Fluence and wavelength dependent ultrafast differential transmission dynamics in graphene

We performed degenerate pump-probe transmission measurements of graphene supported on glass for a range of pump fluences that enable us to observe both positive and negative deferential transmission dynamics. Our results show that at an intermediate pump fluence, where a transition from negative to positive response occurs, the differential transmission dynamics is an order of magnitude faster than at higher and lower pump fluences. This effect can be explained by equal contributions of inter- and intraband transitions with opposite signs to the transient optical conductivity of graphene at an intermediate pump fluence. Moreover, the intermediate threshold pump fluence is shown to increase with decreasing probe energy, which is in agreement with the theoretical model. Furthermore, we show that the relaxation time of the electronic temperature increases monotonically over the range of fluences studied. In perspective, this work is of importance to graphene-based opto-electronic applications such as light modulators.


Introduction
Graphene is a single layer material of carbon atoms with high carrier mobility and unique opto-electronic properties which make it useful for a wide range of applications [1][2][3][4][5][6]. The use of the advantageous properties of graphene in application to opto-electronic devices inevitably entails the generation of hot carriers with energies significantly exceeding the Fermi energy [1,2,4]. Many groups have studied graphene related relaxation dynamics using various techniques such as measurements of photocurrent, ultrafast pump-probe spectroscopy and time resolved Raman spectroscopy [7][8][9]. Ultrafast pump-probe spectroscopy has been particularly fruitful in providing valuable insights into electron-electron, electron-phonon and phonon-phonon interactions [9][10][11][12][13][14][15][16]. Typically, after electrons and holes are excited into a non-thermal distribution by an ultrafast laser pulse, they thermalize into a Fermi-Dirac distribution through Coulomb interactions in tens of femtoseconds [17]. The cooling of the hot thermal population of carriers occurs through the emission of optical phonons. When the temperatures of the electron and phonon systems equilibrate, the hot phonon bottleneck occurs, which significantly lessens the rate of cooling [14,18,19]. Subsequent cooling primarily arises from the hot optical phonons undergoing anharmonic decay into acoustic phonons. However, in the case of supported graphene, direct coupling of the charge carriers to surface phonons in polar substrates is a possible cooling channel [20][21][22][23]. As predicted by theory and measured in experiments, the time constant of hot optical phonon decay in graphene is of the order of a few picoseconds [8, 13-16, 18, 24].
In this paper CVD graphene on a soda-lime glass substrate is studied using degenerate pump-probe spectroscopy at the wavelengths of 775, 800, 825 and 850 nm. In our measurements, we examine the transition from complex DTD to fully positive DTD as a function of pump fluence. At low pump fluences, after the initial positive spike arising from the bleaching of the interband transitions, the differential transmission crosses zero and its slow recovery tail is negative due to primarily intraband absorption processes. A paper by F. Kadi et al [40] explains the observation of transient negative differential transmission, i. e. a zero-crossing of the differential transmission at about 300 fs, using a microscopic treatment of intraband absorption in graphene. At high pump fluences, the slow recovery tail of the differential transmission is positive due to predominant interband transitions. In contrast, at intermediate fluences, we observed the characteristic relaxation time of differential transmission to be an order of magnitude smaller as compared to the results observed at low and high fluences. While such an effect could be intuitively inferred from known theoretical models (for example, see A. Tomadin et al [41]), no experimental demonstration, to the best of our knowledge, has been reported. This effect can be explained by equal magnitude and opposite sign contributions of intra-and interband transitions to the differential optical conductivity of graphene. Thus, this work provides a novel experimental observation of an order of magnitude faster relaxation of differential transmission at the intermediate pump fluences, where the intraband and interband contributions are equal in magnitude and opposite in sign. It is important to note, as shown later in this paper, that by fitting the experimental data with our model, we find the relaxation time of the electron temperature to increase monotonically over the range of pump fluences used in this study. Comparison with the measured relaxation times of the differential transmission implies that pump fluence greatly modifies time-dependent optical properties, while the electron and phonon relaxation processes remain unaffected.

Results and discussion
2.1. Experimental data Single layer CVD graphene (Graphene Supermarket) was transferred onto a soda-lime glass substrate by a wet transfer method (see Methods). The quality and uniformity of the transferred graphene layer was confirmed by Raman spectroscopy using a Thermo Scientific DXR Raman Microscope. Figure 1 shows Raman spectrum averaged over ten different points on the sample with positions of G and 2D peaks being 1591.6±1.5 cm −1 and 2681±3 cm −1 , respectively. The Fermi level (n-type) was estimated to be 306 meV [42]. Ultrafast degenerate pump-probe transmission measurements were carried out using a Ti:Sapphire oscillator producing 120 fs pulses at a 76MHz repetition rate. Pump and probe wavelengths were initially both set to 800 nm (1.55 eV) and subsequently changed to 775 nm (1.60 eV), 825 nm (1.51 eV) and 850 nm (1.46 eV). Both beams were focused onto the specimen with spot diameters of 80 μm and 50 μm for pump and probe, respectively. The pump beam was chopped using a SR540 optical chopper operating at about 4 kHz. The differential transmission signal was detected using a Lock-In amplifier.
Figures 2(a) and 3(a), (c), (e) show the pump fluence dependent DTD at 800 nm, 775 nm, 825 nm and 850 nm, respectively. After the initial positive spike in the differential transmission due to Pauli blocking of the interband transitions upon photoexcitation, the slow recovery differential transmission dynamics can be either positive or negative depending on the pump fluence. At high fluences (above 20 μJ cm −2 , 25 μJ cm −2 , 33 μJ cm −2 and 39 μJ cm −2 for 775 nm, 800 nm, 825 nm and 850 nm, respectively), the differential transmission is fully positive due to predominant contribution of interband transitions [40], while for lower fluences (below 20μJ cm −2 , In order to characterize the relaxation dynamics of the DT, we fit a bi-exponential decay convoluted with Gaussian pump and probe pulses to our data using the following expression [43]: where D 1 , D 2 , t 0 , τ 1 , τ 2 are the fitting parameters. s = w 2 is the width of the autocorrelation function with σ being the width of the Gaussian pump and probe pulses. The time constants τ 1 and τ 2 refer to the fast and slow decay processes, respectively. We focus mostly on the slow relaxation time constant τ 2 , as it largely characterizes the total relaxation times of DT at most fluences. The fitting results using expression (1)

