Conductivity of graphene with resonant adsorbates: beyond the nearest neighbor hopping model

Electronic transport in graphene is sensitive in particular to local defects such as vacancies or adsorbates [1–5]. These defects are interesting in the context of functionalization which aims at controlling the electronic properties by attaching groups of atoms to graphene [6–13]. Therefore, a theoretical understanding of conductivity in the presence of such defects is needed. So far, the case of resonant local defects has attracted much attention because these defects strongly scatter electrons and therefore profoundly affect the electronic transport properties. Yet most of the studies are done in the standard nearest neighbor hopping model of graphene. In particular, in a recent work [12] we detailed some universal regimes of transport close to the Dirac point. The aim of the present study is to analyze the effect of hopping beyond nearest neighbors in the graphene plane. As we show, the universal regimes presented in the previous work [12] still exist but their domain of existence in terms of electron concentration for example is affected, which may be important for precise comparison with experiments.

We have developed a simple tight‐binding (TB) scheme that reproduces ab initio electronic structure in the energy range ±2 eV around the Dirac energy E_D [14, 15]. Only p_z orbitals are taken into account since we are interested in electronic properties at the Fermi energy E_F. The Hamiltonian is

$$\begin{equation} \hat H = \sum\limits_{\left\langle {i;j} \right\rangle } {t_{ij} \left( {c_i^{*} c_j + c_j^{*} c_i } \right)} , \end{equation} \tag{ 1 }$$

where the coupling element matrix t_ij depends on the distance r_ij between orbital i and orbital j,

$$\begin{equation} t_{ij} = - \gamma _0\, {\rm exp}\,\left( {q\left( {1 - \frac{{r_{ij} }}{a}} \right)} \right) \end{equation} \tag{ 2 }$$

with the first neighbor distance, a = 0.142 nm, and the first neighbors interaction in a graphene plane, γ₀ = 2.7 eV. The constant q is fixed to have a second neighbors interaction equal to 0.1γ₀ [10, 13]. We consider that resonant adsobates (such as H, OH, CH₃, etc) can create a covalent bond with some carbon atoms of the graphene sheet. Then a generic model is obtained by removing the p_z orbital of the carbon that is just below the adsorbate [6, 13].

In our calculations resonant adsorbates (mono‐vacancies) are distributed at random with a finite concentration c. Preliminary results of quantum diffusion for these models have been published in [10]. The conductivity is computed by the Mayou–Khanna–Roche–Triozon (MKRT) method [16–19]. This method allows very efficient numerical calculations by recursion in real‐space. Our calculations are performed on a sample containing up to 10⁸ atoms. That corresponds to typical size of about 1 μm² which allows studying systems with inelastic mean‐free length of the order of a few hundreds of nanometers. MKRT method has also been used to study other kinds of defects in graphene sheet [9, 11, 20, 21].

In the relaxation time approximation, we introduce an inelastic scattering time τ_i beyond which the propagation becomes diffusive due to the destruction of coherence by inelastic process (see [12] and references therein). One finally gets the conductivity

$$\begin{equation} \sigma (E_F ,\tau _{\rm{i}} ) = e^2 n(E_{\rm{F}} )D(E_{\rm{F}} ,\tau _{\rm{i}} ) \end{equation} \tag{ 3 }$$

and the diffusivity

$$\begin{equation} D(E_{\rm{F}} ,\tau _{\rm{i}} ) = \frac{{L_{\rm{i}}^2 (E_{\rm{F}} ,\tau _{\rm{i}} )}}{{2\tau _{\rm{i}} }}, \end{equation} \tag{ 4 }$$

where n is the density of states (DOS) and L_i is the inelastic mean‐free path. L_i (E,τ_i ) is the typical distance of propagation during the time interval τ_i for electrons at energy E.

The total density of states (DOS) for concentrations c = 1, 2 and 3% is shown in figure 1. With only first neighboring coupling a resonance of the DOS is found at the Dirac energy (E = 0). This is reminiscent of the midgap state produced by just one missing orbital [6, 13]. With coupling beyond nearest neighbors, the resonance of the density of states is enlarged and displaced by the effect of the hopping beyond nearest neighbors [13, 10]. This changes of course the relation between the density of states, the electronic mean‐free path L_e and the Fermi energy or the filling factor (electron density) of the band.

