Skyrmion phase and competing magnetic orders on a breathing kagomé lattice

Magnetic skyrmion textures are realized mainly in non-centrosymmetric, e.g. chiral or polar, magnets. Extending the field to centrosymmetric bulk materials is a rewarding challenge, where the released helicity/vorticity degree of freedom and higher skyrmion density result in intriguing new properties and enhanced functionality. We report here on the experimental observation of a skyrmion lattice (SkL) phase with large topological Hall effect and an incommensurate helical pitch as small as 2.8 nm in metallic Gd3Ru4Al12, which materializes a breathing kagomé lattice of Gadolinium moments. The magnetic structure of several ordered phases, including the SkL, is determined by resonant x-ray diffraction as well as small angle neutron scattering. The SkL and helical phases are also observed directly using Lorentz-transmission electron microscopy. Among several competing phases, the SkL is promoted over a low-temperature transverse conical state by thermal fluctuations in an intermediate range of magnetic fields.

constant-Q z cuts of the neutron scattering data, which approximately correspond to intensity distributions in the (H, K, 2) plane. Note that the origin of the Q x -Q y plane is shifted to the half-way point between the two (0, 0, 2) reections attributed to dierent crystallographic domains. In this conguration, energetic considerations lead us to expect that the helical domain with q H will grow in volume as the eld is applied. We follow the incommensurate scattering intensity at three magnetic reections starting from the zero-eld cooled state at T = 2.4 K. The relevant reections are marked by white circles in Supplementary Fig. 2 (a-g). The integrated scattering intensity at these reections, shown in Supplementary Fig.   2 (i), exhibits the expected enhancement of intensity for q H, as well as nite hysteresis of the intensities after cycling back to zero eld. This hysteretic behavior indicates that the zero-eld ground state is multi-domain. We note that again, additional reections appear in  (c), Sketch of scattering geometry as used in this experiment, exploiting the high-angle detector bank at TAIKAN for the in-plane magnetic eld conguration. k i and k f are the wave vectors of incoming and outgoing beams, respectively. a * and b * label directions in reciprocal space. Supplementary Fig. 3 (a,b) shows observed and calculated neutron Laue diraction patterns at T = 2.4 K in zero magnetic eld using the high-angle detector bank of TAIKAN ( Supplementary Fig. 3 (c)). Magnetic reections were absent at the (q, 0, (2n − 1)/2) and (q, 0, 2n − 1) positions (n = 1, 2), suggesting that the magnetic modulations in neighboring kagomé layers are coupled in-phase. This analysis conrms that in Gd 3 Ru 4 Al 12 , antifer-romagnetic coupling between bilayers is not present in the ground state. Note that in a recent neutron scattering experiment for the sister compound Dy 3 Ru 4 Al 12 , commensurate q = (1/2, 0, 1/2) order was rened in the zero eld ground state, indicating a doubling of the magnetic unit cell along the c-axis [6]. In the case of Gd 3 Ru 4 Al 12 , we start from an eective spin model and search for the ground state conguration under the mean-eld approximation. Such a Hamiltonian has the well-known general form (assuming Heisenberg moments) where S i are local moment spins, J ij are coupling constants, and the sum runs over all pairs of local moments at lattice sites i, j. In our quest for a minimal eective model reproducing the experimentally observed q-vector in this structure, we found that nearest neighbor J 1 = −1 (ferromagnetic coupling on Gd triangles, T CW > 0), J 2 = 0.25, and J 7 = 0.125 yield the correct magnitude q = 0.275 r.l.u., but aligned along the a-axis as q = (0.167, 0.167, 0) (Sup- The normal Hall eect is estimated by the black dashed line. (f), Identical analysis for another sample, which was cut so that H a * could be applied. The temperature dependence of the Hall eect is much weaker in this geometry, due to smaller value of the AHE for in-plane magnetic eld.
(g), Longitudinal electrical conductivity σ xx is quadratic at high magnetic eld, where the magnetic moments are roughly co-aligned. Dashed line is a quadratic t to the data above µ 0 H = 11 T, from which estimates of carrier mobility µ and carrier density n were obtained. out at the onset of long-range magnetic order. We observed weak hysteresis below T N 1 between ρ xx (T ) recorded with increasing and decreasing temperature, suggesting that this boundary between sinusoidal and helical states is weakly rst order. This is consistent with weak hysteresis in the M − T curves ( Fig. 1 (e)). We could not resolve hysteresis in the measured observables at T N 2 . In Supplementary Fig. 5 (c,d), we show raw data of ρ xx (H int ) and the Hall resistivity ρ yx (H int ), from which the Hall conductivity traces σ xy (H int ) of Fig.   2 (b) were calculated according to σ xy = ρ yx /(ρ 2 xx + ρ 2 yx ). Strong changes of the carrier lifetime, related to a series of magnetic transitions, aect these observables: While ρ xx deep inside the transverse conical state is comparable to the zero-eld ground state, it is possible that domain wall scattering and phase mixing around the metamagnetic critical elds cause enhanced elastic scattering. The high value of ρ xx inside the fan state, similar in magnitude and shape to the paramagnetic state, is notable as well. These changes of the carrier mobility are also reected in the Hall resistivity. For example, in the case of a purely intrinsic mechanism for the anomalous Hall conductivity, we have σ AHE xy = S H M and ρ AHE yx ∼ σ AHE xy · ρ 2 xx (limit of small Hall angle, ρ xx ρ yx ) [10]. The resulting non-linearities of ρ yx complicate the separation of the topological Hall resistivity ρ THE yx arising from the scalar spin chirality of magnetic skyrmions in the SkL phase. We address this issue by using σ xy to subtract the smooth background and isolate the topological signal, instead of directly analyzing ρ yx .
In order to discuss the normal Hall coecient R 0 , we consider the temperature dependent Hall resistivity for two samples of dierent geometry in Supplementary Fig. 5. These samples were cut to be thin plates. Consider a Cartesian coordinate frame where the x − y plane is parallel to the face of the plate, and the current density is J x. The magnetic eld is H z. Sample 2 was cut with x = a * and y = b (H c), while we prepared sample 7 with x = a and y = c (H −b * ). Supplementary Fig. 5 (e,f) shows ρ yx (T )/(µ 0 H), a quantity which emphasizes the onset of non-linear Hall resistivity by a divergence of curves measured at dierent values of the magnetic eld. This point of departure is found at lower T in sample 7 (H −b * ), where the anomalous Hall eect (AHE) is suppressed as compared to the case of H c. We infer that R 0 , which should be comparable for the two sample geometries in the case of a roughly three-dimensional Fermi surface, and which is expected to dominate at T = 300 K, is only weakly dependent on T . In the case of H c, the strong temperature dependence of ρ yx at µ 0 H int > 3 T thus arises mostly due to the AHE from spin-orbit coupling, and possibly from scalar spin chirality on isolated Gd plaquettes [2]. As expected in the framework of the intrinsic mechanism, which has ρ AHE yx strongly dependent on the scattering time τ [10], the AHE dies o at the lowest temperatures. The nature of the high-eld AHE in R 3 Ru 4 Al 12 (R: rare earth) and its anisotropy will be the subject of a future study. Exploiting the weak temperature dependence of R 0 , we use the extrapolated Hall coecient R 0 = 0.2 µΩcm/T (dashed line in Supplementary Fig. 5 (e)) in our analysis of the emergent magnetic eld B em of the SkL (c.f. main text).
What follows is an order-of-magnitude estimation of charge carrier densities and carrier mobilities in Gd 3 Ru 4 Al 12 . Starting from the semiclassical Drude model [11], we assume very crudely that the various bands may be described by an eective mobilityμ, so that (electron charge e, magnetic eld B = µ 0 H, carrier densities n j , where the index j runs over all partially occupied bands) σ xx = eμ j |n j | /(1 + (μB) 2 ) and σ xy = eμ 2 B j n j /(1 + (μB) 2 ).
Expanding these expressions to rst order inμB 1 and using a quadratic t to σ xx (H) at µ 0 H > 11 T, T = 2 K we obtainμ = 311 cm 2 /(Vs) and j |n j | = 1.3·10 21 cm −3 . Let us crosscheck using the Hall resistivity: We have ρ yx = σ xy /(σ 2 xy and experimentally ρ yx /(µ 0 H) ∼ 0.20 µΩcm/T at T = 2 K and high magnetic eld ( Supplementary Fig. 5 (e)). The demagnetization correction is not included in these backof-the-envelope estimates. We arrive at j n j = 5.5·10 20 cm −3 ; provided the approximations made in the derivation, we cannot draw a conclusion about the number and character of bands contributing to transport. However, weak temperature dependence of R 0 is in principle consistent with conduction from a single carrier type, as is the relatively low absolute value of the carrier density (c.f. 8.5·10 22 cm −3 for Cu). Under the assumption of a single band with spherical Fermi surface, the Fermi wave vector is estimated as k F = (6π 2 n/d s ) 1/3 = 3.4 nm −1 , in agreement with the analysis of Ref. [2]. Here, d s = 2 is the spin degeneracy of the band, and n is the total carrier density. Supplementary Note 6.

