Abstract
Applying a temperature gradient in a magnetic material generates a voltage that is perpendicular to both the heat flow and the magnetization. This phenomenon is the anomalous Nernst effect (ANE), which was long thought to be proportional to the value of the magnetization. However, more generally, the ANE has been predicted to originate from a net Berry curvature of all bands near the Fermi level (E_{F}). Subsequently, a large anomalous Nernst thermopower (\({\boldsymbol{S}}_{{\boldsymbol{yx}}}^{\boldsymbol{A}}\)) has recently been observed in topological materials with no net magnetization but a large net Berry curvature [Ω_{n}(k)] around E_{F}. These experiments clearly fall outside the scope of the conventional magnetization model of the ANE, but a significant question remains. Can the value of the ANE in topological ferromagnets exceed the highest values observed in conventional ferromagnets? Here, we report a remarkably high \({\boldsymbol{S}}_{{\boldsymbol{yx}}}^{\boldsymbol{A}}\)value of ~6.0 µV K^{−1} in the ferromagnetic topological Heusler compound Co_{2}MnGa at room temperature, which is approximately seven times larger than any anomalous Nernst thermopower value ever reported for a conventional ferromagnet. Combined electrical, thermoelectric, and firstprinciples calculations reveal that this highvalue of the ANE arises from a large net Berry curvature near the Fermi level associated with nodal lines and Weyl points.
Introduction
Conventional thermoelectric devices are based on the Seebeck effect in a twoterminal geometry, where two electronic reservoirs at different temperatures result in an electric voltage that arises in the direction of the imposed temperature gradient^{1,2}. For applications, however, this configuration has considerable drawbacks because the heat reservoir is forced to be part of the electrical circuit. The problem is achieving good thermal insulation while maintaining good electrical conduction. To circumvent this issue, multiterminal thermoelectric devices have recently attracted increasing attention^{3,4,5,6,7,8,9,10,11,12,13} because they allow for the spatial separation of the heat reservoir from the electric circuitry. One possible implementation of this kind is based on the Nernst effect, which is the generation of a transverse thermoelectric voltage by a temperature gradient and a magnetic field, applied normal to each other^{14,15,16}. Nernst devices are, in principle, simpler to integrate technologically than conventional thermoelectrics because there is no need for both p and ntype materials.
A ferromagnetic material that generates an anomalous transverse voltage mutually perpendicular to both heat current and magnetization demonstrates the socalled anomalous Nernst effect (ANE)^{3,4,17,18,19,20}. In contrast to the ordinary Nernst effect, the ANE increases rapidly in a low magnetic field. Thus, ANE can give rise to a large Nernst thermopower at a low magnetic field. The ANE is the thermoelectric counterpart of the anomalous Hall effect (AHE) and was first observed in topologically trivial ferromagnets. Therefore, it was believed for a long time that the ANE is proportional to the magnetization of a material itself. Despite many efforts, the maximum value of anomalous Nernst thermopower in ferromagnetic systems is limited and bound to below ~1 µV K^{−1}^{9}. However, it was recently found that the AHE is not directly related to the magnetization of a material but is derived more generally from the Berry curvature [Ω_{n}(k)] of the bands near the Fermi level (E_{F})^{5,6}. This outcome was realized along with the discovery that the AHE is linked to the summation of Ω_{n}(k) over all the occupied bands below E_{F}^{21}.
The Ω_{n}(k) is a local gauge field associated with the Berry phase of the band structure and determines the topological aspects of a material. In contrast to previous understanding, it is possible to control the Berry curvature and thus the intrinsic AHE and ANE by suitable manipulations of symmetries and band structures, independent of the finite value of the magnetization. As a result of these considerations, the AHE and ANE have recently been tuned in a distinct manner by tailoring the distribution of Ω_{n}(k) in topological matter, such as in Dirac metals or chiral antiferromagnets^{5,6,7,8,9,22,23,24,25,26}. These investigations have led to the observation of an AHE and an ANE in the absence of ferromagnetism. Furthermore, it has been demonstrated that by manipulating the crystal and magnetic space group symmetry and details of the band structure, the anomalous Hall conductivity can be tuned from zero to a large value^{27}. The Berry curvaturedriven anomalous Hall and Nernst effects have been measured in the noncollinear antiferromagnets Mn_{3}Sn and Mn_{3}Ge without sizeable magnetization, where Berry curvature effects are generated from the noncollinear spin structure^{9,22,23,24,25}. A large enhancement in ANE has been observed in ultrathin films of ferromagnetic metal, including Fe, Co, Ni, and Ni_{80}Fe_{20} due to intrinsic effects and skew scattering through spinorbit coupling^{28}. Additionally, in the Dirac semimetal Cd_{3}As_{2,} the AHE and ANE were measured due to Berry curvature effects produced by the separation of the Weyl nodes in a magnetic field^{7,8}. The results of both experiments clearly fall outside the scope of the conventional model that scales the ANE with the magnetization of the system, but a worthy question remains. Is it possible to push the limit of ANE beyond the magnetization scaling relation in a ferromagnet? Hence, this work desires to go beyond the recent experiments, identify a material with a large Ω_{n}(k) at E_{F} and perform Nernst measurements on this system.
