic field; g = 2 is the g-factor for S = 3 / µ B the Bohr magneton and µ = gSµ B = 3 µ B is themagnetic moment of Cr atoms. TABLE I: Structural and electronic parameters of monolayerCrI and CrBr . The charge transfer per atom ( ρ Cr and ρ X )was calculated using Bader charge technique. d b a θ ρ Cr /ρ X E g (˚A) (˚A) ( ◦ ) ( e − / e − ) (eV)CrI − .
37 0.55CrBr − .
44 1.59
III. RESULTS AND DISCUSSIONA. Structural and electronic properties
We start from the structural properties of monolayerCrX , where X stands for ligand I and Br atoms. CrX crystallizes in the trigonal P m space group. Thehexagonal planar lattice of Cr atoms is sandwiched be-tween triangular planar lattices of ligand atoms as shownin Figs. 1 (a) and (d) for respective materials. OneCr atom bonds to 6 ligand atoms and each ligand atombonds to 2 Cr atoms. The structures have 3-fold in-planesymmetry, where the triangular lattices of ligand atomsare 180 ◦ rotated with respect to each other, which cre-ates octahedral coordination around each Cr atom. Aslisted in Table I, the Cr-I and Cr-Br bond lengths arefound to be 2.78 ˚A and 2.55 ˚A, respectively, which is di-rectly proportional with the radius of the bonding orbitalof the ligand, I-5 p and Br-4 p . Consequently, the latticeconstants of monolayer CrI and CrBr are also different,6.94 ˚A and 6.40 ˚A, respectively, consistent with the ex-perimentally measured 6.95 ˚A and 6.50 ˚A. Note thatboth monolayers exhibit slightly larger lattice constantthan their bulk or few-layer counterparts. The covalentbonding character consists of 1.10 e − and 1.33 e − dona-tion of Cr and 0.37 e − and 0.44 e − gain of I and Br atoms,respectively. This bonding charge is visualized in Figs. 1(b) and (e), as obtained by subtracting bare atom chargedistributions from the charge distribution of the crystals.Charges obviously accumulate to the bonding sites andaround Cr atoms. Observed charge accumulation aroundeach ligand clearly shows three ’charge clouds’ that canbe divided in two groups, representing two types of bond-ing between Cr and ligand atoms: the facial ( σ ) bonding,represented by two charge clouds facing the Cr atoms,and the lateral ( π ) bonding, represented by the thirdcharge cloud, in direction orthogonal to the Cr-X-Cr-Xplane.To depict the energetic arrangements of the orbitalstates, we calculated the partial density of states for two considered materials in Figs. 1 (c) and (f). Both materi-als are semiconductors, in which the valence band max-imum is dominated by p -orbital of the ligand. On theother hand, d -orbital of Cr resides much deeper in thevalence band, with similar energetic delocalization forboth CrI and CrBr . d x (cid:48) y (cid:48) , d x (cid:48) z (cid:48) and d y (cid:48) z (cid:48) are degen-erate, and are plotted together. d z (cid:48) and d x (cid:48) − y (cid:48) orbitalsare also degenerate and appear at the conduction band. p x (cid:48) and p y (cid:48) orbitals are degenerate as well, and domi-nate the valence band maximum. p z (cid:48) orbital mostly re-sides in the middle of the valence band. One should notethat in order to obtain orbital states compatible withthe octahedral coordination, the general coordinates arerotated as suggested by Rassekh et al. . Therefore weconsider xy -plane and z -axis of the general Cartesian co-ordinates aligned with the local coordinates, where theCr-X-Cr-X plane is considered as x (cid:48) y (cid:48) -plane, and z (cid:48) axisis orthogonal to that plane. Orbital decomposition showsthat both materials exhibit very similar orbital delocal-izations. This is also visible in the charge density varia-tions shown in Figs. 1 (b) and (e). Purple regions showthe depletion of charges of isolated atoms through thecharge density of the crystal. It is obvious that the or-bitals laying along the Cr-X bonds exhibit most delocal-ization. We understand from the charge depletion andthe orbital decomposition of DOS that d x (cid:48) y (cid:48) , d x (cid:48) z (cid:48) and d y (cid:48) z (cid:48) orbitals are localized and do not show any variationfrom their single-atom form. d z (cid:48) and d x (cid:48) − y (cid:48) orbitalsform a spd hybridization with the p x (cid:48) and p y (cid:48) orbitalsas expected for an octahedral coordination. Beside thesesimilarities, the particular difference between CrI andCrBr is found between 4 p of Br and 5 p of I, leading toenergetic differences of the spd hybridization. Since 4 p orbital of Br is more confined as compared to 5 p of I, thestates of CrBr are shifted to higher energies. Therefore,the band gap values are different, found as 0.55 eV and1.59 eV for CrI and CrBr , respectively. B. Magnetic properties
