Radiative Thermal Rectification between SiC and SiO2
Karl Joulain, Younes Ezzahri, Jérémie Drevillon, Benoit Rousseau, Domingos De Sousa Meneses
aa r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Radiative thermal rectification betweenSiC and SiO Karl Joulain, , ∗ Youn`es Ezzahri, J´er´emie Drevillon, BenoˆıtRousseau, and Domingos De Sousa Meneses, , ∗ Institut Pprime, Universit´e de Poitiers-CNRS-ENSMA, TSA 41105, F-86073 Poitiers, France LTN, CNRS UMR 6607, B.P 90604, F-44306 Nantes, France CNRS, CEMHTI UPR3079, Universit´e d’Orl´eans, F-45071 Orl´eans, France ∗ [email protected], [email protected] Abstract:
By means of fluctuational electrodynamics, we calculateradiative heat flux between two planar materials respectively made of SiCand SiO . More specifically, we focus on a first (direct) situation where oneof the two materials (for example SiC) is at ambient temperature whereasthe second material is at a higher one, then we study a second (reverse)situation where the material temperatures are inverted. When the two fluxescorresponding to the two situations are different, the materials are saidto exhibit thermal rectification, a property with potential applications inthermal regulation. Rectification variations with temperature and separationdistance are reported here. Calculations are performed using materialoptical data experimentally determined by Fourier transform emissionspectrometry of heated materials between ambient temperature (around 300K) and 1480 K. It is shown that rectification is much more important inthe near-field domain, i.e. at separation distances smaller than the thermalwavelength. In addition, we see that the larger is the temperature difference,the larger is rectification. Large rectification is finally interpreted due toa weakening of the SiC surface polariton when temperature increases, aweakening which affects much less SiO resonances. © 2018 Optical Society of America OCIS codes: (160.6840) Thermo-optical materials; (260.2160) Energy transfer; (290.6815)Thermal emission; (350.5610) Radiation
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Nishikawa, H. Izuka and H. Toshiyoshi, “Experimental investigation of radiative thermal rectifier usingvanadium dioxyde,” Appl. Phys. Lett. , 191907 (2013). . Introduction Since the seminal works of Rytov [1] and Polder and Van Hove [2], it has been known thatradiative heat transfer (RHT) can be much larger than classical Planck’s prediction when theseparation distance between two solid bodies are small compared to the thermal wavelength atwhich heat radiation is exchanged. Enhancement is particularly large in the case of dielectricssupporting phonon-polariton such as SiC or SiO [3] that have probably been the most stud-ied materials in the framework of near-field heat transfer. In the case of materials supportingpolaritons, these non radiative modes can couple so that heat is transferred in vacuum throughphoton tunneling. However, the magnitude of the enhancement strongly depends on the ampli-tude and the width of the resonance in the material optical response which can be affected by itstemperature [4, 5]. Indeed, when the temperature increases, one knows that phonons in generalincluding phonon-polaritons are affected by several processes like phonon-phonon interactionsso that theses modes see their lifetime decreases [6]. But phonons in different materials are notequally affected by temperature changes.A question which naturally arises is whether the heat flux between a material at ambient tem-perature and another one different at a higher temperature will be the same when temperaturesare inverted. If it is not the case, the two materials couple is said to exhibit thermal rectification,a thermal analog of electrical rectification better known as the diode.Thermal rectification has recently received a growing interest due to the emergence of ther-mal flow management needs related to limited energy and global warming issues. If thermalequivalent of the diode and the transistor would be designed and fabricated, this could open theway to new passive temperature regulation and thermal circuits with no needs of electronicsleading to very reliable thermal management.In the last decade, thermal conductors exhibiting thermal rectification have been proposed. Inthese devices, heat carriers