Theoretical framework for dimensional reduction-amplified transverse TE effects
The thermopower tensor (\(\overleftrightarrow{S}={\overleftrightarrow{\sigma }}^{-1}\cdot \overleftrightarrow{\alpha }\)) is determined by both the electrical conductivity tensor \(\overleftrightarrow{\sigma }\) and the TE conductivity tensor \(\overleftrightarrow{\alpha }\). For an isotropic perfectly compensated semimetal (ne = nh and μe = μh), with a temperature gradient applied along the y direction, the induced electric voltage measured along the x direction, and a magnetic field applied along the z direction, the two-carrier Nernst response can be expressed as:17
$${{{\rm{S}}}}_{{{\rm{xy}}}}=\left({{{{\rm{S}}}}_{{{\rm{xy}}}}}^{{{\rm{e}}}}{\sigma _{{{\rm{yy}}}}}^{{{\rm{e}}}}+{{{{\rm{S}}}}_{{{\rm{yx}}}}}^{{{\rm{h}}}}{\sigma _{{{\rm{yy}}}}}^{{{\rm{h}}}}\right)/{\sigma }_{{{\rm{yy}}}}+1/2{\mu }_{{{\rm{yy}}}}{{{\rm{B}}}}_{{{\rm{z}}}}\left({{{{\rm{S}}}}_{{{\rm{yy}}}}}^{{{\rm{h}}}}-{{{{\rm{S}}}}_{{{\rm{yy}}}}}^{{{\rm{e}}}}\right)$$
(1)
where μ denotes the carrier mobility and superscripts e and h refer to electrons and holes, respectively (Supplementary material, Note I). This expression consists of two principal contributions: a diffusive term and an ambipolar term. The diffusive term involves the off-diagonal thermopowers of electrons and holes and scales as tanΘ/ɛ, where tanΘ is the Hall angle, and ɛ is the Fermi energy. In highly conductive metals, the combination of a small Hall angle and a large Fermi energy renders this contribution negligible. The ambipolar term, which is coupled to the Seebeck electric field, typically dominates in topological semimetals7,8,9,10,11,12,13,14,15,16,17,18,19,20 because high mobility bands coexist with large partial diffusion (Syyh ≈ −Syye ~ 1/ɛ) and substantial phonon-drag Seebeck components. Notably, because the partial Seebeck coefficients have opposite signs, the coexistence of electrons and holes actually amplifies the total Nernst thermopower (Sxy), acting as the crucial source of large Sxy in topological semimetals. In low-dimensional systems, the Mott relation shows that a strong energy dependence of conductivity markedly enhances the Seebeck effect. The partial Seebeck coefficients, defined by the energy derivative of conductivity, become maximized under quasi-1D conditions owing to nearly δ-shaped energy-dependent density of states2,4,5. To further enhance the Nernst thermopower, we turn our focus to the mobility term (μyy), which is underexplored in 3D semimetals. Low-dimensional semimetals exhibit pronounced mobility anisotropy; consequently the ambipolar contribution can be reformulated as (Supplementary material, Note I):
$${{{{\rm{S}}}}_{{xy}}}^{{{{\rm{ambipolar}}}}}=1/2\left({{S}_{{{{{\rm{yy}}}}}}}^{{{\rm{h}}}}-{{S}_{{{{{\rm{yy}}}}}}}^{{{\rm{e}}}}\right){B}_{z}{\mu }_{H}^{2}/{\mu }_{{{{\rm{xx}}}}}$$
(2)
where the band mobility μyy is reformulated by a reduced mobility\(\tilde{\mu }\) = μH2/μxx, with μH the Hall mobility. As the dimensionality is reduced, carrier mobility becomes increasingly anisotropic, remaining high along the conducting direction while being strongly suppressed in the transverse directions. This pronounced transport anisotropy leads to a substantial enhancement of the reduced mobility.