1. Spin-orbit coupling and magnetic anisotropy energy
It is already known that monolayer CrI exhibits fer-romagnetism at finite temperature due to the magneticanisotropy originating from SOC on I atoms. It is naturalto assume that a similar mechanism is responsible for fer-romagnetism in CrBr , as proposed in previous studies. In what follows we validate that assumption, but moreimportantly, we analyze deeper the differences in individ-ual atomic contribution to the total SOC in monolayerCrI and CrBr , and the consequences thereof.First of all, we calculate the total energy leading tomagnetic anisotropy energy (MAE) and the contributedSOC energy (∆ E SO ), relative to the respective ground-state energies, depending on the spin direction angle( θ ) with respect to the out-of plane direction. In otherwords, MAE( θ ) = E ( θ ) − E ( θ = 0 ◦ ) and ∆ E SO ( θ ) = FIG. 2: (a) Magnetic anisotropy energy (MAE) and SOCenergy as a function of the spin alignment angle with theout-of-plane direction. (b) Schematic representation of thetop view of the structure. Atoms are enumerated in orderto track the variation of the corresponding SOC energies forspin-alignment in x -direction ( θ = 90 ◦ and φ = 0 ◦ ) (c), andin y -direction ( θ = 90 ◦ and φ = 90 ◦ ) (d). The energy ofthe out-of-plane spin alignment, the ground state, is set to0 eV. Dashed lines represent total ∆ E SO , while square dots(connected by solid lines) represent contributions of each atomto the total ∆ E SO . E SO ( θ ) − E SO ( θ = 0 ◦ ), where angle θ is measured from z -axis (see Fig. 2(b)). MAE is found to be 0.67 (0.11)meV for for CrI (CrBr ) favoring the out-of-plane di-rection. In the literature, the reported MAE values varyfrom ∼ . .
18) meV to ∼ . .
30) meV for CrI (CrBr ) depending on the approximations used. However, MAE of CrI is 4-5 times larger than that ofCrBr in each previous report, as is in the present study.As shown in Fig. 2(a), MAE( θ ) and ∆ E SO ( θ ) exhibitsinusoidal functional behavior. It is further remarkablethat MAE( θ ) is not equal to ∆ E SO ( θ ), which indicatesthat the collective rotation of the spins allows further re-laxation of the spatial part of the wave function, to alower energy. The ratio between MAE( θ ) and ∆ E SO ( θ )is around 0.5 for either monolayer, at each angle θ .Furthermore, we calculate the atomic contributionsto the total SOC energies for the spins pointing in x -( θ = 90 ◦ and φ = 0 ◦ ) and y -direction ( θ = 90 ◦ and φ = 90 ◦ ), as shown in Figs. 2 (c) and (d). The totalcontribution from Cr atoms is significantly lower thanthat from ligand atoms, which confirms results of previ-ous works. More importantly, the atomic contribu-tions of both I and Br vary depending on the directionof the bond coordination of the ligand with respect tothe spin direction. To be more direct, in Fig. 2(c) theligands X , X , X , and X energetically prefer the spinpointing in the z -direction, while X and X prefer thespin in x -direction. On the other hand, in Fig. 2(d), the contributions from X , X , X , and X are almostzero, however, X and X prefer the spins aligned with z -direction. Briefly, the contribution (cid:126)σ · (cid:126)L depends on thedirection of the orbital angular momentum of the ligand L x (cid:48) , L y (cid:48) , which lay on the respective Cr-X bonding direc-tions as p x (cid:48) and p y (cid:48) . There is interplay with the d z (cid:48) and d x (cid:48) − y (cid:48) orbitals through the spd hybridization. On theother hand, L z (cid:48) is orthogonal to Cr-X-Cr-X plane. Theseresults not only confirm the previous works suggestingthe main contribution of MAE originates from SOC ofthe ligand atoms, but also reveal that the main contri-butions are stemming from the bonding orbitals. Previ-ously, Lado et al. reported MAE of monolayer CrI as afunction of SOC strength on Cr and I (ligand) separatelyand concluded that the SOC on ligand is the dominantfactor on the MAE. Here we further reveal that notevery ligand has positive contribution to MAE - thosewith a bond with Cr atom aligned with magnetizationdirection of interest, will have negative (or no) contribu-tion. The apparent difference between CrI and CrBr ,on the other hand, is a direct consequence of the relativestrength of the SOC of 5 p orbital of the I atom comparedto the 4 p orbital of the Br atom.
