Tunable magnetic anisotropy in Cr–trihalide Janus monolayers

Achieving a two-dimensional material with tunable magnetic anisotropy is highly desirable, especially if it is complemented with out-of-plane electric polarization, as this could provide a versatile platform for spintronic and multifunctional devices. Using first principles calculations, we demonstrate that the magnetic anisotropy of Cr–trihalides become highly sensitive to mechanical strain upon structural inversion symmetry breaking through the realization of Janus monolayers. This remarkable feature, absent in pristine Cr–trihalide monolayers, enables mechanical control of the direction of the easy axis: biaxial compressive/tensile strain supports in-plane/out-of-plane orientation of the easy axis. The magnetic exchange itself shows higher sensitivity to compressive than to tensile strain, while in general the Janus monolayers maintain ferromagnetic ordering in the studied range of strain.


Introduction
Two dimensional (2D) materials can host inspiring features [1,2], from spin-orbit physics leading to ultra-thin magnetic memories [3] to more exotic quantum spin liquid states [4] that could nurture futuristic topological quantum computing applications [5]. Relative ease of manipulating the mechanical, electronic, magnetic, and optical properties increases the possibilities to design 2D materials for a new generation of nanodevices [6][7][8]. In this context, it is of great interest to theoretically explore the potential of 2D ferromagnetic (FM) materials. Among the wide range of material candidates for 2D magnetism, transition metal trihalides stand out as a promising platform: following the prediction of FM ordering in monolayers of CrX 3 (X = Cl, Br, and I) [9], an experimental breakthrough confirmed FM ordering in a monolayer of CrI 3 with Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. a Curie temperature of 45 K [10] and boosted the interest in computational studies for this class of materials [11][12][13][14][15][16].
Monolayers of CrX 3 are non-metallic and based on biocompatible trivalent chromium (Cr 3+ ) [11]. It is attractive to achieve out-of-plane electric polarization in addition to the intrinsic magnetism, because this will open a new horizon in the field of multifunctional devices [17]. One way to reach this goal is by breaking the inversion symmetry of the monolayers, in which the Cr atoms are arranged in a honeycomb lattice sandwiched between planes of halide atoms forming a slightly distorted octahedral environment. When the two planes consist of different kinds of halide atoms the inversion symmetry is broken and out-of-plane electric polarization is obtained, leading to spin-momentum locking and thereby to an electricallydriven inverse spin galvanic effect, a photogalvanic effect, and Dzyaloshinski-Moriya interaction. This X 3 -Cr 2 -Y 3 (X, Y = Cl, Br, and I) structure is known as Janus monolayer. The Janus monolayer transition metal dichalogenide MoSSe was recently demonstrated experimentally [18]. Janus Cr-trihalide monolayers are interesting for field-effect spin-filter applications [19] and possess a topological spin texture [20].

Effect of the onsite Coulomb interaction
Bulk CrI 3 is an FM insulator with a Curie temperature of 61 K and has a layered rhombohedral structure, space group R3, below 210-220 K. At higher temperature it has a monoclinic structure, space group C2/m, and is a paramagnetic insulator. A non-magnetic calculation using the generalized gradient approximation (GGA) yields a metallic behaviour, contradicting the experimental result mentioned above. This indicates that the material is a Mott insulator, because the bandgap is not due to magnetic order but rather to electron-electron interaction [32]. Moreover, the FM ordering of bulk CrBr 3 and CrI 3 was reproduced by the GGA while the antiferromagnetic (AFM) ordering of bulk CrCl 3 can only be attained by taking into account an onsite Coulomb interaction within the around mean-field approach [33]. It is known that the local density approximation and GGA fail to capture Mott insulator properties, because they over-delocalize valence electrons [34]. Mott systems have strongly localized electrons and an onsite Coulomb interaction U thus can introduce a bandgap between the occupied and empty d states [23]. If U affects the structure relaxation, it causes significant changes in the hybridization between the d orbitals and their bonding partners, which directly influences the MA. As a matter of fact, this scenario applies to our Janus monolayers, as important features of the density of states (DOS) are affected by U, see the case of Cl 3 -Cr 2 -I 3 in figures 1(a) and (b): for the GGA the bandgap is between the occupied Cr t 2g and empty Cr e g states, whereas in the GGA + U DOS it is between the occupied Br/I p states for Cl 3 -Cr 2 -Br 3 /(Cl 3 -Cr 2 -I 3 , Br 3 -Cr 2 -I 3 ) and the empty Cr e g states (the Cr t 2g states have shifted below −1 eV). This may modify the effect of spin orbit coupling (SOC), which usually is determined by the orbitals near the Fermi level [35,36], and thereby influences the magnetic properties. Thus, a proper treatment of the electronic states requires applying GGA + U calculations. Using the linear response approach, U was calculated for monolayers of CrCl 3 , CrBr 3 , and CrI 3 to be 2.65, 2.68, and 2.63 eV, respectively [11]. We use the average value for our Janus monolayers, U = 2.65 eV, which is justified, because the three values are very close to each other.
