Abstract

By means of fluctuational electrodynamics, we calculate radiative heat flux between two planar materials respectively made of SiC and SiO2. More specifically, we focus on a first (direct) situation where one of the two materials (for example SiC) is at ambient temperature whereas the second material is at a higher one, then we study a second (reverse) situation where the material temperatures are inverted. When the two fluxes corresponding to the two situations are different, the materials are said to exhibit thermal rectification, a property with potential applications in thermal regulation. Rectification variations with temperature and separation distance are reported here. Calculations are performed using material optical data experimentally determined by Fourier transform emission spectrometry of heated materials between ambient temperature (around 300 K) and 1480 K. It is shown that rectification is much more important in the near-field domain, i.e. at separation distances smaller than the thermal wavelength. In addition, we see that the larger is the temperature difference, the larger is rectification. Large rectification is finally interpreted due to a weakening of the SiC surface polariton when temperature increases, a weakening which affects much less SiO2 resonances.

© 2015 Optical Society of America

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  8. B. Li, L. Wang, and G. Casati, “Thermal diode : rectification of heat flux,” Phys. Rev. Lett. 93, 184301 (2004).
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  9. B. Li, J. Lan, and L. Wang, “Interface thermal resistance between dissimilar anharmonic lattices,” Phys. Rev. Lett. 95, 104302 (2005).
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  11. B. Hu, L. Yang, and Y. Zhang, “Asymmetric heat conduction in nonlinear lattices,” Phys. Rev. Lett. 97, 124302 (2006).
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  12. J. Hu, X. Ruan, and Y. P. Chen, “Thermal conductivity and thermal rectification in graphene nanoribbons: a molecular dynamics study,” Nano Letters 9, 2730–2735 (2009).
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  13. N. Yang, N. Li, L. Wang, and B. Li, “Thermal rectification and negative differential thermal resistance in lattices with mass gradient,” Phys. Rev. B 76, 020301(R) (2007).
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  14. D. Segal, “Single mode heat rectifier: controlling energy flow between electronic conductors,” Phys. Rev. Lett. 100, 105901 (2008).
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  15. N. Yang, G. Zhang, and B. Li, “Thermal rectification in asymmetric graphene ribbons,” Appl. Phys. Lett. 95, 033107 (2009).
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  16. L. Wang and B. Li, “Thermal logic gates: computation with phonons,” Phys. Rev. Lett. 99, 177208 (2007).
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  17. W. C. Lo, L. Wang, and B. Li, “Thermal transistor: heat flux switching and modulating,” Journal of the Physical Society of Japan 77, 054402 (2008).
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  18. C. R. Otey, W. T. Lau, and S. Fan, “Thermal rectification through vacuum,” Phys. Rev. Lett. 104, 154301 (2010).
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    [Crossref]
  20. P. Ben-Abdallah and S.-A. Biehs, “Phase-change radiative thermal diode,” Appl. Phys. Lett. 103, 191907 (2013).
    [Crossref]
  21. E. Nefzaoui, J. Drevillon, Y. Ezzahri, and K. Joulain, “Simple far-field radiative thermal rectifier using Fabry–Perot cavities based infrared selective emitters,” Appl. Opt. 53, 3479 (2014).
    [Crossref] [PubMed]
  22. E. Nefzaoui, K. Joulain, J. Drevillon, and Y. Ezzahri, “Radiative thermal rectification using superconducting materials,” Appl. Phys. Lett. 104, 103905 (2014).
    [Crossref]
  23. P. Ben-Abdallah and S.-A. Biehs, “Near-field thermal transistor,” Phys. Rev. Lett. 112, 044301 (2014).
    [Crossref] [PubMed]
  24. K. Joulain, Y. Ezzahri, J. Drevillon, and P. Ben-Abdallah, “Modulation and amplification of radiative far field heat transfer:Towards a simple radiative thermal transistor,” Appl. Phys. Lett. 106, 133505 (2015).
    [Crossref]
  25. P. van Zwol, L. Ranno, and J. Chevrier, “Emissivity measurements with an atomic force microscope,” J. Appl. Phys. 111, 063110 (2012).
    [Crossref]
  26. J. Huang, Q. Li, Z. Zheng, and Y. Xuan, “Thermal rectification based on thermochromic materials,” International Journal of Heat and Mass Transfer 67, 575–580 (2013).
    [Crossref]
  27. H. Iizuka and S. Fan, “Rectification of evanescent heat transfer between dielectric-coated and uncoated silicon carbide plates,” J. Appl. Phys. 112, 024304 (2012).
    [Crossref]
  28. D. Olego and M. Cardona, “Temperature dependence of the optical phonons and transverse effective charge in 3C-SiC,” Phys. Rev. B 25, 3889 (1982).
    [Crossref]
  29. D. De Sousa Meneses, P. Melin, L. del Campo, L. Cosson, and P. Echegut, “Infrared Physics & Technology,” Infrared Physics & Technology 69, 96–101 (2015).
    [Crossref]
  30. D. De Sousa Meneses, J.-F. Brun, P. Echegut, and P. Simon, “Contribution of semi-quantum dielectric function models to the analysis of infrared spectra,” Appl. Spectrosc. 58(8), 969–974 (2004).
  31. D. De Sousa Meneses, M. Malki, and P. Echegut, “Optical and structural properties of calcium silicate glasses,” Journal of Non-Crystalline Solids 352, 5301–5308 (2006).
    [Crossref]
  32. D. De Sousa Meneses, M. Eckes, L. del Campo, and P. Echegut, “Phase transformations of crystalline SiO2 versus dynamic disorder between room temperature and liquid state,” Journal of Physics: Condensed Matter 26, 255402 (2014).
  33. S. A. Biehs, E. Rousseau, and J. J. Greffet, “Mesoscopic description of radiative heat transfer at the nanoscale,” Phys. Rev. Lett. 105, 234301 (2010).
    [Crossref]
  34. P.-O. Chapuis, S. Volz, C. Henkel, K. Joulain, and J.-J. Greffet, “Effects of spatial dispersion in near-field radiative heat transfer between two parallel metallic surfaces,” Phys. Rev. B 77, 035431 (2008).
    [Crossref]
  35. F. Singer, Y. Ezzahri, and K. Joulain, “Near field radiative heat transfer between two nonlocal dielectrics,” J. Quant. Spectr. and Rad. Transf. 154, 55–62 (2015).
    [Crossref]
  36. K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57, 59–112 (2005).
    [Crossref]
  37. A. Volokitin and B. Persson, “Near-field radiative heat transfer and noncontact friction,” Rev. Mod. Phys. 79, 1291–1329 (2007).
    [Crossref]
  38. E. Rousseau, M. Laroche, and J.-J. Greffet, “Asymptotic expressions describing radiative heat transfer between polar materials from the far-field regime to the nanoscale regime,” J. Appl. Phys. 111, 014311 (2012).
    [Crossref]
  39. K. Ito, K. Nishikawa, H. Izuka, and H. Toshiyoshi, “Experimental investigation of radiative thermal rectifier using vanadium dioxyde,” Appl. Phys. Lett. 103, 191907 (2013).

