Nonlinear Nernst & Seebeck Effects

Updated 1 February 2026

Nonlinear Nernst and Seebeck Effects are second-order thermoelectric phenomena where a quadratic temperature gradient induces charge currents even in systems with symmetry constraints.
Intrinsic mechanisms rely on band geometric properties like Berry connection polarizability, while extrinsic scattering such as skew-scattering enhances responses in high-mobility materials.
Experimental studies show voltage signals scaling with (ΔT)², offering promising avenues for advanced thermoelectric energy conversion and caloritronic device applications.

The nonlinear Nernst and Seebeck effects (NNE and NSE) are second-order thermoelectric phenomena in which a temperature gradient drives charge (and, in some cases, valley or spin) currents with a quadratic dependence on the gradient, mediated by intrinsic band geometric properties or extrinsic disorder mechanisms. Unlike the canonical linear Nernst and Seebeck effects, which typically require symmetry breaking (e.g., magnetic order or inversion breaking), NNE and NSE can be prominent even in nonmagnetic, parity–time-symmetric materials, interface-coupled systems, and those exhibiting strong Berry connection polarizability or asymmetric scattering. These nonlinear responses have experimental and theoretical significance for next-generation thermoelectric energy conversion and caloritronic device architectures.

1. Fundamental Principles and Definitions

A static temperature gradient $\nabla T$ generically induces a second-order charge current in the form

$J_a^{(2)} = \chi_{abc} (-\nabla_b T)(-\nabla_c T) = T^2\, \alpha_{a;bc}\, E_b^T E_c^T,$

where $E^T = -\nabla T / T$ is the thermal field. The second-order thermoelectric tensor $\alpha_{a;bc}$ encodes both longitudinal (Seebeck-type) and transverse (Nernst-type) responses (Varshney et al., 2024). In open-circuit geometry, longitudinal and transverse voltage responses define the nonlinear Seebeck coefficient $S^{(2)}$ and nonlinear Nernst coefficient $N^{(2)}$ :

$S^{(2)} = -\frac{\psi_{x;xx}^{(2)}}{\sigma_{xx}^{(1)}}, \qquad N^{(2)} = -\frac{\psi_{y;xx}^{(2)}}{\sigma_{yy}^{(1)}},$

where $\psi^{(2)}$ is the pure thermal second-order tensor and $\sigma^{(1)}$ the linear conductivity (Bhalla, 2020). NNE is characterized by a transverse current proportional to $(\nabla T)^2$ , while NSE is the corresponding longitudinal quadratic response.

2. Intrinsic Band-Geometric Mechanisms

Intrinsic contributions arise due to band geometric or quantum metric effects and depend fundamentally on the Berry connection polarizability (BCP) tensor and quantum metric derivatives. In PT-symmetric bipartite antiferromagnets (e.g., CuMnAs), the Berry connection polarizability is given by

$P_{ab,c}(k) = \sum_{p \neq n} \left(\frac{G_{np}^{ab}}{\varepsilon_{np}}\right)_{,c},$

with $G_{np}^{ab}$ the quantum metric and $\varepsilon_{np}$ the band energy difference. The intrinsic (scattering-independent) NNE and NSE appear in the clean limit ( $\hbar/\tau \to 0$ ), dominating over Drude-type or anomalous (Berry curvature dipole) mechanisms. The relevant tensor coefficients remain finite even when the chemical potential $\mu$ lies in a gap and display pronounced enhancement near band extrema due to divergences in the quantum metric (Varshney et al., 2024).

In energy harvesting contexts, the resulting current is robust against disorder and mobility reduction. Devices leveraging both longitudinal and transverse nonlinear responses can simultaneously optimize power extraction in multiple thermal directions (Varshney et al., 2024).

3. Extrinsic Effects: Asymmetric Scattering

Disorder-induced extrinsic mechanisms, particularly side-jump and skew-scattering processes, generate large nonlinear thermoelectric responses in non-centrosymmetric, time-reversal-symmetric materials. The semiclassical Boltzmann treatment decomposes the second-order current into six contributions: nonlinear Drude (ND), Berry curvature dipole (NA), nonlinear side-jump (NSJ), nonlinear skew-scattering (NSK), and mixed anomalous side-jump/skew terms. Skew-scattering, scaling with $\tau^3$ (scattering time cubed), can dominate in high-mobility systems and is greatly enhanced by strong Berry curvature "hot-spots" near van Hove singularities (Varshney et al., 25 Jan 2026).

In ABA-stacked trilayer graphene, the measured NNE/NSE signals agree quantitatively with theoretical extrinsic predictions, showing the superiority of extrinsic (particularly NSK) channels over intrinsic band geometry alone, especially under realistic disorder regimes (Varshney et al., 25 Jan 2026).