In the present study, we carried out thermoelectric experiments on the ferromagnetic topological Heusler compound Co_{2}MnGa, which has previously been shown to exhibit a large AHE and a large Berry curvature distribution around E_{F}^{27,29}. Co_{2}MnGa is a full Heusler compound that crystallizes in the L2_{1} structure with space group Fm\(\bar 3\)m and a lattice constant a = 5.77 Å (Fig. 1(a)). The structure consists of four interpenetrating fcc sublattices, of which two are formed by Co atoms. The other two sublattices are formed by the Mn and Ga atoms. The structure can also be implicitly viewed as a zincblendetype sublattice, formed by one Co and Ga in which tetrahedral and octahedral holes are occupied by the second Co and Mn atom, respectively. The Co atoms occupy the Wyckoff position 8c (1/4,1/4,1/4), whereas the Mn and Ga atoms occupy the Wyckoff positions 4a (0, 0, 0) and 4b (1/2,1/2,1/2), respectively. Without spinorbit coupling, there are three nodal lines near E_{F} (Fig. 1(b), (c)). The spinorbit coupling lifts the degeneracy in these lines by breaking mirror symmetry depending on the magnetization direction, and a small gap develops (Fig. 1(c) and Fig. S8). The gapping out of nodal lines induces a nonzero Ω_{n}(k) into the system, which makes Co_{2}MnGa a promising candidate for the study of Berry curvature related effects. A recent angleresolved photoemission spectroscopy (ARPES) study in a Co_{2}MnGa singlecrystal shows the position of the nodal lines close to the Fermi level^{30}.
Materials and methods
Co_{2}MnGa singlecrystal growth
Single crystals of Co_{2}MnGa were grown using the Bridgman–Stockbarger crystal growth technique. For growing a singlecrystal, we first synthesized a polycrystalline sample by induction melting from the highquality stoichiometric amounts of elemental cobalt (99.999%), manganese (99.999%), and gallium (99.999%) in an alumina crucible. The inductionmelted sample was then crushed into powder form and packed in a customdesigned sharpedged alumina tube that was sealed in a tantalum tube. The sample was heated to 1523 K and soaked for 10 h to ensure homogeneity of the melt and then slowly cooled to 1023 K. The crystallinity of the asgrown crystal was checked by whitebeam backscattering Laue Xray diffraction at room temperature. The aligned crystal was cut into barshaped samples for transport and magnetization measurements. The typical dimension of the crystals used for electrical and thermal transport measurements was 7.5 × 2 × 0.5 mm^{3}.
Magnetization and electrical transport measurement
Magnetization measurements were performed using a Quantum Design vibrating sample magnetometer (MPMS). The electrical transport properties were measured in a Quantum Design physical property measurement system (PPMS, ACT option). The standard fourprobe method was used for electrical transport measurement. To correct for contact misalignment, the measured raw data were field symmetrized and antisymmetrized.
