Direct observation of cycloidal spin modulation and field-induced transition in N\'eel-type skyrmion-hosting VOSe$_2$O$_5$

We investigate the spin rotational structure of magnetic skyrmions in a tetragonal polar magnet VOSe$_2$O$_5$ via polarized small-angle neutron scattering (SANS). Spin polarization analysis of the scattered neutrons reveals the cycloidal spin modulation in all the incommensurate phases at zero and non-zero magnetic field along the $c$ axis, providing unambiguous evidence for N\'eel-type skyrmions in this system. In the vicinity of the triangular skyrmion-lattice phase, extensive SANS measurements unravel a field-induced spin texture renouncing square skyrmion-lattice state, suggesting the relevance of thermal fluctuations to the topology of spin textures in this polar magnet.

Magnetic domain walls (DW), chiral soliton-lattices, and skyrmions are spatiallyvarying noncollinear and/or noncoplanar arrangement of spins in magnets characterized by nontrivial winding number in 1D or 2D space [1][2][3]. These spin structures, when isolated in ferromagnetic background, can be viewed as a particle, and mobile via strong coupling with conduction electrons, hence being promising candidates for current-driven information carriers in spintronics devices [4,5]. These structures are characterized by spin helicity  which represents how the magnetic moments are twisted with respect to the spatial coordinates [1]. Spin axes rotate in the plane parallel to the propagation vector (q vector) in a Néel-type DW/skyrmion with = 0, and π (Figs. 1(a, c)), which is in contrast with the Bloch-type spin configurations for = ± π 2 (the signs define the handedness), where the spin rotation plane is perpendicular to the q vector ( Figs. 1(b, d)).
Recent studies have revealed a significant role of  in the current-induced motion of DW/skyrmions through spin-orbit torques and spin-transfer torques [5][6][7][8][9][10]. These features provoke pressing interest in experimental techniques to determine  of these topological spin solitons.
Since the discovery of skyrmion-lattice (SkL) state in chiral magnets [11,12], recent experimental efforts have revealed that various classes of materials including bulk compounds [13] and thin-films/heterostructures host skyrmions with different helicities as well as vorticities [14]. Internal degrees of freedom are mostly selected by Dzyaloshinskii-Moriya interaction [15,16] imprinted from the crystal structure with broken inversion symmetry. Recent studies have discovered that these materials harbor versatile magnetic phase diagrams including both equilibrium and metastable skyrmions with different lattice forms such as triangular and square SkL states [17][18][19][20][21]. These rich features are generated from subtle balance among thermal fluctuations, Zeeman energy, magnetic anisotropy, and perhaps multi-spin exchange interactions [17,[22][23][24][25][26]. Development of widely applicable methods to determine the spin configurations as well as the SkL form is essential for understanding the link among spin textures and their stability.
In order to determine the orientations of the magnetic moments composing skyrmions and to monitor the phase transitions in external fields, multiple experimental techniques are available, including Lorentz transmitting electron microscopy (LTEM) [12,27,28], spin polarized scanning tunneling microscopy (SPSTM) [17,29], spin-polarized low-energy electron microscopy (SPLEEM) [30], nitrogen-vacancy (NV) center based microscopy [31,32], and circular-dichroism in magnetic x-ray scattering [33,34]. Although LTEM has been well established to detect Bloch-type DW/skyrmions, it has only a limited sensitivity to Néel-type configuration due to geometrical condition for the electron beam with respect to magnetic moments [27,28]. The other techniques have been proved effective to distinguish the types of skyrmions and their helicities while they are mostly available at, or sensitive to the surfaces of specimens.
We here focus on polarized small-angle neutron scattering (PSANS), which distinguishes the helicity of spin textures, and is broadly applicable to monitor the magnetic phase transitions in bulk compounds [35]. Up to now, Bloch-type magnetic modulations in magnetic fields were partly confirmed in the previous study with MnSi [36]. As for Néel-type skyrmions, previous PSANS study in polar GaV4S8 [37] reported the evidence of cycloidal spin modulation in magnetic phases, while the intensity for skyrmions is hard to separate from that for the multi-domain state of the single-q cycloidal phase. Direct observation of Néel-type skyrmions via PSANS has remained to be established.
