Abstract
Physics of Weyl electrons has been attracting considerable interests and further accelerated by recent discoveries of giant anomalous Hall effect (AHE) and topological Hall effect (THE) in several magnetic systems including noncoplanar magnets with spin chirality or smallsize skyrmions. These AHEs/THEs are often attributed to the intense Berry curvature generated around the Weyl nodes accompanied by band anticrossings, yet the direct experimental evidence still remains elusive. Here, we demonstrate an essential role of the band anticrossing for the giant AHE and THE in MnGe thin film by using the terahertz magnetooptical spectroscopy. The lowenergy resonance structures around ~ 1.2 meV in the optical Hall conductivity show the enhanced AHE and THE, indicating the emergence of at least two distinct anticrossings near the Fermi level. The theoretical analysis demonstrates that the competition of these resonances with opposite signs is a cause of the strong temperature and magneticfield dependences of observed DC Hall conductivity. These results lead to the comprehensive understanding of the interplay among the transport phenomena, optical responses and electronic/spin structures.
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Introduction
Understanding of unconventional electromagnetic phenomena stemming from the quantum topological natures is a central issue in contemporary condensed matter physics. The anomalous Hall effect (AHE) caused by spontaneous breaking of time reversal symmetry invokes a long debate regarding the microscopic origin in the past decades^{1}. Several mechanisms classified into either intrinsic or extrinsic ones have been proposed so far. The intrinsic mechanism based on the band structure affected by relativistic spinorbit coupling (SOC) was first proposed by Karplus and Luttinger, and subsequently formulated in terms of the Berry phase arising from the electronic band topology^{1,2,3,4,5}. The enhanced Berry curvature owing to the crossing point, i.e., Weyl node^{6,7}, is considered to cause the large AHE; this character is contrasted to the extrinsic origin such as skew scattering and sidejump mechanisms, which are irrelevant to the electronic band structure^{8,9,10}. In addition to the topologically protected Weyl point, the accidental crossing points of electronic band structure also generate the Berry curvature where the SOC opens a finite gap (band anticrossing). The subtle balance of the competition among these crossing points near the Fermi level can produce intriguing behaviors such as nonmonotonic temperature dependence and giant magnitude of intrinsic AHE^{11,12,13,14,15,16,17}. On the other hand, when the magnetic structure, to which conduction electron is coupled, is composed of noncoplanar spins with scalar spin chirality (SSC) or finite skyrmion number, the socalled topological Hall effect (THE) emerges even without SOC^{18,19,20,21,22,23,24,25,26}. The THE is often argued in terms of emergent magnetic field in analogy to ordinary Hall effect, when the spinchirality period is large enough compared with the lattice spacing like the conventional skyrmionlattice phase^{22}. However, the electron meanfree path exceeds the size of spin modulation period, e.g., skyrmionlattice spacing, the THE should be considered in the momentum space in terms of Weyl electrons as in the case of AHE, rather than the realspace picture of emergent magnetic field^{18,19,26}. Thus, the direct experimental evidence for the essential role of the Weyl nodes for such nontrivial behaviors of both AHE and THE remains to be experimentally elucidated.
The spectroscopic study in terms of the optical Hall conductivity will be a powerful approach to this issue. Similarly to the ordinary Hall effect arising from the cyclotron resonance, the AHE and perhaps the THE as well should exhibit the characteristic resonance structures in the optical Hall conductivity spectrum σ_{xy}(ω)^{27,28,29,30,31,32}. The interband optical transitions across each bandcrossing point lead to the resonance structures in σ_{xy}(ω), whose DC limit approaches the Hall (AH and/or TH) conductivity observed as the transport phenomena. Accordingly, the σ_{xy}(ω) spectra quantitatively clarify the contribution from each band crossing point distinctly to AHE and THE. Since the enhanced Berry curvature of the crossing points near the Fermi level causes the large AHE and THE, the lowenergy optical transition should play the essential role.
