Giant anomalous Hall effect from spin-chirality scattering in a chiral magnet

The electrical Hall effect can be significantly enhanced through the interplay of the conduction electrons with magnetism, which is known as the anomalous Hall effect (AHE). Whereas the mechanism related to band topology has been intensively studied towards energy efficient electronics, those related to electron scattering have received limited attention. Here we report the observation of giant AHE of electron-scattering origin in a chiral magnet MnGe thin film. The Hall conductivity and Hall angle, respectively, reach \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$40,000$$\end{document}40,000 Ω−1 cm−1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$18$$\end{document}18% in the ferromagnetic region, exceeding the conventional limits of AHE of intrinsic and extrinsic origins, respectively. A possible origin of the large AHE is attributed to a new type of skew-scattering via thermally excited spin-clusters with scalar spin chirality, which is corroborated by the temperature–magnetic-field profile of the AHE being sensitive to the film-thickness or magneto-crystalline anisotropy. Our results may open up a new platform to explore giant AHE responses in various systems, including frustrated magnets and thin-film heterostructures.

, due the uniaxial anisotropy from the substrate. The consequence spin texture is the hedgehog lattice with a dilute density of hedgehogs and anti-hedgehogs. c, The effect of the filmthickness on the direction of the q-vectors revealed by the SANS experiment S6 . is defined as the angle between the q-vectors and the film-normal direction, which is reduced with decreasing the film-thickness. This suggests the fact that the in-plane anisotropy of spin is enhanced in thinner films. d, The magnetic phase diagram of the MnGe thin film (t = 160 nm), which consists of the ferromagnetic (FM) phase, helical state, and hedgehog lattice state S6 .
Magnetic structures of topological spin crystals, such as skyrmion-and hedgehog-lattice states, can be described by the superposition of spin spirals with propagation vectors q, i.e., the multiple-q states S1,S2 . Therefore, the small-angle neutron scattering (SANS) experiment has been one promising way to identify the topological spin texture, by detecting the q-vectors in reciprocal space. In bulk MnGe, three orthogonal q-vectors along <100> crystal axes have been observed by the SANS experiment S2 as well as in the real-space by Lorentz TEM S3 . The consequent spin texture becomes the dense array of spin hedgehogs and anti-hedgehogs connected by the skyrmion strings (i.e., hedgehog lattice) ( Supplementary Fig. 1a), which serve as the source (monopole) and sink (anti-monopole) of the emergent magnetic field, respectively S4,S5 . Meanwhile, in the thin films of MnGe, the presence of uniaxial anisotropy modifies the mutual angles of the three q-vectors, resulting in the hedgehog lattice with dilute monopole/anti-monopole density S6 (Supplementary Fig. 1b). In specific, the q-vectors are tilted to the film-normal direction from the <100> direction, showing the easy-plane anisotropy of spins in thin films ( Supplementary Fig. 1b).
Moreover, this in-plane anisotropy can be enhanced by decreasing the film-thickness ( Supplementary Fig. 1c). The magnetic phase diagram of the MnGe thin film is shown in Supplementary Fig. 1d, where the hedgehog lattice is stabilized below 50 K and 4 T.
It should be noted here that recent STM study observed the multi domains with differently-oriented helical structures at the surface of a MnGe thin film S7 . Since the internal magnetic structure has not been directly observed, we cannot know whether the multi-domain state may be maintained into the inside of the film or the multiple helical structures may interfere with each other and form a hedgehog lattice beneath the surface.
Nevertheless, the observed multi-domain state also contains chains of hedgehogs and anti-hedgehogs at the disclination lines where helical structures with different propagation directions meet.
