Conductivity minimum of graphene with smooth particle-hole-asymmetric disorder in the intermediate-disorder regime

The minimal conductivity of graphene and the doping level at which it occurs are standard characterization observables, yet for smooth disorder that is statistically particle-hole asymmetric — the generic case for charged-impurity disorder — their behavior is understood only in two asymptotic limits. Hering, Schneider, and Brouwer (Phys. Rev. B 92, 075415 (2015)) showed by perturbation theory that weak asymmetric disorder ($\alpha\ll1$) shifts the conductivity minimum away from the Dirac point to $\varepsilon_{\min}=c_{\min}\beta\hbar v/\lambda$ with $c_{\min}\approx0.011$, and numerically that for strong disorder ($\alpha\gg1$) the minimum migrates to the opposite side of the Dirac point, tracking the energy at which electron and hole puddles occupy equal areas. The experimentally relevant window $1\lesssim\alpha\lesssim4$ lies between these limits and remains without a quantitative theory: the position and value of the conductivity minimum, the sign change of the shift near $\alpha\sim1$, the crossover from perturbative to percolation-dominated transport, and the ensemble-universality of these behaviors are all unsettled. The problem is to determine this intermediate-regime behavior, analytically or by controlled numerical simulation.

Background

Near its charge-neutrality (Dirac) point, the low-energy carriers of graphene behave as massless Dirac fermions, and the conductivity exhibits a finite minimum instead of vanishing. In a clean sample this minimal conductivity arises from evanescent quantum modes and equals $4e^2/\pi h$. Real samples additionally carry disorder that is smooth on the lattice scale, typically originating from charged impurities in the substrate; such a potential creates spatially alternating electron-doped and hole-doped regions, called electron-hole puddles, and increases the minimal conductivity further.

Conventional model potentials are Gaussian random fields, which are necessarily statistically particle-hole (electron-hole) symmetric; this symmetry pins the conductivity minimum to the Dirac point $\varepsilon=0$, so only its magnitude can be studied. Realistic impurity ensembles — for example impurities with a preferred charge sign — generate asymmetric potentials, and the position of the conductivity minimum then becomes a nontrivial observable that diagnoses the transport mechanism near the Dirac point: percolation through a network of puddles (minimum expected near the energy $\varepsilon_{\mathrm{size}}$ at which electron and hole puddles occupy equal total areas), a mean-field Boltzmann picture in which disorder is replaced by an effective average carrier density (minimum near $\varepsilon_{\mathrm{charge}}$ where the mean density vanishes), or genuinely coherent quantum transport.

Hering, Schneider, and Brouwer (2015) analyzed a continuum model of massless Dirac fermions, $H=v\mathbf{p}\cdot\boldsymbol{\sigma}+V(\mathbf{r})$ with $\boldsymbol{\sigma}$ the sublattice pseudospin Pauli matrices, in a smooth zero-mean disorder potential with a Gaussian second-moment correlator of strength $\alpha$ and correlation length $\xi$,

$$\langle V(\mathbf{r})V(\mathbf{r}')\rangle=\frac{\alpha(\hbar v)^2}{2\pi\xi^2}\,e^{-|\mathbf{r}-\mathbf{r}'|^2/2\xi^2},$$

plus a nonzero third moment of strength $\beta$ and correlation length $\lambda$ — the lowest-order ingredient of statistical particle-hole asymmetry —

$$\langle V(\mathbf{r})V(\mathbf{r}')V(\mathbf{r}'')\rangle=\frac{3\beta(\hbar v)^3}{4\pi^2\lambda^3}\,e^{-(|\mathbf{r}-\mathbf{r}'|^2+|\mathbf{r}'-\mathbf{r}''|^2+|\mathbf{r}''-\mathbf{r}|^2)/2\lambda^2}.$$

For weak disorder they used the matrix Green function formalism (a Keldysh-type circuit representation of the conductance in a short-and-wide sample of length $L$ and width $W\gg L$, in which conductance follows from a trace over the retarded-advanced Green function matrix and the conductivity is $\sigma=\lim_{W\gg L}GL/W$, in units of $e^2/h$) and found the minimum at

$$\varepsilon_{\min}=\frac{c_2}{2c_1}\frac{\beta\hbar v}{\lambda},\qquad \frac{c_2}{2c_1}\approx0.011,$$

for $\alpha\ll1$. For arbitrary disorder strength they performed Landauer transport simulations (conductance obtained from the transmission matrix via $G=(4e^2/h)\,\mathrm{tr}\,t^\dagger t$, the factor 4 counting spin and valley degeneracy, using the slicing method of Bardarson et al., a transfer-matrix recursion that builds the sample up slice by slice) for a specific particle-hole-asymmetric ensemble built by squaring a Gaussian random field, which fixes $\beta=(8/3)\sqrt{2\pi\alpha^3}$ and $\lambda=\xi\sqrt{2}$. The numerics showed that $\varepsilon_{\min}(\alpha)$ is non-monotonic, changes sign near $\alpha\sim1$, and for $\alpha\gg1$ approaches the percolation value $\varepsilon_{\mathrm{size}}\approx-0.154\,\hbar v\sqrt{\alpha}/\xi$ (opposite in sign to the weak-disorder shift and to the Boltzmann estimate $\varepsilon_{\mathrm{charge}}\approx+0.156\,\hbar v\sqrt{\alpha}/\xi$).

