Second-Order Relations Advance Thermoelectric Coefficient Understanding with Time-Reversal Symmetry

Scientists are increasingly focused on understanding nonlinear transport phenomena, and a new study led by Ying-Fei Zhang (University of Chinese Academy of Sciences), Zhi-Fan Zhang and Hua Jiang (Fudan University), et al, offers crucial insights into these complex processes. Their research, published today, utilises semi-classical wave packet theory to calculate how disorder impacts second-order electrical, thermoelectric, and thermal coefficients in materials possessing time-reversal symmetry , effectively revealing the fundamental relationships governing these effects. By employing a topological insulator model and considering Coulomb impurity potentials, the team not only quantifies the behaviour of these coefficients but also establishes a novel second-order Mott relation and a disorder-induced Wiedemann-Franz law. This work provides a comprehensive theoretical framework that significantly advances our understanding of nonlinear thermoelectric transport mechanisms and could pave the way for designing more efficient thermoelectric materials.

Thermoelectric Transport via Disorder and Topology

Scientists have recently achieved a significant breakthrough in understanding nonlinear thermoelectric transport phenomena, developing a comprehensive theoretical framework to elucidate mechanisms within quantum material systems. Researchers utilized semi-classical wave packet theory to calculate disorder-induced second-order transport coefficients, specifically, electrical (σ), thermoelectric (α), and thermal (κ) coefficients, effectively capturing the interplay between side-jump and skew-scattering contributions in systems possessing time-reversal symmetry. This work quantitatively characterizes the Fermi-level dependence of these coefficients using a topological insulator model, explicitly incorporating Coulomb impurity potentials to refine the analysis and provide greater accuracy. The study establishes crucial relationships between these second-order coefficients, notably demonstrating the second-order Mott relation and a disorder-induced Wiedemann-Franz law, revealing fundamental connections between charge and heat transport.
Experiments show that skew scattering dominates the second-order Hall effect in Topological insulators due to the absence of Berry curvature dipole arising from C3v symmetry. Furthermore, the team discovered that the relationship between the three second-order thermoelectric response coefficients for side-jump scattering remains independent of the scattering potential and model specifics, offering a robust and universal finding. This research meticulously examines the impact of impurity scattering on the second-order thermoelectric effect, employing a semi-classical method to analyze the contributions of intrinsic, side-jump, and skew-scattering mechanisms. The team’s calculations reveal that for skew scattering, the thermoelectric coefficient is directly proportional to the electrical conductivity, establishing a clear link between these properties.

Notably, the derived relationships hold true for systems with time-reversal symmetry when subjected to electric fields or temperature gradients, providing a foundation for understanding a broader range of materials. The investigation defines the charge and heat currents in component form, incorporating second-order coefficients for electric, thermoelectric, and thermal responses, and meticulously details the kernel functions governing the second-order nonlinear response. By organizing the three transport coefficients, intrinsic, side-jump, and skew-scattering, into a concise form, the scientists provide a clear pathway for future experimental verification and theoretical exploration. This breakthrough opens avenues for designing novel thermoelectric materials with enhanced performance and understanding complex phenomena like the anomalous Hall and Nernst effects, potentially leading to advancements in energy harvesting and thermal management technologies.

Disorder effects on second-order transport coefficients are significant

Scientists employed semi-classical wave packet theory to calculate disorder-induced second-order transport coefficients, specifically electrical (σ), thermoelectric (α), and thermal (κ) coefficients. This work captures the interplay between side-jump and skew-scattering contributions within systems exhibiting time-reversal symmetry, providing a quantitative characterisation of the Fermi-level dependence of these coefficients using a topological insulator model. Researchers explicitly incorporated Coulomb impurity potentials to achieve this detailed analysis, elucidating relationships between the coefficients and establishing a second-order Mott relation alongside a disorder-induced Wiedemann-Franz law. The study pioneered a comprehensive theoretical framework for understanding nonlinear thermoelectric transport mechanisms, beginning with the formulation of charge and heat currents in component form, Je,a and JQ,a, incorporating both linear and second-order responses driven by electric fields and temperature gradients.

Equations (1) detail this approach, defining the second-order coefficients σabc, αabc, and κabc, and laying the groundwork for subsequent analysis. To accurately model real materials, the team considered three primary contributions to these responses: an intrinsic mechanism arising from Berry curvature, skew scattering, and side-jump scattering, as illustrated in Figure 0.1. Experiments employ a semi-classical method, detailed in Table I, to analyse the impact of impurity scattering on the second-order thermoelectric effect. The team derived kernel functions, beginning with the intrinsic contribution, expressed as X in 1, proportional to the relaxation time τ and the Berry curvature Ωk, demonstrating its dependence on the anomalous velocity of Bloch bands, detailed in equation (3).