4. Symmetry Constraints and Tunability

The existence and magnitude of NNE and NSE are governed by symmetry:

PT symmetry (combined inversion and time reversal) suppresses linear anomalous Nernst, but allows a finite nonlinear response via even-in- $k$ BCP terms.
In 2D time-reversal symmetric crystals, intrinsic NNE from the Berry curvature dipole is allowed only where a single in-plane mirror symmetry exists. External fields (e.g., applied DC electric field) can lift this constraint, activating a nonlinear anomalous Nernst effect (NANE) even in systems of higher symmetry (e.g., monolayer graphene with $C_{6v}$ ) (Wu et al., 24 Jan 2026).
In the valley Nernst context, the nonlinear valley Nernst effect (valley-polarized current at second order) is permitted for both inversion-symmetric and inversion-asymmetric materials. Inversion forbids the linear valley Nernst effect but allows the nonlinear one (Zhang et al., 27 Aug 2025).

Symmetry considerations thus dictate design rules for realizing large NNE/NSE: broken inversion, presence of hot-spot Berry curvature, and high scattering asymmetry favor maximal responses (Varshney et al., 25 Jan 2026).

5. Experimental Realizations and Scaling Laws

Recent experiments demonstrate the robust detectability and scaling behavior of nonlinear thermoelectric effects:

In Ni $_{81}$ Fe $_{19}$ |Pt bilayers, Hirata et al. observed NSE at room temperature, with voltage scaling as $(\Delta T)^2$ and inversely with bar length $l$ ( $V_{2\omega} \propto 1/l$ ) (Hirata et al., 4 Jul 2025). The nonlinear coefficient $S_2$ reverses sign with magnetization and with the direction of $\Delta T$ .
Second-harmonic lock-in techniques, employing alternating temperature gradients, afford direct measurement of NNE/NSE signals (e.g., $V^{2\omega}$ in graphene and topological insulators) and separate second-order effects from linear and mixed Joule contributions (Wu et al., 24 Jan 2026).
Nonlocal voltage detection in Hall-bar geometries exploits the quadratic scaling of the signal with sheet resistivity ( $\rho^2$ ) and distinct exponential decay with valley diffusion length, as demonstrated for bilayer WTe $_2$ (Zhang et al., 27 Aug 2025).

These scaling laws offer practical guidance for nanoscale device engineering and optimization of nonlinear power extraction.

6. Theoretical Frameworks: Microscopic and Macroscopic Formulations

The general microscopic expansion of the charge current to second order yields

$j_i = \sigma^{(1)}_{ij} E_j + \alpha^{(1)}_{ij}(-\nabla_j T) + \chi^{(2)}_{i;jk} E_j E_k + \phi^{(2)}_{i;jk} E_j (-\nabla_k T) + \psi^{(2)}_{i;jk} (-\nabla_j T)(-\nabla_k T) + \cdots.$

Nonlinear Seebeck and Nernst coefficients directly follow as ratios of second-order tensors to linear conductivities (Bhalla, 2020). In topological insulator surfaces with hexagonal warping, these coefficients exhibit parameter dependence:

$S^{(2)} \sim -\frac{\Delta\,\lambda}{\tau\,v_F^2\,T\,\mu}, \qquad N^{(2)} \sim -\frac{\lambda}{v_F^2\,T},$

where $\lambda$ is the warping strength, $\Delta$ the surface gap, and $v_F$ the Fermi velocity. Increasing warping enhances both nonlinear coefficients, while the gap impacts only $S^{(2)}$ (Bhalla, 2020).

Generalizations to valleytronics utilize BCP dipole tensors and establish Mott-type relations between nonlinear valley Nernst and nonlinear valley Hall conductivities, with proportionality constant given by the Lorenz number at low temperature (Zhang et al., 27 Aug 2025).

7. Implications and Outlook for Thermoelectric Applications

NNE and NSE offer robust, disorder-insensitive avenues for thermoelectric power conversion and caloritronic device design, particularly in situations where the linear response is symmetry-forbidden or fundamentally limited. Intrinsic, quantum-metric-based contributions are especially valuable for low-mobility or gapped systems, while extrinsic scattering mechanisms enable magnetic-field-free nonlinear thermopower in high-mobility materials with engineered disorder asymmetry.

Applied research directions include thermopile arrays, local temperature fluctuation sensors, and multi-directional energy harvesting modules utilizing PT-symmetric antiferromagnets, topological insulators, and 2D multilayer graphene. Open questions persist around the role of spin, magnon, and phonon-drag contributions to the nonlinear effects, the influence of interfacial spin-mixing conductance, and the extension to bulk noncentrosymmetric or lateral heterostructures (Hirata et al., 4 Jul 2025). The development of lock-in detection, nonlocal probes, and optical methods such as valley pumping further expands the experimental toolkit for investigating and harnessing NNE/NSE effects.

Table: Key Second-Order Thermoelectric Contributions in NNE/NSE

Channel	Symmetry Required	Scaling (in $\tau$ )	Dominance Regime
Quantum Metric	PT-symmetric antiferromagnets, nonmagnetic	$\tau^0$	Band-edge, low disorder, gapped
Berry Dipole	Inversion broken, 1 mirror	$\tau^1$	Fermi-surface, clean limit
Side-jump (NSJ)	Inversion broken	$\tau^2$	Moderate disorder, weak skewness
Skew-scattering (NSK)	Inversion broken	$\tau^3$	High mobility, strong disorder skew
Extrinsic Mixed	Noncentrosymmetric magnets	$\tau^2$ – $\tau^3$	Direct diagnosis of PT breaking