Introduction

Elucidating the quantum nature and transport behaviors of topological magnets is at the frontier of condensed matter physics, and provides indispensable information on the application of quantum material research1,2,3,4,5. The interplay between topology and magnetism can naturally occur in kagome lattice systems6,7. Owing to its special lattice geometry, the electronic structure of kagome lattice inherently features both Dirac cones and band singularities including flat bands and van Hove singularities. The former provides the source of topological nontrivial band structure, while the latter can drive magnetic instabilities. In this regard, several transition metal based kagome magnets are of current interest, which exhibit many-body interplays and Chern quantum phases8,9,10,11,12,13,14,15,16,17,18. Among these kagome magnets, TbMn6Sn6 stands out owing to its pristine kagome lattice (without other atoms in the kagome lattice plane) and strong out-of-plane magnetization (persisting to above room temperature). Spatial and momentum resolved spectroscopies have uncovered the topological electronic structure that can be described by a simple Chern-gapped Dirac model16,19,20,21. The tight-binding model of kagome lattice considering the nearest neighbouring hopping hosts Dirac cones at the Brillouin zone corners. Magnetic exchange interaction splits the spin-up and spin-down Dirac cones, while the combination of out-of-plane magnetization and Kane-Mele type spin-orbit coupling further opens the Chern gap for the spin-polarized Dirac fermions. Though the spectroscopic and some preliminary electric transport data have been reported16,17, the quantum transport behavior of this kagome Chern magnet remains largely unveiled, which questions the key topological transport features of Chern-gapped Dirac fermions. Here we use the combination of electric, thermoelectric and thermal transport to characterize this quantum magnet and to discover its topological charge-entropy scaling.

Results

Parametrization of the Chern-gapped Dirac fermion

TbMn6Sn6 crystallizes in the space group of P6/mmm, featuring a pure Mn-based kagome lattice. Below its Curie temperature (~ 420 K), it possesses a ferrimagnetic state in which the magnetic moment of Tb is anti-parallel aligned with the ferromagnetic ordered Mn lattice due to the strong exchange coupling between Tb and Mn moments. The ferrimagnetic state changes its anisotropy from an easy-plane to an out-of-plane configuration when the temperature is below 313 K. The out-of-plane ferromagnetic Mn kagome lattice, which is stable over a wide range in the phase diagram (Fig. 1a), is crucial to support the fully spin-polarized Dirac fermions with a large Chern gap16,17. Figure 1b shows the quantum oscillations resolved in the thermoelectric and electric transport measurements at low temperatures. In the Seebeck signals, strong quantum oscillations with one dominant frequency (α, 96 T) can be observed just above 5 T, which is the smallest field among all known kagome magnets. The overall oscillatory patterns between thermopower and resistance show a π/2 phase shift, indicating that the α orbit is from a hole-like band22, consistent with spectroscopic observations16. The cyclotron mass of α orbit is found to be 0.14 me by fitting to the Lifshitz-Kosevich formula (See Method section). According to the Onsager relation23, this frequency corresponds to a Fermi vector kF of 0.05 Å−1. The (Dirac cone) velocity νD = ℏkF/m* is as large as 4.2 × 105 m/s, and the (Dirac cone) energy \({E}_{D}\simeq {m}^{* }{\nu }_{D}^{2}\) is estimated to be around 140 meV, in remarkable accordance with the hole branch of the Dirac dispersion observed in the previous spectroscopic experiments16. We further subtract the Berry phase from oscillatory peak positions in Fig. 1c. The intercept in the Landau fan diagram is found to be around − 1/8, highlighting a nontrivial band topology of the α orbit24,25.