Thermoelectric transport measurements
The zerofield temperaturedependent Seebeck coefficient was measured by the oneheater and twothermometer configuration using the thermal transport option (TTO) of the PPMS (Quantum Design). Thermal conductivity measurements were performed in PPMS using the TTO option. MagnetoSeebeck and Nernst thermoelectric measurements were carried out using a oneheater and twothermometer configuration on a PPMS. The instrument was controlled by software programmed using LabVIEW. The measurement was performed in the temperature range of 60–340 K, and magnetic fields were swept in both directions. The temperature gradient was generated using a resistive heater connected to the goldcoated copper plate at one end of the sample. The thermal gradient, ∇T, was applied along the [111] direction of the sample, and a magnetic field was applied along the [01\(\bar 1\)] direction. For the heat sink, a goldplated copper plate was attached to the puck clamp. To measure the temperature difference, two goldplated copper leads were attached directly to the sample using silver epoxy along the thermal gradient direction. The distance between the thermometers was ~4 mm. The ∇T was typically set to 1−3% of the sample temperature. The transverse voltage due to the thermal gradient was measured by attaching two copper wires orthogonal to the heat gradient direction of the sample using silver epoxy. The Seebeck thermopower was estimated using the relation S_{xx} = V_{x}/∇T_{x}, where V_{x} is the longitudinal voltage. The Nernst thermopower was estimated as S_{yx} = L_{x}V_{y}/(L_{y}∇T_{x}), where V_{y} is the transverse voltage, L_{x} is the distance between two temperature leads, and L_{y} is the distance between two voltage wires.
Nernst data survey
We reproduced the magnetizationdependent anomalous Nernst thermopower value for different ferromagnets well below their Curie temperatures from a previous report by Ikhlas et al.^{9}. Specifically, Mn_{3}Sn (200 K; ref. ^{9}.), Fe_{3}O_{4} (300 K, B < 0.8 T; ref. ^{20}.), Co/Ni films (300 K; ref. ^{31}.), L1_{0}FePt (300 K; ref. ^{31}.), D0_{22}Mn_{2}Ga (300 K; ref. ^{31}.), L1_{0}MnGa (300 K; ref. ^{31}.), L1_{0}FePd (300 K; ref. ^{31}.), Nd_{2}Mo_{2}O_{7} (T < Tc = 73 K, B = 1 T [111]; ref. ^{32}.), Fe (300 K; ref. ^{33}.), Co (300 K; ref. ^{33}.), MnGe (140 K, B > 2 T; ref. ^{34}.), MnGe (100 K, B > 5 T; ref. ^{34}.), MnGe (20 K, B < 14 T; ref. ^{34}) and Pt/Fe multilayer with N = 1–9 (300 K, B < 5 T; ref. ^{35}) were reproduced.
Ab initio calculations
For our ab initio calculations, we employed densityfunctional theory (DFT) using VASP^{36}. The exchangecorrelation energy is described via PBE. For integrations in kspace, we used a grid of 19 × 19 × 19 points. We extracted Wannier functions from the resulting band structure using WANNIER90^{37} to set up a tightbinding Hamiltonian, which reproduced the DFT band structure within a few meV. With this Hamiltonian, we calculated the Berry curvature \({\vec{\mathrm \Omega }}_n\) via:
where m and n are the eigenstates and ε are the eigenenergies of the Hamiltonian H.
The anomalous Hall conductivity \((\sigma _{xy}^A)\) was calculated using the equation^{5}:
where f_{nk} is the FermiDirac distribution function with the band index n and the wave vector k. Parameter e denotes the elementary charge and \(\hbar\) is the reduced Planck constant. The integral was performed across the whole Brillouin zone for all bands below E_{F}. The anomlous Nernst conductivity (\(\alpha _{yx}^A\)) was calculed as^{5}:
where ε_{nk} is the band energy, k_{B} is the Boltzmann constant and β = k_{B}T.
To realize integrations over the whole Brillouin zone for AHE and ANE, we employed a mesh of 251 × 251 × 251 kpoints, and this mesh was carefully checked to ensure convergence.
Results and discussion
The single crystals used in this study were synthesized via the Bridgman–Stockbarger crystal growth technique. Prior to the transport experiments, the asgrown crystals were characterized by Laue Xray diffraction (XRD), where sharp spots in the pattern indicate high crystal quality (see Materials and methods for details, Supplementary Fig. S1). The transport experiments were performed in a temperaturevariable cryostat (Physical Property Measurement System from Quantum Design) in a helium atmosphere equipped with a 9 T magnet. All transport experiments were performed within the [111] plane of the Co_{2}MnGa crystal with a magnetic field μ_{0}H applied in the [01\(\bar 1\)] direction. In particular, two measurement configurations were used in our study. First, longitudinal measurements were done, where the electric voltage response was measured in the direction of the applied electrical current (I) or heat current (Q) (the corresponding transport parameters are marked with the subindex xx). Second, transverse measurements were done, where the electric voltage response was measured normal to the applied electrical current or temperature gradient (the corresponding transport parameters are marked with the subindex yx, Fig. 1(d)). To gain deeper insights into the experimental results, ab initio calculations were performed using densityfunctional theory as implemented in VASP^{36} as a first step, followed by the extraction of a Wannier tightbinding Hamiltonian via WANNIER90^{37}. With this Hamiltonian, we can evaluate the Berry curvature and consecutively evaluate the AHE and ANE (for further details, see the Materials and methods section).
