Raman spectroscopy of electrochemically gated graphene transistors: Geometrical capacitance, electron-phonon, electron-electron, and electron-defect scattering

We report a comprehensive micro-Raman scattering study of electrochemically gated graphene field-effect transistors. The geometrical capacitance of the electrochemical top-gates is accurately determined from dual-gated Raman measurements, allowing a quantitative analysis of the frequency, linewidth, and integrated intensity of the main Raman features of graphene. The anomalous behavior observed for the G-mode phonon is in very good agreement with theoretical predictions and provides a measurement of the electron-phonon coupling constant for zone-center ($\mathit{}\mathbf{\Gamma }$ point) optical phonons. In addition, the decrease of the integrated intensity of the 2D-mode feature with increasing doping, makes it possible to determine the electron-phonon coupling constant for near zone-edge ($\mathbf{K}$ and ${\mathbf{K}}^{\prime }$ points) optical phonons. We find that the electron-phonon coupling strength at $\mathit{}\mathbf{\Gamma }$ is five times weaker than at $\mathbf{K}\left({\mathbf{K}}^{\prime }\right)$, in very good agreement with a direct measurement of the ratio of the integrated intensities of the resonant intra- (2D′ mode) and intervalley (2D mode) Raman features. We also show that electrochemical reactions, occurring at large gate biases, can be harnessed to efficiently create defects in graphene, with concentrations up to approximately $1.4\times {10}^{12}{\mathrm{cm}}^{-2}$. At such defect concentrations, we estimate that the electron-defect scattering rate remains much smaller than the electron-phonon scattering rate. The evolution of the G- and 2D-mode features upon doping remain unaffected by the presence of defects and the doping dependence of the D mode closely follows that of its two-phonon (2D mode) overtone. Finally, the linewidth and frequency of the G-mode phonon as well as the frequencies of the G- and 2D-mode phonons in doped graphene follow sample-independent correlations that can be utilized for accurate estimations of the charge carrier density.

Figure 1

(a) Optical image of a dual-gated graphene field-effect transistor prior deposition of the polymer electrolyte. The source and drain electrodes are covered with photoresist (SU8) to prevent them from being in contact with the polymer electrolyte. (b) A schematic cross-section of our dual-gated graphene field-effect transistor, with ${\mathrm{Li}}^{+}$ (green) and ${\mathrm{ClO}}_4^{-}$ (red) ions and the electrical double layers near each electrode. The Si substrate is used as a back-gate. (c) Schematic energy diagrams of the electronic states of the gate electrode and of graphene. Occupied states are represented in grey. At zero gate voltage ($V=0$), the electrochemical potentials of the gate electrode $\mu$ and the graphene layer ${\mu }_{\mathrm{SLG}}$ are equal. The Fermi energy of graphene is $E_{\mathrm{F}}^0$. Applying a gate voltage $V$ results in an electrostatically induced shift $\left(e\varphi \right)$ and in a change of the Fermi energy of graphene $\left(E_{\mathrm{F}}\right)$. The electrochemical potential difference is equal to $eV$, leading to Eq. (1) with $e{V}_0=E_{\mathrm{F}}^0$. (d) Equivalent electrical circuit of our device at steady state. $C_{\mathrm{BG}}$ is the geometrical capacitance of the Si/${\mathrm{SiO}}_2$ back-gate, $C_{\mathrm{G}}$ is the geometrical capacitance of the electrical double layer at the graphene/polymer electrolyte interface, and $C_{\mathrm{Q}}$ is the quantum capacitance of graphene.