In three-dimensional (3D) semimetals, the strongest Nernst response typically emerges within a narrow temperature window just below the onset of the phonon-phonon Umklapp process. In this region, the coexistence of high carrier mobility and phonon drag-enhanced partial Seebeck coefficients can yield Nernst thermopower on the order of millivolts per Kelvin17,18,19,20. Beyond this range, electron-phonon interactions, which are strongly constrained by the Debye temperature, further suppress any enhancement of the Nernst thermopower. By contrast, quasi-1D systems, owing to their sharp density of states, inherently sustain large partial Seebeck coefficients that scale with temperature4. When coupled to the phonon drag contributions, this results in enhanced Seebeck responses persisting over a broader temperature range. Furthermore, dimensional reduction amplifies both the effective-mass anisotropy with inter-chain effective masses approaching infinity, which may markedly enhance the reduced mobility \(\tilde{\mu }\) (see Fig. 1a, b). Collectively, these effects yield a substantially amplified Nernst thermopower in quasi-1D LMO, with enhancements exceeding 160-fold compared with 3D counterparts (see Fig. 1d), and sustain large responses over extended temperature ranges, surpassing 8 mV/K from 20 to 41 K and 4 mV/K from 10–61 K (see Fig. 3h).
To quantitatively evaluate the role of reduced dimensionality, we employ a tight-binding model described by H± = c0 -txcoskx - tycos ky - tzcos kz ± Δ, where ti represents hopping parameters, Δ controls the energy separation between the electron and hole bands, and c0 adjusts the band filling and carrier compensation. Because the multiband Nernst response is highly sensitive to both the difference between the electron and hole Seebeck coefficients and the degree of carrier compensation, we impose perfect compensation between the two bands with identical carrier fillings for all anisotropy ratios (Supplementary Fig. 1c, d). This construction eliminates variations arising from carrier imbalance, allowing the effect of dimensionality-induced Fermi-surface reconstruction to be isolated. We first fix ty and set tx = tz = Ktty with Kt varying from 1 to 0 to model the dimensional crossover from 3D to 1D. For each Kt, the Fermi energy is determined self-consistently at fixed carrier filling, and the Nernst thermopower is calculated by solving the Boltzmann transport equation. As shown in Fig. 1c, the Nernst response increases monotonically with decreasing Kt, demonstrating that the evolution toward a more anisotropic Fermi surface geometry can substantially amplify the Nernst response. To further distinguish the contribution of transport anisotropy from that of the Seebeck coefficient, we introduce an additional anisotropy by setting tx = 2tz. For the same longitudinal Seebeck coefficient Syy, Szy exhibits a much stronger enhancement than Sxy, indicating that the enhanced Nernst response cannot be attributed solely to an increased Seebeck coefficient. Instead, the anisotropic Fermi-surface geometry enhances the Nernst response through the reduced mobility. Within this framework, the Nernst thermopower is governed by the dimensionality-dependent parameters Sxxh – Sxxe and Km. Reduced dimensionality, therefore, enhances the transverse thermoelectric response through two synergistic mechanisms: increasing the partial Seebeck coefficient and strengthening transport anisotropy, thereby providing additional degrees of freedom for boosting the Nernst response.
Quasi-one-dimensional LMO with half-filled band and sharp quasi-linear dispersion
Based on the derivation above, low-dimensional systems with perfect compensation and high mobility can be decent transverse TE systems. The quasi-1D LMO serves as an ideal platform for demonstrating the role of dimensional reduction in enhancing transverse TE effects. It crystallizes in a monoclinic structure (space group of P21/m1; see Fig. 2a) with its primitive cell consisting of two stacks of four corner-sharing MoO6 octahedra and two MoO4 tetrahedra. The weak orbital overlap between the adjacent stacks within the ac-plane results in quasi-1D electronic transport characteristics, with a conductive zigzag chain along the crystallographic b-axis. Only two of the four MoO6 units participate in electric conduction, conferring strong quasi-1D properties to the system. This quasi-1D transport is clearly manifested in the temperature-dependent resistivity (see Fig. 2b), with the resistivity ratio along the three principal axes (ρbb: ρaa: ρcc) reaching ~ 1: 30: 262 at 300 K.
