Universality in quantum critical flow of charge and heat in ultra-clean graphene

TL;DR

This work investigates quantum-critical transport in ultra-clean graphene by jointly analyzing dc charge and heat flow near the Dirac point. By combining electrical conductivity $σ$ with thermally driven conductivity $κ_\mathrm{e}$ through Johnson-noise thermometry and a hydrodynamic framework, the authors extract a universal quantum-critical conductivity $\sigma_Q \approx 4 e^2/h$ with minimal device-to-device variation, and observe strong violations of the Wiedemann–Franz law near charge neutrality. They also show that the thermal viscosity $η_\mathrm{th}$ divided by the thermal entropy density $s_\mathrm{th}$ approaches the holographic bound $ħ/(4π k_B)$ within a factor of four in the cleanest devices, highlighting graphene as a platform to test relativistic hydrodynamics and holographic limits in a quantum-critical Dirac fluid. The results unify dc transport and hydrodynamic concepts, demonstrating universality controlled by the universality class of the Dirac point and revealing Planckian-limited dissipation in a solid-state Dirac fluid. Overall, the study provides a benchmark for quantum-critical transport in graphene and informs the design of experiments probing universal bounds in strongly interacting electron systems.

Abstract

Close to the Dirac point, graphene is expected to exist in quantum critical Dirac fluid state, where the flow of both charge and heat can be described with a dc electrical conductivity $σ_\mathrm{Q}$, and thermodynamic variables such as the entropy and enthalpy densities. Although the fluid-like viscous flow of charge is frequently reported in state-of-the-art graphene devices, the value of $σ_\mathrm{Q}$, predicted to be quantized and determined only by the universality class of the critical point, has not been established experimentally so far. Here we have discerned the quantum critical universality in graphene transport by combining the electrical ($σ$) and thermal ($κ_\mathrm{e}$) conductivities in very high-quality devices close to the Dirac point. We find that $σ$ and $κ_\mathrm{e}$ are inversely related, as expected from relativistic hydrodynamics, and $σ_\mathrm{Q}$ converges to $\approx (4\pm 1)\times e^2/h$ for multiple devices, where $e$ and $h$ are the electronic charge and the Planck's constant, respectively. We also observe, (1) a giant violation of the Wiedemann-Franz law where the effective Lorentz number exceeds the semiclassical value by more than 200 times close to the Dirac point at low temperatures, and (2) the effective dynamic viscosity ($η_\mathrm{th}$) in the thermal regime approaches the holographic limit $η_\mathrm{th}/s_\mathrm{th} \rightarrow \hbar/4πk_\mathrm{B}$ within a factor of four in the cleanest devices close to the room temperature, where $s_\mathrm{th}$ and $k_\mathrm{B}$ are the thermal entropy density and the Boltzmann constant, respectively. Our experiment addresses the missing piece in the potential of high-quality graphene as a testing bed for some of the unifying concepts in physics.

Figures (10)

