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Graphene Carrier Transport Theory

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Graphene Nanoelectronics

Part of the book series: NanoScience and Technology ((NANO))

Abstract

This chapter describes the theory of carrier transport in two-dimensional graphene sheets. At high carrier density, the conductivity of graphene depends on carrier density, the dielectric constant of the substrate, and the properties of the impurity potential, which all can be treated using the Boltzmann transport formalism. At low carrier density, disorder causes the local random fluctuations in carrier density to exceed the average density. As a consequence, the carrier transport at the Dirac point is highly inhomogeneous. The ensemble-averaged properties of these puddles of electrons and holes are described by a self-consistent theory, and the conductivity of this inhomogeneous medium is given by an effective medium theory. Comparing this transport theory with the results of representative experiments rigorously tests it validity and accuracy.

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Notes

  1. 1.

    The Born approximation treats the scattering potential to leading order. It is not obvious that attempts to go beyond the approach presented here have provided a more accurate solution (see discussion in [15] and [41]). A critique of the Born approximation can be found in [17] and [47].

  2. 2.

    Here, κ is the average of the dielectric constants of the medium above and below the graphene sheet, and the numerical coefficient (max[r s ] ≈ 2) is a material parameter set by the overlap of the carbon π-orbitals in the honeycomb lattice and the separation between the carbon atoms set by theσ-bonds.

  3. 3.

    Historically, the RPA was introduced by Bohm and Pines when discussing the plasma oscillations of the electron gas in the high density limit. In that context, the approximation corresponds to looking at the Fourier transform of the potential energy and showing that after subtracting the term that was linear in carrier density, the subleading term had sums over the phase of electrons that depended on their position. Averaging over position gave a highly oscillatory summand that would be negligible or equivalent to the vanishing of that sum for random electron phases.

  4. 4.

    The integral can be done by making the substitution \(x = \vert k\vert -\vert k + q\vert \) and noticing that

    $$\begin{array}{rcl} kdk = \frac{1} {2} \frac{{x}^{2} - {q}^{2}} {x + q\cos {\theta }_{kq}} \frac{dx} {1 -\cos {\theta }_{kk\prime }}.& & \end{array}$$
    (12.6)
  5. 5.

    The fact that Π(q) = ν(E F) for q ≤ 2k F implies that for high-density transport properties, the Thomas–Fermi approximation and the RPA give the same results (notice that (12.3) only integrates the dielectric function from 0 to 2k F), where \(\epsilon (q \leq 2{k}_{\mathrm{F}}) = 1 + V (q)\Pi (q) = 1 + {q}_{\mathrm{s}}/q\).

  6. 6.

    To fully discuss the role of Klein tunneling in graphene transport would require a larger discussion than is possible here. For the semiclassical calculation presented here, we ignore the additional contribution to the resistance arising from p-n junctions. The validity of this assumption is rigorously tested in Sect. 12.3.4 below. For a more complete discussion on the role of p-n junctions in graphene transport, as well as an explanation for the remarkable property of perfect transmission of carriers at normal incidence, see Chap. 12 and [15].

  7. 7.

    To illustrate somewhat simplistically how one could get into trouble at the Dirac point, consider the Einstein relation that was discussed earlier σ = e 2ν(E F )D. At the Dirac point, ν(E F ) vanishes but for short-range impurities, D → ∞, which gives rise to a disorder-dependent minimum conductivity at the Dirac point (see Fig. 12.2). Similar cancellation of divergences gives rise to the following puzzle [19]. If one calculated the conductivity by first taking the clean limit while keeping either temperature or frequency finite, one would obtain the universal value \({\sigma }_{\mathrm{min}} = (\pi /2){e}^{2}/h\). However, taking frequency and temperature to zero first, then taking the clean limit gives \({\sigma }_{\mathrm{min}} = (4/\pi ){e}^{2}/h\). At the time of writing, the crossover between these two universal limits remains an unsolved problem. However, for the purposes of understanding current graphene dc transport experiments, we maintain that none of this “universal” physics is relevant.

  8. 8.

    We have been slightly sloppy with language, using the term Gaussian approximation to refer to both when the disorder potential has Gaussian two-point spatial correlation function (see (12.17)) and when the disorder probability distribution function is determined only from the second moment (see (12.22)). From the context, it should be clear which case we mean, although we should caution that the two approximations can be quite different. For example, (12.24) describes a Gaussian two-point correlation function, but is equivalent to a Gaussian distribution function only in the limit when n impπξ2 ≫ 1.

  9. 9.

    Here, we assume that the temperature dependence arises only from thermal smearing of the Fermi distribution function. While this assumption that the temperature dependence occurs only from this activation-like behavior is an excellent approximation for bilayer graphene (see [40]), for monolayer graphene, additional physics such as the degradation of the conductivity due to phonons in dirty samples, and the crossover to the ballistic regime for suspended samples restrict the temperature range for which this thermal broadening picture dominates the conductivity.

  10. 10.

    As an aside, we should mention that the two-carrier model above (and assuming Coulomb impurities, (12.12)) relates field-effect mobility to the carrier mobility as

    $$\begin{array}{rcl}{ \mu }_{H} \equiv \frac{{\rho }_{xy}} {{\rho }_{xx}B} = \frac{{\sigma }_{xy}} {{\sigma }_{xx}B} \approx \frac{A[{r}_{s}]} {{n}_{\mathrm{imp}}} \left (\frac{{n}_{e} - {n}_{h}} {{n}_{e} + {n}_{h}}\right ),& & \end{array}$$
    (12.35)

    where only in the very limited carrier density range n  ∗  ∼ n imp ≪ n ≪ B[r s ]σ0 ∕ n is the Hall mobility the same as the field-effect mobility \({\mu }_{c} = \sigma /ne\) (where the specific boundaries of this window depend on the number of short-range and long-range impurities and the dielectric environment).

  11. 11.

    The Landauer approach gives the universal value only for W ≫ L, where the transport is primarily through evanescent modes. In the opposite limit, the conductivity depends strongly on the boundary conditions and is not universal. When comparing the quantum and semiclassical models, we will assume that W ≫ L. For further discussion, see [15].

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Acknowledgements

Most of the research leading up to the material covered in this chapter was done while I was a postdoctoral research associate with the Condensed Matter Theory Center at the University of Maryland under the mentorship of Sankar Das Sarma, where it was supported by U.S. ONR, and the NSF-NRI-SWAN. During this time, I also benefited enormously from collaborations with Michael Fuhrer, Euyheon Hwang, Enrico Rossi, Victor Galitski and Ellen Williams. Some of the more recent work was done with Piet Brouwer, Mark Stiles and Parakh Jain. I am also grateful to Mark Stiles, Michael Fuhrer, Jabez McClelland, Joseph Stroscio, Hongki Min, Hassan Raza, and Nikolai Zhitenev for their valuable comments.

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  1. Center for Nanoscale Science and Technology, National Institute of Standards and Technology, Gaithersburg, MD, 20899-6202, USA

    Shaffique Adam

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  1. , Electrical and Computer Engineering, University of Iowa, IATL 116, IOWA City, 52246, USA

    Hassan Raza

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Adam, S. (2011). Graphene Carrier Transport Theory. In: Raza, H. (eds) Graphene Nanoelectronics. NanoScience and Technology. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22984-8_12

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