Hot carrier dynamics and electron-optical phonon coupling in photoexcited graphene via time-resolved ultrabroadband terahertz spectroscopy

Electron-electron (e-e) interaction is known as a source of logarithmic renormalizations for Dirac fermions in quantum field theory. The renormalization of electron-optical phonon coupling (EPC) by e-e interaction, which plays a pivotal role in hot carrier and phonon dynamics, has been discussed since the discovery of graphene. We investigate hot carrier dynamics and EPC strength using time-resolved ultrabroadband terahertz (THz) spectroscopy combined with numerical simulation based on the Boltzmann transport equation and a comprehensive temperature model. The numerical simulation demonstrates that the extrinsic carrier scatterings by the Coulomb potential of the charged impurity and surface polar phonons are significantly suppressed by the carrier screening effect and have negligible contributions to the THz photoconductivity in heavily doped graphene on polyethylene terephthalate (PET) substrate. The large negative photoconductivity and the non-Drude behavior of THz conductivity spectra appear under high pump fluence and can be attributed to the temporal variation of the hot carrier distribution and scattering rate. The transient reflectivity well reflects the EPC strength and temporal evolution of the hot carrier and optical phonon dynamics. We successfully estimate the EPC matrix element of the $A_1^{\prime }$ optical phonon mode near the $\mathbf{K}$ point as ${\langle D_{\mathbf{K}}^2\rangle }_{\mathrm{F}}\approx 450$ ${(\mathrm{eV}/\AA )}^2$ from the fitting of THz conductivity spectra and temporal evolution of transient THz reflectivity. The corresponding dimensionless EPC constant ${\lambda }_{\mathbf{K}}\approx 0.09$ at Fermi energy ${\varepsilon }_{\mathrm{F}}=0.43\mathrm{eV}$ is slightly larger than the prediction of the renormalization group approach including the dielectric screening effect of the PET substrate. This leads to a significant difference in hot carrier and phonon dynamics compared with those without the renormalization effect by the e-e interaction. This approach can provide a quantitative understanding of hot carrier and optical phonon dynamics and support the development of future graphene optoelectronic devices.

Figure 1

Simulation results of heavily doped graphene with $|{\varepsilon }_F|=0.43\mathrm{eV}$ for $F_0=100$ $\mu \mathrm{J}/{\mathrm{cm}}^2$. Temporal evolutions of $T_e$ and $T_{\eta }$ for (a) ${\langle D_{\mathbf{K}}^2\rangle }_{\mathrm{F}}=193$ and (b) ${\langle D_{\mathbf{K}}^2\rangle }_{\mathrm{F}}=703$ ${(\mathrm{eV}/\AA )}^2$. (c) ${\langle D_{\mathbf{K}}^2\rangle }_{\mathrm{F}}$ dependence of $\mathrm{\Delta }{E}_{\mathrm{r}}\left({\tau }_1\right)/E_0$ of graphene calculated using temporal waveforms of the THz probe pulse expressed by the second derivative of the Gaussian function $exp(-t^2/{\tau }_{\mathrm{pump}}^2)$ with pulse durations of $2{\tau }_{\mathrm{pump}}=300\mathrm{fs}$. (d) $\mathrm{\Delta }{E}_{\mathrm{r}}\left({\tau }_1\right)/E_0$ at ${\langle D_{\mathbf{K}}^2\rangle }_{\mathrm{F}}=193$ ${(\mathrm{eV}/\AA )}^2$ for $n_{\mathrm{i}}=0$ and $1.7\times {10}^{12}{\mathrm{cm}}^{-2}$.

Figure 2

Simulation results of lightly doped graphene with $|{\varepsilon }_F|=0.15\mathrm{eV}$. Temporal evolutions of $T_e$ and $T_{\eta }$ for (a) ${\langle D_{\mathbf{K}}^2\rangle }_{\mathrm{F}}=92.0$ and (b) ${\langle D_{\mathbf{K}}^2\rangle }_{\mathrm{F}}=703$ ${(\mathrm{eV}/\AA )}^2$ for $F_0=100$ $\mu \mathrm{J}/{\mathrm{cm}}^2$. (c) ${\langle D_{\mathbf{K}}^2\rangle }_{\mathrm{F}}$ dependence of $\mathrm{\Delta }{E}_{\mathrm{r}}\left({\tau }_1\right)/E_0$ of graphene. (d) $\mathrm{\Delta }{E}_{\mathrm{r}}\left({\tau }_1\right)/E_0$ at ${\langle D_{\mathbf{K}}^2\rangle }_{\mathrm{F}}=193$ ${(\mathrm{eV}/\AA )}^2$ for $n_{\mathrm{i}}=0$ and $0.17\times {10}^{12}{\mathrm{cm}}^{-2}$.

Figure 3

$T_e$ dependence of (a) chemical potential $\mu \left(T_e\right)$ and (b) Drude weight $D\left(T_e\right)$ of graphene with ${\varepsilon }_{\mathrm{F}}=0.01$, 0.15, and $0.43\mathrm{eV}$.

