MnGe thin film with topological spin texture

Here we report the observation of a giant AHE in a chiral magnet MnGe thin film, where the giant Hall angle ($$ 18$$%) leads to a Hall conductivity reaching $$ {\sigma }_{xy}~40,000$$ Ω⁻¹ cm⁻¹, being two orders of magnitude larger than the intrinsic AHE, even with a moderate longitudinal conductivity ($$ {\sigma }_{xx}~2\times 10^5$$ Ω⁻¹ cm⁻¹). The target material is the epitaxially grown thin films of MnGe²², a member of the B20-type chiral magnets hosting topological spin textures^23,24. The crystal structure belongs to the non-centrosymmetric space group P2₁3, where the lattice chirality is characterized by the stacking direction of the atoms as viewed from [111] axis (only one enantiomeric form is shown in Fig. 1a). Since the lack of inversion symmetry allows Dzyaloshinskii–Moriya interaction (DMI), twisted spin structures such as helical or non-coplanar structures are often observed in B20-type magnets^23,24. Among them, MnGe hosts unique magnetic textures composed of spin hedgehogs and anti-hedgehogs. In a bulk sample, those magnetic defects form a short-period lattice (~2.8 nm) with large magnetic fluctuations even persisting in the nominally ferromagnetic (FM) region²⁴. The hedgehog lattice is deformed to a rhombohedral form (Supplementary Fig. 1)²² or decomposed into multi-domains with differently oriented helical structures²⁵ in strained thin films (see Supplementary Note 1 for comment on the magnetic texture in thin films). In either case, there exist chains of hedgehogs and anti-hedgehogs connected by skyrmion-strings. Such a non-coplanar magnetic order produces a THE in metallic systems including MnGe ($$ {\sigma }_{xx}=2.0\times 10^5$$ Ω⁻¹ cm⁻¹ at 2 K) (Fig. 1b); the large negative THE below $$ T<50$$ K (Fig. 1c) is ascribed to emergent magnetic field from the skyrmion-strings (inset in Fig. 1c)¹⁴.

Fig. 1

Observation of the giant anomalous Hall effect (AHE) in MnGe thin film.

a B20-type chiral crystal structure (space group P2₁3) of MnGe. The chirality is characterized by the stacking direction of atoms as viewed from [111] direction. b Temperature dependence of conductivity ($$ {\sigma }_{xx}$$) at zero magnetic field for the film-thickness of t = 160 nm. c Magnetic-field dependence of Hall conductivity ($$ {\sigma }_{xy}$$) at various temperatures for t = 160 nm. The negative dip structures are attributed to topological Hall effect arising from the formation of spin hedgehogs and anti-hedgehogs bridged by skyrmion-strings (shown in the inset). d High-magnetic-field data of $$ {\sigma }_{xy}$$ at various temperatures for t = 160 nm, showing a large enhancement. The value of the quantization limit ($$ e^2/ha$$) in three-dimensions is denoted by a dashed line. The observed giant AHE far exceeds the value allowed by the intrinsic AHE.

Hall conductivity in the ferromagnetic region

The main focus of this work is on the high-magnetic-field Hall response shown in Fig. 1d, where $$ {\sigma }_{xy}$$ shows a striking enhancement reaching $$ ~40,000$$ Ω⁻¹ cm⁻¹ at the temperature $$ T=2$$ K, with a sharp peak structure at around the magnetic field $$ B=14$$ T (see Supplementary Fig. 2 for the resistivity data). Whereas $$ {\sigma }_{xx}=2.02\times 10^5$$ Ω⁻¹ cm⁻¹ at 2 K (Fig. 1b) belongs to the empirical intrinsic regime where the Berry curvature mechanism is dominant, the observed $$ {\sigma }_{xy}$$ far exceeds the threshold value of the intrinsic AHE from the momentum space, which is roughly estimated from the quantization limit ($$ ~e^2/ha=800$$ Ω⁻¹ cm⁻¹ for MnGe) in three-dimensions (denoted by the dashed line in Fig. 1d)¹⁷. Here, $$ a=4.795$$ Å for MnGe. Upon increasing the temperature, $$ {\sigma }_{xy}$$ is rapidly suppressed while showing a broader peak structure. Above $$ T~70$$ K, $$ {\sigma }_{xy}$$ follows the conventional behavior of the intrinsic AHE, which scales with the magnetization² (Supplementary Fig. 2).