Estimates for charged-impurity disorder in typical exfoliated graphene samples place the effective disorder strength in the range $1\lesssim\alpha\lesssim4$ — precisely the intermediate window in which the weak-disorder perturbation expansion has already failed but the strong-disorder percolation asymptotics has not yet set in. The source paper itself flags this window as unsettled.

Problem Statement

Determine, quantitatively, the behavior of the conductivity minimum of graphene with smooth particle-hole-asymmetric disorder in the intermediate-disorder regime $1\lesssim\alpha\lesssim4$ that is experimentally relevant but lies between the two controlled asymptotic limits established in the source study. Specifically: give the position $\varepsilon_{\min}$ and the value $\sigma_{\min}$ of the minimum as functions of the disorder parameters ($\alpha$, $\beta$, $\lambda$, $\xi$; for instance for the squared-Gaussian asymmetric ensemble used in the source), including the non-monotonic dependence and the sign change of $\varepsilon_{\min}$ near $\alpha\sim1$; and characterize the crossover from the perturbative regime ($\alpha\ll1$, where $\varepsilon_{\min}=(c_2/2c_1)\beta\hbar v/\lambda$ with $c_2/2c_1\approx0.011$) to the percolation regime ($\alpha\gg1$, where the minimum sits at the equal-puddle-area energy $\varepsilon_{\mathrm{size}}$). These deliverables are facets of one objective — a quantitative theory of the conductivity minimum $\sigma(\varepsilon;\alpha)$ across the full experimentally relevant disorder range — and may be met either by a controlled analytical description (e.g. an interpolation, effective-medium, renormalization-group, or percolation-based theory) or by controlled numerical simulation with finite-size and statistical-error analysis.

References

1. M. Hering, M. Schneider, P. W. Brouwer, Electronic transport in graphene with particle-hole-asymmetric disorder, Phys. Rev. B 92, 075415 (2015), DOI: 10.1103/PhysRevB.92.075415, arXiv:1504.05410, https://arxiv.org/abs/1504.05410
2. A. Tagliacozzo, G. Campagnano, D. Giuliano, P. Lucignano, B. Jouault, Thermal transport driven by charge imbalance in graphene in a magnetic field close to the charge neutrality point at low temperature: Nonlocal resistance, Phys. Rev. B 99, 155417 (2019), DOI: 10.1103/PhysRevB.99.155417, arXiv:1904.03046, https://arxiv.org/abs/1904.03046
3. A. Schuessler, P. M. Ostrovsky, I. V. Gornyi, A. D. Mirlin, Full counting statistics in disordered graphene at the Dirac point: From ballistics to diffusion, Phys. Rev. B 82, 085419 (2010), DOI: 10.1103/PhysRevB.82.085419, https://doi.org/10.1103/PhysRevB.82.085419
4. A. Schuessler, P. M. Ostrovsky, I. V. Gornyi, A. D. Mirlin, Analytic theory of ballistic transport in disordered graphene, Phys. Rev. B 79, 075405 (2009), DOI: 10.1103/PhysRevB.79.075405, https://doi.org/10.1103/PhysRevB.79.075405
5. V. V. Cheianov, V. I. Fal'ko, B. L. Altshuler, 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://doi.org/10.1103/PhysRevLett.99.176801
6. J. H. Bardarson, J. Tworzydło, P. W. Brouwer, 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://doi.org/10.1103/PhysRevLett.99.106801
7. S. Adam, E. H. Hwang, V. M. Galitski, S. Das Sarma, A self-consistent theory for graphene transport, Proc. Natl. Acad. Sci. USA 104, 18392 (2007), DOI: 10.1073/pnas.0704772104, https://doi.org/10.1073/pnas.0704772104
8. Y.-W. Tan, Y. Zhang, K. Bolotin, Y. Zhao, S. Adam, E. H. Hwang, S. Das Sarma, H. L. Stormer, P. Kim, Measurement of Scattering Rate and Minimum Conductivity in Graphene, Phys. Rev. Lett. 99, 246803 (2007), DOI: 10.1103/PhysRevLett.99.246803, https://doi.org/10.1103/PhysRevLett.99.246803
9. E. Zion, A. Butenko, A. Sharoni, M. Kaveh, I. Shlimak, Charge carrier transport asymmetry in monolayer graphene, Phys. Rev. B 96, 245143 (2017), DOI: 10.1103/PhysRevB.96.245143, arXiv:1801.03565, https://arxiv.org/abs/1801.03565
10. W. Chen, Y. Shen, B. Fu, Q. Shi, W. Zhu, On the sample-dependent minimal conductivity in weakly disordered graphene, arXiv:2108.13787 (2021), https://arxiv.org/abs/2108.13787
11. R. N. Sajjad, F. Tseng, K. M. Habib, A. W. Ghosh, Quantum transport at the Dirac point: Mapping out the minimum conductivity from pristine to disordered graphene, Phys. Rev. B 92, 205408 (2015), DOI: 10.1103/PhysRevB.92.205408, arXiv:1507.01257, https://arxiv.org/abs/1507.01257