In the first set of transport experiments, we investigated the longitudinal thermoelectric magnetotransport of Co_{2}MnGa. Figure 2(a) shows the electrical resistivity (ρ_{xx}) and Seebeck coefficient (S_{xx}) as a function of temperature T. The decreasing ρ_{xx} with decreasing T reveals the metallic behavior of the compound with a value of 130 μΩ cm at 300 K. The temperaturedependent Seebeck coefficient (S_{xx}) for Co_{2}MnGa shows a negative sign throughout the measurement temperature. This observation indicates electrondominated transport in the thermoelectric signal. Furthermore, the magnitude of S_{xx} increases with increasing T and reaches a value of approximately −26 μV K^{−1} at 300 K. We would like to note that the temperaturedependent S_{xx} does not show any suppression at high temperatures. This lack of suppression is a clear indication of the absence of a bipolar effect in the system (Supplementary discussion, Figs. 2a and S2). The magnetoSeebeck data show almost no fielddependence of S_{xx} within the measured temperature range (Supplementary Fig. S3).
Next, we measured the magnetization (M) of our sample. The compound is known to be a ferromagnet with a relatively high Curie temperature of (T_{C} = 687 K)^{38}. In Fig. 2(b), we present temperaturedependent magnetization data for Co_{2}MnGa. In agreement with the literature, we observed ferromagnetic behavior. From the data taken as a function of the sweeping magnetic field, a saturation magnetization value of approximately M_{sat} = 3.9 μ_{B} f.u.^{−1} was measured at 50 K, which slightly decreased with increasing temperature to M_{sat} = 3.7 μ_{B} f.u.^{−1} at 300 K.
Furthermore, we measured the anomalous Hall effect of Co_{2}MnGa. We estimated the total Hall conductivity (σ_{xy}) from the measured Hall resistivity ρ_{yx} and the longitudinal resistivity ρ_{xx} as:
In Fig. 2(c), we present σ_{xy} as a function of the magnetic field at different base temperatures. As expected for a ferromagnetic material, σ_{xy} increases with μ_{0}H and saturates when the saturation magnetization is reached despite a small slope contributed by the ordinary Hall effect (compare with Fig. 2(b)). The anomalous Hall conductivity (AHC) \(( {\sigma _{xy}^A} )\) was estimated by extrapolating the slope of the highfield data points (Supplementary Fig. S4) to the zerofield value. The observed values of \(\sigma _{xy}^A\) and its temperature dependence are consistent with previous reports^{27}. As shown in the inset of Fig. 2(c), a value of \(\sigma _{xy}^A\)~ 1260 S cm^{−1} was estimated at 60 K and gradually decreases with increasing temperature and reaches ~845 S cm^{−1} at 300 K. From the magnetic fielddependence, one would naively expect \(\sigma _{xy}^A\) to only scale with M. However, as shown in our ab initio calculation in Fig. 2(d), the AHC conductivity in Co_{2}MnGa strongly depends on the position of the Fermi level, which is due to the Berry curvature effect discussed above. In the case of the intrinsic AHE, Ω_{n}(k) adds a term to the group velocity of the electron^{5} (see Eq. 2, Materials and methods).
Recent results of ARPES measurements on our sample show that the actual Fermi level E_{F} in our sample is slightly shifted toward the valence bands compared to that of the theoretical charge neutral point E_{0}, which is estimated by counting the intrinsic electron number by E_{F }– E_{0} = −80 meV^{30}. Based on the ab initio calculations and ARPES results, we predicted an AHC value of \(\sigma _{xy}^A\left( {E_{\mathrm{F}}} \right) =\)1242 S cm^{−1} at T = 0 K. Extrapolating the measured \(\sigma _{xy}^A\) linear to T = 0 K, we obtained a value of \(\sigma _{xy}^A\) = 1362 S cm^{−1,} which is reasonably close to the experimentally measured value.
