Abstract
Quantum states characterized by nontrivial topology produce interesting electrodynamics and versatile electronic functionalities. One source for such remarkable phenomena is emergent electromagnetic field, which is the outcome of interplay between topological spin structures with scalar spin chirality and conduction electrons. However, it has scarcely been exploited for emergent function related to heatelectricity conversion. Here we report an unusually enhanced thermopower by application of magnetic field in MnGe hosting topological spin textures. By considering all conceivable origins through quantitative investigations of electronic structures and properties, a possible origin of large magnetothermopower is assigned to the strong energy dependence of chargetransport lifetime caused by unconventional carrier scattering via the dynamics of emergent magnetic field. Furthermore, highmagneticfield measurements corroborate the presence of residual magnetic fluctuations even in the nominally ferromagnetic region, leading to a subsisting behavior of fieldenhanced thermopower. The present finding may pave a way for thermoelectric function of topological magnets.
Introduction
Highperformance thermoelectric materials provide a viable solution towards environmental issues since they realize efficient electricity generation from waste heat without greenhouse gas emissions^{1}. In long history of the research, extensive efforts have been made to enhance Seebeck coefficient (S) with minimal increase in electrical resistivity (ρ) to improve thermoelectric figure of merit ZT = S^{2}T/ρκ, where T and κ represent temperature and thermal conductivity, respectively.
Seebeck coefficient, i.e., electromotive force per unit temperature gradient, can be interpreted as averaged entropy flow per charge carrier on the basis of the Onsager relations^{2}. In the framework of the band structure picture, semiclassical treatment using the Boltzmann transport equation provides a guiding principle for obtaining the efficient entropy flow, which is described by Mott formula^{1}; by neglecting the Tdependence of chemical potential μ, i.e., setting μ = ε_{F}, it reads
where k_{B} and e are Boltzmann constant and elementary charge, respectively. The first and second terms in the brackets, respectively, represent energy derivatives of density of states D(ε) and relaxation time τ(ε) at Fermi energy ε_{F}. It commonly occurs in metals that entropy flows of electrons with their potential energy above and below ε_{F} cancel out with each other, resulting in small S on the order of a few μV K^{−1}. This is because the Fermi distribution function permits only electrons within their energy range approximately between ε_{F} ± k_{B}T to be involved in heat transport phenomena, and electrons above and below ε_{F} carry heat (product of entropy and temperature) with opposite signs. The Mott formula suggests that such cancellation can be avoided in the presence of difference in number, velocity, and scattering rate between electrons above and below ε_{F}, the first two of which are measured by the derivative \(\left. {\frac{{\partial \ln D\left( \varepsilon \right)}}{{\partial \varepsilon }}} \right_{\varepsilon = \varepsilon _{\mathrm{F}}}\) and the last of which by \(\left. {\frac{{\partial \ln \tau \left( \varepsilon \right)}}{{\partial \varepsilon }}} \right_{\varepsilon = \varepsilon _{\mathrm{F}}}\). Indeed, asymmetric band structures around ε_{F} generate large S (e.g., pseudogap structures in Heusler compounds^{3}), whereas Kondo scattering creates strong energy dependence in τ and the consequent exotic Seebeck effect in rareearth compounds^{4}.
Partly because materials with good thermoelectric properties (e.g., Bi_{2}Te_{3}) have recently been identified as topological insulators^{1,5}, the nontrivial electronic topology is expected to offer a unconventional mechanism to enhance thermoelectric effects. However, their good performance does not seem to be related to their topology, but can be understood in the context of the semiclassical model^{6,7}. It also remains largely unexplored to utilize the emergent magnetic field acting on conduction electron, which is generated by topological spin structures in real or momentum space^{8,9}, for improving thermoelectric efficiency. For one thing, that would be because existence of magnetic field would rather suppress S basically by quenching the internal degrees of freedom of carriers, such as spin and orbital entropy, spindependent scatterings, and so on, as exemplified by the observed reduction of S in Na_{ x }CoO_{2}, where spin and orbital degeneracy is lifted by magnetic fields^{10,11}. Suppression of S by magnetic field is also observed as a hallmark of chiral anomaly in Weyl semimetals such as GdPtBi where charge pumping causes dramatic decrease in S^{12}.