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The conductivity σ  and the inelastic scattering length L_i are shown in figure 2. At short times the propagation is ballistic and the conductivity increases when τ_i increases. For large τ_i, the conductivity decreases with increasing τ_i due to quantum interference effects, and it goes to zero due to Anderson localization in two‐dimension (2D) [22]. We defined the microscopic conductivity σ_M  as the maximum value of σ (τ_i) values (figure 2). The microscopic conductivity is shown in figure 3 for different values concentration c. According to the renormalization theory [22] this value is obtained when the inelastic mean free path L_i and the elastic mean free path L_e are comparable. Here we compute L_e by

$$\begin{equation} L_{\rm{e}} (E) = \frac{{2D_{\rm{M}} (E)}}{V}, \end{equation} \tag{ 5 }$$

where D_M(E) is the maximum value of the diffusivity and V is the velocity of electrons at energy E. L_e versus E is shown figure 4. Finally we define the localization length ξ  as the value for which the extrapolated conductivity (see below) cancels, i.e. σ (L_i ≃ ξ ) = 0 (figure 5).

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For Fermi energy E_F very close to the Dirac energy E_D a strong fine peak in the microscopic conductivity is found for the model with nearest‐neighbor coupling only (inset of figure 3) [11, 12, 23]. This particular behavior, which is a consequence of the resonance in the DOS, is specific to this model and it is not found when hopping beyond nearest neighbors is included. Indeed in the last case, the microscopic conductivity is slightly higher but still close to the universal plateau [9–12, 20, 21] of microscopic conductivity (figure 3), $\sigma _{\rm M} \simeq 4e^2 /(\pi h)$. Thus, our results with hopping beyond nearest neighbors show that the plateau of the microscopic conductivity near the Dirac energy is robust. In contrast, the high central peak of the conductivity and the anomalous behavior at the Dirac energy (see [12]) are not robust and are specific to the model with nearest neighbor hoping only.

For L_e < L_i < ξ , the conductivity is not equal to the microscopic conductivity and quantum interference have to be taken into account. As shown in figure 5, σ (L_i) follows the linear variation with the logarithm of the inelastic mean free path L_i that was found in our previous work for smaller concentration of adsorbates [12]

$$\begin{equation} \sigma (L_{\rm{i}} ) = G_0 \left( {2 - \alpha \log \left( {{{L_{\rm{i}} }/{L_{\rm{e}} }}} \right)} \right)\quad {\rm{with}}\quad G_0 = {{2e^2 }/h} \end{equation} \tag{ 6 }$$

and α ≃ 0.25, which is close to the result of the perturbation theory of 2D Anderson localization for which α = 1/π [22]. As discussed in [12, 24] this could be tested through magneto‐conductance measurements.

We propose a unified description of transport in graphene with resonant adsorbates simulated by simple vacancies [12]. Sufficiently far from the Dirac energy and at sufficiently small concentration of adsorbates, the semi‐classical theory is a good approximation. For Fermi energy E_F close to the Dirac energy E_D different quantum regimes are found.

Some universal aspects of the conductivity are present with or without the hoping beyond nearest neighbors. For small inelastic scattering length L_i such as $L_{\rm{i}} \simeq L_{\rm{e}}$, the conductivity σ  is almost equal to the universal minimum (plateau) of microscopic conductivity $\sigma _{\rm{M}} \simeq {{4e^2 }/{(\pi h)}}$, except for $E_{\rm{F}} \simeq E_{\rm{D}}$ when the model only takes into account nearest neighbor hopping. For larger L_i, L_e < L_i, the conductivity follows a linear variation with the logarithm of L_i [12] for both models, with nearest neighbor hopping only and with hopping beyond nearest neighbors. In contrast, the high central peak of the conductivity and the anomalous behavior at the Dirac energy (see [12]) are not robust and are specific to the model with nearest neighbor hoping only. Therefore, we conclude that a precise comparison of conductivity with experiments requires a detailed description of the electronic structure and in particular of that of graphene [25].

We thank L Magaud, C Berger and W A de Heer for fruitful discussions and comments. The computations have been performed at the Centre de Calcul of the Université de Cergy‐Pontoise. We thank Y Costes and D Domergue for computing assistance.