Fermiology-driven goniopolar transverse thermoelectricity in kagome metals

Abstract

The exotic geometry of the kagome lattice drives emergent quantum states and advances energy technologies; however, the anomalous Nernst effect (ANE)-based magnetic systems are fundamentally limited by low thermopowers (<6μ V K⁻¹) and stray-field interference. Here we propose goniopolarity (axis-dependent carrier polarity) to achieve high zero-field transverse thermoelectric responses in kagome systems. By exploiting flat-band- and van Hove singularity-driven electronic states, we uncover exceptionally large goniopolar thermoelectric responses in LuCo₆Ge₆, including a transverse thermopower of 18.4 μV K⁻¹ and a transverse Peltier conductivity of 105 A m⁻¹ K⁻¹ at room temperature and zero field. The synergy of flat bands with high electrical conductivity yields values an order of magnitude greater than those achieved in conventional ANE-based systems. Our findings establish goniopolar kagome metals as promising candidates for thermoelectrics.

Axis-dependent carrier polarity, in combination with flat-band and van Hove singularity states, enable transverse thermopower of 18.4 μV K⁻¹ and transverse Peltier conductivity of 105 A m⁻¹ K⁻¹ at room temperature and zero field in LuCo₆Ge₆.

Main

The kagome lattice, a two-dimensional network of corner-sharing triangles, serves as an archetypal platform for probing correlated electron physics and emergent quantum states¹. In kagome materials, the structural geometry naturally hosts Dirac fermions, flat bands and van Hove singularities (VHSs) in their electronic structure, and a hierarchy of exotic quantum-ordered phases has been uncovered^2–4. Beyond their pivotal role in quantum matter exploration, kagome materials exhibit transformative potential for energy technologies, particularly in thermoelectrics^5,6. Thermoelectricity is a solid-state energy conversion mechanism that enables the direct interconversion of thermal and electrical energy^7–9. It addresses key challenges in waste heat recovery and advanced thermal management, offering a pathway towards carbon-neutral industrial operations¹⁰.

Recent studies have highlighted the potential of transverse thermoelectric effects for energy harvesting due to their ability to efficiently cover curved heat sources across large areas in a flexible manner^8,11–13. However, high-efficiency transverse thermoelectrics typically require an external magnetic field for operation, which inevitably induces substantial electromagnetic interference^14,15. This interference poses a challenge for integrating transverse thermoelectrics with other electronic devices. Currently, strategies leveraging the anomalous Nernst effect (ANE) and goniopolarity have been developed to achieve zero-field transverse thermoelectrics^12,16–18. The ANE originates from time-reversal symmetry breaking and the Berry curvature in magnetic topological systems^19–21. The ANE thermopower remains an order of magnitude lower than the Seebeck coefficient²², indicating that the merit still falls short of commercial applications. Because strong stray fields and the inherent difficulty of harnessing intrinsic Berry curvature in magnetic materials severely limit their development, it is essential to explore alternative mechanisms for zero-field transverse energy conversion.

Fortunately, goniopolarity offers a promising approach to achieving high-efficiency transverse thermoelectric conversion in nonmagnetic systems under zero magnetic field^18,23–25. Goniopolarity, also known as axis-dependent carrier polarity, enables opposite carrier polarities (p-type or n-type) in a single material along different crystallographic axes. Notably, it can contribute to higher transverse thermopowers compared with ANE. Moreover, goniopolarity allows broad tunability of thermoelectric properties through optimization of carrier concentration and mobility, constituting a critical advantage for adaptive application requirements²⁶. Despite its promise, current design frameworks and material options remain limited, leaving plenty of scope for further exploration. However, the pioneering work by Mahan and Sofo proposed that thermoelectricity would be enhanced with a δ-shape transport distribution²⁷; the kagome lattice hosting flat bands and VHSs at varied fillings exactly fulfils this criterion, which is featured by the enhanced electronic density of states (DOS), suggesting substantial potential for large thermopowers. Despite the characteristic flat bands and VHSs in kagome systems, whether kagome-specific electronic structures can generate goniopolarity remains unknown. Here, motivated by the intriguing electronic structure, we propose a fermiology-driven goniopolar transport framework involving two distinct mechanisms: a flat-band-enabled two-band mechanism and a VHS-driven single-band mechanism. Guided by theoretical models, we experimentally realize high zero-field transverse thermoelectric responses in LuCo₆Ge₆ and its siblings Lu(Ho)Cr₆Ge₆. Our results establish kagome materials as a natural platform for goniopolar transport and provide a conceptual route for engineering transverse thermoelectricity through electronic-structure design.

