## 4 Concluding Remarks  
In this paper, we attempted to gain further insight into the intriguing behavior of hot-electron dynamics in graphene, a subject of fundamental importance in understanding its behavior under high-intensity optical fields. We presented a comprehensive model that captures the interplay between photogenerated non-thermalized and hot carriers, clarifying their temporal evolution as a function of various experimental parameters.  
The conventional perturbative theory used to examine nonlinearities in graphene is suitable only for low pumping intensities. However, when dealing with stronger optical pulses, this approach fails as the electrons attain temperatures and Fermi levels significantly deviating from equilibrium values. To be specific, we can use a simple criterion, which provides a quantitative measure of pump intensity at which the hot electron effect becomes significant, based on the comparison between the effective temperature of the hot-electron gas and that of the lattice (\(T_{0}\)), \((T(t)-T_{0})/T_{0}>1\). Section S6 of the Supplemental Material shows that, even for low \(I_{p}\sim 0.1\) GW/cm\({}^{2}\), the perturbative approach is not suitable for the description of the nonlinearities in graphene as the electronic temperature is about twice as high as \(T_{0}\), so that the hot-electron effects become very important. Moreover, the developed electron relaxation model with little modifications can be employed for understanding other types of experiments, including, for instance, dynamics of electrically injected electrons out of equilibrium which is important for graphene-based photodetectors [10; 51].  
We applied the model to the results obtained in pump-probe experiments aimed at an all-optical generation of graphene surface plasmons [7] and achieved a fairly good agreement for the differential reflectance in the off-resonance situation (where no plasmons are generated). It has been crucial to verify the reliability of the model for a better understanding and theoretical description of the more complex situation where surface plasmons are indeed generated by two beams of slightly unequal frequencies. As suggested in a recent work [52], the resonant frequency-difference process [30; 31; 53] is supported by the hot-electrons-mediated generation of hot plasmons [46] to yield the otherwise unexplainable strong effect observed by [PERSON] et al. [7]. We notice that the distinction and separate consideration of non-thermalized and thermalized (hot) photocarriers is important for considering the situation where the pump and probe pulses have equal or near-central wavelengths, as was the case of the considered experiments.  
The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app14031250/s1](https://www.mdpi.com/article/10.3390/app14031250/s1)]([https://www.mdpi.com/article/10.3390/app14031250/s1](https://www.mdpi.com/article/10.3390/app14031250/s1)). References [54; 55; 56] are cited in the supplementary materials.  
Conceptualization, M.I.V.; Software, D.F.P.C. and R.D.; Validation, M.J.L.F.R.; Investigation, R.D.; Data curation, M.J.L.F.R.; Writing--original draft, D.F.P.C.; Writing--review & editing, M.I.V.; Supervision, M.I.V. All authors have read and agreed to the published version of the manuscript.  
We are thankful to the Portuguese Foundation for Science and Technology (FCT) for funding in the framework of the Strategic Funding UIDB/04650/2020. Additionally, R.D. and D.C. acknowledge their respective MSc and PhD grants from the FCT.  
Not applicable.  
Data Availability Statement: Data are contained within the article.  
Discussions with [PERSON], [PERSON] and [PERSON] are appreciated.  
The authors declare no conflicts of interest.  
References