Conductivity of suspended graphene at the Dirac point

We study transport properties of clean suspended graphene at the Dirac point. In the absence of the electron-electron interaction, the main contribution to resistivity comes from interaction with flexural (out-of-plane deformation) phonons. We find that the phonon-limited conductivity scales with the temperature as $T^{-\eta },$ where $\eta$ is the critical exponent (equal to $\approx$$0.7$ according to numerical studies) describing renormalization of the flexural phonon correlation functions due to anharmonic coupling with the in-plane phonons. The electron-electron interaction induces an additional scattering mechanism and also affects the electron-phonon scattering by screening the deformation potential. We demonstrate that the combined effect of both interactions results in a conductivity that can be expressed as a dimensionless function of two temperature-dependent dimensionless constants, $G\left[T\right]$ and $G_e\left[T\right]$, which characterize the strength of electron-phonon and electron-electron interactions, respectively. We also discuss the behavior of conductivity away from the Dirac point as well as the role of the impurity potential and compare our predictions with available experimental data.

Figure 1

Schematic plot of the dimensionless electron-phonon scattering rate ${\gamma }_{\mathrm{ph}}\left(x\right)$ at $G_e\ll 1$ for four different values of $G$ ($G$ increases from I to IV). Also shown are the transport scattering rate ${\gamma }_{\mathrm{ee}}\left(x\right)$ (thick solid) and the energy relaxation rate ${\gamma }_E\left(x\right)$ (dashed) induced by electron-electron scattering.

Figure 2

Conductivity of clean suspended graphene at the Dirac point. Transport regimes characterized by different behavior of dimensionless conductivity $\Sigma$ are shown in the parameter plane of effective (temperature-dependent) dimensionless couplings $G$ and $G_e$. The analytical expressions for dashed lines separating different regimes are given in the white boxes.

Figure 3

Dimensionless conductivity $\Sigma (G,G_e)$ at the Dirac point as a function of effective phonon coupling $G$ for fixed small electron-electron coupling $G_e$. According to the definition of $G$, Eq. (42), the shown dependence can be also understood as temperature dependence of the resistivity. In the low $G$ (low-temperature) region, the conductivity is governed by impurity scattering (shown by dashed line).

Figure 4

Velocity renormalization for $G_e\ll 1$ and $G/N{G}_e^{1-\eta }>1$.

Figure 5

Regions A, B, and C where a “nonperturbative” interval of energies exists.

Figure 6

Temperature dependence of dimensionless conductivity $\Sigma$ for $G_e\ll 1$ and relatively small $\mu$ such that $G\left[\mu \right]\ll G_e$. Dashed lines show conductivity limited by charged impurities for different impurity concentration increasing from the curve (1) to the curve (4).

Figure 7

Conductivity at the Dirac point ($\mu =0$) for different values of the impurity concentration ($n_i/{10}^{10}{\text{cm}}^{-2}={10}^{-4},{10}^{-3},{10}^{-2},{10}^{-1},0.5,3,10$) increasing from the top to the bottom. Dashed line corresponds to SCBA limit $\sigma =4e^2/\pi h.$ Calculations are controlled only well above this line.

Figure 8

Resistivity as a function of electron concentration at $n_i=5\times {10}^9{\text{cm}}^{-2}$ and different temperatures ($T/1\text{K}=5,40,90,150,230$) increasing from the bottom to the top at large $n$. Within the grey area temperature dependence is “insulating,” while outside this region it is “metallic.”

Figure 9

Conductivity at fixed impurity concentration ($n_i=5\times {10}^9{\text{cm}}^{-2}$) for different values of chemical potential ($\mu /1\text{K}=0$, 50, 70, 90, 110, 130, 150, 170, 190, 210, 230, 250, 270, 290, 310, 330, 350, 400, 450, 500, 550, 600) increasing from the bottom to the top. Dashed line corresponds to SCBA limit $\sigma =4e^2/\pi h.$

Figure 10

Conductivity at fixed impurity concentration ($n_i={10}^9{\text{cm}}^{-2}$) for different values of chemical potential ($\mu /1\text{K}=0,10,20,30,35,40,45,50,60,70,80,90$) increasing from the bottom to the top. Dashed line corresponds to SCBA limit $\sigma =4e^2/\pi h.$

Figure 11

Conductivity at $\mu =50$ K for different values of the impurity concentration ($n_i/{10}^{10}{\text{cm}}^{-2}=0.07,0.085,0.1,0.125,0.15,0.2,0.25,0.3,0.4,0.5,0.7,0.9$) increasing from the top to the bottom. Dashed line corresponds to SCBA limit $\sigma =4e^2/\pi h.$

Figure 12

Conductivity as a function of electron concentration at $T=40$ K and two different values of impurity concentration: $n_i=5\times {10}^9{\text{cm}}^{-2}$ and $n_i=3\times {10}^{10}{\text{cm}}^{-2}$.