Fermiology models of goniopolar kagome lattices

The electronic structure of a two-dimensional (2D) kagome lattice with the nearest-neighbour hopping exhibits two defining features: perfectly flat bands and logarithmically divergent VHSs²⁸. In contrast, extending to a three-dimensional (3D) kagome lattice fundamentally alters these properties. Incorporation of the nearest-neighbour out-of-plane hopping imparts dispersion to the formerly flat bands along the k_z direction, enabling out-of-plane charge transport. Concurrently, VHSs lose their divergent characters in three dimensions, leading to suppressed electron instabilities^29–31. Figure 1a shows the characteristic band structure and DOS of a 3D kagome lattice, considering only the nearest-neighbour in-plane and out-of-plane hoppings. The DOS reveals two key signatures: (1) prominent peaks arising from the dispersive flat bands and (2) smaller features attributed to non-divergent VHSs. On the basis of these electronic distinctions, we propose two fermiology-related mechanisms driving goniopolar transport in kagome lattices: (1) flat-band-enabled two-band mechanism and (2) single-band mechanism enabled by VHS-driven asynchronous particle–hole asymmetry.

Fig. 1. Goniopolarity in kagome lattice.

a, Band structure of single-orbital kagome models featuring exclusively in-plane and out-of-plane nearest hoppings. The DOS g(ε) and its derivative g′(ε) are shown on the right side, where ε is the chemical potential. The parameters are set as a = 1, c = 1, t_i = −0.5 and t_o = −0.2, where a and c are the in-plane and out-of-plane lattice constants, respectively, and t_i and t_o denote the in-plane and out-of-plane hopping. Inset: the typical kagome lattice. b, Corresponding in-plane S_IP (blue) and out-of-plane S_OP (red) Seebeck coefficients. The single-band and two-band goniopolar zones, marked in blue and red shading, respectively, show opposite signs for in-plane (IP) and out-of-plane (OP) Seebeck response. These zones are positioned near flat bands and VHSs. c,d, Typical Fermi surface at Fermi level E_F = 0.9 eV (c), illustrating the exact flat bands (red) and VHS (blue), whereas E_F = 0.1 eV (blue), 0.2 eV (green), 0.3 eV (red) (d), showing the Fermi surface around the VHS. e, Band-resolved Seebeck coefficients with shaded areas for single-band (blue) and two-band (red) goniopolar zones. f, Two-dimensional Fermi surface for VHS with a saddle point. g, Three-dimensional Fermi surface for VHS at E_F = 0 (green), +Δ (red) and −Δ (blue). Δ denotes the energy deviation from zero energy. The parameters for VHS are set as: c_i = 1 and |k_i| < 0.5, where c_i is a model parameter and k_i is the momentum. h, Goniopolarity in VHS, including the in-plane and out-of-plane Seebeck response. Normalized DOS g(ε) and its derivative g′(ε) are also presented. A strong t_o case is shown in Supplementary Fig. 1.

Source data

In 3D flat bands, transport primarily occurs in the out-of-plane direction, because the in-plane transport behaviours are strongly suppressed by vanishing group velocity or diverging effective mass (Supplementary Fig. 2). Consequently, an additional dispersive band dominates the in-plane transport, inducing anisotropy in this two-band system. When these bands comprise one electron-like and one hole-like band, the system exhibits goniopolar transport. As shown in Fig. 1c (Fermi level E_F = 0.9 eV), the Fermi surface reveals an in-plane flat band for out-of-plane transport and an isotropic hole pocket for in-plane transport. This results in goniopolar Seebeck responses within the red shaded zone (Fig. 1b,e). The enhanced DOS of flat bands contributes substantially to the Seebeck response peak.

VHSs show distinct particle–hole symmetry characteristics across dimensions. In 2D systems, conventional VHSs maintain strict particle–hole symmetry²⁸. Conversely, 3D VHSs are inherently particle–hole asymmetric. Such asymmetry directly leads to the observable thermoelectric responses, which can be described by S = σ⁻¹α $={-\frac{{\mathrm{\pi }}^2}{3}\frac{{\kappa }_{\text{B}}^{2}T}{\mid e\mid }\left(\frac{\mathrm{d}\ln g}{\mathrm{d}\epsilon }+\frac{\mathrm{d}\ln v^2}{\mathrm{d}\epsilon }+\frac{\mathrm{d}\ln \tau }{\mathrm{d}\epsilon }\right)\mid }_{\epsilon =E}$, where σ is the electrical conductivity, α is the Peltier conductivity, k_B is the Boltzmann constant, e is the elementary charge, T is the temperature, g is the density of states, v is the group velocity and τ is the relaxation time. The contributions can be ascribed to three components: the DOS, the kinetic energy and the relaxation time, quantifying the competition between electron and hole energy flux. When the electron and hole contributions are kept in balance, the thermal responses vanish, which means that the imbalance determines both the magnitude and sign of the response. Thus, thermoelectric measurements serve as direct probes of particle–hole symmetry at finite temperatures. Transport anisotropy of VHSs generates asynchronous particle–hole asymmetry along the in-plane and out-of-plane directions, allowing for single-band goniopolarity. The single-band goniopolar zones near VHSs are highlighted in Fig. 1b,e (blue-shaded zone). The corresponding Fermi surface around the VHS is illustrated in Fig. 1d.