Figure 4

(a) $(r_{\mathrm{p}}/r_{\mathrm{s}})$ of graphene at equilibrium measured by THz-TDSE. (b) $\sigma \left(\omega \right)$ at equilibrium. The dashed curves are the fitting curves of the simple Drude model.

Figure 5

(a) Schematic of reflection-type OPTP setup. Pump, pump pulse; Trigger, trigger pulse; SHG, second-harmonic generation; HV, high voltage; PMT, photomultiplier tube; sub., substrate. (b) Temporal waveforms of THz probe pulse measured at ${\tau }_1=$ 0.1 ps. (c) Frequency dependence of $[r_{\mathrm{s}}^{\prime }(\omega ,{\tau }_1)/r_{\mathrm{s}}\left(\omega \right)]$ at ${\tau }_1$ = 0.1, 3.1, 6.6, and 8.6 ps at $F_0=200$ $\mu \mathrm{J}/{\mathrm{cm}}^2$. (d) Pump fluence dependence of $\mathrm{\Delta }E\left({\tau }_1\right)/E_0$.

Figure 6

Comparison of $\mathrm{\Delta }{E}_{\mathrm{r}}\left({\tau }_1\right)/E_0$ between experiment and calculations for different EPCs for pump fluence (a $F_0=50$, (b) $F_0=100$, and (c) $F_0=200\mu \mathrm{J}/{\mathrm{cm}}^2$. The red open circles represent the experimental $\mathrm{\Delta }{E}_{\mathrm{r}}\left({\tau }_1\right)/E_0$. The solid curves correspond to the $\mathrm{\Delta }{E}_{\mathrm{r}}\left({\tau }_1\right)/E_0$ calculated using ${\langle D_{\mathrm{\Gamma }}^2\rangle }_{\mathrm{F}}=92.0(\mathrm{DFT},\mathrm{black})$, $193(\mathrm{GW},\mathrm{red})$, $325\left(\mathrm{green}\right)$, 450 (magenta), 703 (dark blue), and 946 (light blue) ${(\mathrm{eV}/\AA )}^2$, respectively, under the maximum $J_{\mathrm{sc}}$ condition.

Figure 7

Pump fluence dependence of $\sigma (\omega ,{\tau }_1)={\sigma }_1(\omega ,{\tau }_1)+i{\sigma }_2(\omega ,{\tau }_1)$ (orange symbols) of heavily doped graphene obtained from for (a) and (b) $F_0=100$, (c) and (d) $F_0=200$, and (e) and (f) $F_0=300$ $\mu \mathrm{J}/{\mathrm{cm}}^2$. The gray solid and dashed curves correspond to the fitting curve of $\sigma (\omega ,{\tau }_1)$ at equilibrium shown in Fig. 4. The black, red, magenta, green, and blue curves correspond to $\sigma (\omega ,{\tau }_1)$ at ${\tau }_1=0.1$ ps calculated using ${\langle D_{\mathbf{K}}^2\rangle }_{\mathrm{F}}=92.0$, 193, 450, 703, and 946 ${\left(\mathrm{eV}{\AA }^{-1}\right)}^2$, respectively, under the maximum $J_{\mathrm{sc}}$ condition.

Figure 8

Temporal evolution of $T_e$ and $T_{\eta }$ calculated for (a) ${\langle D_{\mathbf{K}}^2\rangle }_{\mathrm{F}}=92.0$ and (b) ${\langle D_{\mathbf{K}}^2\rangle }_{\mathrm{F}}=450$ ${(\mathrm{eV}/\AA )}^2$ under the maximum $J_{\mathrm{sc}}$ condition. The magenta curve is the absorbed pump intensity ${\mathcal{I}}_{\mathrm{ab}}$, calculated considering the SA effect.

Figure 9

Flow of dimensionless coupling constants ${\lambda }_{\mathrm{\Gamma }}$ and ${\lambda }_{\mathbf{K}}$ (three dashed and solid curves, respectively) for three values of ${\epsilon }_{\mathrm{av}}=1$, 2, and 5. The red and blue symbols correspond to the ${\lambda }_{\mathbf{K}}$ determined in this study (${\epsilon }_{\mathrm{av}}=2$) and by Raman studies (${\epsilon }_{\mathrm{av}}=5$) from Ref. [49] (blue open square and circle) and Ref. [127] (blue open triangle). The black open square and circle correspond to ${\lambda }_{\mathbf{K}}$ by the DFT [44, 49] and GW [47, 49, 55] calculations (${\epsilon }_{\mathrm{av}}=1$), respectively.

Figure 10

Schematic of systems considered in the THz-TDSE and OPTP measurements, showing the incident THz probe or optical pump pulse.

