Low-field magnetic anisotropy of Sr2IrO4

Magnetic anisotropy in strontium iridate (Sr2IrO4) is essential because of its strong spin–orbit coupling and crystal field effect. In this paper, we present a detailed mapping of the out-of-plane (OOP) magnetic anisotropy in Sr2IrO4 for different sample orientations using torque magnetometry measurements in the low-magnetic-field region before the isospins are completely ordered. Dominant in-plane anisotropy was identified at low fields, confirming the b axis as an easy magnetization axis. Based on the fitting analysis of the strong uniaxial magnetic anisotropy, we observed that the main anisotropic effect arises from a spin–orbit-coupled magnetic exchange interaction affecting the OOP interaction. The effect of interlayer exchange interaction results in additional anisotropic terms owing to the tilting of the isospins. The results are relevant for understanding OOP magnetic anisotropy and provide a new way to analyze the effects of spin–orbit-coupling and interlayer magnetic exchange interactions. This study provides insight into the understanding of bulk magnetic, magnetotransport, and spintronic behavior on Sr2IrO4 for future studies.


Introduction
Currently, 5d transition metal oxides have emerged as a captivating class of materials arising from strong spin-orbit or electron-lattice interactions [1][2][3]. This strongly interacting many-body system provides a viable platform for studying unique frustrations and anisotropic magnetic interactions [4][5][6][7]. The strong spin-orbit coupling (SOC) in these materials makes them most suitable for high-temperature superconductors, topological insulators, and devices with spintronics functionality [8]. Recent advances have opened a new paradigm for achieving spintronic functionality, namely, * Author to whom any correspondence should be addressed.
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. antiferromagnetic spintronics [1,2,[4][5][6][9][10][11][12][13]. Specifically, antiferromagnetic materials are of great interest because they have advantages such as no stray fields, insensitivity to magnetic fields, and fast antiferromagnetic (AFM) dynamics [14][15][16][17].
Strontium iridate (Sr 2 IrO 4 ) is a representative of a 5d strongly correlated electronic system with a space group I41/acd [142]. In a free Ir atom, the 5d states are degenerated because of the rotational symmetry of the atomic Hamiltonian. However, the crystal field removes the degeneracy and splits the 5d orbital. The 5d states are split into highenergy two-fold-degenerate (e g ) states and low-energy threefold-degenerate (t 2g ) states. The total effective orbital angular momentum (l eff ) of the t 2g state is 1, and the total spin sums up to 1/2. Because a strong SOC essentially entangles the spin and orbital momenta, a unique J eff = 1/2 quantum state is formed by the pure spins and spatially anisotropic orbitals. This novel J eff = 1/2 state can provide a comprehensive understanding of the correlation between the anisotropic magnetoresistance and magnetocrystalline anisotropy and also help explore possible controllable AFM spintronics [16,[18][19][20][21][22]. Magnetic anisotropy is a key phenomenon driven by a strong electron correlation. Magnetic anisotropy interactions and anisotropy energy are used in several industrial and technical fields [23]. The magnetocrystalline anisotropy in antiferromagnets arises from relativistic SOC. Moreover, microscopic observations have shown that the first nearestneighbor basal-plane exchange constant is higher (60 meV) than the interlayer exchange constant (1 μeV), indicating a strong magnetic anisotropy [15,24,25].
The J eff = 1/2 isospins in Sr 2 IrO 4 can be modulated by applying an external magnetic field [5,15,20,26]. An in-plane canted AFM ordering arises below T N ∼ 240 K owing to the antisymmetric exchange interaction known as the Dzyaloshinskii-Moriya interaction (DMI) [27,28]. DMI competes with the symmetric Heisenberg interaction and favors spin-canting in an otherwise AFM system. A spin-canting angle of ∼12 • results in a net magnetic moment in each domain. However, AFM ordering along the c axis leads to a zero net magnetization [29]. Moreover, the fascinating feature of Sr 2 IrO 4 is that it exhibits both weak ferromagnetic (WFM) and AFM characteristics. The magnetic anisotropy and isospin reorientation substantially affect the electronic transport properties, making the material ideal for AFM-based spintronics functionality [20,30,31]. The magnetocrystalline contribution to the magnetic anisotropy in single-crystal Sr 2 IrO 4 has been demonstrated based on torque magnetometry up to 9 T [3,30]. Furthermore, a two-fold sawtooth shape of the out-of-plane (OOP) rotation torque was evaluated. This strong OOP magnetic anisotropy shows that the magnetic easy axis is along the in-plane direction. In addition, the field-induced WFM order is attributed only to the in-plane component of the external magnetic field. Thus, we conducted a detailed study on the in-plane magnetic anisotropy in single-crystal Sr 2 IrO 4 and found the b axis to be the easy axis of magnetization [23]. However, there is a lack of mapping of the explicit evolution of the isospins and the magnetic anisotropy for different OOP rotations. Moreover, investigating the magnetic anisotropy of single-crystal Sr 2 IrO 4 is essential, particularly in the low-magnetic-field region before the isospins are fully ordered.