To elucidate the single-band goniopolarity, we analyse the Hamiltonian $H=c_1{k}_x^2+c_2{k}_y^2-c_3{k}_z^2$. Figure 1f,g depicts the Fermi surfaces near the saddle point in two and three dimensions, respectively: hyperbolic along k_z, yet circular within the plane. The Fermi surface consists of two separate parabolic sheets below the saddle point, which merge into a single hyperbolic surface above it, emphasizing the transition occurring at the saddle point. When the Fermi level is below the saddle point, the energy flux increases as the Fermi level increases in both the in-plane and out-of-plane directions, contributed by the increasing DOS, indicating a negative Seebeck response. When the Fermi level is above the saddle point, the derivative of the DOS exhibits a sudden sign reversal, leading to negative contributions in both the in-plane and out-of-plane directions. However, the kinetic energy is still positive and increasing in the in-plane direction while it is always negative in the out-of-plane direction due to the opposite masses, leading to the goniopolar response (Fig. 1h and Supplementary Fig. 3). The theoretical analysis suggests that the sign reversal of the out-of-plane response can be captured by the evolution of boundaries of the Fermi surface due to the depletion of high-kinetic-energy states. When the Fermi level is high enough, the energy flux decreases in both the in-plane and out-of-plane directions due to the decreasing DOS, indicating a positive Seebeck response. Therefore, the goniopolar behaviour in single-band systems arises from asynchronous particle–hole asymmetry caused by anisotropic kinetic distributions of opposite masses^18,32,33.

Characterization of kagome single crystals

Guided by fermiology models, we selected three 166-type kagome metals (LuCo₆Ge₆, LuCr₆Ge₆ and HoCr₆Ge₆) to experimentally investigate the goniopolar effect. Three paramagnetic metals (>10 K) were crystallized in a typical kagome lattice structure (Supplementary Figs. 4–6) with a space group of P6/mmm (No. 191). The clear diffraction spots observed in the Laue diffraction patterns of these metals serve as evidence for a perfect single-crystal structure. Furthermore, the heat capacity measurements of the three kagome metals reveal no discernible phase or magnetic transitions between 10 K and 300 K.

Scanning transmission electron microscopy (STEM) was used to investigate the atomic structures. High-angle annular dark-field STEM (HAADF-STEM) images (Supplementary Figs. 7–9) reveal well-ordered atomic arrangements of LuCo₆Ge₆, LuCr₆Ge₆ and HoCr₆Ge₆, with distinct kagome lattices formed by Co or Cr atoms. HAADF-STEM images of LuCo₆Ge₆ along the [001], [210] and [010] zone axes are presented in Fig. 2a–c, where Lu and Co atoms are highlighted by blue and red circles, respectively. Figure 2d shows the energy-dispersive spectroscopy (EDS) maps of Lu and Co elements from the same field of view as Fig. 2a. These results indicate the high crystal quality and uniform elemental distribution of the LuCo₆Ge₆ sample. In addition, the Co atomic columns show elongation along in-plane directions when viewed along the [010] axis, as evidenced by measuring ellipticity. As defined in Fig. 2e, the ellipticity can be determined by the ratio between c_x and c_y, where the c_x and c_y represent the standard deviation along two orthogonal directions when fitting the 2D Gaussian peak. Figure 2f shows the measured ellipticity of the Co site in LuCo₆Ge₆ when viewed along the [010] zone axis, demonstrating elongation along the in-plane direction. This elongation suggests a slight in-plane disorder shift of Co atoms from their ideal lattice positions, leading to a minor distortion of the in-plane lattice. It is noteworthy that slight in-plane distortion might cause in-plane carrier scattering, reducing carrier relaxation time and thereby inducing anisotropic resistivities in LuCo₆Ge₆ (ref. ³⁴; Supplementary Fig. 10). However, the impact of disorder shift on the kagome structure is negligible, as indicated by the powder X-ray diffraction data (Supplementary Fig. 4).

Fig. 2. Atomic-resolution STEM results of the single-crystalline LuCo₆Ge₆ kagome metal.

a–c, HAADF-STEM images of LuCo₆Ge₆ viewed along the [001] (a), [210] (b) and [010] (c) zone axes. The Lu and Co sites are highlighted with blue and red circles, respectively. d, EDS mappings of Lu and Co elements. The field of view is same as in a. e, Top: schematic graph of the definition of ellipticity in Co atomic columns when viewed along the [010] zone axis. The c_x and c_y represent the standard deviation along two orthogonal directions when fitting the 2D Gaussian peak (see Methods for more details). Bottom: HAADF-STEM image overlapped with measured ellipticity. The colour represents the ellipticity and the arrow represents the elongated direction. f, Ellipticity of Co atomic columns measured from 8 × 8 nm².