flux is made asymmetric by nanostructuring the material or by guid-ing the carriers flux differently whether they are moving in one direction or the other [7–15].These physical ideas can be used to conceive and fabricate a thermal transistor leading to thepossibility of logical thermal circuits [16, 17]. More recently, radiative thermal rectifiers havebeen proposed [18–22] as well as radiative thermal transistors [23, 24]. Very promising deviceshave been imagined based on materials exhibiting phase change transition such as Vanadiumdioxyde (VO ) [20, 23, 25], superconductors [22] or thermochromic materials [26]. The goalof this article is to show that radiative thermal rectification can also be achieved with the moststudied polaritonic supported materials like SiC and SiO .We first present optical properties measurements showing how both SiC and SiO dielectricfunctions change with temperature. These measurements performed using Fourier TransformEmission Spectroscopy allow us to make near-field radiative heat transfer calculations takinginto account the actual material dependance with temperature. Similar near-field calculationtaking into account the temperature dependance of optical properties have already been takenin the past [27] with measurement on 3-C SiC of [28] but in a different situation between SiCand metallic coated-SiC. Here, we calculate heat transfer between two planar interfaces of SiCand SiO . We show that these two materials exhibit almost no thermal rectification when theexchange occurs in the far-field i.e. when the main contribution to the flux comes from thepropagative waves. On the contrary, at distances below 100 nm, we show that rectification ratiodrastically increases in order to reach values as high as the highest predicted and measuredones in the literature. We finally show that rectification is mainly due to the SiC resonanceattenuation at high temperature. . SiC and SiO optical datas In this section, we report SiC and SiO optical properties temperature dependance with tem-perature. The dielectric functions of SiC and vitreous SiO have been extracted from measure-ments of their spectral emittance at near normal incidence. The SiC sample provided by MTICorporation is a single crystal with a (0001) orientation and two epi polished sides. The stackingsequence is of 6H type and its electric resistivity lies between 0.02 and 0.2 ohm.cm. The SiO material is a polished plate of vitreous silica with low OH content ( < ppm ). The emittancespectra were acquired with a homemade spectrometer built around two Bruker infrared spec-trometers, a Vertex 80v working under vacuum and a Vertex 70 purged with dry air (see [29]).The first and second devices have been used to probe the far- and mid-infrared ranges respec-tively, at a spectral resolution of 4 cm − . In order to avoid parasite fluxes during the measure-ments, a CO laser has been used to heat the samples up to 1500 K. The retrieval of the opticalfunctions has been performed in a second step by fitting the experimental data with adequatephysical expressions. A semi-quantum dielectric function model, well adapted for crystallinemedia, was selected to reproduce the infrared response of the silicon carbide sample [30]. Thedisordered nature of vitreous silica needed instead the use of a causal Gaussian dielectric func-tion model able to take account for the inhomogeneous broadening of the absorption bands ofglasses [31]. The whole process allowed to obtain the temperature dependences of the opti-cal properties for both materials in a wide range, the results are reported in Fig. 1. For SiO ,we note that the resonance around 2 × rad s − slightly shifts and broadens when the tem-perature increases. For SiC, we note a similar behavior (shift and broadening) but even morepronounced for the resonance around 1.5 × rad s − . This is due to the fact that anharmonic-ity effects responsible of the phonon-phonon interaction increases with the temperature [6]. Wewill see that this behavior is responsible for the important rectification between SiO and SiCin the near-field. Note however that the second resonance in SiO does not change a lot withtemperature as long as temperature is below the vitreous transition which occur around 1500K [32]. This is probably related to the very weak SiO thermal dilatation.