In either case, the main discussions on the SSC excitations and the consequent skew scatterings in the ferromagnetic region are unaffected by the incomplete understanding of ground magnetic state. The temperature dependence of = 1⁄ at zero magnetic field for the film thickness of 80 nm and 300 nm is shown in Supplementary Fig. 3a. The sample quality seems to become better (i.e., the residual resistivity decreases) with increasing the film thickness in MnGe. The magnetic field dependence of and are summarized in Supplementary Figs. 3b,c. Importantly, the maximum value of the Hall angle (~ 20 %) is nearly independent of the film-thickness, suggesting the underlying common mechanisms (i.e., spin-chirality skew-scattering).
We also note that the Hall angle is the intrinsic quantity, which is independent of the relaxation time or the sample quality in the framework of skew-scattering mechanism.
Therefore, we have discussed the variation of the Hall angle against the film thickness as shown in Fig. 3 of the main text. Since the normal Hall effect satisfies the relation = 0 2 , where 0 = 1⁄ is the normal Hall coefficient, the scaling relation of ∝ 2 is expected under fixed . As shown in Supplementary Fig.4, however, plotted against with varying temperature (T = 2-40 K ) shows the complex behavior, and does not follow any kind of scaling relations (i.e., ∝ 2 for the normal Hall effect and ∝ for the conventional skew-scattering). We note that this result rather corroborates the spinchirality skew-scatteirng mechanism, where the SSC excitation responds sensitively to the temperature and magnetic-field variation. If we take the data points where the Hall angle becomes the maximim at each temperature, the linear scaling relation ∝ holds true irrespective of the film-thickness or sample quality (see Fig. 2a in the main text). This is because the Hall angle reflects the magnitude of the SSC excitation (i.e. tilting angles of spins or density of monopole/anti-monopole exciation), and hence the data points used in Fig. 2a may share the same situations regarding the SSC excitation.

Supplementary Note 5: The analytical calculation for the thermal excitation of scalar-spin chirality (SSC).
We have reproduced the overall B-T profile of the SSC excitation by performing analytical calculations. For the low-temperature region, we describe the thermal fluctuation in terms of the low-energy excitations. At higher temperatures, we calculated the SSC using the high-temperature expansion of a chiral magnet. Based on these specific examples, we discuss the general behavior irrespective of model parameters.
• Low-temperature region The low-energy excitations of the chiral magnets are spin-wave and the monopole/anti-monopole pair (skyrmion string) excitations. The existence of the lowenergy skyrmion excitation owes to the fact that the spin hedgehog lattice phase exists in the vicinity of the field-forced ferromagnetic phase. The phase transition between the spin hedge hog lattice and the ferromagnetic orders are described by the condensation of skyrmion strings. Therefore, skyrmion strings exists as low-energy excitations even in the ferromagnetic state. In the continuum model, the skyrmion string excitations contribute to the scalar spin chirality while the spin wave contribution vanishes in the linear order.
Therefore, we focus on the skyrmion string excitations in the rest of this section.
We assume a monopole/anti-monopole string with energy where 0 (1 ∓ ) is the energy of the monopoles and anti-monopoles for skyrmion (+) and anti-skyrmion (-) strings, is the length of the skyrmion string connecting the monopole and anti-monopole, ℎ is the magnetic field, (1 ∓ ) is the energy of skyrmion/anti-skyrmion string per unit length (we assume the string runs parallel to the magnetic field), 2 is the energy difference between the skyrmion and anti-skyrmion strings. The energy difference reflects the difference of skyrmion and antiskyrmion spin textures. Microscopically, it is related to DM interaction and the magnetic field. We neglect the interaction between the monopoles because they are negligible if the monopole density is sufficiently small. Under these assumptions, the distribution function ℎ is introduced for a technical purpose which will be clear in the following. We note that the transformation from the second to the third line holds only when The chirality of the ferromagnetic phase is proportional to the density of the skyrmion strings. Therefore, ) .