PHASE
We have reproduced the topological Hall eect and transport data for a second sample of Gd 3 Ru 4 Al 12 from the same growth, with slightly dierent sample geometry (J a, H c for sample 8 instead of J a * , H c for sample 2). The results are in good qualitative agreement ( Supplementary Fig. 6), although the boundary between the TC and SkL phases is located at slightly lower temperature in sample 8. Pinning from defects may aect the exact character of the rst-order boundary between these two eld-induced phases.
7.  Fig. 2 (a). For T = 2 K in (d), several signatures associated with phase boundaries in Fig. 2 Fig. 7 (c)). For the SkL phase, we can conclude that it is bounded by rst order phase transitions on all sides. In addition to the magnetic phase boundaries discussed in the main text, we here mention two further anomalies: First, a weaker peak in χ DC is present at elevated temperatures (e.g. T = 10 K in Supplementary Fig. 7 (d)) within the boundaries of the SkL phase. This signature is labeled as H 2 in Supplementary Fig. 7 (d) and indicated by solid, small green symbols in the phase diagrams of Supplementary Fig. 7 (g,h). This may be related to a possible, yet at the time of writing unconrmed, subdivision of the SkL phase. Furthermore, a shoulder emerges in χ DC (H int ) for T = 2 K, H c ( Supplementary Fig. 7 (d), third red arrow from the right hand side). Magnetoresistance data shows a kink at the same value of the internal magnetic eld H 4 , and we can track this critical eld up to about T = 7 K, where it grows very weak. We have marked H 4 with small, solid blue circles in Supplementary Fig. 7 (g,h).
As we were unable to determine H 4 at high temperature, it remains unclear whether this eld scale corresponds to a thermodynamic phase boundary or merely to a cross-over eld.
Finally, we note that the signatures that characterize the transition between the fan state and phase V are rather weak as compared to other phase transitions. This transition is most readily observed in the M − T curves, while it leaves no obvious trace in transport experiments. We have also found that signatures associated with phase V are sensitive to disorder. For example, our 160 Gd enriched sample with enhanced impurity concentration did not show any anomalies for phase V, while phases H, TC, SkL, and F were clearly identied.
Similarly, phase V was not observed in thin-plate samples used for TEM measurements.
Note that the thin-plate preparation process involves mechanical thinning.