2015 (3)

K. Joulain, Y. Ezzahri, J. Drevillon, and P. Ben-Abdallah, “Modulation and amplification of radiative far field heat transfer:Towards a simple radiative thermal transistor,” Appl. Phys. Lett. 106, 133505 (2015).
[Crossref]

D. De Sousa Meneses, P. Melin, L. del Campo, L. Cosson, and P. Echegut, “Infrared Physics & Technology,” Infrared Physics & Technology 69, 96–101 (2015).
[Crossref]

F. Singer, Y. Ezzahri, and K. Joulain, “Near field radiative heat transfer between two nonlocal dielectrics,” J. Quant. Spectr. and Rad. Transf. 154, 55–62 (2015).
[Crossref]

2014 (4)

D. De Sousa Meneses, M. Eckes, L. del Campo, and P. Echegut, “Phase transformations of crystalline SiO2 versus dynamic disorder between room temperature and liquid state,” Journal of Physics: Condensed Matter 26, 255402 (2014).

E. Nefzaoui, K. Joulain, J. Drevillon, and Y. Ezzahri, “Radiative thermal rectification using superconducting materials,” Appl. Phys. Lett. 104, 103905 (2014).
[Crossref]

P. Ben-Abdallah and S.-A. Biehs, “Near-field thermal transistor,” Phys. Rev. Lett. 112, 044301 (2014).
[Crossref] [PubMed]

E. Nefzaoui, J. Drevillon, Y. Ezzahri, and K. Joulain, “Simple far-field radiative thermal rectifier using Fabry–Perot cavities based infrared selective emitters,” Appl. Opt. 53, 3479 (2014).
[Crossref] [PubMed]

2013 (4)

K. Ito, K. Nishikawa, H. Izuka, and H. Toshiyoshi, “Experimental investigation of radiative thermal rectifier using vanadium dioxyde,” Appl. Phys. Lett. 103, 191907 (2013).

P. Ben-Abdallah and S.-A. Biehs, “Phase-change radiative thermal diode,” Appl. Phys. Lett. 103, 191907 (2013).
[Crossref]

J. Huang, Q. Li, Z. Zheng, and Y. Xuan, “Thermal rectification based on thermochromic materials,” International Journal of Heat and Mass Transfer 67, 575–580 (2013).
[Crossref]

E. Nefzaoui, Y. Ezzahri, J. Drevillon, and K. Joulain, “Maximal near-field radiative heat transfer between two plates,” The European Physical Journal Applied Physics 63, 30902 (2013).
[Crossref]

2012 (3)

H. Iizuka and S. Fan, “Rectification of evanescent heat transfer between dielectric-coated and uncoated silicon carbide plates,” J. Appl. Phys. 112, 024304 (2012).
[Crossref]

P. van Zwol, L. Ranno, and J. Chevrier, “Emissivity measurements with an atomic force microscope,” J. Appl. Phys. 111, 063110 (2012).
[Crossref]

E. Rousseau, M. Laroche, and J.-J. Greffet, “Asymptotic expressions describing radiative heat transfer between polar materials from the far-field regime to the nanoscale regime,” J. Appl. Phys. 111, 014311 (2012).
[Crossref]

2011 (1)

S. Basu and M. Francoeur, “Near-field radiative transfer based thermal rectification using doped silicon,” Appl. Phys. Lett. 98, 113106 (2011).
[Crossref]

2010 (3)

S. A. Biehs, E. Rousseau, and J. J. Greffet, “Mesoscopic description of radiative heat transfer at the nanoscale,” Phys. Rev. Lett. 105, 234301 (2010).
[Crossref]

P. Ben-Abdallah and K. Joulain, “Fundamental limits for noncontact transfers between two bodies,” Phys. Rev. B 82, 121419(R) (2010).
[Crossref]

C. R. Otey, W. T. Lau, and S. Fan, “Thermal rectification through vacuum,” Phys. Rev. Lett. 104, 154301 (2010).
[Crossref] [PubMed]

2009 (2)

N. Yang, G. Zhang, and B. Li, “Thermal rectification in asymmetric graphene ribbons,” Appl. Phys. Lett. 95, 033107 (2009).
[Crossref]

J. Hu, X. Ruan, and Y. P. Chen, “Thermal conductivity and thermal rectification in graphene nanoribbons: a molecular dynamics study,” Nano Letters 9, 2730–2735 (2009).
[Crossref] [PubMed]

2008 (3)

D. Segal, “Single mode heat rectifier: controlling energy flow between electronic conductors,” Phys. Rev. Lett. 100, 105901 (2008).
[Crossref] [PubMed]

W. C. Lo, L. Wang, and B. Li, “Thermal transistor: heat flux switching and modulating,” Journal of the Physical Society of Japan 77, 054402 (2008).
[Crossref]

P.-O. Chapuis, S. Volz, C. Henkel, K. Joulain, and J.-J. Greffet, “Effects of spatial dispersion in near-field radiative heat transfer between two parallel metallic surfaces,” Phys. Rev. B 77, 035431 (2008).
[Crossref]

2007 (3)

A. Volokitin and B. Persson, “Near-field radiative heat transfer and noncontact friction,” Rev. Mod. Phys. 79, 1291–1329 (2007).
[Crossref]