Next, we investigated the Nernst effect of Co_{2}MnGa. In Fig. 3(a) (also see Supplementary Fig. S5), the fielddependence of the Nernst thermopower (S_{yx}) is shown at different T. At all T, the Nernst signal increases rapidly at low magnetic fields (μ_{0}H) and saturates above μ_{0}H ~ 1 T despite a small remaining increase due to ordinary Nernst signal contribution. These features are consistent with the M and \(\sigma _{xy}^A\) data. We, therefore, attribute the saturation value of the Nernst signal to the ANE and the small increasing contribution beyond 1 T to the ordinary Nernst effect. In a similar procedure as that for the Hall data, the ordinary and anomalous contributions to the Nernst thermopower can then be separated by fitting the highfield slope of the total Nernst thermopower up to μ_{0}H = 9 T (Supplementary Fig. S4). In doing so, we found that the ordinary Nernst effect reached values of \(S_{yx}^O\)= 50 nV K^{−1} at room temperature and μ_{0}H = 1 T. The ANE, on the other hand, reached remarkably high values of \(S_{yx}^A\) ~6.0 µV K^{−1} at 300 K (inset of Fig. 3(a)), for example. Similar to the AHE, the ANE of conventional ferromagnets is proportional to the magnetization, that is, \(S_{yx}^A\)=\(N_{yx}^A\)μ_{0}M, where \(N_{yx}^A\)is the anomalous Nernst coefficient, ranging from 0.05 to 1 μV K^{−1}T ^{−1} at 300 K. According to this relation, Co_{2}MnGa should maximally exhibit \(S_{yx}^A\) = 0.9 µV K^{−1} (see Materials and methods for details). This value is ~7 times lower than that observed in the experiment and indicates that the ANE arises from a mechanism that is distinct from the linear scaling of conventional ferromagnets. We also plotted the temperature (T) normalized Nernst effect (S_{yx}/T) at different temperatures for the Co_{2}MnGa samples. The anomalous Nernst amplitude shows nearly Tindependent behavior at high temperatures (Supplementary Fig. S6).
In the Nernst effect and thermal conductivity, the electron and hole contributions toward the total transport are additive in nature. Therefore, to understand the multicarrier or bipolar effect in the system, we performed temperaturedependent thermal conductivity measurements of Co_{2}MnGa up to room temperature. The analysis of thermal conductivity data shows that the κ_{lat} at high temperatures follows the T^{−1} relationship (for details, see Supplementary Fig. S7 and discussion). Hence, the temperaturedependent Seebeck coefficient (discussed above and in Supplementary information) and thermal conductivity data provide important experimental evidence to exclude the multicarrier effect and, therefore, verify the role of topological effects for the large anomalous Nernst signal in Co_{2}MnGa.
To gain deeper insight into the large ANE in Co_{2}MnGa, we calculated the transverse thermoelectric conductivity (α_{yx}) plotted in Fig. 3(b)). Using the measured components of ρ_{xx}, ρ_{yx}, S_{xx}, and S_{yx}, we estimated α_{yx} via:
Based on the linear scaling relation for conventional ferromagnetic metals, one would expect that α_{yx} increases linearly with decreasing T and follows the temperature dependence of M^{17}. However, in Co_{2}MnGa, we observed the inverse effect, whereby \(\alpha _{yx}^A\) increased with increasing temperature, although the saturation magnetization decreased. This outcome is another indication that the anomalous Nernst signal in ferromagnetic Co_{2}MnGa arises from a different mechanism than the ANE in conventional ferromagnets. It is noteworthy that the AHC in Co_{2}MnGa also does not follow the conventional scaling relation. In topological ferromagnetic metals, the anomalous signal is purely intrinsic, is universal and arises due to the Berry curvature [Ω_{n}(k)] effect rather than the magnetization of the material^{21}. These characteristics suggest that the ANE and the AHE in Co_{2}MnGa may share a common origin, which is the Berry curvature effect.
