What sets the minimum conductivity of graphene at the charge-neutrality point: electron–hole hydrodynamic friction versus disorder-induced electron–hole puddles

At graphene's charge-neutrality point (the Dirac point, where electron and hole densities balance), the dc conductivity saturates at a finite 'minimum conductivity' whose microscopic mechanism has been contested for two decades. The hydrodynamic picture holds that, when interparticle collisions dominate, charge-current relaxation occurs through electron–hole friction in a quantum-critical Dirac fluid, predicting (Svintsov et al., 2012) a temperature-independent minimum conductivity $G_0\propto\hbar v_F^2\kappa^2/e^2$ controlled by the dielectric permittivity $\kappa$ of the environment and containing no disorder-fitting parameter. The disorder picture holds that substrate charged impurities fragment the nominally neutral sheet into electron- and hole-doped 'puddles', with percolative/Boltzmann transport predicting a minimum conductivity set by impurity density (e.g., the dirty-limit saturation $\sigma_{min}\approx 128q^2/\pi^3h\approx 4.12\,q^2/h$, with $q$ the elementary charge, of Sajjad et al., 2015). The open problem is a predictive, experimentally validated account of which mechanism sets the charge-neutrality-point dc conductivity as a function of disorder density and sample platform (suspended, hBN-encapsulated, SiO2-supported), dielectric environment, and temperature, across the full hydrodynamic-to-percolative crossover.

Background

Graphene is a two-dimensional semimetal whose low-energy band structure is a Dirac cone: carrier energy grows linearly with momentum, $E=\hbar v_F|k|$, with Fermi velocity $v_F\approx 10^6$ m/s, and the electron and hole bands touch at the Dirac point. Gating the sheet moves the chemical potential $\mu$ away from this point; at $\mu=0$ the sheet is charge neutral (the charge-neutrality point, CNP). Experiments since 2005 have consistently found that the dc conductivity at the CNP does not diverge or vanish but saturates at a value of order $4e^2/h$, where $e^2/h$ is the conductance quantum — the famous 'minimum conductivity' of graphene.

Two broad theoretical pictures compete to explain it. The disorder picture starts from charged impurities in the substrate (typically SiO2): their screened Coulomb potentials fluctuate spatially and exceed the thermal scale near the CNP, so the sheet breaks into electron-doped and hole-doped regions — electron–hole puddles, imaged directly with a scanning single-electron transistor (SET) by Martin et al. (2008). Transport then proceeds through thermally generated carriers in a puddle-averaged density of states; using the Einstein relation (conductivity $\sigma=q^2\nu D$, with $q$ the elementary charge, $\nu$ the density of states, and $D$ the diffusivity) together with Fermi's golden rule for impurity scattering, Sajjad et al. (2015) derived $\sigma_{min}=(8q^2v_F^2\hbar/n_{imp})D_{puddle}^2$, where $D_{puddle}$ is the Gaussian-broadened puddle-averaged density of states at the Dirac point (distinct from the diffusivity $D$ above), and showed by non-equilibrium Green's function (NEGF, a quantum-transport simulation method) plus disorder sampling that $\sigma_{min}$ evolves from ballistic to diffusive behavior depending on device aspect ratio and impurity density $n_{imp}$, saturating near $4.12\,q^2/h$ in the dirty limit. This picture is consistent with the experimentally observed nonuniversality: Tan et al. (2007) found $\sigma_{min}$ varies from sample to sample in correlation with impurity-limited mobility, and the self-consistent screening theory of Adam et al. (2007) explains why SiO2 devices nevertheless cluster near $4e^2/h$ ('apparent universality'). In suspended graphene, Gornyi, Kachorovskii and Mirlin (2012) showed puddles and quantum interference control the low-temperature Dirac-point conductivity. The authoritative review by Das Sarma et al. (RMP 2011) devotes a central section to the 'minimum conductivity puzzle', defined there as the dichotomy between a predicted universal Dirac-point conductivity of intrinsic graphene and the observed sample-dependent nonuniversal minimum, for which the review presents the puddle-based self-consistent screening account as an acceptable solution; the aspect that remains unresolved is the later-documented rivalry between that disorder picture and the electron–hole-friction picture of the Dirac fluid in clean devices.