The result of the SSC as a function of temperature at various magnetic fields is shown in Supplementary Fig. 5a, which is consistent with the experimental result in the low-temperature limit shown in Fig. 2c of the main text. SSC is zero at zero temperature because it is the ferromagnetic state. At a finite temperature, the chirality becomes nonzero due to the imbalance of the population between skyrmion and anti-skyrmion strings.
• High-temperature region The description based on the skyrmion/anti-skyrmion strings fails at a high temperature because of the weak spin correlation. Therefore, we employ the hightemperature expansion to discuss the B-T profile of the SSC in the high-temperature region ≫ ( is the energy scale of the ferromagnetic exchange interaction).
For this purpose, we here consider an fcc lattice model with weak Dzyaloshinskii-Moriya interaction ( Supplementary Fig. 5b), Here, ̂[ 111] is the unit vector along the [111] axis. When 2 = = 0, this model is an fcc version of the effective model often used for MnSi S8 . We also note that a related twodimensional model has been studied by Leonov et al S9 ., which finds a skyrmion crystal phase at = 0.
We first focus on the = 0 case. In this case,

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The result is the same for the downward triangles shown in Supplementary Fig. 5b.
The result of the SSC as a function of temperature at various magnetic fields is shown in Supplementary Fig. 5c, which is consistent with the experimental result of the high temperature shown in Fig. 2c of the main text. The SSC increases with increasing the magnetic field, which is related to the fact that the terms increases with ℎ, such as 〈( ) 3 〉 = We note that this is a general trend for the models with a triangular network of magnetic ions. Therefore, we expect a larger chirality at higher magnetic field, when ≪ 1.
The expected overall behavior of the SSC connecting the low-temperature and hightemperature regions is shown in Supplementary Fig. 5d, showing a peak structure at a finite temperature as observed in the experiment.
• Effect of easy-plane anisotropy We consider the effect of anisotropy on the SSC excitation in this section. The singleion anisotropy is incorporated in the thermal average term 〈⋯ 〉 as 〈⋯ 〉 ≡ ∫ ∅ sin ℎ cos + 2 (cos ) 2 (⋯ ) .
Therefore, the formula in Supplementary Eq. (13) holds with only difference in the definition of the thermal average. The leading-order effect from the anisotropy appears in the linear order of , which reads Physically, the easy-plane anisotropy cause frustration between the magnetic field; the anisotropy tries to keep the spins in plane while the magnetic field prefers to point the spins in the perpendicular direction. The competition of anisotropy and magnetic field enhances the noncollinearlity. This observation is consistent with the results discussed in with a sharp peak structure ("dispersive-resonance" profile), can arise from the normal Hall effect when the carrier mobility is sufficiently large, as typically observed in Dirac or Weyl semimetals S12 . Here, the profile of follows the where the peak position (magnetic field) is determined by the inverse of ( Supplementary Fig. 7a). In the case of MnGe, one possibility is that the ferromagnetic phase transition entails the emergence of a highmobility carrier pocket, such as the magnetic Weyl points. Although MnGe is a multiband system with a relatively large carrier density, the existence of one high-mobility pocket might dominate the low-field transport, resulting in the "dispersive-resonance" profile. Therefore, we first tried to fit with a single-carrier Drude model while fixing the peak position, assuming that the peak structure were produced by such high-mobility carriers. As shown in Supplementary Fig. 7b, the observed largely deviates from the Drude curve. The large enhancement of the Hall response at low temperature is observed only in the thin-films and bulk single-crystals S13 , but NnT in bulk poly-crystals of MnGe S4 ( Supplementary Fig. 9). Because the value of , which reflects the carrier density or mobility, is almost identical between thin-films ( = 2.0 × 10 5  − cm -1 at 2 K) and poly-crystals ( = 1.6 × 10 5  − cm -1 at 2 K), this result suggests that the observed large Hall response may NnT be dominated by the normal Hall effect.
In terms of the spin-chirality skew-scattering mechanism, we speculate that the presence of random crystal domains in poly-crystals may weaken the total SSC excitation.