To evaluate this possibility, we calculated \(\alpha _{yx}^A\), employing Eq. 3^{5} (see Materials and methods). The {(ε_{nk} − μ)f_{nk} + k_{B}T ln [1 + e^{−β(ε}_{nk}^{ − μ)}]} term has finite values only around the Fermi energy. Therefore, while \(\sigma _{xy}^A\) is related to the summation of Ω_{n}(k) of all the occupied bands below E_{F}, \(\alpha _{yx}^A\) is related to Ω_{n}(k) of all bands near E_{F}. As discussed earlier, the Fermi level is ~ − 80 meV shifted from neutral point E_{0}, which provided access to all three nodal lines. Thus, a suitable position of E_{F} results in a strong Berry curvature effect in Co_{2}MnGa. Our theoretical calculations show a very large ANC in the system, which is linked to the high Berry curvature induced by the gapping of the nodal lines (Fig. 3(c)). Like the AHC, the ANC shows a strong Fermi level dependence, which is not captured by the conventional theory for ferromagnets. As the suitable position of the Fermi level is important to observe the Berry curvature effect in anomalous transport, it would be interesting to explore the anomalous signal by tuning the Fermi level position by doping.
Importantly, the calculations reproduced and explained the Tdependence of the experimental data, as shown in the inset of Fig. 3(d). At E_{F} – E_{0} = −80 meV, the Berry curvature in the model led to the same characteristic decrease of \(\alpha _{yx}^A\) with decreasing T that was also observed in the experiment. We note that only slight shifts in the position of E_{F} leads to a strong modification of the Tdependence of \(\alpha _{yx}^A\). Thus, \(\alpha _{yx}^A(T)\) is a sensitive probe for the underlying mechanism for the ANE in Co_{2}MnGa. The coincident observation of the high values of AHE and ANE, together with the same characteristic Tdependence of \(\alpha _{yx}^A\) in experiment and theory, go beyond the conventional model for the ANE in ferromagnets and can be considered as fingerprints of the Berry curvature effect associated with emerging nodal lines and Weyl points.
Finally, we turn to the initial question of whether the value of the ANE in such topological ferromagnets can exceed the highest values observed in conventional ferromagnets. Therefore, in Fig. 4, the anomalous Nernst signal of Co_{2}MnGa is compared with the values of various other ferromagnetic metals^{9,20,31,32,33,34,35}, extending the plot of Ikhlas et al.^{9}. We plotted the absolute values of the Nernst thermopower in the magnetically ordered states, where μ_{0}H and T were kept as absolute parameters. The shaded area of the graph represents the linear relation of the anomalous Nernst thermopower as measured for conventional ferromagnets. The red dotted line marks the limit of the highest value of \(S_{yx}^A\) ever measured in a ferromagnet. As explained above, according to the conventional scaling relation, the expected value of \(S_{yx}^A\) for Co_{2}MnGa at room temperature is ~0.05 to 0.9 µV K^{−1}. Nevertheless, the Nernst signal of Co_{2}MnGa shows a much larger value ∼6.0 µV K^{−1} at room temperature and further increased to ~6.6 µV K^{−1} at 340 K.
Conclusions
In summary, we have observed a remarkably large ANE in the ferromagnetic topological Heusler compound Co_{2}MnGa, which is ~7 times larger than any value reported for conventional ferromagnets to date in the literature. Combined electrical, thermoelectric and firstprinciples calculations revealed that this high value of the ANE arises from a large net Berry curvature near the Fermi energy associated with nodal lines and Weyl points. Our results show that the ANE in Co_{2}MnGa originates from a net Berry curvature of all bands near E_{F} rather than directly from the magnetization. The large Nernst signal at room temperature and high Curie temperature also make Co_{2}MnGa an ideal candidate for further investigation for use in transverse thermoelectric devices. Moreover, it would be interesting to explore the anomalous signal by tuning the Fermi level position by doping. In a broader sense, the flexible electronic structure of Heusler compounds can open up adequate room for the realization of a giant Nernst effect via a suitable Berry curvature design. We believe that the present study can act as a guide in the search for new magnetic topologically nontrivial materials with a large nonzero Berry curvature for the observation of a giant anomalous Nernst effect.
Note added to proof
During the peer review process of our paper, we noticed that another independent group also found a large anomalous Nernst effect in Co_{2}MnGa^{39}.
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Acknowledgements
This work was financially supported by the ERC Advanced Grant No. 742068 “TOPMAT”, and Deutsche Forschungsgemeinschaft DFG under SFB 1143. S.N.G. and Ch.F. acknowledges financial support from the Alexander von Humboldt Foundation.
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Guin, S.N., Manna, K., Noky, J. et al. Anomalous Nernst effect beyond the magnetization scaling relation in the ferromagnetic Heusler compound Co_{2}MnGa. NPG Asia Mater 11, 16 (2019). https://doi.org/10.1038/s414270190116z
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DOI: https://doi.org/10.1038/s414270190116z
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