B20type compounds of the present focus, e.g., MnSi, (Fe,Co)Si, FeGe, and MnGe, form the family of chirallattice magnets hosting the magnetic skyrmion^{13,14}. Noncentrosymmetric crystal structure allows Dzyaloshinskii–Moriya interaction beside ferromagnetic and other competing magnetic interactions, which in consequence gives rise to the universal phase diagram composed of helical/conical (Fig. 1d), ferromagnetic, and skyrmion phases^{13}. MnGe shows a unique topological spin texture and occupies a unique position among those B20type compounds. The skyrmion crystal is in general described by the multipleq state, where q is the modulation vector of the helix state. The ordinary skyrmion crystal, e.g., in MnSi, is composed of the three qvectors, which lie in the same plane and make the angle of 120° with each other, forming the twodimensional triangular lattice. By contrast, the spin texture in MnGe is approximately described by the orthogonal three qvectors along <100> directions and hence forms a threedimensional lattice of spin hedgehogs and antihedgehogs (Fig. 1a). The spin hedgehog and antihedgehog therein serve as source (monopole) and sink (antimonopole) of the emergent magnetic field acting on conduction electrons^{8,9,15}. The large emergent magnetic field (~40 T) due to its short magnetic period (~3 nm) is already detected by transport measurements as the large topological Hall effect^{16} and the topological Nernst effect^{17}. Another key difference from other B20type chiral magnets is the extraordinarily large temperature (T)magneticfield (H) range (T < T_{N} ~ 170 K and μ_{0} H < μ_{0}H_{c} ~ 12 T), including at 0 T, where the topological spin texture is realized as the thermodynamically stable magnetic state^{16}. These features make MnGe a special material for seeking unprecedented physics originating in the topological spin texture.
Here we report the observation of magneticfieldinduced large thermopower in a chirallattice magnet MnGe with hedgehogtype topological spin textures^{15,16,18}. The unprecedented Hdependence of S is highlighted by a comparative study of isostructural and isovalent magnet MnSi with helical/conical or skyrmion spin textures^{19,20}, which exhibits usual monotonous decrease in S with application of magnetic fields^{21}. Energy dependence of D(ε), which is revealed by photoemission spectroscopy (PES) and band calculations, shows some structures contributing to S, however, which alone is not enough to explain the observed magnitude. Along with other striking contrast between magnetoresistivity (MR) and specific heat of MnGe and MnSi, a unique scattering mechanism originating from strong fluctuations of emergent fields in MnGe may cause the strong dependence of transport lifetime, leading to the enhanced S even at low temperatures. This proposed scenario is corroborated by highmagneticfield measurements, where we demonstrate the close correlation between MR and S in terms of magnetic fluctuations.
Results
Fieldinduced large thermopower in MnGe
We have observed unprecedented magnetothermopower in the polycrystalline MnGe, which exhibits strong enhancement with increasing H below T_{N} (Fig. 1b). Its increment ratio becomes prominent at low temperatures: S shows a typical value for ordinary metals at zero field (e.g., −5.5 μV K^{−1} at 15 K), and it develops an order of magnitude larger at 14 T (e.g., 26 μV K^{−1} at 15 K). Phase transition from hedgehog lattice (HL, Fig. 1a) to ferromagnetic (FM) state at the critical magnetic field H_{c} is recognized as a kink in the S–H curve (indicated by the black triangle in Fig. 1b), followed by a saturating behavior. This clearly indicates the strong correlation between thermoelectric property and spin texture in this compound. By contrast, the thermopower in MnSi decreases with H (Fig. 1e), which is a behavior generically expected for magnetic materials; this highlights the unconventional magnetothermopower in MnGe.
