**4. Discussion**

The amplitude and phase spectra courses for the ideal semiconductor are known [25]. To measure against them, the temperature distribution in the tested sample and the waveforms for the absorption coefficient below and above the band gap were required. The expression for the absorption coefficient is one of the parameters of the temperature distribution. The expression for the phase and amplitude of the photothermal signal depends on this distribution. Details are presented in prior work [25].

The changes observed in the experimental spectra were interpreted as originating from defects located on the surface of the samples, as they were observed to be strongly dependent on the method of surface preparation. To simulate the changes, it was assumed that the defects could be simulated in temperature distributions with Gauss-shaped expressions. In the simulations, parameters such as energy, thermal diffusivity, and thermal conductivity were determined, and their courses were compared with the obtained experimental spectra.

The modified Blonskij's model [26,29,30] was applied to interpret the obtained spectra. The heat conduction equation describes the sample's temperature distribution as previously described [30]. The modified model considers a damaged subsurface layer on both surfaces of the sample with significantly different thermal parameters from the main part of the sample. The absorption coefficient due to the presence of the defects has the Gaussian character.

$$\beta = A\_d \cdot \exp\left(\frac{E - E\_d}{\beta\_1}\right)^2 \tag{1}$$

where *Ed* is the value of the energy of the defect, *β*<sup>1</sup> is the parameter describing the width of Gaussian shape maximum, and *Ad* is the amplitude of the maximum.

The defect is located at the surfaces of the sample, and its nature can be associated with the quality of the surface after the preparation process (grounding, polishing, etching). For the energy *Ed* of radiation, the layer strongly absorbs, while the volume of the sample is transparent. The temperature distribution in the sample is the sum of temperatures generated on the surface and the volume of the sample:

$$T'(\mathbf{x}) = T(\mathbf{x}) + T\_1^d(\mathbf{x}) + T\_2^d \tag{2}$$

The piezoelectric signal is given by the expression [24]

$$V \sim \left(\frac{1}{l} \int\_{-l/2}^{l/2} T'(\mathbf{x}) d\mathbf{x} \pm \frac{6}{l^2} \int\_{-l/2}^{l/2} T'(\mathbf{x}) \mathbf{x} d\mathbf{x} \right) \tag{3}$$

Two terms in the expression represent piston and drum effects [26]; they add up in the case of front detection and subtract in the case of rear mode.

Assuming the proper expressions for the absorption coefficient in the direct bandgap semiconductors (Urbach tail thermal broadening and absorption connected with the bandto-band electron transitions) [26], one can simulate the amplitude and phase spectra for the semiconducting sample with the presence of the defects localized on the subsurface damaged layer.

The nature of the amplitude and phase spectra is influenced by thermal bulk and layer parameters, the most important of which are thermal diffusivities, energy gaps, and the thicknesses of the sample and the near-surface layer. These basic parameters characterizing the material were used to simulate both amplitude and phase spectra.

Figure 9 presents amplitude (a) and phase (b) spectra of ground Cd0.95Be0.05Te mixed crystal. Experimental data are shown together with simulations according to the abovegiven model. The simulation was performed for the presence of two defects with the location *Ed*<sup>1</sup> = 1.51 eV and *Ed*<sup>2</sup> = 1.55 eV, thermal diffusivity of 0.05 cm2/s, thermal conductivity of 0.07 W/(cm·K) for the sample, and conductivity of the layer, ten times smaller than the bulk: *Ad*<sup>1</sup> = 15 cm−1, *Ad*<sup>2</sup> = 7 cm−1, *β*<sup>1</sup> = 0.2 cm−1, *β*<sup>2</sup> = 0.06 cm−1. The thickness of the defective layer was 0.01 mm. The simulations qualitatively confirm the presence of defects on the surface, but they are not entirely consistent with the experiment data.

**Figure 9.** Amplitude (**a**) and phase (**b**) spectra of the ground Cd0.95Be0.05Te mixed crystal for 126 Hz modulation frequency. Simulated spectra are presented by black lines, experimental amplitude by red lines, and experimental phase by blue lines.

Figure 10 presents amplitude (a) and phase (b) spectra of etched Cd0.95Be0.05Te mixed crystal. Experimental data are shown together with simulations. The simulation was performed for the presence of two defects with a location different than the previous case: *Ed*<sup>1</sup> = 1.6 eV and *Ed*<sup>2</sup> = 1.62 eV, thermal diffusivity of 0.05 cm2/s, thermal conductivity of 0.07 W/(cm·K) for the sample, and conductivity of the layer, 12 times smaller than the bulk: *Ad*<sup>1</sup> = 110 cm−1, *Ad*<sup>2</sup> = 80 cm−1, *β*<sup>1</sup> = 0.045 cm−1, *β*<sup>2</sup> = 0.06 cm−1. The thickness of the defective layer was 0.001 mm, ten times smaller than in the previous case. In this case, lower compatibility in experiment and simulation was achieved.

**Figure 10.** Amplitude (**a**) and phase (**b**) spectra of the etched Cd0.95Be0.05Te mixed crystal for 126 Hz modulation frequency. Simulated spectra are presented by black lines, experimental amplitude by red lines, and experimental phase by blue lines.

The applied theory assumes the generation of a photothermal signal only due to the thermoelastic effect. As shown before, the fall of the signal above the gap may be due to free carrier contribution. Their participation should also be considered when generating the signal in the defective subsurface layer. This assumption can be supported by the most recent results obtained by Aleksi´c et al. [31]. In their article, the dependence of the photoacoustic signal on the frequency of the excitation beam was used to analyze the influence of thin transparent foil on the thermal and elastic properties of a two-layer sample consisting of a silicon substrate and a thin TiO2 foil. A thin layer can significantly affect the substrate's thermal state and increase the sample's bending. This is related to a slight change in the number of photogenerated carriers in the transparent layer, which strongly influences the temperature differences between the illuminated and non-illuminated sides of the sample [31]. The influence of the carriers in the defective layer probably extinguished the signal for the first sample etching procedure.

#### **5. Conclusions**

Using piezoelectric photothermal spectroscopy, a new Cd1−xBexTe material was investigated, and its basic optical parameters were determined (Eg primary and thermal diffusivity). The influence of the sample preparation method on the amplitude and phase of the photothermal signal was observed and interpreted. High-quality sample surfaces have yet to be achieved, and further work on surface treatment procedures is needed. Further research and development of an appropriate etching method are required to obtain a defect-free surface of the samples. The next step will be to measure the different etching mixtures and etching times. The extinction of the signal-etching procedure may have been caused by excessive etching time. Despite not achieving good surface quality, piezoelectric spectroscopy is a sensitive method and can help choose the appropriate surface treatment method to obtain the desired quality.

In this paper, we presented a preliminary interpretation of our experimental data using the modified Blonskij model. As not all courses of the amplitude and phase could be interpreted with the proposed theory, it is necessary to expand it. We have shown that considering only the thermoelastic effect for analyzing and simulating experimental spectra cannot fully reproduce them. Effects related to carriers should also be considered in the interpretation, both in terms of nonradiative bulk recombination of carriers that diffuse in the crystal and the nonradiative surface recombination of carriers.

**Author Contributions:** Methodology, D.M.K.; Investigation, J.Z., K.S., M.B., A.M., A.A. and D.M.K.; Writing—original draft, J.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Emerging Fields Team IDUB of Nicolaus Copernicus University—Material Science and Technology, 2023–2026.

**Institutional Review Board Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Conflicts of Interest:** The authors declare no conflict of interest.