First, we test the influence of U for the pristine Cr-trihalide monolayers. We notice in table 1 as general pattern that U elongates all bonds, the easy axis becomes more in-plane, and the energy difference between FM and AFM ordering increases. This trend is particularly important when the energy difference is small, such as 13 meV in the case of monolayer CrBr 3 compressed by −5% strain, which is AFM when calculated without U but FM when calculated with U. Therefore, U can cause not only quantitative but also qualitative differences. The Janus monolayers turn out to follow the same trends as a function of U as the pristine monolayers, see table 2.

Mechanical properties
The elastic moduli are important physical parameters to evaluate whether a monolayer is mechanically stable, the equilibrium structure is planar or bended, and how much strain a monolayer can withstand. For sufficiently small stresses, the stress−strain (σ−ε) relationship is given by the generalized Hooke's law, which can be expressed for a 2D hexagonal crystal as [6] ⎛ ⎝ σ 11 Here A 0 is the equilibrium area, E s is the strain energy (energy difference between the strained and unstrained systems, which depends quadratically on the applied strain, see figure 2), and the applied strain is defined as a−a 0 a 0 , where a and a 0 are the lattice constants of the strained and unstrained systems, respectively. According to the Born criteria, C 11 > 1 N m −1 and C 11 − C 12 > 1 N m −1 implies mechanical stability [38]. Our obtained values, as stated in table 3, satisfy these conditions.
To determine the stiffness of the Cr-trihalide Janus monolayers, Young's modulus is calculated from the elastic constants as [6] According to table 3, Young's modulus of Cl 3 -Cr 2 -I 3 , Br 3 -Cr 2 -I 3 , and Cl 3 -Cr 2 -Br 3 is less than 10% of that of graphene (340 N m −1 ) [39]. These small values imply that the Cr-trihalide Janus monolayers can be strained rather easily. Moreover, the out-of-plane deformation induced by gravity can be estimated by elastic theory as [40,41] where L is the length of the flake, set to 100 μm, and ρ is the areal density, listed in table 3. We find h L = 4.3 × 10 −4 , 4.8 × 10 −4 , and 4.1 × 10 −4 for Cl 3 -Cr 2 -I 3 , Br 3 -Cr 2 -I 3 , Table 2. Calculated lattice constant (a), Cr-X/Cr-Y/Cr-Cr bond lengths (d Cr-X /d Cr-Y /d Cr-Cr ), exchange energy (E ex ), and magnetic anisotropy energy (MAE) for Y 3 -Cr 2 -X 3 (X, Y = I, Br, and Cl) under +5%, 0%, and −5% strain, using the GGA and GGA + U methods.    Table 3. Elastic constants C 11 and C 12 , Young's modulus Y 2D , and areal density ρ.  and Cl 3 -Cr 2 -Br 3 , respectively. These values are of the same order of magnitude as in the case of graphene flakes of the same size [41], implying that Cr-trihalide Janus monolayers can withstand their own weight and keep a planar structure without bending. Finally, figure 2 shows a small asymmetry of the strain energy as a function of the applied strain, implying that the application of tensile strain requires less energy than the application of compressive strain.