Figure 2

(a)–(c) Color maps of the Raman spectra of a pristine graphene monolayer (sample 1), measured using a 532-nm laser beam, as a function of the top-gate voltage $V_{\mathrm{TG}}$. The G- and 2D-mode features appear prominently and no defect-induced D-mode feature is observed. (b) and (c) show a clear evolution of the G- and 2D-mode features with varying $V_{\mathrm{TG}}$. The black dashed lines correspond to the central frequency of each Raman feature. The charge neutrality point (CNP) is indicated by an arrow. (d) Raman spectra at values of $V_{\mathrm{TG}}$ between $-0.5$ and +1 V. The circles are the experimental data and the solid lines are fits (see text for details). The CNP is reached at $V_{\mathrm{TG},0}=+0.5\mathrm{V}$ (see green line). (e)–(g) Schematic representation, in the momentum-energy space, of the main Raman features in graphene [11]. $\hslash {\omega }_{\mathrm{L}}$ ($\hslash {\omega }_{\mathrm{S}}$) denote the incoming laser (Raman scattered) photon energy. The G mode (e) is a nonresonant process [15, 46]. In (f), the 2D mode is represented as the dominant fully resonant inner process [59, 60], involving two near zone-edge transverse optical phonons with frequency ${\omega }_{\mathrm{D}}$ and opposite momentum $\pm \mathbf{q}$. The defect-induced D mode [11, 60, 61, 62] (g) involves one electron-phonon (solid arrow) scattering process ($\mathbf{q}, {\omega }_{\mathrm{D}}$) and one electron-defect (dashed arrow) scattering process. One of these two processes is resonant. In (g), a resonant electron-phonon scattering process is represented.

Figure 3

(a) Frequency ${\omega }_{\mathrm{G}}$ and (b) relative FWHM $\Delta {\Gamma }_{\mathrm{G}}$ of the G-mode feature as a function of the top-gate voltage, recorded at various back-gate voltages on sample 1. The curves are vertically offset by 10 ${\mathrm{cm}}^{-1}$ for clarity. The symbols are experimental data. (c) Top-gate voltage $V_{\mathrm{TG},\mathrm{neutral}}$, corresponding to the CNP in dual-gated graphene, as a function of the applied back-gate voltage $V_{\mathrm{BG}}$. A top-gate capacitance $C_{\mathrm{TG}}=3.3\mu {\mathrm{F}\mathrm{cm}}^{-2}$ is deduced from a linear fit of the data (solid line). The solid lines in (a) and (b) are fits based on Eqs. (5) and (6), respectively, with $C_{\mathrm{TG}}=3.3\mu {\mathrm{F}\mathrm{cm}}^{-2}$.

Figure 4

Frequency ${\omega }_{\mathrm{G}}$ (red squares, left axis) and relative FWHM $\Delta {\Gamma }_{\mathrm{G}}$ (blue circles, right axis) of the G-mode feature, extracted from the measurements in Fig. 2, as a function of Fermi energy or doping. The corresponding Feynman diagrams are shown as insets. The left inset represents the renormalization of the G-mode phonon frequency due to interactions with virtual electron-hole pairs. The right inset represents lifetime broadening due to the resonant decay of a G-mode phonon into an electron-hole pair. The solid blue and red lines are fits based on Eqs. (5) and (6), respectively. The fitting parameters are $C_{\mathrm{TG}}=3.9\mu {\mathrm{F}\mathrm{cm}}^{-2}, {\lambda }_{\Gamma }=0.027$, and $\delta E_{\mathrm{F}}=35\mathrm{meV}$.

Figure 5

Fermi energy $E_{\mathrm{F}}$ as a function of the relative frequency of the G mode $\Delta {\omega }_{\mathrm{G}}$. Measurements on five different devices are represented with different symbols. The dashed and solid lines correspond to Eqs. (9) and (10), respectively.

Figure 6

Doping dependence of the frequency ${\omega }_{2\mathrm{D}}^{}$ (red squares, left axis) and FWHM ${\Gamma }_{2\mathrm{D}}^{}$ (blue open circles, right axis) of the 2D mode-feature. The measurements are performed on sample 2, before the creation of defects.

Figure 7

Left axis: doping dependence of the ratio between the integrated intensity of the 2D-mode feature and that of the G-mode feature (red squares). Right axis: doping dependence of the square root of the ratio between the integrated intensity of the G mode and that of the 2D mode (blue open squares). The dashed and solid lines are fits based on Eq. (12), without and with Fermi energy fluctuations, respectively. A value of ${\gamma }_{\mathrm{e}-\mathrm{ph}}=51\mathrm{meV}$ is deduced from the fit. The measurements are performed on sample 2, before the creation of defects.

Figure 8

Raman spectra at $V_{\mathrm{TG}}=0$ recorded at the same location on sample 1, before any gate bias has been applied (black line) and after an electrochemical reaction has occurred (red line). The spectra are vertically offset for clarity. The dotted lines correspond to the baseline.