In this work, we investigate the optical transitions associated with the giant AHE and THE in MnGe thin film in terms of the terahertz optical Hall conductivity. MnGe is a member of the B20type alloys with the chiral crystal structure (Fig. 1a) and below 170 K exhibits the helimagnetically ordered state, which is viewed as composed of the hybridized three orthogonal spin helices with helical pitch of 3 nm to form spin hedgehogantihedgehog lattice^{33,34,35,36}, as shown in Fig. 1b. Each hedgehog and antihedgehog behave as emergent magnetic monopole and antimonopole acting on the conduction electron, which are bridged over by skyrmion strings, as shown in Fig. 1c. When external magnetic field is applied on this hedgehog lattice, the bridging skyrmionstrings are elongated or shrunk, thereby producing emergent magnetic field or nonzero skyrmion number density in total and leading to the large THE in the magneticfield range in between the zerofield state and the fieldaligned ferromagnetic state^{35}. On the other hand, the inherently large SOC in this compound also produces the conventional AHE approximately in proportion to the fieldinduced magnetization. Therefore, MnGe provides a good arena to spectroscopically identify and distinguish the Berry curvatures generated around the Weyl nodes by the SOC as well as by the SSC (skyrmion number), which respectively show up as AHE and THE. We demonstrate that the terahertz resonances of the band anticrossing points (Weyl nodes) close to the Fermi level are assigned distinctly to the origins of AHE and THE and that the competition of these resonances causes the strong temperature and magneticfield dependence of DC Hall conductivity as observed.
Results
Magentotransport properties
Figure 1d summarizes the Hall conductivity σ_{xy} of the MnGe thin film used in this study when magnetic field H is applied normal to the film plane, i.e., H[111]. At relatively high temperatures, e.g., above 30 K, σ_{xy} increases with H, followed by the saturation, which well corresponds to the magnetization curve as a hallmark of AHE (dashed lines in Fig. 1d; see also the discussion in Supplementary Note 1). With decreasing temperature below 30 K, however, the magneticfield dependence of σ_{xy} deviates from that of the conventional AHE being proportional to the magnetization; this is due to the increased negative component of σ_{xy}, as indicated by shaded regions in Fig. 1d. This diplike structure has been ascribed to the THE^{21,33,35,36}, which is expected to appear in the presence of the SSC of the noncoplanar spin texture. More specifically, the THE in MnGe originates from the threedimensional topological spin crystal consisting of the hedgehog and antihedgehog spin textures (Fig. 1b)^{33,34,35,36}. These spin textures act as monopole and antimonopole of the emergent magnetic field which is canceled out in total at zero external magnetic field. Upon the application of external magnetic field, however, the emergent monopole and antimonopole are forced to move in opposite directions; these movements of emergent magnetic monopoles/antimonopoles are accompanied by deformation of the bridging skyrmionstrings (Fig. 1c), giving rise to the uncancelled emergent magnetic field or the relevant Berrycurvature generation in the momentum space. We ascribe these topological magnetic orders to the origin of THE with negative σ_{xy} values, which show a negative peak around a middle field between the zerofield state and the forced ferromagnetic transition^{21,33,35,36}.
Terahertz polarimetry
We performed the phase sensitive timedomain terahertz polarimetry in the normal incidence for the MnGe(111) thin film with the thickness of 85 nm fabricated on the insulating Si(111) substrate (see also Methods). We obtained the complex spectra of the magnetooptical Faraday rotation in the magnetic field applied along the MnGe [111] axis, i.e., normal to the film plane (Fig. 1e). We note that the hedgehog lattice is sensitive to the magnetic anisotropy depending on the film thickness and deformed in the thin film form^{36}. The three q vectors i.e., the magnetic basis vectors of hedgehog lattice state, tilt toward the normal direction in the thin film; accordingly, the hedgehog and antihedgehog are sparsely distributed as compared with the bulk system. The dataset obtained by this technique directly provides the optical Hall conductivity σ_{xy}(ω) in addition to the longitudinal optical conductivity σ_{xx}(ω) (see Methods). The real part of σ_{xx}(ω) shows a steep increase towards zero frequency at low temperatures (Fig. 1f). In the terahertz region, the σ_{xx}(ω) spectra represent the conduction electron dynamics as expressed by the Drude model \({\sigma }_{{xx}}\left(\omega \right)=\frac{{\varepsilon }_{0}{\omega }_{p}^{2}\tau }{1i\omega \tau }\); ε_{0}, 1/τ and ω_{p} are the dielectric constant in vacuum, scattering rate and plasma frequency, respectively. The scattering rate 1/τ, i.e., the width of the zerofrequency peak in Re σ_{xx}(ω), decreases monotonically with decreasing temperature and reaches ~ 1 meV at 4.5 K, indicating good metallic conduction and high quality of the sample (Supplementary Fig. 4).