Fig. 2: Crystal structure, electronic structure, and quasi-1D transport properties of LMO.
a Crystal structure of LMO (emerald green, Li; steel blue, Mo; rose brown, O) together with the Nernst measurement configuration, where the heat current is applied along the conducting chains and the transverse voltage is detected along the inter-chain a-axis (Sab) and c-axis (Scb), respectively. b Axis-dependent resistivities. c Band structure along Γ-Y, showing a sharp quasi-linear band in the vicinity of the Fermi level EF (red dashed line). d Quasi-1D Fermi surface of LMO within the Brillouin zone, comprising four nearly flat sheets. Both positive and negative curvature of sheets yield near-perfect compensation between electrons and holes. e The temperature dependence of Seebeck coefficients. A remarkably small net Seebeck coefficient (Sbb ~ 5 μV/K) is observed at 25 K, indicative of nearly perfect electron-hole compensation. The total partial Seebeck coefficients, extracted from extended Nernst Eq. (2) calculations (see Note 5 of the Supplementary Material for details), peak at 25 K with pronounced contributions from both phonon drag and diffusion. The partial diffusion contributions are obtained through DFT calculations. The partial phonon-drag contributions are then obtained by subtracting the diffusion component from the total contribution. The diffusion term, strongly enhanced by reduced dimensionality, dominates above 70 K and extends the temperature window of large Nernst response. Even at 25 K, the diffusion contribution remains significant, reaching ~50 μV/K, owing to the quasi-1D properties. f Temperature-dependent carrier concentration extracted from two-carrier model fitting.
Electronic band structure (DFT) calculations reveal that the Fermi surface of LMO consists primarily of two half-filled quasi-1D bands (see Fig. 2c). These generate four nearly flat Fermi surface sheets (see Fig. 2d), mirrored by inversion symmetry. Each sheet contains both positive and negative curvature23, yielding near-perfect compensation between electrons and holes. Experimental Hall conductivity fitting by the two-carrier model and temperature dependence of Seebeck coefficient further confirm this compensation across the measured temperature range (see Fig. 2e, f). Above 20 K, the carrier densities remain nearly constant. By contrast, a sharp decline in both carrier types occurs below 20 K, which potentially indicates the onset of possibly correlated effects associated with the metal-insulator transition in ρ(T)24,25,26. Moreover, the sharp band dispersions near the Fermi level are quasi-linear, resembling those of topological semimetals and enabling relatively high mobility. The Hall mobility reaches 0.3 m2 V−1 s−1 at 2 K (Supplementary Fig. 8c).
Difference in giant Nernst and Ettingshausen signals after fixing heat and electric currents
The combination of carrier compensation and relatively high mobility in LMO provides a robust foundation for achieving large transverse TE signals. Moreover, the quasi-1D properties make it possible to verify the correlation between transverse geometry and transverse TE coefficients. In order to exclude the influence of Seebeck electric field (based on the extended Nernst Eq. (2)), we design the Nernst thermopower measurement under a fixed thermal gradient applied along the conductive b-axis (see Figs. 2a and 3a–g). The Sab is proportional to the applied magnetic field and reached a maximum of 2760 μV/K at 25 K and 9 T. Remarkably, upon rotation of the transverse voltage probe towards a more quasi-1D orientation, Scb exhibits a substantial enhancement, reaching a peak value of 11430 μV/K under identical measurement conditions. This value is second only to that of pure Bi27,28, yet occurs at a higher temperature (25 K vs 8 K), resulting in a substantial transverse TE signal across a broader temperature range (with values exceeding 1000 μV/K below 120 K, see Fig. 3h), highlighting the strong potential of LMO for transverse TE cooling applications.
Fig. 3: Dimensional reduction enhanced Nernst and Ettingshausen responses.
a–g Magnetic field-dependent Nernst signals measured between 20 K and 100 K, with the heat current applied along the b-axis and the transverse voltage detected along the a-axis (Sab) and c-axis (Scb), respectively. h Temperature dependence of Nernst signals at 9 T with a significant difference between Sab and Scb. i–o Magnetic-field-dependent Ettingshausen signals recorded between 24.5 K and 102.2 K, with the temperature difference measured along the b-axis and electric current applied along the a-axis (∆Tba/jaa) and c-axis (∆Tbc/jcc), respectively. p Comparison of the Ettingshausen coefficient at 9 T obtained from the Bridgman relation with that determined by direct measurement.