Fig. 1: Topological quantum oscillations.
figure 1

a Magnetic phase diagram of TbMn6Sn6 when the field B is applied along the crystallographic c axis. Three main regions can be resolved, including the out-plane ferrimagnetic state where Chern-gapped Dirac states are supported, in-plane ferrimagnetic state, and paramagnetic state above 420 K. Inset shows the magnetic Mn kagome lattice and corresponding Chern-gapped Dirac cone at the Brillouin Zone corner of the momentum space. b Quantum oscillations revealed in magneto-Seebeck signal Sxx and magnetoresistance ρxx. There is a π/2 phase shift between oscillatory parts of Sxx and ρxx. c Landau fan diagram for the oscillations, suggesting a non-trivial Berry phase. d Fast Fourier transform spectrum of the oscillatory component in Sxx and ρxx, showing dominant contribution of the α orbit. e Cyclotron mass fitting of oscillatory amplitude to the Lifshitz-Kosevich formulas. f Oscillatory parts in Seebeck signals when the field is tilted away from the c direction. g Angle dependent oscillatory frequencies and corresponding cyclotron mass for orbit α, both showing an inverse cosine behavior.

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To confirm that the main frequency α is stemming from the quasi-two-dimensional electronic pocket from Mn kagome lattice, we performed magneto-Seebeck measurements when the magnetic field is tilted away from the c axis. As shown in Fig. 1g, both the oscillatory frequency F and the cyclotron mass m* change in a \(1/\!\cos \theta\) manner with respect to the tilted angle θ. For a gapped Dirac dispersion satisfying \({E}_{D}=\sqrt{{(\hslash {\nu }_{D}{k}_{F})}^{2}+{({{\Delta }}/2)}^{2}}\), the oscillatory frequency and cyclotron mass should follow the relations26,

$$F=\frac{{E}_{D}^{2}-{({{\Delta }}/2)}^{2}}{2e\hslash {\nu }_{D}^{2}\cos \theta },\,\,\,{m}^{* }=\frac{{E}_{D}}{{\nu }_{D}^{2}\cos \theta },$$
(1)

where Δ is the gap size, νD is the Fermi velocity of the Dirac cone. Because TbMn6Sn6 is a hard magnet with the easy axis pinned to the c axis at low temperatures27,28,29,30, its ED, Δ and νD change little when the external magnetic field is tilted away from the c direction. The ratio F/m* is always close to ED/2eℏ, strongly supporting that the main quantum oscillation orbit α in transport is stemming from the Chern-gapped Dirac fermion. This is unlike the case of Fe3Sn2 whose electron mass and gap size can be easily tuned by an external magnetic field15,26,31. The strong coupling between 4f and 3d electrons in TbMn6Sn6, which is absent in other transition-metal-bearing kagome magnets, guarantees a stable out-of-plane ferromagnetic Mn sublattice even in an extremely large external field and temperature range30. This large anisotropy makes the topological band of Chern-gapped Dirac fermion robust against the change of magnetic field and the elevated temperature, providing an ideal platform for studying Chern related topology.

Topological transverse transport

To further elucidate the quantum topology of this kagome magnet, we study the anomalous Hall effect, anomalous Nernst effect and anomalous thermal Hall effect. These three effects, stemming from the electron’s anomalous velocity endowed by the Berry curvature field in magnets3, are important fingerprints of the topological band structures residing near the Fermi energy. Figure 2a–c shows the Hall conductivity σxy, Nernst thermopower Sxy and thermal Hall conductivity κxy, which behave consistently with the M(H) loops at various temperatures shown in Supplementary Fig. S1. Below 220 K, sharp and large hysteresis loops can be observed in all three off-diagonal signals, corresponding to the spontaneous, out-of-plane ferrimagnetic state. The hysteresis loops become narrow gradually and show additional bending as the in-plane magnetic components develop at higher temperature. Above around 310 K, the three off-diagonal signals show the anomalous effects as long as the external field forces the in-plane moments to align along the c axis27,28,29. The temperature dependence of the anomalous effects is summarized in Fig. 3d, e. An anomalous Nernst coefficient as large as −2.4 μV/K is observed at 330 K, that is comparable to the largest values reported in other topological magnets like Co2MnGa and Co3Sn2S232,33,34,35. The anomalous thermal Hall conductivity on the other hand, reaches 0.155 W/Km at room temperature, much larger than those in topological magnets like Mn3Sn (0.04 W/Km)36 and Mn3Ge (0.015 W/Km)37 at 300 K. These large anomalous transport terms at room temperature make TbMn6Sn6 a promising candidate for thermoelectric applications38,39,40.

