Intrinsic nonlinear valley Nernst effect

TL;DR

The work establishes intrinsic nonlinear valley Nernst transport as a fundamental, band-structure–driven phenomenon arising from Berry connection polarizability, enabling a transverse valley current in response to a second-order temperature gradient. It derives a compact intrinsic expression for the nonlinear valley Nernst conductivity $\beta^v_{abc}$, and proves a nonlinear Mott relation $\beta^v=\mathcal{L}\chi^v$ linking it to the intrinsic nonlinear valley Hall conductivity $\chi^v$ at low $T$. Through symmetry analysis and model calculations (tilted Dirac) and first-principles results for bilayer WTe$_2$, the paper demonstrates sizable intrinsic signals and provides nonlocal transport signatures with distinct $\rho^2$ scaling and a thermopower–Lorenz-number ratio as a diagnostic. It also discusses experimental detection strategies (nonlocal measurements, valley pumping, and magneto-optical probes) and clarifies the role of extrinsic contributions, laying a foundation for valley caloritronics and geometry-governed nonlinear thermoelectric phenomena.

Abstract

We investigate the intrinsic nonlinear valley Nernst effect, which induces a transverse valley current via a second-order thermoelectric response to a longitudinal temperature gradient. The effect arises from the Berry connection polarizability dipole of valley electrons and is permissible in both inversion-symmetric and inversion-asymmetric materials. We demonstrate that the response tensor is connected to the intrinsic nonlinear valley Hall conductivity through a generalized Mott relation, with the two being directly proportional at low temperatures, scaled by the Lorenz number. We elucidate the symmetry constraints governing this effect and develop a theory for its nonlocal measurement, revealing a nonlocal second-harmonic signal with a distinct $ρ^2$ scaling. This signal comprises two scaling terms, with their ratio corresponding to the square of the thermopower normalized by the Lorenz number. Key characteristics are demonstrated using a tilted Dirac model and first-principles calculations on bilayer WTe$_2$. Possible extrinsic contributions and alternative experimental detection methods, e.g., by valley pumping and by nonreciprocal directional dichroism, are discussed. These findings underscore the significance of band quantum geometry on electron dynamics and establish a theoretical foundation for nonlinear valley caloritronics.

Figures (5)

• Figure 1: Schematic illustration of VNE. Under a temperature gradient, carriers from two valleys (denoted as $K$ and $K'$) are deflected to opposite transverse directions. For nonlinear VNE, the resulting contribution to Nernst valley current $j^v$ is of second order in the temperature gradient.
• Figure 2: Intrinsic nonlinear VNE and nonlinear Mott relation for the tilted Dirac model. (a) The spectrum of the model, which consists of two $\mathcal{T}$-connected tilted Dirac valleys. (b) Calculated intrinsic nonlinear valley Nernst conductivity $\beta_{xyy}^{v}$ at three different temperatures. (c) Calculated intrinsic nonlinear valley Hall conductivity $\chi_{xyy}^{v}$ at three different temperatures. Here, to check nonlinear Mott relation, $\beta$ and $\chi$ are put in units of $\pi ek^2_Bw/(96\Delta^2)$ and $e^3w/(32\pi\Delta^2)$, respectively. The two units differ by the Lorenz number, so that $\beta$ and $\chi$ in these figures can be directly compared. (d,e) show such a comparison at (d) low and (e) high temperatures, where the solid (dashed) line is for $\beta_{xyy}^{v}$ ($\chi_{xyy}^{v}$). In the calculation, we take $v=1\times10^6$ m/s, $w=0.4v$, and $\Delta=0.1$ eV.
• Figure 3: Schematic illustration of a nonlocal measurement setup. See the main text for a description.
• Figure 4: Intrinsic nonlinear VNE in bilayer WTe$_2$. (a) Crystal structure of bilayer WTe$_2$. It has a mirror $\mathcal{M}_x$ symmetry, as shown in the right panel. (b) Calculated low-energy band structure, which has two valleys on the $k_x$ axis. (c) Fermi surface at 0.05 eV, which shows the two valleys (labeled as $V_1$ and $V_2$ here). (d) Calculated intrinsic nonlinear VNE conductivity as a function of chemical potential at 10 K. (e) displays the dependence of the ratio $c_2T^2/c_1$ in Eq. (\ref{['ratio']}) on the chemical potential.
• Figure 5: Calculated extrinsic nonlinear VNE conductivity in Eq. (\ref{['extrinsic']}) as a function of chemical potential for bilayer WTe$_2$. In the calculation, we take $\tau=10$ fs ma2019Observation.