Magnetotransport near the minimum-conductivity point of inhomogeneous graphene: the field-independent residual conductivity, the quantum-regime breakdown of effective-medium theory, and the puddle-versus-homogeneous origin of the $\nu=0$ Hall plateau

Near graphene's charge-neutrality point (CNP), charged-impurity disorder breaks the sheet into coexisting electron and hole puddles. Cho and Fuhrer (Phys. Rev. B 77, 081402(R) (2008)) showed that the magnetoresistivity near the CNP is well described by the exact effective-medium result for an equal-mobility electron-hole mixture only after adding an unexplained field-independent conductivity term $\sigma_{xx,1}\approx0.8\,e^{2}/h$, that the description fails in the quantum regime ($B\gtrsim0.8$ T, $T\lesssim100$ K) where $\rho_{xx}$ saturates near $0.4\,h/e^{2}$ with $\sigma_{xy}\approx0$, and proposed — without proof — localized $\nu=+2$ and $\nu=-2$ quantum Hall puddles as the origin. In the subsequent literature, the puddle picture is confirmed for disordered samples, homogeneous symmetry-broken (ferromagnetic/canted-antiferromagnetic) $\nu=0$ states with counterpropagating spin-filtered edge conduction are established in clean samples and now imaged directly, but a quantitative microscopic theory of CNP magnetotransport through the classical-to-quantum crossover, including the residual field-independent term, and a spatially resolved test of the puddle picture in the quantum regime on disorder-dominated samples remain open.

Background

Graphene's electronic spectrum is that of massless Dirac fermions: the conduction and valence bands touch at the Dirac point, where the density of states vanishes. Gating a graphene device tunes the carrier density $n$ through this point, called the charge-neutrality point (CNP); experiments consistently show a finite minimum conductivity $\sigma_{xx,min}$ of order $4\,e^{2}/h$ there instead of an insulating state. A widely accepted explanation is that long-range charged-impurity disorder shifts the local Dirac point up and down across the sample, so that at zero average density the sheet breaks into mesoscopic electron-doped and hole-doped regions ("electron-hole puddles"), directly imaged by scanning single-electron-transistor microscopy (imaging with a nanoscale ultrasensitive charge sensor) and by scanning-tunneling microscopy (puddles of order 20-30 nm on SiO$_2$). In a perpendicular magnetic field $B$ this spatially inhomogeneous two-carrier system has unusual magnetotransport. For a uniform conductor containing both electrons and holes of equal mobility $\mu$ (the "two-fluid" model), $\rho_{xx}(B)$ grows as $1+(\mu B)^{2}$. For a macroscopic mixture of electron and hole regions of equal mobility and equal concentration, exact effective-medium theory (EMT; the theory of transport in spatially inhomogeneous composites, here the exact solution of Guttal and Stroud) instead predicts $\sigma_{xx}(B)=\sigma_{xx}(0)(1+(\mu B)^{2})^{-1/2}$ with vanishing Hall conductivity $\sigma_{xy}=0$, i.e. a resistivity growing only linearly in $B$ at high field. Cho and Fuhrer measured $\rho_{xx}(B)$ and $\rho_{xy}(B)$ near the CNP of graphene on SiO$_2$ from 1.6 to 300 K and found the two-fluid model fails while the inhomogeneous EMT form fits well — but only after adding a phenomenological field-independent term: $\sigma_{xx}(B)=\sigma_{xx,0}(1+(\mu B)^{2})^{-1/2}+\sigma_{xx,1}$, with $\sigma_{xx,1}\approx0.8\,e^{2}/h$ (the arXiv versions state $0.81\,e^{2}/h$; the published version is quoted as $0.88\,e^{2}/h$), whose microscopic origin they could not identify. At low temperature and high field ($B\gtrsim0.8$ T, where the magnetic length $\ell_{B}=\sqrt{\hbar/eB}$, the characteristic radius of cyclotron motion, becomes shorter than the puddle size, and $T\lesssim100$-160 K) the data depart from even this modified semiclassical description: $\rho_{xx}$ saturates near $0.4\,h/e^{2}$ and $\sigma_{xy}$ shows a plateaulike region near zero. In graphene's quantum Hall effect the filling factor $\nu$ (carrier density times magnetic flux quantum) takes half-integer values $\nu=4(n+1/2)$ because the $n=0$ Landau level (the discrete cyclotron orbit band shared by electrons and holes) is shared by both carrier types. Exactly at the CNP the Fermi level lies in this $n=0$ Landau level, and the $\nu=0$ state is special: candidate descriptions include homogeneous symmetry-broken states — a spin-polarized ferromagnet or canted antiferromagnet (CAF) whose opposing valley/spin components support counterpropagating, spin-filtered edge modes — and, in disordered samples, a percolating network of localized $\nu=+2$ (electron) and $\nu=-2$ (hole) quantum Hall puddles, with the dissipation occurring along the boundaries between them. Cho and Fuhrer's data were consistent with the puddle picture but could not distinguish it from the homogeneous broken-symmetry states.

Problem Statement

Provide a complete microscopic account of magnetotransport near the minimum-conductivity (charge-neutrality) point of spatially inhomogeneous graphene, as posed by Cho and Fuhrer (2008), with three jointly defining components. (1) Identify the microscopic origin of the small field-independent longitudinal conductivity term $\sigma_{xx,1}\approx0.8\,e^{2}/h$ that must be added to the exact equal-mobility effective-medium result $\sigma_{xx}(B)=\sigma_{xx,0}(1+(\mu B)^{2})^{-1/2}$, $\sigma_{xy}=0$ to fit the measured magnetoresistivity of graphene on SiO$_2$ near the charge-neutrality point at temperatures where the semiclassical description otherwise holds — deciding among, or refuting, the candidate mechanisms of imperfect electron-hole balance, unequal electron and hole mobilities, and non-ideal sample geometry/contact effects. (2) Extend the theory of electron-hole-puddle transport into the quantum regime (magnetic length $\ell_{B}$ shorter than the puddle size, roughly $B\gtrsim0.8$ T at low temperature), reproducing quantitatively the measured $\rho_{xx}(B)$ and $\rho_{xy}(B)$ through the classical-to-quantum crossover, including the low-temperature high-field saturation of $\rho_{xx}$ to a value of order $h/e^{2}$ and the plateaulike $\sigma_{xy}\approx0$ region. (3) Determine, by spatially resolved measurement correlated with transport or by an equivalent microscopic theory, whether this saturation and plateau regime in disorder-dominated graphene arises from localized $\nu=+2$ and $\nu=-2$ quantum Hall puddles separated by regions in which the fourfold-degenerate $n=0$ Landau level crosses the Fermi level, or from homogeneous symmetry-broken quantum Hall states. The common objective is a single quantitative description of the conductivity tensor near the charge-neutrality point of disordered graphene that remains valid from the semiclassical puddle regime into the quantum Hall regime; the three components are linked pieces of that description, not independent questions.

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