## Near-field thermal radiation between graphene-covered doped silicon plates |

Optics Express, Vol. 21, Issue 19, pp. 22173-22185 (2013)

http://dx.doi.org/10.1364/OE.21.022173

Acrobat PDF (2092 KB)

### Abstract

The present work describes a theoretical investigation of the near-field thermal radiation between doped Si plates coated with a mono-layer of graphene. It is found that the radiative heat flux between doped Si plates can be either enhanced or suppressed by introducing graphene layer, depending on the Si doping concentration and chemical potential of graphene. Graphene can enhance the heat flux if it matches resonance frequencies of surface plasmon at vacuum-source and vacuum-receiver interfaces. In particular, significant enhancement is achieved when graphene is coated on both surfaces that originally does not support the surface plasmon resonance. The results obtained in this study provide an important guideline into enhancing the near-field thermal radiation between doped Si plates by introducing graphene.

© 2013 OSA

## 1. Introduction

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

2. J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophys. Eng. **6**, 209–222 (2002). [CrossRef]

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

2. J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophys. Eng. **6**, 209–222 (2002). [CrossRef]

4. 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]

9. T. Kralik, P. Hanzelka, M. Zobac, V. Musilova, T. Fort, and M. Horak, “Strong Near-Field Enhancement of Radiative Heat Transfer between Metallic Surfaces,” Phys. Rev. Lett. **109**, 224302 (2012). [CrossRef]

10. F. Marquier, K. Joulain, J.-P. Mulet, R. Carminati, and J.-J. Greffet, “Engineering infrared emission properties of silicon in the near field and the far field,” Opt. Commun. **237**, 379–388 (2004). [CrossRef]

12. S. Basu, B. J. Lee, and Z. M. Zhang, “Near-field radiation calculated with an improved dielectric function model for doped silicon,” J. Heat Transfer **132**, 023302 (2010). [CrossRef]

13. S. Basu, B. J. Lee, and Z. M. Zhang, “Infrared radiative properties of heavily doped silicon at room temperature,” J. Heat Transfer **132**, 023301 (2010). [CrossRef]

14. A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. **6**, 183–191 (2007). [CrossRef] [PubMed]

15. P. Avouris, “Graphene: Electronic and photonic properties and devices,” Nano Lett. **10**, 4285–4294 (2010). [CrossRef]

16. F. Rana, “Graphene optoelectronics: Plasmons get tuned up,” Nat. Nanotechnol. **6**, 611–612 (2011). [CrossRef] [PubMed]

17. B. N. J. Persson and H. Ueba, “Heat transfer between graphene and amorphous SiO_{2},” J. Phys. Condens. Matter **22**, 462201 (2010). [CrossRef]

20. O. Ilic, M. Jablan, J. D. Joannopoulos, I. Celanovic, H. Buljan, and M. Soljačić, “Near-field thermal radiation transfer controlled by plasmons in graphene,” Phys. Rev. B **85**, 155422 (2012). [CrossRef]

21. O. Ilic, M. Jablan, J. D. Joannopoulos, I. Celanovic, and M. Soljačić, “Overcoming the black body limit in plasmonic and graphene near-field thermophotovoltaic systems,” Opt. Express **20**, A366–A384 (2012). [CrossRef] [PubMed]

22. R. Messina and P. Ben-Abdallah, “Graphene-based photovoltaic cells for near-field thermal energy conversion,” Scientific Reports **3**, 1383 (2013). [CrossRef] [PubMed]

17. B. N. J. Persson and H. Ueba, “Heat transfer between graphene and amorphous SiO_{2},” J. Phys. Condens. Matter **22**, 462201 (2010). [CrossRef]

18. A. I. Volokitin and B. N. J. Persson, “Near-field radiative heat transfer between closely spaced graphene and amorphous SiO_{2},” Phys. Rev. B **83**, 241407 (2011). [CrossRef]

20. O. Ilic, M. Jablan, J. D. Joannopoulos, I. Celanovic, H. Buljan, and M. Soljačić, “Near-field thermal radiation transfer controlled by plasmons in graphene,” Phys. Rev. B **85**, 155422 (2012). [CrossRef]

21. O. Ilic, M. Jablan, J. D. Joannopoulos, I. Celanovic, and M. Soljačić, “Overcoming the black body limit in plasmonic and graphene near-field thermophotovoltaic systems,” Opt. Express **20**, A366–A384 (2012). [CrossRef] [PubMed]

19. V. B. Svetovoy, P. J. van Zwol, and J. Chevrier, “Plasmon enhanced near-field radiative heat transfer for graphene covered dielectrics,” Phys. Rev. B **85**, 155418 (2012). [CrossRef]

22. R. Messina and P. Ben-Abdallah, “Graphene-based photovoltaic cells for near-field thermal energy conversion,” Scientific Reports **3**, 1383 (2013). [CrossRef] [PubMed]