In contrast to the conventional metallic behavior observed in the σ_{xx}(ω) spectra, the Faraday rotation and Hall conductivity spectra show prominent features at low temperatures and at moderate magnetic fields. At 40 K and 7 T, the Faraday rotation spectrum θ_{F}(ω) is almost constant, and the magnitude is consistent with the DC Hall angle (a red curve in Fig. 1g). With decreasing temperature, the diplike resonance structure develops at ~ 2.0 meV, and shows a sign reversal at 4.5 and 10 K. To examine the correlation with the AHE and THE quantitatively, the complex optical Hall conductivity spectra, Re σ_{xy}(ω) and Im σ_{xy}(ω), are deduced from σ_{xx}(ω), θ_{F}(ω) and ellipticity η_{F}(ω) (see Methods). The dramatic spectral changes in the Hall conductivity spectra σ_{xy}(ω) are observed with the changes of magnetic field (Fig. 2a, b) and temperature (Fig. 2c, d). In the course of increasing the magnetic field at 4.5 K, the sharp dip structure at ~1.5 meV develops in addition to the increase of the flat background above 3 meV in Re σ_{xy}(ω) (Fig. 2a). On the other hand, the Im σ_{xy}(ω) shows a shallow dip at ~2.3 meV and the steep rise towards lower frequency, which are enhanced at higher magnetic fields (Fig. 2b). Since the Im σ_{xy}(ω) should converge to zero at ω = 0 due to the causality constraint (see ω = 0 plane in Fig. 2b), the emergence of the peak structure below 1 meV is suggested in high magnetic fields.
The DC Hall conductivity at 7 T shows the anomaly at low temperatures (the filled circles at the ω = 0 plane in Fig. 2c); it increases up to 680 Ω^{−1}cm^{−1} at 10 K from 380 Ω^{−1}cm^{−1} at 40 K with decreasing temperature, which is followed by the decrease towards the lowest temperature (4.5 K). This lowtemperature anomaly in DC Hall conductivity coincides with the appearance of the resonance structure in σ_{xy}(ω) (Fig. 2c, d). At 40 K, both Re σ_{xy}(ω) and Im σ_{xy}(ω) are flat and consistent with the DC Hall conductivity (7 T; blue curves). With decreasing temperature, the resonance structures rapidly develop while keeping σ_{xy}(ω) above 4 meV almost unchanged. It is thus concluded that the dramatic change of transport Hall conductivity at low temperatures and moderate magnetic fields (at least up to 7 T) stems from the resonance structures in terahertz region (< 4 meV). These lowenergy resonances in σ_{xy}(ω) indicate the interband optical transitions across the band (anti)crossing points or Weyl nodes, which suggests the dominant role of the Berrycurvature mechanism for the both AHE and THE in MnGe. We note that the ordinary Hall effect hardly contributes to the σ_{xy}(ω) in this energy region (see the discussion in Supplementary Note 2).