Scaling relation and temperature–magnetic-field profile of the Hall response

To understand the characteristics of this large Hall response observed at low temperatures, we first plotted the data of $$ {\sigma }_{xy}$$ (Fig. 1d) against $$ {\sigma }_{xx}$$ (Supplementary Fig. 2) to investigate their scaling relation, which has been typically used to identify the mechanism of the AHE^2,16,17. Figure 2a shows the plot for various samples with different film-thicknesses ($$ t=80,160,$$ and $$ 300$$ nm) for T = 2–30 K (see Supplementary Fig. 3 for the complete data set). The displayed data points ($$ {\sigma }_{xx}$$, $$ {\sigma }_{xy}$$) are those exhibiting the maximum Hall angle at each temperature. In every sample, the scaling relation between $$ {\sigma }_{xy}$$ and $$ {\sigma }_{xx}$$ is linear ($$ {\sigma }_{xy}\propto {\sigma }_{xx}$$), which is a typical feature expected for skew-scattering mechanism (see Supplementary Fig. 4 and Supplementary Note 4 for detailed discussions). To overview this Hall response in the $$ B$$–$$ T$$ plane, the contour map of $$ {\sigma }_{xy}$$ is shown with the magnetic phase diagram in Fig. 2b. The striking enhancement of the Hall response appears at low temperatures below 50 K, in proximity to the FM phase boundary. The constant-$$ B$$ cut of $$ {\sigma }_{xy}$$ shows a peak structure at finite temperature, and the peak structure shifts to higher temperatures under higher magnetic fields (Fig. 2c).

Fig. 2

Characteristics of the AHE observed in MnGe thin film, suggesting a spin-chirality skew-scattering.

a Plot of Hall conductivity ($$ {\sigma }_{xy}$$) versus conductivity ($$ {\sigma }_{xx}$$) for the thickness of t = 80, 160, and 300 nm. The data are taken from the peak positions of the Hall angle below $$ 30$$ K. The linear relation ($$ {\sigma }_{xy}\propto {\sigma }_{xx}$$) appears in a certain $$ B$$–$$ T$$ region with a large Hall angle of $$ ~18\%$$. b Contour plot of $$ {\sigma }_{xy}$$ in the $$ B$$–$$ T$$ space with the magnetic phase diagram consisting of the hedgehog lattice (HL), helical, and the ferromagnetic (FM) phases. c Constant-$$ B$$ cut of $$ {\sigma }_{xy}$$ showing a peak structure at a finite temperature. The triangles denote the peak positions, which shifts to a higher temperature at a higher magnetic field. These results are suggestive of the relation to the thermal excitation of the scalar spin chirality (inset in Fig. 2c).

The above features of the skew-scattering in MnGe are distinct from the conventional skew-scattering induced by the non-magnetic chemical defects or single-spin impurity^2,16–21, while showing the similar linear ($$ {\sigma }_{xy}\propto {\sigma }_{xx}$$) scaling relation. In the conventional skew-scattering, the perturbatively small SOC compared to the electron bandwidth results in a small Hall angle of 0.1–1%. Also, $$ {\sigma }_{xy}$$ should be monotonously increased to lower temperature with higher $$ {\sigma }_{xx}$$. Hence, not only the giant Hall angle of 18~22% (Fig. 2a), but also the non-monotonous $$ B$$–$$ T$$ profile (Fig. 2b, c) are both beyond the conventional understanding of skew-scattering. One possibility for this unconventional skew-scattering is the recently proposed “spin-chirality skew-scattering” mechanism²⁶, where the thermally fluctuating spins in the FM state act as the spin-clusters with the SSC (inset in Fig. 2c). Hence, the magnetic scattering contributes to the AHE. In principle, the scattering process there does not involve SOC and the Hall angle is predicted to be significantly large compared to the conventional skew-scattering. In particular, the Hall angle in the order of $$ 10$$% is anticipated to show up when the size of the spin-cluster is comparable to the inverse of the Fermi-wave-vector (~$$ \frac{1}{k_{\mathrm{F}}}$$), which corresponds to the resonance condition of the interference effect²⁶. Moreover, the observed $$ B$$–$$ T$$ profile of the AHE is also consistent with the spin-cluster scenario. Under fixed magnetic field, the spin-cluster AHE is expected to be maximized at a finite temperature as shown in Fig. 2c. Two types of spin excitations with opposite SSC are thermally excited, being responsible for skew-scatterings with opposite Hall angles. Because those SSC excitations have different activation energies due to the presence of DMI, the cancellation of AHE signal only occurs above the temperature where the SSC excitations with the higher activation energy start to proliferate. Although the temperature dependence of the Hall signal under fixed magnetic field is consistent with our analytical calculations and Monte Carlo simulations (see Supplementary Figs. 5 and 6, and Supplementary Notes 5 and 6 for details), the reason why the Hall signal is enhanced at high magnetic field under fixed temperature remains elusive and needs further theoretical investigations. This may be attributed to the divergent spin correlation and the criticality around the FM transition. The former invalidates the short-correlation-length assumption used in the skew-scattering theory, while the latter may affect the transport phenomena in a nontrivial way. Study on such effects is interesting but requires intensive theoretical investigations that go beyond the scope of this paper.