Figure 9

Doping dependence of the Raman features in sample 1, after the creation of defects. (a) Doping dependence of the frequency ${\omega }_{\mathrm{G}}$ (red squares, left axis) and relative FWHM $\Delta {\Gamma }_{\mathrm{G}}$ (blue circles, right axis) of the G-mode feature. The blue and red solid lines are fits based on Eqs. (5) and (6), as in Fig. 4, respectively. The fitting parameters are $C_{\mathrm{TG}}=3.4\mu {\mathrm{F}\mathrm{cm}}^{-2}, {\lambda }_{\Gamma }=0.035$, and $\delta E_{\mathrm{F}}=30\mathrm{meV}$. (b) Doping dependence of the frequencies of the 2D- (red squares, left axis) and D-mode (blue open circles, right axis) features (denoted ${\omega }_{2\mathrm{D}}$ and ${\omega }_{\mathrm{D}}$, respectively) in defective graphene.

Figure 10

(a) Integrated intensity ratio $\sqrt{\frac{I_{\mathrm{G}}}{I_{2\mathrm{D}}}}$ as a function of $E_{\mathrm{F}}$ in sample 1 before (red open squares) and after (red circles) the creation of defects. (b) Integrated intensity ratios $\frac{I_{2\mathrm{D}}}{I_{\mathrm{G}}}$ (red circles), $\frac{I_{2\mathrm{D}}}{I_{\mathrm{D}}}$ (green triangles), and $\frac{I_{\mathrm{D}}}{I_{\mathrm{G}}}$ (blue squares), as a function of $E_{\mathrm{F}}$ in sample 1, after the creation of defects. (c) $\sqrt{\frac{I_{\mathrm{G}}}{I_{2\mathrm{D}}}}$ (red circles, left axis) and $\sqrt{\frac{I_{\mathrm{G}}}{I_{\mathrm{D}}}}$ (blue squares, right axis) as a function of $E_{\mathrm{F}}$, in sample 1, after the creation of defects. As described in Sec. 6d, a defect concentration $n_{\mathrm{D}}\approx 9\times {10}^{11}{\mathrm{cm}}^{-2}$ is estimated from these measurements.

Figure 11

Doping dependence of the integrated intensity ratio $\frac{I_{\mathrm{D}}}{I_{\mathrm{G}}}$, for five sets of measurements on defective graphene samples. Each data set is normalized by the value ${\left(\frac{I_{\mathrm{D}}}{I_{\mathrm{G}}}\right)}_0$ measured near the charge neutrality point. The measured ${\left(\frac{I_{\mathrm{D}}}{I_{\mathrm{G}}}\right)}_0$ are indicated in the legend.

Figure 12

Correlation between the relative FWHM $\Delta {\Gamma }_{\mathrm{G}}$ and relative frequency $\Delta {\omega }_{\mathrm{G}}$ of the G-mode feature in doped graphene, for the five samples introduced in Fig. 5. The solid and dashed lines correspond to theoretical calculations [based on Eqs. (5) and (6)], for electron and hole doping, respectively.

Figure 13

(a) Correlation between the frequencies of the 2D- and G-mode features (relative to their values near the charge neutrality point) in doped graphene, for the five samples introduced in Fig. 5. The continuous and short-dashed lines are global fits performed on the linear portions of the hole doping and electron doping branches, respectively. The dashed line corresponds to the evolution of ${\omega }_{2\mathrm{D}}$ versus ${\omega }_{\mathrm{G}}$ under pure strain [76, 80]. Statistical distribution of the measured slopes $\frac{\partial {\omega }_{2\mathrm{D}}}{\partial {\omega }_{\mathrm{G}}}$ under hole and electron doping are shown in (b) and (c), respectively.

Figure 14

Charge carrier density $n$ as a function of the shift ${[{({\omega }_{\mathrm{G}}-{\omega }_{\mathrm{G}}^0)}^2+{({\omega }_{2\mathrm{D}}-{\omega }_{2\mathrm{D}}^0)}^2]}^{1/2}$ from the reference point corresponding to undoped graphene. Data are shown for the five samples introduced in Fig. 5. The continuous and dashed lines are global linear fits, performed in the low doping regime on the electron and hole branches, respectively.