Analysis model for Hall conductivity spectrum
To understand the spectral feature of the terahertz resonances, we introduce the twoband model describing the intrinsic AHE and THE with minimal assumptions (Fig. 3a): The model Hamiltonian H(k) is expressed as follows^{27,28,30};
where σ_{0} and σ_{i} (i = x, y, z) are the identity and Pauli matrices, respectively, and μ is the chemical potential measured from the band crossing point. Accordingly, we assume that the band dispersion is quasi twodimensional and (h_{x}, h_{y}, h_{z}) = (k_{x}, k_{y}, m), so that it has the level splitting of 2m at the crossing point (Fig. 3a, inset). We calculate the optical Hall conductivity σ_{xy}(ω) with use of the general expression given by the Kubo formula;
where the J_{x(y)} is the current operator given by \(\frac{\hslash }{e}\mathop{\sum}\limits_{k}{c}^{{{\dagger}} }(k)\frac{\partial H\left(k\right)}{\partial {k}_{x\left(y\right)}}c(k)\). f(ε_{n}) is the Fermi distribution function, \({\varepsilon }_{n}\) and n〉 are the energy and the Bloch wave function of the nth band, respectively, and δ is the phenomenological damping constant. c(k) (\({c}^{{{\dagger}} }\)(k)) is the annihilation (creation) operator. The energydependent Hall conductivity σ_{xy}(ω) is thus given by^{27,28};
where e, h and a are the elementary electric charge, Planck constant and lattice constant, respectively. Re σ_{xy}(ω) and Im σ_{xy}(ω) in Eq. (3) have sharp resonance peaks at the energy of 2μ without the sign change as shown in Fig. 3a. Since the observed spectra show the sign change for both of Re σ_{xy}(ω) and Im σ_{xy}(ω) (Fig. 2), it is reasonable to assume two resonances with opposite signs of m. This assumption is consistent with the coexistence of AHE and THE with opposite signs in the DC Hall conductivity (Fig. 1d). Accordingly, the total Hall conductivity spectrum consisting of the vertical transitions on two anticrossing points near the Fermi level is expressed as,
where α and β are band indices. Two anticrossing points of α and βbands give rise to σ_{xy}^{α}(ω) and σ_{xy}^{β}(ω), respectively. The constant term σ_{xy}^{const.} represents the lowenergy tail of higherenergy resonances above the present energy window. We note that in the chiral metal MnGe the band crossing with intense Berry curvature is the (threedimensional) Weyl point, which is robust against some perturbation^{6,7}; the minimal Hamiltonian is given by, \(H=\mathop{\sum}\limits_{i=x,y,z}{v}_{i}{k}_{i}{\sigma }_{i}\). In the present model analysis, we consider the anticrossing points present along the certain k vector interconnecting two Weyl points (Fig. 3b). Here, we introduced f_{i} as a free parameter representing the spectral weight in Eq. (4), which effectively describes the averaged gap magnitude; the δ represents the spectral width of the resonance, which can also reflect the threedimensional band dispersion (Supplementary Note 3). In the results shown below, we adopted the fitting parameters producing the DC limit value less than the typical upper limit of the intrinsic AH conductivity^{37}, \(\frac{{e}^{2}}{{ha}}\) ~1000 Ω^{−1}cm^{−1}. As shown in Fig. 3c, the calculated spectra (black), which are composed of σ_{xy}^{α}(ω) (green), σ_{xy}^{β}(ω) (orange) and σ_{xy}^{const.} (light blue), quantitatively reproduces the experimental spectral features; the dispersive shape with the sign change in Re σ_{xy}(ω) (red) and the large positive peak (~1 meV) in Im σ_{xy}(ω) (blue). The similar results were obtained for the spectra at other magnetic fields and temperatures. The fitting parameters at 4.5 K are summarized in Fig. 3d. The spectral weight f_{α,β} and σ_{xy}^{const.} monotonically increase by applying the magnetic field, while the resonance energies of αband (2μ_{α} ~ 0.8 meV) and βband (2μ_{β} ~ 1.4 meV) show little magneticfield dependence. These small energy scales of resonance energies (1 meV ~ 11.6 K) are consistent with the strong temperature dependence of the Hall responses below 30 K, because the lowenergy structures should be smeared out by thermal agitations at high temperatures.
