## 3 Analysis and Discussion  
PA measurements with a minimal volume cell record pressure fluctuations in the gas column located on the non-illuminated side of the sample. The measurement configuration is transmission-based, as the source and the detector are on opposite sides of the sample being examined [51]. The literature indicates that in this type of measurement configuration, the spectral function of the resulting pressure fluctuations in the closed cell can be described by a composite piston model [52]. Specifically, these pressure fluctuations are a consequence of the thermal piston effect, which involves the expansion and contraction of the thin layer of air adjacent to the illuminated surface of the sample [53; 54]. Additionally, the bending of the sample surface due to the generation of a thermal moment within the sample [52] acts as a mechanical piston, producing the thermoelastic (TE) component of the PA response. In semiconductors, due to the photogeneration of charge carriers, there exists a plasma-elastic component resulting from the bending of the sample surface caused by the concentration gradient of excess charge carriers in illuminated semiconductor samples [18; 39]. If the sample is thin, both the TE and plasma-elastic components can be neglected [22; 55]. In such cases, it can be assumed that the pressure fluctuations are caused solely by the thermal piston effect and are consequently proportional to the surface temperature variations [55; 56; 57]. However, the thermal piston model is not suitable for time-domain photoacoustics [22; 36; 58; 59; 60]. In this case, it is more appropriate to use models derived in [22; 36; 58; 59; 60], which indicate that temperature changes at the non-illuminated surface generate pressure variations in a closed PA cell. However, the thermal thickness of the gas column in the closed PA cell [36] and the transfer characteristics of the microphone [22; 36] can alter the time profile of the recorded time-resolved electrical signal. \[\overline{p}_{th}(s)=\overline{\theta}(l_{S})\overline{T}_{gas-column}(s) \tag{67}\]  
\[S_{mic}(t)=L^{-1}\{\overline{p}_{th}(s)\overline{T}_{mic}(s)\} \tag{68}\]  
As can be seen from Equations (67) and (68), the recorded photoacoustic signal is a convolution of the temperature variations at the non-illuminated side of the sample and the system's transfer characteristics, described by \(\overline{T}_{gas-column}(s)\) and \(\overline{T}_{mic}(s)\). This implies that a complete understanding of the measured time-domain signal [23] requires the deconvolution of these effects. However, in order to determine the electronic properties of the semiconductor, it is essential to first investigate the influence of photogenerated charge carriers on the surface temperature variations \(\overline{\theta}(l_{S})\). In this study, we analyze the evolution of temperature changes at the non-illuminated surface and the influence of photogenerated charge carriers on the magnitude and slope of these changes based on the derived model (hereafter referred to as the time-domain temperature signal). However, modeling the transfer function of the gas column (\(\overline{T}_{gas-column}(s)\)) and the microphone (\(\overline{T}_{mic}(s)\)), which is essential for a complete understanding and accurate processing of the recorded time-resolved PA signal in semiconductors [23], remains the focus of our future research. In the calculations, it was assumed that the optical source has a nondestructive irradiance, with the power of the employed LED diodes typically below 20 mW and the illuminated surface area approximately \(25\pi\,\mathrm{mm}^{2}\) (see Figure 1) [18; 22]. Furthermore, the excitation takes place in the visible part of the electromagnetic spectrum, where the photon energy is approximately equal to the energy bandgap of narrow-gap semiconductors such as silicon. Under these conditions, \(E/E_{\mathrm{G}}\) is close to unity [18]. The normalized components were evaluated relative to a constant reference value, \(K=S_{0}D_{T}/kl_{s}\).