3. Radiative heat transfer between SiC and SiO We now come to the calculation of the RHT between two planar interfaces made of SiC(medium 1) and SiO (medium2). RHT between two planar interfaces can easily be calcu-lated using fluctuational electrodynamics formalism. Heat flux appears as a semi-analyticalexpressions which is nothing else than the summation of individual plane wave contributions tothe heat transfer. These plane waves are labelled by a triplet ( w , K , a ) where w is the angularfrequency of the wave, K is the wavector component parallel to the interfaces whereas a isthe wave polarization ( s or p ). Let us call T temperature of medium 1 and T temperature ofmedium 2. Heat flux expression between media 1 and 2 reads [4, 33] j ↔ = (cid:229) a = s , p Z ¥ [ Q ( w , T ) − Q ( w , T )] d w Z ¥ KdK p t a ( w , K ) (1)where Q ( w , T ) = ¯ h w / [ exp [ ¯ h w / k b T ] − ] is the mean energy of a photon at temperature T .In the preceding expression, t a ( w , K ) appears as a transmission coefficient for a plane wave ( w , K , a ) between medium 1 and medium 2 or medium 2 and medium 1. Note that expres-sions of transmission coefficient are different whether the wave is propagative ( K < w / c orevanescent ( K > w / c ). If we call medium 3, the medium separating the two other media andintroducing Fresnel reflection coefficients, the expression for the transmission coefficient readsfor propagating waves t a ( w , K ) = ( − | r a | )( − | r a | ) | − r a r a e i g d | (2) ig. 1. Measured real and Imaginary part of the dielectric function e variation with angularfrequency w for different temperatures. Top 2 figures : SiO . Bottom 2 figures : SiC. nd for evanescent waves t a ( w , K ) = ` ( r a ) ` ( r a ) e − ` ( g ) d | − r a r a e − ` ( g ) d | (3)In these expressions, ` denotes the imaginary part, g i is the perpendicular wave vector compo-nent in medium i which for non magnetic materials reads g i = ( e i k − K ) / . The usual fresnelreflection coefficient are given by r si j = ( g i − g j ) / ( g i + g j ) and r pi j = ( e j g i − e i g j ) / ( e j g i + e i g j ) when one considers local materials which is the case at nanometric separation distance [34,35].The flux expression only depends on the material temperatures, their local optical response andthe separation distance between the bodies. Note that the symmetry of Eq. (1) implies that recti-fication will occur if and only if the two materials are different and have their optical propertiesthat change with temperature in a different manner.Optical properties variations with temperature have rarely been taken into account, mostlyin the case of phase change materials. Here, our main goal is to perform these calculations inthe case of two very well studied materials but for which optical properties variations with tem-perature are taken into account. The input properties in our calculation are the optical propertiespresented in the preceding section. It can be noted in these measurements that they have beentaken at close temperatures for both materials but not exactly at the same one. As explainedin the introduction, our goal is also to explore how a system constituted of SiC and SiO canexhibit radiative thermal rectification. Thus, a difficulty arises since rectification compares asituation where the materials are at two different temperatures with a second situation wherethe temperatures considered before have been inverted. For example, measurements at ambienttemperature for SiC have been performed at 295 K whereas they have been performed at 298 Kfor SiO . In our calculation, we choosed ambient temperature to be 297 K and we assumed thatoptical properties at this temperature were the ones at 295 K for SiC and 298K for SiO . Othertemperatures considered in the paper are 471 K, 671 K, 981 K, 1102 K and 1470 K. To performheat flux calculations, optical properties for SiC and SiO have been respectively taken at 464K and 479 K, 686 K and 672 K, 996 K and 966 K, 1105 K and 1098 K, 1460 K and 1480 K(See Fig. 1).We report in Fig. 2 heat flux variations as a function of the separation distance between thetwo materials. We compare the situation where SiO is at 297 K and SiC at 1470 K with thesituation where the temperatures are inverted. We note that for separation distances larger thana few microns the two fluxes are constant and almost superimposed each other. When SiO is atambient temperature, the value of the heat flux is 1.872 × W m − whereas it is 1.896 × Wm − in the reverse case. These two values are around 71% of the heat exchanged between twoblackbodies for which Fresnel reflection factors are equal to 0 in the fluxes expression. Whenthe separation distance is decreased, the heat flux increases for both situations. For distanceslarger than 200 nm, heat fluxes are very close in both situations. For distances smaller than 200nm, we see that the situation where SiC remains at ambient temperature exhibits a higher fluxthan the one corresponding to the reverse one.We define rectification R as the relative variation of the heat flux [20,21] in the two situationsso that R = | j ↔ ( T , T ) − j ↔ ( T , T ) | Max [ j ↔ ( T , T ) , j ↔ ( T , T )] (4)With such a definition, R = R = for different temperatures. One material is always at 297 K whereas the secondone is taken at 471 K, 671 K, 981 K, 1102 K or 1470 K. We note a very similar behavior for ig. 2. Computed radiative heat transfer between a plane interface of SiC and a second oneconstituted of SiO versus their separation distances. Two situations are compared. In thefirst one (plain) SiO is at 297 K and SiC at 1470 K whereas in the second one (dashed)SiC is at 297 K and SiO is at 1470 K.Fig. 3. Computed rectification variations as a function of the separation distance betweentwo planar interfaces made of SiC and SiO when one material is at 297 K and the secondone at 471 K, 671 K, 981 K, 1102 K or 1470 K. ll the cases presented. For distances larger than 200 nm rectification remains very low. It goesfrom 10 − for the smallest temperature difference to 10 − for the largest. One can thereforesay that SiC and SiO exhibit very low radiative thermal rectification in the far field and moregenerally for separation distances larger than 200 nm. Below 200 nm, rectification increasesdrastically to reach values up to 0.7 for a temperature of 1470 K. Rectification saturates whenthe separation distance goes below 10 nm. Note also that the higher temperature difference is,the higher is the rectification in the near-field.