Figure 11

(a)–(f) Pump intensity $I_{\mathrm{pump}}$ dependence of ${\alpha }_{\mathrm{inter}}$ and $A_{\mathrm{ij}}^s$ at $\theta ={60}^{\circ }$ in heavily doped graphene with $|{\varepsilon }_{\mathrm{F}}|=0.43\mathrm{eV}$, assuming ${\epsilon }_2=2.4$ for various ${\tau }_{\mathrm{ie}}$ and $T_e$. The $F_0=I_0\times 2{\tau }_{\mathrm{pump}}$ on the upper horizontal axis was calculated by assuming $2{\tau }_{\mathrm{pump}}=220$ fs in the OPTP experiment.

Figure 12

$T_e$ dependence of saturation pump intensity $I_s$ of heavily doped graphene with $|{\varepsilon }_{\mathrm{F}}|=0.43\mathrm{eV}$, assuming ${\epsilon }_2=2.4$ for (a) ${\alpha }_{\mathrm{inter}}$ and (b) $A_{\mathrm{ij}}^s$ at $\theta ={60}^{\circ }$. The $F_{\mathrm{s}}=I_{\mathrm{s}}\times 2{\tau }_{\mathrm{pump}}$ on the right vertical axis was calculated by assuming $2{\tau }_{\mathrm{pump}}=220$ fs in the OPTP experiment.

Figure 13

(a) Envelope function of pump fluence $F_0$ incident on graphene at $\theta ={60}^{\circ }$. Absorbed pump intensity ${\mathcal{I}}_{\mathrm{ab}}$ in graphene with (b) $|{\varepsilon }_{\mathrm{F}}|=0.15$ and (c) $|{\varepsilon }_{\mathrm{F}}|=0.43$ eV at $T_e=295$ K using ${\epsilon }_2=2.4$.

Figure 14

(a) Temporal evolution of $T_e$ and $T_{\eta }$ of heavily doped graphene with $|{\varepsilon }_{\mathrm{F}}|=0.43\mathrm{eV}$. (b) Temporal waveform of THz probe pulse with $2{\tau }_{\mathrm{prob}}=100$, 300, and $600\left(\mathrm{fs}\right)$ used in the simulation. (c) Corresponding Fourier spectrum of THz probe pulse. Temporal evolution of $\sigma (\omega ,{\tau }_1)$ at ${\tau }_1=-1.0$, 0.1, 1.0, 2.0, and $4.0\mathrm{ps}$ calculated using the THz probe with (d) $2{\tau }_1=600$, (e) $2{\tau }_1=300$, and (f) $2{\tau }_1=100$ fs.

Figure 15

${\langle D_{\mathbf{K}}^2\rangle }_{\mathrm{F}}$ dependence of $\mathrm{\Delta }{E}_{\mathrm{r}}\left({\tau }_1\right)/E_0$ of heavily doped graphene calculated using the THz probe pulse with $2{\tau }_{\mathrm{prob}}=600$, 300, and $100\mathrm{fs}$ for $F_0=100$ $\mu \mathrm{J}/{\mathrm{cm}}^2$.

Figure 16

${\langle D_{\mathbf{K}}^2\rangle }_{\mathrm{F}}$ dependence of $T_e$ of heavily doped graphene for $F_0=200$ $\mu \mathrm{J}/{\mathrm{cm}}^2$ under different parameter conditions. (a) $J_{\mathrm{a}}=10\mathrm{eV}$, ${\rho }_{\mathrm{s}}=40\mathrm{\Omega }$, and ${\tau }_{\mathrm{ph}}=0.5\mathrm{ps}$. (b) $J_{\mathrm{a}}=30\mathrm{eV}$, ${\rho }_{\mathrm{s}}=100\mathrm{\Omega }$, and ${\tau }_{\mathrm{ph}}=0.57\mathrm{ps}$.

Figure 17

Fitting of $\mathrm{\Delta }{E}_{\mathrm{r}}\left({\tau }_1\right)/E_0$ of heavily doped graphene under different parameter conditions. (a) $J_{\mathrm{a}}=10\mathrm{eV}$, ${\rho }_{\mathrm{s}}=40\mathrm{\Omega }$, and ${\tau }_{\mathrm{ph}}=0.5\mathrm{ps}$. (b) $J_{\mathrm{a}}=30\mathrm{eV}$, ${\rho }_{\mathrm{s}}=100\mathrm{\Omega }$, and ${\tau }_{\mathrm{ph}}=0.57\mathrm{ps}$.

Figure 18

Comparison of ${\sigma }_1(\omega ,{\tau }_1)$ of heavily doped graphene for $F_0=200$ $\mu \mathrm{J}/{\mathrm{cm}}^2$ under different parameter conditions. (a) $J_{\mathrm{a}}=10\mathrm{eV}$, ${\rho }_{\mathrm{s}}=40\mathrm{\Omega }$, and ${\tau }_{\mathrm{ph}}=0.5\mathrm{ps}$. (b) $J_{\mathrm{a}}=30\mathrm{eV}$, ${\rho }_{\mathrm{s}}=100\mathrm{\Omega }$, and ${\tau }_{\mathrm{ph}}=0.57\mathrm{ps}$.