[Paper Review] Graphene - a nearly perfect fluid

Markus Mueller, Jörg Schmalian+1 more|ArXiv.org|Mar 24, 2009

Electrohydrodynamics and Fluid Dynamics3 references18 citations

TL;DR

This paper demonstrates that clean, undoped graphene exhibits an extremely low shear viscosity-to-entropy density ratio (η/s), approaching the theoretical lower bound of 1/(4π)ℏ/kB, due to its quantum criticality and strong electron-electron interactions. Using quantum kinetic theory, the authors show that this near-perfect fluid behavior arises from logarithmically running Coulomb interactions, leading to potential pre-turbulent electronic flow at low temperatures and nanoscale geometries.

ABSTRACT

Hydrodynamics and collision dominated transport are crucial to understand the slow dynamics of many correlated quantum liquids. The ratio η/s of the shear viscosity ηto the entropy density s is uniquely suited to determine how strongly the excitations in a quantum fluid interact. We determine η/s in clean undoped graphene using a quantum kinetic theory. As a result of the quantum criticality of this system the ratio is smaller than in many other correlated quantum liquids and, interestingly, comes close to a lower bound conjectured in the context of the quark gluon plasma. We discuss possible consequences of the low viscosity, including pre-turbulent current flow.

Motivation & Objective

• To understand the transport properties of clean, undoped graphene in the collision-dominated regime.
• To determine the shear viscosity-to-entropy density ratio (η/s) in graphene using quantum kinetic theory.
• To investigate whether graphene's low η/s indicates near-perfect fluid behavior analogous to the quark-gluon plasma.
• To explore experimental signatures of viscous and hydrodynamic effects, including non-local conductance and potential electronic turbulence.

Proposed method

• Employed a quantum kinetic theory to compute the shear viscosity η in clean, undoped graphene, accounting for electron-electron interactions.
• Used the Boltzmann transport equation with a collision integral derived from the effective Coulomb interaction, including logarithmic renormalization due to quantum criticality.
• Calculated the entropy density s from the thermal excitation spectrum, assuming a marginal Fermi liquid-like behavior near the Dirac point.
• Evaluated the ratio η/s as a function of temperature, incorporating the effective fine structure constant α(T) = e²/(εℏv) with logarithmic running.
• Assessed the Reynolds number Re to estimate the onset of turbulent flow, using velocity gradients and viscous dissipation in a Navier-Stokes framework.
• Considered experimental geometries with split-source/drain contacts to model viscous drag effects on conductance, predicting scale-dependent resistance.

Experimental results

Research questions

• RQ1 What is the value of the shear viscosity-to-entropy density ratio η/s in clean, undoped graphene, and how does it compare to the lower bound conjectured in relativistic quantum fluids?
• RQ2 How does quantum criticality—specifically the logarithmic running of the Coulomb interaction—impact the viscosity and hydrodynamic transport in graphene?
• RQ3 What experimental signatures can be expected for viscous electron flow in graphene, particularly in nanoscale device geometries?
• RQ4 Under what conditions might electronic turbulence emerge in graphene, and how does the Reynolds number depend on temperature and system size?

Key findings

• The ratio η/s in clean, undoped graphene is extremely small, approaching the theoretical lower bound of 1/(4π)ℏ/kB, especially in a broad temperature window where the effective coupling α(T) remains of order O(1).
• The low η/s arises from quantum criticality, where marginally irrelevant Coulomb interactions lead to strong correlations and enhanced scattering, resulting in hydrodynamic transport.
• Viscous effects reduce conductance in split-contact geometries, with resistance increasing as the contact separation L decreases, deviating from Ohmic scaling due to non-local viscous forces.
• The Reynolds number Re increases with decreasing temperature, suggesting that even moderate fluid velocities (u ~ 0.1v) can lead to complex, chaotic flow at L ~ 1 μm, indicating potential electronic turbulence.
• Away from zero doping, η/s increases as (|μ|/T)^3, indicating a crossover from quantum critical to Fermi liquid-like behavior.
• The system's near-perfect fluidity is more pronounced than in other systems like ultracold atoms with diverging scattering length, making graphene a more perfect fluid.