L. Wang and B. Li, “Thermal logic gates: computation with phonons,” Phys. Rev. Lett. 99, 177208 (2007).
[Crossref] [PubMed]

N. Yang, N. Li, L. Wang, and B. Li, “Thermal rectification and negative differential thermal resistance in lattices with mass gradient,” Phys. Rev. B 76, 020301(R) (2007).
[Crossref]

2006 (3)

C. W. Chang, D. Okawa, A. Majumdar, and A. Zettl, “Solid-state thermal rectifier,” Science 314, 1121–1124 (2006).
[Crossref] [PubMed]

B. Hu, L. Yang, and Y. Zhang, “Asymmetric heat conduction in nonlinear lattices,” Phys. Rev. Lett. 97, 124302 (2006).
[Crossref] [PubMed]

D. De Sousa Meneses, M. Malki, and P. Echegut, “Optical and structural properties of calcium silicate glasses,” Journal of Non-Crystalline Solids 352, 5301–5308 (2006).
[Crossref]

2005 (2)

K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57, 59–112 (2005).
[Crossref]

B. Li, J. Lan, and L. Wang, “Interface thermal resistance between dissimilar anharmonic lattices,” Phys. Rev. Lett. 95, 104302 (2005).
[Crossref] [PubMed]

2004 (2)

2002 (2)

M. Terraneo, M. Peyrard, and G. Casati, “Controlling the energy flow in nonlinear lattices: a model for a thermal rectifier,” Phys. Rev. Lett. 88, 094302 (2002).
[Crossref] [PubMed]

J. P. Mulet, K. Joulain, R. Carminati, and J. J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophysical Engineering 6, 209–222 (2002).
[Crossref]

1982 (1)

D. Olego and M. Cardona, “Temperature dependence of the optical phonons and transverse effective charge in 3C-SiC,” Phys. Rev. B 25, 3889 (1982).
[Crossref]

1971 (1)

D. Polder and M. van Hove, “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B 4, 3303–3314 (1971).
[Crossref]

Basu, S.

S. Basu and M. Francoeur, “Near-field radiative transfer based thermal rectification using doped silicon,” Appl. Phys. Lett. 98, 113106 (2011).
[Crossref]

Ben-Abdallah, P.

K. Joulain, Y. Ezzahri, J. Drevillon, and P. Ben-Abdallah, “Modulation and amplification of radiative far field heat transfer:Towards a simple radiative thermal transistor,” Appl. Phys. Lett. 106, 133505 (2015).
[Crossref]

P. Ben-Abdallah and S.-A. Biehs, “Near-field thermal transistor,” Phys. Rev. Lett. 112, 044301 (2014).
[Crossref] [PubMed]

P. Ben-Abdallah and S.-A. Biehs, “Phase-change radiative thermal diode,” Appl. Phys. Lett. 103, 191907 (2013).
[Crossref]

P. Ben-Abdallah and K. Joulain, “Fundamental limits for noncontact transfers between two bodies,” Phys. Rev. B 82, 121419(R) (2010).
[Crossref]

Biehs, S. A.

S. A. Biehs, E. Rousseau, and J. J. Greffet, “Mesoscopic description of radiative heat transfer at the nanoscale,” Phys. Rev. Lett. 105, 234301 (2010).
[Crossref]

Biehs, S.-A.

P. Ben-Abdallah and S.-A. Biehs, “Near-field thermal transistor,” Phys. Rev. Lett. 112, 044301 (2014).
[Crossref] [PubMed]

P. Ben-Abdallah and S.-A. Biehs, “Phase-change radiative thermal diode,” Appl. Phys. Lett. 103, 191907 (2013).
[Crossref]

Brun, J.-F.

Cardona, M.

D. Olego and M. Cardona, “Temperature dependence of the optical phonons and transverse effective charge in 3C-SiC,” Phys. Rev. B 25, 3889 (1982).
[Crossref]

Carminati, R.

K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57, 59–112 (2005).
[Crossref]

J. P. Mulet, K. Joulain, R. Carminati, and J. J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophysical Engineering 6, 209–222 (2002).
[Crossref]

Casati, G.

B. Li, L. Wang, and G. Casati, “Thermal diode : rectification of heat flux,” Phys. Rev. Lett. 93, 184301 (2004).
[Crossref]

M. Terraneo, M. Peyrard, and G. Casati, “Controlling the energy flow in nonlinear lattices: a model for a thermal rectifier,” Phys. Rev. Lett. 88, 094302 (2002).
[Crossref] [PubMed]

Chang, C. W.

C. W. Chang, D. Okawa, A. Majumdar, and A. Zettl, “Solid-state thermal rectifier,” Science 314, 1121–1124 (2006).
[Crossref] [PubMed]

Chapuis, P.-O.

P.-O. Chapuis, S. Volz, C. Henkel, K. Joulain, and J.-J. Greffet, “Effects of spatial dispersion in near-field radiative heat transfer between two parallel metallic surfaces,” Phys. Rev. B 77, 035431 (2008).
[Crossref]

Chen, Y. P.

J. Hu, X. Ruan, and Y. P. Chen, “Thermal conductivity and thermal rectification in graphene nanoribbons: a molecular dynamics study,” Nano Letters 9, 2730–2735 (2009).
[Crossref] [PubMed]

Chevrier, J.

P. van Zwol, L. Ranno, and J. Chevrier, “Emissivity measurements with an atomic force microscope,” J. Appl. Phys. 111, 063110 (2012).
[Crossref]

Cosson, L.

D. De Sousa Meneses, P. Melin, L. del Campo, L. Cosson, and P. Echegut, “Infrared Physics & Technology,” Infrared Physics & Technology 69, 96–101 (2015).
[Crossref]

De Sousa Meneses, D.

D. De Sousa Meneses, P. Melin, L. del Campo, L. Cosson, and P. Echegut, “Infrared Physics & Technology,” Infrared Physics & Technology 69, 96–101 (2015).
[Crossref]

D. De Sousa Meneses, M. Eckes, L. del Campo, and P. Echegut, “Phase transformations of crystalline SiO2 versus dynamic disorder between room temperature and liquid state,” Journal of Physics: Condensed Matter 26, 255402 (2014).

