Quantum kinetic theory of nonlinear thermal current

We investigate the second-order nonlinear electronic thermal transport induced by the temperature gradient. We develop the quantum kinetic theory framework to describe thermal transport in the presence of a temperature gradient. Using this, we predict an intrinsic scattering time-independent nonlinear thermal current in addition to the known extrinsic nonlinear Drude and Berry curvature dipole contributions. We show that the intrinsic thermal current is determined by the band geometric quantities and is nonzero only in systems in which both the space inversion and time-reversal symmetries are broken. We employ the developed theory to study the thermal response in tilted massive Dirac systems. We show that besides the different scattering time dependencies, the various current contributions have distinct temperature dependencies in the low-temperature limit. Our systematic and comprehensive theory for nonlinear thermal transport paves the way for future theoretical and experimental studies on intrinsic thermal responses.

Figure 1

Schematic experimental setup for measuring the nonlinear anomalous thermal Hall effect. Here, the dark pink shows the hot end of the sample, while the light blue shows the cold end. This sets a temperature gradient ($-\mathbf{\nabla }T$) along the $x$ direction. We probe the nonlinear thermal Hall response ($J^Q$) along the $y$ (Hall) and the $x$ (longitudinal) directions.

Figure 2

Energy dispersion of the tilted massive Dirac system specified by Eq. (28) is plotted against $k_x$ and $k_y$ in (a) for the $K$ valley and in (b) for the $K^{\prime }$ valley. Parts (c) and (d) display the momentum space distribution of the band-resolved Berry curvature ${\mathrm{\Omega }}_{+-}^{xy}$ for the two valleys, respectively. The momentum space distribution of the band-resolved quantum metric ${\mathcal{G}}_{+-}^{xy}$ for the two valleys is shown in panels (e) and (f), respectively. The different energy contours in (c)–(f) are indicated by yellow ($\mu =0.2$ eV), brown ($\mu =0.3$ eV), and magenta ($\mu =0.45$ eV) lines. Here, $k_x$ and $k_y$ are in units of Å${}^{-1}$ and $\varepsilon \left(k\right)$ in eV. For these plots, we have used $\hslash v_F=1.55$ eV Å, $v_t=0.4v_F$, and $\mathrm{\Delta }=0.1$ eV.

Figure 3

Variation of the (a) longitudinal and (b) transverse nonlinear Drude conductivity with chemical potential. The solid line represents the numerical results, while the circles convey the analytical results calculated up to linear order in tilt velocity $v_t$. For numerical analysis, we have used $\hslash v_F=1.0$ eV Å, $v_t=0.1v_F, \mathrm{\Delta }=0.1$ eV, and $\hslash /\tau =0.16$ eV. In addition, we have considered the temperature to be 50 K.

Figure 4

(a) Variation of the nonlinear anomalous Hall conductivity with the chemical potential $\mu$, where solid line represents the numerically obtained result. In (b), we have shown the variation of the longitudinal (left) and transverse (right) total nonlinear intrinsic conductivity with chemical potential ($\mu$). The total nonlinear intrinsic conductivity ${\kappa }^{\mathrm{tot}}$ is the sum of the intrinsic contributions arising from ${\kappa }^{\mathrm{do}}, {\kappa }^{\mathrm{od}}$, and ${\kappa }^{\mathrm{oo}}$. Therefore, in (b), the solid line represents the total intrinsic conductivity, while the dashed lines are the different thermal current contributions giving rise to intrinsic conductivity. Additionally, in this figure, the circles denote the analytical result calculated up to linear order in tilt velocity $v_t$. The model parameters, scattering time, and temperature considered here are the same as in Fig. 3. For the numerical and analytical calculation of the nonlinear intrinsic thermal current, we considered the cutoff energy of the valence band to be $2\mathrm{eV}$.