Modelling
Applying the optical boundary conditions at the air/graphene/substrate interfaces, the optical transmission t(ω) through the single layer graphene on a substrate normalized to the transmission through the substrate can be expressed as [44,45] where σ(ω) is the optical conductivity of graphene, Z 0 is the vacuum impedance and n s is the refraction index of the substrate. Since | ( )| ( )  s w + Z n 1 1 s 0 , the contribution of the imaginary part of the optical conductivity is negligible.
The optical conductivity of single layer graphene is given by [ The Fermi level E F e and temperature T of electrons in equation (4) are in general time dependent quantities. The dynamics of the electron temperature T is modelled by a bi-exponential decay (see Methods): where T 0 is the equilibrium lattice temperature, T 1 is the temperature upon photo-excitation, t t k, , e ph op serve as fitting parameters. For graphene, the heat capacity of phonons is much larger than the heat capacity of electrons. This indicates that te ph can be thought of as the electron-phonon thermalization time constant, while τ op is the hot phonon relaxation time constant (see Methods).
The initial electron temperature T 1 and the Fermi level E F e are estimated from energy conservation [37,46]:  Figure 4(b) shows the pump fluence dependence of the phenomenological optical phonon relaxation time constant τ op . Clearly, τ op does not show a negative peak at the intermediate pump fluences and increases monotonically. This indicates that the observed fast relaxation dynamics of DT is not due to a change in the physical relaxation mechanism, but due to equal magnitude contributions with opposite signs of inter-and intraband transitions to the differential optical conductivity of graphene at the intermediate pump fluences. The monotonic changes in τ op as a function of pump fluence occur as a result of ignoring the temperature dependence of the electron-phonon coupling constant and the electron and phonon heat capacities. While microscopic theories describing ultrafast dynamics in graphene without invoking any fitting parameters have been developed [40,47], the present approach is sufficient to demonstrate that an order of magnitude faster DTD at the intermediate pump fluences is well described by the model applied here accounting for intra-and interband transitions. Furthermore, an order of magnitude difference between the characteristic relaxation time of DT at the intermediate pump fluences τ 1 and the characteristic relaxation time of the electron temperature τ op indicates significant changes in the optical properties without altering the electron and phonon relaxation mechanisms. Figure 5 shows the dependence of the threshold pump fluence on the wavelength as measured in our experiments. At the threshold pump fluence, a perfect balance between intra-and interband transitions leads to an order of magnitude faster DT relaxation. Beyond this fluence interband transitions dominate the signal. As seen from figure 5, the threshold pump fluence as well as the corresponding excited carrier density are greater for lower probe energies. This behavior can be qualitatively understood based on equation (4) describing the contributions of intra-and interband transitions to the optical conductivity of graphene. The first Drude-like intraband term increases with decreasing probe energies. In other words, intraband absorption is more significant at lower probe energies. The only way to make the interband term have greater contribution is to reach higher electron temperatures so that the changes in the optical conductivity due to that interband term become greater. Hence, one needs to excite more electrons in order to reach higher temperatures and, consequently, overcome intraband absorption.

Conclusion
We have investigated ultrafast differential transmission dynamics in supported graphene on a soda-lime glass substrate using transmission pump-probe spectroscopy. The range of applied pump fluences distinctly shows different regimes of the DTD. The main result is an experimental measurement of an order of magnitude faster relaxation of the DT at the intermediate pump fluences as compared to low and high fluences. Such unconventional differential transmission relaxation times at the intermediate pump fluences are explained well within the model of intra-and interband transitions contributing with opposite signs to the differential optical conductivity of graphene. Fitting the measured differential transmission spectra with our model, which is generally based on previous models [16,37,39,46], reveals that τ op characterizing the relaxation of the electronic temperature does not show a negative peak with respect to the pump fluence, while the differential transmission relaxation time has a minimum at the intermediate pump fluences. Approximately equal contributions in magnitude and opposite in sign of inter-and intraband processes to the differential optical conductivity at the intermediate pump fluences result in an order of magnitude faster DTD as compared to high and low fluences. The value of the threshold pump fluence and the critical excited electron density, at which the faster relaxations are observed, increase with decreasing probe energies as the intraband absorption contributes more at lower probe energies. In perspective, this work demonstrates an order of magnitude faster DTD as well as the diversity of the differential transmission dynamics that can be controlled by pump fluence, which is of interest to graphene-based opto-electronic devices such as light modulators.  T  T  T  T  T   T  T T  T  T  T  T   T  T  T   ,  with G and C denoting coupling constants and heat capacities of electrons and phonons. Assuming the coefficients are constant with respect to temperature, this system of differential equations (A.1) can be solved analytically. The solution for the electron temperature is