Mounir Zahir
email: mounir.zahir@univ-poitiers.fr

Jeremie Drevillon

Younes Ezzahri

Improved thermophotovoltaic system

Improved thermophotovoltaic system performance with a novel metamaterial thermal emitter combined with a passive radiative cooling approach

Mounir Zahir, Jérémie Drevillon, Younes Ezzahri

To cite this version:

Mounir Zahir, Jérémie Drevillon, Younes Ezzahri. Improved thermophotovoltaic system performance with a novel metamaterial thermal emitter combined with a passive radiative cooling approach. Solar Energy Materials and Solar Cells, 2023, 263, pp.112587. 10.1016/j.solmat.2023.112587. hal-

1-Introduction

Radiative cooling is a promising strategy to dissipate excess heat in many energy conversion technologies such as the solar cell (photovoltaic (PV)) system [START_REF] An | Radiative cooling of solar cells: opto-electro-thermal physics a modeling[END_REF][START_REF] Underwood | A thermal model for photovoltaic systems[END_REF][START_REF] Safi | Improving photovoltaic performance through radiative cooling in both terrestrial and extraterrestrial environments[END_REF][START_REF] Li | A comprehensive photonic approach for solar cell cooling[END_REF][START_REF] Zhu | Radiative cooling of solar cells[END_REF], the thermophotovoltaic (TPV) system [START_REF] Zhou | Radiative cooling for thermophotovoltaic systems[END_REF], and the thermoelectric generator [START_REF] Fan | Maximal nighttime electrical power generation via optimal radiative cooling[END_REF], directly into the clear sky via mid-infrared radiation. The radiative cooling devices consist of a selective emitter that operates at ambient temperature and can achieve below-ambient cooling by using the atmospheric transparency window (8-direct sunlight [START_REF] Raman | Passive radiative cooling below ambient air temperature under direct sunlight[END_REF][START_REF] Catalanotti | The radiative cooling of selective surfaces[END_REF][START_REF] Mabchour | Daytime radiative cooling purposes with selective multilayer design based on Ta2O5[END_REF]. The broadband cooling emitters are usually used above ambient conditions, where they can overcome the absorbed atmospheric radiation with radiated power due to their blackbody emission [START_REF] Zahir | Daytime radiative cooler using porous TiO2: new approach[END_REF][START_REF] Fu | Daytime passive radiative cooler using porous alumina[END_REF][START_REF] Kong | Ultra-broadband all-dielectric metamaterial thermal emitter for passive radiative cooling[END_REF]. To achieve high performance daytime radiative cooling, several studies have been realized with different materials, Chae et al. [START_REF] Chae | Spectrally selective inorganic-based multilayer emitter for daytime radiative cooling[END_REF] proposed a daytime radiative cooler with a simple multilayer structure of inorganic materials of , , and , the experimental test showed that the optimized radiative cooler can reduce the temperature by up to 8.2 °C under direct sunlight. Kim et al. [START_REF] Kim | Optimization and performance analysis of a multilayer structure for daytime radiative cooling[END_REF] elaborated an optimized multilayer structure of repeatedly stacked , , and , to enhance the mid-IR emission and reduce the absorption in the wavelength range ( ), the optimized radiative cooler achieves a net cooling power of , °C in daytime at air-mass 1.5 conditions. Yang et al. [START_REF] Yang | A dual-layer structure with record-high solar reflectance for daytime radiative cooling[END_REF] developed a two-layer structure based on 0.24mm thick PTFE and an Ag reflector, this structure can achieve a very high reflectance of about 0.991 in the solar spectrum, and a high emittance of about 0.9 in the mid-infrared region. Kecebas et al [START_REF] Kecebas | Spectral Emissivity Profiles for Radiative Cooling[END_REF] investigated and analyzed the role of environmental factors (ambient temperature and degree of thermal insulation from the environment) on spectral emissivity distributions by using different optical materials to maximize the cooling power and minimize the equilibrium temperature. Due to its optical properties, vanadium dioxide has been used for adaptive radiative cooling applications. Taylor et al [START_REF] Taylor | Vanadium dioxide based Fabry-Perot emitter for dynamic radiative cooling applications[END_REF] proposed a Fabry-Perot emitter made of a thin film and an opaque aluminum substrate to achieve a high emissivity in the first atmospheric window (8-13µm). Didari et al [START_REF] Didari-Bader | Adaptive plasmonic metasurfaces for radiative cooling and passive thermoregulation[END_REF] investigated the performance of two nanophotonic structures. A thin layer of unpatterned or patterned is incorporated in each design to enable the desired thermal switching between the hot state (T >68°C) and the cold state (T < 68°C). The structures achieve high reflectivity within the solar spectrum ( ) and simultaneously high absorption within the atmospheric transparency window ( ) in the hot state. In the other work, Didari et al [START_REF] Didari | A biomimicry design for nanoscale radiative cooling applications inspired by Morpho didius butterfly[END_REF], designed a metamaterial system by modifying the palm tree-like structure with periodic multilayer films separated by nanoscale gaps. The spectrally selective enhancement resulting from geometric variations, the spatial location of the source, and the material properties, increased the heat flux in the infrared region of the spectrum. Another approach used a single material with a surface morphology with a specific periodic pattern. The Saharan silver ant is a typical biological example of diurnal radiative cooling [START_REF] Shi | Keeping cool: Enhanced optical reflection and radiative heat dissipation in Saharan silver ants[END_REF]. The hierarchical photonic architectures of the hairs improve the reflectivity in the visible and near-infrared, and the emittance in the mid-infrared spectral range, thereby reducing the body temperature of the silver ants through radiative cooling. In recent years, several studies have shown that the radiative cooling approach can be applied to solar cells to achieve sub-ambient operating temperatures. In a recent work [22], one of us contributed to the development of a solar cell/radiative cooling hybrid design based on porous TiO2 to improve the solar cell performance. The reduction of the operating temperature in such a design was about 18K. Another theoretical study conducted by Zhu et al. [START_REF] Zhu | Radiative cooling of solar cells[END_REF], based on a photonic crystal showed that the radiative cooling design achieves cooling up to 18.3K. Experimental cooling of 13K has also been measured [START_REF] Zhu | Radiative cooling of solar absorbers using a visibly transparent photonic crystal thermal blackbody[END_REF]. Wu et al., [START_REF] Wu | Solar heating of GaAs nanowire solar cells[END_REF] demonstrated that the GaAs nanowire-based PV cell can enhance the radiative cooling and reduce the operating temperature by nearly . The operating principle of a solar cell is the same as that of a TPV cell. The two devices differ only in the source of the irradiance spectrum; the sun for the solar cell and a thermal emitter at high-temperatures for the TPV cell. The TPV systems have attracted much interest as a potential energy conversion technology and could indeed benefit greatly from the radiative cooling approach to reduce their operating temperature. To date, few studies have investigated the effect of passive radiative cooling on the performance of a TPV system. Zhou et al. proposed a low-iron soda-lime photonic crystal-based passive radiative cooler (PRC) to improve the conversion efficiency of the TPV cell [START_REF] Zhou | Radiative cooling for thermophotovoltaic systems[END_REF]. The area ratio between the PV cell and the PRC was a key factor in this study to achieve an 18% improvement over the reference. Blandre et al., [START_REF] Blandre | New insights into the thermal behavior and management of thermophotovoltaic systems[END_REF] studied the effect of the PRC and the radiator emission (high temperature thermal emitter) on the performance of the TPV system with mitigating thermal effects. It was concluded that radiative cooling ca the TPV system without adjusting the emissivity of the radiator. Other studies, both theoretical and experimental, have succeeded in improving the efficiency of the TPV systems by increasing and controlling the emittance of the thermal emitter in the TPV system, using photonic crystals [START_REF] Chirumamilla | Multilayer tungsten-alumina-based broadband light absorbers for high-temperature applications[END_REF][START_REF] Blandre | Spectrally shaping high-temperature radiators for thermophotovoltaics using Mo-HfO2 trilayer-on-substrate structures[END_REF], selective emitters [START_REF] Pfiester | Selective emitters for thermophotovoltaic applications[END_REF], and metamaterial structures [START_REF] Chirumamilla | Metamaterial emitter for thermophotovoltaics stable up to 1400 C[END_REF]. In order to improve the performance of the TPV system, we have designed a TPV-PRC system which can be used as an attractive way to achieve a high efficiency of the proposed TPV cell. Our design consists of a high temperatures selective emitter based on a metamaterial placed opposite to a GaSb-based PV cell that is characterized by a bandgap energy of 0.72eV [START_REF] Ferguson | Theoretical study of GaSb PV cells efficiency as a function of temperature[END_REF]. The metamaterial thermal emitter (MTE) emits very close to the ideal emitter in the wavelength range of 0.8the GaSb PV cell, and generates a huge photon flux through the vacuum to the TPV cell, especially at high temperatures. We will show that the newly developed MTE generates high power density that can be converted into electrical energy. In addition, we will develop a new PRC that emits strong mid-infrared radiation through the 8--Section 5 why the PRC could be a more practical and scalable solution to achieve high conversion efficiency. For example, when the area ratio between the PV cell and the PRC is 20, our simulation has demonstrated a thermalto-electrical power conversion efficiency of at a realistic MTE temperature of . In addition, the operating temperature of the PV cell was reduced by .

