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# Giant anomalous Nernst effect and quantum-critical scaling in a ferromagnetic semimetal

## Abstract

In conducting ferromagnets, an anomalous Nernst effect—the generation of an electric voltage perpendicular to both the magnetization and an applied temperature gradient—can be driven by the nontrivial geometric structure, or Berry curvature, of the wavefunction of the electrons1,2. Here, we report the observation of a giant anomalous Nernst effect at room temperature in the full-Heusler ferromagnet Co2MnGa, an order of magnitude larger than the previous maximum value reported for a magnetic conductor3,4. Our numerical and analytical calculations indicate that the proximity to a quantum Lifshitz transition between type-I and type-II magnetic Weyl fermions5,6,7 is responsible for the observed –Tlog(T) behaviour, with T denoting the temperature, and the enhanced value of the transverse thermoelectric conductivity. The temperature dependence of the thermoelectric response in experiments and numerical calculations can be understood in terms of a quantum critical-scaling function predicted by the low-energy effective theory over more than a decade of temperatures. Moreover, the observation of an unsaturated positive longitudinal magnetoconductance, or chiral anomaly8,9,10, also provides evidence for the existence of Weyl fermions11,12 in Co2MnGa.

## Main

Recent studies of phenomena arising from the coupling between spin and heat currents13,14,15 as well as types of anomalous Hall effects in various magnets1,2,16,17,18 have triggered renewed interest in the anomalous Nernst effect (ANE) as one of the topologically nontrivial phenomena and for its potential application to thermoelectric devices14,15,1923. The ANE is known to generate an electric voltage perpendicular to the applied temperature gradient $$\mathrm{\nabla} T$$ and magnetization $${\mathbf{M}}$$, namely $${\bf{E}}_{{\rm{NE}}} = Q_s\left(\mu{\mathrm{0}{\mathbf{M}} \times \boldsymbol{\nabla} T} \right)$$, where Qs is the anomalous Nernst coefficient and μ0 is the vacuum permeability. This transverse geometry enables a lateral configuration of the thermoelectric modules to efficiently cover a heat source even with a curved surface22, having a much simpler structure than the modules using the conventional Seebeck effect24. The ANE is not as thoroughly studied as the Seebeck effect, so there is an enormous scope for understanding the mechanism and controlling the size of ANE through new material synthesis. This should open a new avenue for identifying efficient energy-harvesting materials.

On the other hand, the size of ANE in generic magnetic materials is too small for practical applications, and it is essential to overcome this hurdle. Promisingly, recent theoretical and experimental investigations have indicated that the intense Berry curvature near Weyl points residing in the vicinity of the Fermi energy EF can potentially enhance the intrinsic ANE (refs 2,4,23,25,26). However, there is still no clear analytical framework and guiding principle for estimating and systematically increasing the size of the ANE for magnetic Weyl fermions by a few orders of magnitude. Therefore, experimental and theoretical studies of thermoelectric properties of Weyl magnets are critically important for both basic science and technological applications.

Recent first-principles calculations have shown that Co2TX (T = transition metal, X = Si, Ge, Sn, Al, Ga) are potential magnetic Weyl metals, where multiple Weyl points exist in the momentum space near the Fermi energy, EF (refs 11,12). In particular, first-principles calculations were performed to explain the experimentally observed giant anomalous Hall effect in the ferromagnet Co2MnAl27. However, the large anomalous Hall effect does not guarantee a large ANE, because the ANE at low temperature is given by the Berry curvature at the Fermi energy, whereas the anomalous Hall effect is determined by the sum of the Berry curvature for all the occupied states2,4,23. Therefore, to find a comprehensive understanding of the ANE of Weyl fermions and its correlation with the anomalous Hall effect over a few decades of temperature, we investigate the full Heusler ferromagnet Co2MnGa, which has a Curie temperature TC ≈ 694 K (Fig. 1a)28.

