Graphene Electron Hydrodynamics

• Electron hydrodynamics in graphene is a regime where charge carriers form a viscous electron–hole fluid, yielding signatures like negative vicinity resistance and Poiseuille flow.
• Theoretical frameworks using kinetic and quasi-relativistic models capture the breakdown of the Wiedemann–Franz law and an enhanced Lorenz ratio near the Dirac point.
• Mutual drag, high shear viscosity, and nonlinear effects in electron–hole interactions drive collective modes with implications for terahertz plasmonics and device innovation.

Electron hydrodynamics in graphene is a regime of transport in which charge carriers behave as a viscous electron–hole fluid, fundamentally distinct from impurity- or phonon-limited conduction. In high-purity graphene at appropriate temperature and doping, frequent electron–electron collisions establish local equilibrium, enabling well-defined experimental signatures—such as negative vicinity resistance, Poiseuille flow profiles, and marked breakdown of the Wiedemann–Franz law—captured by a hydrodynamic theoretical framework blending kinetic and quasi-relativistic fluid models (Nayak et al., 14 Sep 2025).

1. Experimental Evidence for Electron Hydrodynamics

Key observations underpinning hydrodynamic behavior in graphene include the following (Nayak et al., 14 Sep 2025):

• Negative vicinity resistance: Voltage probes placed proximal to a current injector detect negative (local) resistance, indicating vorticity and backflow typical of a viscous fluid (Bandurin et al., 2015). This phenomenon arises from whirlpool formation in the flow, serving as a direct signature of collective, momentum-conserving transport.
• Poiseuille flow profiles: Direct visualization studies of constricted graphene have resolved parabolic current density profiles (Poiseuille flow), consistent with viscous shear and momentum-conserving boundary scattering (Danz et al., 2019).
• Violation of the Wiedemann–Franz (WF) law: Thermal and electrical conductivity measurements, e.g., via Johnson noise thermometry, reveal sharply enhanced Lorenz ratios (L = κ/(σT)), reaching values up to 22 L₀ (L₀ = (π²/3)(k_B/e)²) or more near the Dirac point, reflecting the decoupling between heat and charge transport in the Dirac fluid regime (Lucas et al., 2017, Ho et al., 2017, Nayak et al., 14 Sep 2025).

These signatures differentiate the hydrodynamic regime from both ballistic (edge scattering-dominated) and diffusive (impurity-limited) transport.

2. Theoretical Frameworks for Hydrodynamic Transport

Two complementary regimes and corresponding theoretical frameworks are distinguished in the analysis of electron transport in graphene (Nayak et al., 14 Sep 2025):

Regime	Governing Description	Key Observables
Fermi liquid (μ ≫ k_BT)	Kinetic theory with linear E = p v_F dispersion	σ ≈ n e² τ_c / m*, standard L₀
Dirac fluid (μ ≪ k_BT)	Quasi-relativistic hydrodynamics (v_F as c analog)	Enhanced Lorenz ratio, Poiseuille flow

In the Fermi liquid regime, the electronic transport follows conventional kinetic theory with the conductivity $\sigma_{\text{non-fluid}} = n e^2 \tau_c / m^*$. The Lorenz number aligns with the Sommerfeld value: $L_{\text{non-fluid}} = (\pi^2/3)(k_B/e)^2$. Here, scattering is dominated by impurities or phonons, and electron–electron interactions mainly influence thermalization without collective flow.

In the Dirac fluid regime, the strong electron–electron scattering rate ($\tau_{ee}^{-1}$) exceeds all other scattering rates, enabling a hydrodynamic description. The theoretical framework employs microscopic conservation laws for energy–momentum and charge, leading to a stress–energy tensor

$$T_0^{\mu\nu} = \frac{E}{v_F^2} u^{\mu} u^{\nu} - P \Delta^{\mu\nu}$$

where $u^{\mu}$ is the fluid four-velocity, $E$ the energy density, $P$ the pressure, and $$\Delta^{\mu\nu} = \eta^{\mu\nu} - u^{\mu}u^{\nu}/v_F^2$$ (Nayak et al., 14 Sep 2025). Dissipative corrections, calculated via the relaxation time approximation to the Boltzmann equation, yield expressions for the electrical and thermal conductivities involving Fermi–Dirac integrals and the transport collision time $\tau_c$.

Notably, the Lorenz ratio in the hydrodynamic regime diverges near charge neutrality: $$L = \left(\frac{\mathfrak{h}}{k_B T}\right)^2 \frac{k_B^2}{e^2},\quad \mathfrak{h} = \frac{E + P}{n}$$ with $\mathfrak{h}$ diverging as $n \rightarrow 0$, accounting for the observed breakdown of the WF law.

3. Mutual Drag, Viscosity, and Nonlinear Effects

A defining characteristic of graphene hydrodynamics is the strong coupling between electrons and holes due to Coulomb interactions. The effect of mutual drag is formalized as a viscous friction force between the two components (Svintsov et al., 2012, Chen et al., 2022). The frictional dissipative term, parameterized by a drag coefficient $\beta_{eh}$ proportional to the electron–hole collision frequency $\nu_{eh}$, substantially impacts the dc conductivity, particularly at the Dirac point where charge neutrality prevails: $$G_0 = \frac{e^2 \Sigma_0^2}{\beta_{eh}} \propto \frac{\hbar v_F^2 \varkappa^2}{e^2}$$ This accounts for the near-temperature independence of minimal conductivity.