Fig. 2: Topological transverse transport.
figure 2

a–c The electric Hall conductivity σxy, the Nernst thermopower divided by temperature − Sxy/T and thermal Hall conductivity divided by temperature κxy/T at representative temperatures, showing dominant contribution of anomalous terms. Curves are shifted vertically for clarity. The resemblance of σxy, Sxy and κxy profiles indicates a shared origin from Berry curvature contributions. Insets show the sketches for anomalous Hall, Nernst and thermal Hall effect. d–f The temperature dependence of \({\sigma }_{xy}^{A}\), \({S}_{xy}^{A}\) and \({\kappa }_{xy}^{A}\), respectively.

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Fig. 3: Topological charge-entropy scaling.
figure 3

a Scaling of the anomalous Hall conductivity. The longitudinal conductivity σxx for TbMn6Sn6 lies within the good metal region, suggesting a dominant intrinsic contribution. Inset shows a polynomial fitting of the intrinsic Hall conductivity, amounting to 0.13 e2/h per kagome layer. b The pondering function for anomalous Hall conductivity \({\sigma }_{xy}^{A}\), anomalous thermoelectric Hall conductivity \({\alpha }_{xy}^{A}\), and the anomalous thermal Hall conductivity \({\kappa }_{xy}^{A}\) at 100 K and 300 K, respectively, together with the \(2\pi {\tilde{\sigma }}_{xy}(\varepsilon )\) for the Chern gapped Dirac fermion with a gap size of Δ. For clarity, the pondering functions at 300 K are multiplied by 3 times. Top-right sketch shows the Chern-gapped Dirac cone with gap size Δ ~ 34 meV and Dirac cone energy ED ~ 130 meV. c The ratio \({\alpha }_{xy}^{A}/{\sigma }_{xy}^{A}\) at different temperatures, approaching kB/e at around 330 K. This ratio scales with the linear function of kBT/ED, which is obtained from the Chern-gapped Dirac model. The error bars reflect the uncertainty in determining the sample size and low temperature anomalous Nernst signals. d The ratio \({\kappa }_{xy}^{A}/{\sigma }_{xy}^{A}\) at different temperatures. Above 100 K, this ratio significantly enhances over the T-linear function expected by the Wiedemann-Franz law at elevated temperatures, which matches the \({L}_{0}T(1+\eta {k}_{B}^{2}{T}^{2}/{E}_{D}^{2})\) behavior for the Chern-gapped Dirac model. The error bars come from the uncertainty in the sample’s geometric factor.

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In order to further study the intertwined relationships between these topological transport behaviors, we analyse their scaling relations. A conventional version in solids is the widely known Mott formula and the Wiedemann-Franz law, which associate the charge and entropy of the electron via1,2

$${{{{{{{\boldsymbol{\alpha }}}}}}}}=\frac{{\pi }^{2}}{3}\frac{{k}_{B}^{2}T}{e}\frac{\partial {{{{{{{\boldsymbol{\sigma }}}}}}}}}{\partial \varepsilon }{| }_{{\varepsilon }_{F}},$$
(2)
$${{{{{{{\boldsymbol{\kappa }}}}}}}}=\frac{{\pi }^{2}}{3}\frac{{k}_{B}^{2}}{{e}^{2}}{{{{{{{\boldsymbol{\sigma }}}}}}}}T={L}_{0}{{{{{{{\boldsymbol{\sigma }}}}}}}}T,$$
(3)

where ε is the energy, L0 = 2.44 × 10−8 V2/K2 is the Sommerfeld value, σ, α and κ are the electric, thermoelectric and thermal conductivity tensor, respectively. In the context of the anomalous transverse transport in topological magnets, however, their relevance remains much less explored.

