Experimental Evidence of a change of Exchange Anisotropy Sign with Temperature in Zn-Substituted Cu2OSeO3

We report small-angle neutron scattering from the conical state in a single crystal of Zn-substituted Cu2OSeO3. Using a 3D vector-field magnet to reorient the conical wavevector, our measurements show that the magnitude of the conical wavevector changes as a function of crystallographic direction. These changes are caused by the anisotropic exchange interaction (AEI), whose magnitude transitions from a maxima to a minima along the<111>and<100>crystallographic directions respectively. We further find that the AEI constant undergoes a change of sign from positive to negative with decreasing temperature. Unlike in the related compound FeGe, where similar behaviour of the AEI induces a reorientation of the helical wavevector, we show that the zero field helical wavevector in (Cu0.98Zn0.02)2OSeO3 remains along the<100>directions at all temperatures due to the competing fourth-order magnetocrystalline anisotropy becoming dominant at lower temperatures.

The lack of inversion symmetry in the family of B20 chiral cubic materials allows a non-vanishing Dzyaloshinskii-Moriya interaction (DMI), which competes with numerous other magnetic interactions to stabilize a wide variety of incommensurate magnetic states [1][2][3][4][5][6][7][8]. At zero field and at temperatures below T C , the isotropic exchange interaction and DMI (which prefer spin-aligned with constant A and spin-perpendicular with constant D respectively), twist the magnetic texture into a helix with a periodicity determined by the ratio of the strengths of the two interactions, q ≈ D/A. Typically, q is close to 0.1 nm −1 , suitable for measurement with magnetic small angle neutron scattering (SANS) experiments. The non-centrosymmetric crystal structure allows for non-zero off-diagonal terms in the exchange tensor [9], which gives rise to the so-called anisotropic exchange interaction (AEI) with a strength γ, where |γ| < |A|. In MnSi, the helical wavevector was shown to align along the 1 1 1 directions [10] due to a negative AEI constant [11][12][13][14]. γ is also known to be temperature dependent, with a change of sign with temperature being responsible for the helical wavevector reorientation in the related material, FeGe [15,16].
The orientation of these incommensurate textures can also be manipulated by the application of a magnetic field, which induces a transition from the multi-domain magnetic helix into a single-domain magnetic cone state, whose wavevector typically follows the direction of the applied magnetic field [10]. As shown in Fig. 1a, in the cone state the spins cant towards the field direction to minimize the Zeeman interaction. Applying a higher field reduces the conical angle θ, but maintains a constant wavevector, q, until the cone angle reaches zero to form a forced ferromagnetic state [17].
The behaviour of chiral magnets adopting a B20 structure materials near T C is further enriched by thermal fluctuations, which induce a hexagonal lattice of mag-netic skyrmions within small region at non-zero field. Magnetic skyrmions are vortex-like whirls of magnetization that are the 2D solitonic solutions to systems hosting a number of competing magnetic interactions [1,[18][19][20][21][22]. Their particle-like nature and topological properties make magnetic skyrmions appealing for applications within a variety of spintronic devices, ranging from racetrack memory schemes and logic devices, to stochastic and neuromorphic computing [23][24][25][26][27][28][29].
The effects of anisotropic interactions such as magnetocrystalline anisotropy (MCA) and the AEI on these magnetic textures are typically overlooked, as the inherent cubic symmetry of the B20 materials limits the strength of these energy terms [30].
Despite this, anisotropy-stabilised magnetic textures within the multiferroic B20 material Cu 2 OSeO 3 have been found to exist at low temperatures, such as the low-temperature skyrmion (LTS) [31][32][33][34] and tilted conical (TC) phase, where the conical wavevector deviates from the direction of the magnetic field applied along a [1 0 0], with q canting towards the 1 1 1 [35]. The forms of the MCA and AEI for a magnetic cone with angle θ and wavevectorangle φ are shown in Fig. 1b. In (ii), the local minima along the 1 1 1 directions for K > 0 explain the origin of the TC, but an additional AEI shown in (i) is essential to prevent the texture collapsing into a fieldpolarized state. Currently, the effects of the AEI have not been investigated on the LTS, and decoupling the effects of MCA and the AEI remains a challenge as the conventional method of torque magnetometry only investigates the field-polarized state where the AEI vanishes [36]. Distinguishing and quantifying these anisotropic interactions is therefore important to better understand the low-temperature behaviour of Cu 2 OSeO 3 .
