Controlling Electron-Phonon Interactions in Graphene at Ultrahigh Carrier Densities

We report on the temperature dependent electron transport in graphene at different carrier densities $n$. Employing an electrolytic gate, we demonstrate that $n$ can be adjusted up to $4\times {10}^{14}{\mathrm{cm}}^{-2}$ for both electrons and holes. The measured sample resistivity $\rho$ increases linearly with temperature $T$ in the high temperature limit, indicating that a quasiclassical phonon distribution is responsible for the electron scattering. As $T$ decreases, the resistivity decreases more rapidly following $\rho (T)\sim T^4$. This low temperature behavior can be described by a Bloch-Grüneisen model taking into account the quantum distribution of the two-dimensional acoustic phonons in graphene. We map out the density dependence of the characteristic temperature ${\Theta }_{\mathrm{BG}}$ defining the crossover between the two distinct regimes, and show that, for all $n$, $\rho (T)$ scales as a universal function of the normalized temperature $T/{\Theta }_{\mathrm{BG}}$.

Figure 1

(a) Resistivity as a function of applied electrolyte gate voltage $V_{\mathrm{eg}}$ at $T=300\mathrm{K}$. Right inset: a schematic view of the electrolyte gated device. The Debye layers are formed $d\sim 1\mathrm{nm}$ above the graphene surface. The left inset shows an optical microscope image (false colors) of a typical etched Hall bar device. (b) The inset shows the Hall voltage $V_H$ as a function of the magnetic field $B$ for different $V_{\mathrm{eg}}$ (current across the sample $I=100\mathrm{nA}$). The main panel shows the extracted densities $n$ from the various Hall measurements as a function of $V_{\mathrm{eg}}$ for four different samples. The slope of the line fit represents the capacitive coupling of the electrolyte gate to the graphene.

Figure 2

(a) Temperature dependence of the resistivity for different charge carrier densities of sample G8A4. (b) The temperature dependent part of the resistivity $\Delta \rho (T)$ scales as $T^4$ in the low $T$ range and smoothly crosses over into a linear $T$ dependence at higher $T$. The dashed lines represent fits to the linear $T$ and $T^4$ dependencies, respectively. The inset shows the mobility ${\mu }_0$ at $T=2\mathrm{K}$ as a function of the density $n$. The gray line is the theoretically expected mobility due to short and long range impurity scattering.

Figure 3

(a) Scaling of the prefactors $\alpha (n)$ [$\Delta \rho \approx \alpha (n)T^4$]. Data points were obtained from $T^4$ fits of the $\rho (T)$ traces at different carrier densities and for different samples. The dashed line represents a theoretically predicted fit $\propto |n{|}^{-(3/2)}$. (b) ${\Theta }_{\mathrm{BG}}$ at different carrier densities [symbols are defined as in (a)]. The gray line is a fit to the theoretically predicted ${\Theta }_{\mathrm{BG}}=2\hslash v_s\sqrt{\pi n}/k_B$.

Figure 4

Universal scaling of the normalized resistivity $\Delta \rho (T)/\Delta \rho (\xi {\Theta }_{\mathrm{BG}})$ as a function of the normalized temperature $T/{\Theta }_{\mathrm{BG}}$, explicitly using the constant $\xi =0.2$. Data points correspond to $\Delta \rho (T)$ of sample G8A4 at different $n$ and ${\Theta }_{\mathrm{BG}}$ and are normalized with respect to $\Delta \rho (\xi {\Theta }_{\mathrm{BG}})$—the resistivity at $T=\xi {\Theta }_{\mathrm{BG}}$. The dashed line trace represents the theoretically predicted scaling of $f_s({\Theta }_{\mathrm{BG}}/T)/f_s({\xi }^{-1})$ without the use of fitting parameters.