D. De Sousa Meneses, M. Malki, and P. Echegut, “Optical and structural properties of calcium silicate glasses,” Journal of Non-Crystalline Solids 352, 5301–5308 (2006).
[Crossref]

D. De Sousa Meneses, J.-F. Brun, P. Echegut, and P. Simon, “Contribution of semi-quantum dielectric function models to the analysis of infrared spectra,” Appl. Spectrosc. 58(8), 969–974 (2004).

del Campo, L.

D. De Sousa Meneses, P. Melin, L. del Campo, L. Cosson, and P. Echegut, “Infrared Physics & Technology,” Infrared Physics & Technology 69, 96–101 (2015).
[Crossref]

D. De Sousa Meneses, M. Eckes, L. del Campo, and P. Echegut, “Phase transformations of crystalline SiO2 versus dynamic disorder between room temperature and liquid state,” Journal of Physics: Condensed Matter 26, 255402 (2014).

Drevillon, J.

K. Joulain, Y. Ezzahri, J. Drevillon, and P. Ben-Abdallah, “Modulation and amplification of radiative far field heat transfer:Towards a simple radiative thermal transistor,” Appl. Phys. Lett. 106, 133505 (2015).
[Crossref]

E. Nefzaoui, K. Joulain, J. Drevillon, and Y. Ezzahri, “Radiative thermal rectification using superconducting materials,” Appl. Phys. Lett. 104, 103905 (2014).
[Crossref]

E. Nefzaoui, J. Drevillon, Y. Ezzahri, and K. Joulain, “Simple far-field radiative thermal rectifier using Fabry–Perot cavities based infrared selective emitters,” Appl. Opt. 53, 3479 (2014).
[Crossref] [PubMed]

E. Nefzaoui, Y. Ezzahri, J. Drevillon, and K. Joulain, “Maximal near-field radiative heat transfer between two plates,” The European Physical Journal Applied Physics 63, 30902 (2013).
[Crossref]

Echegut, P.

D. De Sousa Meneses, P. Melin, L. del Campo, L. Cosson, and P. Echegut, “Infrared Physics & Technology,” Infrared Physics & Technology 69, 96–101 (2015).
[Crossref]

D. De Sousa Meneses, M. Eckes, L. del Campo, and P. Echegut, “Phase transformations of crystalline SiO2 versus dynamic disorder between room temperature and liquid state,” Journal of Physics: Condensed Matter 26, 255402 (2014).

D. De Sousa Meneses, M. Malki, and P. Echegut, “Optical and structural properties of calcium silicate glasses,” Journal of Non-Crystalline Solids 352, 5301–5308 (2006).
[Crossref]

D. De Sousa Meneses, J.-F. Brun, P. Echegut, and P. Simon, “Contribution of semi-quantum dielectric function models to the analysis of infrared spectra,” Appl. Spectrosc. 58(8), 969–974 (2004).

Eckes, M.

D. De Sousa Meneses, M. Eckes, L. del Campo, and P. Echegut, “Phase transformations of crystalline SiO2 versus dynamic disorder between room temperature and liquid state,” Journal of Physics: Condensed Matter 26, 255402 (2014).

Ezzahri, Y.

K. Joulain, Y. Ezzahri, J. Drevillon, and P. Ben-Abdallah, “Modulation and amplification of radiative far field heat transfer:Towards a simple radiative thermal transistor,” Appl. Phys. Lett. 106, 133505 (2015).
[Crossref]

F. Singer, Y. Ezzahri, and K. Joulain, “Near field radiative heat transfer between two nonlocal dielectrics,” J. Quant. Spectr. and Rad. Transf. 154, 55–62 (2015).
[Crossref]

E. Nefzaoui, K. Joulain, J. Drevillon, and Y. Ezzahri, “Radiative thermal rectification using superconducting materials,” Appl. Phys. Lett. 104, 103905 (2014).
[Crossref]

E. Nefzaoui, J. Drevillon, Y. Ezzahri, and K. Joulain, “Simple far-field radiative thermal rectifier using Fabry–Perot cavities based infrared selective emitters,” Appl. Opt. 53, 3479 (2014).
[Crossref] [PubMed]

E. Nefzaoui, Y. Ezzahri, J. Drevillon, and K. Joulain, “Maximal near-field radiative heat transfer between two plates,” The European Physical Journal Applied Physics 63, 30902 (2013).
[Crossref]

Fan, S.

H. Iizuka and S. Fan, “Rectification of evanescent heat transfer between dielectric-coated and uncoated silicon carbide plates,” J. Appl. Phys. 112, 024304 (2012).
[Crossref]

C. R. Otey, W. T. Lau, and S. Fan, “Thermal rectification through vacuum,” Phys. Rev. Lett. 104, 154301 (2010).
[Crossref] [PubMed]

Francoeur, M.

S. Basu and M. Francoeur, “Near-field radiative transfer based thermal rectification using doped silicon,” Appl. Phys. Lett. 98, 113106 (2011).
[Crossref]

Greffet, J. J.

S. A. Biehs, E. Rousseau, and J. J. Greffet, “Mesoscopic description of radiative heat transfer at the nanoscale,” Phys. Rev. Lett. 105, 234301 (2010).
[Crossref]

J. P. Mulet, K. Joulain, R. Carminati, and J. J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophysical Engineering 6, 209–222 (2002).
[Crossref]

Greffet, J.-J.

E. Rousseau, M. Laroche, and J.-J. Greffet, “Asymptotic expressions describing radiative heat transfer between polar materials from the far-field regime to the nanoscale regime,” J. Appl. Phys. 111, 014311 (2012).
[Crossref]

P.-O. Chapuis, S. Volz, C. Henkel, K. Joulain, and J.-J. Greffet, “Effects of spatial dispersion in near-field radiative heat transfer between two parallel metallic surfaces,” Phys. Rev. B 77, 035431 (2008).
[Crossref]

K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57, 59–112 (2005).
[Crossref]

Henkel, C.

