# Transient Changes of Optical Properties in Semiconductors in Response to Femtosecond Laser Pulses
## 2 Model  
For tracing electronic and atomic dynamics we use an in-house-developed simulation tool XTANT (DESY, Hamburg, Germany, X-ray induced Thermal And Non-thermal Transitions) [13]. This is a hybrid code which combines several simulation techniques to achieve computational efficiency. It has been recently extended to treat also polyatomic systems (XTANT-2). Following the so-called \"bump-on-hot-tail\" distribution [14; 15], XTANT separates all electrons into high-energy and low-energy fraction. The high-energy electrons are modeled with Monte Carlo scheme tracing trajectories of individual particles event by event, while the low-energy electrons are traced with a temperature model including Boltzmann collision integral for the electron-ion energy exchange [13]. Matrix elements for this non-adiabatic coupling are calculated with the tight-binding method [13]. Thus, the low-energy electrons are characterized by the Fermi distribution function evolving in time, according to the changes of the electron chemical potential and temperature. The atomic motion is tracked with Ehrenfest-like molecular dynamics technique (MD). In this scheme atoms additionally exchange energy with electrons via electron-phonon coupling. The band structure of the material and corresponding potential energy surface are calculated with the transferable tight-binding (TB) method. The transferable TB method is sensitive to the changing geometry of atomic ensemble. Therefore, it can be applied for different phases or states of material. Tight-binding parameterizations for C are taken from [16]; for Si from [17]; and for GaAs from [18]. The tight-binding Hamiltonian is recalculated at every time step in accordance with the new atomic positions. In that way the model becomes complete, with a definite cycle of calculations performed at every time step [12]. For the considered pump-probe scheme with the optical probe pulses, we can apply the dipole approximation, neglecting momentum transfer to the lattice. This can be done as the wave vector of the incoming light is much smaller than any typical wave vector of electrons in the system. Frequency-dependent complex dielectric function (CDF) [19] describes optical properties of solids. In our approach we calculate CDF within a TB-based model. The tensor of complex dielectric function in the random-phase approximation at \(q=0\) (the dipole limit) reads [20]:  
\[\epsilon^{\alpha\beta}(\omega)=\delta_{\alpha,\beta}+\frac{e^{2}\,\hbar^{2}}{m ^{2}\Omega\,e_{0}}\sum_{nn^{\prime}}\frac{F_{nn^{\prime}}}{E_{nn^{\prime}}^{2 }}\frac{f_{n^{\prime}}-f_{n}}{\hbar w-E_{nn^{\prime}}+i\,\gamma}. \tag{1}\]In this equation the fundamental constants are \(e\) (the electron charge), \(\hbar\) (Planck constant), \(m\) (the mass of a free electron) and \(e_{0}\) (the vacuum permitivity; SI units). The constant \(\gamma\) is assumed to be \(\gamma=1.5\times 10^{13}\) s\({}^{-1}\) throughout the calculation [10]. Required parameters can be obtained with the tight-binding scheme, i.e., \(E_{m^{\prime}}=E_{n^{\prime}}-E_{n}\) (transition energy between two energy states); \(f_{n}\) and \(f_{n^{\prime}}\) (the corresponding transient electron occupation numbers obtained assuming their transient Fermi distribution). Parameter \(\Omega\) is the volume of the simulation box. Parameters \(F_{m^{\prime}}\) are elements of the oscillator strength matrix. Although this oscillator strength is not directly available in the tight-binding schemes, according to [PERSON] et al., it can be approximately expressed via momentum matrix elements as [20]:  
\[P(\mathbf{R_{a}},\mathbf{R^{\prime}_{a}})=\ \frac{m}{i\hbar}[\mathbf{R_{a}}- \mathbf{R^{\prime}_{a}}]H(\mathbf{R_{a}},\mathbf{R^{\prime}_{a}}), \tag{2}\]  
where \(H(\mathbf{R_{a}},\mathbf{R^{\prime}_{a}})\) is the Hamiltonian matrix, and \(R_{a}\) and \(R^{\prime}_{a}\) are the coordinates of atoms calculated by the molecular dynamics module within XTANT.