In this paper, we report the PSANS observation of the triangular-lattice state of the Néel-type skyrmion in VOSe2O5. This material forms a square lattice of magnetic vanadium ions (V 4+ : S = 1/2) ( Fig. 1(e)) belonging to a unique tetragonal polar group (space group: P4cc) [38][39][40], and has recently been identified to host the SkL state via unpolarized SANS [41]. Magnetic phase diagram for the magnetic field (H) along the c axis is reproduced in Fig. 1(f), which contains incommensurate spin states for cycloid (IC-1), triangular SkL (A), and IC-2 states together with paramagnetic (PM) and field-induced ferrimagnetic (FM) states. The spin structure for IC-2 state was provisionally assigned to the square SkL state based on the theories [22,23] and the observed four-fold SANS pattern in terms of the in-plane magnetic modulation [41]. We performed further experiment to address this hypothesis, and find that the square SkL is to be excluded, instead, an anisotropic double-q [22,42] structure is newly proposed here. The mechanism to stabilize this spin texture is also discussed.
Polarized and unpolarized SANS measurements were performed at KWS-1 with the previously used assembled single crystals [41]. Here, we introduce a Cartesian coordinate xyz as shown in the schematic experimental configuration of Fig. 1(g). The sample was mounted in a closed-cycle 4 He refrigerator, so that the crystallographic a, b, and c axes were parallel to the x, y, and z axes, respectively. An incident spin-polarized neutron beam was obtained by a supermirror polarizer. The direction of incident neutron spins was controlled to be parallel or antiparallel to the z axis by a spin flipper, and was maintained by guide fields. The wavelength of the incident neutron beam was tuned to 10 Å. We employed polarized 3 He spin filter to analyze the spin-flip (SF) and non-spin-flip (NSF) scattering [43,44]. Flipping ratio and transmission were 20 to 15 % and 17 to 9 % at maximum to minimum, respectively. The decay time of the 3 He cells used was approximately 100 hours, which was monitored through the transmission of the direct neutron beam. Two different cells were used, J1 and Puck, which contain 5.2 and 5.0 bar cm of 3 He, respectively [45,46]. To extract the magnetic correlations below TC = 7.5 K, the spectrum measured above TC, typically at 8 K, was chosen as background. Imperfection of beam polarization was measured periodically and corrected following Refs. [47,48]. The H on the sample was produced by a Helmholtz coil.
Here, we describe the spatially-modulated magnetization with a wave vector q as (1) Here, we define ( ) as the Fourier transformation of the ( ) and ⊥ ( ) as the projection on the plane perpendicular to the Q, i.e., where ̂ is the unit vector parallel to . The unified expression for the cycloidal and proper-screw spin configurations propagating along the x axis ( Figs. 1(a,b)) is given by Note that the component does not contribute to ⊥ ( ). By substituting Eq. (3) into Eqs. (1) and (2), one can find that the difference between cycloidal and proper-screw spin configurations are clearly seen in the SF , which is absent in the former, but present in the latter. NSF channel is produced by the component, and present in both cases [50].
These relationships are available to determine the type of helicity of a SkL state, which is described by the superposition of q vectors in the plane perpendicular to the neutron beam.
Figures 2(a) and 2(b) show the observed NSF and SF scattering, respectively, for the triangular SkL state in the A phase. Presence of intensity in NSF channel ( Fig. 2(a)) and absence of that in SF channel ( Fig. 2(b)) provide clear evidence for the Néel-type skyrmion state with  = 0 or π [51]. We also confirmed that the NSF channel reproduces the twelvefold intensity peaks at |q| = 0.048 nm -1 as observed in Ref. [41]. This is due to the coexistence of two domains of triangular SkL state with one of the q vectors locked along either a or b axis ( Fig. 2(g)). This feature enables us to observe the skyrmion state in the present system separably from the single-q cycloidal state, which produces four-fold Bragg peaks (Fig. 2(c)). As for GaV4S8 [37], on the other hand, the SANS intensity for the ( ) = cos( )̂+ ( coŝ+ sin̂) sin( ).
skyrmion state isotropically distributes for the q vector in the plane perpendicular to the trigonal crystal axes to form a ring-like scattering pattern. Due to this situation, it was difficult to exclude the possibility of the multidomain of cycloidal state by the PSANS measurement alone. Note that the observation of the NSF channel in the A phase indicates the mz component, excluding the multiple-q state with spins lying in the ab-plane such as being recently reported for polar YCo8Sn4 [52].