Therefore, in this study, we verified OOP magnetic anisotropy before and after WFM transition and analyzed the ac and bc plane anisotropies separately. Here, we present a comprehensive study of the low-field isospin interactions that generate in-plane anisotropy, where the static magneticdomain structure is related to the orientation of the magnetic easy axis. Because Sr 2 IrO 4 is a layered material, we cannot ignore the interfacial anisotropy and interlayer exchange. Thus, this study considered the various factors contributing to the anisotropy, such as the shape, interfacial direction, interlayer exchange, and crystal symmetry, in the typical singlecrystal Sr 2 IrO 4 exhibiting strong SOC.

Materials and methods
Sr 2 IrO 4 crystals were prepared using a flux method with SrCl 2 as the flux [3]. The crystallinity was confirmed using x-ray diffraction measurements. Single-crystal Sr 2 IrO 4 was mounted on a piezoresistive cantilever with a crystallographic c axis perpendicular to the lever plane. The change in the torque was measured through the change in the resistance of the piezo material comprising a Wheatstone bridge circuit [3,23]. This is a nondestructive and reliable technique for detecting spin canting and domain magnetization. The magnetic field direction with respect to the crystal axis was controlled using a rotator. Moreover, we applied the OOP angulardependent torque of Sr 2 IrO 4 , i.e., τ (θ), for various crystal orientations by varying ψ. Here, θ is the angle between the applied magnetic field and the ab plane, and ψ is the crystal orientation with respect to the reference axis in the in-plane direction, see figure 1(a).

Results and discussion
Figure 1(b) shows the magnetization at 30 K when a magnetic field is applied along the ab plane. The magnetization gradually increases and saturates above a certain critical field at approximately 0.3 T with a small saturated moment [30]. A low degree of hysteresis with a negligible amount of residual net magnetization can be observed, indicating the WFM nature of Sr 2 IrO 4 along the ab plane. Two distinguishable domains are present: one with isospins along the b axis and the other with isospins along the a axis. A correlation exists between the isospin direction and the stacking pattern, implying the presence of anisotropic interlayer coupling [30]. The regions can be divided into a low field (<0.3 T) with a mixed-domain state and a high field (>0.3 T) with the WFM state. Figure 1(c) shows the various crystal orientations with respect to the reference axis marked by a red dotted line. Note that ψ is a measure of the angle between the reference line and one specific corner of the crystal. A solid red line is displayed at one corner of the crystal to clarify the Ψ crystal orientation. Crystals usually grow in high symmetry direction and provide a clue regarding the crystallographic a and b axes. Based on the crystalline morphology of SIO, we designated one corner as the a axis, later confirmed after x-ray diffraction measurement. We performed the angle-dependent torque τ (θ) at Ψ = 0 • , 25 • , 90 • , 120 • , and 180 • . By comparing τ (θ) at Ψ = 0 • and 180 • , as shown in figure 1(d), we determine whether there is a difference in the magnetization strengths along the positive and negative a axes in the basal plane. Identical τ (θ) values for Ψ = 0 • (solid line) and 180 • (dotted line), including hysteresis around θ = 90 • , are observed. This confirms that the crystal is accurately placed on the cantilever without torsion. The result represents the uniaxial magnetic behavior of the crystal along the in-plane direction. Hence, the two-fold torque curves come from the domain magnetization and AFM ordering in Sr 2 IrO 4 . Furthermore, the  sawtooth response of the torque in high magnetic fields is a dominant characteristic of Sr 2 IrO 4 with a strong SOC. τ (θ) vanishes at θ = 0 • (180 • ) and θ = 90 • , indicating that the easy axis is either parallel or perpendicular to the crystal surface. As θ increases, the gradually increasing torque changes rapidly