2. Magnetic exchange interaction and SIA
To characterize the magnetic properties of the twomonolayers under study, we calculate the magnetic ex-change interactions matrix ( J ) in the first nearest-neighbor (NN) approximation, using a 2 × , once the exchange matrix of one of the threefirst NN pairs is obtained, such as for the pair (1-2), thematrices of other first NN pairs, (2-3) and (2-5), can becalculated by rotation operation on the exchange matrixof pair (1-2) around the out-of-plane axis. The so ob-tained pairwise results are listed in Table II. Dependingon the considered coordinate axes, the exchange matrix ofa pair can change. Here (1-2) pair lays on the x -axis, con-sequently (2-3) and (2-5) pairs make 60 ◦ and − ◦ anglewith the x -axis. Each pair of CrI and CrBr has a sym-metric exchange matrix, due to preserved inversion sym-metry, hence no Dzyaloshinskii-Moriya interaction (DMI)is present. All diagonal elements show that both mono-layer materials exhibit ferromagnetic interaction in thefirst NN consideration. As an effective exchange interac-tion for Cr, the mean of the matrices of (1-2), (2-3), and(2-5) pairs is listed in Table II as (cid:104) J (cid:105) , and is identicalfor each magnetic site in the respective monolayer. Inthis form of representation, it is clearly seen that CrI exhibits stronger ferromagnetic exchange as compared toCrBr . It is also rather remarkable that the out-of-planeexchange anisotropy, ∆ = (cid:104) J (cid:105) xx − (cid:104) J (cid:105) zz = (cid:104) J (cid:105) yy − (cid:104) J (cid:105) zz of CrI is 0.22 meV, much larger than 0.04 meV of CrBr .Finally, we also calculated SIA, which as a result of three-fold symmetry is represented by a single parameter, A zzii ,out of nine elements of the A ii matrix. A zzii of CrI is TABLE II: Magnetic exchange parameters of monolayer CrI and CrBr . J xx , J yy , and J zz are diagonal elements, and J xy = J yx , J xz = J zx , J yz = J zy are off-diagonal elements of the exchange matrix. The mean value (cid:104) J (cid:105) of the exchangeparameters is also given. Out-of-plane anisotropy ∆ is calculated as (cid:104) J xx (cid:105) − (cid:104) J zz (cid:105) . A ii is SIA parameter, same for each Crsite. MAE and E SO are magnetic anisotropy and total SOC energies, respectively.pair J xx J yy J zz J xy = J yx J xz = J zx J yz = J zy ∆ A zzii MAE ∆ E SO ( i - j ) (meV) (meV) (meV) (meV) (meV) (meV) (meV) (meV) (meV/f.u.) (meV/f.u.)CrI -0.22 -0.07 0.67 1.21(1-2) -5.10 -3.72 -4.63 0.00 0.00 0.83(2-3) -4.07 -4.76 -4.63 -0.60 0.72 -0.42(2-5) -4.07 -4.76 -4.63 0.60 -0.72 -0.42 (cid:104) J (cid:105) -4.41 -4.41 -4.63 0.00 0.00 0.00CrBr -0.04 -0.01 0.11 0.22(1-2) -3.45 -3.29 -3.42 0.00 0.00 0.09(2-3) -3.33 -3.41 -3.42 -0.07 0.08 -0.05(2-5) -3.33 -3.41 -3.42 0.07 -0.08 -0.05 (cid:104) J (cid:105) -3.37 -3.37 -3.42 0.00 0.00 0.00 found to be − .
07 meV, much larger than that of CrBr , − .