First, we describe our main result, namely the observation of a giant ANE in Co2MnGa at room temperature. Figure 2a shows the magnetic field dependence of the Nernst signal −Syx for $$\mathbf{B}$$ || [100], [110] and [111] and the heat current $$\mathbf{Q}$$ along [001] or [10$$\bar 1$$]. Clearly, −Syx increases with elevating temperature, reaching a record high value of $$\left| {S_{yx}} \right|$$ ≈ 6 μV K−1 at room temperature and approaching 8 μV K−1 at 400 K (Fig. 2b), which is more than one order of magnitude larger than the typical values known for the ANE (refs 3,4,23). The observed value of −Syx is large in comparison with the Seebeck coefficient Sxx (Supplementary Fig. 2). For example, $$\left| {S_{yx}{\mathrm{/}}S_{xx}} \right|$$ = 0.2, an unprecedented value for the Nernst angle θN ≈ tan θN = Syx/Sxx (Fig. 2a, right axis). In addition, we found there is almost no anisotropy in Syx within an error bar (Methods).

Similar to the ANE, the Hall resistivity is found to be very large, reaching approximately 15 μΩ cm at room temperature and its maximum of approximately 16 μΩ cm at around 320 K (Fig. 2c,d). The Hall angle θH ≈ tan θH = ρyx/ρxx is also large and exceeds 0.1 at room temperature. Figure 2c,e shows the field dependence of the Hall resistivity ρyx and the magnetization M. Both the Hall and Nernst effects show nearly the same magnetic field dependence as the magnetization curve, indicating that the anomalous contribution (proportional to M) to the Hall and Nernst effects is dominant and the normal contribution (proportional to B) is negligibly small at 300 K. The saturated magnetization, which is Ms ≈ 3.8μB at 300 K, gradually grows on cooling and reaches Ms ≈ 4μB at 5 K (Fig. 2f), consistent with the predicted value based on the Slater–Pauling rule. The anisotropy for M is negligibly small at 300 K, which is fully consistent with the cubic structure.

The observed $$\left| {\rho _{yx}} \right|$$ ≈ 15 μΩ cm is one of the largest known for AHE. Likewise, the Hall conductivity is also exceptionally large. Figure 3a shows the temperature dependence of the Hall conductivity, $$\sigma _{yx} = - \rho _{yx}{\mathrm{/}}\left( {\rho _{xx}^2 + \rho _{yx}^2} \right)$$, obtained at B = 2 T. Here, ρxx is the longitudinal resistivity, which is found to be isotropic as expected for a cubic system (Supplementary Fig. 2). Also −σyx increases monotonically on cooling and reaches −σyx ≈ 2,000 Ω−1 cm−1. This large value is of the same order of magnitude as that known for the layered quantum Hall effect. Namely, the anomalous Hall conductivity can reach a value as large as $$\sigma _{\mathrm{H}} = \frac{{e^2}}{{ha}}\sim 670\,{\kern 1pt} {\mathrm{\Omega }}^{ - 1}$$ cm−1, a value expected for a three-dimensional quantum Hall effect with a Chern number of unity, where h is Planck’s constant and a is the lattice constant6.

We have also evaluated the temperature dependence of the anomalous, transverse thermoelectric conductivity αyx as shown in Fig. 3b (Supplementary Information). Up to T ≈ 25 K, −αyx increases almost linearly with T, displaying a maximum around T ≈ 140 K, followed by a gradual decrease. Notice that the −αyx versus T curve closely resembles the functions −Tlog(T), and by plotting −αyx/T against log(T), as shown in Fig. 3c, we find the crossover between two distinct scaling behaviours αyx ~ T (at low temperatures) and αyx ~ −T log(T) (at high temperatures). The T-linear behaviour is consistent with the Mott relation between αyx at low temperature $$\left( {k_{\mathrm {B}}T \ll E_{\mathrm {F}}} \right)$$ and the energy derivative of σyx at T = 0, which predicts $$\alpha _{yx} \approx - \frac{{\pi ^2}}{3}\frac{{k_{\mathrm{B}}^2T}}{\left | e \right |}\frac{{\partial \sigma _{yx}}}{{\partial E_{\mathrm{F}}}}$$. However, the observed αyx ~ −T log(T) behaviour over a decade of temperature between approximately 30 K and 400 K constitutes a clear violation of the Mott relation. In the inset of Fig. 3b, we have also presented the numerical calculations for -αyx as a function of temperature. Although the maximum value of |αyx | and the overall functional dependence on T are in considerable agreement, the temperature scales where |αyx | attains its maximum value have an order of magnitude difference. Now we show how these results can be understood in terms of Weyl fermions.