Goniopolarity and transverse thermoelectric effect

The goniopolar behaviour is experimentally examined by transport measurements. Evident anisotropy in the electrical resistivity is observed for LuCo₆Ge₆, which shows metallic behaviour with increasing temperature (Supplementary Fig. 10). Temperature-dependent Seebeck and Hall coefficients are depicted in Fig. 3a. The in-plane Seebeck coefficient S_IP remains consistently negative, hovering around −15 μV K⁻¹ across the temperature range. Conversely, the out-of-plane Seebeck coefficient S_OP shows positive values that increase with temperature, ranging from 0 to 9.1 μV K⁻¹ between 60 K and 300 K. The in-plane and out-of-plane Hall resistivities and coefficients show the same signs as the Seebeck coefficients (Supplementary Fig. 10 and inset of Fig. 3a), demonstrating electron transport in the in-plane direction and hole transport in the out-of-plane direction. We have also examined the transport properties of LuCr₆Ge₆ and HoCr₆Ge₆ kagome metals (Supplementary Figs. 11 and 12). They show the opposite goniopolar behaviour—hole conduction along the in-plane axis and electron conduction along the out-of-plane axis.

Fig. 3. Thermoelectric transport properties of the single-crystalline LuCo₆Ge₆ kagome metal.

a, Temperature-dependent in-plane (S_IP) and out-of-plane (S_OP) Seebeck coefficients of the LuCo₆Ge₆ single crystal. Inset: corresponding Hall coefficients (R_H) as a function of temperature. b, Longitudinal thermopower S_L, transverse goniopolar thermopower S_xz and transverse Peltier conductivity |α_xz| versus temperature. Theoretical S_L and S_xz are represented by green and grey dashed lines, respectively. Inset: experimental set-up for transverse thermoelectric measurement. c, Comparison of the transverse thermopower and |α_xz| with those of reported ANE-based materials and the mixed-dimensional conductor LaPt₂B under zero magnetic field (B = 0 T)^{11–13,16,17,36–38}. d, Temperature dependence of power factors and thermal conductivities (inset). e, Thermoelectric figure of merit zT as a function of temperature. f, Transverse thermoelectric figure of merit (z_xzT) for our kagome metals compared to ANE-based systems, the artificially tilted multilayers Bi₈₈Sb₁₂/Bi_0.2Sb_1.8Te₃, the goniopolar Re₄Si₇, and the mixed-dimensional conductor LaPt₂B under zero magnetic field^{11–13,16,17,25,38,41,42}. The data for ANE-based materials were estimated under zero-magnetic-field conditions, based on transverse thermopower measured in the remanent magnetization state.

Source data

Goniopolarity can induce a transverse thermoelectric response under zero magnetic field¹⁷. Upon applying a temperature gradient along the longitudinal direction (inset of Fig. 3b), longitudinal S_L and goniopolar transverse S_xz thermopowers were generated in LuCo₆Ge₆ (Fig. 3b). The negligible transverse temperature difference suggests that S_xz primarily originates from goniopolarity, with negligible contributions from off-diagonal thermal conductivity and Seebeck effect to S_xz (Supplementary Fig. 13). S_xz exhibits a temperature-dependent increase, achieving 7 μV K⁻¹ at 60 K and 18.4 μV K⁻¹ at 300 K, surpassing the highest recorded room-temperature ANE thermopower in ANE-based materials¹². The low S_L values remain consistent across the temperature range, and both the experimental S_L and S_xz align well with the calculated values (Fig. 3b). LuCr₆Ge₆ and HoCr₆Ge₆ also demonstrate exceptional S_xz of 7.3 μV K⁻¹ and 8.1 μV K⁻¹ at 300 K, respectively. Furthermore, a large transverse Peltier conductivity |α_xz| of 105 A m⁻¹ K⁻¹ at 300 K was achieved, primarily due to the high electrical conductivity and large S_xz. The parameter |α_xz| is crucial, as it quantifies the induction of an electric current along the x axis in response to a temperature gradient applied along the z axis³⁵. It is worth noting that both S_xz and |α_xz| in LuCo₆Ge₆ are comparable to those reported in ANE-based systems under zero field^{11–13,16,17,36–38} (Fig. 3c). Figure 3d shows the temperature-dependent longitudinal and goniopolar transverse power factors. Across a broad temperature range, the goniopolar transverse power factor surpasses the longitudinal one, attaining 8.2 μW cm⁻¹ K⁻² at room temperature, comparable to some leading thermoelectric semiconductors^39,40. LuCo₆Ge₆ shows notable anisotropy in total thermal conductivities, as evidenced by the inset in Fig. 3d, where the out-of-plane and in-plane thermal conductivities show values of 69 W m⁻¹ K⁻² and 11 W m⁻¹ K⁻² at 300 K, respectively. On the basis of the Wiedemann–Franz law, the lattice thermal conductivity was determined. Across the entire temperature range, the in-plane lattice thermal conductivity remains lower than that of the out-of-plane direction. The slight in-plane lattice distortion may suppress phonon propagation (Fig. 2f), thereby contributing to a low in-plane lattice thermal conductivity in LuCo₆Ge₆. Nevertheless, the in-plane lattice thermal conductivity remains higher than that of several thermoelectric semiconductors, which can be decreased by incorporating defect structures. Combining the high goniopolar transverse power factor and anisotropic thermal conductivities, we obtained a high goniopolar transverse thermoelectric figure of merit z_xzT value of 1.15 × 10⁻² at 300 K for LuCo₆Ge₆ (Fig. 3e), exceeding those of all previously reported ANE-based systems by an order of magnitude under zero field^{11–13,16,17,25,38,41,42} (Fig. 3f). Obviously, outstanding goniopolar kagome metals are expected to exhibit high electrical conductivities characterized by low carrier concentration and high mobility, along with large S_xz and low thermal conductivities (Supplementary Table 1). The reproducibility of the goniopolar behaviour and thermoelectric performance has been verified across different batches of samples (Supplementary Figs. 14 and 15). The differences in sample performance may be due to differences in carrier concentrations between crystals. Moreover, LuCr₆Ge₆ and HoCr₆Ge₆ also demonstrate exceptional transverse thermoelectric performance (Fig. 3f and Supplementary Figs. 16 and 17).

