Diffusion and localization in carbon nanotubes and graphene nanoribbons

We study transport length scales in carbon nanotubes and graphene ribbons under the influence of Anderson disorder. We present generalized analytical expressions for the density of states, the elastic mean free path and the localization length in arbitrarily structured quantum wires. These allow us to analyze the electrical response over the full energy range, including the regions around van Hove singularies, traditionally difficult to access by alternative approaches. Comparing with the results of numerical simulations, we demonstrate that both the diffusive and the localized regime are well represented by the analytical approximations over a wide range of the energy spectrum. The approach works well for both metallic and semiconducting nanotubes and nanoribbons but breaks down near the edge states of zigzag ribbons. ar X iv :0 80 4. 48 33 v1 [ co nd -m at . m es -h al l] 3 0 A pr 2 00 8 Diffusion and localization in carbon nanotubes and graphene nanoribbons 2 The exceptionally high electrical conductivity of carbon nanotubes (CNTs) can be attributed to a combination of various factors. The quasi-one-dimensional (quasi-1D) crystalline structure allows the preparation of samples with a very low defect density [1]. The rigidity of the sp-hybridized carbon lattice minimizes the effects of thermal vibrations [2]. One truely unique feature of metallic CNTs, however, is their electronic structure [3]: independent of the tube diameter, there are exactly two massless bands of high velocity crossing at the Fermi energy, resulting in a very low density of states (DOS) which effectively suppresses scattering even in the presence of disorder [4] and allows ballistic transport over hundreds of nanometers [5, 6]. Planar, finite-width graphene nanoribbons (GNRs) display many similarties to their rolled-up counterparts. The band gap of armchair-edge GNRs depends on the width just as it depends on the circumference in zigzag CNTs. Zigzag-edge GNRs, on the other hand, are always metallic just as armchair CNTs. A remarkable difference, however, lies in the edge states generally found at zigzag edges in graphene [7, 8] that are nearly localized by their extremely low dispersion and exist beside the conducting channels at the band center [9]. Energetically, these states lie exactly around the Fermi energy, so it would be expected to have an important impact on the low-energy transport properties of GNRs. The low velocity of the edge channels results in a large DOS, which should by itself lead to strong scattering. However, the local DOS (LDOS) is strongly concentrated on the edge atoms and protecting the conduction channel from scattering into the edge states. The theory of quantum transport in disordered systems has been studied for many decades [10]. Typically, the underlying model for such studies is that of free electrons or, if a discretization is desired, electrons on a (square) lattice. Quantities such as effective electron mass, density of states or Fermi velocity are then viewed as free parameters that can be adjusted to fit the properties of metals [11]. Graphene and CNTs, however, have an electronic structure that is very different from any effective-mass free-electron approximation. In two-dimensional graphene, the electrons have a Diraclike massless dispersion [12, 13], which results in equally massless bands of high velocity in metallic CNTs [3]. Tuning the Fermi energy away from the charge neutrality point (CNP) via electronic gating or chemical doping, CNTs of all chiralities go through a sequence of additional bands that lead to van-Hove singularities in the DOS which are typically not covered by simple models based on a smooth or fixed DOS and number of conductance channels. In this paper, we will present a generalized expression for the DOS in arbitrarily structured, Anderson-disordered quantum wires that allows to access the whole energy range including the regions around the van Hove singularities. Using this representation of the DOS, we then show that a general expression, originally devised for metallic quantum wires with a fixed number of channels still holds true near band edges if used correctly. The various expressions will be applied to both CNTs and GNRs, including the special case of pure edge disorder in the latter case. For 1D systems such as CNTs, it was argued by Mott and Twose [11] that all quantum states should become localized at arbitrarily low disorder, leading to an exponential suppression of the conductance at zero-temperature. For truly 1D quantum wires, Thouless demonstrated that the localization length `loc is identical to the mean free path `el [14]. Later, he derived a generalized expression for metallic quantum wires of finite diameter [15], `loc ≈ 2Ak F 3π2 `el, (1) where the prefactor consisting of cross section A and Fermi wave vector kF can be identified as the number of conduction channels Nch in the modern language of quantum transport theory. A more accurate expression derived by Beenakker from random matrix theory reads [16] `loc ≈ [β (Nch − 1) /2 + 1] `el, (2) where β = 1 for time-reversal invariant systems and β = 2 otherwise. In either case, however, `el and Nch are viewed as fixed parameters and it is not at all clear, how these relations should be applied to systems where these quantities vary strongly with the energy and may not even be well defined near band edges where Nch is discontinuous. An alternative approach based on a perturbative expression for the Lyapunov exponents takes into account the true band structure of a quantum wire, but it still fails to reproduce the correct behavior near band edges [17]. Diffusion and localization in carbon nanotubes and graphene nanoribbons 3 0.0 0.5 1.0 1.5 2.0 2.5 E(eV) 0 20 40 60 80 ` l o c ( n m ) (5,5) CNT 3⁄4" =0:5 eV g( E ) Figure 1. Density of states g(E) and localization length `loc of an armchair (5,5) CNT under the influence of Anderson disorder. Dashed line: g (E) of the clean system displaying the van Hove singularities. Solid lines: g (E) from Eq. (11) and `loc obtained from it via Eqs. (2) and (9). Data with errorbars: values obtained numerically by averaging over 100 samples of length 1000 to 5000`uc. We model CNTs based on an orthogonal first-nearest-neighbor tight-binding approximation [3] with the hopping parameter γ0 = 2.66 eV, which accurately reproduces the Fermi velocity of armchair CNTs and gives reasonable agreement with the correct band structure for several low energy bands. For zigzag GNRs, this approximation is insufficient, as it results in a completely dispersion-free edge state while precise calculations show that this state in fact has a small but non-vanishing band width [18, 9]. Likewise, this approximation predicts every third armchair GNR to be metallic, while precise calculations predict the opening a small gap [19, 20]. The true electronic structure of a GNR would have to take into account the relaxed electronic structure at the edges [21]. As it turns out, a third-nearest neighbor parameterization [22] of an unrelaxed GNR gives a good approximation of the bands near the Fermi energy and will thus be used here. Anderson disorder is defined as uncorrelated potential fluctations [10]: 〈εiεj〉 − 〈εi〉 〈εj〉 = δijσ εi , (3) usually in terms of a uniform random distribution εi − ε0 ∈ [−W/2,W/2], which has a standard deviation σε = W/ √ 12. For perturbatively weak disorder, however, the exact shape of the distribution is irrelevant. Numerically, we use a Gaussian distribution for practical reasons. Experimentally measured values of `el are of the order of 10 μm in CNT of 1.5 nm diameter [6], which translates to an equivalent model disorder of strength σε ∼ 0.05 eV. The values used in Figs. 1 and 2 are chosen significantly larger to enhance the visibility of the qualitative effects. The diffusive transmission through a quantum wire with a disordered section of length L, embedded in an otherwise disorder-free, infinite quantum wire, is defined as