Conductivity of suspended and non-suspended graphene at finite gate voltage

We compute the dc and the optical conductivity of graphene for finite values of the chemical potential by taking into account the effect of disorder, due to midgap states (unitary scatterers) and charged impurities, and the effect of both optical and acoustic phonons. The disorder due to midgap states is treated in the coherent-potential approximation (a self-consistent approach based on the Dyson equation) whereas that due to charged impurities is also treated via the Dyson equation with the self-energy computed using second-order perturbation theory. The effect of the phonons is also included via the Dyson equation with the self-energy computed using first-order perturbation theory. The self-energy due to phonons is computed both using the bare electronic Green’s function and the full electronic Green’s function although we show that the effect of disorder on the phonon propagator is negligible. Our results are in qualitative agreement with recent experiments. Quantitative agreement could be obtained if one assumes water molecules under the graphene substrate. We also comment on the electron-hole asymmetry observed in the dc conductivity of suspended graphene.

Figure 1

(Color online) Phonon spectrum of the honeycomb lattice, using $\alpha =500\text{N}/\text{m}$ and $\beta =10\text{N}/\text{m}$, such that the optical frequency at the $\Gamma$ point is of the order of $1600{\text{cm}}^{-1}$. Also shown are the spectra of the simplified model discussed in the text (dashed lines) and the effective model discussed in Appendix (dashed-dotted lines).

Figure 2

(Color online) Density of states of graphene in the presence of midgap states. The CPA calculation is compared with a numerical exact method. The concentration of impurities is $n_i=0.005$ and $n_i=0.01$. Here and in the following figures, we use $t=3\text{eV}$ and a cut-off energy of $D=7\text{eV}$.

Figure 3

(Color online) CPA calculation of the self-energy $\hslash {\Sigma }_{\text{unit}}^{\text{ret}}\left(\omega \right)$ for two impurity concentrations, $n_i=0.001$ and $n_i=0.0005$. The left panel shows the imaginary part and the right one shows the real part of the self-energy.

Figure 4

(Color online) Electronic self-energy due to unitary scatterers, and acoustic and optical phonons. Only the imaginary part is represented. The impurity concentration is $n_i=0.0002$. The self-energy due to optical phonons depends on temperature and on the chemical potential $\mu$. We have chosen $\mu =0.29\text{eV}$ and $T=45\text{K}$. In the central panel we show the calculation for self-energy due to optical phonons both using the bare electronic Green’s function (solid line) and Green’s function with midgap states (dashed line). The acoustic self-energy is independent of the chemical potential and is represented for a temperature of $T=45\text{K}$.

Figure 5

(Color online) Electronic self-energy due to electron-optical phonon interaction $\mathfrak{I}{\Sigma }_{\text{opt}}^{\text{ret}}$ (left) and optical conductivity $\sigma$ (right) for various phonon propagators. The electronic propagator is dressed by the self-energies due to Coulomb $\left(n_C=0.00013\right)$ and impurity scatterings $\left(n_i=0.00004\right)$. The temperature is $T=45\text{K}$ and the applied gate voltage $V=71\text{V}$.

Figure 6

(Color online) Left panel: Experimental data (circles) from Ref. 26, conductivity with the full self-energy (solid line) due to phonons, midgap states, and charged impurities, and conductivity (dashed line) with self-energy due to midgap states and charged impurities only. The parameters are $T=45\text{K}$, $n_i=4\times {10}^{-5}$, $n_i^C=7.5\times {10}^{-5}$, and $d=0$. The dashed-dotted line is the same as the solid line but with $n_i^C=1.3\times {10}^{-4}$. Right panel: Imaginary part of the self-energy, $\hslash \mathfrak{I}{\Sigma }_{\text{Coul}}^{\text{ret}}\left(\mathit{K},\omega \right)$, due to charged impurities, the most dominant contribution.

Figure 7

(Color online) Left: Conductivity $\sigma \left(\mu \right)$, in units of ${\sigma }_0=\pi e^2/\left(2h\right)$, considering both short-range and charged scatterers. Right: Conductivity $\sigma \left(\mu \right)$ when the influence of charge scatterers is removed.

Figure 8

(Color online) Optical conductivity of graphene at two different temperatures $T=45$ and 300 K. Each panel has two curves: the solid curve is the conductivity with only short-ranged impurities while the dashed curve is the conductivity with both short-ranged impurities and phonons, with the self-energy due to phonons computed with full Green’s function. The chemical potential is that associated with a gate voltage of $V_g=71\text{V}$. The vertical, dashed line marks twice this value.

Figure 9

(Color online) Optical conductivity of graphene at temperature $T=45\text{K}$. The left-hand side shows curves for various short-ranged impurity densities $n_i$. The right-hand side shows curves for various gate voltage $V_g$. The vertical, dashed lines mark twice the chemical potential.

Figure 10

(Color online) Real part of the optical conductivity including the effect of phonons, midgap states, and charged impurities. The parameters are $T=45\text{K}$, $n_i=4.0\times {10}^{-5}$, and $n_i^C=1.3\times {10}^{-4}$. The dashed vertical lines correspond to $\hslash \omega =2\mu$ and the shorter dotted-dashed ones to $\hslash \omega =2\hslash {\omega }_0+2\mu$.

Figure 11

(Color online) Real part of the optical conductivity including the effect of phonons, midgap states, and charged impurities, compared with the experimental data from Ref. 26. The parameters are $T=45\text{K}$, $n_i=4.0\times {10}^{-5}$, $n_i^C=1.3\times {10}^{-4}$, and $d=0$.