01 meV. Both monolayers exhibit negative SIA, whichindicates the energetic preference of the spin alignmentin the out-of-plane direction. It is worth to mention thatXu et al. previously calculated 3 × and therefore reported different values, however,our results are qualitatively consistent with their ones.Namely, they found the ferromagnetic diagonal elementswith anisotropy favoring the out-of-plane direction, sym-metric off-diagonal elements, and SIA parameter of Cratoms favoring out-of-plane as well. They also analysedthe exchange and SIA parameters as a function of SOCstrength and concluded the main contribution to thoseparameters comes from the SOC in I atoms as Lado etal. did.In order to reveal the effect of SOC from another per-spective, we next calculated the exchange matrix and SIAparameter in the non-collinear scheme without SOC. Asexpected, the exchange matrix for both monolayers is di-agonal, with equal diagonal elements. SIA parameter iszero, which is also expected. The results confirm thatthe origin of the magnetic anisotropy, consequently theorigin of the magnetization of these monolayer materi-als, is the spin-orbit interaction. However the compari-son between exchange parameters with and without SOCfurther indicates the difference between CrI and CrBr .The exchange parameter of CrI without SOC is found tobe − .
53 meV which is between (cid:104) J (cid:105) xx = (cid:104) J (cid:105) yy = − . (cid:104) J (cid:105) zz = − .
63 meV when SOC is included.For CrBr , on the other hand, it is found to be − . (cid:104) J (cid:105) xx = (cid:104) J (cid:105) yy = − .
37 meV. Therefore, one understandsthat the SOC in CrI not only enhances the exchange in-teraction of the out-of-plane spin components but alsoreduces the interaction of the in-plane components. ForCrBr , the SOC contributes to the interaction of out-of-plane spins only. C. Effect of strain
The responses of monolayer CrI and CrBr to biaxialstrain are further revealing the influence of SOC on themagnetic exchange interaction. Principally, the struc-tural changes upon straining are similar for CrI andCrBr . Cr-Cr distance changes according to the degreeof strain applied ( ± ∼ ± ∼ ± spd bonds of one ligand due to bong-angle modification andleads to modification of the SOC contribution in both Crand the ligand.We next calculated the total energy difference and thetotal SOC energy difference between out-of plane and in-plane spin alignments under biaxial strain. As shownin Figs. 4 (b) and (c), E SO of monolayer CrI [orangecurve in Fig. 4(b)] linearly increases under increasingtensile strain, which is similar to its behavior in CrBr [green curve in Fig. 4(c)]. However, the behavior ofthe two materials under compressive strain is completely FIG. 3: 2 × FIG. 4: (a) Schematic illustration of the distortion caused on the octahedral unit of the material under tensile and compressivestrain. Panels (b) and (c) show total SOC energy and MAE as a function of biaxial strain for CrI and CrBr , respectively. Theatomic contributions are also plotted. (d) and (e) show the charge density variation (compared to bare atoms) in the strainedcrystal of CrI and CrBr , respectively. One notes the overlap of the bonding charges under compressive strain, zoomed outfor facilitated visualization, and absent for the tensile strain. different. CrI exhibits parabolic increase with increas-ing compressive strain, while CrBr maintains linear be-havior and decreases with increasing compressive strain!The atomic contributions reveal the source of these be-haviors. As shown by magenta curve in Fig. 4(b), io-dine contribution dominates the behavior for both com-pressive and tensile strain. For CrBr in Fig. 4(c), incase of the compressive strain, the contributions from Crand Br are almost equal but have opposite sign, meaningthat Br atoms favor to have spin in out-of-plane directionwhile Cr atoms prefer in-plane spin direction. For tensilestrain, Br contribution dominates and both Cr and Brprefer out-of-plane spin alignment. Since the structuralchanges of both monolayers under strain are similar, oneexpects similar variation of Cr energies, however, Cr ofCrI exhibits three times larger energy variation as com-pared to Cr of CrBr . This indicates that the anisotropyrelated with Cr also originates from the bonding electronsrather than Cr-only electrons. In Figs. 4 (d) and (e) weshow the charge density variation for 5% and −
5% strainin both CrI and CrBr . It is clearly seen that in caseof compressive −
5% strain the bonding charges of twodifferent bonds of one ligand overlap. This reveals theorigin of the interaction under compressive strain. For5% tensile strain, the bonding charges are clearly sepa-rated and exhibit no overlap.Our results for behavior of MAE are generally in goodagreement with results reported by Ref.17. In case ofCrBr we also obtain that MAE is smallest in case of −