P.-O. Chapuis, S. Volz, C. Henkel, K. Joulain, and J.-J. Greffet, “Effects of spatial dispersion in near-field radiative heat transfer between two parallel metallic surfaces,” Phys. Rev. B 77, 035431 (2008).
[Crossref]

Hu, B.

B. Hu, L. Yang, and Y. Zhang, “Asymmetric heat conduction in nonlinear lattices,” Phys. Rev. Lett. 97, 124302 (2006).
[Crossref] [PubMed]

Hu, J.

J. Hu, X. Ruan, and Y. P. Chen, “Thermal conductivity and thermal rectification in graphene nanoribbons: a molecular dynamics study,” Nano Letters 9, 2730–2735 (2009).
[Crossref] [PubMed]

Huang, J.

J. Huang, Q. Li, Z. Zheng, and Y. Xuan, “Thermal rectification based on thermochromic materials,” International Journal of Heat and Mass Transfer 67, 575–580 (2013).
[Crossref]

Iizuka, H.

H. Iizuka and S. Fan, “Rectification of evanescent heat transfer between dielectric-coated and uncoated silicon carbide plates,” J. Appl. Phys. 112, 024304 (2012).
[Crossref]

Ito, K.

K. Ito, K. Nishikawa, H. Izuka, and H. Toshiyoshi, “Experimental investigation of radiative thermal rectifier using vanadium dioxyde,” Appl. Phys. Lett. 103, 191907 (2013).

Izuka, H.

K. Ito, K. Nishikawa, H. Izuka, and H. Toshiyoshi, “Experimental investigation of radiative thermal rectifier using vanadium dioxyde,” Appl. Phys. Lett. 103, 191907 (2013).

Joulain, K.

F. Singer, Y. Ezzahri, and K. Joulain, “Near field radiative heat transfer between two nonlocal dielectrics,” J. Quant. Spectr. and Rad. Transf. 154, 55–62 (2015).
[Crossref]

K. Joulain, Y. Ezzahri, J. Drevillon, and P. Ben-Abdallah, “Modulation and amplification of radiative far field heat transfer:Towards a simple radiative thermal transistor,” Appl. Phys. Lett. 106, 133505 (2015).
[Crossref]

E. Nefzaoui, K. Joulain, J. Drevillon, and Y. Ezzahri, “Radiative thermal rectification using superconducting materials,” Appl. Phys. Lett. 104, 103905 (2014).
[Crossref]

E. Nefzaoui, J. Drevillon, Y. Ezzahri, and K. Joulain, “Simple far-field radiative thermal rectifier using Fabry–Perot cavities based infrared selective emitters,” Appl. Opt. 53, 3479 (2014).
[Crossref] [PubMed]

E. Nefzaoui, Y. Ezzahri, J. Drevillon, and K. Joulain, “Maximal near-field radiative heat transfer between two plates,” The European Physical Journal Applied Physics 63, 30902 (2013).
[Crossref]

P. Ben-Abdallah and K. Joulain, “Fundamental limits for noncontact transfers between two bodies,” Phys. Rev. B 82, 121419(R) (2010).
[Crossref]

P.-O. Chapuis, S. Volz, C. Henkel, K. Joulain, and J.-J. Greffet, “Effects of spatial dispersion in near-field radiative heat transfer between two parallel metallic surfaces,” Phys. Rev. B 77, 035431 (2008).
[Crossref]

K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57, 59–112 (2005).
[Crossref]

J. P. Mulet, K. Joulain, R. Carminati, and J. J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophysical Engineering 6, 209–222 (2002).
[Crossref]

Lan, J.

B. Li, J. Lan, and L. Wang, “Interface thermal resistance between dissimilar anharmonic lattices,” Phys. Rev. Lett. 95, 104302 (2005).
[Crossref] [PubMed]

Laroche, M.

E. Rousseau, M. Laroche, and J.-J. Greffet, “Asymptotic expressions describing radiative heat transfer between polar materials from the far-field regime to the nanoscale regime,” J. Appl. Phys. 111, 014311 (2012).
[Crossref]

Lau, W. T.

C. R. Otey, W. T. Lau, and S. Fan, “Thermal rectification through vacuum,” Phys. Rev. Lett. 104, 154301 (2010).
[Crossref] [PubMed]

Li, B.

N. Yang, G. Zhang, and B. Li, “Thermal rectification in asymmetric graphene ribbons,” Appl. Phys. Lett. 95, 033107 (2009).
[Crossref]

W. C. Lo, L. Wang, and B. Li, “Thermal transistor: heat flux switching and modulating,” Journal of the Physical Society of Japan 77, 054402 (2008).
[Crossref]

L. Wang and B. Li, “Thermal logic gates: computation with phonons,” Phys. Rev. Lett. 99, 177208 (2007).
[Crossref] [PubMed]

N. Yang, N. Li, L. Wang, and B. Li, “Thermal rectification and negative differential thermal resistance in lattices with mass gradient,” Phys. Rev. B 76, 020301(R) (2007).
[Crossref]

B. Li, J. Lan, and L. Wang, “Interface thermal resistance between dissimilar anharmonic lattices,” Phys. Rev. Lett. 95, 104302 (2005).
[Crossref] [PubMed]

B. Li, L. Wang, and G. Casati, “Thermal diode : rectification of heat flux,” Phys. Rev. Lett. 93, 184301 (2004).
[Crossref]

Li, N.

N. Yang, N. Li, L. Wang, and B. Li, “Thermal rectification and negative differential thermal resistance in lattices with mass gradient,” Phys. Rev. B 76, 020301(R) (2007).
[Crossref]

Li, Q.

J. Huang, Q. Li, Z. Zheng, and Y. Xuan, “Thermal rectification based on thermochromic materials,” International Journal of Heat and Mass Transfer 67, 575–580 (2013).
[Crossref]

Lo, W. C.

W. C. Lo, L. Wang, and B. Li, “Thermal transistor: heat flux switching and modulating,” Journal of the Physical Society of Japan 77, 054402 (2008).
[Crossref]

Majumdar, A.

C. W. Chang, D. Okawa, A. Majumdar, and A. Zettl, “Solid-state thermal rectifier,” Science 314, 1121–1124 (2006).
[Crossref] [PubMed]

Malki, M.