As shown in Fig. 3a, we have confirmed that an intrinsic mechanism dominates in the anomalous Hall effect of TbMn6Sn6 by scaling \({\sigma }_{xy}^{A}\) versus σxx in the presented temperature range16,17. A polynomial fitting41,42 as shown in the inset of Fig. 3a gives an intrinsic contribution of \({\sigma }_{xy}^{int} \sim 0.13\,{e}^{2}/h\) per kagome layer. In accordance to the anomalous Hall conductivity and quantum oscillations of the α orbit, all parameters of the Chern-gapped Dirac fermion (ED ~ 130 meV, \({{\Delta }}=2{\sigma }_{xy}^{int}{E}_{D}h/{e}^{2} \sim 34\,{{{{{{{\rm{meV}}}}}}}}\)) can been experimentally nailed down and its Berry spectrum can be depicted. We can therefore trace the Berry curvature field to the Chern-gapped Dirac fermion in the stable out-of-plane ferrimagnetic state in TbMn6Sn6, which gives a unique opportunity to test the scaling relation of the \({\sigma }_{xy}^{A}\), \({\alpha }_{xy}^{A}\) and \({\kappa }_{xy}^{A}\) of the Chern-gapped Dirac fermion in a topological magnet.

The anomalous thermoelectric Hall conductivity is calculated as \({\alpha }_{xy}^{A}={\sigma }_{xy}^{A}{S}_{xx}+{\sigma }_{xx}{S}_{xy}^{A}\). As shown in Fig. 3c, the ratio \({\alpha }_{xy}^{A}/{\sigma }_{xy}^{A}\) in TbMn6Sn6 is small at low temperatures, and then shows an overall increase at elevated temperatures, reaching − 67 μV/K at 330 K. For a Chern-gapped Dirac model as shown in Supplementary Notes 2, it can be treated analytically that \(\partial {\sigma }_{xy}^{A}/\partial \varepsilon ={\sigma }_{xy}^{A}/{E}_{D}\). Substituting this into the Mott formula in Eq. (2), we have the relation between \({\alpha }_{xy}^{A}\) and \({\sigma }_{xy}^{A}\),

$${\alpha }_{xy}^{A}/{\sigma }_{xy}^{A}=\frac{{\pi }^{2}}{3}\frac{{k}_{B}}{e}\frac{{k}_{B}T}{{E}_{D}}.$$
(4)

The experimental data fit well into this simple linear line. The deviation around 100 K is currently unclear, and we speculate its origin from spin excitations43,44 as also evident in the magnetization susceptibility in Supplementary Fig. S1 which deserves future study. It has been suggested empirically that the ratio \({\alpha }_{xy}^{A}/{\sigma }_{xy}^{A}\) should be a sizable fraction of \(\left|{k}_{B}/e\right| \sim 86\,{{{{{{{\rm{\mu V/K}}}}}}}}\) in topological magnets33,35. Our analysis shows that for the kagome Chern magnet, the physics behind the kB/e threshold is actually a competition between two energies: the thermal energy kBT and the Dirac cone energy ED.

We now study the relation between the anomalous thermal Hall conductivity \({\kappa }_{xy}^{A}\) and anomalous Hall conductivity \({\sigma }_{xy}^{A}\). As shown in Fig. 3d, the ratio \({\kappa }_{xy}^{A}/{\sigma }_{xy}^{A}\) deviates from the linear T dependence expected by the standard Wiedemann-Franz law. It apparently surpasses L0T above 100 K, reaching 1.8 times L0T at 320 K and then the slope weakly damps at higher temperature. This substantial deviation can not be attributed to experimental errors, as the heat loss and geometric factors would only introduce at most 10% uncertainty in our study. It was reported that \({\kappa }_{xy}^{A}/{\sigma }_{xy}^{A}T\) remains close to L0 in Mn3Sn and Co2MnGa when the Berry spectrum is smooth in the vicinity of the Fermi level33,36. In Ni and Fe, the \({\kappa }_{xy}^{A}/{\sigma }_{xy}^{A}T\) deviates the Sommerfeld value downwards at higher temperatures due to the inelastic scattering36,45. A particular Berry spectrum also causes a suppressed \({\kappa }_{xy}^{A}/{\sigma }_{xy}^{A}T\) at high temperatures in Mn3Ge and Co3Sn2S237,46. To the best of our knowledge, TbMn6Sn6 is the first topological magnet that possesses an enhanced \({\kappa }_{xy}^{A}/{\sigma }_{xy}^{A}T\) ratio at elevated temperatures. The inelastic scattering mechanism is self-evidently ruled out as it should cause an inverse effect. Considering the phonon Hall and magnon Hall effect is usually in the order of 10−3 W/Km43,44,47,48, they are too small to account for this substantial enhancement. Below we show this anomaly in TbMn6Sn6 is due to the Berry curvature field effect hosted in the Chern-gapped Dirac fermions.