2-System modeling

The proposed structure consists of a TPV cell combined with a PRC. Figure 1 shows our design of the TPV-PRC system. The latter consists of three main elements as follows: 1) A MTE composed of seven cavities that are put between top and bottom layers of 75nm thickness each. Each cavity is made of two 15nm thick layers of Tantalum ( ) separated by a 150nm thick layer. The ensemble is supported by a Ta substrate. has a melting temperature point above and is widely used for high temperature applications [START_REF] Dh | A review on the joining of SiC for high-temperature applications[END_REF]. On the other hand, has has been shown to be thermally stable under vacuum conditions and at high temperatures [START_REF]High-temperature stability and selective thermal emission of polycrystalline tantalum photonic crystals[END_REF]. The optical constants of and are taken from Palik s handbook [START_REF] Palik | Handbook of Optical Constants of Solids[END_REF]. We arrived at this design for the MTE by trial and error approach, taking advantage of our experience in manipulating the radiative properties of selective emitters [START_REF] Drevillon | Ab initio design of coherent thermal sources[END_REF][START_REF] Nefzaoui | Selective emitters design and optimization for thermophotovoltaic applications[END_REF][START_REF] Hervé | Radiative cooling by tailoring surfaces with microstructures: Association of a grating and a multi-layer structure[END_REF]. The MTE absorbs waste heat and re-emits thermal radiation towards to a PV cell. 2) A 10µm thick layer with a low bandgap of eV at room temperature is chosen as the PV cell for this study. Its optical properties are measured by normal incidence reflectance using ellipsometry, are taken from reference [37], and modeled as being independent of temperature. Its external quantum efficiency (EQE) at 300K, shows a very good agreement with the MTE emissivity (Fig 2(a)), which will allow it to receive a higher radiant power density. The PV cell absorbs thermal radiation (photons) from the and converts it into electrical energy.

3) The PRC can dissipate the heat generated by the operating TPV cell, by emitting its infrared radiation. Thus, it helps to further push the limits of the TPV cell power conversion efficiency. Given that TPV systems typically operate in a vacuum, the MTE and the PV cell are assumed to be enclosed in a vacuum chamber [START_REF] Zhou | Radiative cooling for thermophotovoltaic systems[END_REF]. In the following, we use the matrix formulation method [START_REF] Yeh | Optical Waves in Layered Media[END_REF][START_REF] Macleod | Thin Film Optical Filters[END_REF] to calculate the optical properties of one single Ta/SiC/Ta cavity, the proposed MTE, and the flat emitter. Figure 2(a) shows the emissivity/absorptivity of the designed MTE under normal incidence compared with the emissivity of the single cavity and the flat . Our MTE exhibits a good spectral emissivity close to the one of an ideal emitter in the wavelength range and shows an excellent match with the EQE of the GaSb cell [START_REF] Fraas | Fundamental characterization studies of GaSb solar cells[END_REF]. The high average emissivity below the cutoff wavelength is about and the low reflectivity is calculated using the energy balance principle and according to Kirchhoff law of thermal radiation [START_REF] An | Radiative cooling of solar cells: opto-electro-thermal physics a modeling[END_REF] . Most of the photons emitted by the designed MTE should be absorbed by the PV cell. The radiated power density of each thermal emitter is expressed by [START_REF] Silva-Oelker | High-temperature tungsten-hafnia optimized selective thermal emitters for thermophotovoltaic applications[END_REF]:

where is the spectral hemispherical emissivity, = is the solid angle, and is the reflectance of the PV cell. The spectral radiance of a blackbody (BB) at the emitter temperature , is given by: where is the speed of light, is Boltzmann constant and is Planck constant. As can be seen in Fig 2(b), the radiated power density at high temperature is much higher in the case of the MTE than in the case of the flat emitter and each curve presents a radiated power density peak below the cutoff wavelength . The maxima are and for the MTE and flat emitter, respectively. We note that the radiated power density observed after can increase both the heat generation and the operating temperature of the PV cell. This impact of the heat source on the operating temperature can be reduced using a PRC placed on the bottom of the TPV cell.

There are other factors that affect the performance of the TPV cell such as the polarization and the directional behavior of the thermal radiation. We have investigated the effect of the incident angle on the spectral emissivity of our designed MTE. Figure 3 shows contour plots of the spectral emissivity(absorptivity) of the proposed MTE for both transverse electric (TE) and transverse magnetic (TM) polarizations, as functions of the incident angle and wavelength. For both TE and TM polarizations, the maximum of the spectral emissivity can be found in the range of and decreases abruptly beyond the cutoff wavelength of , which gives a perfect match with the EQE of the GaSb cell at 300K. In addition, the designed MTE also exhibits high spectral emissivity in the range even at large incident angles up to 80° in the case of TM polarization and up to 60° for the case of TE polarization. Consequently, our designed selective MTE can maximize the power conversion efficiency of the TPV system.

3-Electrothermal model and TPV Performance

To evaluate the performance of the TPV system, we consider the GaSb PV cell to be placed in contact with a silver plate substrate for heat dissipation. As mentioned above, this cell has a bandgap ( at and an infrared absorption spectrum extending up to . decreases as a function of temperature. However, the rate of decrease is less than 5% for a 100K increase in the PV cell temperature [START_REF] Vadiee | Temperature dependence of GaSb and AlGaSb solar cells[END_REF]. Therefore, it is reasonable to assume that remains constant over such a temperature variation range, and this is what we consider in the present study. According to the energy balance of the system, the power conversion efficiency of the TPV cell is defined as the ratio of the output electrical power ( ) to the total input thermal power ( ):

Where the total input heat power is:

The output electrical power density per unit area from the TPV cell is:

is the short circuit current density and is the open-circuit voltage, given, respectively, by [START_REF] Vadiee | Temperature dependence of GaSb and AlGaSb solar cells[END_REF]:

where is the fill factor [START_REF] Breitenstein | Understanding the current-voltage characteristics of industrial crystalline silicon solar cells by considering inhomogeneous current distributions[END_REF], is the elementary charge of an electron, is the EQE. The saturation current density is estimated using the expression [START_REF] Fraas | Fundamental characterization studies of GaSb solar cells[END_REF]:

where is the PV cell temperature.

The PV cell is characterized not only by a voltage or a current , but also by a full current-voltage characteristic, where the operating point is determined by the required current and by the load, the electrical power for a constant current is:

where is the sum of the electron and hole currents in a PV cell. In the presence of the series resistance and the shunt resistance , the PV cell equation is given as [START_REF] Franzitta | Assessment of the usability and accuracy of the simplified one-diode models for photovoltaic modules[END_REF]:

Where is the ideality factor of the cell.

In the case of a short circuit, the cell current is equal to and the power is zero. As the load increases, the also increases reaches its maximum at maximum power located the current and the voltage . At zero load, the current decreases rapidly to zero, and the voltage . Figures 4 (a,b)

show the characteristics of the PV cell and the corresponding electrical power evolution as the voltage increases for a MTE temperature ranging from 1000K to 1800K. increases from to when the temperature of the MTE varies from to . To compare the performance of our MTE with that of the Ta/SiC/Ta single cavity and flat emitters, the key performance parameters such as , , , and the power conversion efficiency were calculated at

. Table 1 summarizes the obtained values of these parameters. We find that the MTE has values of , and that are greater than those of the single cavity and flat emitters. This result was expected since the emissivity of the MTE is very close to unity in the wavelength range. We illustrate in Figs 5(a) and 5(b), the computed behaviors as functions of the emitter temperature between 1000K and 1800K, of the output electrical power density and the power conversion efficiency of the TPV cell in the case of the designed MTE, the single Ta/SiC/Ta cavity and the flat emitters. The MTE leads to a higher than the single cavity and the flat emitter over the whole temperature range considered. In our MTE structure, we have a very good Pout at higher temperatures because this structure has a very good spectral matching with the EQE of the GaSb PV cell, which results in a significant improvement of of the TPV system.

Structure

4-Optical properties of the passive radiative cooler

The PRC we designed consists of an optical structure which is composed of a double-layer coating; 30µm of and 50µm of deposited on a Silver (Ag) film. As for the case of the MTE, the design of the PRC was achieved by using a trial-and-error procedure. Our design can reflect solar radiation and strongly emit mid-infrared radiation through the atmospheric window under direct sunlight. The optical constants of and are taken from [START_REF] Palik | Handbook of Optical Constants of Solids[END_REF] and those of Ag are taken from reference [START_REF]Optical dielectric function of silver[END_REF]. and show an auxiliary thermal radiation emissivity in the atmospheric window 8-13µm which ensures the overall cooling effect of the TPV system. They are also characterized by a small extinction coefficient in the solar spectral range of 0.3-2.5µm with a low parasitic heat absorption. The Bruggeman medium theory [START_REF] Bruggeman | Dielectric constant and conductivity of mixtures of isotropic materials[END_REF] is applied to estimate the effective dielectric function of over the wavelength range . The model is based on the following equation: where and are the complex dielectric functions of and , respectively. denotes the volume fraction of in the matrix and is the depolarization factor related to the shape of the inclusions. In this study, we assume which describes spherical shaped MgF2 inclusions [START_REF] Bruggeman | Dielectric constant and conductivity of mixtures of isotropic materials[END_REF]. We use the transfer matrix method [START_REF] Yeh | Optical Waves in Layered Media[END_REF][START_REF] Macleod | Thin Film Optical Filters[END_REF] to calculate the optical properties and optimize the volume fraction for the proposed PRC. The validity of this method with the experimental data was presented in our previous work [START_REF] Zahir | Daytime radiative cooler using porous TiO2: new approach[END_REF]. A comparison between the calculated and measured emissivity/absorptivity of a thin film coatings was demonstrated, and a strong correlation between the experimental and simulation results was found. Figure 6 shows the calculated emissivity (absorptivity) spectrum of the proposed PRC structure as a function of the incident for different volume fractions f of in the SiO2 matrix. We also show the result of an silica layer on the same Ag substrate. Figure 6(a) shows the case of a normal incidence and TE polarization while Figure 6(b) shows the case of an oblique incidence (