The electron–hole drag also strongly damps oppositely phased collective excitations (plasma waves) while leaving in-phase (electron–hole sound) modes weakly damped. The explicit form for the drag coefficient is

$$\beta_{eh} = \nu_{eh} v_F \frac{\langle p_e \rangle \langle p_h \rangle}{\langle p_e \rangle + \langle p_h \rangle}$$

At the Dirac point, mutual drag dominates over impurity/phonon scattering.

The shear viscosity, crucial for the hydrodynamic response, achieves large values (ν ≈ 0.1 m²/s), exceeding that of ordinary liquids, and is further enhanced by electron–hole collisions, especially near charge neutrality (Chen et al., 2022). The viscosity enters the hydrodynamic equations and sets the scale for phenomena such as Poiseuille flow and whirlpool (vorticity) formation. Nonlinear hydrodynamics in graphene enables phenomena such as soliton and shock-wave propagation, governed by dissipative Kadomtsev–Petviashvili or KdV–Burgers-type equations, and is sensitive to effects from quantum potential and odd viscosity (Cosme et al., 2021, Svintsov et al., 2013).

4. Collective Modes and Plasmonics

The coupled hydrodynamic–Poisson system in graphene supports two classes of neutral collective excitations (Svintsov et al., 2012):

• Plasma waves: Density waves with accompanying electric field oscillations, damped strongly by electron–hole friction; the phase velocity can exceed v_F due to electrostatic effects, particularly at high gating or density.
• Electron–hole sound waves: Quasineutral modes, corresponding to in-phase fluctuations of electron and hole densities; these propagate with velocity $s_- \approx 0.6 v_F$ at charge neutrality and are only weakly damped by mutual drag, thus offering potential low-loss channels for signal propagation.

The distinct nature of these modes underpins graphene plasmonics, with direct implications for terahertz devices, detectors, and modulators (Svintsov et al., 2012, Cosme et al., 2021).

5. Thermoelectric and Viscous Effects Near Charge Neutrality

Graphene devices near charge neutrality exhibit remarkably enhanced effective conductivities owing to the coupling between hydrodynamic flow and charge transport in the presence of inhomogeneity (disorder, gate-induced density variation). In a Hall-bar geometry, the convective (hydrodynamic) mode augments the charge and thermal transport, so that the observed conductivity can dramatically exceed the intrinsic fluid value (Li et al., 2022): $\sigma_e = \sigma + \frac{d^2 n^2}{3 \eta}$ where $\sigma$ is the intrinsic conductivity, $n$ the carrier density, $d$ the channel half-width, and $\eta$ the shear viscosity. Long-range disorder or gate-induced inhomogeneity can further enhance this effect by mixing convective and dissipative contributions.

The effective Lorenz ratio near charge neutrality is sharply peaked, reflecting the strong coupling between energy and charge current and the breakdown of the standard Fermi liquid paradigm. This feature is robust against weak disorder and is considered a "smoking gun" for Dirac fluid hydrodynamics (Ho et al., 2017, Li et al., 2022, Nayak et al., 14 Sep 2025).

6. Dissipation, Energy Relaxation, and Extensions

Energy relaxation in nearly neutral graphene is controlled by disorder-assisted electron–phonon processes, or "supercollisions," which introduce weak decay terms even at the ideal hydrodynamic level (Narozhny et al., 2021). These terms enter the energy and imbalance continuity equations, modifying heat and charge transport, especially at high temperatures above the Bloch–Grüneisen threshold. The resultant decay rates are essential for realistic thermoelectric modeling and for understanding the limits of hydrodynamic transport.

Beyond conventional hydrodynamics, theoretical advances have incorporated additional degrees of freedom, such as microrotation (for valley hydrodynamics in gapped graphene (Sano et al., 2022)), or have generalized to other systems, including topological materials and Weyl semimetals (Narozhny, 2022). The connection to relativistic hydrodynamics is formal but not exact, due to non-Lorentz-invariant Coulomb interactions and dissipative corrections required in graphene (Narozhny, 2019, Narozhny, 2022).

7. Outlook and Open Directions

Recent progress has established graphene as the canonical platform for electron hydrodynamics, with experimental and theoretical work detailing the transition between ballistic, hydrodynamic, and disorder-dominated regimes. Outstanding challenges include (Nayak et al., 14 Sep 2025):

• Precisely resolving the crossover boundaries as a function of temperature, doping, and device geometry.
• Quantitative modeling of microscopic scattering rates, including detailed electron–electron, electron–phonon, and disorder contributions, for accurate computation of transport coefficients.
• Extending hydrodynamic theory to capture non-linear, vorticity-driven, and spin- or valley-coupled phenomena.
• Exploring hydrodynamic behavior in related materials and identifying universal features across distinct quantum fluids.
• Investigating the implications of viscosity enhancement, Hall viscosity, and supercollision-limited energy relaxation for device applications and fundamental quantum criticality.

Graphene thus remains at the forefront of research into hydrodynamic transport, serving as a testbed for advancing both experimental techniques and theoretical frameworks to describe collective electronic phenomena beyond the Fermi liquid paradigm.