Comparison with transport results
The signs of the DC AH and TH conductivities are observed to be positive and negative, respectively (Fig. 1d). The terahertz Hall conductivity spectra reveal the presence of two positive components, σ_{xy}^{α}(ω) and σ_{xy}^{const.}, and negative one, σ_{xy}^{β}(ω). Therefore, the negative resonance, σ_{xy}^{β}(ω), can be ascribed to the THE, while the positive resonance σ_{xy}^{α}(ω) to the positive AHE. The positive σ_{xy}^{const.} may possibly have both contributions from the AHE and THE whose resonances locate at higher energies, but appears to be mostly dominated by the AHE process in the present case because of its nearly M(H)linear behavior (lower panel of Fig. 3d). These assignments are further verified by the quantitative comparison with the transport measurement. We calculated the DClimit values of these resonances, i.e., Re{σ_{xy}^{α}(ω = 0)+σ_{xy}^{const.}} and Re{σ_{xy}^{β}(ω = 0)} as indicated by filled and open circles at 4.5 K (Fig. 4a) and 20 K (Fig. 4b), and plotted as functions of magnetic field for each temperature in Fig. 4c–f. The σ_{xy}^{α}(ω = 0) + σ_{xy}^{const.} (filled circles) quantitatively reproduces the monotonic increase of the AHE estimated from the transport data (a blue line; see also Supplementary Note 1) with increasing the magnetic field at each temperature. The enhancement of the magnitude of the estimated THE (red line) as a function of magnetic field is also described by the σ_{xy}^{β}(ω = 0) (open circles). In addition, the rapid diminishment of the THE at higher temperatures (T = 20, 30 K) is consistent with the σ_{xy}^{β}(ω = 0) obtained from the terahertz measurement. These observations confirm the whole consistency of our theoretical analysis and assignment, thus demonstrating that the emergence and competition of these two anticrossing points are a major origin of the strong magneticfield dependence and temperature dependence of DC Hall conductivity.
Discussion
It should be emphasized that the observation of the resonance of the THE indicates that the formation of the topological spin texture modifies the electronic band structure. For MnGe thin film, the magnetic period (~3 nm) is shorter than the meanfree path (~3.6 nm at 2 K) estimated from the simple freeelectron model (Supplementary Note 4). In this regime, the noncoplanar spin texture with SSC, as exemplified by the ordered skyrmion strings bridging the emergent magnetic monopoles and antimonopoles (Fig. 1c), can induce the anticrossing points in the band dispersions, which gives rise to the THE as reported for pyrochlore Mooxides^{18,19,26}. In such the shortperiod spin order with SSC, the THE is ascribed to the Berry curvature arising from the band crossings similarly to AHE, while the AHE and THE can be associated with different band crossings as suggested by the recent study^{26}; the Berry curvature for the AHE arises from the band crossings composed of the equal and oppositespin band pairs impartially while that for the THE only of the oppositespin band pairs. This momentumspace picture of the THE is not applicable to the archetypal skyrmionhosting material MnSi with much longer magnetic period (~18 nm). In MnSi, the magnitude of the THE, that is much smaller than the present case, is well explained in terms of the realspace picture that the conduction electrons are deflected by the emergent magnetic field carried by skyrmions^{20,22,23}; namely, there is no necessity to invoke the momentumspace Weyl nodes.
We note that the presence of lots of Weyl nodes can be confirmed by the density functional theory calculation for the forced ferromagnetic state (Supplementary Fig. 6). We also calculate the optical Hall conductivity spectra. The sharp resonance structure with positive sign can show up at certain Fermi level (Supplementary Fig. 7), which well captures the spectral characteristics of the terahertz resonance producing the AHE. Nevertheless, it is difficult to directly compare these calculated spectra for the ferromagnetic state with the experimental one for the hedgehog lattice because the band structure may be significantly modified by the band folding due to the formation of the shortperiod topological spin texture. Such a reconstructed band structure probably changes the AHE features of Berry curvature origin. The clear elucidation of the modified band structure by the formation of smallsize skyrmions will be the important future task.
