# Optical Study on Temperature-Dependent Absorption Edge of \(\gamma\)-InSe-Layered Semiconductor
## 3 Results and Discussion
From here on, we use the term absorption coefficient (\(\alpha\)) freely in place of normalized PC intensity. It is known that \(\gamma\)-InSe is a direct bandgap material [1,3]. [PERSON] et al. and [PERSON] et al. have used the PC spectroscopy for extraction and estimation of bandgap by using Tauc-relation [16,17], which is given as follows:  
\[(\alpha hv)^{2}=\text{A}\big{(}hv-E_{g}\big{)} \tag{1}\]  
Figure 2: The normalized PC spectra of InSe at selected temperatures between 15 and 300 K.  
where _Inv_ is the photon energy, \(\alpha\) is the absorption coefficient, A is a constant, and \(E_{g}\) is the bandgap for the \(\gamma\)-InSe sample. The optical bandgap of the \(\gamma\)-InSe at 15 and 300 K can be extracted by extrapolating the straight line of the plot (_alv_)\({}^{2}\) versus energy (_Inv_) to the energy axis, as shown in Figure 3a,b. In this way, the bandgap of \(\gamma\)-InSe at 15 and 300 K is estimated to be about 1.220 \(\pm\) 0.002 and 1.152 \(\pm\) 0.003 eV, respectively. It is observed that the obtained bandgap value is less than the reported results [1; 3], which may be caused by the band tailing effect frequently observed in absorption measurement methods used to estimate bandgap energy [16]. The extracted bandgaps of \(\gamma\)-InSe at different temperatures are displayed as an open circle in Figure 4. The temperature-dependent \(E_{g}\) of \(\gamma\)-InSe are further analyzed by the empirical Varshni's relationship and Bose-Einstein empirical equation. The [PERSON] empirical relationship is written as follows [13]:  
\[E_{g}(T)=E_{g}(0)-\frac{aT^{2}}{T+\beta} \tag{2}\]  
where \(E_{g}\) (0) is the transition energy at 0 K and \(a\) and \(\beta\) are constants referred to as Varshni coefficients. The constant \(a\) is related to the electron/exciton-phonon interaction and \(\beta\) is approximately equal to the Debye temperature (\(\Theta_{D}\)) [18]. The [PERSON] empirical equation is written as shown in [14]:  
\[E_{g}(T)=E_{g}(0)-a_{B}\bigg{\{}1+\frac{2}{[\exp(\Theta/T)-1]}\bigg{\}} \tag{3}\]  
where \(a_{B}\) represents the strength of the electron/exciton-phonon interaction and \(\Theta\) corresponds to the average phonon temperature. The theoretical curve, shown as a red solid line in Figure 4a,b, from the empirical [PERSON]'s relationship and [PERSON] empirical equation, respectively, can accurately describe the variations in the temperature-dependent \(E_{g}\) of \(\gamma\)-InSe. The parameters fitted to the empirical [PERSON]'s relationship and [PERSON] empirical equation are obtained and listed in Table 1. The obtained \(\beta\) is about 279 K for [PERSON]'s equation and the average phonon temperature \(\Theta\) is about 205 K for the Bose-Einstein relation. It is known that the \(\Theta\) is closely related to the Debye temperature (\(\Theta_{D}\)) by the relation \(\Theta\ \approx\frac{3}{6}\Theta_{D}\)[18]. Hence, the estimated Debye temperature in this work is about 273 K, which is in close agreement with the fitted parameter \(\beta\). The good agreement  
Figure 3: Bandgap extraction using extrapolation at (**a**) 15 K and (**b**) 300 K.  
indicates consistency of our fitted parameters using both [PERSON]'s and [PERSON]'s empirical equation. Additionally, all PC spectra of the bulk \(\gamma\)-InSe sample at different temperatures shown in Figure 2 displayed a significant absorption tail below the absorption coefficient, which is known as Urbach's tail. Numerous theoretical and experimental studies about the origin of this tail have been presented. [PERSON] et al. have indicated that the width of the absorption tail is related to localized states in the optical gap of material, which are associated with structural and thermal disorder or a low-quality crystalline structure [PERSON] et al. (2019). [PERSON] et al. have attributed the absorption tail to the electron-exciton interaction with phonons in the layered semiconductors [PERSON] et al. (2018).