Crystal and electronic structure
Cr-trihalide Janus monolayers are structurally stable [14] and share a hexagonal structure, see figures 1(c-e). As the octahedral crystal field splits the Cr d orbitals into three occupied t 2g orbitals and two empty e g orbitals, the Cr magnetic moments turn out to be approximately 3 μ B (4s 0 3d 3 electronic configuration), where the halide ions show a residual countermagnetization of at most 0.16 μ B . The crystal field splitting is larger when the Cr oxidation state is higher and/or the electronegativity of the halide is higher [42]. This is the reason why the bandgaps of Cl 3 -Cr 2 -I 3 and Br 3 -Cr 2 -I 3 are smaller than the bandgap of Cl 3 -Cr 2 -Br 3 , as I has lower electronegativity than Br and Cl. In each case the bandgap is indirect, as shown in    An interesting feature is noted for the size of the bandgap E g in the strain range from −2% to +2%: for Cl 3 -Cr 2 -I 3 and Br 3 -Cr 2 -I 3 it decreases under compression and increases under tension, whereas for Cl 3 -Cr 2 -Br 3 it follows the opposite trend, as elucidated by figure 4. SOC cannot explain this observation, because it only decreases E g but does not alter the general behaviour under strain. However, the observation can be explained by competing effects of the band width (orbital overlap) and crystal field splitting. Under compression/tension the former decreases/increases and the latter increases/decreases E g. . Accordingly, one can conclude that Cl 3 -Cr 2 -I 3 and Br 3 -Cr 2 -I 3 are dominated by the alteration of the band width, whereas Cl 3 -Cr 2 -Br 3 is dominated by the modification of the crystal field splitting.

Magnetic properties
In Cr-trihalide Janus monolayers the two magnetic Cr atoms interact through a non-magnetic halide atom by superexchange. The Goodenough-Kanamori-Anderson rules establish the sign of the superexchange constant [43]. All monolayers under study satisfy these rules for FM superexchange, because the d orbitals of the magnetic atoms are half occupied and form an angle of approximately 90 • with the halide atoms. A recent study [16] has shown, using the Green's function method, that the halide atoms play a vital role in the FM ordering of CrCl 3 and CrI 3 , because when their spin-polarization is neglected the ordering becomes AFM. In the strain range from +5% to −5% all the Cr-trihalide Janus monolayers maintain FM ordering. However, we find that compression decreases the energy difference between FM and AFM ordering (exchange energy E ex = E FM − E AFM ), see table 5. Therefore, we expect a transition to AFM ordering when the compression is further increased. On the other hand, tension alters E ex only little. This can be explained by competition between direct exchange through Cr-Cr bonding, which leads to AFM ordering, and superexchange through Cr-X-Cr bonding, which leads to FM ordering. The Cr-Cr bonding is more sensitive to the strain than the Cr-X bonding, compare the influence of strain on the bond lengths given in table 4. Moreover, the Cr-X-Cr bond angle barely changes under strain, see table 4, which points to a low impact of strain on the superexchange. To calculate E ex , we consider the energies of FM and AFM orderings and fit them to the nearest neighbour Heisenberg spin Hamiltonian (localized spin system) where J is the exchange constant, S i / S j is the spin at lattice site i/j, and the sum includes only nearest neighbours [22]. This , and fur- The obtained values of J are summarized in table 5.
The magnetic anisotropy energy (MAE) depends on the SOC, known as magneto crystalline anisotropy, and the dipole−dipole interaction, known as shape anisotropy [24]. The MAE is the total energy difference between in-plane and out-of-plane magnetization. When it is negative/positive the easy axis is oriented in-plane/out-of-plane. As non-collinear calculations performed by the Vienna ab initio simulation package (VASP) do not distinguish between easy and hard axes when SOC is not included [44], it is not possible to obtain the dipole-dipole contribution to the MAE. Fortunately, the magneto crystalline anisotropy dominates the MAE in low dimensional systems [22], i.e., we can relate our results to the effect of SOC only. The contribution of SOC to the Hamiltonian is [23] where L is the total orbital angular momentum and Φ (r) = Ze 4πε o r the electrostatic potential of the nucleus. The SOC constant ξ r is considered to be an atomic property (hardly changes from the free atom to a crystal), because the derivative is largest close to the nucleus [45]. It is roughly proportional to the square of the atomic number Z [22]. Table 5 indicates that compression/tension decreases/increases the MAE, i.e., compression destabilizes an out-of-plane easy axis and promotes an in-plane