) and TM polarization. For both TE and TM polarizations, a significant enhancement of the emissivity spectrum is achieved by increasing of the in the range . In the wavelength range, the emissivity is close to unity since both the and exhibit a high emissivity in this range. In the case of a pure silica layer deposited on a Ag substrate, two dips are observed at and corresponding to the phonon polariton resonances of silica [START_REF] Zhu | Radiative cooling of solar cells[END_REF]. On the other hand, and according to Kirchhoff's law, our design reflects most incident photons coming from the irradiance of the sun for both TE and TM polarizations in the solar spectral range . The second parameter that can affect the spectral emissivity of our PRC is the angle of incidence of the solar radiation. Figures 7(a) and 7(b) show the calculated spectral emissivity (absorptivity) as a function of the wavelength for different angles of incidence for both polarizations TE and TM at the optimal volume fraction . The designed PRC manifests very low absorptivity in the wavelength range which is also independent of the angle of incidence of the solar radiation. On the other hand, we obtain a high emissivity over the first sky window at normal incidence. However, this emissivity decreases significantly and rapidly in the second sky window for both polarizations. In the case of TE polarization (Fig 7(a)), we see that increasing the angle of incidence leads to a degradation of the emissivity in both sky windows, since this emissivity is proportional to the cosine of the angle between the direction of the incident photon and the normal direction of our design [START_REF] Macleod | Thin Film Optical Filters[END_REF]. Therefore, the normal incidence is the optimal angle at which the maximum cooling could be provided by the double coating. In the case of TM polarization (Fig 7(b)), the PRC shows a similar behavior as in case of TE polarization case but with an emissivity that remains close to unity in the range of for incidence angles between and . We also notice that the effect of the angle of incidence on the second sky window is relatively less pronounced in comparison to the TE polarization case. Due to the small temperature variation in the horizontal direction of our designed TPV-PRC system, heat transfer process occurs mainly in the cross-plane direction. Hence, in order to analyze the cooling performance of such a system, we use a 1D thermal model. Fig 8 illustrates the schematic representation of this model. The Ag plate acts as a substrate for the TPV-PRC system and is considered a perfect thermal conductor with an optimized thickness [START_REF] Hervé | Radiative cooling by tailoring surfaces with microstructures: Association of a grating and a multi-layer structure[END_REF]. The MTE temperature inside the TPV system is . The PRC is exposed to daytime at ambient temperature . The solar irradiance spectrum and the atmospheric transmittance data where is the radiative power density from the MTE absorbed by the PV cell and is the radiative power density emitted by the PV cell [START_REF] Safi | Improving photovoltaic performance through radiative cooling in both terrestrial and extraterrestrial environments[END_REF].

5-Thermal model of the TPV-PRC system

is the maximum electrical power density generated by the PV cell. We first investigate the effect of the parameter area ratio between the PRC and the PV cell. is defined for an equilibrium state of the TPV-PRC system, where the net power is . By rearranging the power densities and surfaces and , we easily show that is given by Eq. ( 20). As we shall see next, impacts tremendously the operating temperature and power conversion efficiency of the TPV cell. Large values of this parameter can indeed, lead to a below ambient temperature cooling and improve the TPV conversion efficiency. Besides, by considering the small thickness of the Ag plate and its high thermal conductivity, we could assume that the PRC and the PV at the same temperature ( . Figure 10 reports the calculated power conversion efficiency of TPV-PRC system a function of the area ratio where we assume that the convection heat exchange coefficient to be non negligeable on the top surface of the system and on the bottom surface (Fig 8). We can observe that the TPV-PRC conversion efficiency increases monotonically with and its maximum would be reached for . Figures 11(a) and 11(b) report, respectively, the calculated behaviors of the operating temperature of the PV cell and the TPV power conversion efficiency with and without the PRC for an area ratio at fixed . By increasing the area , the operating temperature of the PV cell both cases, without and with the PRC, decreases which leads to an increase of the power conversion efficiency of the TPV system. When the area ratio is , the operating temperature difference between the PV cell without and with the approaches and the power conversion efficiency increases from to . Our designed PRC emits mid-infrared radiation through the atmospheric windows in a close manner to an ideal cooler and achieves below-ambient cooling when the area ratio . At this ratio, the temperature of the PV cell drops by 19K compared to the case without the PRC, whereas the power conversion efficiency increases by approximately and exceeds the optimal value calculated when the operating temperature is set equal to the ambient temperature . The performance of the TPV system tends to stabilize for values of the area ratio Since the top and the bottom surfaces of our designed TPV-PRC system are exposed to sky, the performance of the device can be influenced by the weather conditions: humidity and wind. The convection cooling due wind is expressed by the convection heat exchange coefficient . It can lower the operating temperature of the TPV system. Figures 12(a) and 12(b) illustrate, respectively, the calculated behaviors of the operating temperature of the PV cell and the power conversion efficiency of the TPV system with and without the PRC, as functions of at a fixed area ratio . and manifest similar tendencies to those of Fig 11. By increasing , decreases in both cases with and without the PRC and consequently increases. The results of Fig 12 show clearly how important is the effect of the convection cooling on the whole performance of the TPV-PRC system. By increasing , the convection cooling becomes dominant and overshadows the effect of the PRC. The difference between the cases with and without the PRC reduces by increasing and the results seem to start stabilizing for values of .

Conclusion

We presented a TPV-PRC energy conversion system that includes a high temperature metamaterial thermal emitter (MTE) and a broadband passive radiative cooler (PRC). The MTE emits very close to the ideal emitter in the wavelength range and matches well with the external quantum efficiency the GaSb PV cell at 300K. The simulated results also indicate that the TPV power conversion efficiency reaches without cooling the system at . To solve the problem of the conversion efficiency loss due to the heating effect of the GaSb cell, a new PRC placed in contact with the PV cell substrate was developed that is characterized with a thermal emissivity close to ideal in the atmospheric windows. The impact of the area ratio ( between the PRC and the PV cell was studied. The analysis has shown a reduction of 44K in the operating temperature and an increase of 1.89% in the conversion efficiency of the TPV-PRC system when . It is also possible to achieve below ambient temperature cooling using values of . These results clearly show that improvement of the TPV performance can indeed be achieved using a good high temperature thermal emitter design and the passive radiative cooling mechanism.

Declaration of Interest

The authors declare that there are no conflicts of interest.

Figure 1 :

 1 Figure 1: (a) Schematic of the suggested TPV-PRC system design (b) Schematic of the designed MTE structure with the thicknesses of each layer (c) Schematic of the PRC with thicknesses of each layer.

Figure 2 :

 2 Figure 2: (a) Calculated absorptivity/emissivity, at normal incidence of one single Ta/SiC/Ta cavity, designed MTE, flat and EQE of GaSb at 300K [37] (b) The radiated power density at high-temperature and at normal incidence of one single cavity, MTE and flat as a function of the wavelength. represents the cut-off wavelength at , corresponding to the bandgap of the GaSb PV cell at .

Figure 3 :

 3 Figure 3: Contour plots of the emissivity of the designed selective thermal emitter under (a) TM and (b) TE polarizations as a function of angle of incidence and wavelength.

Figure 4 :

 4 Figure 4: Characteristics of the TPV system for different temperatures of the MTE, showing (a) the current-voltage (J-V) characteristics and (b) the ( -V) characteristics of the GaSb cell at .

Figure 5 :

 5 Figure 5: Calculated behaviors as functions of the emitter temperature of (a) the output electrical power density and (b) power conversion efficiency of the TPV cell for the case of the MTE (blue), flat Ta (green), and single Ta/SiC/Ta cavity (red) emitters.

Figure 6 :

 6 Figure 6: Spectral emissivity of the proposed PRC for different volume fractions f of at (a) normal incidence and TE polarization case and (b) an angle of incidence of and TM polarization case.

Figure 7 :

 7 Figure 7: Spectral emissivity of the proposed PRC for different angles of incidence at the optimal volume fraction f=0.7: (a) TE polarization and (b) TM polarization.

Figure 8 :

 8 Figure 8: Schematic of the TPV-PRC system; the MTE is at a temperature of and is enclosed with the PV cell in a vacuum chamber.

Figure 10 :

 10 Figure 10: TPV-PRC power conversion efficiency as a function of the area ratio at a given convective hat exchange coefficient . The ambient temperature is set at .

Figure 11 :

 11 Figure 11: (a) Operating temperature of the PV Cell and (b) TPV power conversion efficiency as functions of the area ratio at given convective coefficients . The ambient temperature is set at .

Figure 12 :

 12 Figure 12: (a) Operating temperature of the PV Cell and (b) TPV power conversion efficiency as functions of the convective heat exchange coefficient for both cases with and without the PRC at a fixed area ratio The ambient temperature is set at

have been taken from references [49,[START_REF]IR Transmission Spectra, Gemini Observatory Kernel Description[END_REF] and the water vapor of the atmospheric transmittance was considered . The spectral and angular emissivity of the PRC is calculated by assuming the optimal volume fraction .

(

) is the absorbed power density due to the atmospheric emissivity:

where is the spectral radiance of a blackbody at the ambient temperature as given by Eq. ( 2) above, is the emissivity of the and is the angle-dependent emissivity of the atmosphere [START_REF] Zahir | Daytime radiative cooler using porous TiO2: new approach[END_REF]. The power loss due convection and conduction in the top and bottom surfaces of the PRC are given, respectively by: and We assume . The radiative power density from the sun PRC expressed as: is the absorbed power that dissipates into heat within the PV cell. It is given by:


domain: phys spi

text:
Younès Ezzahri
email: younes.ezzahri@univ-poitiers.fr

Thermal decay in the one-dimensional transient thermal grating experiment using modified Debye-Callaway model

We calculate the thermal decay in the one-dimensional Transient Thermal Grating (TTG) experiment by solving the transient Boltzmann-Peierls transport equation (BPTE) within the framework of the single mode relaxation time approximation and using modified Debye-Callaway model in which both longitudinal and transverse phonon modes are included explicitly. We consider surface heating of an opaque thick semiconductor (SC) crystal film that we assume to have a cubic symmetry and is treated as a continuum, elastic, isotropic and dispersionless medium. We obtain a non-universal spectral suppression function (SSF) in the integrand of the effective apparent thermal conductivity that is similar to the one obtained by Chiloyan et al., [Phys. Rev. B 93, 155201 (2016)] using the standard single mode relaxation time approximation (RTA) model. Therefore, the non-universal character of the SSF in the TTG experiment does not depend on the form of the collision operator approximation in the BPTE; Callaway's or standard. Moreover, the analysis of the behavior of the thermal decay rate, shows how the peculiar crystal momentum shuffling effect of phonon-phonon scattering Normal processes (N-processes) that is captured by Callaway's model, influences the onset of the nondiffusive (quasi-ballistic) regime in the phonon transport process in SC crystals. This effect tends independently from the other phonon scattering processes to favor the maintenance of the phonon diffusive regime over a large length scale range. A remarkable feature that cannot be put into light with the standard RTA model used in previous works. Hence, the implicit effect of N-processes has certainly an important impact on the extraction of the phonon mean free path spectrum distribution, especially in the high temperature regime.

