Asymptotic zero-temperature conductivity at the Dirac point of disordered graphene: mechanism, value, and symplectic-scaling versus electron-hole-puddle percolation

At the charge-neutrality (Dirac) point of graphene, the conductivity never falls below a minimum value of order $e^2/h$, but the mechanism and the exact asymptotic ($T\to 0$, system size $L\to\infty$) value of this 'minimal conductivity' were left unresolved by the finite-size Kubo-formula study of Nomura and MacDonald (Phys. Rev. Lett. 98, 076602 (2007)), which found $\sigma^{\min}\simeq e^2/h$ for screened-Coulomb (charged-impurity) disorder in the massless Dirac fermion model and a slowly increasing, size-dependent conductivity at zero energy. The open problem is to determine the thermodynamic-limit, zero-temperature conductivity of the charge-neutral Dirac point: whether symplectic-class weak-antilocalization scaling (diverging conductivity at large $L$ in the absence of intervalley scattering) or percolation through interaction- and disorder-induced electron-hole puddles controls transport; what happens once intervalley scattering always becomes relevant and drives localization; and how electron-electron interactions and screening beyond the Thomas-Fermi approximation set the precise value of $\sigma^{\min}$. Later literature has resolved the classical part of this question — the minimal conductivity of real (extrinsic) graphene is nonuniversal, set by the charged-impurity density through self-consistently screened puddle percolation — while a complete theory of the interacting, quantum-coherent, $T\to 0$ charge-neutral point, and a definitive adjudication of the Dirac-point scaling flow of the idealized model, remain open.

Background

Graphene, a single atomic layer of carbon on a honeycomb lattice, has a low-energy band structure of two Dirac cones located at the inequivalent Brillouin-zone corners (valleys) $K$ and $K'$, so its charge carriers behave as massless Dirac fermions (MDF): two-component spin-1/2-like wavefunctions with linear (conical) dispersion $E=\pm\hbar v|k|$, where $v$ is the Fermi velocity. A key early puzzle of graphene transport was the 'minimal conductivity' $\sigma^{\min}$: experimentally the conductivity never falls below roughly $4e^2/h$ (with $e$ the electron charge and $h$ Planck's constant) as the carrier density $n$ is gated through zero, whereas the self-consistent Born approximation (SCBA, a self-consistent perturbative treatment of disorder broadening) predicted the smaller universal value $(1/\pi)e^2/h$ per valley-degenerate channel, and Boltzmann transport theory for Coulomb scatterers predicted a conductivity that vanishes as $n\to 0$. Nomura and MacDonald (Phys. Rev. Lett. 98, 076602 (2007)) addressed this numerically by evaluating the Kubo formula — the exact linear-response expression for the conductivity as a sum over eigenstates of the disordered Hamiltonian — for finite systems of the MDF model with either short-range (delta-function) or screened-Coulomb random scatterers, using $T=0$ Thomas-Fermi screening (the approximation in which the carrier gas is treated as a static gas that screens the impurity potential through its local density of states). They found that for Coulomb scatterers the Dirac-point conductivity stays finite at $\sigma^{\min}\simeq e^2/h$, a few times the SCBA value, depends only on the ratio $n/n_i$ of carrier to impurity density, and increases very slowly with system size $L$ over the accessible range. Three physical issues define what they left open. First, intervalley versus intravalley scattering: a potential smooth on the lattice scale does not mix the two valleys, which is the condition for the single-valley MDF model to describe graphene; atomically sharp disorder mixes the valleys. In the presence of intervalley scattering, two-dimensional (2D) systems localize (Anderson localization: quantum interference drives the conductance to zero at sufficiently large size), whereas without it each valley retains an effective time-reversal symmetry, placing the system in the symplectic symmetry class, where interference is antilocalizing — the conductivity tends to grow with size (weak antilocalization) and scaling arguments predict an ultimately divergent 2D conductivity. Whether the MDF model at zero energy actually flows to this symplectic divergence at very large $L$, or whether the observed weak size dependence instead reflects growing spatial fluctuations of the carrier density — 'puddles' of electrons and holes through which current percolates (percolation: transport along a continuously connected path through a spatially inhomogeneous medium) — was left as an explicit either/or. Second, screening and interactions: the Thomas-Fermi approximation strictly fails at the Dirac point (the clean density of states vanishes there), so the precise value of $\sigma^{\min}$ was expected to depend on electron-electron interactions, for which no self-consistent many-body treatment was available. Third, the finite-size Kubo numerics cover only a limited range of $L$ (minimum simulated size roughly $0.1\,\mu$m in physical units), so convergence to the true thermodynamic ($L\to\infty$, $T\to 0$) conductivity remained to be established.

