## 2 Materials and Methods
### Electron Relaxation Model
#### 2.2.2 Thermalized Carriers  
After the creation of the non-thermalized population, \(n_{\text{nt}}+p_{\text{nt}}\), photocarriers undergo electron-electron, electron-hole, and hole-hole scattering processes leading to a quasi-equilibrium of the (hot) electron and hole gases. The electron and hole distributions are approximated by quasi-Fermi-Dirac functions with transient chemical potentials (quasi-Fermi levels) \(F_{e}\) and \(F_{h}\), respectively, and a unique electronic temperature \(T\):  
\[f_{e}(\varepsilon,F_{e},T)=f_{c}(\varepsilon)=\frac{1}{1+\exp\biggl{(}\frac{ \varepsilon-F_{e}}{k_{B}T}\biggr{)}}\quad,\quad\varepsilon>0\ ; \tag{16}\]  
\[f_{h}(\varepsilon,F_{h},T)=1-f_{v}(\varepsilon)=\frac{1}{1+\exp\biggl{(}\frac{ F_{h}-\varepsilon}{k_{B}T}\biggr{)}}\quad,\quad\varepsilon<0. \tag{17}\]  
This assumption is justified if the electron-electron, electron-hole, and hole-hole scattering processes (characterized by the typical scattering time \(\tau_{ee}\)) are more probable than the electron-phonon and electron-impurity ones. Experimental results [4; 5] indicate that this situation is usually the case in typical graphene samples. The distribution functions (16) and (17) are considered slowly varying functions of time on the scale \(t\gg\tau_{ee}\). In principle, they are also functions of \(\vec{\tau}\), but we apply an averaging procedure explained below in order to avoid complications with local (spatially-dependent) chemical potentials and temperature. To model the evolution of thermalized carriers, we write down balance equations for \(n_{t}\) and \(p_{t}\) as  
\[\dot{n}_{\text{t}}(\vec{\tau},t) =\frac{n_{\text{nt}}(\vec{\tau},t)}{\tau_{ee}}+D\
abla^{2}n_{\text {t}}(\vec{\tau},t)-\frac{n_{t}(\vec{\tau},t)-n_{0}}{\tau_{r}}\ ; \tag{18}\] \[\dot{p}_{\text{t}}(\vec{\tau},t) =\frac{p_{\text{nt}}(\vec{\tau},t)}{\tau_{ee}}+D\
abla^{2}p_{\text {t}}(\vec{\tau},t)-\frac{p_{t}(\vec{\tau},t)-p_{0}}{\tau_{r}}. \tag{19}\]  
Equations (18) and (19) include a diffusion term (\(D\
abla^{2}\)) to take into account the departure of thermalized carriers from the pump spot. The importance of diffusion of hot carriers for a very sharp spatial profile of excitation has been shown experimentally [40]. Figure 2: Temporal variation of the maximum concentration of non-thermalized electrons derived from the solution of Equation (15). The inset illustrates how the peak concentration of non-thermalized carriers varies across different intensities. The parameters used in this calculation are as follows: \(\alpha_{b}=40^{\circ}\), \(\lambda_{b}=615\) nm, \(\tau_{b}\approx 85\) fs, \(w_{0}=300\) μm, \(r=0\), and \(\tau_{ee}=10\) fs. At a timescale of \(\sim\tau_{\varepsilon}\), the thermalized electrons, scattered mostly by phonons, give their energy, \(\varepsilon_{t}\), to the lattice, and the electron energy relaxation is described by the following balance equation:  
\[\dot{\varepsilon}_{t}(\vec{r},t)=\frac{\varepsilon_{\text{nt}}(\vec{r},t)}{ \tau_{ee}}+D_{\varepsilon}\
abla^{2}\varepsilon_{\text{t}}(\vec{r},t)-\frac{ \varepsilon_{t}(\vec{r},t)-\varepsilon_{0}^{\prime}(t)}{\tau_{\varepsilon}}\, \tag{20}\]  
where \(\varepsilon_{0}^{\prime}\) is the energy density at the equilibrium temperature but with a chemical potential out of equilibrium, reflecting the rapid decay of temperature compared to the quasi-Fermi levels. In (20), \(D_{\varepsilon}\) denotes the thermal diffusivity, which can be related to the diffusion coefficient; actually, with a good approximation, \(D_{\varepsilon}\approx D\), as demonstrated in Section S3 of the Supplementary Material.