Microscopic origins of the terahertz carrier relaxation and cooling dynamics in graphene

The ultrafast dynamics of hot carriers in graphene are key to both understanding of fundamental carrier–carrier interactions and carrier–phonon relaxation processes in two-dimensional materials, and understanding of the physics underlying novel high-speed electronic and optoelectronic devices. Many recent experiments on hot carriers using terahertz spectroscopy and related techniques have interpreted the variety of observed signals within phenomenological frameworks, and sometimes invoke extrinsic effects such as disorder. Here, we present an integrated experimental and theoretical programme, using ultrafast time-resolved terahertz spectroscopy combined with microscopic modelling, to systematically investigate the hot-carrier dynamics in a wide array of graphene samples having varying amounts of disorder and with either high or low doping levels. The theory reproduces the observed dynamics quantitatively without the need to invoke any fitting parameters, phenomenological models or extrinsic effects such as disorder. We demonstrate that the dynamics are dominated by the combined effect of efficient carrier–carrier scattering, which maintains a thermalized carrier distribution, and carrier–optical–phonon scattering, which removes energy from the carrier liquid.

exponential relaxation at all substrate temperatures and all pump fluences (dashed black lines).
g-h, Experimental differential THz transmission, ∆t/t, as a function of pump-probe delay recorded at a substrate temperature of 300 K for a few different pump fluences (g) and at a pump fluence of 23.4 µJ cm −2 for a few different substrate temperatures (h) for a lightly doped MEG sample with ∼ 63 layers. The THz carrier dynamics in all lightly doped graphene samples evolve from a faster mono-exponential relaxation at room temperature to a slower bi-exponential relaxation at cryogenic temperatures (dashed black lines). samples. c-d, Maximum absolute value of theoretical differential THz transmission, maximum |∆t/t|, as a function of pump fluence at a substrate temperature of 300 K (c) and as a function of substrate temperature for a few different pump fluences (d) for disorder-free undoped graphene (|ε F | = 0 meV). Experiment and theory are in excellent agreement under all conditions. Supplementary Figure 5. THz carrier dynamics in mono-and bi-layer pCVDG. a, Carrier relaxation times extracted from fits to experimental differential THz transmission, ∆t/t, as a function of pump fluence at a substrate temperature of 300 K for highly doped mono-and bilayer pCVDG samples. b, Maximum absolute value of experimental differential THz transmission, maximum |∆t/t|, as a function of pump fluence at a substrate temperature of 300 K for highly doped mono-and bi-layer pCVDG samples.
Supplementary Figure 6. Carrier temperature and Fermi level dynamics in the microscopic theory. a-b, Carrier temperature dynamics (a) and Fermi level dynamics (b) calculated within the microscopic theory at a substrate temperature of 300 K and a pump fluence of 12.5 µJ cm −2 for disorder-free highly doped graphene (|ε F | = 300 meV). c-d, Carrier temperature dynamics (c) and Fermi level dynamics (d) calculated within the microscopic theory at a substrate temperature of 300 K and a pump fluence of 1.5 µJ cm −2 for disorder-free undoped graphene (|ε F | = 0 meV). Here, we present additional detailed description of the hot-carrier relaxation and cooling dynamics in graphene with various doping densities. A schematic illustration of the hotcarrier dynamics in graphene following ultrafast photoexcitation appears in Figure 1 in the main text, which shows how they depend critically on the Fermi level. It is important to distinguish between carrier thermalization via carrier-carrier scattering, which only redistributes the deposited energy within the electron gas, and carrier cooling via carrier-phonon scattering, which takes energy from the electron gas into the phonon system (that is, the lattice). Figure 1a-b in the main text show the hot-carrier dynamics in highly doped graphene.

Supplementary
Initially, the ultrafast optical pump pulse injects high-energy non-equilibrium electrons in the conduction band and holes in the valence band. Strong intraband and interband carriercarrier scattering lead to ultrafast carrier relaxation and thermalization which establish a single uniform hot-carrier Fermi-Dirac distribution within ∼ 100 − 200 fs after photoexcitation [1][2][3]. Hot-carrier cooling proceeds further via optical phonon emission on a timescale of a few picoseconds. It is essentially facilitated by these extraordinarily efficient Coulomb interactions which continuously rethermalize the hot-carrier distribution and replenish the carriers at high energies which emit optical phonons.
The situation in undoped (very lightly doped) graphene is very different, as shown in Figure 1c in the main text. Immediately after photoexcitation, the phase space for impact ionization is large, while Auger recombination is inhibited, which leads to significant carrier multiplication in the conduction band up to a moderate excitation regime [4,5]. Again, the strong intraband and interband carrier-carrier scattering establish a single uniform hotcarrier Fermi-Dirac distribution on an ultrafast timescale [1][2][3] followed by hot-carrier cooling via optical phonon emission. At later times, the small phase space near the Dirac point strongly reduces the efficiency of carrier rethermalization via the Coulomb interactions, which slows down optical phonon emission and hot-carrier cooling which last on a timescale exceeding hundreds of picoseconds at cryogenic temperatures.
The hot-carrier relaxation and cooling processes illustrated in Figure 1 in the main text are directly accessible via the dynamic response of the system to a THz probe pulse. Qualitatively, for all types of graphene having high doping density, we observe in both experiment and theory a fast and substrate-temperature-independent THz carrier dynamics. On the other hand, for graphene having very low doping density, we observe a comparatively slower and strongly substrate-temperature-dependent THz carrier dynamics. A second feature of our experiments is that we find that graphene samples with very different degrees of disorder show essentially the same hot-carrier relaxation times; this suggests that disorder-assisted electron-phonon (supercollision) scattering might not play a crucial role in the hot-carrier dynamics as has been proposed in previous literature [6][7][8]. We show that the dependence of the dynamic THz response on Fermi level, substrate temperature and initial carrier temperature, and its independence of disorder, can be quantitatively understood in the microscopic theory.

