# An Analytical Multiple-Temperature Model for Flash Laser Irradiation on Single-Layer Graphene
## 4 Simulations  
The input parameters given in Refs. [7; 20] have been used for simulations: heat capacity \(C_{\mathrm{e}}\) of \(3.6\times 10^{2}\) J/m\({}^{3}\)K and thermal conductivity \(K_{e}\) of 50 W/mK (see also Table 1 from Ref. [20]). The only exception is the laser intensity factor \(S_{1}\), which is equal to unity, in order to have arbitrary units. A laser spot radius equal to 0.5 \(\upmu\)m and pulses of 25 fs have been taken into account. For the electron temperature, according to Equation (15), the 3D graphic, simulated via Mathematica software, is presented in Figure 1. One should notice that here, and in sequel, \(T\) stands for temperature variation rather than the absolute temperature. Figure 1 is in very good agreement with 2D graphics in Ref. [20]. In current simulation, the electrons are heated for a very short time (tenth of fs) to the maximum temperature, similar to data in Ref. [7], but the cooling time took, more realistically, about 0.5 ps, compared to about 0.15 ps in Ref. [7]. Figure 2a shows the phonon's temperature (using Equation (21)), based upon current model, for transverse optical (TO) phonons branch. It can be observed that where the laser beam intensity is zero, the temperature distribution is zero. This shows the perfect coupling between [PERSON]'s formalism and Multiple-Temperature Model (MTM). However, thermal effects can be observed clearly beyond laser beam spot size (=1 \(\upmu\)m), as a very short laser-graphene interaction time has been choosen (of ps order). The same phenomena can be analyzed in-case of Figure 2b,c related to longitudinal optical phonons(LO) and longitudinal acoustic phonons (LA) braches, respectively. As the simulation time has been increased, from ps to ns, the heat-wave finds enough time to transfer from the irradiated spot  
Figure 1: The electrons temperature versus current coordinate and time during flash laser irradiation of a single-layer graphene. to the rest of the target. This result has been presented in Figure 2d, in case of TA branch. These results prove the trustworthiness of our modeling approach. Figure 2a-d shows the coupling of laser beam in different branches (TO, LO, LA, and TA): higher the coupling factor, the greater the temperature field intensity. It divides the results into two groups, mathematically, as follows: (i) TO and LO, and (ii) LA and TA. The output of current model is in good agreement with the one presented in Refs. [7; 20]. The observation from Ref. [7] that electrons cool-down faster than all kinds of phonons (not only optical ones) is also confirmed by current simulation. It is due to similar reasons, related to \(K_{e}\) and \(C_{e}\) values, which are higher or similar (\(K_{e}\)) to and much smaller (\(C_{e}\)) than the phonon ones. The current simulation has also the advantage that it accurately indicates the thermal decrease of acoustic phonons, which is not very pronounced in Ref. [7]. The cooling time for acoustic phonons is higher, of the order of ns vs. hundred of ps. Accordingly, the present simulational analysis is more accurate than the one presented in Refs. [7; 20]. It can be identified that the higher the coupling constant between electrons and phonons of kind \(i\), the better the concordance between current model and the one presented in Refs. [7; 20]. When the coupling constant is zero, the present model gives, as expected, a zero temperature variation, which is not the case in Refs. [7; 20]. These results have been identified as classical results, while in Refs. [7; 20], because of considering the Quantum-field effect, the temperature is not zero despite having the coupling-constant equal to zero. This is due to the Quantum-fluctuations of the vaccum and the laser beam. This study extended the original TTM into a MTM, by using phonon branch-resolved electron to phonon coupling factors. The results showed that different phonon branches are in strong nonequilibrium.