The anomalous Hall conductivity at T = 0 is given by $$\sigma _{yx}\left( {E_{\mathrm{F}}} \right)$$= $$-\frac{{e^2}}{\hbar}\mathop {\sum}\nolimits_{n,{\boldsymbol{k}}} {\kern 1pt} {\mathrm{\Omega }}_{n,z}({\boldsymbol{k}}) \theta \left( {E_{\mathrm{F}} - \epsilon _{n,{\boldsymbol{k}}}} \right)$$, involving the summation over all occupied states, where Ωn,z(k) is the Berry curvature along the $$\hat{\mathbf{z}}$$ direction, n is the band index, and θ(x) is the unit step function. Consequently, $$\frac{{\partial \sigma _{yx}}}{{\partial E_{\mathrm{F}}}}$$= $$-\frac{{e^2}}{\hbar}\mathop {\sum}\nolimits_{n,{\boldsymbol{k}}} {\kern 1pt} {\mathrm{\Omega }}_{n,z}({\boldsymbol{k}})\delta \left( {E_{\mathrm{F}} - \varepsilon _{n,{\boldsymbol{k}}}} \right)$$, where δ(x) is the Dirac delta function, and only partially occupied bands can contribute to αyx. Therefore, to obtain a large value of yx/T|, it is essential to simultaneously enhance the density of states (DOS) and the Berry curvature around the Fermi pockets.

Thus, we focus on the largest Fermi surface (Fig. 1b, Supplementary Information, Supplementary Fig. 4) and find that the pertinent bands in the numerical calculations produce Weyl points around E0 ≈ 20 meV above the Fermi surface12 (Fig. 1c). Indeed, this Fermi surface has a large Berry curvature owing to its proximity to the Weyl points (as shown in Fig. 1d) and also a large DOS (Supplementary Fig. 5) due to the flatness of the dispersion. These Weyl points are located on the zone boundary along the U–Z–U line (Fig. 1d) at $$\pm {\mathbf{k}}_0$$ = $$\pm \frac{{2\pi }}{a} \times 0.15 [110]$$ and along the nodal direction they can be modelled with the low-energy Hamiltonians, Hj ≈ sgn(j)[v2(k − sgn(j)k0) · $$\widehat {\mathbf{k}}_0$$ + v1(k − sgn(j)k0) · $$\widehat {\mathbf{k}}_0$$σz], where j = ±1 denote the chirality of the right- and left-handed Weyl points, v1 and v2 are two velocity parameters with the tilt parameter v2/v1 = 0.99, and v1 ≈ 105 m s−1. The strength of the tilt parameter v2/v1 is extremely close to the critical value v2/v1 = 1, which describes a Lifshitz quantum critical point, separating the type-I Weyl fermions (v2/v1 < 1, with a vanishing DOS at the Weyl points, causing only an electron or a hole pocket) from the type-II Weyl fermions (v2/v1 > 1, with a finite DOS at the Weyl points, where the electron and hole pockets touch). At the Lifshitz quantum critical point, the energy derivative of the Hall conductivity displays the following singular behaviour

$$\begin{array}{l}\frac{{\partial \sigma _{yx}}}{{\partial E}} \approx -\frac{{e^2}}{{4\pi ^2\hbar ^2v_1}}{\mathrm{log}}\left( {\frac{{\left| {E - E_0} \right|}}{{C\left( {k_0} \right)\hbar v_1k_0}}} \right),\cr C\left( {k_0} \right) = \frac{{8{\kern 1pt} {\mathrm{sin}}\left( {k_0a{\mathrm{/}}2} \right)}}{{k_0a}}{\mathrm{exp}}\left[ { - 4{\kern 1pt} {\mathrm{tan}}\left( {k_0a{\mathrm{/}}4} \right)} \right]\end{array}$$
(1)

Away from the critical point (inside type-I or type-II phases) this log divergence gets cut off by the distance from the critical point defined as δ = 1 − v2/v1, leading to $$\frac{{\partial \sigma _{yx}}}{{\partial E}} \propto {\mathrm{log}}\left| \delta \right|$$ (Supplementary Information). Since δv1k0 < $$\left| {E_{\mathrm{F}} - E_0} \right|$$ < v1k0, these Weyl fermions belong to the quantum critical regime. The numerically calculated $$\frac{{\partial \sigma _{yx}}}{{\partial E}}$$ is shown in Fig. 3e. By analysing the sharp peak around E0 ≈ +20 meV, we have clearly identified the log-divergent behaviour in Fig. 3f.

