Transport Properties of Graphene in the High-Current Limit

We present a detailed study of the high-current transport properties of graphene devices patterned in a four-point configuration. The current tends to saturate as the voltage across graphene is increased but never reaches the complete saturation as in metallic nanotubes. Measurements are compared to a model based on the Boltzmann equation, which includes electron-scattering processes due to charged and neutral impurities, and graphene optical phonons. The saturation is incomplete because of the competition between disorder and optical phonon scattering.

Figure 1

(a) Raman spectrum of a single layer. (b) Atomic force microscopy image of a graphene device. The channel width is $1\mu \mathrm{m}$. (c) Schematic of the four-point measurement. The device is symmetrically voltage biased. $I$ is measured through electrode 1 and $V_{4\mathrm{pt}}$ between electrodes 2 and 3.

Figure 2

(a) Conductivity as a function of $V_g$ measured at zero voltage ($e{V}_{4\mathrm{pt}}<\mathrm{kT}$) at 300 K and ${10}^{-6}\mathrm{mbar}$. $W=350\mathrm{nm}$ and $L=1.3\mu \mathrm{m}$. The mobility is $14000{\mathrm{cm}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$. The line is the theoretical calculation with the model “$C\delta$.” Similar agreement is obtained with the model “$C$” (b) $I\mathrm{\text{−}}{V}_{4\mathrm{pt}}$ characteristics for different $V_g$ (from 0 V to $-24\mathrm{V}$ in 3 V steps). (c) Numerically differentiated curves $dI/d{V}_{4\mathrm{pt}}$ as a function of $V_{4\mathrm{pt}}$ for different $V_g$ (from 0 V to $-18\mathrm{V}$ in 6 V steps). The dashed line is the reference value for the evaluation of $I_{\mathrm{SR}}$. (d) Current in the saturation regime $I_{\mathrm{SR}}$ as a function of $V_g$. The ${\mathrm{I}}_{\mathrm{FS}}$ line corresponds to the full saturation, Eq. (1), with $\hslash \Omega =149\mathrm{meV}$. The other lines are the results of the Boltzman calculation with the models “$C\delta$” and “$C$,” respectively.

Figure 3

(a),(b) Current in the saturation regime $I_{\mathrm{SR}}$ as a function of width and length for 7 different graphene samples at $V_g$ about $-25\mathrm{V}$ from the Dirac point. The mobility varies from 4500 to $15000{\mathrm{cm}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$. The straight lines are guides to the eye.

Figure 4

(a) Electron population (green $f=1$, white $f=0$) on the Dirac cone in the ideal saturation regime [Eq. (1)]. (b) Electron population in the ideal saturation regime (upper half) and according to the Boltzmann treatment with the model “$C\delta$” (lower half) at $V_g=+24\mathrm{V}$ and $V_{4\mathrm{pt}}=2\mathrm{V}$. ${\mathbf{k}}_{\parallel }$ is the direction parallel to the electric field and $\mathbf{k}=\mathbf{0}$ is the Dirac point. (c) Current as a function of voltage at $V_g=\pm 24\mathrm{V}$. Lines are calculations done in the ideal saturation regime ($I_{\mathrm{FS}}$) and with the two Boltzmann models ($I_C$, $I_{C\delta }$). Full (open) dots are measurements for $V_g=-24(+24)\mathrm{V}$. (d) Electron transport scattering lengths: elastic ($l_{\mathrm{el}}$), electron-phonon ($l_{\mathrm{ph}}$), and optical phonon activation ($l_{\Omega }^1=198\mathrm{nm}$) at $V_{4\mathrm{pt}}=1\mathrm{V}$. Notice that $l_{\Omega }$ varies with $V_{4\mathrm{pt}}$ as $l_{\Omega =}{l}_{\Omega }^1$ ($\mathrm{Volt}/V_{4\mathrm{pt}}$). $l_{\mathrm{el}}$, $l_{\mathrm{ph}}^{\Gamma }$, and $l_{\mathrm{ph}}^K$ are obtained taking into account the angular dependence of the scattering rate as in Eq. (1) of 20. $l_{\mathrm{ph}}^{\Gamma }$ and $l_{\mathrm{ph}}^K$ are computed for the electrons at ${\epsilon }_F+\hslash \Omega /2$, supposing that the states at ${\epsilon }_F-\hslash \Omega /2$ are empty (with $\hslash \Omega$ corresponding to the $\mathbf{\Gamma }$ or $\mathbf{K}$ phonon). Notice that $l_{\mathrm{ph}}$ diverges for ${\epsilon }_F-\hslash \Omega /2=0$ (in contrast to the case of nanotubes) since the density of final states vanishes at the Dirac point.