{'fraction_non_alphanumeric': 0.015828795745219705, 'fraction_numerical': 0.00531847537039382, 'mean_word_length': 4.907255048616305, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Comparison between different Photothermal Deflection methods to determine thermal properties of bulk semiconductor samples

In this study we will describe and compare different methods based on the Photothermal Deflection Technique (PTD) which permits the determination of thermal diffusivity for bulk semiconductors. The two first methods proposed here consist in drawing the experimental amplitude and phase variation of the photothermal signal versus square root modulation frequency. The sample placed in air is heated thanks to a modulated uniform light beam. The difference between these two methods is that in the second one the sample is covered by a thin graphite layer. We notice that the first method is only sensitive to the thermal diffusivity however the second method is sensitive for both thermal diffusivity and thermal conductivity. Finally the third method which is a spectroscopic one and where the sample is immersed in a CCl4 filled cell consists to draw the phase variation of the photothermal signal versus wavelength at a fixed modulation frequency. The phase difference between the two saturated zone (high and low absorption coefficient) is sensitive to the thermal diffusivity.


Introduction
During the past few years, the application of the photothermal deflection technique which have the advantage of being non destructive has been considerably involved [1,2]. The aim of this technique is in the first time to detect the thermal wave generated by the sample's optical absorption of a modulated light beam which will propagate into the sample and in the surrounding fluid inducing a temperature gradient then a refractive index gradient and in a second time to study the deflection of a laser probe beam skimming the sample surface and crossing the inhomogeneous refractive index region. This deflection may be related to the thermal properties of the sample. Several methods based on the Photothermal Deflection Technique or the so-called "Mirage Effect" have been developed to determine the thermal diffusivity of materials with high precision [3,4]. The most used technique consists to heat the sample by a modulated laser pump beam and to draw the experimental in phase signal versus the distance x between the pump beam and the probe beam axis at a fixed modulation frequency ν . The obtained curves cut the x axis in two points distant from d. The linear curve d = f ( ν 1 ) whose slope depends on the thermal diffusivity will allow determining it. In this work, we present and compare three different methods applied to bulk semi conductor such as GaSb samples in order to determine their thermal diffusivity. The two first methods deal with the analysis of the photothermal signal variation versus modulation frequency where the sample is placed in air and In the second method the sample is covered with a thin graphite layer where interest will be discussed later. The third method is a spectroscopic one and consists to draw the photothermal deflection signal versus wavelength at a fixed modulation frequency where the sample is immersed in a CCl 4 filled cell. The thermal diffusivity of bulk GaSb samples is obtained by fitting the experimental curves.

Theoretical model
The theoretical model is described in detail in [5,6].

Experimental set-up
For the thermal study, the experimental set-up is described in [5]. For the optical study we interpose between the halogen lamp and the mechanical chopper a Monochromator (Jobin Yvon HR250) and we plot the amplitude and phase variation of the photothermal signal versus wave length at a fixed modulation frequency.

Experimental results
The thermal study in the two first methods consists in drawing the logarithm of the amplitude and phase variation versus square root modulation frequency. The best coincidence between experimental and theoretical curves will give the best values of thermal properties.

First method
If we plot the theoretical logarithm of the amplitude and phase variations versus modulation frequency at a fixed value of the thermal diffusivity and for different values of the thermal conductivity one can notice that the obtained curves are confused so the photothermal signal is insensitive to the thermal conductivity in this case. However, if we vary the thermal diffusivity at a fixed value of thermal conductivity (figure 1), one can notice that both the logarithm of amplitude and phase variations are very sensitive to D s . Then increasing the thermal diffusivity of the sample we remark that the theoretical phase maximum moves towards high frequencies. Therefore, the best value of the thermal diffusivity D s is only obtained for the best coincidence between the experimental curves and the corresponding theoretical ones. In figure 2 are represented the experimental logarithm of the amplitude and phase variation obtained respectively for an undoped, Te-doped and Zn-doped GaSb sample. The corresponding theoretical curves which best coincide with experimental ones are obtained for values of thermal diffusivity reported in table 1.

Second method
Here, the samples of Gallium Antimonide are covered by a graphite layer of thickness 1.6µm for the undoped and Te-doped GaSb and 4µm for the Zn-doped. To determine the sensitivity of our experimental set-up towards the thermal properties we have plotted in figures 3-a and 3-b respectively the theoretical logarithm of amplitude and phase variations for different couples (K s , D s ) of an undoped GaSb sample. We notice from these curves that both the logarithm of amplitude and phase, unlike in the first method, are very sensitive as well as to K s and to D s which prove the interest of this method. We notice also and as in the first method that the theoretical phase maximum moves towards high frequencies when the thermal diffusivity increases.  The theoretical curves which fit best the experimental ones are obtained for the couple (k s =35.2 W.m -1 .K -1 , D s = 2.1×10 -5 m 2 .s -1 ). A similar study was made for the Te-doped and the Zn-doped. In figures 4-a and 4-b are represented the experimental and the corresponding theoretical curves of respectively the logarithm of the amplitude and the phase versus square root modulation frequency for each samples at a same distance z 0 =215µm. We notice from these figures the good agreement between the experimental and the theoretical curves for both the logarithm of amplitude and phase variation. The three samples thermal diffusivity and thermal conductivity values so deduced are reported in Table 1.

Third method
Our study in this method consists to plot the phase variation of the photothermal signal near the band gap region versus wavelength. The experimental corresponding curves obtained for each sample are shown in figure 5. From this figure, one can notice that the phase saturates above and below the gap energy. The phase difference ∆Φ between this two saturated zones is a function of the thermal diffusivity which may be determined by comparison of the theoretical phase difference with the corresponding experimental ones. The obtained thermal diffusivity values are listed in table1.

Conclusion
In this work, we have investigated the thermal diffusivity for an undoped GaSb, Te-doped and Zndoped using three different methods based on the photothermal deflection technique by fitting the experimental curves and we have compared their sensitivity to the thermal properties especially the thermal diffusivity. In fact: while the first and the third methods are sensitive only to the thermal diffusivity, the second method depends on both thermal diffusivity and thermal conductivity. By depositing a thin graphite layer on the sample (second method), we have shown one simple and chip method applied for bulk semiconductors which allow to determine simultaneously the thermal diffusivity and the thermal conductivity with good precision.