. (iii) The third approach is a variational one that was first used by Chiloyan et al., to solve the transient BPTE using also the standard RTA model [12]. The authors found a different SSF than the one obtained earlier [10, 11, 13]. The authors argued afterwards about the non-universality of the SSF, as the one they obtained depends explicitly on the material properties. The SSF obtained by Maznev et al., [10], Collins et al., [13] and Hua and Minnich [11] is a universal one; i.e., it depends only on the ratio of the phonon MFP with

INTRODUCTION

For the purpose of heat transport in semiconductor (SC) and dielectric materials, the most important characteristic intrinsic length is the mean free path (MFP) of phonons that represent the main energy (heat) carriers in these materials [1][2][3][4].

Transient Thermal Grating (TTG) spectroscopy technique has been proven to be a robust method to study length scale based quasi-ballistic (non-diffusive) phonon transport, including the transition from the quasi-ballistic regime to the diffusive regime, in dielectric and SC crystals [5][6][7][8][9]. Using a reconstruction method, the TTG spectroscopy technique has allowed probing the phonon MFP spectrum distribution and obtaining valuable information about the contribution weight and role of low and high frequency phonons in the heat transport process in these materials [5].

In this technique, two crossed laser pulses are shone on the surface of a thin or thick film material. The film can be either suspended or deposited on a supporting substrate. The interference pattern of the pulses results in a spatially sinusoidal temperature profile that constitutes the thermal grating with a spatial period that can be varied by adjusting the angle between the crossed laser beams. Once heated, the sample is allowed to relax and the timely decay of the thermal profile is measured to yield information about the in-plane phonon transport process within the material [5][6][7][8][9]. Because of the great importance TTG has gained in the study of phonon spectroscopy, its modeling has become critical. So far, three approaches have been suggested to mathematically model the TTG technique in a one-dimensional case.

(i) The first approach was performed in the framework of a "two-fluid" model by Maznev et al., [10]. An analytical spectral suppression function (SSF) in the integrand of the effective apparent thermal conductivity was obtained using simplifying assumptions about the scattering of high and low frequency phonons. This SSF was later utilized in the reconstruction of the phonon MFP spectrum distribution by Minnich [5]. As mentioned by Hua and Minnich [11] and Chiloyan et al., [12], there is a concern as the extent of validity of the "two-fluid" model is not clear. (ii) The second approach is based on solving the transient Boltzmann-Peierls transport equation (BPTE) in the framework of the standard single mode relaxation time approximation (RTA) model. In the standard RTA model, all phonon-phonon scattering processes are treated similarly regarding the onset of thermal conduction, with no distinction whatsoever between Normal and Umklapp anharmonic processes [11,13]. This second approach was first explored by Collins et al., where they used analytical and numerical methods to solve the problem and obtained the exact solution of the BPTE, in both the gray case and the full spectral case for Si respect to a characteristic length and not otherwise on the material properties. A more detailed and extended analysis of the application of the variational approach to include the effect of the two-dimensional case of phonon transport in a TTG configuration, was later presented by Chiloyan et al., [14] and Hubermann et al., [15]. The authors showed that the optical penetration depth could have an important effect for thin film materials that are transparent or semitransparent to the laser wavelength used in the TTG experiment [14,15]. Two recent works by Hua and Lindsay [16] and Chiloyan et al., [17] went beyond the standard RTA model and applied a more elaborate method using the full linearized collision matrix. The authors obtained closed forms expressions for the temperature profile and the effective apparent thermal conductivity and highlighted the limits of the standard RTA model in treating the results of the TTG spectroscopy technique, especially for high thermal conductivity materials.

The differences in the outcomes of all these approaches show clearly how critical and crucial the modeling of the TTG spectroscopy technique is, for a better and rigorous understanding of phonon transport regimes in this experimental configuration. Besides, no analysis so far has considered Callaway's model or a detailed analysis of the effect of temperature.

The motivation behind the present work is to calculate and analyze the thermal decay in the one-dimensional (1D) TTG experiment that corresponds to the case of a surface heating of an opaque thick SC crystal film, by solving the transient BPTE using the Callaway's approximation of the collision operator. This will allow us to address and highlight more respectfully and simply the critical role of phonon-phonon scattering Normal processes. The analysis will show how considering separately these processes, affects the expression of the SSF of the in-plane effective apparent thermal conductivity and the onset of the non-diffusive (quasi-ballistic) regime in the phonon transport process. This latter effect cannot readily be inferred from rigorous treatment using the full linearized collision matrix [16,17]. In addition, the analysis will also allow us to examine the important impact on the phonon MFP spectrum distribution that could be reconstructed based on the obtained thermal decay rate.

We present the main steps of theoretical modeling in section 2. In section 3, we discuss the results of this approach in application to the 1D TTG experiment by analyzing the effect of varying the ambient temperature. We summarize and establish our concluding remarks in section 4.

II. THEORY

TTG spectroscopy technique is an experimental technique that is particularly sensitive to in-plane phonon transport [5][6][7][8][9][10][11][12][13][14][15]. In this section, we present the key elements of the method that allows deriving the full expression of the thermal decay rate that would be measured in this technique, by analyzing the case of a surface heating of an opaque thick SC crystal film that we can consider as a bulk material [13]. In contrast to previous theoretical works [11][12][13][14][15], we will consider phonon-phonon scattering Normal processes (N-processes) and phonon-phonon scattering Umklapp processes (U-processes) separately. Once the expression of the thermal decay rate is obtained, one can get straightforwardly the expression of the in-plane effective apparent thermal conductivity. The integrand of the latter contains the phonon SSF.

A. Callaway's approach of the Boltzmann-Peierls transport equation

The starting point of our modeling is the transient BPTE in the framework of the single mode relaxation time approximation and using modified Debye-Callaway model in which both longitudinal and transverse phonon modes are included explicitly [18][19][20][21][22][23][24][25][26]. The SC system is assumed to have a cubic symmetry and is treated as a continuum, elastic and isotropic medium characterized by a linear (Debye-like) phonon spectrum for each phonon branch polarization so that one considers heat transport due only to acoustic phonons and ignore any contribution from optical phonons [18][19][20][21][22][23][24][25][26]. Callaway's approximation of the collision operator in BPTE captures quite fairly and respectfully the peculiar effect of phonon-phonon N-processes that distinguishes them from the rest of phonon scattering processes including phonon-phonon Uprocesses. Thus, it allows a simple separation of N-processes and U-processes [18][19][20][21].

Despite its simplicity, Debye-Callaway model has been proven to be very robust and effective in the study and prediction of the steady-state temperature behavior of the thermal conductivity of SC crystals within the conventional local/linear non-equilibrium thermodynamics theory [18][19][20][21][22][23][24][25]. Besides, it constitutes one of the first models to be used to study the second sound phenomenon in SC and dielectric crystals [27]. We recently used the model to analyze the modulation frequency behavior of the reduced effective thermal conductivity of SC crystals that is observed in Time Domain Thermoreflectance (TDTR) and

Frequency Domain Thermoreflectance (FDTR) experiments. We obtained an expression of the effective apparent thermal conductivity of the SC crystal that is characterized by a universal SSF that captures and describes the role, the weight and the contribution of quasi-ballistic and non-diffusive phonons. The SSF only depends on the ratio between the spectral phonon MFP and the thermal penetration depth as defined based on the diffusive Fourier's law [26]. Indeed, the thermal penetration depth constitutes the central characteristic length scale in TDTR and FDTR configurations.

We follow the same procedure as Collins et al., [13] and Hua and Minnich [11] to derive the expression of the thermal decay in a TTG experiment. In addition, we will consider local thermal equilibrium throughout, which is required to define a temperature [18][19][20][21][22][23][24][25][26][27]. Surface heating of an opaque bulk SC crystal in a TTG experiment results in a one-dimensional spatially periodic temperature profile that we assume to be established along the direction x



. Under the single mode relaxation time approximation, the transient Callaway's form of the BPTE along the x-axis can be written as the following [11][12][13]26]:

𝜕𝑈 , 𝜕𝑡 + 𝑚𝑣 𝜕𝑈 , 𝜕𝑥 = - 𝑈 , -𝑈 ,, 𝜏 , + 𝑔 , 𝜏 , (1)
where we have introduced the deviational spectral energy density per phonon mode (phonon wave-packet) of wave-vector q and polarization p as 𝑈(𝑥, 𝑡, 𝑚, 𝑞, 𝑝) ≡ 𝑈 , = ℏ𝜔 𝑛 , -𝑛 , . 𝑛 , is the phonon distribution function at the absolute local thermal equilibrium temperature T. 𝑈 , = ℏ𝜔 𝑛 , -𝑛 , is therefore the deviational equilibrium spectral energy density per phonon mode with 𝑛 , and 𝑛 , denoting the equilibrium phonon Planck distribution functions, at temperatures T and T0, respectively. T0 represents an absolute reference temperature [11][12][13]26].