Intriguingly, the low-energy theory suggests that the effects of the Lifshitz quantum critical point over a wide range of temperatures can be captured in terms of a scaling function

$$\alpha _{yx}(T,\mu ) = \frac{{k_{\mathrm{B}}^2\left | e \right |T_0}}{{12\hbar ^2v_1}}G\left( {\frac{T}{{T_0}},\frac{{\mu - E_0}}{{k_{\mathrm{B}}T_0}}} \right)$$
(2)

where T0 ≈ exp(1) × Tm, and Tm is the temperature where yx| attains its maximum value $$\alpha _{yx}^{{\mathrm{max}}}$$. By setting μ = E0, we have obtained the scaling function at the critical point as

$$\begin{array}{l}\alpha _{yx}(T,0) = -\alpha _{yx}^{{\mathrm{max}}}\left({k_0}\right){\mathrm{exp}}(1)\frac{T}{{T_0}}{\mathrm{log}}\left( {T{\mathrm{/}}T_0} \right),k_{\mathrm{B}}T_0 \approx C\left( {k_0} \right)\hbar v_1k_0\cr \alpha _{yx}^{{\mathrm{max}}}\left( {k_0} \right) = -\frac{{k_{\mathrm{B}}^2\left |e \right | T_0}}{{12\hbar ^2v_1{\kern 1pt} {\mathrm{exp}}(1)}} \approx \frac{{50}}{a}{\mathrm{sin}}\left({\frac{{k_0a}}{2}} \right){\mathrm{exp}}\left[{-4{\kern 1pt} {\mathrm{tan}}\left( {\frac{{k_0a}}{2}} \right)} \right]{{{\rm{A}}\,{\rm{K}}}}^{ - 1}{\kern 1pt} {\mathrm{m}}^{ - 1}\end{array}$$
(3)

where the lattice constant is measured in ångstroms. Notice that the maximum value $$\alpha _{yx}^{{\mathrm{max}}}\left( {k_0} \right)$$ does not depend on the velocity v1 of the Weyl fermions. In contrast, the slope $$\alpha _{yx}{\mathrm{/}}T \approx -\left( {k_{\mathrm{B}}^2\left| e \right|} \right){\mathrm{/}}\left( {12\hbar ^2v_1} \right)$$ explicitly depends on v1. We have found that $$\alpha _{yx}^{{\mathrm{max}}}\left( {k_0} \right)$$ is maximized for k0 ≈ 0.14 × (2π/a) and the numerically determined location of Weyl points is indeed very close to this value. After substituting a = 5.8 × 10−10 m and k0 = 0.15 × (2π)/a we find $$\alpha _{yx}^{{\mathrm{max}}}$$ ≈ 1.5 A K−1 m−1, which is about three times smaller than the experimentally and numerically obtained values. We note that the estimation for T0 can be slightly modified by microscopic details and even after ignoring the factor C(k0) we would obtain $$\alpha _{yx}^{{\mathrm{max}}}$$ ≈ 1 A K−1 m−1. Such O(1) uncertainty regarding the value of T0 and the presence of a few additional pairs of Weyl points in the vicinity of the Fermi level can lead to the O(1) difference between this estimate and the experimentally and numerically determined values of $$\alpha _{yx}^{{\mathrm{max}}}$$.