To further examine the role of transverse geometry with the heat flow direction held constant, we measured the Ettingshausen signals, i.e., the magnetic field induced transverse temperature gradient with the temperature difference recorded along the b-axis (see Fig. 3i–o). The Ettingshausen response shows a similar field dependence to that of the Nernst thermopower, displaying an approximately linear increase with increasing field strength across 24.5–102.2 K range. Maximum Ettinghausen signals reach 1.18 × 10−3 Km/A and 6.3 × 10−3 Km/A at 52.5 K and 43.1 K for electric currents along the a- and c-axis, respectively, which are comparable to or even surpass that of Bi (0.45 × 10−3 Km/A at 6.8 K)27,28. While Bi exhibits extremely high mobility and conductivity at zero magnetic field, its conductivity is strongly suppressed under a magnetic field due to the Lorentz force, reaching the same order as that of LMO27,28. These results further underscore the Ettingshausen cooling capacity of LMO. Additionally, the Ettingshausen coefficient (E) and the Nernst thermopower (Syx) are related via the Bridgman relation, Eyx = SxyT/κyy29, where κ denotes the thermal conductivity and is independent of magnetic field in LMO (Supplementary Fig. 3c). Figure 3p compares the Ettinghausen coefficients obtained via direct measurement with those calculated from the Nernst thermopower. The excellent agreement between the two provides compelling evidence for the pronounced difference in transverse TE response of LMO under fixed directions of heat flow and output temperature difference.
The correlations between dimensional reduction and transverse TE signals
The giant Nernst thermopower and the pronounced disparity between Sab and Scb indicate that reduced dimensionality plays a crucial role in enhancing the transverse TE response. To elucidate its origin, we employ the extended Nernst Eq. (2). Similar to the case of topological semimetals, the carrier-diffusion contribution plays only a negligible role in enhancing the transverse TE response. Although the Hall angle is large, the Fermi energy is also comparatively high (ɛ ≈ 500 meV). Consequently, the enhancement associated with the large Hall angle is largely offset by the high Fermi energy, resulting in a carrier-diffusion contribution of only ~10 μV/K at 25 K. This value is nearly three orders of magnitude smaller than the measured Nernst thermopower of 11,470 μV/K. Therefore, the giant Nernst response is dominated by the ambipolar contribution (Supplementary Fig. 7b). Earlier studies have reported large Sab and anisotropic transport properties focusing on the ab plane30,31, but overlooked the even larger Scb, attributing the enhancement to a phonon drag-induced Seebeck electric field within an isotropic Nernst framework. However, this simplified isotropic formulation fails to capture the giant difference between Sab and Scb when the direction of heat flow is fixed. In our results, the dimensional reduction and phonon drag together yield a large partial Seebeck electric field that persists over a wide temperature range (see Figs. 2e and 3h). At low temperatures, the phonon-drag Seebeck component dominates, particularly near the phonon peak. As the temperature increases, the contribution from the Seebeck effect induced by the low-dimensional density of states gradually becomes more significant (Sbbdrag ≈ Sbbdiffusion at 120 K; Supplementary Fig. 8f). Moreover, quasi-1D transport anisotropy further enhance Nernst thermopower via a extended Nernst Eq. (2) (Supplementary Fig. 8e), in which the reduced mobility\(\tilde{\mu }\) embodies both intrinsic contributions from Fermi surface curvature Km.
Figure 4a presents the temperature dependence of the Hall mobility. Both μHab and μHcb increase upon cooling, with μHab consistently exceeding μHcb, reflecting the lower value of μcc compared to μaa. We applied the particular subscription of a and b to represent the effective mass ratio maa*/mbb*. Hall resistivity measurements at 2 K (see inset, Fig. 4a) further reveal a pronounced difference between ρcb and ρab, accompanied by nonlinear field dependence arising from multiband transport. To quantitatively assess the effect of dimensional reduction (manifested as the reduced mobility \(\tilde{\mu }\)) on Nernst signals, we compare the experimental ratio Scb/Sab with the theoretical expression \({\tilde{\mu }}_{{{{\rm{cb}}}}}/{\tilde{\mu }}_{{{{\rm{ab}}}}}\) (Fig. 4b). The near identical temperature dependence and magnitude of these ratios provide strong evidence that the difference in Nernst response originates from the substantial difference in the reduced mobility induced by quasi-1D properties of LMO. The pronounced anisotropy in Fermi-surface geometry, characteristic of low-dimensional transport, thus governs the observed amplification of the transverse TE response.
