# Optical Study on Temperature-Dependent Absorption Edge of \(\gamma\)-InSe-Layered Semiconductor
## 3 Results and Discussion
To understand the effects of defect states or electronic/exciton-phonon interaction in the bulk \(\gamma\)-InSe sample, the absorption coefficient near the absorption edge is further investigated by the Urbach's tail model as an exponential relation form, as shown in [PERSON] et al. (2019); [PERSON] et al. (2020):  
\[\alpha\propto\alpha_{0}\exp(\hbar v/E_{\mathrm{U}}) \tag{4}\]  
where \(\alpha_{0}\) is a constant, and \(E_{\mathrm{U}}\) is the Urbach energy that has the dimensions of energy to describe the distribution of tail states extending into the forbidden gap. As shown in Figure 5, the Urbach's tail in the absorption edge for the \(\gamma\)-InSe at 15 K and 300 K can be well described by the red solid line calculated using Equation (4). According to the fits, the Urbach energy \(E_{\mathrm{U}}\) as a function of temperature is extracted and plotted in Figure 6a. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{[PERSON]} & \(E_{g}(0)\) (eV) & \(a\) (eV/K) & \(\beta\) (K) \\ \cline{2-4}  & 1.221 \(\pm\) 0.002 & 4.6 \(\times\) 10\({}^{-4}\) & 279 \(\pm\) 10 \\ \hline \multirow{2}{*}{Bose–Einstein} & \(E_{g}(0)\) (eV) & \(a_{B}\) (meV) & \(\Theta\)(K) \\ \cline{2-4}  & 1.220 \(\pm\) 0.002 & 20.6 \(\pm\) 0.5 & 205 \(\pm\) 10 \\ \hline \end{tabular}
\end{table} 
Table 1: Values of the parameters obtained from a fit to the [PERSON] and Bose–Einstein equation. Figure 4: The experimental values of the temperature dependence of \(E_{g}(T)\) for InSe. Solid lines represent the best fit to (**a**) Varshni and (**b**) the Bose–Einstein equation. Furthermore, the Urbach energy can be expressed by an inverse logarithmic slope relation \(E_{\text{U}}=k_{B}T/\sigma(T)\), where \(k_{B}\) is the Boltzmann constant and the \(\sigma(T)\) is called the steepness parameter [19; 20]. Figure 6b shows the temperature-dependent steepness parameter, which is derived from the Urbach energy as plotted in Figure 6a. The relationship between the steepness parameter \(\sigma\), temperature, and phonon energy is discussed in detail by [19; 20] and is written as follows:  
\[\sigma(T)=\sigma_{0}\frac{2k_{B}T}{h\
u_{p}}\text{tanh}(\frac{h\
u_{p}}{2k_{B} T}) \tag{5}\]  
Figure 5: Urbach’s absorption tail in the PC spectrum at 15 and 300 K for InSe. Solid lines represent the best fit to Equation (4). Figure 6: (**a**) Urbach energy (\(E_{\text{U}}\)) and (**b**) steepness parameter (\(\sigma\)) as a function of the temperature for InSe. The solid line represents the best fit to Equations (5) and (6) for the temperature-dependent Urbach energy and steepness parameter, respectively. where \(\sigma_{0}\) is a temperature-independent but material-dependent constant. The \(h\upsilon_{p}\) is attributed to the effective phonon energy in the single-oscillator model describing certain information about the interaction of electrons and/or excitons with phonons in semiconductors. The red solid line displayed in Figure 6b represents the best fit to the derived \(\sigma(T)\) data and the estimated value of \(h\upsilon_{p}\) is about 52 meV, which is the effective average energy of phonons contributing to the Urbach tails. It has to be noted that the obtained value of \(h\upsilon_{p}\) is almost double that of the actual phonon energy of about 28 meV, which is calculated using the observed highest optical mode (A\({}_{1g}^{2}\) at 229 cm\({}^{-1}\)) in the Raman spectrum shown in Figure 1b. The larger \(h\upsilon_{p}\) indicates that in addition to the electron/exciton-phonon interaction, extra effects also contribute significantly to the effective phonon energy. Similar results have been investigated in previous reports [10; 20; 21]. [PERSON] et al. have attributed the origin in the higher value of \(h\upsilon_{p}\) observed in mixed chalcogenides to the structural disorders caused by cation-cation disorder, cation vacancies, and interstitials [10]. [PERSON] et al. and [PERSON] et al.