In this letter, we utilize a 3D vector-field magnet together with small-angle neutron scattering (SANS) to show that the AEI changes sign from positive to negative with decreasing temperature in (Cu 0.98 Zn 0.02 ) 2 OSeO 3 . Unlike in the related compound FeGe, this change of sign does not induce a reorientation of the zero-field magnetic helix. This demonstrates that the 4 th order MCA also increases in magnitude with decreasing temperature, greater that the AEI in order to stabilize the zero-field magnetic helices along the 1 0 0 directions at all temperatures [37]. Our findings provide insight into the low temperature behavior of Cu 2 OSeO 3 that will be essential for complete understanding of the creation and stabilization mechanisms of the recently discovered LTS and TC phases within pristine Cu 2 OSeO 3 . A 20.5 mg single crystal of (Cu 0.98 Zn 0.02 ) 2 OSeO 3 was grown at the University of Warwick utilizing the chemical vapour transport technique, see [38] for details. For the SANS experiment, the LARMOR instrument at the ISIS Pulsed Neutron and Muon Source was used. The sample was aligned with an x-ray Laue camera (Multi-wire Laboratories) such that the [1 1 0] direction was vertical in the laboratory frame and the [1 1 0] direction was parallel with the incident neutron beam. This orientation allows the magnetic textures whose periodic components lie within a plane spanned by the all 3 high cubicsymmetry directions ( 1 0 0 , 1 1 0 , and 1 1 1 ) to simultaneously satisfy the Bragg condition [39], see Fig. 1c. A zinc-substituted sample was chosen for our study due to the reduction of the exchange interaction compared to pristine Cu 2 OSeO 3 , resulting in a greater value of q [37]. Previously, we have shown strong similarities of the magnetic behavior between Zn-substituted and pristine samples [37], and hence we expect our results here will extend to pristine samples. The time-of-flight neutron diffraction data was reduced with Mantid [40], using scattered neutron wavelengths between 0.9 and 13.5 A.
To observe the effects of the anisotropic interactions on the magnetic textures within Cu 2 OSeO 3 , we performed field scans by rotating the direction of the field at a constant magnitude. Initially, the field was applied vertically in the laboratory frame after zero-field cooling. The field orientation was then rotated 90 degrees in 46 steps about an axis parallel with the neutron beam, such that the [1 1 0], [1 1 1], [0 0 1] directions were parallel with the field at angles of 0, 35.3 and 90 degrees respectively. This procedure was performed at 5, 12 and 50 K at magnetic field magnitudes of 70, 60 and 40 mT respectively. These fields were chosen to avoid phase coexistence between the magnetic conical and helical states, see Supplementary Material [URL will be inserted by publisher]. A selection of frames from the field scans are shown in Fig. 2a.
In the first frame of the 50 K dataset, Fig. 2a(i), the vertically applied magnetic field induces a magnetic conical state with wavevector along the [1 1 0] direction, which we detect as a single pair of vertical peaks with q ≈ 0.011 A −1 , corresponding to a real space length of 57 nm. Upon rotating the field, Fig. 2a(ii-v), the conical wavevector rotates in an attempt to follow the direction of the magnetic field. This behaviour is expected in the limit in which the AEI and MCA are much smaller than the isotropic Dzyaloshinskii-Moriya and exchange interactions (γ, K << A, D), allowing the conical wavevector to rotate freely to minimize the Zeeman interaction. However, this behaviour is not replicated within the data taken at lower temperatures in Fig. 2b(i-v), where angle of the conical wavevector (φ) lags behind the magnetic field angle (H φ ), particularly after the field passes through the [1 1 1] direction. This offset between the field direction and q increases in magnitude with further rotation of the magnetic field, up to a maxima just before the field is applied along the [0 0 1] direction. This magnetic state is characteristic of the tilted conical state seen previously in pristine Cu 2 OSeO 3 [35], and shows that while isotropic interactions dominate at temperatures near T C , the anisotropic interactions become increasingly important at low temperatures and prevent free rotation of the  magnetic conical state. In order to quantify the effects of the anisotropic interactions, the conical peaks collected during the field scans shown in Fig. 2 and an additional set at 12 K were fit using two dimensional Gaussian functions in polar coordinates, allowing the extraction of the magnetic wavevector angle, φ, and magnitude, q. The results of this fitting for field angle scans at the three different temperatures are shown in Fig. 3, with q normalized to the value at H φ = 90 degrees (q 0 ) for each dataset.