We also noticed some anomalous behaviors of S at low temperatures in MnGe; it keeps on increasing even above 14 T, although the spin texture should have almost turned into the FM state. The suppression of S is, however, eventually realized in the lowT and highH region, along with the reduction of positive MR, which we attribute to the suppression of magnetic fluctuations caused by annihilation of HL. This will be discussed in a later section. Another point is that nonmonotonous structure appears around 6–8 T (shown as gray triangles in Fig. 1b), however, these anomalies gradually disappear with the elevation of T. We speculate that this anomalous structure may be related to the Hdependence of emergent magnetic field in MnGe, whose magnitude becomes the maximum around the corresponding magnetic field^{16}.
Comparison between T–H variations of S in MnGe and MnSi gives a clear summary of the features listed above, as shown in Fig. 1c, f. In MnGe, we can confirm the increasing behavior of thermopower towards the phase boundary between HL and FM (a white line in Fig. 1c) at every T and the saturating behavior in the FM state. The profile of S forms a broad peak structure around the phase boundary; this suggests a widespread effect of the large fluctuations around the HL–FM phase boundary, where the topological transition occurs as accompanied by the annihilation of hedgehog and antihedgehog magnetic textures^{15}. A former study showed that hedgehog (antihedgehog) is viewed as emergent magnetic monopole (antimonopole) and that the HL–FM topological transition corresponds to the pair annihilation of monopole and antimonopole^{15}. In contrast, MnSi shows nearly featureless profile of S (Fig. 1f) and no discernible structure in its narrow skyrmion phase region (see Supplementary Figure 1, Supplementary Note 1, and ref. ^{21}).
Electronic structure and thermopower
To discuss what physical parameters mainly contribute to the large S in terms of the Mott formula, electronic structures and other magnetotransport properties have been investigated for MnGe. First, we examine the band structure of MnGe by performing photoemission spectroscopy (PES) and band calculation. Figure 2a shows Tdependence of photoemission spectra of MnGe near the Fermi level ε_{F}. When a metallic system has a large Fermi surface with tiny energy dependence of D(ε), the spectrum obeys Fermi distribution function with respect to ε_{F}. In the case of MnGe, deviation from the typical Fermi distribution function profile (represented by the spectrum in Au, indicated by a black line in Fig. 2a) becomes discernible with decreasing temperature. To further evaluate this Tdependence, we divided the spectra by resolutionconvoluted Fermi distribution function to obtain the effective D(ε) (Fig. 2b). Here the formation of narrow pseudogap (~40 meV) is clearly observed especially below the transition temperature of MnGe (T_{N} ~ 170 K), suggesting its relationship with magnetic ordering in this system. We estimated the upper limit of its contribution to S by assigning the steepest downward slope of D(ε) curve at 11 K to the first term of Mott formula \( \frac{{\pi ^2k_{\mathrm{B}}^2T}}{{3e}}\frac{{\partial \ln D\left( \varepsilon \right)}}{{\partial \varepsilon }}\) (see the dashed line in Fig. 2b for the corresponding slope). It turns out that the narrow pseudogap generates thermopower of S ≈ 0.13 T (μV K^{−1}). Under the assumption that application of H effectively causes a shift in ε_{F} while keeping the pseudogap structure robust, our estimation of the upper limit of S should be also valid for the FM state; the estimated value of S is far short of the experimental one. We also calculated the band structure in the FM state using the density functional theory (DFT) to theoretically derive S (see Methods). As shown in Fig. 2c, D(ε) does not present any distinct structures like the pseudogap detected by PES. Therefore the corresponding S calculated with the approximation of constant relaxation time represents only a small value of –0.9 μV K^{−1} at 50 K (Fig. 3d). Thus, electronic structure of MnGe alone cannot be the dominant source for the unconventional thermoelectric response as observed.