We have also computed αyx(T, μ) for μ ≠ E0, in order to demonstrate its crossover from T log(T) behaviour to $$\alpha _{yx} \approx -\alpha _{yx}^{{\mathrm{max}}}\frac{T}{{T_0}}{\mathrm{log}}\left( {\frac{{\left| {\mu - E_0} \right|}}{{k_{\mathrm{B}}T_0}}} \right)$$ when $$k_{\mathrm{B}}T \le \left| {\mu - E_0} \right|$$ (the solid black line in Fig. 3c). In the real material, the scale T0 (which is proportional to v1) is an order of magnitude smaller than the numerical one, which we can attribute to the correlation and disorder-driven suppression of the bandwidth or the velocity of Weyl fermions. This order-of-magnitude suppression is consistent with the enhancement of the Sommerfeld coefficient for the specific heat by a factor of seven observed in the experiment (Supplementary Information). Therefore, the experimental and numerical values for αyx/T differ by an order of magnitude. Finally in Fig. 3c we have compared the dimensionless scaling function G obtained from the experimental and numerical data with the predictions of low-energy theory. After selecting T0 = 550 K and T0 = 6,000 K, respectively, for the experimental and numerical results, the G function matches with the low-energy predictions for more than a decade of temperature. The crossover to the Mott regime occurs when (μ − E0)/(kBT0) ≈ −0.05. Thus, our analysis indicates that the proximity of the Weyl fermions to a quantum Lifshitz transition is responsible for the large values of $$\alpha _{yx}^{{\mathrm{max}}}$$, the slope αyx/T at low temperatures, as well as the logarithmic violation of the Mott relation at high temperatures.

To provide further evidence for the existence of Weyl fermions in the vicinity of the Fermi energy, we have performed angle-dependent magnetoresistance measurements to reveal the chiral anomaly. In Weyl metals, the number imbalance between the Weyl nodes with opposite chirality is expected to cause a negative longitudinal magnetoresistance when the electric current $$\mathbf{I}$$ and magnetic field $$\mathbf{B}$$ are parallel $$\left( {\mathbf{I}|\,|\mathbf{B}} \right)$$, while the transverse $$\left( {\mathbf{I} \bot \mathbf{B}} \right)$$ magnetoresistance remains positive8,9,10. However, such behaviour can be masked in a ferromagnetic Weyl metal for weak magnetic fields, since field-induced suppression of magnetic fluctuations (which give a decrease in scattering rate) can cause positive magnetoconductance for an arbitrary angle between $$\mathbf{I}$$ and $$\mathbf{B}$$. Therefore, it is imperative to perform the measurements at sufficiently low temperatures and in the presence of strong enough magnetic fields to minimize the effects of magnetic fluctuations. Figure 4 and Supplementary Fig. 6 show the B-dependence of the longitudinal and transverse magnetoconductivity σxx(B) at T = 0.1 K and 5 K, and in B ≤ 16 T with different current directions (Supplementary Information). We find positive and negative magnetoconductivities, respectively, for $$\mathbf{I}|\,|\mathbf{B}$$ and $$\mathbf{I} \bot \mathbf{B}$$ (Fig. 4a), and a cos2(θ) dependence on the angle θ between $$\mathbf{I}$$ and $$\mathbf{B}$$ (Fig. 4b) in high magnetic fields. This provides a clear signature of the chiral anomaly down to T = 0.1 K, giving evidence that the Weyl excitations remain gapless at very low temperatures, even in the presence of strong electronic interactions.

In summary, our comprehensive study indicates that the giant Nernst effect in Co2MnGa originates from the large intrinsic transverse thermoelectric conductivity of Weyl points occurring near the Fermi energy. In fact, $$\left| {\alpha _{yx}} \right|$$ ≈ 4 A K−1 m−1 for Co2MnGa is very large compared to the other typical ferromagnets $$\left| {\alpha _{yx}} \right|$$  ≈ 0.01–1 A K−1 m−1 as shown in Fig. 4c (Supplementary Information). Our combined experimental, numerical and effective theory based analysis provides guidance for further increasing αyx in Weyl magnets by decreasing the lattice constant. The observed large ANE with small anisotropy for Co2MnGa indicates that Weyl magnets are potentially useful for creating efficient thermopile devices for thermoelectric power generation and heat current sensor as well as for studying the quantum critical effects of Weyl fermions in spintronics (Supplementary Information).

## Methods

### Co2MnGa single crystal and experimental

Single crystals of Co2MnGa were prepared by the Czochralski method after making polycrystalline samples by arc-melting Co, Mn and Ga in an appropriate ratio. Co2MnGa is known to be highly resistive to oxidation and we found that the sample is stable in air29. As-grown single crystals were used for all the measurements except powder X-ray diffraction, as described below. Our analyses using both inductively coupled plasma spectroscopy and energy dispersive X-ray analysis indicate that our single crystals are stoichiometric within a resolution of a few per cent.