D. De Sousa Meneses, M. Malki, and P. Echegut, “Optical and structural properties of calcium silicate glasses,” Journal of Non-Crystalline Solids 352, 5301–5308 (2006).
[Crossref]

Marquier, F.

K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57, 59–112 (2005).
[Crossref]

Melin, P.

D. De Sousa Meneses, P. Melin, L. del Campo, L. Cosson, and P. Echegut, “Infrared Physics & Technology,” Infrared Physics & Technology 69, 96–101 (2015).
[Crossref]

Mulet, J. P.

J. P. Mulet, K. Joulain, R. Carminati, and J. J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophysical Engineering 6, 209–222 (2002).
[Crossref]

Mulet, J.-P.

K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57, 59–112 (2005).
[Crossref]

Nefzaoui, E.

E. Nefzaoui, K. Joulain, J. Drevillon, and Y. Ezzahri, “Radiative thermal rectification using superconducting materials,” Appl. Phys. Lett. 104, 103905 (2014).
[Crossref]

E. Nefzaoui, J. Drevillon, Y. Ezzahri, and K. Joulain, “Simple far-field radiative thermal rectifier using Fabry–Perot cavities based infrared selective emitters,” Appl. Opt. 53, 3479 (2014).
[Crossref] [PubMed]

E. Nefzaoui, Y. Ezzahri, J. Drevillon, and K. Joulain, “Maximal near-field radiative heat transfer between two plates,” The European Physical Journal Applied Physics 63, 30902 (2013).
[Crossref]

Nishikawa, K.

K. Ito, K. Nishikawa, H. Izuka, and H. Toshiyoshi, “Experimental investigation of radiative thermal rectifier using vanadium dioxyde,” Appl. Phys. Lett. 103, 191907 (2013).

Okawa, D.

C. W. Chang, D. Okawa, A. Majumdar, and A. Zettl, “Solid-state thermal rectifier,” Science 314, 1121–1124 (2006).
[Crossref] [PubMed]

Olego, D.

D. Olego and M. Cardona, “Temperature dependence of the optical phonons and transverse effective charge in 3C-SiC,” Phys. Rev. B 25, 3889 (1982).
[Crossref]

Otey, C. R.

C. R. Otey, W. T. Lau, and S. Fan, “Thermal rectification through vacuum,” Phys. Rev. Lett. 104, 154301 (2010).
[Crossref] [PubMed]

Persson, B.

A. Volokitin and B. Persson, “Near-field radiative heat transfer and noncontact friction,” Rev. Mod. Phys. 79, 1291–1329 (2007).
[Crossref]

Peyrard, M.

M. Terraneo, M. Peyrard, and G. Casati, “Controlling the energy flow in nonlinear lattices: a model for a thermal rectifier,” Phys. Rev. Lett. 88, 094302 (2002).
[Crossref] [PubMed]

Polder, D.

D. Polder and M. van Hove, “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B 4, 3303–3314 (1971).
[Crossref]

Ranno, L.

P. van Zwol, L. Ranno, and J. Chevrier, “Emissivity measurements with an atomic force microscope,” J. Appl. Phys. 111, 063110 (2012).
[Crossref]

Rousseau, E.

E. Rousseau, M. Laroche, and J.-J. Greffet, “Asymptotic expressions describing radiative heat transfer between polar materials from the far-field regime to the nanoscale regime,” J. Appl. Phys. 111, 014311 (2012).
[Crossref]

S. A. Biehs, E. Rousseau, and J. J. Greffet, “Mesoscopic description of radiative heat transfer at the nanoscale,” Phys. Rev. Lett. 105, 234301 (2010).
[Crossref]

Ruan, X.

J. Hu, X. Ruan, and Y. P. Chen, “Thermal conductivity and thermal rectification in graphene nanoribbons: a molecular dynamics study,” Nano Letters 9, 2730–2735 (2009).
[Crossref] [PubMed]

Rytov, S.M.

S.M. Rytov, Principle of Statistical Radiophysics 3. Elements of Radiation Fields (Springer Verlag, 1989).

Segal, D.

D. Segal, “Single mode heat rectifier: controlling energy flow between electronic conductors,” Phys. Rev. Lett. 100, 105901 (2008).
[Crossref] [PubMed]

Simon, P.

Singer, F.

F. Singer, Y. Ezzahri, and K. Joulain, “Near field radiative heat transfer between two nonlocal dielectrics,” J. Quant. Spectr. and Rad. Transf. 154, 55–62 (2015).
[Crossref]

Terraneo, M.

M. Terraneo, M. Peyrard, and G. Casati, “Controlling the energy flow in nonlinear lattices: a model for a thermal rectifier,” Phys. Rev. Lett. 88, 094302 (2002).
[Crossref] [PubMed]

Toshiyoshi, H.

K. Ito, K. Nishikawa, H. Izuka, and H. Toshiyoshi, “Experimental investigation of radiative thermal rectifier using vanadium dioxyde,” Appl. Phys. Lett. 103, 191907 (2013).

van Hove, M.

D. Polder and M. van Hove, “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B 4, 3303–3314 (1971).
[Crossref]

van Zwol, P.

P. van Zwol, L. Ranno, and J. Chevrier, “Emissivity measurements with an atomic force microscope,” J. Appl. Phys. 111, 063110 (2012).
[Crossref]

Volokitin, A.

A. Volokitin and B. Persson, “Near-field radiative heat transfer and noncontact friction,” Rev. Mod. Phys. 79, 1291–1329 (2007).
[Crossref]

Volz, S.

P.-O. Chapuis, S. Volz, C. Henkel, K. Joulain, and J.-J. Greffet, “Effects of spatial dispersion in near-field radiative heat transfer between two parallel metallic surfaces,” Phys. Rev. B 77, 035431 (2008).
[Crossref]

Wang, L.

