Transport and Elastic Scattering Times as Probes of the Nature of Impurity Scattering in Single-Layer and Bilayer Graphene

Transport and elastic scattering times, ${\tau }_{\mathrm{tr}}$ and ${\tau }_e$, are experimentally determined from the carrier density dependence of the magnetoconductance of monolayer and bilayer graphene. Both times and their dependences on carrier density are found to be very different in the monolayer and the bilayer. However, their ratio ${\tau }_{\mathrm{tr}}/{\tau }_e$ is found to be close to 1.8 in the two systems and nearly independent of the carrier density. These measurements give insight on the nature (neutral or charged) and range of the scatterers. Comparison with theoretical predictions suggests that the main scattering mechanism in our samples is due to strong (resonant) scatterers of a range shorter than the Fermi wavelength, likely candidates being vacancies, voids, adatoms or short-range ripples.

Figure 1

Gate voltage dependence of the conductance at several magnetic fields, 1 $T$ apart. The contact resistances have been subtracted. Top panel: monolayer $A$. Bottom panel: bilayer $B$. Inset: electron micrographs of the samples.

Figure 2

Analysis of the magnetoresistance. Left panel: Magnetoresistance of monolayer sample $A$. Dots: experimental points at $T=1\mathrm{K}$; Continuous line: fit according to Eqs. (2, 1). Inset: $B^2$ dependence of the low-field magnetoresistance for different gate voltages (Curves shifted along the $Y$ axis for clarity). ${\tau }_{\mathrm{tr}}$ is extracted from the slopes of these curves according to Eq. (2). Notice that the slope increases in the vicinity of the Dirac point reflecting the divergence of the inverse effective mass. Right panel: ShdH oscillations of the longitudinal component of the resistivity in bilayer sample $B$ for different temperatures after subtraction of the quadratic background. The Fermi wave vector $k_F$ and the elastic time ${\tau }_e$ are deduced from the period and the decay of the oscillations with $1/B$ at low temperature. Inset: Temperature dependence of the oscillations amplitude normalized to $T=0$. Solid line: fit according to the Lifshitz-Kosevich formula $D_T=\gamma /\sinh (\gamma )$ with $\gamma =2{\pi }^2{k}_{B}T/\hslash {\omega }_c$ [15]. The effective mass determined from this fit is $m_{\mathrm{eff}}=0.035\pm 0.002m_e$ in the whole range of gate voltage investigated.

Figure 3

$k_F$ dependence of ${\tau }_{\mathrm{tr}}$ and ${\tau }_{\mathrm{tr}}/{\tau }_e$ ratio. Left panel: monolayers $A$, $C$, $D$, and $E$. Right panel: bilayer $B$. The continuous lines are the fits for samples $A$, $B$, and $D$ according to the resonant impurity model, Eq. (5). For samples $A$ $B$ and $C$ (two-terminal configuration) ${\tau }_{\mathrm{tr}}$ was extracted from the low-field magnetoresistance (crosses), whereas it was extracted from the zero-field conductivity for samples $D$ and $E$. Positive (negative) values of $k_F$ correspond to electron (hole) doping. Lower panels: ratio ${\tau }_{\mathrm{tr}}/{\tau }_e$ where ${\tau }_e$ is deduced from the fit of the low temperature decay of the ShdH oscillations. Dotted lines figure the average value ${\tau }_{\mathrm{tr}}/{\tau }_e=1.8$. Interestingly, although the mobilities and accordingly ${\tau }_{\mathrm{tr}}$ vary substantially from one sample to the other (from 5000 to $800{\mathrm{cm}}^2/\mathrm{V}\mathrm{s}$ from samples $A$ to $E$) the ratio ${\tau }_{\mathrm{tr}}/{\tau }_e$ is similar for all samples.

Figure 4

Comparison of $G(\nu )$ at 5 T for samples $A$ and $B$ with the expression of the conductance derived in Ref. 13, taking the aspect ratio $L/W$ of each sample and using Eqs. (3, 4) with ${\tau }_e(k_F)$ determined above. The dashed vertical lines indicate the positions of ${\nu }_n$ which, as expected, are different for the mono ${\nu }_n=\pm 4(n+1/2)$ and the bilayer ${\nu }_n=\pm 4n$. The conductance quantization is well obeyed for the monolayer but not for the bilayer. This is well explained by the aspect ratio of the bilayer sample [13, 14].