Origin of the goniopolarity in LuCo₆Ge₆ and its siblings

To elucidate the origin of goniopolarity in LuCo₆Ge₆ and its sibling compounds Lu(Ho)Cr₆Ge₆, we investigated their band structures using density functional theory (DFT) calculations and angle-resolved photoemission spectroscopy (ARPES). As depicted in Fig. 4a and Supplementary Fig. 18, the band structure of LuCo₆Ge₆ shows hallmark features of kagome lattice such as Dirac points, VHSs, and flat bands near the Fermi level. The theoretical energy-dependent Seebeck coefficients demonstrate evident goniopolar behaviour in various energy zones due to the presence of VHSs and flat bands (Fig. 4b). On the basis of experimental and theoretical Seebeck coefficients, we conclude that the system is hole-doped, with the band dispersion shifted by about 60 meV below the Fermi level E_F. This estimation was further corroborated by low-temperature specific heat analysis (Supplementary Fig. 4). The derived electronic specific-heat coefficient of γ = 0.01754 J mol⁻¹ K⁻² corresponds to an experimental DOS of g_exp = 7.44 (states per eV). DFT calculations within the proposed energy range indicate a DOS of ~7.05 states per eV, aligning well with the experimental approximation.

Fig. 4. Theoretical analysis of goniopolarity.

a, Band structure of LuCo₆Ge₆. b, The energy-dependent Seebeck coefficients for LuCo₆Ge₆. The red dashed lines represent the real Fermi level. c,d, The hole-dominated Fermi pocket A (c) and the electron-dominated pocket C (d) of LuCo₆Ge₆. e, Calculated Seebeck coefficients for all pockets (top) and each individual pocket (bottom) in LuCo₆Ge₆. f, Second derivatives of the ARPES spectra of LuCr₆Ge₆ (T = 15 K) along the $\overline{\mathrm{\Gamma }}$−$\overline{\mathrm{K}}$−$\overline{\mathrm{M}}$−$\overline{\mathrm{\Gamma }}$ lines measured by the 55-eV (top) and 75-eV (bottom) photons with linear horizontal (LH) polarization. The red solid curves are the DFT calculated bulk bands in k_z = 0 (top) and π (bottom) planes. As highlighted by the black arrows, multiple Dirac points and VHSs revealed in experiments are well reproduced by the calculations. g, The energy-dependent Seebeck coefficients for LuCr₆Ge₆. h, The hyperbolic pocket D of LuCr₆Ge₆. L and H denote low and high Fermi velocities, respectively. i, The constant-energy ARPES image of LuCr₆Ge₆ (hν = 75 eV, LH polarization) obtained by integrating the photoemission intensity within E_F = ± 20 meV. The red solid curves indicate the (001)-projected Brillouin zone. L and H denote low and high DOS, respectively. j, Calculated Fermi surface counter. k, Calculated Seebeck coefficients for all pockets (upper) and pocket D (lower) in LuCr₆Ge₆.