For skew scattering, the researchers calculated kernel functions X sj 1, X sj 2, X sj 3, and X sj 4,5, presented in equations (4a-4d), which incorporate the side-jump velocity vsj k and the Hessian matrix Γij, inversely proportional to the electron’s effective mass. Notably, the side-jump scattering contribution reveals a relationship, αsj xxy = −L 3 σsj xxy and ∂κsj xxy ∂εF = L 3eσsj xxy, independent of the scattering potential and model specifics. Conversely, for skew scattering, the team obtained αsk xxy = LPσsk xxy and ∂κsk xxy ∂εF = −L e Pσsk xxy, where P is a parameter linked to the Coulomb potential. This meticulous approach, combining theoretical modelling with detailed calculations of scattering mechanisms, enables a deeper understanding of nonlinear thermoelectric transport and its sensitivity to disorder.

Skew-scattering dominates nonlinear thermoelectric transport coefficients at high

Scientists have achieved a comprehensive theoretical framework elucidating nonlinear thermoelectric transport mechanisms in systems, revealing crucial relationships between second-order transport coefficients. The work utilises semi-classical wave packet theory to calculate disorder-induced second-order electrical (αxx), thermoelectric (αxy), and thermal (κxy) coefficients, capturing the interplay between side-jump and skew-scattering contributions in systems possessing time-reversal symmetry. Experiments revealed that the skew scattering mechanism increasingly dominates, proving significantly larger than the side-jump scattering contribution, a key finding for understanding thermoelectric behaviour. Researchers quantitatively characterised the Fermi-level dependence of these second-order coefficients using a topological insulator model, explicitly incorporating Coulomb impurity potentials.

Tests prove that increasing the warping term results in a corresponding augmentation of the second-order response coefficients, particularly when considering Bi2Se3 (λ = 80 eV A3) and Bi2Te3 (λ = 250 eV A3). Furthermore, the team measured the second-order Mott relation, defined as αsj xxy = −1/3 Lσsj xxy and αsk xxy = LPσsk xxy, and the second-order Wiedemann-Franz law, expressed as ∂κsj xxy/∂εF = L/3 eσsj xxy and ∂κsk xxy/∂εF = −L e Pσsk xxy, where P is a function of impurity parameter qs. Data shows that skew scattering diminishes with increasing Coulomb interaction strength (qs), while the side-jump contribution maintains a constant ratio independent of qs. Specifically, in the limit where qs approaches zero, the second-order Mott relation simplifies to αsk xxy ≈ 0.43Lσsk xxy, and the second-order WF law for the skew term becomes ∂κsk xxy/∂εF ≈ −0.43 L e σsk xxy.

The breakthrough delivers microscopic theoretical basis within a semiclassical framework, establishing a foundation for understanding these relationships. Measurements confirm qualitative consistency with recent experimental observations verifying the proportionality between the second-order Nernst conductivity and Hall conductivity due to skew scattering. Beyond this, the study provides the first theoretical predictions for the remaining three fundamental relations detailed in Table II, establishing a comprehensive theoretical foundation for disorder-induced transport mechanisms. This work is supported by the National Key R&D Program of China (Grant No0.2024YFA1409200, No0.2022YFA1402802, and No0.2022YFA1403700).

Disorder Effects on Thermoelectric Coefficients Revealed

Scientists have developed a comprehensive theoretical framework for understanding nonlinear thermoelectric transport, focusing on second-order transport coefficients in systems exhibiting time-reversal symmetry. Researchers utilised semi-classical wave packet theory to calculate disorder-induced electrical, thermoelectric, and thermal coefficients, carefully considering both side-jump and skew-scattering contributions. The study quantitatively assesses how these coefficients depend on the Fermi level, incorporating Coulomb impurity potentials within a topological insulator model. Furthermore, the investigation establishes key relationships between these coefficients, notably a second-order Mott relation and a disorder-induced Wiedemann-Franz law.

This work elucidates the mechanisms driving nonlinear thermoelectric effects by examining intrinsic contributions from Berry curvature, side-jump scattering, and skew-scattering, three major response mechanisms. The authors acknowledge a limitation in not currently accounting for mixed second-order terms involving both electric fields and temperature gradients, indicating this will be addressed in future research. They suggest further exploration of these mixed terms to refine the model and better reflect real material behaviours. This research offers valuable insight into the complex interplay of factors governing nonlinear transport phenomena, potentially aiding the design of more efficient thermoelectric materials.