Emergence of Fourier's law of heat transport in quantum electron systems

TL;DR

Fourier's law in quantum electronic heat transport is shown to emerge when the energy-level broadening $\Gamma$ exceeds the level spacing $\Delta E$, so that many nondegenerate states contribute to transport. The study uses a nonequilibrium Green's function framework with a floating thermoelectric probe to map local temperatures in graphene nanojunctions and demonstrates a crossover from quantum-interference dominated profiles to near-classical linear gradients. It then constructs a thermal resistor-network model, predicting universal contact resistances and a sample resistance scaling with the density of states and length, in agreement with semiclassical intuition. These results clarify the microscopic origin of Fourier's law in nanoscale electronic systems and have implications for designing heat management in nanoelectronics.

Abstract

The microscopic origins of Fourier's venerable law of thermal transport in quantum electron systems has remained somewhat of a mystery, given that previous derivations were forced to invoke intrinsic scattering rates far exceeding those occurring in real systems. We propose an alternative hypothesis, namely, that Fourier's law emerges naturally if many quantum states participate in the transport of heat across the system. We test this hypothesis systematically in a graphene flake junction, and show that the temperature distribution becomes nearly classical when the broadening of the individual quantum states of the flake exceeds their energetic separation. We develop a thermal resistor network model to investigate the scaling of the sample and contact thermal resistances, and show that the latter is consistent with classical thermal transport theory in the limit of large level broadening.

Figures (4)

• Figure 1: Classical (panels a,d) and quantum temperature profiles of a graphene flake under thermal bias for two contact geometries. The hot electrode (red) is held at 110K and the cold electrode (dark blue) is held at 90K, where red and blue squares indicate the carbon atoms covalently bonded to the hot and cold electrodes, respectively. In contact type I (upper panels), only the left and right edges of the flake couple to the electrodes, while in contact type II, the coupling to the electrodes wraps around three edges each, leading to three times stronger coupling to the electrodes. The quantum calculations are at Fermi energies $\mu_0=-0.1$eV (b,e) and $-0.6$eV (c,f), relative to the Dirac point. The quantum temperature distributions for contact type I exhibit strong oscillations that depend sensitively on $\mu_0$, while for contact type II, the temperature distributions resemble pixelated versions of the classical distribution.
• Figure 2: The calculated density of states (DOS) $g(E)$ of a graphene flake junction for two different contact geometries, defined in Fig. \ref{['fig:temperature_profile']}. The DOS for contact type I exhibits a sequence of sharp peaks corresponding to the energies of individual energy eigenstates [or manifolds of (nearly) degenerate eigenstates] of the flake, broadened by coupling to the electrodes. Contact type II, for which the broadening is three times as large, exhibits a smooth, nearly featureless DOS for $E<1.2$eV.
• Figure 3: Top panel: The heat current density $\vec{J}_Q$ for contact type I at $\mu_0 = -2.2$eV is calculated using classical, and quantum transport theories, indicated with the red and blue arrows, respectively. Bottom panel: $\vec{J}_Q$ for contact type II at $\mu_0 = -0.1$eV calculated using classical, and quantum transport theories. As highlighted by the swirling blue arrows, the heat current profile of the junction shown in the top panel is highly non-classical, while the heat transport profile of the bottom panel's junction is better represented by a classical description.
• Figure 4: Thermal resistance values as a function of Fermi energy $\mu_0$ for four different sized hexagonal graphene flake junctions with contact type II, where $N$ is the number of atoms in the flake. $R_s$ is the sample thermal resistance and $R_1$ and $R_2$ are the contact thermal resistances, defined in Eqs. (\ref{['eq:Rs']})--(\ref{['eq:R2']}), respectively. The contact resistances are nearly universal in this transport regime: $R_1,\,R_2\approx R_0/N_c$, where $R_0$ is the thermal resistance quantum and $N_c$ is the number of atoms bonded to each contact. The sample thermal resistance $R_s$ is inversely correlated with the density of states per unit area times the sample length, $g(\mu_0)L$, as expected based on semiclassical Boltzmann transport theory.