in the opposite direction near θ = 90 • , where the magnetic field passes through the c axis. This is characterized by a spin floplike structure, indicating that the c axis is a hard magnetization axis and confirming the presence of an easy axis along the ab plane when the sample surface is parallel to the field direction, i.e., when θ = 0 • (180 • ). This coincides with the reported AFM alignment of magnetic-domain magnetization along the a axis [32]. This dominant in-plane anisotropy at a low field becomes isotropic at higher fields. The magnetic field-dependent amplitude of τ /H at θ = 45 • , shown in figure 2(c), represents the relative amplitude of the magnetization. The difference in the amplitudes for different Ψ values solely arises from the difference in the magnetization strengths along the a and b axes, given that the c axis magnetization is insignificant in Sr 2 IrO 4 [30]. The hysteresis is significant in the low-H regime (H < 0.3 T), reported as canted AFM-domain states [23]. Moreover, the hysteresis disappears at 0.3 T, known as the field value where the WFM state is established [26,30]. The jumplike trait approximately at θ = 90 • is prominent only under bc rotation (Ψ = 90 • ). This difference in the torque curves might be due to the difference in the domain populations along the a and b axes. Figure 3 shows the τ (θ) values for Ψ = 0 • under various magnetic fields at 60 K. The sin 2θ angle dependence of the torque curves in the low-H regime is a characteristic of the linear response regime, while the deviation from this behavior at higher fields is direct evidence of the strong magnetic correlation. The sawtooth shape of the τ (θ) curve at higher fields (>1 T) indicates the AFM ordering of the spins [3]. Generally, applying a high magnetic field overcoming all the magnetic correlations helps restore the sin 2θ dependence due to the contribution of the Zeeman energy. However, in our case, the persistence of the sawtooth behavior above 0.3 T confirms the existence of a strong spin correlation [33].
To investigate the OOP magnetic anisotropies, we fitted the τ (θ) curves. We introduced a mathematical model to determine the strong uniaxial magnetic anisotropy for multilayered systems [34]. Because of the layered structure of Sr 2 IrO 4 , acting as a natural thin film, and electrons whose isospin orientation are strongly coupled to the pure spin magnetic moment in the plane, the model can be modified as follows: The magnetic free energy can be written as F = F A + F D + F H , which is the summation of the anisotropy energy (F A ), demagnetization energy (F D ), and free energy under an applied magnetic field (F H ), such that Here, k 1 and k 2 are the second-order uniaxial anisotropy terms, and θ and ϕ are the angles of the magnetic field (H ) and magnetization axis (M) from the ab plane, respectively.
Reducing the total energy of the system under equilibrium conditions ∂F ∂θ = 0 and rearranging the equations yield: In equation (2), we set the coefficients of sin 2θ and sin 4θ as A 2 and A 4 , respectively. The third term corresponds to the torque (τ ); therefore, τ becomes τ (θ) = A 2 sin 2θ + A 4 sin 4θ. (3) A 2 characterizes an easy plane if A 2 > 0 and the easy axis if A 2 < 0. τ will be zero when ϕ = θ, i.e., when H and M are parallel. For Sr 2 IrO 4 , the demagnetization energy contribution was negligible owing to the insignificant shape anisotropy [35]. Thus, extracting information regarding magnetic anisotropy is easy because of the crystal symmetry in the presence of H. The red solid lines in figure 3 indicate the fitting result obtained using equation (3). Up to 0.3 T, the fitting equation yields a relatively good fit. Nevertheless, the higher the magnetic field, the worse the fitting. To obtain a more accurate fitting curve, we added a higher-order anisotropy term to equation (3) as follows. τ = A 2 sin 2θ + A 4 sin 4θ + A 6 sin 6θ + A 8 sin 8θ The red dotted line indicates the fitting results on τ (θ) at 1 T using equation (4). In this case, the result improves but still not perfect.
