Nonlinear heat transport in mesoscopic conductors: Rectification, Peltier effect, and Wiedemann-Franz law

We investigate nonlinear heat properties in mesoscopic conductors using a scattering theory of transport. Our approach is based on a leading-order expansion in both the electrical and thermal driving forces. Beyond linear response, the transport coefficients are functions of the nonequilibrium screening potential that builds up in the system due to interactions. Within a mean-field approximation, we self-consistently calculate the heat rectification properties of a quantum dot attached to two terminals. We discuss nonlinear contributions to the Peltier effect and find departures from the Wiedemann-Franz law in the nonlinear regime of transport.

Figure 1

Sketch of a dot system driven out of equilibrium by voltage biases $V_1$ and $V_2$ and thermal differences $T_1$ and $T_2$. ${\Gamma }_1$ and ${\Gamma }_2$ denote the reservoir-dot tunneling rates to the external reservoirs and $E_d$ represents the dot energy level. $U$ is the internal dot potential illustrated as a shift of the dot bottom band.

Figure 2

Heat current $\mathcal{J}$ (in units of ${\mathcal{J}}_0=2{\Gamma }_1{\Gamma }_2/h$) for a quantum dot system with ${\mu }_1=E_F+eV$, ${\mu }_2=E_F$, ${\theta }_1=T+\theta$, and ${\theta }_2=T$ for three different values of the tunneling asymmetry $\eta$ and $E_d=10\Gamma$ with $T=0.5\Gamma$. (a) $\mathcal{J}$–$V$ characteristics (full lines) for $\theta =0$ along with the leading-order nonlinearity $\mathcal{J}\simeq R_{11}V+R_{111}{V}^2$ (dashed lines) for the $\eta$ values indicated in the right panel. Upper inset: We show with solid lines the self-consistent potential $U$ as a function of $eV/\Gamma$. Dotted lines correspond to the approximated values $U\approx uV={\Gamma }_1/\Gamma$ to leading order in an expansion in powers of $V$. Lower inset: temperature dependence of $R_{111}$. (b) $\mathcal{J}$–$\theta$ characteristics (full lines) for $V=0$ along with the leading-order nonlinearity $\mathcal{J}\simeq K_{11}\theta +K_{111}{\theta }^2$ (dashed lines) for the indicated values of $\eta$. Upper inset: We show with solid lines the self-consistent potential $U$ as a function of the thermal shift. Lower inset: temperature dependence of the nonlinear thermal conductance $K_{111}$.

Figure 3

(a) Linear-response Peltier coefficient ${\Pi }_0$ for a current biased quantum dot at $E_d=10\Gamma$. We show both the low and high temperature limits. (b) Full Peltier coefficient $\Pi$ including contributions beyond linear response for three different background temperatures for $\eta =0.5$. We show with colored dotted lines the leading-order expansion calculated from Eq. (34).

Figure 4

Nonlinear departures of the Wiedemann-Franz law as expressed by Eq. (39). Nonlinear transport responses are calculated for a two-terminal quantum dot with voltage bias $V=0.25\Gamma$, background temperature $T=0.1\Gamma$, and different values of thermal shift $\theta$ as a function of the dot level position $E_d$.