𝜏 , = 𝜏 , + 𝜏 , is the "combined" phonon scattering rate [18,19] with 𝜏 , representing the single relaxation time with which all resistive phonon scattering processes (all scattering processes that change the total phonon wave-vector: Umklapp, boundary, defects, imperfections) tend to return the phonon system to its thermal equilibrium state. 𝜏 , is the single relaxation time due to N-processes (scattering processes that don't change the total phonon wave-vector). As pointed out by Callaway [18] and others [19][20][21], N-processes tend to return the phonon system to a displaced (drifted) Planck distribution function 𝑛 , = 𝑒𝑥𝑝 ℏ ( ) 𝝀 𝒑 .𝒒 -1 . By symmetry consideration in cubic SC crystals, p is a constant vector in the direction of the applied temperature disturbance, which has the dimension of a velocity times the reduced Planck constant ħ. 𝜔 (𝑞), vp and m are, respectively, the dispersion relation of the phonon in state (q, p), group velocity of a p-polarization phonon and directive cosine; cosine of the angle between the x-axis and the phonon wave-vector q [18][19][20][21][22][23][24][25][26][27].

By using Callaway's analysis and Debye-like phonon dispersion 𝜔 (𝑞) = 𝑣 𝑞 [18,19,26],

we can easily show that the term 𝑔 , is given by:

𝑔 , = -𝛽 𝜏 , 𝜏 , 𝑚𝑣 𝐶 𝑑𝑇 𝑑𝑥 (2)
p is Callaway's parameter that has the dimension of a relaxation time [18][19][20][21][22][23][24][25][26][27] and 𝐶 = , = ℏ𝜔 , is the specific heat or heat capacity per phonon normal mode [24,26]. The Callaway pseudo-relaxation time p describing the effect of N-processes is calculated as in the conventional steady-state local/linear treatment, by recalling that N-processes cannot change the total phonon wave-vector (total crystal momentum) [18][19][20][21][22][23][24][25][26][27]. The term 𝑔 , as given by Eq.

(2) represents the spectral energy density per phonon mode associated to the phonon gas drift [18,19].

B. Application to the 1D TTG configuration

The one-dimensional spatially periodic temperature profile is of the form ∆𝑇(𝑥, 𝑡) = 𝑇(𝑥, 𝑡) -𝑇 = ∆𝑇 (𝑡)𝑒 where 𝜂 = is the wavenumber of the thermal grating of spatial period d. Therefore, we seek a spatially periodic solution for 𝑈 , of the form 𝑈 , (𝑥, 𝑡) = 𝑈 , (𝑡)𝑒 . By noting that = (∆ ) , Eq. (1) becomes:

𝑑𝑈 , 𝑑𝑡 + 𝛾𝑈 , = 𝐶 𝜏 ∆𝑇 (3)
where 𝛾 = 𝜏 , + 𝑖𝜂𝑚𝑣 and 𝜏 = 𝜏 , -𝑖𝜂 𝛽 𝑝 𝜏 𝑞,𝑝 𝑁 𝑚𝑣 .

To obtain Eq. (3), we used the fact that for small ∆𝑇, we can write 𝑈 , = 𝐶 ∆𝑇 .

Eq. ( 3) is a simple one variable first-order inhomogeneous ordinary differential equation for where 𝑞 denotes Debye's cutoff wave-vector of the acoustic branch polarization p [18][19][20][21][22][23][24][25][26][27].

By considering the expression of 𝜏 , we can straightforwardly show that Eq. ( 5) leads to:

∆𝑇 (𝑡) = ∑ ∫ ∫ 𝑈 , (t) 𝜏 , 𝑞 𝑑𝑞 𝑑𝑚 2 ∑ ∫ 𝐶 𝜏 , 𝑞 𝑑𝑞 (6)
By inserting Eq. ( 4) into Eq. ( 6), we can directly extract the full expression of the normalized thermal decay 𝑌 (𝑡) = ∆ ( ) ∆ ( ) as:

𝑌 (𝑡) = 1 2 ∑ ∫ 𝐶 𝜏 , 𝑞 𝑑𝑞 𝐶 𝜏 , 𝜏 𝑒 𝑌 (𝑡′)𝑑𝑡′ + 𝐶
𝜏 , e 𝑞 𝑑𝑞 𝑑𝑚 (7) The exploitation of the expression of 𝜏 allows us to rearrange Eq. ( 7) in a form that clearly highlights the effects of phonon-phonon scattering N-processes and Callaway's model in comparison to the standard RTA model. 𝑌 (𝑡) can be written as:

𝑌 (𝑡) = 1 2 ∑ ∫ 𝐶 𝜏 , 𝑞 𝑑𝑞 𝐶 𝜏 , 𝑒 𝑌 (𝑡′)𝑑𝑡′ + 𝐶 𝜏 , e 𝑞 𝑑𝑞 𝑑𝑚 - 𝑖𝜂 2 ∑ ∫ 𝐶 𝜏 , 𝑞 𝑑𝑞 𝛽 𝑣 𝑚 𝐶 𝜏 , 𝜏 , 𝑒 𝑌 (𝑡′)𝑑𝑡′ 𝑞 𝑑𝑞 𝑑𝑚 (8)
The first term in Eq. ( 8) is the result we obtain using the standard RTA model [11][12][13][14][15], while the additional second term represents the effect of taking separately phonon-phonon scattering N-processes and U-processes in the framework of the modified Debye-Callaway model [18][19][20][21][22][23][24][25][26][27].

We apply Laplace transform to Eq. ( 7) in order to isolate the expression of the normalized thermal decay 𝑌 (𝑠) in Laplace domain. After performing simple calculations of the different integrals with respect to the directive cosine m and rearranging the different terms, we can straightforwardly obtain the following result:

𝑌 (𝑠, 𝜂) = ∑ ∫ 𝐶 𝜓 , (𝑠, 𝜂)𝑞 𝑑𝑞 ∑ ∫ 𝐶 𝜏 , 1 + 𝛽 𝜏 , -1 + 𝛽 𝜏 , 1 + 𝜏 , 𝑠 𝜓 , (𝑠, 𝜂) 𝑞 𝑑𝑞 (9)
where the function 𝜓 , (𝑠, 𝜂) is given by:

𝜓 , (𝑠, 𝜂) = 1 2𝜏 , 𝑑𝑚 𝛾 + 𝑠 = 𝑖 2𝐾𝑛 , Log 1 + 𝐾𝑛 , -𝑖𝐾𝑛 , 1 + 𝐾𝑛 , + 𝑖𝐾𝑛 , = 𝐴𝑟𝑐𝑡𝑎𝑛 𝐾𝑛 , 1 + 𝐾𝑛 , 𝐾𝑛 , (10)
where we introduce two non-dimensional Knudson numbers; a spatial one 𝐾𝑛 , = 𝜂𝑣 𝜏 , that compares the spectral phonon MFP to the thermal grating period and a temporal one 𝐾𝑛 , = 𝑠𝜏 , that compares the phonon combined relaxation time to the thermal decay time [11]. These two parameters specify the nature of the phonon transport regime completely in the spatio-temporal domain. In the diffusive limit, 𝐾𝑛 , ≪ 1 and 𝐾𝑛 , ≪ 1, while in the ballistic limit 𝐾𝑛 , ≫ 1 and 𝐾𝑛 , ≫ 1. In addition, we can obtain more insights onto these regimes based on the full expression of the normalized thermal decay given by Eq. ( 7) as we shall see later.

The behavior of 𝑌 (𝑠, 𝜂) is conditioned by the one of 𝜓 , (𝑠, 𝜂) depending on the phonon transport regime 𝐾𝑛 , → ; 𝐾𝑛 , → .

The conventional Fourier's form of the normalized thermal decay in the herein 1D TTG configuration under consideration is given by: 𝑌 (𝑡, 𝑥) = 𝑒 -Γ 𝐹 𝑡 𝑒 (11) where the thermal decay rate Γ = 𝛼 𝜂 = 𝜂 ; F, F and C are the thermal diffusivity, conductivity and total specific heat, respectively.

If one notes 𝑌 (𝑡) = 𝑒 -Γ 𝐹 𝑡 , we easily remark that the thermal decay rate will be given by:

Γ = 𝑌 (𝑡)𝑑𝑡 =𝑌 (0) (12)
We apply this formula to our result that is given by Eq. ( 9) and we set Γ = 𝜂 where 𝜅 represents the effective apparent thermal conductivity that would be extracted from the measured thermal decay in the considered 1D TTG configuration. We easily obtain after some algebra, the full expression of 𝜅 as:

𝜅 (𝜂) = 𝐶 ∑ ∫ 𝐶 𝜒 , 𝑞 𝑑𝑞 𝑣 𝐶 𝜏 , 1 + 𝛽 𝜏 , 1 -𝜒 , 𝐾𝑛 ,
𝑞 𝑑𝑞 (14) where the function 𝜒 , is given by:

𝜒 , (𝜂) = 𝜓 , (0, 𝜂) = 𝐴𝑟𝑐𝑡𝑎𝑛 𝐾𝑛 , 𝐾𝑛 , (15)

III. RESULTS AND DISCUSSION

In the theory section, we assumed that the Callaway pseudo-relaxation time p, describing the implicit effect of phonon-phonon scattering N-processes, does not depend on space and that this approximation should preserve the essential features of thermal conduction by phonons. This means that the dependence of the phonon gas drift on space is contained in the expression of the drift velocity 𝝀 𝒑 ℏ only through the temperature gradient 𝜵𝑇(𝑥) [19]. p is a complicated quantity, depending on 𝜏 , and 𝜏 , . This complication is necessary because of the behavior of N-processes, which shuffle crystal momentum back and forth between phonon normal modes, and then contribute implicitly to the lattice thermal conduction (resistance) process of a given SC crystal material [18,19].