• Figure 1: Viscous electron flow in ultra-clean graphene devices: (a) Different regimes of electron transport in graphene, based on the interplay between momentum-conserving and momentum-relaxing collisions. (b) (Top) Optical micrograph of the device along with the circuit diagram used for conducting the electrical transport measurements. (Bottom) Schematic of the heterostructure. (c) Electrical conductance of device D4S4 as a function of the carrier density $n$ for $90$ K $\leq T \leq 260$ K. The dashed line labelled $G_\mathrm{ball}$ indicates the ballistic conductance for the measured channel. (d) Charge inhomogeneity ($n_\mathrm{min}$) as a function of $T$ for five different devices. The colour gradient in the background indicates the transition from a disorder-driven regime to a thermally-driven transport regime. (e) Electrical conductance (normalised by the ballistic conductance, $G_\mathrm{ball}$) of D4S4 as a function of $l_\mathrm{ee}/W$ for different carrier densities, varying from $n = 1 \times 10^{11}$ cm$^{-2}$ to $5 \times 10^{11}$ cm$^{-2}$. (f) Normalised conductance as a function of the Knudsen number for five different devices. (g) Electrical conductivity as a function of width $W$ at $n = 8 \times 10^{10}$ cm$^{-2}$ and $T = 160$ K. For the chosen values of $n$ and $T$, we have observed that $l_\mathrm{ee}/W \leq 0.5$. The solid line scales as $W^2$ and serves as a guide to the eye. Inset shows the same dataset in double logarithmic scale where the straight line represents quadratic dependence. (h) Kinematic viscosity $\nu$ as a function of $n$ for four different devices, measured at $T = 180$ K.
• Figure 1: Differential resistance measurements showing crossover from Knudsen to Poiseuille regime: (Left side) Colour plot showing the variation of $dV/dI$ for D3S5 as a function of $n$ and applied electrical current density ($J$) at four different $T$. (Right side) Line plots, obtained from the respective colour plots on the left, depicting the variation of $dV/dI$ as a function of applied electrical current density ($J$) for two distinct number densities: $n = 10^{10}$ cm$^{-2}$ (marked in black) and $n = 10^{11}$ cm$^{-2}$ (marked in red).
• Figure 2: Violation of the WF Law for hydrodynamic electrons: (a) Circuit diagram for measuring Johnson-Nyquist noise of hot electrons in our graphene devices, when subjected to Joule heating by in-plane electric fields. The LC network acts as a tank circuit for impedance matching the graphene device to the $50$$\Omega$ noise measurement circuit. (b) Normalised Lorentz number for device D2S1 as a function of $n$ for $T = 19$ K and $170$ K. The solid lines are theoretical fits of the experimental data, as per Eqn. \ref{['L']}. The electron-doped and hole-doped data points have been fitted independently using different values for the fitting parameters. (c) Electronic component of thermal conductivity for device D2S1 as a function of $T$ for 3 different number densities, from the charge neutrality point to a highly electron-doped regime. The dashed lines indicate the variation of thermal conductivity with $T$, assuming the normalised Lorentz number ${\cal L}/{\cal L}_\mathrm{WF} = 1$. (d) Enthalpy density ($\cal{H}$) for D2S1 as a function of $T$. The dashed lines (black - hole doped, red - electron doped) highlight $T^3$-like asymptotic behaviour. The colour gradient in the background signifies a transition from a disorder-driven regime to a thermally-driven transport regime. [Inset] Entropy density for D2S1 as a function of $T$ for $T > 80$ K. The black dashed line is the $T^2$ fit of the experimentally obtained data and is within a factor of 2 of what is expected from the theoretical expression in Ref. yudhistira2023non. (e) Schematic showing the interplay between two different types of electrical transport - disorder-driven and thermal excitation-driven. For $T<T_\mathrm{imp}$, the thermal energy of the impurities is greater than the thermal energy of electrons and hence the electrical transport is dominated by charged impurities and defects, whereas for $T>T_\mathrm{imp}$, the electronic thermal energy dominates and electron-electron interactions drive electrical transport in this regime.
• Figure 2: Electronic temperature across the graphene channel at low electric fields: (a) $T_\mathrm{e}$ in D2S1 as a function of the applied electric field ($E$) at different electron densities, recorded at $T = 19$ K. The dashed lines are theoretically fitted curves for the experimental data, following Eqn. \ref{['Te']}. [Inset] Transfer characteristics of the device D2S1 at $19$ K, with colored dots indicating the resistance at the different $n$ at which $T_\mathrm{e}$ vs $E$ data has been recorded. (b) $T_\mathrm{e}$ as a function of the applied electric field ($E$) at different temperatures, for $n = 5 \times 10^{11}$ cm$^{-2}$.
• Figure 3: Universality of the quantum critical conductivity: (a) Electronic component of thermal conductivity ($\kappa_\mathrm{e}$) for device D2S1 as a function of $n$ for $T = 110$ K, $170$ K and $260$ K. (b) $\kappa_\mathrm{e}$ as a function of electrical conductivity $\sigma$ for device D2S1 at three different temperatures from the region indicated by the bounding box in panel (a). The dashed lines indicate a $1/\sigma$ dependence and serve as a guide to the eye. (c) $\sigma_\mathrm{Q}$ for D2S1 as a function of $T$ for a range of densities from $n = -3 \times 10^{11}$ cm$^{-2}$ (hole-doped) to $n = 5 \times 10^{11}$ cm$^{-2}$ (electron-doped). The shaded region corresponds to temperatures greater than the scale of background chemical potential fluctuations. (d) $\sigma_\mathrm{Q}$ as a function of the ratio of Fermi temperature to absolute temperature ($T_\mathrm{F}/T$), calculated at different temperatures for four different devices. The dashed line is based on theoretical calculations performed in Ref. muller2008quantum. (e) Comparison of the quantum critical conductivity $\sigma_\mathrm{Q}$ (obtained from thermal transport measurements) and the minimum electrical conductivity $\sigma_\mathrm{min}$ (obtained from electrical transport measurements at $T<60$ K), as a function of the intrinsic charge inhomogeneity $n_\mathrm{min}(0)$. The dashed line scales as $n^2_\mathrm{min}(0)$ and serves as a guide to the eye.