Specific heat and magnetoresistivity
The fieldinduced behavior of S is suggestive of enhancement of entropy, which can be best illustrated by the fielddependent specific heat (C). Temperature dependence of C under various H in MnSi and MnGe are shown as the curves of C/T vs. T^{2} in Fig. 3a and b, respectively. We also convert these data to the change in ratio of specific heat [ΔC(H)/C(0) = (C(H)—C(0))/C(0)] as functions of normalized magnetic field (H/H_{c}) at various temperatures (Fig. 3c for MnSi and Fig. 3d for MnGe). There is again stark contrast between these two systems. As to MnSi, we observe a monotonous decrease of specific heat with H at every temperature. Here we note that the significant decrease in ΔC(H)/C(0) is due to release of latent heat associated with the firstorder transition from the conical to collinear (ferromagnetic) spin structure (see Fig. 3c, 28 K and ref. ^{22}). In contrast, specific heat of MnGe shows a clear increase around H_{c}, where the spin hedgehogs and antihedgehogs undergo the pair annihilation and the topological transition into the FM state occurs^{15}. There obviously exist strong fluctuations unique to the topological phase transition in MnGe. Now we can estimate the maximum of thermoelectric contribution from the entropy enhancement in thermal equilibrium. If we can take full advantage of the increase in specific heat ΔC(H), S can change by ΔS(H) = ΔC(H)/ne, whatever the mechanism is (e.g., phonon or magnon drag^{23}). With the largest ΔC (0.181 J K^{−1} mol^{−1} at 25 K) and carrier density n (~1.3 × 10^{23} cm^{−3})^{16}, we obtain ΔS(H) ≤ 0.52 μV K^{−1}, which is again quantitatively insufficient to be a dominant origin of the large magnetothermopower in MnGe.
As all the above possible origins have failed to explain the large Hinduced S in MnGe, some scattering processes are likely to make a major contribution, producing a large value of the second term in Mott formula \( \frac{{\pi ^2k_{\mathrm{B}}^2T}}{{3e}}\frac{{\partial \ln \tau \left( \varepsilon \right)}}{{\partial \varepsilon }}\). In other words, the carriers should acquire the large entropy by getting scattered. With increasing H, such Henhanced scattering of carriers manifests itself in a large positive MR around the Hinduced HL–FM topological transition, as observed in a previous study^{15} and reproduced for the present sample in Fig. 3f. Note that enhanced spin fluctuations accompanied by those of the emergent magnetic field around the topological phase transition, are evidenced also by the result of specific heat (Fig. 3d). The characteristic Hdependence of MR in MnGe (Fig. 3f) shows a broad peak structure around H_{c} at every temperature below T_{N}. This positive MR is also unconventional since external magnetic fields basically suppress spindependent scattering, which leads to the monotonously decreasing negative MR as observed in MnSi (Fig. 3e) and other related B20type compounds^{24}. The anticipated strong energy dependence of τ may also be rooted in such a Hdependent enhancement of the fluctuating emergent magnetic field.