W. C. Lo, L. Wang, and B. Li, “Thermal transistor: heat flux switching and modulating,” Journal of the Physical Society of Japan 77, 054402 (2008).
[Crossref]

L. Wang and B. Li, “Thermal logic gates: computation with phonons,” Phys. Rev. Lett. 99, 177208 (2007).
[Crossref] [PubMed]

N. Yang, N. Li, L. Wang, and B. Li, “Thermal rectification and negative differential thermal resistance in lattices with mass gradient,” Phys. Rev. B 76, 020301(R) (2007).
[Crossref]

B. Li, J. Lan, and L. Wang, “Interface thermal resistance between dissimilar anharmonic lattices,” Phys. Rev. Lett. 95, 104302 (2005).
[Crossref] [PubMed]

B. Li, L. Wang, and G. Casati, “Thermal diode : rectification of heat flux,” Phys. Rev. Lett. 93, 184301 (2004).
[Crossref]

Xuan, Y.

J. Huang, Q. Li, Z. Zheng, and Y. Xuan, “Thermal rectification based on thermochromic materials,” International Journal of Heat and Mass Transfer 67, 575–580 (2013).
[Crossref]

Yang, L.

B. Hu, L. Yang, and Y. Zhang, “Asymmetric heat conduction in nonlinear lattices,” Phys. Rev. Lett. 97, 124302 (2006).
[Crossref] [PubMed]

Yang, N.

N. Yang, G. Zhang, and B. Li, “Thermal rectification in asymmetric graphene ribbons,” Appl. Phys. Lett. 95, 033107 (2009).
[Crossref]

N. Yang, N. Li, L. Wang, and B. Li, “Thermal rectification and negative differential thermal resistance in lattices with mass gradient,” Phys. Rev. B 76, 020301(R) (2007).
[Crossref]

Zettl, A.

C. W. Chang, D. Okawa, A. Majumdar, and A. Zettl, “Solid-state thermal rectifier,” Science 314, 1121–1124 (2006).
[Crossref] [PubMed]

Zhang, G.

N. Yang, G. Zhang, and B. Li, “Thermal rectification in asymmetric graphene ribbons,” Appl. Phys. Lett. 95, 033107 (2009).
[Crossref]

Zhang, Y.

B. Hu, L. Yang, and Y. Zhang, “Asymmetric heat conduction in nonlinear lattices,” Phys. Rev. Lett. 97, 124302 (2006).
[Crossref] [PubMed]

Zheng, Z.

J. Huang, Q. Li, Z. Zheng, and Y. Xuan, “Thermal rectification based on thermochromic materials,” International Journal of Heat and Mass Transfer 67, 575–580 (2013).
[Crossref]

Ziman, J.M.

J.M. Ziman, Electrons and Phonons (Oxford University Press, 1960).

Appl. Opt. (1)

Appl. Phys. Lett. (6)

E. Nefzaoui, K. Joulain, J. Drevillon, and Y. Ezzahri, “Radiative thermal rectification using superconducting materials,” Appl. Phys. Lett. 104, 103905 (2014).
[Crossref]

S. Basu and M. Francoeur, “Near-field radiative transfer based thermal rectification using doped silicon,” Appl. Phys. Lett. 98, 113106 (2011).
[Crossref]

P. Ben-Abdallah and S.-A. Biehs, “Phase-change radiative thermal diode,” Appl. Phys. Lett. 103, 191907 (2013).
[Crossref]

K. Joulain, Y. Ezzahri, J. Drevillon, and P. Ben-Abdallah, “Modulation and amplification of radiative far field heat transfer:Towards a simple radiative thermal transistor,” Appl. Phys. Lett. 106, 133505 (2015).
[Crossref]

N. Yang, G. Zhang, and B. Li, “Thermal rectification in asymmetric graphene ribbons,” Appl. Phys. Lett. 95, 033107 (2009).
[Crossref]

K. Ito, K. Nishikawa, H. Izuka, and H. Toshiyoshi, “Experimental investigation of radiative thermal rectifier using vanadium dioxyde,” Appl. Phys. Lett. 103, 191907 (2013).

Appl. Spectrosc. (1)

Infrared Physics & Technology (1)

D. De Sousa Meneses, P. Melin, L. del Campo, L. Cosson, and P. Echegut, “Infrared Physics & Technology,” Infrared Physics & Technology 69, 96–101 (2015).
[Crossref]

International Journal of Heat and Mass Transfer (1)

J. Huang, Q. Li, Z. Zheng, and Y. Xuan, “Thermal rectification based on thermochromic materials,” International Journal of Heat and Mass Transfer 67, 575–580 (2013).
[Crossref]

J. Appl. Phys. (3)

H. Iizuka and S. Fan, “Rectification of evanescent heat transfer between dielectric-coated and uncoated silicon carbide plates,” J. Appl. Phys. 112, 024304 (2012).
[Crossref]

P. van Zwol, L. Ranno, and J. Chevrier, “Emissivity measurements with an atomic force microscope,” J. Appl. Phys. 111, 063110 (2012).
[Crossref]

E. Rousseau, M. Laroche, and J.-J. Greffet, “Asymptotic expressions describing radiative heat transfer between polar materials from the far-field regime to the nanoscale regime,” J. Appl. Phys. 111, 014311 (2012).
[Crossref]

J. Quant. Spectr. and Rad. Transf. (1)

F. Singer, Y. Ezzahri, and K. Joulain, “Near field radiative heat transfer between two nonlocal dielectrics,” J. Quant. Spectr. and Rad. Transf. 154, 55–62 (2015).
[Crossref]

Journal of Non-Crystalline Solids (1)

D. De Sousa Meneses, M. Malki, and P. Echegut, “Optical and structural properties of calcium silicate glasses,” Journal of Non-Crystalline Solids 352, 5301–5308 (2006).
[Crossref]

Journal of Physics: Condensed Matter (1)

D. De Sousa Meneses, M. Eckes, L. del Campo, and P. Echegut, “Phase transformations of crystalline SiO2 versus dynamic disorder between room temperature and liquid state,” Journal of Physics: Condensed Matter 26, 255402 (2014).