5% strain and it is linearly growing, reaching the max-imum for +5% strain. In case of CrI , however, our re-sults agree with Ref.17 only for the compressive strain -with increased compressive strain, anisotropy is growing.On the other hand, with tensile strain, we report com-pletely different behavior of MAE in CrI , compared toRef.17. There, anisotropy was decreasing with increasedtensile strain, while in our study, the behavior is different- anisotropy grows with increased tensile strain.The variations of exchange parameters and SIA un-der biaxial strain are also examined based on the con-siderations related to Table II and Fig. 3. The meanvalues (cid:104) J (cid:105) of the exchange matrices of three NNs areplotted as a function of strain in Fig. 5(a). The fer-romagnetic exchange interaction increases with the ten-sile strain and decreases with compressive strain for bothCrI and CrBr , for all components. The variation on thecurves of CrI is much larger such that the compressivestrain almost equalizes CrI and CrBr in terms of ex-change interaction, and the difference between two mono-layers increases with tensile strain. These results agreewith Ref.17 only in case of compressive strain. Namely,both our study and mentioned previous work suggest thatwith compressive strain, two materials become less FM.However in case of tensile strain, our results suggest thatmaterials are more FM than in pristine case, while inRef.17, the opposite was suggested. In Fig. 5(b), theanisotropy between in-plane and out-of-plane exchangeparameters, ∆, is plotted as a function of strain. ∆ of FIG. 5: The magnetic exchange parameters (a), out-of-planeanisotropy (b), and SIA parameters (c) of monolayer CrI andCrBr as a function of the biaxial strain. CrI increases for increasing either tensile or compres-sive strain. It is important to note that the behaviorof ∆ is consistent with the behavior of MAE in Fig. 4,since most of the contribution to MAE comes from theanisotropy of the exchange interaction of the first NN.For CrBr , the anisotropy slightly increases (decreases)under tensile (compressive) strain, which is also consis-tent with the behavior of MAE. In Fig. 5(c), SIA ispresented as a function of strain. SIA in CrI exhibitsa gradually slowing decrease (in absolute value) whenmoving from compressive to tensile strain. In CrBr onthe other hand, SIA exhibits opposite behavior to thatof CrI . It is significant that for compressive strain be-yond −
3% SIA parameter becomes positive, indicatingpreference for in-plane direction of magnetization. No-tice that SIA is an order of magnitude smaller than theexchange parameter, therefore its contribution to overallmagnetic properties is limited. However, MAE of CrBr under high compressive strain is of the same order as SIA,indicating that the drop of MAE is due to the decreaseof SIA parameter, while ∆ stays almost constant. D. Temperature-dependent magnetization andhysteresis in applied magnetic field
The above-indicated differences in the strength of themagnetic exchange parameters, exchange anisotropy, andSIA parameter between two magnetic monolayers un-der investigation can be monitored via temperature-dependent magnetization and hysteretic behavior in ap-plied magnetic field, which are both readily experimen-tally accessible. Having obtained all the parameters toconstruct the Heisenberg spin Hamiltonian in Eq. (1) forstrained monolayers, we next calculate the temperature-dependent magnetization of CrI and CrBr and the cor-responding T C where the magnetic phase transits fromparamagnetic to ferromagnetic state. We used stochas-tic LLG simulation to obtain the temperature-dependentmagnetization M z /M s ( T ), where M z is the out-of-planemagnetization and M s is the saturation magnetization.As shown in Figs. 6 (a) and (b), the obtained T C ofthe unstrained monolayers of CrI and CrBr of 56 Kand 38 K is reasonably close to the experimentally ob-tained values of 45 K and 21 K, respectively. These T C values are also consistent with those found in previ-ous works, obtained using renormalization spin-wavetheory combined with classical MC calculations . Fur-ther, as shown in Fig. 6(c), T C decreases (increases) un-der compressive (tensile) strain, mainly due to the previ-ously described strong variation of the exchange param-eters with strain. In case of CrBr , contrary to CrI , theinfluence of ∆ and SIA is very small since the variationof those parameters under strain can be considered neg-ligible. One should note that the behavior of T C as afunction of strain presented here is completely differentthan that reported in Ref.17 where T C is calculated usingmean-field theory. Such disagreement is hardly a surprisesince in Ref.17 a single exchange parameter is used to de-termine T C in absence of information on anisotropy. Fur-ther we note that two monolayer materials have almostequal T C at compressive −
5% strain. In the case of ten-sile strain on the other hand, the difference between T C ofCrI and CrBr increases with strain. CrBr reaches thesaturation of T C ≈
45 K at around 3% strain while CrI exhibits T C ≈
74 K at 5% strain and tends to furtherincreased T C with further straining.The behavior of M z /M s ( T ) curves is also illustrativeof the difference between the two materials. In gen-eral, all curves are well fitted by the functional behav-ior (1 − T /T C ) β , where β is the critical exponent. InFig. 6(d), obtained exponents β are plotted as a functionof strain, together with β values from different availablemodels for comparison. In our results, the critical expo-nent of monolayer CrI at all strains can be approximatedby β ≈ .