Magnetoresistivity and thermopower at the lowtemperature and highfield regime
The positive MR due to the fluctuations of emergent field only gradually falls down and still remains well above H_{c} as shown in Fig. 3f. This implies that there may exist robust or pinned excitations of spin states with noncoplanar spin arrangements like hedgehogs even in the FM phase, causing large magnetic fluctuations as a source of the nondiminishing behavior of S. To corroborate this interpretation, we performed highmagneticfield measurements on MR (up to 33 T) and S (up to 24 T) at low temperatures (below 10 K), where such magnetic fluctuations is anticipated to be sufficiently suppressed. Figure 4a–d shows the results for MR and M. As the resistivity is generally expressed as \(\left[ {W \cdot \tau } \right]\)^{−1} with W being Drude weight, the MR can be decomposed to the fieldinduced respective changes of W and τ. The MR due to the fieldchange of W is well known for the doubleexchange system (e.g., colossal magnetoresistance manganites^{25}) and can be well scaled with the magnetization M, as confirmed also for the present case of MnGe^{15}. Here, with use of the corresponding magnetization data, we estimate the conventional negative MR due to the fieldincrease of W, as shown with black lines in Fig. 4a–d. Then, we can deduce the effect of magnetic fluctuations on MR, i.e., the fieldinduced change of τ^{−1}, as the deviation from the conventional negative MR^{15}. The estimated deviations, which correspond to the colorshaded regions of Fig. 4a–d, are displayed in Fig. 4e–h. They clearly show the residual magnetic fluctuations to scatter the conduction electrons in the FM state, which appear to be steeply enhanced with increasing temperature across T = 10 K. As for S in the highfield regime, we found a strong correlation with the observed MR. A decreasing behavior of S with the field is clearly identified across the HL to FM transition (shown in dashed lines) for T = 2 K and 5 K (Fig. 4i, j) along with the strong suppression of magnetic fluctuations as evidenced by the MR measurement (Fig. 4e, f). When the temperature is elevated, by contrast, the nondecreasing behavior of S (Fig. 4j–l) takes over, due to the finite magnetic fluctuations surviving in the FM phase (Fig. 4f–h). Even a larger magnetic field (more than 24 T) seems to be required to fully suppress the enlarged magnetic fluctuations above 5 K. Here we note that the enhancement of S does not measure the variation of scattering rate τ^{−1} itself but its energy dependence \(S \propto \partial {\mathrm{ln}}\tau /\partial \varepsilon\) as described by Mott’s formula. Hence, the enhancement of S can happen in principle as long as there exist any finite magnetic fluctuations (Fig. 4f–h) affecting τ, although the quantitative connection between τ and \(\partial \tau /\partial \varepsilon\) is difficult to verify at the moment.
Thermoelectric power factor
We lastly note that S and thermoelectric conversion efficiency [power factor S^{2}/ρ (μW K^{−2 }cm^{−1})] largely vary in different samples. The maximum power factor obtained in our study reaches as large as S^{2}/ρ = 65 μW K^{−2 }cm^{−1} at H = 14 T, T = 19 K, although it shows the large sample dependence and is apparently related to the value of residual resistivity or to the background (H = 0) transport lifetime (Supplementary Figures 2 and 3).
Discussion
We have unraveled an unusual magnetoSeebeck effect in MnGe, which shows a large enhancement by applying external magnetic field. Through examining its origin from every possible aspect by photoemission spectroscopy, band calculation, specific heat, and magnetotransport measurements, we propose that the anomalous enhancement is rooted in strong energy dependence of transport lifetime τ, which may arise from the Hdependent dynamics of emergent magnetic field. In addition, highfield measurements on MR and S verifies the presence of surviving magnetic fluctuations to scatter the conduction electrons even in the FM phase. The most important integrant for the observed thermoelectric phenomena in MnGe should be the dense lattice of magnetic singularities like spin hedgehogs and antihedgehogs, where their large emergent fields and fluctuations critically affect the motion of electrons. The paradigm presented in this paper, that is the efficient heatelectricity conversion of topological origin, may lead to new guiding principles of achieving high thermoelectric performance in topological magnets.
Methods
Sample preparation
Polycrystalline samples of MnGe were prepared by highpressure synthesis technique. Mn and Ge were first mixed with an atomic ratio of 1:1 and then melted in an arc furnace under an argon atmosphere. Afterwards, it was heated at 1073 K for 1 h under 4 GPa in a cubicanviltype highpressure apparatus. Powder Xray analyses confirmed B20type crystal structure (P2_{1}3) with no detectable impurity content (Supplementary Figure 4). A single crystal of MnSi was grown by the Czochralski method in tetraarc furnace under an argon atmosphere. Powder Xray diffraction pattern of the pulverized single crystal indicated that the sample was of single phase.