Fig. 4: Hall mobility and transverse TE performance.
a Temperature-dependent Hall mobility extracted from two-carrier model fitting. The inset shows the Hall resistivity measured at 2 K with the electric current applied along the b-axis and the transverse voltage detected along the a-axis (ρab) and c-axis (ρcb), respectively. b Comparison between the ratios Scb/Sab and \({\tilde{\mu }}_{{{{\rm{cb}}}}}/{\tilde{\mu }}_{{{{\rm{ab}}}}}\) as a function of temperature at 9 T with \({\widetilde{\mu }}_{{{\mathrm{cb}}}}\) = μHcb²/μcc and \({\tilde{\mu }}_{{{{\rm{ab}}}}}\) = μHab²/μaa. In this analysis, differences in charge carriers between the a-axis and c-axis were neglected, with ρcc/ρaa used as a proxy for μaa/μcc. c Comparison of the Nernst response and carrier mobility in LMO with those of representative transverse TE materials8,9,10,11,12,13,14,15,16,17,18,19,20,27,28. d Temperature-dependent transverse power factors of LMO benchmarked against other typical transverse TE systems and Bi2Te38,9,10,11,12,13,14,15,16,17,18,19,20,34.
Crucially, these findings broaden the conventional framework for understanding large transverse TE effects. In addition to the established roles of high mobility and large compensated Seebeck coefficients, we identify reduced dimensionality as a decisive factor that can dramatically amplify transverse TE responses. At 25 K, the reduced mobilities (\({\tilde{\mu }}_{{{{\rm{ab}}}}}\) and \({\tilde{\mu }}_{{{{\rm{cb}}}}}\)) reach 4.1 and 19.1 m2 V−1 s−1, corresponding to 22- and 160-fold enhancements in transverse TE signals compared with those expected from the bare Hall mobility and Seebeck electric fields (see Fig. 1d). Figure 4c benchmarks these mobility-dependent Nernst thermopower against other representative transverse TE materials. Despite its relatively modest Hall mobility, LMO yields a large Nernst signal second only to that of Bi, underscoring the critical role of dimensional reduction in enabling superior transverse TE responses.
Giant transverse TE power factors below the liquid nitrogen temperature region
From an application perspective, the orthogonal configuration of heat flow and voltage output in transverse TEs not only simplifies device architecture but also enables high power density. Beyond this structural advantage, our results reveal that such an orthogonal geometry intrinsically provides a pathway to enhance both the transverse response and overall TE performance. Achieving a substantial transverse TE signal requires the heat-flux direction to sustain high carrier mobility and a large compensated Seebeck electric field, while the voltage-output direction maintains comparatively lower mobility. This yields a large reduced mobility, naturally realized in quasi-1D systems with anisotropic Fermi-surface geometry. This requirement directly challenges the conventional view that high resistivity suppresses the TE power factor. Instead, we demonstrate that dimensional reduction amplifies the transverse TE response by simultaneously enhancing the Seebeck electric field and the reduced mobility, thereby boosting the transverse power factor (PFcb = Scb2/ρcc ~ μHcb4ρcc) through geometric effects. As illustrated in Fig. 4d, the temperature-dependent power factor PFab reaches a maximum of ~139 μW cm−1 K−2 at 25 K and 9 T. The transverse TE response is further amplified along stronger quasi-one-dimensional direction, yielding a sixfold increase in power factor. Specifically, PFcb attains 814 μW cm−1 K−2 at the same temperature and field, comparable to those of other reported transverse TE materials, including WTe2 and NbSb2. Notably, this high performance persists throughout the entire liquid nitrogen temperature range. At sub-liquid nitrogen temperature, the transverse power factor of LMO significantly exceeds that of Bi2Te3, the benchmark longitudinal TE material in commercial applications. This comparison underscores the substantial promise of LMO for high-efficiency transverse TE cooling in cryogenic environments.