Film-thickness dependence of the Hall response

To elaborate on the possibility of the spin-cluster mechanism, we have controlled the magneto-crystalline anisotropy by changing the film-thickness; the SANS experiment on MnGe thin films has revealed the enhanced in-plane magnetic anisotropy with decreasing film-thickness²² (Supplementary Fig. 1). Therefore, the spins can “tilt” easier from the field-polarized direction in the FM state in thinner films and we expect that the SSC excitation in the collinear spin background has a corresponding lower energy. In other words, larger SSC can be produced due to the enhanced in-plane anisotropy in the thinner films as schematically illustrated in the insets of Fig. 3. We note that the insets of Fig. 3 are overdrawn to emphasize the difference induced by the film-thickness. The $$ B$$–$$ T$$ profile of the AHE shows a clear variation with film-thickness of $$ t=80,160,$$ and $$ 300$$ nm (Fig. 3, see also Supplementary Figs. 2 and 3 for the complete data set). While the maximum value of the Hall angle is almost independent of the film-thickness, the $$ B$$–$$ T$$ profile of the AHE changes dramatically as shown in Fig. 3. In thinner films with enhanced in-plane anisotropy, a large Hall angle shows up at lower temperature and higher applied magnetic field, where the SSC excitation costs more energy. We also provide analytical calculations on the effect of easy-plane anisotropy in Supplementary Note 5.

Fig. 3

Film-thickness dependence of the temperature–magnetic-field profile of the AHE in MnGe thin film.

Contour plot of $$ {\tan \mathrm{\Theta }}_{\mathrm{H}}\left(={\sigma }_{xy}/{\sigma }_{xx}\right)$$ for the thickness of $$ t=80$$ (a), $$ 160$$ (b), and $$ 300$$ nm (c). The solid diamond markers connected by a dashed line represent the maximum point of $$ \tan {\theta }_{\mathrm{H}}$$ at each temperature. The insets are the intuitive schematics to explain the fact that the larger scalar spin chirality excitation is more favorable in thinner films with enhanced in-plane magnetic anisotropy.

Comparison with other materials

The discovery of this new type of the AHE, showing giant Hall conductivity and Hall angle simultaneously, provides a distinct exception in the universal scaling curve of $$ {\sigma }_{xy}$$ versus $$ {\sigma }_{xx}$$ established for various ferromagnets^2,17 (Fig. 4). The Hall angle of $$ 18\%$$ in MnGe gives one- or two-orders of magnitude larger $$ {\sigma }_{xy}$$ for a given $$ {\sigma }_{xx}$$, compared to the conventional skew-scattering, resulting in an upward shift of the scaling plot with a linear relation ($$ {\sigma }_{xy}\propto {\sigma }_{xx}$$). We also emphasize that $$ {\sigma }_{xy}$$ from the intrinsic AHE cannot reach this regime, due to the limitations set by the Berry curvature in momentum space. Recently, the spin-cluster AHE was also reported in a frustrated magnet KV₃Sb₅, with large Hall conductivity of 15,507 Ω⁻¹ cm⁻¹ and Hall angle of 1.8%²⁷. We assume that the difference in the Hall angle between MnGe and KV₃Sb₅ may be related to the resonance condition for the spin-chirality skew-scattering. For MnGe, the unusually short magnetic period ($$ \lambda ~2.8$$ nm) has been discussed in terms of the conduction-electron mediated mechanisms^24,28,29 such as the Rudermann–Kittel–Kasuya–Yosida (RKKY) interaction; this can automatically tune the size of the spin cluster to the typical size of $$ ~\frac{1}{k_{\mathrm{F}}}$$, satisfying the resonance condition. Hence the RKKY magnets or the metallic spin-glass systems would be the promising candidates for achieving a large Hall angle. However, the true nature of the excitation modes for the SSC in MnGe remains to be a challenge for future study. For instance, the origin of the SSC can be either the slightly tilted spin clusters or the pairwise excitations of spin hedgehogs and anti-hedgehogs connected by the skyrmion-strings. We speculate that the latter may be the case, given the fact that the magnetic ground state of MnGe at low temperature ($$ T<50$$ K) is an ordered state of these spin singularities.