We also determined the type of helicity of the spin modulations in the IC-1, and IC-2 state. Figure 2(c) shows the intensity pattern for the NSF channel in the IC-1 state, reproducing that in the unpolarized SANS study [41], i.e., four-fold peaks for q||a* and q||b* superposed with weak ring-like intensity due to diffusive q-domains. This indicates that two kinds of domain with qa and qb dominates as shown in Fig. 2(h). Absence of the intensity in the SF channel ( Fig. 2(d)) confirms the cycloidal spin configuration in the IC-1 state as assigned in Ref. [41]. We also proved the cycloidal spin configuration in the IC-2 state by the presence and absence of the PSANS intensity for NSF ( Fig. 2(e)) and SF channel ( Fig. 2(f)), respectively. The IC-2 state was previously proposed as the squarelattice of the Néel-type skyrmions. This hypothetical spin texture is expressed by the equivalent superposition of two cycloidal spin modulations propagating along a and b axes, respectively. We did not confirm whether the magnitude of these two modulations are equivalent or contain some sort of anisotropy because the possible multidomain state of the latter would be indistinguishable from the former. In the latter case, the spin texture is not the square SkL state, but is rather termed an anisotropic double-q state proposed in theoretical works [22,42]. Schematic multidomain configuration is depicted in Fig. 2(i).
Diffusive q-domains as observed by the ring-like pattern in Fig. 2(e) are not shown here.
In order to investigate the multiple-q nature for the IC-2 phase, we monitor the field dependence of the unpolarized SANS intensity through the phase transition from IC-1 state to IC-2 state for H||c at T = 6.67 K. Prior to the measurement, we rotated the Helmholtz coil in the horizontal (zx) plane to apply H along the x (a) axis. The single domain of the IC-1 state with q||b* (Fig. 3(a)) was selected due to the anisotropy of the Zeeman energy for the cycloidal spin configuration [41]. After eliminating the H, we rotated the coil back to H||z(c) configuration. We subsequently applied the field along the c axis to take the minor loop as shown in Figs. 3(a)-(h). We find that even at 0H = 5 and 6 mT (Figs. 3(cd)), in the center of the IC-2 phase (Fig. 1(f)), the SANS intensity for q||a* remains considerably weaker than that for q||b*. This excludes the formation of the square SkL state in the IC-2 phase. We note that at the end of the minor loop ( Fig. 3(h), 0H = 0 mT) the intensity for q||a* shows up, suggesting the multidomain formation for the IC-1 state.
For more quantitative perspective, we plotted the field dependence of the averaged wavelength  (= 2 ) (Fig. 4(a)) and the integrated intensities for the respective vertical and horizontal areas ( Fig. 4(b)). As increasing the field,  increases and shows a kink at around 2.5 mT signaling the transition into the IC-2 state. We find that  shows a hysteresis in the backward scan for the minor loop (open symbols in Fig. 4(a)) only in the IC-2 region. This reflects the different spin textures in IC-1 and IC-2 state separated by a first-order phase boundary. Furthermore, the intensity for the horizontal direction (q||a*) appears in the IC-2 state as shown in Fig. 4(b). This feature for the IC-2 state is hard to be reconciled with the single-q model as the IC-1 state.
Here, we propose an anisotropic double-q state for the spin structure of the IC-2 state.
This was initially proposed as a stable spin texture in a chiral magnet with tetragonal single-site anisotropy and/or compass-type anisotropic exchange interaction [22,42].  Fig. 4(b)). Inequality of these intensities can be ascribed to the remnant domain for the IC-2 state, which does not lose the memory from the initial single domain state. Eventually, the system comes back to the ( , ) = (sin , 0, cos ) + (0, cos , sin ), IC-1 state (5) in zero field with contamination of the domain with qa, as observed in Fig.   3(h).
The anisotropic double-q model can qualitatively explain the observation as mentioned above, while we point out a deviation from the theories [22,42], where the double-q state is predicted to continuously connect to zero field. The IC-2 state is stable only in a nonzero field ( Fig. 1(f)). We hypothesize that this difference would originate from the thermal fluctuations, which were not considered in the previous theoretical studies. Square SkL state is supposed to position, if present, at relatively low temperatures from the fact that the corresponding square lattice states of the Bloch-type skyrmions have been observed only as the metastable state close to the zero temperature [19,20] or thermodynamically stable only at low temperatures [21]. In VOSe2O5, on the other hand, easy-plane type anisotropy destabilizes the spin modulation below 4 K [41]; this perhaps makes the square SkL state difficult to form in the present compound. To clarify this point, further theoretical investigation considering a delicate effect of thermal fluctuations would be desirable.
In conclusion, we directly determined the orientation of the magnetic moments composing of the SkL state via PSANS measurement to confirm the Néel-type skyrmions in the tetragonal polar magnet VOSe2O5. We also obtained the evidence that the spin configuration for the other modulated magnetic phases surrounding the SkL phase, i.e., the IC-1 and IC-2 states, are both described with the cycloidal modulations. On the basis of the field evolution of the SANS pattern from the IC-1 (single-q cycloid) to the IC-2 states, we proposed the anisotropic double-q state for the IC-2 phase. The present study provides a useful guiding principle to apply this procedure to determination of complex or multiple helical modulations anticipated to occur in skyrmion-hosting materials.