In conclusion, we have investigated the lowenergy electron dynamics associated with the AHE and THE in MnGe thin film. We have found that the prominent lowenergy resonance structures unveiled by the terahertz Hall conductivity spectra dominantly contribute to the DC Hall conductivity. This observation experimentally demonstrates the band anticrossing points or Weyl nodes just above or beneath the Fermi level indeed realize the large AHE and THE as anticipated. Moreover, spectral characteristics of optical Hall conductivity clearly point out that the competition of lowenergy anticrossing points causes the strong temperature and magneticfield dependence of the DC Hall conductivity. The present terahertz magnetooptical spectroscopy establishes the Berrycurvature generation in the electronic bands, relevant to AHE as well as THE with shortperiod SSC, which will further stimulate exploration of giant or even quantized AHE/THE^{41,42} and promote understanding of topological physics and promising functions.
Methods
Thin film growth
An 85nmthick MnGe thin film was fabricated on Si(111) substrates with use of the molecular beam epitaxy technique. The details of the growth procedure and characterization are described in Supplementary Note 6.
Transport measurements
The magnetoresistivity and Hall resistivity were measured by using Physical Property Measurement System (Quantum Design). Concerning the analysis for the decomposition of the AHE and THE, see Supplementary Note 1.
Terahertz time domain spectroscopy (THzTDS)
In the THzTDS, laser pulses with the duration of 100 fs from a modelocked Ti: sapphire laser were split into two paths to generate and detect THz pulses with using the photoconductive antenna. Transmittance spectra of the thin film were obtained by measuring the transmission of both the sample and bare substrate. We used the following standard formula to obtain the complex conductivity σ_{xx}(ω) = Re σ_{xx}(ω) + i Im σ_{xx}(ω) of the thin film;
where t (ω), d, Z_{0} and n_{s} are the complex transmittance, the thickness of the film, the impedance of free space (377 Ω), and the refractive index of the Si substrate, respectively.
Faraday rotation measurements
The rotatory component of the transmitted THz pulses E_{y}(t), which is polarized perpendicular to the incident light E_{x}(t), was measured in the crossedNicols configuration by using wiregrid polarizers. To eliminate the background signal, we antisymmetrized E_{y}(t) with respect to positive and negative magnetic fields. The Fourier transformation of the THz pulses E_{x}(t) and E_{y}(t) gives the Faraday rotation θ_{F}(ω) and ellipticity η_{F}(ω); E_{y}(ω)/E_{x}(ω) ≃ θ_{F}(ω) + iη_{F}(ω) for the small rotation angles.
Terahertz Hall conductivity
By using the longitudinal conductivity spectra σ_{xx}(ω) and Faraday rotation and ellipticity spectra θ_{F}(ω) + iη_{F}(ω), the Hall conductivity spectra were calculated from the following formula; \({\sigma }_{xy}(\omega)=(\theta_{{{{{\rm{F}}}}}}+i\eta _{{{{{\rm{F}}}}}})\frac{1+{n}_{s}+{Z}_{0}d{\sigma }_{xx}(\omega)}{{Z}_{0}d}\) ^{28}. To estimate the DClimit rotation angle θ_{F}(ω = 0) in Fig. 1g, we used this formula by substituting σ_{xx}(0) and σ_{xy}(0) obtained from the transport measurements.
Data availability
The data that support the plots of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank N. Nagaosa for valuable comments and Y. Kozuka and R. Watanabe for experimental help and A. Kitaori for providing the transport data of single crystals. This work was partially supported by JSPS KAKENHI (Grant no. 19K14653) and by JST CREST (Grant no. JPMJCR1874).
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Y.H. performed the terahertz measurement and analyzed the data with help of Y.O. and Y.Ta. Y.H. and N.K. fabricated the thin film and measured the transport property with supervision of A.T., M.I., and M.K. T.Y., T.K. and R.A. performed the firstprinciples calculation. Y.To and Y.Ta conceived the project. Y.H., Y.O. and Y.Ta wrote the manuscript with assistance of other authors. All authors discussed and interpreted the results.
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Hayashi, Y., Okamura, Y., Kanazawa, N. et al. Magnetooptical spectroscopy on Weyl nodes for anomalous and topological Hall effects in chiral MnGe. Nat Commun 12, 5974 (2021). https://doi.org/10.1038/s41467021252761
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DOI: https://doi.org/10.1038/s41467021252761
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