Anisotropic magnetism can be estimated by comparing the amplitude of the nth-fold symmetric contribution, A n (see figure 4(a)). Two-fold symmetry contributions are the major contributions, as evident from the τ (θ) curves. The amplitude of sin 2θ (A 2 ) is positive at all magnetic fields, confirming that the anisotropy in Sr 2 IrO 4 is an easy-plane type. A 2 is higher for Ψ = 90 • than for Ψ = 0 • , indicating that the bc plane is an easy plane. A 2 gradually changes in the low-H regime and then increases linearly. This demonstrates the weak spin responses because of the lower Zeeman energy contributions in the low-H regime. The nonsaturation of all A n up to 1 T is because the applied magnetic field is insufficient to overcome the high anisotropic energy barrier between the ab plane and the c axis. A 2 encodes all possible anisotropy contributions, including both intrinsic characteristics, such as the crystal structure and magnetic interaction, and extrinsic characteristics, such as the shape and surface effects. As reported in the literature, the extrinsic contributions are insignificant in Sr 2 IrO 4 because of the small crystal dimensions (0.1 mm) and minor spin-glass phenomena [35]. The superexchange interaction between Ir moments in Sr 2 IrO 4 is mediated by non-magnetic oxygen. The responsible strong SOC for microscopic antisymmetric exchange interaction, which competes with isotropic Heisenberg exchange, leads to a tilted Ir-O-Ir bond angle of 178 • along the in-plane. Thus, the main anisotropic effect arises from a spin-orbit-coupled magnetic exchange interaction. The higher-order term is essential in systems exhibiting interfacial anisotropy. The 4 1 screw-axis symmetry about the c axis and strong SOC in Sr 2 IrO 4 necessitates the inclusion of this interfacial anisotropy. In our OOP measurements of Sr 2 IrO 4 , interfacial anisotropy cannot be a premise for total magnetic anisotropy; however, it is a prominent factor along with SOC in the OOP magnetic anisotropy. Thus, the main anisotropic effect arises from a spin-orbit-coupled magnetic exchange interaction. The higher-order term is essential in systems exhibiting interfacial anisotropy. This interfacial anisotropy is most common in magnetic thin films. In our OOP measurements of Sr 2 IrO 4 , interfacial anisotropy cannot be a premise for total magnetic anisotropy; however, it cannot be excluded. Nonzero A 4 appears near the high-field regime (H > 0.3 T) when τ (θ) deviates from the sin 2θ pattern. Since SIO has weak OOP magnetic coupling or magnetic exchange interaction [15,32], nonzero A 4 pertains to the weakly coupled regions whose high prospect is in the OOP anisotropy measurements. In the low-H regime (H < 0.3 T), A 4 is negative for Ψ = 0 • and positive for Ψ = 90 • , as shown in the inset of figure 4. The positive value of A 2 (A 2 > 0) and the negative value of A 4 (A 4 < 0) for Ψ = 0 • (ac rotation) can be interpreted as the canted state, indicating AFM ordering along the a axis in the ab plane [32]. This result is also associated with a metamagnetic transition, where the system first becomes AFM a (AFM ordering along the a axis) and then transitions to an FM state above 0.3 T by applying a magnetic field along the in-plane [36]. Because the b axis is an easy axis of magnetization, no such A 4 < 0 can be observed for Ψ = 90 • (bc rotation) in the absence of any metamagnetic transition. Coupling along the c axis (J c ) is weaker than in-plane coupling (J ab ) [15]. As a correlation exists between the isospin direction and the stacking pattern, the anisotropic interlayer coupling is present depending on the field applied along the a axis or the b axis. An evolution of the domains with increasing applied magnetic field is shown in figure 4(b), where the system magnetizes along the easy b axis in a right-right-right-right pattern above the critical field value >0.3 T. The shift toward positive values above 0.3 T is because the FM state cannot be realized by applying H along the hard axis, i.e., c axis. The application of a high field along the OOP direction will induce magnetization along the c axis owing to the tilting of the isospins. Such slight tilting of the isospins generates additional anisotropic terms (A 4 and higher-order parameters) and provides important information about the system. This will lead to the emergence of higher orders at fields >0.3 T, and off-diagonal interactions will lead to a sawtooth behavior in τ (θ). The reason for the increase in the bifurcation of the A 4 curves for ac and bc rotations in the high-field regime remains unknown.

Conclusions
Based on torque magnetometry measurements of singlecrystal Sr 2 IrO 4 for different OOP rotations, we identified the in-plane magnetic anisotropy, which coincided with the AFM alignment of the magnetic domain along the a axis. This dominant in-plane anisotropy at low fields with a mixeddomain state becomes isotropic in higher fields with the WFM state. The sawtooth response of the torque in a high magnetic field is characterized by a spin flop-like change when the field passes through the c axis, which is a hard magnetization axis. The deviation from the sin 2θ dependence of the torque is direct evidence of a strong magnetic correlation. By performing a fitting analysis of the strong uniaxial magnetic anisotropy, we compared the amplitudes of the two-fold (A 2 ) and four-fold (A 4 ) symmetric contributions. The higher value of A 2 for bc rotation than for ac rotation indicates that the bc plane is an easy plane. The nonzero value of A 4 in the high-field regime is attributed to a weakly coupled interlayer direction along with a slight tilting of the isospins generates additional anisotropic terms. Our results provide complete mapping of OOP magnetic anisotropy, verifying that the major anisotropic effects arise from spin-orbit-coupled magnetic exchange interactions. The results are relevant to understanding the giant magnetoresistance effect in a single crystal Sr 2 IrO 4 reported recently [15]. This study provides insights into the OOP magnetic anisotropy in a low (<1 T) magnetic field regime.