24. This value suggests that strong out-of-planeanisotropy of CrI separates its behavior from those ex-pected by 3D models and sets it closer to the 2D limit.Our value of β is also comparable with the value of ∼ . since the intralayer in-teractions dominate the magnetic behavior even in bulk FIG. 6: The temperature-dependent magnetizationM z /M s ( T ) for different amounts of strain applied to mono-layer CrI (a) and CrBr (b). Simulation data are fitted by(1 − T /T C ) β , where β is the critical exponent and T C the Curietemperature. (c) and (d) panels plot the thereby obtained T C and β as a function of strain, respectively. For comparison, β values from different models are shown as dashed lines in(d). CrI . CrBr with β ≈ .
3, on the other hand, is muchcloser to the 3D Ising value ( β = 0 . β = 0 . ± . et al. . Finally, motivated by its accessibility by advancedmagnetometry, we calculated the hysteretic behavior ofthe monolayers under magnetic field B tilted by angle θ B from the z -axis, for both CrI and CrBr .In the simulations, the spin system is initialized at highapplied field, where all the spins are aligned to the mag-netic field direction. The field is then looped in stepsof 0 .
01 T and 0 .
05 T for CrBr and CrI respectively,where the magnetization is relaxed by 5 × time stepsfor each value of field. Dipole-dipole interactions havebeen included in the simulations. In Fig. 7(a,b), weshow the obtained ∆M z = | M z ( → ) − M z ( ← ) | as a color-map plot where shades of red (CrI ) and blue (CrBr )color show the deviation of M z ( B ) hysteresis loop fromthe rectangular shape. In Figs. 7 (c) and (d), we plotthe corresponding hysteresis curves for selected tilt an-gles of the applied magnetic field. One clearly sees thathysteresis loops evolve from sharp rectangular to an ovalform as the tilt angle is increased, as was recently shownexperimentally by Kim et al. for the case of monolayerCrBr .In absence of dipolar interactions, the behavior of mag-netic spins in external field is captured by minimization of the energy E T OT ( θ ) = − MAE cos(2 θ ) − E B cos( θ − θ B ),where E B is the energy associated with the magneticfield, and grows linearly with increasing B . The value ofthe critical, ’switching’ field for the given angle θ B , scaledto the switching field for θ B = 0 ◦ [Γ = B cr ( θ B ) /B cr (0 ◦ )]is then analytically obtained from the condition that theenergy extrema coalesce, leading to equation(Γ − Γ = −
274 sin (2 θ B ) . (3)This functional dependence of critical field on the tiltangle θ B perfectly reproduces the numerically calculatedswitching field, shown by dashed lines in Figs. 7(a,b).Notably, the switching field in absence of dipolar interac-tions (Γ( θ B )) shows symmetric behavior with respect to θ B = 45 ◦ . However, with dipolar interactions included,that symmetry is broken in case of CrBr , and switch-ing field for θ B = 0 ◦ becomes significantly lower thanfor θ B → ◦ . The latter was indeed validated experi-mentally, in Ref.8. However, dipolar interactions causeno changes in the critical field of CrI for any θ B , whichis another important distinction between two materialsthat could be verified by Hall micromagnetometry.One should however note that in our considerationswe do not involve finite size effects nor demagnetization,or temperature fluctuations, likely playing an importantrole in experiment (next to the ever-present defects inthe monolayers, that can facilitate magnetic reversal lo-cally). Our simulations explore primarily the effect of themicroscopic parameters on the apparent magnetic behav-ior, with a goal of capturing the intrinsic differences be-tween CrI and CrBr . As a consequence, the switchingfields in our simulations are significantly larger than ex-perimentally reported values of ≈ for CrI and ≈ for CrBr , for θ B = 0. Having said that, our sim-ulations capture the large ratio between switching fieldsof the two monolayer materials (approximately 8 and 5for simulation and experiment, respectively).To feature yet another experimentally verifiable differ-ence between monolayer CrI and CrBr , we also cal-culated the hysteresis curves for the strained magneticmonolayers under out-of-plane magnetic field. We recallthat MAE and SIA, plotted in Fig. 4(b,c) and Fig. 5(c),respectively, showed distinctively different behavior intwo materials when under strain. As shown in Figs. 7 (e)and (f), the width of the hysteresis loop strongly changeswith the strain. However, while the switching magneticfield of monolayer CrI increases under both compressiveand