Density-dependent electrical conductivity in suspended graphene: Approaching the Dirac point in transport

We theoretically consider, comparing with the existing experimental literature, the electrical conductivity of gated monolayer graphene as a function of carrier density, temperature, and disorder in order to assess the prospects of accessing the Dirac point using transport studies in high-quality suspended graphene. We show that the temperature dependence of graphene conductivity around the charge neutrality point provides information about how closely the system can approach the Dirac point, although competition between long-range and short-range disorder as well as between diffusive and ballistic transport may considerably complicate the picture. We also find that the acoustic phonon scattering contribution to the graphene resistivity is always relevant at the Dirac point, in contrast to higher density situations where the acoustic phonon contribution to the resistivity is strongly suppressed under the low-temperature Bloch-Grüneisen regime. We provide detailed numerical results for temperature- and density-dependent conductivity for suspended graphene.

Figure 1

The Dirac point conductivity ${\sigma }_D$ as a function of temperature. ${\sigma }_i$ (${\sigma }_{\delta }$) indicates the conductivity due to screened charged Coulomb disorder with an impurity density $n_i=0.3\times {10}^{10}{\mathrm{cm}}^{-2}$ [short-range disorder with $n_{\delta }{V}_{\delta }^2=1.5{\left(\mathrm{eV}\AA \right)}^2$]. ${\sigma }_{\mathrm{ph}}$ represents the conductivity limited by acoustic phonon scattering. ${\sigma }_{\mathrm{tot}}$ is the total conductivity including all scattering mechanisms. The dashed line indicates the conductivity due to the Coulomb disorder and the short-range disorder.

Figure 2

Temperature-dependent electron density $n\left(T\right)$ [Eq. (38)] and energy $\varepsilon \left(T\right)$ [Eq. (36)] as a function of temperature, $T/T_F$.

Figure 3

Conductivity of SG corresponding to the experimental data of Bolotin et al.[3] (a) Calculated conductivity as a function of density for different temperatures—$T=0,$ 100, 200, and 300 K (from top to bottom)—with $n_i=0.85\times {10}^{10}$ cm${}^2$ and $n_{\delta }{V}_{\delta }^2=1.5$ (eV Å)${}^2$. (b) Mobility and (c) mean free path. Solid lines are calculated with temperature-dependent $n\left(T\right)$ and $\varepsilon \left(T\right)$, and dashed lines are calculated with the zero-temperature density $n_0$ and energy $E_F$. (d) $\sigma$, (e) $\mu$, and (f) $l$ at low densities (down to $n={10}^7$ cm${}^{-2}$). Note that as $n\to 0$ or $T/T_F\to \infty$ for a fixed temperature, $\mu \left(T\right)\propto \sigma \left(T\right)/T^2$ and $l\left(T\right)\propto \sigma \left(T\right)/T$. Thus, both $\mu (n\to 0)$ and $l(n\to 0)$ saturate at a finite temperature.

Figure 4

Conductivity corresponding to the experimental data of Du et al.[2] (a) Calculated conductivity as a function of density for different temperatures—$T=0,$ 100, 200, and 300 K (from top to bottom)—with $n_i=5.0\times {10}^{10}$ cm${}^2$ and $n_{\delta }{V}_{\delta }^2=14.7$ (eV Å)${}^2$. (b), (c) Mobility (b) and mean free path (c) are shown as a function of density, respectively. Solid lines are calculated with the temperature-dependent $n\left(T\right)$ and $\varepsilon \left(T\right)$, and dashed lines are calculated with a zero-temperature density $n_0$ and an energy $E_F$. (d) $\sigma$, (e) $\mu$, and (f) $l$ at low densities.

Figure 5

Conductivity corresponding to the experimental data of Mayorov et al.[5] (a) Calculated conductivity as a function of density for different temperatures—$T=0,$ 100, 200, and 300 K (from top to bottom)—with $n_i=0.3\times {10}^{10}$ cm${}^2$ and $n_{\delta }{V}_{\delta }^2=1.5$ (eV Å)${}^2$. (b), (c) Mobility (b) and mean free path (c) as a function of density. Solid lines are calculated with the temperature-dependent $n\left(T\right)$ and $\varepsilon \left(T\right)$, and dashed lines are calculated with a zero-temperature density $n_0$ and an energy $E_F$. (d) $\sigma$, (e) $\mu$, and (f) $l$ at low densities.

Figure 6

Calculated conductivity as a function of density for different temperatures: (a) For a long-range Coulomb potential with $n_i={10}^{10}$ cm${}^{-2}$ and (b) for a neutral short-range potential with $n_{\delta }{V}_{\delta }^2=5$ (eV Å)${}^2$. Solid lines indicate $\sigma$ vs $n\left(T\right)$, and dashed lines indicate $\sigma$ vs $n_0=n(T=0)$ or density induced by only the gate voltage.

Figure 7

(a), (b) Calculated mobility as a function of density for different temperatures and for a long-range Coulomb potential with $n_i={10}^{10}$ cm${}^{-2}$. Here $n_0$ indicates the density induced by the gate voltage and $n\left(T\right)$ indicates the total density, i.e., the density from the gate plus the density from thermal excitations. Solid lines represent Eq. (40) with $n\left(T\right)$ and dashed lines represent Eq. (42) with $n_0=n(T=0)$ or density induced by only gate voltage. (c), (d) Mean free path shown as a function of density. Solid (dashed) lines indicate Eq. (41) [Eq. (43)].

Figure 8

(a), (b) Calculated mobility as a function of density for different temperatures and for a short-range neutral potential with $n_{\delta }{V}_{\delta }^2=5$ (eV Å)${}^2$. Solid lines represent Eq. (40) with $n\left(T\right)$ and dashed lines represent Eq. (42) with $n_0=n(T=0)$ or density induced by only the gate voltage. (c), (d) Mean free path shown as a function of density. Solid (dashed) lines indicate Eq. (41) [Eq. (43)].

Figure 9

Temperature-dependent conductivity of SG corresponding to the experimental data of (a) Du et al.[1] and (b) Bolotin et al.[3] The same parameters used in Figs. 3 and 4 are used in this calculation. Solid (dashed) lines indicate the results with (without) phonon scattering.

Figure 10

Temperature-dependent conductivity of SG corresponding to the experimental data of (a), (b) Bolotin et al.[3] and (c), (d) Du et al.[1]. The same parameters used in Figs. 3 and 4 are used in this calculation. (a), (c) Solid (dashed) lines indicate the results without (with) phonon scattering, and black (red) lines indicate the results for a density $n={10}^{10}$ cm${}^{-2}$ ($n={10}^9$ cm${}^{-2}$). (b), (d) Individual contribution to the total conductivity (black line) is shown for a density $n={10}^{10}$ cm${}^{-2}$. In (d) the nonmonotonic behavior at high densities does not appear due to the strong short-range potential scattering, but in high-mobility samples (b) the nonmonotonic behavior shows up due to the much weaker neutral impurity scatterings.