In transport theories, \({\sigma }_{xy}^{A}\) and \({\kappa }_{xy}^{A}\) can be expressed in the form of the Berry curvature \({{{\Omega }}}_{z}^{n}({{{{{{{\boldsymbol{k}}}}}}}})\)49,

$${\sigma }_{xy}^{A}=\frac{{e}^{2}}{\hslash }\int d\varepsilon \left(-\frac{\partial f}{\partial \varepsilon }\right){\tilde{\sigma }}_{xy}(\varepsilon ),$$
(5)
$${\kappa }_{xy}^{A}=\frac{1}{\hslash T}\int d\varepsilon \left(-{\varepsilon }^{2}\frac{\partial f}{\partial \varepsilon }\right){\tilde{\sigma }}_{xy}(\varepsilon ),$$
(6)
$${\tilde{\sigma }}_{xy}(\varepsilon )={\int}_{BZ}\frac{d{{{{{{{\boldsymbol{k}}}}}}}}}{{(2\pi )}^{3}}\mathop{\sum}\limits_{{\varepsilon }_{n} < \varepsilon }{{{\Omega }}}_{z}^{n}({{{{{{{\boldsymbol{k}}}}}}}}),$$
(7)

where f is the Fermi-Dirac distribution. In a Chern-gapped Dirac model, \({\tilde{\sigma }}_{xy}(\varepsilon )\) can be calculated as50

$${\tilde{\sigma }}_{xy}(\varepsilon ) = \frac{1}{2\pi }\frac{{{\Delta }}}{\sqrt{{{{\Delta }}}^{2}+4{\hslash }^{2}{k}_{F}^{2}{v}_{D}^{2}}} \\ = \left\{\begin{array}{rlr}\frac{1}{2\pi }\frac{{{\Delta }}/2}{\left|\varepsilon -{E}_{D}\right|},&\left|\varepsilon -{E}_{D}\right| \, > \, {{\Delta }}/2\\ \frac{1}{2\pi },\hfill& \left|\varepsilon -{E}_{D}\right| \, < \, {{\Delta }}/2.\end{array}\right.$$
(8)

As shown in Fig. 3b, the pondering functions in \({\sigma }_{xy}^{A}\) and \({\kappa }_{xy}^{A}\), namely − ∂f/∂ε and − ε2∂f/∂ε have different shapes. The former is a delta-like function, while the latter adopts a double-peak-like profile and is more extended in the energy scale. The Wiedemann-Franz law would only hold when \({\tilde{\sigma }}_{xy}(\varepsilon )\) is antisymmetric around zero energy. However in the case of a Chern-gapped Dirac fermion, Berry curvature maximizes around the Dirac gap, and contributes more in the latter integration as − ε2∂f/∂ε lies more close to the gap with increasing temperature. We note that it is the singularity of Berry curvature from the Chern Dirac gap that results in the enhancement of \({\kappa }_{xy}^{A}/{\sigma }_{xy}^{A}\). Using the Sommerfeld expansion as shown in Supplementary Notes 2, it can be deduced that

$${\kappa }_{xy}^{A}/{\sigma }_{xy}^{A}={L}_{0}T\left[1+\eta {\left(\frac{{k}_{B}T}{{E}_{D}}\right)}^{2}\right],\,\eta =\frac{7{\pi }^{2}}{5}.$$
(9)

The experimental ratios of \({\kappa }_{xy}^{A}/{\sigma }_{xy}^{A}\) fit well into the curves from the simple model.