Journal of the Physical Society of Japan (1)

W. C. Lo, L. Wang, and B. Li, “Thermal transistor: heat flux switching and modulating,” Journal of the Physical Society of Japan 77, 054402 (2008).
[Crossref]

Microscale Thermophysical Engineering (1)

J. P. Mulet, K. Joulain, R. Carminati, and J. J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophysical Engineering 6, 209–222 (2002).
[Crossref]

Nano Letters (1)

J. Hu, X. Ruan, and Y. P. Chen, “Thermal conductivity and thermal rectification in graphene nanoribbons: a molecular dynamics study,” Nano Letters 9, 2730–2735 (2009).
[Crossref] [PubMed]

Phys. Rev. B (5)

N. Yang, N. Li, L. Wang, and B. Li, “Thermal rectification and negative differential thermal resistance in lattices with mass gradient,” Phys. Rev. B 76, 020301(R) (2007).
[Crossref]

D. Polder and M. van Hove, “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B 4, 3303–3314 (1971).
[Crossref]

P. Ben-Abdallah and K. Joulain, “Fundamental limits for noncontact transfers between two bodies,” Phys. Rev. B 82, 121419(R) (2010).
[Crossref]

D. Olego and M. Cardona, “Temperature dependence of the optical phonons and transverse effective charge in 3C-SiC,” Phys. Rev. B 25, 3889 (1982).
[Crossref]

P.-O. Chapuis, S. Volz, C. Henkel, K. Joulain, and J.-J. Greffet, “Effects of spatial dispersion in near-field radiative heat transfer between two parallel metallic surfaces,” Phys. Rev. B 77, 035431 (2008).
[Crossref]

Phys. Rev. Lett. (9)

P. Ben-Abdallah and S.-A. Biehs, “Near-field thermal transistor,” Phys. Rev. Lett. 112, 044301 (2014).
[Crossref] [PubMed]

S. A. Biehs, E. Rousseau, and J. J. Greffet, “Mesoscopic description of radiative heat transfer at the nanoscale,” Phys. Rev. Lett. 105, 234301 (2010).
[Crossref]

B. Hu, L. Yang, and Y. Zhang, “Asymmetric heat conduction in nonlinear lattices,” Phys. Rev. Lett. 97, 124302 (2006).
[Crossref] [PubMed]

M. Terraneo, M. Peyrard, and G. Casati, “Controlling the energy flow in nonlinear lattices: a model for a thermal rectifier,” Phys. Rev. Lett. 88, 094302 (2002).
[Crossref] [PubMed]

B. Li, L. Wang, and G. Casati, “Thermal diode : rectification of heat flux,” Phys. Rev. Lett. 93, 184301 (2004).
[Crossref]

B. Li, J. Lan, and L. Wang, “Interface thermal resistance between dissimilar anharmonic lattices,” Phys. Rev. Lett. 95, 104302 (2005).
[Crossref] [PubMed]

D. Segal, “Single mode heat rectifier: controlling energy flow between electronic conductors,” Phys. Rev. Lett. 100, 105901 (2008).
[Crossref] [PubMed]

C. R. Otey, W. T. Lau, and S. Fan, “Thermal rectification through vacuum,” Phys. Rev. Lett. 104, 154301 (2010).
[Crossref] [PubMed]

L. Wang and B. Li, “Thermal logic gates: computation with phonons,” Phys. Rev. Lett. 99, 177208 (2007).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

A. Volokitin and B. Persson, “Near-field radiative heat transfer and noncontact friction,” Rev. Mod. Phys. 79, 1291–1329 (2007).
[Crossref]

Science (1)

C. W. Chang, D. Okawa, A. Majumdar, and A. Zettl, “Solid-state thermal rectifier,” Science 314, 1121–1124 (2006).
[Crossref] [PubMed]

Surf. Sci. Rep. (1)

K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57, 59–112 (2005).
[Crossref]

The European Physical Journal Applied Physics (1)

E. Nefzaoui, Y. Ezzahri, J. Drevillon, and K. Joulain, “Maximal near-field radiative heat transfer between two plates,” The European Physical Journal Applied Physics 63, 30902 (2013).
[Crossref]

Other (2)

J.M. Ziman, Electrons and Phonons (Oxford University Press, 1960).

S.M. Rytov, Principle of Statistical Radiophysics 3. Elements of Radiation Fields (Springer Verlag, 1989).

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Figures (4)

Fig. 1
Fig. 1 Measured real and Imaginary part of the dielectric function ε variation with angular frequency ω for different temperatures. Top 2 figures : SiO2. Bottom 2 figures : SiC.

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Fig. 2
Fig. 2 Computed radiative heat transfer between a plane interface of SiC and a second one constituted of SiO2 versus their separation distances. Two situations are compared. In the first one (plain) SiO2 is at 297 K and SiC at 1470 K whereas in the second one (dashed) SiC is at 297 K and SiO2 is at 1470 K.

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Fig. 3
Fig. 3 Computed rectification variations as a function of the separation distance between two planar interfaces made of SiC and SiO2 when one material is at 297 K and the second one at 471 K, 671 K, 981 K, 1102 K or 1470 K.

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Fig. 4
Fig. 4 Computed spectral radiative heat transfer between two planar interfaces made of SiC and SiO2 in the direct and the reverse situation for 4 different separation distances: SiO2 is at 297 K and SiC is at 1470 K in the first situation (plain) whereas SiO2 is at 1470 K and SiC is at 297 K in the second situation (dashed).

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Equations (4)

Equations on this page are rendered with MathJax. Learn more.

φ 1 ↔ 2 = ∑ α = s , p ∫ 0 ∞ [ Θ ( ω , T 1 ) − Θ ( ω , T 2 ) ] d ω ∫ 0 ∞ K d K 4 π 2 τ α ( ω , K )
τ α ( ω , K ) = ( 1 − | r 31 α | 2 ) ( 1 − | r 32 α | 2 ) | 1 − r 31 α r 32 α e 2 i γ 3 d | 2
τ α ( ω , K ) = 4 ℑ ( r 31 α ) ℑ ( r 32 α ) e − 2 ℑ ( γ 3 ) d | 1 − r 31 α r 32 α e − 2 ℑ ( γ 3 ) d | 2
R = | φ 1 ↔ 2 ( T 1 , T 2 ) − φ 1 ↔ 2 ( T 2 , T 1 ) | Max [ φ 1 ↔ 2 ( T 1 , T 2 ) , φ 1 ↔ 2 ( T 2 , T 1 ) ]

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