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Saadallah et al. [29] have developed a model to find the expression for the photothermal

deflection caused by a carbon black film for which the amount of generated heat was known from the knowledge of the optical absorption coefficient. In this way they could write the expression of photothermal deflection angle as a function of the thermal diffusivity of specific different liquids in which the carbon black reference sample was embedded. They applied this model to find the thermal diffusivity of paraffin oils. Salazar et al. [7] developed a complete theoretical model for interpreting the PDS data and extracting the thermal diffusivity of solids under specific conditions. They classified thin film materials as thermally thick and thermally thin depending upon the thermal and optical properties of the samples and their geometrical thickness. A thermally thick

sample can be defined as a sample for which the thermal diffusion length LD (related to

thermal diffusivity) is much smaller than the geometrical thickness ‘d’ of the sample, like in Figures 1.4a and 4b. A thermally thick sample can be either optically thick (like in

Figure 1.4a) or optically thin (like in figure 1.4b) depending if the attenuation length of light (defined as the reciprocal of the optical absorption coefficient, l =1/) is much smaller than (l << d) or comparable with (l ≥ d) the geometrical thickness of the sample. For optically thick, thermally thin samples the PDS signal is in phase with the pulses of the “pump” beam, the thermal properties cannot be measured and only surface optical properties can be measured. For optically thin and thermally thick samples heat is uniformly generated along the sample cross-section and slowly diffuses to the surface of the sample. In this case, the thermal wave has a phase difference with respect to the “pump” light pulse and allows the measurement of the thermal properties, provided the substrate is sufficiently thermally insulating. In all of the other situations, in which samples are thick or thin both thermally and optically, the PDS signal depends on both thermal and optical properties of the sample, so that knowledge on one type of properties is necessary for measuring the other type of properties. On the other hand, samples that are optically thin at specific wavelength of illumination (i.e. below the optical band gap) can be optically thick at other wavelengths (i.e. above the optical band gap). Instead, the thermal thickness of a sample depends not only on the thermal diffusivity but also on the modulation frequency of the “pump” light beam. This makes PDS a flexible technique for which the thermal and optical properties of solids can be very often simultaneously measured.

Fournier et al. [39] have developed a theoretical model to investigate optically thin and optically thick semiconductors and used it to measure the thermal diffusivity of silicon-based materials. Suher et al. [40] used transverse PDS to measure the thermal diffusivity of aluminum oxide and investigated the effect of porosity on the thermal

properties of this material. Kou et al. [41,42] have measured the thermal diffusivity of a number of pure materials, compounds and semiconductors, including silicon carbide, silicon nitride ceramics, and metal alloys. In these experiments, the samples surfaces

Figure 1.4Schematic ofthermal and optical thicknesses. (a) Thermally and optically thick sample whose thickness d is greater than both thermal diffusion length LD and optical attenuation length l, (b) thermally thick and

optically thin sample whose thickness d is greater than thermal diffusion length LD but smaller than the optical attenuation length l, (c) thermally

thin and optically thick sample whose thickness d is less than thermal diffusion length LD and greater optical attenuation length l, and (d)

thermally and optically thin sample whose thickness d is smaller than both thermal diffusion length LD and optical attenuation length l of the

were scanned by moving the probe beam away from the heating “pump” beam along the sample surface at a constant height from the sample surface. For a given modulation frequency, the scan effectively measured the thermal wave with the wavelength equal to the distance between the probe beam positions on the sample where the phase of PDS signals changes by 180°. By repeating the scan at different modulation frequencies, a number of wavelengths were measured. Thermal diffusivity was calculated from the

slope of the plot of wavelength versus square root of frequency. Ranalta et al. [43] used

the transverse PDS to measure the thermal diffusivity of soda lime glass and polypropylene. Bertolotti et al. [44] have measured the thermal conductivity of thin polycrystalline diamond films by PDS. Hurler et al. [45] used transverse PDS with a modulated light with line heating source instead of point source to determine the thermal properties of hydrogenated amorphous carbon thin films. The line source heating method was used to reduce the power density to which the sample was exposed, to avoid damage of the sample, and to obtain a good signal-to-noise ratio. Another advantage of the line source heating method is that it can average the response of small heterogeneities along

the heating line. Chen et al. [20] used photothermal reflectance with a pulsed light beam

to measure the thermal conductivity of tetrahedral amorphous carbon (ta-C) films coated with metal thin films.

Gharib et al. [46] have used transverse PDS to measure thermal diffusivity and thermal conductivity simultaneously by depositing a layer of graphite on top of the film samples so that the measured signal is sensitive to both thermal diffusivity and thermal conductivity and the quantity of heat deposited in graphite is known, since its optical and thermal properties are available in the literature. Bertolotti et al. [47] used PDS to

measure the thermal diffusivity of porous silicon thin films deposited on silicon wafer. Jeon et al. [48] measured by PDS and modeled the thermal conductivity of anisotropic materials and studied the effect on the PDS signal of the position of the probe beam with respect to the pump beam. They also studied the effect of the angle that was imposed between the probe beam and the crystallographic c-axis of Pyrolytic graphite and its effects on the measurement of the thermal conductivity along specific lattice directions in this thermally anisotropic solid. They found that their measured values agreed well with the theoretical prediction for isotropic iron and copper films but had significant deviations for pyrolytic graphite. N. A. George [49] used fibre optics to efficiently couple the PD signal to the position detector and determined the thermal diffusivity of indium phosphide wafers from the phase of the PDS signal. Saadallah et al. [50] used PDS to measure the thermal properties of thin layers of β-In2S3 grown on glass substrates by

spray pyrolysis and investigated the effect of aluminum doping on the thermal properties of such films.