## 2 Theoretical Model  
### Ultrafast Dynamics  
It is known that during laser irradiation of a material, various physical processes occur on a femtosecond timescale. In this work, excitation of a semiconducting material, rutile TiO\({}_{2}\) with (001) and (100) surface orientation was considered through a laser beam of wavelength \(\lambda_{L}\) = 248 nm that corresponds to photon energies equal to \(\sim\)5 eV which is higher than the band gap of the material (\(\sim\)3.0 eV, at 300 K); thus, it is assumed that predominantly one- and (secondly and with a substantially smaller probability) two-photon absorption mechanisms contribute to excitation of the charge carriers in the valence and conduction bands, while higher order photon processes are less likely to occur. Furthermore, linear free carrier photon absorption on the intraband transitions are possibly increase the carrier energy (but not the carrier density), while Auger recombination and impact ionization processes lead to a decrease and increase, respectively, of the number of carriers in the conduction band. The relaxation time approximation to [PERSON]'s transport equation [22; 26; 56; 57; 58] is employed to determine the spatial (\(\overrightarrow{r}~{}=~{}(x,y,z)\), in Cartesian coordinates) and temporal dependence (\(t\)) of the carrier density number, carrier energy and lattice energy; more specifically, the following set of coupled energy and particle balance equations are used to describe the temporal variation of the carrier density number \(N_{c}\), carrier temperature \(T_{c}\) and lattice temperature \(T_{L}\):  
\[ \begin{array}{l}C_{c}\frac{\partial T_{c}}{\partial t}=-\frac{C_{c}}{\tau_{ c-l}}(T_{c}-T_{L})+S(\overrightarrow{r},t)\\ C_{L}\frac{\partial T_{L}}{\partial t}=\vec{\
abla}\cdot\left(K_{L} \overrightarrow{r}T_{L}\right)+\frac{C_{c}}{\tau_{c-l}}(T_{c}-T_{L})\\ \frac{\partial N_{c}}{\partial t}=\frac{\alpha_{SPA}}{\hbar\omega_{L}}I( \overrightarrow{r},t)+\frac{\beta_{TPA}}{2\hbar\omega_{L}^{2}}I( \overrightarrow{r},t)-\gamma N_{c}{}^{3}+\theta N_{c}\end{array}  \tag{1}\]  
where \(C_{c}\) (\(C_{L}\)) is the carrier (lattice) specific heat, \(K_{L}\) corresponds to the lattice heat conductivity, \(\hbar\omega_{L}\) stands for the photon energy and \(\alpha_{SPA}\) and \(\beta_{TPA}\) correspond to the single and two-photon absorption coefficients, respectively. \(C_{c}\) is provided by the expression \(C_{c}~{}=~{}3N_{c}k_{B}\), where \(k_{B}\) stands for the Boltzmann constant. The lattice specific heat and heat conductivity for TiO\({}_{2}\) are \(C_{L}\) = 690 J Kgr\({}^{-1}\) K\({}^{-1}\)[59] and \(K_{L}\) = 13 Wm\({}^{-1}\) K\({}^{-1}\)[60], respectively. At \(\lambda_{L}\) = 248 nm, \(\alpha_{SPA}\sim 10^{6}~{}\times~{}\exp(T_{L}/300)~{}\)cm\({}^{-1}\)[61] while \(\beta_{TPA}\sim\)37 cm/GW [62; 46]. In Equation (1), \(\tau_{c-l}\) is the carrier-phonon energy scattering time which is derived from \(1/\tau_{c-l}\simeq 10^{4}N_{c}^{1/3}\) s\({}^{-1}\)[50]. Finally, \(\gamma\)is the coefficient for Auger recombination (\(\gamma\)\(\sim\)10\({}^{-32}\) cm\({}^{6}\)/s [63]) and \(\theta\)is the impact ionization rate coefficient. In this work, an intensity dependent expression is used for the impact ionization parameter \(\theta\)[33; 64] (unlike an approximating formula used for other semiconductors and at higher laser wavelengths, i.e., \(\sim e^{-E_{g}/k_{B}T_{c}}\) s\({}^{-1}\), for Silicon at 800 nm [22; 57]. \[\theta= \frac{e^{2\tau_{c}I}}{c_{\varepsilon_{0}mn_{r}E_{g}\left({\omega_{L}}^{2 }(\tau_{c})^{2}+1\right)(2-m_{r}/m_{c})}} \tag{2}\]  
In Equation (2), \(c\) is the speed of light, \(e\) is the electron charge, \(\varepsilon_{0}\) stands for the vacuum permittivity, \(m_{r}\) (i.e., \(m_{r}=0.5m_{e}\)) is the reduced electron mass and \(n\) is the refractive index of the material, while \(I\) is the peak intensity of the laser beam, and \(\tau_{c}\) is the electron collision time.