In order to have a closer look at the steady-state behavior of the effective apparent thermal conductivity 𝜅 and the thermal decay rate Γ as functions of the thermal grating period and temperature, we consider natural silicon (Si) and Si0.7Ge0.3 alloy as two illustrative examples of single and alloy SC crystals, respectively. Indeed, the first experimental results regarding phonon MFP spectroscopy utilizing TTG technique were carried out on a silicon thin film [5][6][7]. In addition to three phonon-phonon anharmonic scattering U and N-processes, we assume scattering of phonons by the boundaries of the thick opaque SC crystal film and by impurities (defects). All geometrical and physical properties of Si and Si0.7Ge0.3 materials can be found in tables I and II of reference [24]. We address a particular attention to the case of Si, for which we use two different sets of expressions for the relaxation times of phonon-phonon scattering U and N-processes depending on the temperature regime. In the low temperature regime, up to 100K, the expressions of the relaxation times of the different phonon scattering processes considered are the conventionally and widespread ones we used in our early works [23,24]. In the high temperature regime (>100K), we exploit the expressions suggested by Ward and Broido based on an ab-initio approach using first-principles density-functional perturbation theory (DFPT) [28,29]. For the case of Si0.7Ge0.3 and because of the lack of finding expressions of the relaxation times of phonon-phonon scattering U and N-processes using firstprinciples DFPT calculations, we assume the conventional expressions to be valid over the entire temperature range considered [23,24].

We assume all physical properties of these SC crystals to be independent of temperature.

The total resistive phonon scattering rate 𝜏 , is obtained via Matthiessen's rule [22][23][24].

To simplify more the expression of 𝜅 in Eq. ( 14), we express it, as it is customary in the modified Debye-Callaway model, using a sum over one longitudinal (𝜅 ) and two degenerate transverse (𝜅 ) phonon acoustic branch polarizations [22][23][24][25][26]. We also make use of the usual change of variable 𝑦 = ℏ which allows us to obtain straightforwardly the final computational expression of 𝜅 .

A. Callaway's model vs standard RTA model

The reason that motivates the use of Callaway's model within the framework of the single mode relaxation time approximation, is the ability of this model to treat more fairly the different phonon-phonon scattering processes than does the standard RTA model. This model captures quite respectfully the peculiar effect of phonon-phonon scattering N-processes that distinguishes them from the rest of the phonon scattering processes including phonon-phonon scattering U-processes [18][19][20][21][22][23][24][25][26][27]. The standard RTA model often used, does not take this difference into consideration, and it treats all phonon scattering processes similarly, as if they were all independent resistive processes. It is well known however, that phonon-phonon scattering N-processes alone do not provide thermal resistance [18,19,[START_REF] Ziman | Electron and Phonons[END_REF]. scattering U and N-processes, were originally derived using approaches that implicitly assumed low frequency phonons and low temperature regime [19,[START_REF] Ziman | Electron and Phonons[END_REF]. Thus, it is not surprising that these expressions sometimes fail to predict the correct result, especially in the high temperature regime, depending on the used scattering term in the BPTE; standard or Callaway's and the nature of the SC crystal [29].

Furthermore, it is interesting to note that Callaway's model yields almost the same prediction of the temperature steady-state behavior of  of Si using the two different sets of expressions for the relaxation times of phonon-phonon scattering U and N-processes.

In the case of Si0.7Ge0.3, the deviation between Callaway's model and the standard RTA model including N-processes, occurs starting at room temperature. Therefore, we could infer that use of the conventional expressions of the relaxation times of phonon-phonon scattering U and N-processes for this alloy SC crystal is potentially valid up to at least, this temperature.

In regard to the above discussion, one could assert that a better analysis, interpretation and exploitation of the results of the TTG technique shall arguably consider Callaway's model.

B. Phonon transport process in the 1D TTG configuration

TDTR/FDTR and TTG are two different but complementary experimental techniques that have different excitation and detection procedures. The former one is more sensitive to the cross-plane phonon transport process while the latter is more sensitive to the in-plane one. The thermal penetration depth as defined based on the diffusive Fourier's law constitutes the characteristic length scale in TDTR/FDTR experiment [26]. On the other hand, and because of the nature of TTG experimental excitation and detection, no Fourier's based thermal penetration appears in the calculation, but instead the optical penetration depth could have an effect as mentioned by Chiloyan et al., [14] and Hubermann et al., [15]. In the 1D TTG configuration we are assuming in the herein work, the only characteristic length scale to be considered is the period of the thermal grating generated as a result of the interference pattern of the two crossed laser pulses shone on the surface of the opaque thick SC crystal film.

In our recent work regarding the analysis of the modulation frequency behavior of the reduced effective thermal conductivity of SC crystals that is observed in TDTR and FDTR experiments, we derived an expression of the effective apparent thermal conductivity of the SC crystal that is characterized with a universal SSF. We obtained the same SSF either using The universality of the SSF we found, was obtained naturally as a function of the only ratio of the spectral phonon MFP to the thermal penetration depth as defined based on the diffusive Fourier's law. The thermal penetration depth constitutes the characteristic length scale in the TDTR/FDTR geometry once an analogy with Fourier's based thermal conductivity is sought [26].

It is interesting to note from Eq. ( 14) that the expression we derived for the effective apparent thermal conductivity 𝜅 that would be extracted from the measured thermal decay in the considered 1D TTG configuration, has the same non-universal SSF, Chiloyan et al., found using the standard RTA model [12]. The only difference is that we have an effective relaxation time 𝜏 , = 𝜏 , 1 + , that is characteristic of Callaway's model in place of just the combined relaxation time 𝜏 , of the standard RTA. The full expression of the SSF is given by:

Ξ , (𝜂) = 6𝜋 𝐶 ∑ ∫ 𝐶 𝜒 , 𝑞 𝑑𝑞 1 -𝜒 , 𝐾𝑛 , (16)
Hua and Minnich found the same universal SSF of Maznev et al., [10] due to a further assumption they used in their mathematical derivation in what they called "weak quasiballistic" regime [11]. If we don't use this additional assumption, the general result we obtain is a non-universal SSF. Hence, the non-universal character of the SSF of the effective apparent thermal conductivity in the 1D TTG configuration, is not related to the form of the scattering term in the BPTE; standard or Callaway's. That means that considering the peculiar effect of phonon-phonon scattering N-processes through Callaway's model does not influence the expression of the SSF. The latter originates from the time integration of the thermal decay that embodies time nonlocal effects as shown in Eq. (7). The appearance of time nonlocal effects is a direct consequence of solving the time dependent BPTE in association with the initial boundary condition. These effects will indeed appear either using the standard RTA model or the full Callaway's model.

Before we tackle the general spectral result of 𝜅 given by Eq. ( 14) and the corresponding result of the thermal decay rate, it is very instructive to discuss first the simple, yet very meaningful, case of the Gray Spectrum Approximation (GSA). In this approximation, all phonon modes belonging to an acoustic branch polarization p have the same relaxation time independent of the wave-vector q for each phonon scattering process. We shall however continue to assume separately phonon-phonon scattering N-processes characterized by a relaxation time 𝜏 and phonon-phonon scattering resistive processes characterized by a relaxation time 𝜏 . In this case, we can easily show that the Callaway pseudo-relaxation time  will exactly be given by 𝜏 . This shows the very fundamental intertwining between anharmonic N-processes and resistive processes; the implicit effect of N-processes in the onset of a noninfinite thermal conductivity is taken account of, through the resisting causing collisions, namely the relaxation time of the resistive processes which effect is explicit [24]. In addition, we assume there is no distinction between the different acoustic phonon polarizations. These assumptions shall not significantly hamper the physical picture. Starting from the general expression of the normalized thermal decay in Laplace domain as given by Eq. ( 9) and applying the above assumptions, one gets straightforwardly the following GSA formula which we express using non-dimensional variables 𝜉 = 𝜂𝜏 𝑣, 𝜛 = 𝜏 𝑠 and

𝜏 = 𝜏 𝜏 ⁄ : ⎩ ⎪ ⎨ ⎪ ⎧𝑌 (𝜛, 𝜉) = Ψ(𝜛, 𝜉, 𝜏) (1 + 𝜏) -[(1 + 𝜏) + 𝜏𝜛]Ψ(𝜛, 𝜉, 𝜏) Ψ(𝜛, 𝜉, 𝜏) = 𝐴𝑟𝑐𝑡𝑎𝑛 𝜉 1 + 𝜏 + 𝜛 𝜉 (17)
Note that 𝜉 and 𝜛 are nothing else than the GSA versions of the non-dimensional spatial and temporal Knudson numbers introduced above, respectively.

We report in figure 2 , [13]. These oscillations are due to spatial nonlocal effects [16] and could also be viewed as a signature of the phonon hydrodynamic transport regime [17]. In this case of the 1D TTG configuration, the phonons travel at a speed determined by the ratio of the thermal grating period to the period of oscillations and would be comparable to the speed of second sound [17].

Here also, the impact of the crystal momentum shuffling effect of the phonon-phonon scattering N-processes is evident as the change in 𝜏 impacts both the amplitude and the period of the oscillations. It is remarkable to note however, from figures 2(c) and 2(d), that the oscillations seem to die off almost at the same moment independent of the value of 𝜏. We note that the consideration of Callaway's model in the framework of the simple GSA is able to exhibit one of the most important features that was recently highlighted using rigorous analysis based on the full linearized collision matrix [16,17].

As we mentioned above, we obtain the same expression of the SSF in the integrand of the effective apparent thermal conductivity that would be extracted from the measured thermal decay in the 1D TTG configuration using both Callaway and the standard RTA models. The non-universal character of this SSF is captured through the term worth mentioning that for the quasi-particle description of the phonon wave packet to be valid in the first place, the minimum phonon MFP cannot be less than 2a where a denotes the lattice constant of the SC crystal [START_REF] Ziman | Electron and Phonons[END_REF]. Therefore, we set the minimum value of d to be 𝑑 = 2𝑎.