The behaviour of q as a function of field angle is shown in Fig. 3a. A clear difference between the high temperature and low temperature regimes is observed. In the 50 K dataset q initially decreases slightly as the applied field is rotated from the [1 1 0] to [1 1 1] direction and then increases as the field is rotated further, up to a maximum when H is along [0 0 1]. This behaviour is reversed at low temperatures (both at 12 K and 5 K), where the magnitude of the conical wavevector instead increases to a maximum when the field is rotated from [1 1 0] to [1 1 1], before quickly decreasing as the field angle approaches the [0 0 1] direction. In Fig. 3b, the effects of the anisotropic interactions can also clearly be seen to be more significant in the low-temperature datasets, as in them φ deviates substantially from linearity after the magnetic field passes through [1 1 1], as compared to the 50 K dataset where φ deviates only slightly from the expected linear trend in the absence of anisotropic interactions. The dependence of q on H φ allows us to decouple the two dominant anisotropic interactions. We start by using the non-constant terms within the free energy expansion derived by Bak and Jensen [11], which is valid for systems of P 2 1 3 crystal symmetry that host slowly-varying magnetization densities m(r) [41]: Where D, A are the familiar Dzyaloshinskii-Moriya and exchange stiffness, and K, γ are the 4 th order MCA and AEI constants respectively. In the case of D = 0, and A, γ > 0, the spin-texture reduces to a simple ferromagnet, with the spin direction given by the sign of the MCA constant K. We approximate a model spin-texture using a conical ansatz: m(r)/M s = sin θ(cos(q · r)ê 1 + sin(q · r)ê 2 ) + cos θê 3 (2) where we follow the convention that θ is the conical angle, and {ê n } define three mutually orthogonal basis vectors, withê 3 q. To compare with our experimental observations, we consider a conical texture with the wavevector restricted to the plane spanned by the three high cubic-symmetry directions, such that q = ( q √ 2 sin φ, q √ 2 sin φ, q cos φ). Using this form for the magnetic wavevector and substituting (2) into (1), integrating over one conical period, λ, and differentiating with respect to q we find a single stable solution: The implications of this result can be seen in Fig. 4(ac), which show the experimentally measured values of the conical wavevector q for different conical wavevector angles, φ, at 50, 12 and 5 K respectively. At 50 K, the increased wavevector at Using the experimentally determined value for the exchange stiffness, A = 4.4 × 10 −13 J/m from T C = 57 K [38,42], and fitting (3) to the data in Fig. 4, we find γ = 2.1(2)×10 −14 J/m at 50 K, −3.4(4)×10 −14 at 12 K, and −6.7(3) × 10 −14 J/m at 5 K. The low-temperature values are consistent with theoretical values of γ required for TC formation [35], in agreement with the TC states observed in this study.
During the refinement of the low temperature datasets, only angles above 55 degrees were used. This is because at low temperatures, the direction of the conical wavevector deviates significantly from the magnetic field direction for φ < 55 degrees, inducing a component of the magnetic field perpendicular to the conical wavevector. Applying transverse fields to a helical state with a pinned q direction is known to deform the helix [43], introducing higher-order components to the Fourier transform of the magnetic state. The presence of higher order peaks in our SANS pattern shown in Fig. 4d suggest that similar deformations are occurring in our tilted conical state. For helices, these deformations are known to reduce the value of q as the perpendicular field strength increases, which would be consistent with our observations of a large q discernibly where the higher order peaks are prominent. Fully accounting for these deformations within our model spin-texture remains an avenue intended for future study. As shown in Fig. 1b(i), in the absence of cubic anisotropy, a change of sign of the AEI from positive to negative with decreasing temperature would induce a helical reorientation from the 1 0 0 to 1 1 1 directions. In order for the orientation of the helical wavevector to remain aligned along the 1 0 0 directions(consistent with experimental observations), we find that K must be negative, with the condition |K| > 4 3 |γ|q 2 . This requires that the magnitude of K increases with decreasing temperatures. This increase in |K| at low temperatures is also in agreement with previous work discussing the origin of the low temperature skyrmion state [33].
In conclusion, we performed SANS on a single crystal of (Cu 0.98 Zn 0.02 ) 2 OSeO 3 , using a novel 3D vector magnet to decouple the anisotropic interactions within the material by investigating the behaviour of the magnetic conical state with a rotating magnetic field of constant magnitude. We observed a change of behaviour of the magnitude of the conical wavevector, q, as a function of wavevector angle, φ, whereby the crystal directions corresponding to a maximum or minimum in q reversed with cooling from 50 K to 12 K. We have explain this using a mean-field model and find that the AEI changes sign between these temperatures, with fitted values consistent with those required for tilted conical state formation. Unlike in the related compound FeGe, helical reorientation does not occur within Cu 2 OSeO 3 with the change in sign of the AEI. This is due to an increasingly negative cubic anisotropy, which increases in magnitude faster than the AEI, causing the helical wavevectors to remain along the 1 0 0 directions. We believe our finding that the AEI changes sign at similar temperatures to the occurrence of the LTS and TC magnetic phases will be highly useful for understanding the formation and stabilisation of these newly discovered magnetic textures within this material.