Supplementary Note 2. THz carrier dynamics in various types of graphene
Here, we present additional experimental data on the THz carrier dynamics in all types of graphene samples that we have studied to further illustrate the conclusions in the main text. Supplementary Figures 1 and 2 show the differential THz transmission signal at the peak of the THz probe pulse normalized to the THz transmission without photoexcitation, ∆t/t, as a function of pump-probe delay, for variable pump fluence and for variable substrate temperature, respectively, for a highly doped MEG sample (a-b), a highly doped sCVDG sample (c-d), a highly doped pCVDG sample (e-f) and a lightly doped MEG sample (g-h).
Based on extensive measurements on many graphene samples under various conditions, we observe that the differential THz transmission is positive for all highly doped graphene samples, which corresponds to a pump-induced increase of the THz transmission or a decrease of the THz absorption, and negative for all lightly doped graphene samples, which corresponds to a pump-induced decrease of the THz transmission or an increase of the THz absorption, under all experimental conditions. In addition, the differential THz transmission of all highly doped graphene samples follows closely a fast and substrate-temperature-independent mono-exponential relaxation (dashed black lines in Supplementary Figures 1 and 2) with relaxation times on the order of a few picoseconds that vary from sample to sample within ∼ 20 − 30%. These THz carrier dynamics are very well explained by the microscopic theory of carrier-carrier and carrier-phonon interactions in a disorder-free graphene system. On the other hand, the differential THz transmission of all lightly doped graphene samples follows closely a comparatively slower and strongly substrate-temperature-dependent bi-exponential relaxation (dashed black lines in Supplementary Figures 1 and 2) with a short and a long relaxation component governed by two distinct cooling mechanisms. The short relaxation times are on the order of a few to a few tens of picoseconds varying similarly from sample to sample within ∼ 20 − 30% and are also very well explained by the microscopic theory.
The long relaxation times extend up to on the order of hundreds of picoseconds monotonically increasing with the number of epitaxial graphene layers and are attributed to a unique cooling mechanism enabled by interlayer energy transfer via screened Coulomb interactions [9]. Supplementary Table 1 summarizes the key characteristics of the THz carrier dynamics in the various graphene samples.

Supplementary Note 3. Maximum signal in the THz carrier dynamics
Here, we present additional evidence to further illustrate the excellent agreement between the experimental data and the theoretical calculations of the THz carrier dynamics in all types of graphene samples that we have studied. Supplementary Figure 3a Figure 6a in the main text. It is apparent (see the dashed black line in Figure 6a in the main text) that if the effect of the optical pump was only to heat the carriers without changing the Fermi level, then a negative sign of the ∆t/t signal would always be observed. We conclude that, for a constant scattering rate Γ, carrier heating alone cannot explain the positive sign of the ∆t/t signal observed in highly doped graphene. Figure 6a in the main text also illustrates the pump-induced temporal evolution of T (t) and ε F (t) obtained by solving the full graphene Bloch equations within this approximation for low (12.5 µJ cm −2 (red line)) and high (80 µJ cm −2 (blue line)) photoexcitation. Interestingly, we find not only carrier heating, but also a transient decrease of the Fermi level which reduces the THz absorption. Obviously, the pump-induced shift of the Fermi level outweighs the impact of carrier heating. As a consequence, we find a positive sign of the ∆t/t signal for the lower pump fluence (red line) at zero time delay and for the higher pump fluence (blue line) just after 0.8 ps. Thus, the positive ∆t/t signal for highly doped graphene observed in the measurements can be explained to a large extent by a standard Drude model with constant Γ, provided that the carrier heating and cooling dynamics and the Fermi level dynamics are described correctly. Previously, the observation of a positive ∆t/t signal in doped graphene has been attributed phenomenologically to a pump-induced increase of the carrier scattering rate, and a corresponding decrease of the THz conductivity within a Drude model [10][11][12], which is essentially a metal-like behavior, although the mechanism responsible for the increase in the carrier scattering rate could not be positively identified. Here, we find that the positive ∆t/t signal can be partly explained within a simple Drude picture as a consequence of the Fermi level shift. However, the simple Drude model is not sufficient to explain the behavior at higher pump fluence, for which the short time dynamics would exhibit a negative ∆t/t signal, unless other phenomenological parameters such as a carrier heating efficiency [13,14] or a non-monotonic carrier-temperature-dependent Drude weight [15] are included.
A complete understanding is obtained by applying the full microscopic formalism including the explicitly time-and momentum-dependent scattering rates Γ 0 λk (t) of the probeinduced carrier dynamics. The results are shown in Figure 2c-d in the main text, exhibiting excellent agreement with the experiment under all conditions. Specifically, we observe, in both experiment and theory, an overall positive ∆t/t signal at all time delays and all pump fluences. Thus, the microscopic model for the pump-induced carrier scattering is essential to capture the very initial THz carrier dynamics correctly. In the regime of high fluence and short time, even if carrier thermalization is complete, a constant carrier scattering rate approximation cannot be trusted. In particular, we find that in the strong excitation regime efficient time-dependent Coulomb scattering provides the main contribution. All results presented in the comparisons with experiment are obtained from the full microscopic formalism beyond the standard Drude model.