Fig. 4

Full logarithmic plot of σ_xy versus σ_xx for various materials.

The shaded region with a red dashed line indicates a linear relation ($$ {\sigma }_{xy}\propto {\sigma }_{xx}$$) in MnGe thin films with a giant Hall angle reaching $$ 18~22\%$$ for the thickness of t = 80, 160, and 300 nm. The data are taken from the peak positions of the Hall angle below $$ 30$$ K. The large Hall conductivity and the Hall angle realized in MnGe thin films result in a clear deviation from the conventional scaling plots established for ferromagnets^2,16,17. The reported data for other materials are cited from literature^{6,7,14,16,17,20,27,38–40}.

Thin-film growth

MnGe thin films were grown epitaxially on Si(111) substrates by the molecular beam epitaxy method²². We employed a 2 nm thick MnSi(111) buffer layer, which was grown by reacting a deposited Mn layer with the Si(111)-(7$$ \times $$7) surface at $$ 250^{\circ }\mathrm{C}$$. Then, Mn and Ge were co-evaporated at approximately $$ 90^{\circ }\mathrm{C}$$, followed by annealing at $$ 250^{\circ }\mathrm{C}$$. The growth of B20-type MnGe was confirmed by $$ \theta $$-$$ 2\theta $$ x-ray diffraction scans.

Electric transport measurement

Resistivities up to 14 T were measured with a conventional four-probe method using a DC option of Physical Properties Measurement System (PPMS, Quantum Design). High-magnetic-field transport measurements up to 56 T were performed utilizing non-destructive pulse magnets energized by capacitor banks installed at the International MegaGauss Science Laboratory of Institute for Solid State Physics (ISSP), University of Tokyo. The magnetic field was applied perpendicular to the film plane, i.e., (111) plane, while the electric field was applied parallel to $$ \left[1\overline{1}0\right]$$ direction. The longitudinal conductivity ($$ {\sigma }_{xx}$$) and the Hall conductivity ($$ {\sigma }_{xy}$$) were calculated as $$ {\sigma }_{xx}={\rho }_{xx}/\left({\rho }_{xx}^2+{\rho }_{yx}^2\right)$$ and $$ {\sigma }_{xy}={\rho }_{yx}/\left({\rho }_{xx}^2+{\rho }_{yx}^2\right)$$. Here, $$ {\rho }_{xx}$$ and $$ {\rho }_{yx}$$ are the longitudinal and Hall resistivity, respectively. We could not perform the transport measurements in thick samples (film-thickness larger than $$ 400$$  nm) due to the formation of microcracks upon cooling.

Dark-field transmission electron microscopy measurement

The epitaxially grown MBE thin films of MnGe are composed of a mixed-domain state of enantiomers, although the different enantiomer (lattice chirality) with an opposite-sign DMI should show an identical contribution to any kind of Hall effect. The size of the chiral domain in MnGe thin films was investigated by the dark-field transmission electron microscopy (TEM) measurement^36,37. First, we prepared the sample by grinding the Si substrate down to $$ ~6$$–$$ 8$$ μm using the multiprep and dimple grinding systems. We then utilized precision ion milling (Pips) to remove the remaining Si substrate, leaving the 80, 160, and 300 nm thick films, which were mounted onto TEM sample holder-ready Mo annuli. We characterized the orientation of each film using electron diffraction. As in Ref. ²², we found the [111] axis incidence using selected area electron diffraction within a JEOL JEM 2100 TEM as shown in Supplementary Fig. 10a. We utilized a LN-holder (Gatan 636) for its high angle tilt capability. As shown in Supplementary Fig. 10a, b, the diffraction peaks corresponding to the two domains of interest (labeled A and B in Supplementary Fig. 10b) emerge when the crystal is tilted around the [$$ 11\overline{2}$$] axis by approximately 22°. The spots were isolated using the objective aperture, and the passing electrons were focused into an image of the domains onto the charge-coupled device (CCD) camera installed within the microscope.