Source data

To further understand the goniopolar responses in LuCo₆Ge₆, we conducted theoretical calculations of Fermi pockets (labelled as A, B and C) at E_F = −60 meV and the corresponding axis-dependent Seebeck coefficients (Fig. 4c–e and Supplementary Fig. 18). The quasi-one-dimensional Fermi pocket A originating from the flat band near E_F, localized at the k_z = π plane, shows hole-like dispersion, while pockets B and C exhibit electron-like characteristics. The calculated in-plane (S_cIP) and out-of-plane (S_cOP) Seebeck coefficients at E_F = −60 meV reveal a notable anisotropy: S_cOP shows positive values (hole-dominated transport), while S_cIP remains negative (electron-dominated behaviour), in agreement with experimental observations (Fig. 4e). The hole-dominated pocket A consistently shows positive values along all crystallographic axes, while the electron-dominated pockets B and C consistently show negative values. The total Seebeck coefficient (S_total) is calculated as a conductivity-weighted average across all Fermi pockets: $S_{\mathrm{total}}=\frac{{\sum }_{i=1}^n{S}_i{\sigma }_i}{{\sum }_{i=1}^n{\sigma }_i}$, where S_i and σ_i represent the Seebeck coefficient and electrical conductivity of the ith pocket, respectively⁴³. Analysis reveals that the contribution of pocket B is negligible due to its low S_i value. In the in-plane direction, pockets A and C exhibit similar conductivity (Supplementary Fig. 18), yet the negative S_C outweighs the positive S_A, yielding a negative S_IP. In the out-of-plane direction, the conductivity of pocket A (σ_A) surpasses that of pocket C by a factor of ~6, amplifying the positive S_A contribution and resulting in a positive S_OP. This synergy between the flat-band-related pocket A and the isotropic pocket C highlights a flat-band-driven two-band goniopolar mechanism in LuCo₆Ge₆.

In contrast to the LuCo₆Ge₆ metal, the band structures of LuCr₆Ge₆ and HoCr₆Ge₆ are similar, which were investigated using synchrotron-based ARPES and DFT calculations (Supplementary Figs. 19 and 20). Figure 4f shows the second derivatives of the ARPES spectra of LuCr₆Ge₆ (T = 15 K) measured along the $\overline{\mathrm{\Gamma }}$ − $\overline{\mathrm{K}}$ − $\overline{\mathrm{M}}$ − $\overline{\mathrm{\Gamma }}$ high-symmetry path by the 55-eV (Fig. 4f, top) and 75-eV (Fig. 4f, bottom) photons with linear horizontal polarization. The theoretical band structure shows typical flat bands and VHSs near the Fermi level, leading to the Seebeck coefficients displaying goniopolar behaviour in various energy zones (Fig. 4g and Supplementary Fig. 19). The goniopolar response of LuCr₆Ge₆ also shows strong E_F dependence, similar to the tunability observed in LuCo₆Ge₆. By correlating the energy of the Dirac point in k_z = π plane with the experimental spectra, the real Fermi level was determined at E_F = −0.1 eV (Fig. 4f and Supplementary Fig. 19). As indicated by the appended Brillouin zone and labelled high-symmetry points ($\overline{\mathrm{\Gamma }}$, $\overline{\mathrm{K}}$ and $\overline{\mathrm{M}}$), the measured Fermi surface exhibits a characteristic hexagonal topology, in agreement with theoretical calculations (Fig. 4i,j). The DFT calculations reveal four anisotropic Fermi pockets (labelled as D, E, F and G) at E_F = −0.1 eV (Fig. 4h and Supplementary Fig. 19), where the hyperbolic pocket D and bowl-like pocket E are more than half-filling, while open pockets F and G are less than half-filling, attributed to the 3d orbitals of the Cr-based kagome lattice. At E_F = −0.1 eV, LuCr₆Ge₆ displays pronounced anisotropy in the calculated Seebeck coefficients, showing positive in-plane values but negative out-of-plane values, in contrast to LuCo₆Ge₆ (Fig. 4k). The axis-dependent Seebeck coefficients were calculated for four Fermi pockets, as depicted in Fig. 4k and Supplementary Fig. 19. The hyperbolic pocket D and bowl-like pocket E yield positive values along the in-plane direction but negative values along the out-of-plane direction. Conversely, open pockets F and G show uniform polarity across all crystallographic axes. The negligible contribution of pockets F and G is attributed to their low S_i values based on the conductivity-weighted equation. Either pocket D or E alone is sufficient to produce the observed goniopolar behaviour, distinct from the flat-band-driven two-band mechanism, and instead points to a VHS-enabled single-band mechanism in LuCr₆Ge₆. A detailed analysis of the Fermi surface evolution shows that no abrupt topological transition or pocket reconnection occurs exactly at the onset or disappearance of goniopolarity (Supplementary Fig. 20). The electronic structure and goniopolar behaviour of HoCr₆Ge₆ closely resemble those of LuCr₆Ge₆, further emphasizing a single-band mechanism (Supplementary Fig. 21). Hence, we can achieve both single-band and two-band mechanisms in kagome systems. These findings underscore the pivotal role of fermiology in tailoring emergent transport phenomena in kagome metals.

Outlook

To summarize, we demonstrate that the electronic band structures of kagome metals inherently support goniopolarity through two distinct mechanisms: a flat-band-enabled two-band mechanism realized in LuCo₆Ge₆ and a VHS-driven single-band mechanism observed in Lu(Ho)Cr₆Ge₆. Goniopolar thermoelectricity highlights the versatility of kagome lattices in tailoring transport phenomena, beneficial to not only energy conversion but also the emergent topological quantum states. The absence of magnetic fields eliminates stray-field interference, positioning the goniopolar thermoelectric technology as a promising route for waste heat recovery and solid-state cooling. Furthermore, the fermiology-driven framework can, in principle, be extended to other kagome materials and potentially to a broader class of topological quantum materials. This work establishes a conceptual bridge between kagome fermiology and goniopolarity, expanding the role of kagome materials from hosting exotic quantum states to enabling emergent transport design.