tensile strain, the switching field of monolayer CrBr is reduced by compressive strain while the tensile strainincreases the switching field stronger than was the casein CrI (measured with respect to the unstrained case).This behavior can thus be mapped on the behavior ofexchange anisotropy and SIA under strain, all rooted inthe known difference in spin-orbit coupling between thetwo materials. FIG. 7: The height of the hysteresis loop ∆M z = | M z ( → ) − M z ( ← ) | as a function of the external magnetic field B tilted byangle θ B from the out-of-plane direction, for (a) CrI and (b) CrBr . Panels (c) and (d) show the corresponding hysteresiscurves obtained for indicated angles θ B . Panels (e) and (f) highlight the difference in the hysteretic response of two materialswhen strained (here θ B = 0). IV. CONCLUSIONS
We compared two of the very first ferromagnetic 2Dmaterials, monolayer CrI and CrBr , belonging to thesame chromium-trihalide CrX family. Although verysimilar qualitatively in structural and electronic proper-ties (with some quantitative differences such as latticeconstant, bond length, and electronic band gap), thesematerials exhibit strong differences in magnetic proper-ties and their behavior with external stimuli. We at-tribute these differences to the spin-orbit interaction notonly on ligand atoms (X=I, Br) and but also at the bond-ing orbitals. We show that the energetic preference ofthe direction of spin of a ligand directly depends on thebonding direction of that particular ligand relative to thespin direction. That means spin-orbit coupling (SOC) en-ergy contribution of an individual ligand can be predictedqualitatively via its coordination and spin direction.We also present the magnetic exchange parameters forthe two monolayers. The mean exchange interaction inCrI is larger that one of CrBr , but the difference be-tween their out-of-plane anisotropy values (∆) as well asbetween single-ion anisotropies (SIA) are more than sig-nificant. We also clearly demonstrated that the origin of both the out-of-plane anisotropy and the SIA is the spin-orbit interaction, since our analogous analysis withoutSOC yielded no exchange anisotropy and no SIA.By applying biaxial strain, we revealed further differ-ences between monolayer CrI and CrBr . The strainmostly changes the Cr-X-Cr angle instead of the bondlength. That results in significant structural distortionof the octahedral units of the monolayers. Magneticanisotropy energy (MAE) of CrI increases under eithercompressive and tensile strain while MAE of CrBr lin-early increases (decreases) under increasing tensile (com-pressive) strain. This difference in the variation of MAEis reflected on the corresponding changes in the out-of-plane anisotropy of the exchange parameters and the SIAparameter. With such differences in obtained parametersfor the two materials, we calculated the temperature-dependent magnetization for pristine and the strainedmonolayers, to reveal much stronger variation with strainof T C in CrI than in CrBr . The found critical expo-nent of our M ( T ) data places CrBr virtually in the 3Dregime, owing to its low out-of-plane anisotropy, contraryto the strong 2D character of CrI .For facilitated direct observation of the reported dif-ferences between monolayer chromium-trihalides, and as0a direct probe of their magnetic anisotropy, fostered byspin-orbit coupling, we also calculated the behavior ofhysteretic magnetization loops as a function of the tiltangle between the applied field and the monolayer plane.We revealed that magnetic behavior of CrBr is far moreaffected by dipolar interactions than is the case in CrI ,but also that the behavior of the switching field withstrain is entirely different in two materials, analogouslyto previously observed differences in MAE and SIA asa function of strain. These findings are yet anotherproof that even subtle differences in atomic contributionsto spin-orbit coupling between two akin materials canlead to rather dissimilar magnetic properties, and canbe broadly tuned by gating, straining and heterostruc-turing of the 2D material. Although sourced in proper-ties at atomistic scale, these differences can clearly man-ifest in macroscopic observables and are verifiable exper- imentally (owing to e.g. recent advances in Hall mag-netometry). Therefore, tailored solutions for spatiallyengineered spin-orbit coupling in magnetic monolayerspresent an attractive roadmap towards advanced spin-tronic and magnonic nanocircuitry. Acknowledgments
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