Discussion

Due to the complex of magnetism and thermoelectric/thermal transport in real magnetic materials, it has been difficult to relate the anomalous Nernst effect and anomalous thermal Hall effect directly with the topological electrons. A common strategy is to calculate the energy dependence of the Berry curvature from ab initio and thoroughly scan over the energy to find a rough match with the anomalous transverse effects in experiment. With the lack of the topological electrons’ detailed information, the validity of this strategy, to a large extent, depends on the accuracy of the ab initio calculation which is known to be difficult for correlated magnetic materials. In this work we have, for the first time, established a direct link between the anomalous effects and a particular topological fermion in a kagome magnet TbMn6Sn6. Starting from the angle-dependent quantum oscillations, we depict the two-dimensional Dirac dispersion. Then by comparing the anomalous transverse electric, thermoelectric and thermal transports, we are able to connect the anomalous effects to the Berry spectrum of the Chern-gapped Dirac fermion. In particular, two ratios, the \({\alpha }_{xy}^{A}/{\sigma }_{xy}^{A}\) and \({\kappa }_{xy}^{A}/{\sigma }_{xy}^{A}\), give inherent scaling relations. The former scales with kBT/ED over an extended temperature range, while the latter, proportional to \(T(1+\eta {k}_{B}^{2}{T}^{2}/{E}_{D}^{2})\), enhances over the linear T-dependence above 100 K. Only two parameters, the Dirac cone energy ED and the gap size Δ both of which are determined from experiments, are involved in our discussion, and the simple Chern-gapped Dirac model is able to capture most of the (anomalous) transport behaviors. Our results, therefore, establish a new model topological magnet for the proof-of-principle elaboration of the topological charge-entropy scaling. In future, it would be interesting to engineer this material down to atomic layers to realize higher temperature quantum anomalous Hall effect and to explore the topological scaling at the quantum-limit.

Methods

Material growth and magnetization measurement

Single crystals of TbMn6Sn6 were grown via a standard tin flux method29. Magnetic measurements on TbMn6Sn6 were performed in a Quantum Design magnetic property measurement system (MPMS-3). When the field was applied along the c axis, a sharp transition could be observed in the M(T) profile at around 313 K (TSR) in Supplementary Fig. S1, corresponding to the spin-reorientation transition. A magnetic anomaly occurs at around 100 K as indicated by the hump in the temperature derivative of the susceptibility, which suggest possible spin excitations in the kagome lattices43,44 and is beyond our current scope of research.

Transport measurements

Transport properties of TbMn6Sn6 crystals were characterized in a Quantum Design physical property measurement system (PPMS-9) and a 14 T Oxford Teslatron PT system. Longitudinal and Hall resistivities were measured using a standard four-probe method. An AC Resistance Bridge (Model 372, Lakeshore) was used to measure the small electric signals. We use the convention Ei = ρijJj, namely ρyx being positive for hole carriers.

For thermoelectric measurements, all samples were finely polished in order to reduce heat radiation. A one-heater-three-thermometer setup was used to monitor both the longitudinal and transverse temperature gradient. Temperature readings were acquired via Cernox thermometers below 50 K and type-E thermocouples above 50 K. Heat flow was applied within the ab plane. Vacuum better than 1 × 10−4 Pa was maintained during the measurement. Thermal conductivity data were collected simultaneously. Seebeck and Nernst voltages were amplified by EM A10 and then collected by nano-voltmeter Keithley 2182A. The sign convention of Ei = Sij∂jT is adopted. In order to reveal the oscillatory components in ρxx and Sxx in Fig. 1, a smooth background has been subtracted.

Angle-dependent Seebeck signals at low temperatures were measured in the same magnets in a homemade vacuum chamber which can be rotated with respect to the magnetic field. The temperature gradient (−∇ T) was applied within the crystallographic ab plane and perpendicular to the a axis. Field is titled perpendicular to −∇ T. The angle between c axis and field direction is defined as θ.

Cyclotron mass fitting

The cyclotron masses (m*) of the carriers were fitted by a standard and a modified Lifshitz-Kosevich formula22,23,

$${R}_{T}^{\rho }\propto \frac{X}{\sinh (X)},$$
(10)
$${R}_{T}^{S}\propto \frac{(\alpha X)\coth (\alpha X)-1}{\sinh (\alpha X)},$$
(11)

where \({R}_{T}^{\rho }\) and \({R}_{T}^{S}\) are the temperature-dependent oscillatory amplitude of resistivity and thermopower, respectively, and α = 2π2kB/eℏ, X = m*T/B.

Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.