4. Discussion
Let us now shed some light on the physical reason that is responsible of the behavior of therectification between SiC and SiO with both distance and temperature. To do so, we plotin Fig. 4 the variation of the spectral radiative heat transfer flux as a function of the angularfrequency for the case where one material is at 297 K and the other one at 1470 K. The spectralflux is represented for 4 different separation distances 1 m m, 100 nm, 10 nm and 1nm. Atlarge distance (here 1 m m), the spectral flux is very similar to the one representative of theexchange between two black bodies except that there exist some frequencies where one canidentify dips. These dips correspond to the frequencies where SiO and SiC exhibit resonances.Indeed, SiO exhibit two resonances and SiC one resonance that had been identified in thefirst section. At these frequencies SiO and SiC are very reflective so that transfer is greatlyreduced. However, the range of frequencies where materials are low absorbent is much smallerthan the one where both materials almost behave as blackbodies. This explains why at largedistance when this spectral flux is integrated over all frequencies, heat flux reaches 70% of theone between two blackbodies. If we examine carefully the region where the spectral flux ismuch lower than between blackbodies, one see that these are the regions where the spectralfluxes differ in the direct and the reverse situations. We notice that the depth of the dips is lesspronounced when the material concerned is at high temperature. For example dips for SiO (around 9 × rad s − and 2 × rad s − ) are less pronounced in the plain curve whereSiO is at high temperature. In the same way, the dip corresponding to the SiC resonancearound 1.5 × rad s − is smaller in the dashed curve corresponding to the case where SiCis at high temperature. Despite of these differences, as most of the contribution comes from theregion where the spectral fluxes are equal, rectification in the far-field is very small and evensmaller at lower temperature when the dielectric function is still close to the one at ambienttemperature.When the distance is reduced, the situation drastically changes. The main contributions tothe spectral fluxes mainly comes from the frequencies where the resonances are present. Thisis a well known behavior when one is treating materials supporting phonon polaritons such asSiC and SiO . Indeed, the heat flux is completely dominated by the contribution of evanescentwaves in p polarization [36, 37] which depends on the imaginary part of the static reflectioncoefficient which reads for medium i ( e i − ) / ( e + ) . The frequency where e i is approaching -1 corresponds to the one where there is the largest density of states of surface polaritons close tothe material. What appears here is that the contribution of the polaritons of a material is reducedwhen the material is at high temperature. Moreover, one notes that SiO and SiC do not behavein the same way. While the SiO resonance contribution is weakly attenuated, the one of SiC ismuch more decreased by almost one order of magnitude above 1000 K. This explains why theheat flux is larger when the SiC is at low temperature. Moreover rectification does not changebelow a certain distance which correspond to the one where the flux is completely dominatedby the p contribution of the evanescent waves. It has been shown in several papers [3, 36, 38],that this flux behaves as 1 / d multiplied by the product of the imaginary parts of the reflectioncoefficients. When the distance decreases, both fluxes increase following the same distance ig. 4. Computed spectral radiative heat transfer between two planar interfaces made ofSiC and SiO in the direct and the reverse situation for 4 different separation distances:SiO is at 297 K and SiC is at 1470 K in the first situation (plain) whereas SiO is at 1470K and SiC is at 297 K in the second situation (dashed). caling law. Therefore, rectification becomes distance independent and saturates.
5. Conclusion