Problem Statement

Determine the conductivity of graphene at the charge-neutrality (Dirac) point in the true thermodynamic limit — the zero-temperature limit $T\to 0$ taken at fixed system size, followed by $L\to\infty$, with $\sigma$ the zero-frequency Kubo (current–current correlator) conductivity averaged over the disorder ensemble — together with its mechanism. Concretely, for the massless Dirac fermion model with randomly distributed (charged-impurity/Coulomb) disorder, and for real graphene in which intervalley scattering is ultimately always present: (1) establish the large-system-size scaling of the zero-energy conductivity — does the symplectic-class weak-antilocalization flow (logarithmically or ultimately diverging conductivity when intervalley scattering is absent) emerge, or does percolation through disorder- and interaction-induced electron-hole puddles control the asymptotic value — and determine the crossover scales and the conductivity in the intervalley-scattering-dominated regime in which localization sets in; (2) determine quantitatively how electron-electron interactions and screening beyond the Thomas-Fermi approximation affect the effective disorder potential and the precise value of the minimal conductivity $\sigma^{\min}$, including a self-consistent many-body treatment at the Dirac point where the clean density of states vanishes; and (3) establish that finite-size Kubo results converge to this thermodynamic-limit conductivity for realistic disorder parameters. These are component results of one overarching objective: a mechanism-resolved, quantitative theory of zero-temperature charge-neutral transport in disordered graphene, predicting the asymptotic conductivity (or its insulating counterpart) and its dependence on impurity density, impurity range, intervalley scattering strength, and interaction strength. A complete resolution of the objective requires all three component results (1), (2), and (3) together; each is independently refutable, but no proper subset of them settles the task.

References

1. K. Nomura and A. H. MacDonald, 'Quantum Transport of Massless Dirac Fermions', Phys. Rev. Lett. 98, 076602 (2007), doi:10.1103/PhysRevLett.98.076602, https://doi.org/10.1103/PhysRevLett.98.076602 (preprint: arXiv:cond-mat/0606589, https://arxiv.org/abs/cond-mat/0606589).
2. P. M. Ostrovsky, I. V. Gornyi, and A. D. Mirlin, 'Quantum criticality and minimal conductivity in graphene with long-range disorder', Phys. Rev. Lett. 98, 256801 (2007), doi:10.1103/PhysRevLett.98.256801, https://arxiv.org/abs/cond-mat/0702115.
3. J. H. Bardarson, J. Tworzydlo, P. W. Brouwer, and C. W. J. Beenakker, 'One-Parameter Scaling at the Dirac Point in Graphene', Phys. Rev. Lett. 99, 106801 (2007), doi:10.1103/PhysRevLett.99.106801, https://arxiv.org/abs/0705.0886.
4. V. V. Cheianov, V. I. Fal'ko, B. L. Altshuler, and I. L. Aleiner, 'Random resistor network model of minimal conductivity in graphene', Phys. Rev. Lett. 99, 176801 (2007), doi:10.1103/PhysRevLett.99.176801, https://arxiv.org/abs/0706.2968.
5. S. Adam, E. H. Hwang, V. Galitski, and S. Das Sarma, 'A self-consistent theory for graphene transport', Proc. Natl. Acad. Sci. USA 104, 18392 (2007), doi:10.1073/pnas.0704772104, https://arxiv.org/abs/0705.1540.
6. J. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J. H. Smet, K. von Klitzing, and A. Yacoby, 'Observation of electron-hole puddles in graphene using a scanning single-electron transistor', Nat. Phys. 4, 144 (2008), doi:10.1038/nphys781, https://arxiv.org/abs/0705.2180.
7. S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, 'Electronic transport in two-dimensional graphene', Rev. Mod. Phys. 83, 407 (2011), doi:10.1103/RevModPhys.83.407, https://arxiv.org/abs/1003.4731.
8. I. F. Herbut, V. Juricic, and O. Vafek, 'Coulomb Interaction, Ripples, and the Minimal Conductivity of Graphene', Phys. Rev. Lett. 100, 046403 (2008), doi:10.1103/PhysRevLett.100.046403, https://arxiv.org/abs/0707.4171.
9. I. Terekhov, A. Milstein, V. Kotov, and O. Sushkov, 'Screening of Coulomb impurities in graphene', Phys. Rev. Lett. 100, 076803 (2008), doi:10.1103/PhysRevLett.100.076803, https://arxiv.org/abs/0708.4263.
10. F. Amet, J. R. Williams, K. Watanabe, T. Taniguchi, and D. Goldhaber-Gordon, 'Insulating Behavior at the Neutrality Point in Single-Layer Graphene', Phys. Rev. Lett. 110, 216601 (2013), doi:10.1103/PhysRevLett.110.216601, https://arxiv.org/abs/1209.6364.
11. J. Crossno, J. Shi, K. Wang, X. Liu, A. Harzheim, A. Lucas, S. Sachdev, P. Kim, T. Taniguchi, K. Watanabe, T. A. Ohki, and K. C. Fong, 'Observation of the Dirac fluid and the breakdown of the Wiedemann-Franz law in graphene', Science 351, 1058 (2016), doi:10.1126/science.aad0343, https://doi.org/10.1126/science.aad0343.
12. C. Yanik, V. Sazgari, A. Canatar, Y. Vaheb, and I. I. Kaya, 'Strong localization in suspended monolayer graphene by intervalley scattering', Phys. Rev. B 103, 085437 (2021), doi:10.1103/PhysRevB.103.085437, https://arxiv.org/abs/2006.11797.
13. W. Chen, Y. Shen, B. Fu, Q. Shi, and W. Zhu, 'On the sample-dependent minimal conductivity in weakly disordered graphene', arXiv:2108.13787 (2021), https://arxiv.org/abs/2108.13787.
14. Y. Takane, 'Finite-Size Scaling Analysis of the Conductivity of Dirac Electrons on a Surface of Disordered Topological Insulators', J. Phys. Soc. Jpn. 85, 094715 (2016), doi:10.7566/JPSJ.85.094715, https://arxiv.org/abs/1608.07873.