The samples were oriented by the Laue backscattering method, and then cut into a bar-shape by spark erosion. All the surfaces were polished to get flat and mirror-like surfaces. Three samples were prepared by the same procedure to perform the measurements in the different geometries, #100 for $$\mathbf{B}$$ || [100], #110 for $$\mathbf{B}$$ || [110] and #111 for $$\mathbf{B}$$ || [111]. All the three samples have nearly the same dimensions, with a typical size of 7.5 × 2.0 × 1.3 mm3. For both Seebeck and Nernst effect measurements, the distance between the thermometers lth was set to be ~4.0 mm. Given the width of the temperature probes, the error bars of the Seebeck and Nernst signals mainly come from the uncertainties of the corresponding geometrical factors and are estimated to be of the order of 10%. All the transport measurements including the electric resistivity, Hall, Seebeck and Nernst effects, as well as thermal conductivity, were measured for each bar-shape sample using a commercial system (PPMS, Quantum Design). Magnetization was measured for a small piece of the single crystal by using a commercial SQUID magnetometer (MPMS, Quantum Design). The sample was reshaped into a cubic-like shape to reduce the shape anisotropy before the magnetization measurements.

### Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

This work was supported by CREST (JPMJCR15Q5) by Japan Science and Technology Agency, by Grants-in-Aid for Scientific Research (grant numbers 16H02209, 25707030), by Grants-in-Aid for Scientific Research on Innovative Areas “J-Physics” (grant numbers 15H05882 and 15H05883) and Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers (grant number R2604) from the Japanese Society for the Promotion of Science. P.G. was supported by JQI-NSF-PFC and LPS-MPO-CMTC (at the University of Maryland) and start-up funds from the Northwestern University. The use of the facilities of the Materials Design and Characterization Laboratory at the Institute for Solid State Physics is appreciated.

## Author information

### Affiliations

1. #### Institute for Solid State Physics, University of Tokyo, Kashiwa, Japan

• Akito Sakai
• , Agustinus Agung Nugroho
• , Rombang Sihombing
• , Rieko Ishii
• , Daisuke Nishio-Hamane
•  & Satoru Nakatsuji
2. #### CREST, Japan Science and Technology Agency (JST), Kawaguchi, Japan

• Akito Sakai
• , Takashi Koretsune
• , Michi-To Suzuki
• , Ryotaro Arita
•  & Satoru Nakatsuji
3. #### Faculty of Mathematics and Physics, Kanazawa University, Kanazawa, Japan

• Yo Pierre Mizuta
4. #### Center for Emergent Matter Science (CEMS), RIKEN, Hirosawa, Wako, Japan

• Yo Pierre Mizuta
• , Takashi Koretsune
• , Michi-To Suzuki
• , Nayuta Takemori
•  & Ryotaro Arita
5. #### Faculty of Mathematics and Natural Sciences, Bandung Institute of Technology, Bandung, Indonesia

• Agustinus Agung Nugroho
•  & Rombang Sihombing
6. #### Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, MD, USA

• Pallab Goswami
7. #### Department of Physics and Astronomy, Northwestern University, Evanston, IL, USA

• Pallab Goswami

### Contributions

S.N. conceived and planned the experimental project. A.N., R.S. and S.N. worked on the single-crystal growth and preparation of samples. A.S. and R.S. carried out the transport and low-temperature measurements and analysed the data. Y.M., T.K., M.S., N.T. and R.A. performed the first-principles calculations. P.G. formulated the quantum critical theory and scaling analysis of the experimental and numerical results. S.N. performed the scaling analysis. R.I. performed the chemical analyses. D.H. acquired the electron diffraction image. S.N., A.S. and P.G. wrote the paper with inputs from Y.M. and R.A. All authors discussed the results and commented on the manuscript.

### Competing interests

The authors declare no competing interests.

### Corresponding author

Correspondence to Satoru Nakatsuji.

## Supplementary information

1. ### Supplementary Information

Supplementary Figures S1–S7, Supplementary Table S1