easy axis. This can be explained qualitatively in terms of the Cr-X and Cr-Y bond lengths. According to table 4, both bond lengths decrease under compression, while the thickness Δ, as defined in figures 1(c-e), increases. This means that the in-plane component of the Cr d orbitals is more sensitive to the strain than the out-of-plane component. Disturbance of the in-plane motion of the electrons causes a reduction of the out-of-plane component of L [23]. Because the easy axis is determined by the direction of L, it is oriented in-plane/outof-plane when L points predominantly in-plane/out-of-plane. Therefore, compression destabilizes an out-of-plane easy axis until an in-plane easy axis is achieved, and tension has the opposite effect. Figure 5 shows the site-projected DOS for Cl 3 −Cr 2 −I 3 in the cases of +5%, 0%, and −5% strain. We observe that the band width of the Cr-dominated conduction band (which consists of both in-plane and out-of-plane components of the Cr d orbitals) increases (0.60 eV, 0.63 eV, and 0.72 eV at +5%, 0%, and −5% strain, respectively). The outof-plane component cannot be the cause of this increase, as Δ increases. Thus, the increased band width must be due to the fact that the in-plane component of the d orbitals interacts more with the halide orbitals, as the bond length decreases. This causes disturbance of the in-plane motion of the electrons. An exception is monolayer CrI 3 where the MAE increases under both compression and tension. The single ion anisotropy of Cr can play a role in explaining this behaviour, because strain increases its contribution to the MAE depending on the type of atoms involved, whereas its contribution to the MAE can be neglected for unstrained CrI 3 , as shown in reference [12], and the MAE is dominated by the SOC of I through superexchange. Single ion anisotropy can also explain why the MAE under −5% strain follows the trend Cl 3 -Cr 2 -Br 3 < Br 3 -Cr 2 -I 3 < Cl 3 -Cr 2 -I 3 , i.e., the same trend as the difference in the electronegativity between the involved atoms, χ Cl −χ Br < χ Br −χ I < χ Cl −χ I . On the other hand, under +5% strain the SOC of I through superexchange seems to be dominant, as the MAE increases with the mass of the involved atoms as Cl 3 -Cr 2 -Br 3 < Cl 3 -Cr 2 -I 3 < Br 3 -Cr 2 -I 3 .
Applying strain to Cr-trihalide Janus monolayers gives rise to three unique features not available in the pristine monolayers. Firstly, the ability to switch the easy axis by a small amount of strain. For example, Cl 3 -Cr 2 -I 3 switches from in-plane to out-of-plane easy axis at +3% strain and Br 3 -Cr 2 -I 3 switches from out-of-plane to in-plane easy axis at −2% strain. On the other hand, monolayer CrI 3 and CrBr 3 do not lose their outof-plane easy axis even under ±10% strain [46]. Secondly, Cl 3 -Cr 2 -I 3 has a very negative (in-plane) MAE under −5% strain, −3.77 meV per formula unit. Thirdly, all the studied Cr-trihalide Janus monolayers maintain FM ordering during the switching, while in the case of the pristine Cr-trihalide monolayers the easy axis switches at the phase transition from FM to AFM ordering, as shown for monolayer CrCl 3 in table 5 and previously in reference [46].
Assuming that there are only two-fold symmetries, which dominate in low-dimensional systems, the MAE of a perfect thin film is given by [22] where θ is the angle between the direction of magnetization and the 2D plane, and K zz is the single-site anisotropy constant (positive/negative for out-of-plane/in-plane easy axis). Figure 6 shows for all three Cr-trihalide Janus monolayers that at maximum strain (±5%) the MAE follows the angular dependence sin 2 θ, except for the case of Cl 3 -Cr 2 -Br 3 at −5% strain, where it is very small and the accuracy of the calculation is not sufficient to decide the functional dependence. Yet, the data do not contradict a sin 2 θ dependence. It is worth to mention an observation that enables us to predict the easy axis without SOC calculations (both with and without U). As shown in figure 4, when the bandgap opens up between states of identical/different spin the preferable direction of the easy axis is out-of-plane/in-plane. Also, we find that for decreasing/increasing spin splitting of the valence band near the Fermi level the value of the MAE decreases/increases. Again there is an exception to this rule: Cl 3 -Cr 2 -Br 3 under compression. The reason can be the strong crystal field splitting or the fact that the SOC in this case has two comparably strong sources, the Br p orbitals and Cr d orbitals, as ξ r is of the same order of magnitude for Br and Cr. Consequently, Br is not the dominating source of the SOC in contrast to Cl 3 -Cr 2 -I 3 and Br 3 -Cr 2 -I 3 , where I dominates over Cr, as shown previously for monolayer CrI 3 [12].