The hydrodynamic picture instead notes that in clean samples electron–electron collisions can be the fastest rate. Near the CNP, electrons and holes then form a strongly interacting, quasirelativistic 'Dirac fluid' — a quantum-critical plasma whose transport scales are set by temperature and fundamental constants alone, with no Fermi-surface scale — in which energy and charge currents decouple and the Wiedemann–Franz law (the metal prescription linking thermal and electrical conductivities) breaks down. Because graphene's lattice lacks Galilean invariance, electron–hole collisions relax the charge current even though they conserve total momentum: Müller, Fritz and Sachdev (2008) computed this quantum-critical conductivity in the clean limit. Svintsov et al. (2012) modeled the electron–hole plasma as two coupled fluids with mutual viscous friction (coefficient $\beta_{eh}$) and obtained for negligible extrinsic scattering $G_0=e^2\Sigma_0^2/\beta_{eh}\propto\hbar v_F^2\kappa^2/e^2$ — with $\Sigma_0$ the thermodynamic equilibrium carrier density at the Dirac point — strikingly temperature-independent and governed by the effective dielectric permittivity $\kappa$ of the environment. Mendoza et al. (2013) simulated an ideal relativistic fluid past impurity obstacles, a hybrid of the two pictures. Svintsov (2018) later developed the kinetic theory of the crossover between hydrodynamic and ballistic transport in Dirac materials.

Modern experiments in hBN-encapsulated devices (hBN: hexagonal boron nitride, an atomically flat, low-disorder substrate) established that the Dirac fluid is real: Crossno et al. (2016) observed Wiedemann–Franz breakdown and an e–h-collision-dominated regime near the CNP at 60–300 K; Gallagher et al. (2019) measured the quantum-critical (Planckian, i.e. of order $k_BT/\hbar$) scattering rate in the THz optical conductivity at the CNP; Ku et al. (2020) imaged viscous Dirac-fluid flow at room temperature. However, the measured dc conductivity at the CNP substantially exceeds clean quantum-critical predictions, and Lucas et al. (2016) — who built a hydrodynamic framework nonperturbative in charge-density inhomogeneity — reached only improved, not parameter-free, agreement with the data. Bal, Lucas and Luskin (2021) constructed a homogenization framework for hydrodynamic transport in inhomogeneous Dirac fluids precisely because real CNP samples remain puddle-laden; it provides no numerical predictions for realistic puddle statistics. A recent review (Narozhny, 2022) still presents theory-versus-experiment comparison in this regime as an open front. The consequence is that the two decades-old competing predictions — a $T$-independent, dielectric-permittivity-controlled $G_0$ from electron–hole friction versus an impurity-density-controlled percolative $\sigma_{min}$ — have never been jointly mapped onto the full range of modern device platforms and temperatures.

Problem Statement

Determine, by predictive theory and/or systematic experiment, what mechanism sets the dc conductivity of graphene at the charge-neutrality point: momentum relaxation by electron-hole (quantum-critical) interparticle friction in a Dirac fluid, or percolative transport through disorder-induced electron-hole puddles. The common objective is a predictive account of the minimum conductivity $\sigma_{min}$ as it depends on temperature $T$, charged-impurity/disorder density $n_{imp}$ (spanning suspended, hBN-encapsulated, and SiO2-supported devices), and dielectric environment $\kappa$, across the hydrodynamic-to-percolative crossover. The documented limiting predictions — the temperature-independent, $\kappa$-controlled $G_0\propto\hbar v_F^2\kappa^2/e^2$ of the electron-hole-friction picture (Svintsov et al., 2012) and the impurity-density-controlled scaling with $\approx 4q^2/h$ dirty-limit saturation of the puddle picture (Sajjad et al., 2015) — stand as benchmark context for the two competing mechanisms against which such an account is read; the open question is which mechanism, or what competition between them, sets the observed charge-neutrality-point conductivity in which regime.

References

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