Methods

Sample synthesis

Single crystals of LuCo₆Ge₆ were synthesized using the Sn-flux growth method. A mixture of Lu_1.5Co₆Ge₁₈Sn₁₀₀ high-purity elements was placed in an alumina crucible, sealed in a quartz ampoule under partial argon pressure, and heated to 1,373 K over 10 h. The temperature was maintained for 48 h before gradually cooling to 973 K at a rate of 2 K h⁻¹. The single crystals were then separated from the flux using a centrifugal process, resulting in bar-shaped crystals with shiny surfaces. The preparation of LuCr₆Ge₆ and HoCr₆Ge₆ single crystals followed the same method.

Sample characterization

Laue X-ray diffraction was conducted to confirm the single-crystalline nature of the as-grown crystals at room temperature. STEM specimens of LuCo₆Ge₆, LuCr₆Ge₆ and HoCr₆Ge₆ were prepared using a Tescan dual beam SEM focused ion beam (SEM-FIB). After FIB thinning, the specimens underwent further low-energy Ar⁺ ion milling using a Fischione 1040 Nanomill. Final surface cleaning and thinning start from 700 eV and gradually decreases to 300 eV. HAADF-STEM and EDS were performed using a JEOL JEM-ARM300F transmission electron microscope at 300 kV. The microscope was equipped with double aberration correctors, Gatan Image Filter Quantum with Gatan K2 Summit and dual 100 mm² Si drift detectors. The probe convergence semi-angle for HAADF-STEM images is 25.7 mrad, with a large inner collection angle of 70 mrad. For EDS measurements, to avoid overlapping between different elements, L-line was used for Lu, with K-lines for Cr, Co and Ge. To measure the ellipticity of the atomic column, we first fit the atomic column with a 2D Gaussian function with Atomap⁴⁴. The standard deviation along two orthogonal directions, c_x and c_y, is then collected, whose ratio determines the ellipticity and elongated direction. Specifically, when c_x > c_y, the ellipticity was calculated as c_x/c_y; when c_x < c_y, the ellipticity was calculated as c_y/c_x.

ARPES measurements

High-resolution ARPES measurements of LuCr₆Ge₆ and HoCr₆Ge₆ were performed at the APE-LE beamline in Elettra Sincrotrone Trieste and the 1²-ARPES end station of UE-112-PGM2 beamline at Helmholtz Zentrum Berlin BESSY-II light source. The energy and angular resolutions were set to better than 10 meV and 0.1°, respectively. Samples were cleaved in situ, yielding flat mirror-like (001) surfaces. During the experiments, the temperature was kept at 15–26 K if not specified otherwise, and the vacuum conditions were maintained better than 7.0 × 10⁻¹¹ torr. We used linear horizontal polarization for all the measurements.

Transport property measurements

The longitudinal and Hall resistivities of kagome metals were measured using a physical property measurement system (Quantum Design) equipped with the electrical transport option. The Hall carrier concentration (n) was determined from the slope of the magnetic-field dependence of the Hall resistivity. Hall mobility (μ) values were calculated from the equation 1/ρ = eμn (where e represents the elementary charge). The in-plane (S_IP) and out-of-plane (S_OP) Seebeck coefficients, as well as the in-plane (κ_IP) and out-of-plane (κ_OP) thermal conductivities of kagome metals, were measured using a standard one-heater, two-thermometer steady-state method in a physical properties measurement system under high vacuum conditions. To create a temperature gradient, the sample was affixed to a heat sink with a heater. Chromel–constantan thermocouples were used to measure temperature and voltage differences. Raw data were field-symmetrized for resistivity, Seebeck coefficient and thermal conductivity, and antisymmetrized for Hall resistivity to account for contact misalignment. In-plane and out-of-plane transport properties were measured by applying a current or temperature gradient along the in-plane (001) and out-of-plane $\left(\overline{1}00\right)$ directions, respectively.