For each of the three computation approaches, the behavior of 𝜅 shows three different Some samples are however thick enough, to be considered as bulk materials. These works show indeed a recovery of the thermal conductivity of Si membranes starting around 7-10µm. Once again, this anomaly is corrected using first-principles DFPT-based expressions of the relaxation times as illustrated in figure 3(c). Here also, we report the results using the two different couples of values for Debye temperatures of the longitudinal and transverse acoustic polarization branches. Callaway's model and the standard RTA model including N-processes predict now similar behaviors where the recovery of the bulk thermal conductivity of Si starts occurring for a thermal grating period around 100µm with a full recovery around 1000µm. This is in total agreement with previous theoretical works in the literature that used the standard RTA model [6,12,16].

There is a small difference between Callaway's model and the standard RTA model including N-processes. We will emphasize next on the impact of this difference on the information we could obtain relative to the transition between the diffusive and non-diffusive (quasi-ballistic) regimes in the phonon transport process, as well as on the phonon MFP spectrum distribution extraction in bulk SC crystals.

As mentioned by Chiloyan et al., the behavior of the thermal decay rate Γ as a function of the square of thermal grating wavenumber 𝜂 constitutes a very efficient metric that allows precise and quantitative determination of the length scale threshold at which the phonon nondiffusive transport regime onset occurs in a TTG experiment [14]. Indeed, 𝛤 is proportional to 𝜂 in the Fourier's diffusive regime and the proportionality coefficient is equal to the thermal diffusivity [13,14]. Any deviation from this scaling law is an indication of a deviation of the phonon transport process from the diffusive regime. transport regime decreases as expected by increasing T. Indeed, the phonon MFP decreases by increasing T [START_REF] Ziman | Electron and Phonons[END_REF]. We note also that in the ballistic limit (𝐾𝑛 , → +∞), (i) the three approaches collapse on each other and (ii) for each approach, all the curves for different T collapse on each other as expected, in accordance with the tendencies in figures 3(b) and 3(c).

In the TTG configuration, the characteristic length scale of the experiment (thermal grating period) does not depend on the model, Callaway's or standard RTA. We define the length scale threshold at which there will be a transition in the phonon transport process from the nondiffusive (quasi-ballistic) regime to the diffusive regime, as the thermal grating period for which 𝛤 𝛤 = 0.99 ⁄ . We summarize in Table 1, the values of these thresholds obtained from figures 4 and 5 for both Si and Si0.7Ge0.3 SC crystals using the three different models at different temperatures. The length scale threshold ranges from cms in the low T regime to mms in the high T regime. At each T and for each SC crystal, the behavior of the threshold for the three different manifest very close length scale thresholds for the onset of non-diffusive phonon transport regime for temperatures in the vicinity of their Top which happen to be around 20K for both SC crystals as can be seen in figure 1. We could attribute this result to the fact that in the neighboring of  peak, all phonon scattering processes, intrinsic and extrinsic, are important and contribute all significantly [28]. Since  of both Si and Si0.7Ge0.3 reaches its peak value at almost the same Top, this means that the interplay mechanism between the different phonon scattering processes in the vicinity of this temperature, would have comparable strengths or magnitudes in both SC crystals.

In order to shed more light on the implications of the difference between Callaway's model and the standard RTA model including N-processes, we report in Table 2, the values of the thermal grating period threshold defined in Table 1 in the case of Si in the high temperature regime. The relative difference between the values of the thermal grating period thresholds obtained using the two approaches remains almost the same over the whole high temperature range and is equal to ~2.3%. It is remarkable to notice that this is very close to the relative difference between the values of the steady-state thermal conductivities  obtained using these approaches 𝜅 -𝜅 𝜅 . Nevertheless, while in the study of the temperature behavior of the steady-state , this difference might be considered negligible, this cannot be the case when investigating the phonon MFP spectrum distribution. This fact, consolidates and comforts the importance of Callaway's model.

It is interesting to remind that in our 1D TTG experiment configuration for an opaque thick film (bulk material) SC crystal, there is no scattering process of phonons associated with the thermal grating period; the thermal grating does not physically reduce the phonon MFP [13].

The boundary scattering process we assumed in our computations corresponds to a fixed effective length scale that is representative of a bulk material [24].

In order to have an approximative yet very instructive physical picture, of the effect of phonon-phonon scattering N-processes in the framework of Callaway's model, on the onset of the non-diffusive phonon transport regime, we could use the simple argument of the GSA approach as we did above. In the light of this approach and based on a combination of Eqs. ( 12) and ( 17), one could straightforwardly get:

𝜏 𝛤(𝜏, 𝜉) = (1 + 𝜏) 𝜉 -(1 + 𝜏)𝐴𝑟𝑐𝑡𝑎𝑛 𝜉 1 + 𝜏 𝐴𝑟𝑐𝑡𝑎𝑛 𝜉 1 + 𝜏 (18)
Moreover, we can easily show that 𝜏 𝛤 = 𝜉 in this case. One sets the real number 0 < 𝑛 < 1 such that 𝛤 𝛤 = 𝑛 ⁄ . Thus, one could show after some algebra, that we obtain the following equation:

𝐴𝑟𝑐𝑡𝑎𝑛(𝑍)= 𝑍 1 + 𝑛𝑍 𝑍 = 𝜉 1 + 𝜏 (19)
If 𝑍 is the solution of Eq. ( 19), then one could write:

𝜉 = 𝑍 𝜏+𝑍 = 2𝜋𝜏 𝑣 𝑑 ( 20
)
The length scale threshold 𝑑 for the onset of the non-diffusive phonon transport regime at 100𝑛% of the Fourier's diffusive regime scales therefore, inversely to 𝜏 = 𝜏 𝜏 ⁄ . The peculiar crystal momentum shuffling effect of phonon-phonon scattering N-processes tends thus, independently from the other phonon scattering processes, to reduce the length scale threshold of occurrence of the transition from the diffusive regime to the non-diffusive (quasi-ballistic) regime in the phonon transport process. Consequently, it favors the maintenance of the phonon diffusive regime over a large length scale range. The standard RTA model excluding phononphonon scattering N-processes is unable to unveil this remarkable feature, which nonetheless will be hid when using the standard RTA model including these processes. Callaway's model sheds light on this key characteristic of N-processes straightforwardly. This simple analysis using the GSA approach confirms very well the results of the general spectral case shown in Tables 1 and2 above.

Phonon-phonon scattering N-processes are treated similarly to all other phonon scattering processes in the standard RTA model [START_REF] Ziman | Electron and Phonons[END_REF]. This does not mean they are not considered important. But rather, we commonly neglect in such a model the particularity these scattering processes have of shuffling the crystal momentum between the different phonon normal modes [18,19,[START_REF] Ziman | Electron and Phonons[END_REF]. The use of Callaway's model captures this peculiar effect twice; first through the combined relaxation time and second through Callaway's parameter, which leads to an additional term to the standard RTA model [18,19]. Therefore, by using the standard RTA model, we miss this effect either by including or excluding phonon-phonon scattering Nprocesses in the total relaxation time. This can be justified in the steady-state low temperature regime but not in the steady-state high temperature regime, where consideration of the peculiar implicit effect of phonon-phonon scattering N-processes becomes more relevant.

All the above results we obtained in the framework of Callaway's approach of the collision operator in the BPTE, shed light on the fundamental role of phonon-phonon scattering Nprocesses in the intrinsic intertwining interaction between low and high frequency phonons that leads to the onset of the non-diffusive (quasi-ballistic)-diffusive transition regime in the phonon transport process in cubic SC crystals in the 1D TTG configuration. As such, the approach would allow obtaining more details and information about this intermediate regime than what we can get based on the standard RTA approach used in previous works [5][6][7][8][9][10][11][12][13][14][15]. The accumulation function and more precisely, the thermal conductivity per phonon MFP are directly influenced.

Furthermore, the contribution weights of phonons with MFP within this quasi-ballisticdiffusive transition regime can be very sensitive to temperature, the SC crystal thin film thickness as well as the depth of the dissipated heat inside the latter. The study of the 1D TTG configuration we performed in the present work can be deepened and enlarged to a multidimensional TTG configuration, in order to evaluate the real impact of Callaway's model in phonon spectroscopy. Indeed, the SSF in the framework of this model will be important for a more accurate prediction of thermal conductivity reduction over the entire phonon spectrum.

This will allow a better understanding of how thermal length scales in the TTG experiment affect which phonons conduct heat in each transport regime, in a rather simpler manner. To a certain point, this will be equivalent to the rigorous treatment based on the full linearized collision matrix used in recent works [16,17]. Of course, these works have the merit of a more accurate physical picture. But Callaway's model big advantage and great usefulness will lie in its simplicity and reduced computational cost with the aim of applying a reconstruction method to obtain the phonon MFP spectral distribution from the experimentally measured thermal decay in the TTG experiment [5,[START_REF] Forghani | [END_REF]32]. The method used in the case of the standard RTA model will have to be modified [5]. This would probably increase the sensitivity of the reconstruction method to hyperparameters [33], but this is beyond the scope of the present work.

IV. CONCLUSIONS

An approach based on solving the transient single mode relaxation time approximated

Figure 1 :

 1 Figure 1 : (a) Computed steady-state behaviors of the thermal conductivity of Si single and Si0.7Ge0.3 alloy SC bulk crystals using the conventional expressions of the relaxation times of phonon-phonon scattering U and N-processes, as functions of temperature. Callaway's model (solid line), the standard RTA model excluding N-processes (dotted line) and including them (dashed line). (b) The case of the high temperature regime for Si using first-principles DFPTbased expressions of the relaxation times of phonon-phonon scattering U and N-processes. The experimental data of natural Si are extracted from reference [23].