Thermoelectric measurement
The samples of MnGe and MnSi were cut into rectangular shape with the size of about 4 × 2 × 0.5 mm^{3}. Magnetic field was applied along the longest side of sample ([111] direction for the MnSi single crystal), which is parallel to the thermal current generated by 1 kΩ chip resistor. The temperature gradient was read by two Cernox thermometers attached to the sample with varnish. Voltage was measured through Manganin wires attached to the sample by solder pastes. Temperature and magnetic field were controlled by Physical Property Measurement System (PPMS), Quantum Design. The highfield measurements of thermopower were performed utilizing 25 T Cryogenfree Superconducting Magnet (CSM) installed at High Field Laboratory for Superconducting Materials of Institute for Materials Research (IMR), Tohoku University, Japan^{26}.
Transport and specific heat measurements
Magnetoresistivity and specific heat capacity were measured by using ACtransport option (AC excitation current of 23 Hz and 20 mA) and heat capacity option, respectively, with Physical Property Measurement System (PPMS). Magnetic field was applied parallel to electrical current for magnetoresistivity measurement. Highfield measurements of magnetization and longitudinal magnetoresistivity were performed utilizing nondestructive pulsed magnets installed at International MegaGauss Science Laboratory of Institute for Solid State Physics (ISSP), University of Tokyo, Japan. Magnetization was measured by the conventional induction method, using coaxial pickup coils. Resistivity was measured by the conventional four probe method with voltage preamplifiers using a numerical lockin technique with an excitation current of 25 kHz and 20 mA.
Photoemission spectroscopy
Photoemission spectroscopy on MnGe was performed with a VGScienta R4000WAL electron analyzer and a helium discharge lamp with the photon energy of 21.2 eV at the University of Tokyo. The energy resolution was set to 8 meV. The Fermi energy was determined from the photoemission spectrum of a gold film evaporated on the substrate, within an accuracy of better than ± 0.3 meV. The MnGe sample was fractured at 11 K in an ultrahigh vacuum better than 1 × 10^{−10} Torr. We confirmed the reproducibility of the temperaturedependent photoemission spectrum by the temperaturecycled measurements.
Calculations of band structure and Seebeck coefficient
Electronic structure calculations for ferromagnetic MnGe were performed using the density functional theory with a generalized gradient approximation^{27} as implemented in the quantumESPRESSO code^{28}. Ultrasoft pseudopotentials^{29} and the planewave basis set with cutoff energies of 50 Ry for wave functions and 400 Ry for charge densities were used. Seebeck coefficients were obtained within the semiclassical Boltzmann theory using wannier90 code^{30,31} (See Supplementary Figure 5 for the calculated band structure).
Data availability
The data sets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
We thank T. Yokouchi for fruitful discussions. We also thank Y. Yoshida and N. Mitsuishi for their cooperation on photoemission spectroscopy. This work was supported by JSPS KAKENHI (Grants Nos. 24224009 and 15H05456) and JST CREST (Grant No. JPMJCR16F1). A part of study was performed at International MegaGauss Science Laboratory of Institute for Solid State Physics, University of Tokyo, and at High Field Laboratory for Superconducting Materials, Institute for Materials Research, Tohoku University.
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Y.F. and N.K. synthesized polycrystalline samples of MnGe and performed transport and specific heat capacity measurements. A.K. grew the single crystal samples of MnSi. Y.F., N.K., A.N., and T.S. conducted photoemission spectroscopy under the supervision of K.I. T.K. and R.A. performed calculations of band structure and Seebeck coefficient. Y.F., N.K., A.M., H.M., K.A., and M.T. performed highfield measurements of magnetization and magnetoresistivity using pulsed magnet at ISSP. Y.F., N.K., J.S., and S.K. conducted highfield measurements of Seebeck effect using 25 T CSM at IMR under the supervision of A.T. and S.A. Y.F., N.K., and Y.To. analyzed the data and wrote the draft with support from T.S., K.I., T.K., R.A., and Y.Ta. All the authors discussed the results and commented on the manuscript. Y.To. conceived and organized the project.
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Fujishiro, Y., Kanazawa, N., Shimojima, T. et al. Large magnetothermopower in MnGe with topological spin texture. Nat Commun 9, 408 (2018). https://doi.org/10.1038/s41467018028571
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