Conclusion
We have used GGA + U calculations to investigate the structural properties, magnetic properties, and electronic states of the Cr-trihalide Janus monolayers X 3 -Cr 2 -Y 3 (X, Y = Cl, Br, and I) under strain in the range from −5% to +5%. All three Janus monolayers maintain a planar (non-bended) structure and have small Young's moduli, which means that the application of large strain is experimentally possible. Both the pristine CrX 3 (X = Cl, Br, and I) and Cr-trihalide Janus monolayers turn out to be Mott insulators, i.e., the onsite Coulomb interaction is essential to be taken into account in the calculations, as it changes the physical properties qualitatively. FM ordering turns out to be the ground state for all three Cr-trihalide Janus monolayers in the studied range of strain, even though the energy difference between FM and AFM ordering decreases for increasing compressive strain. Having a strain-tuneable bandgap, a remarkable feature of the Cr-trihalide Janus monolayers pertains to the high sensitivity of their MAE to mechanical strain. Our calculations show that small strain up to −2% can lead to anisotropy switching from out-of-plane to in-plane, while maintaining the FM ordering. This feature is absent in the pristine Cr-trihalide monolayers. Compressive strain favours an in-plane easy axis, whereas tensile strain favours an out-of-plane easy axis. The MAE of Cl 3 -Cr 2 -I 3 reaches a very negative value of −3.77 meV per formula unit under −5% strain. The fact that the availability of Janus monolayers with lattice constants and material properties different from those of the pristine monolayers increases the choices to engineer 2D van der Waals magnetic heterostructures creates a venue to provide spintronic and valleytronic materials with desired properties. For example, it was demonstrated that the spin and valley pseudospin can be controlled magnetically in a heterostructure consisting of an ultrathin CrI 3 film on top of WSe 2 [47]. In addition, van der Waals stacking can be used to raise the Curie temperature to room temperature, which was achieved experimentally for a EuS-Bi 2 Se 3 heterostructure (Curie temperature of EuS rises from 17 K to 300 K) [48]. This finding was explained theoretically by the presence of topological surface states at the EuS-Bi 2 Se 3 interface [49]. It is also interesting to study Dzyaloshinski-Moriya interaction in Cr-trihalide Janus monolayers, H DM = i, j D i, j · ( S i × S j ), where D i, j is governed by the symmetry of the system and depends on the real space direction given by sites i and j [22], because the Cr-trihalide Janus monolayers combine SOC, absence of time-reversal symmetry, and broken inversion symmetry (gradient of the electric potential normal to the surface). As a consequence, the electron hopping from site i to site j is not equivalent to the hopping from site j to site i, leading to the onset of Dzyaloshinski-Moriya interaction, which can stabilize homochiral, helical, and cycloidal spin spirals [50,51]. All these properties make the Cr-trihalide Janus monolayers promising for spintronic and multifunctional devices.

Computational details
The first principles calculations in this work were performed using the projector augmented wave method as implemented in VASP [52]. We applied the Perdew-Burke-Ernzerhof parametrization [53] of the GGA for the exchange correlation functional, taking into account an onsite Coulomb interaction U [54] to accurately describe the Cr 3d states. The Brillouin zone was sampled on a Γ -centred Monkhorst-Pack 16 × 16 × 1 k-mesh, with a total energy difference of less than 0.002 meV to a 20 × 20 × 1 k-mesh. The plane wave cutoff energy was set to 800 eV, with a total energy difference of less than 0.025 meV (0.058 meV) for the Cr-trihalide Janus monolayers (for the pristine Cr-trihalide monolayers) to a 900 eV cutoff energy. The atomic structures were carefully relaxed until the Hellmann-Feynman forces on each atom became smaller than 0.005 eV Å −1 . We introduced a layer of vacuum of at least 18 Å thickness in the z-direction to avoid interaction between periodic images, i.e., to model a 2D system. The MAE was determined by 1) spin polarized calculations (collinear) for the FM and AFM orderings shown in figure 7 followed by 2) SOC calculations (non-collinear) for the FM ground state, all performed with full atomic relaxation, to obtain the total energy difference between in-plane and out-of-plane magnetization.