  Callaway's model or the standard RTA model. In Callaway's model, we have an effective relaxation time 𝜏 , = 𝜏 , 1 + , in the integrand of the effective apparent thermal conductivity while in the standard RTA model, we have only the combined relaxation time 𝜏 , .

Figure 2 :

 2 Figure 2 : Computed behavior of the normalized thermal decay in the 1D TTG configuration in the framework of the GSA for a bulk SC crystal as a function of the non-dimensional time 𝒕 𝝉 𝑹 ⁄ , for different values of the parameters 𝝉 and 𝝃: (a) 𝝃 = 𝟎. 𝟏, (b) 𝝃 = 𝟏, (c) 𝝃 = 𝟏𝟎 and (d) 𝝃 = 𝟓𝟎.

  the computed behavior of the normalized thermal decay in the 1D TTG configuration in the framework of the GSA for a bulk SC crystal as a function of the nondimensional time 𝑡 𝜏 ⁄ , for different values of the parameters 𝜏 and 𝜉. For large values of the thermal grating period that correspond to small values of the non-dimensional spatial Knudson number 𝜉, the phonon transport regime is diffusive and the normalized thermal decay decreases with time in a Fourier's like exponential fashion. As discussed by Collins et al., based on the standard RTA model [13], the full solution derived from the BPTE seems to decrease slowly compared to the conventional Fourier's solution. Furthermore, and as one can see in figures 2(a) and 2(b), consideration of the peculiar implicit effect of phonon-phonon scattering Nprocesses through Callaway's model tends to increase the decreasing rate of the normalized thermal decay 𝑌 . The curve of 𝑌 falls faster by increasing 𝜏. As 𝜉 increases, the phonon transport regime transitions to a quasi-ballistic (non-diffusive) regime, then to a full ballistic regime for very high values of 𝜉 where the phonon MFP becomes very large in comparison to the thermal grating period. In this regime, 𝑌 manifests an oscillatory behavior as a function of time that is a manifestation of the phonons traveling ballistically as explained by Collins et al.

  this function on the material properties, in addition to the non-dimensional spatial Knudson number 𝐾𝑛 , .In the diffusive regime (𝐾𝑛 , << 1), , , and 𝜒 , tend to 1/3 and 1, respectively.Hence 𝜅 tends logically and naturally to the expression of steady-state Fourier's thermal conductivity within the framework of the modified Debye-Callaway model[18][19][20][22][23][24]. On the other hand, in the ballistic regime limit (𝐾𝑛 , >> 1),

Figure 3 :

 3 Figure 3 : Computed room temperature steady-state behaviors of the 1D TTG effective apparent thermal conductivity of Si single (a) and Si0.7Ge0.3 alloy (b) SC bulk crystals using the conventional expressions of the relaxation times of phonon-phonon scattering U and Nprocesses, as functions of the thermal grating period. Callaway's model (solid line), the standard RTA model excluding N-processes (dotted line) and including them (dashed line). (c) The case of Si using first-principles DFPT-based expressions of the relaxation times of phonon-phonon scattering U and N-processes.

  regimes: (i) 𝜅 increases slowly as function of d for small values of the latter, then (ii) the increasing rate gets faster with d over a certain intermediate interval and finally (iii) 𝜅 saturates to a higher value for long values of d. These three regimes correspond to the quasiballistic (non-diffusive), intermediate (transitional) and diffusive ones, respectively, of the phonon transport process. Nonetheless, we clearly see in figures 3 how the features of these three phonon transport regimes are affected and hugely impacted by the computation model used, the nature of the SC cubic crystal; single or alloy as well as the form of the expressions of the relaxation times of phonon-phonon scattering U and N-processes. The zone of the transition from the quasi-ballistic regime to the diffusive regime seems to be particularly influenced in terms of onset threshold, width, height and increasing rate (slope). Excluding phonon-phonon scattering N-processes in the standard RTA model tends to widen this quasiballistic-diffusive transition zone in addition to its dramatic over-estimation of the value of 𝜅 in the diffusive regime. As one can see in figure 3(a), by using the conventional expressions of the relaxation times of phonon-phonon scattering U and N-processes, Callaway's model seems to recover the bulk thermal conductivity of Si at a shorter thermal grating period threshold (around 1µm) than does the standard RTA model. This is an odd prediction in comparison to experimental TTG results on Si material. It is true that several published works regarding TTG experiment on Si, dealt with Si membranes or thin films [5-9].

Figure 4 :

 4 Figure 4 : Computed behaviors of the 1D TTG thermal decay rate of Si SC bulk crystal as functions of the thermal grating wavenumber squared using Callaway's model and the standard RTA model excluding and including phonon-phonon scattering N-processes; at room temperature (a) and at different temperatures: Callaway (b), standard RTA with N-processes (c) and standard RTA without N-processes (d).

  models used, mirrors to a certain point, the steady-state T behavior of the thermal conductivity  discussed above in figure1. The three models lead to almost the same values of the length scale threshold for temperatures below the optimal Top of the peak value of  and start to depart from each other for temperatures above Top. For each SC crystal above its Callaway's model shows the lowest threshold and the standard RTA model without phonon-phonon scattering Nprocesses included, shows the highest. It is worth noticing also, that both Si and Si0.7Ge0.3

  Boltzmann-Peierls Transport Equation (BPTE) in the framework of modified Debye-Callaway model has been developed to analyze the thermal decay of opaque thick semiconductor (SC) cubic crystal films in the 1D configuration of the Transient Thermal Grating (TTG) experiment. We have obtained a non-universal spectral suppression function (SSF) in the integrand of the effective apparent thermal conductivity that is similar to the one obtained using the standard single mode relaxation time approximation (RTA) model. The only difference is that we have an effective relaxation time 𝜏 , = 𝜏 , 1 + , that is characteristic of Callaway's model in place of just the combined relaxation time , C q p of the standard RTA model. This proves that the non-universal character of the SSF in the TTG experiment does not depend on the form of the collision operator approximation in the BPTE; Callaway's or standard. The SSF captures and describes very well the intertwining interaction between low and high frequency phonons in the onset of the quasi-ballistic (non-diffusive) heat transport regime and as such, the reduction of the effective apparent thermal conductivity of SC crystals observed in TTG experiments.Callaway's approach unveils the central and relevant role that the meticulously implicit shuffling effect of the crystal momentum by phonon-phonon scattering N-processes, plays in the onset of the non-diffusive (quasi-ballistic) regime in the phonon transport process in SC crystals. Thus, the peculiar effect of phonon-phonon scattering N-processes captured by Callaway's model has undoubtedly an impact on the extraction of the phonon MFP spectrum distribution, especially in the high temperature regime.

  𝑈 , that we can easily and readily solve. We get the following solution:

	𝑈 , (t) =	𝐶 𝜏	𝑒	∆𝑇 (𝑡′)𝑑𝑡′ + 𝐶 ∆𝑇 (0)e	(4)
	Another relation between 𝑈 , (t) and ∆𝑇 (𝑡) is obtained through the energy conservation
	relation. The latter takes the form:
		𝐶 𝜏		∆𝑇 (𝑡) -	𝑈 , (t) 𝜏 ,	𝑞 𝑑𝑞 𝑑𝑚 = 0 (5)

Table 1 :

 1 Computed thermal grating period thresholds at which 𝛤 𝛤 = 0.99 ⁄ for both Si and Si0.7Ge0.3 SC bulk crystals at different temperatures, using Callaway's model and the standard RTA model excluding and including phonon-phonon scattering N-processes.

		Thermal grating period threshold at 99% of Fourier's diffusive regime
	Temperature (K)	Si	Callaway Si0.7Ge0.3	Standard RTA with C Si Si0.7Ge0.3	Standard RTA with R Si Si0.7Ge0.3
	3	15.783cm	13.126cm	15.783cm	13.126cm	15.783cm	13.126cm
	10	14.393cm	10.667cm	14.393cm	10.667cm	14.729cm	10.915cm
	20	9.727cm	9.289cm	9.954cm	9.289cm	11.97cm	9.954cm
	300	5.581mm	2.3247mm	5.712mm	2.6087mm	9.506cm	2.277cm
	500	3.52mm	226.6498µm	3.602mm	403.279µm	9.506cm	1.4033cm

Table 2 :

 2 Computed thermal grating period thresholds at which 𝛤 𝛤 = 0.99 ⁄ for Si SC bulk crystal in the high temperature regime, using Callaway's model and the standard RTA model including phonon-phonon scattering N-processes.

		Thermal grating period threshold at 99% of Fourier's diffusive regime (mm)
	Temperature (K)	Callaway 𝑑	Standard RTA with C 𝑑	Relative difference (%) 𝑑 -𝑑 𝑑
	100	15.747	16.114	2.28
	200	8.07	8.259	2.29
	300	5.581	5.712	2.29
	500	3.52	3.602	2.28
	800	2.434	2.491	2.29

We report, respectively, in figures 4 and 5, the computed behaviors of 𝛤 in the 1D TTG configuration as functions of 𝜂 for Si and Si0.7Ge0.3 SC crystals at different temperatures for the three models considered; Callaway's model and the standard RTA model with and without N-processes included.

We remind here that in the case of Si, we use the conventional expressions of the relaxation times of phonon-phonon scattering U and N-processes for the temperatures 3K, 10K and 20K and we use first-principles DFPT-based expressions of these relaxation times for the temperatures 300K and 500K. In the case of Si0.7Ge0.3, we use the conventional expressions of these relaxation times for all temperatures as we mentioned in the beginning of the discussion section. As one can see in these figures, the three models show the deviation of the phonon transport process from the Fourier's diffusive regime to occur at different thresholds depending on ambient temperature T. For each of the approaches, 𝛤 decreases and 𝜂 at which the quasiballistic (non-diffusive) phonon transport regime starts to manifest itself, shifts to higher values, by increasing T. This means that the onset length scale threshold of the non-diffusive phonon