After obtaining S_IP and S_OP, transverse thermopower measurements were conducted using chromel–constantan thermocouples to measure longitudinal thermopowers S_L, longitudinal temperature difference, and transverse temperature difference (shown in the inset of Fig. 3b). Copper wires were used to measure transverse thermopowers S_xz. The sign of the transverse voltage is determined by the electrode definition (V+ and V−) and the measurement geometry (inset of Fig. 3b). Thus, the absolute sign of S_xz holds no physical significance. The temperature gradient is homogeneously oriented at a 45° angle from the in-plane (x) and out-of-plane (z) axes. The experimental transverse thermopowers are calculated as follows: S_xz = E_T/∇T_L = (∆V_T/W)/(∆T_L/L), where ∆V_T, W, ∆T_L and L represent the transverse voltage difference, transverse width, longitudinal temperature difference and longitudinal length, respectively. The theoretical S_xz and S_L (dashed lines) in Fig. 3b are determined through the equations S_xz = (S_OP − S_IP)/2, and S_L = (S_OP + S_IP)/2, respectively. We evaluated the transverse Peltier conductivity |α_xz| using the experimental S_L and S_xz values for kagome metals as follows: $|{\alpha }_{\mathit{xz}}|=\frac{1}{2}({\sigma }_{\mathrm{IP}}+{\sigma }_{\mathrm{OP}})S_{\mathit{xz}}+\frac{1}{2}({\sigma }_{\mathrm{IP}}-{\sigma }_{\mathrm{OP}})S_{\mathrm{L}}$, where σ_IP and σ_OP represent in-plane and out-of-plane electrical conductivities, respectively. The in-plane power factor PF_IP, out-of-plane power factor PF_OP and transverse power factor PF_xz can be calculated as PF_IP = σ_IPS_IP², PF_OP = σ_OPS_OP² and ${\mathrm{PF}}_{\mathit{yx}}=\frac{4S_{\mathit{xz}}^2}{{\left(\sqrt{{\rho }_{\text{IP}}}+\sqrt{{\rho }_{\text{OP}}}\right)}^2}$, respectively. Finally, the in-plane z_IPT, out-of-plane z_OPT and transverse z_xzT figure of merits can be obtained as: z_IPT = PF_IPT/κ_IP, z_OPT = PF_OPT/κ_OP and $z_{\mathit{xz}}T=\frac{4S_{\mathit{xz}}^{2}T}{{\left(\sqrt{{\rho }_{\text{IP}}{\kappa }_{\text{IP}}}+\sqrt{{\rho }_{\text{OP}}{\kappa }_{\text{OP}}}\right)}^2}$, respectively. The low-temperature (2–300 K) heat capacity C_p was measured using a PPMS DynaCool (Quantum Design) instrument.

Magnetization measurements were conducted on our samples using a magnetic property measurement system (MPMS3, Quantum Design) across the temperature range of 2–300 K (Supplementary Figs. 4–6). LuCo₆Ge₆ and LuCr₆Ge₆ exhibited consistent paramagnetic behaviour without any evidence of magnetic ordering or fluctuations. HoCr₆Ge₆ showed a magnetic transition at ~4 K, showing paramagnetic characteristics above this temperature⁴⁵. Thus, all three compounds are unequivocally in the paramagnetic state within the goniopolar temperature ranges.

Theoretical calculations

First-principles calculations are carried out based on DFT^46,47 as implemented in the Vienna ab initio simulation package (VASP)^48,49 with the projector augmented wave method⁵⁰. The generalized gradient approximation (GGA) within Perdew–Burke–Ernzerbof scheme is adopted to describe the exchange–correlation energy functional⁵¹. The cut-off energy is set to be 500 eV. A Γ-centred k-mesh of 12 × 12 × 5 is adopted to sample the Brillouin zone. The energy and force convergence criteria are set to be 10⁻⁶ eV and 10⁻³ eV Å⁻¹, respectively. With the obtained DFT results, the maximally localized Wannier functions are constructed using WANNIER90 package⁵², including the d orbitals of Lu/Ho atoms and kagome Cr/Co atoms, and p orbitals of Ge atoms. The thermoelectric properties are analysed based on Boltzmann module⁵³ and self-made codes. A constant relaxation time of 80 fs and a k-mesh of 200 × 200 × 200 is adopted to ensure the convergence. The Fermi surfaces are processed using WannierTools package⁵⁴ and visualized via Fermisurfer⁵⁵. Without specific notation, the presented results are shown without spin–orbit coupling.

Online content

Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/s41563-026-02678-4.

Author contributions

H.H. and B.H. designed the work. H.H. prepared the single crystals and carried out the thermoelectric and magnetic property measurements. H.H. E.C., X.F., F.S., H.W., B.H. and C.F. discussed the results. Y.J. and X.P conducted the STEM observations and analysis. X.F. carried out the theoretical calculations. R.L. performed the ARPES experiments and analysed the data with the assistance of A.F., O.S., A.J., J.F., I.V., D.V.V. and B.B. W.S. and R.K. conducted the measurement of heat capacity. C.F. supervised the project. All authors conceived of the experiments, analysed the results and co-edited the paper.

Peer review

Peer review information

Nature Materials thanks the anonymous reviewer(s) for their contribution to the peer review of this work.

Funding

C.F., R.L., A.F. and B.B. disclose support for the research of this work from Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the Würzburg-Dresden Cluster of Excellence ctd.qmat–Complexity, Topology and Dynamics in Quantum Matter (EXC 2147, grant 390858490). E.C. and X.F. acknowledge the financial support from the Alexander von Humboldt Foundation. Open access funding provided by Max Planck Society.

Data availability

All relevant data supporting the findings of this study are available within the article and its Supplementary Information. Source data are provided with this paper.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s41563-026-02678-4.