## 2. Theoretical modeling

*p*-type doped Si plates covered by a monolayer of graphene and separated by a vacuum gap width

*d*, as shown in Fig. 1. The doping concentration of Si varies from 10

^{17}to 10

^{21}cm

^{−3}, and the resulting dielectric function

*ε*is obtained from the functional expressions in Basu

*et al.*[13

13. S. Basu, B. J. Lee, and Z. M. Zhang, “Infrared radiative properties of heavily doped silicon at room temperature,” J. Heat Transfer **132**, 023301 (2010). [CrossRef]

*d*, without graphene layers. The induced electric field

**E**(

**x**,

*ω*) at the point

**x**outside of the source (i.e., body 1) is given in terms of the Green’s dyadic function

**G̿**(

**x**,

**x**

*′*,

*ω*) and the fluctuating current density

**j**[2

2. J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophys. Eng. **6**, 209–222 (2002). [CrossRef]

23. B. J. Lee and Z. M. Zhang, “Lateral shifts in near-field thermal radiation with surface phonon polaritons,” Nanoscale Microscale Thermophys. Eng. **12**, 238–250 (2008). [CrossRef]

*ω*is the angular frequency,

*μ*

_{0}is the magnetic permeability of vacuum, and

*V*is volume of the source. As shown in Fig. 1, cylindrical coordinate is used such that space variable

_{s}**x**=

*r*

**r̂**+

*z*

**ẑ**, where

**r̂**and

**ẑ**are unit directional vectors. The dyadic Green’s function is given by [24

24. J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B **4**, 481–489 (1987). [CrossRef]

**ŝ**=

**r̂**×

**ẑ**and

**p̂**

*=(*

_{i}*β/k*)

_{i}**ẑ**−(

*k*)

_{iz}/k_{i}**r̂**are polarization vectors. In Eq. (2),

*β*is the parallel wavevector component, and

*c*

_{0}(i.e.,

**H**(

**x**,

*ω*) obtained using Maxwell’s equation, the spectral energy flux can be expressed by the ensemble average of the Poynting vector,

*z*-component of the Poynting vector at

*z*=

*d*; that is, 〈

*S*〉, which can then be expressed as where superscript

_{z}*γ*indicates a polarization index. Expression of

*S*(

^{γ}*β*,

*ω*) is different for propagating (i.e.,

*β*<

*ω/c*

_{0}) and for evanescent (i.e.,

*β*>

*ω/c*

_{0}) waves in vacuum; that is [12

12. S. Basu, B. J. Lee, and Z. M. Zhang, “Near-field radiation calculated with an improved dielectric function model for doped silicon,” J. Heat Transfer **132**, 023302 (2010). [CrossRef]

*i*−

*j*interface for a given polarization state. In the above equation,

*T*

_{1}is the temperature of body 1 and

*h̄*is the Planck constant divided by 2

*π*and

*k*is the Boltzmann constant. The net heat flux

_{B}*q″*between body 1 and body 2 can be calculated as

_{net}*q″*

_{ω,1→2}in Eqs. (3) and (4) depends on only the Fresnel reflection coefficients at vacuum-source and vacuum-receiver interfaces. As stated in Francoeur

*et al.*[25

25. M. Francoeur, M. P. Mengüç, and R. Vaillon, “Spectral tuning of near-field radiative heat flux between two thin silicon carbide films,” J. Phys. D: Appl. Phys. **43**, 075501 (2010). [CrossRef]

*F*stands for film. In other words, we can regard the thin-film-coated semi-infinite source as a single body with the Fresnel reflection coefficient

25. M. Francoeur, M. P. Mengüç, and R. Vaillon, “Spectral tuning of near-field radiative heat flux between two thin silicon carbide films,” J. Phys. D: Appl. Phys. **43**, 075501 (2010). [CrossRef]

26. S.-A. Biehs, “Thermal heat radiation, near-field energy density and near-field radiative heat transfer of coated materials,” Eur. Phys. J. B **58**, 423–431 (2007). [CrossRef]

19. V. B. Svetovoy, P. J. van Zwol, and J. Chevrier, “Plasmon enhanced near-field radiative heat transfer for graphene covered dielectrics,” Phys. Rev. B **85**, 155418 (2012). [CrossRef]

22. R. Messina and P. Ben-Abdallah, “Graphene-based photovoltaic cells for near-field thermal energy conversion,” Scientific Reports **3**, 1383 (2013). [CrossRef] [PubMed]

*σ*(

*ω*)=

*σ*(

_{I}*ω*)+

*σ*(

_{D}*ω*); that is, a summation of interband and intraband (Drude) contributions of where

*e*is the electron charge,

*τ*and

*μ*represent the relaxation time and chemical potential of graphene, respectively [27

27. L. A. Falkovsky, “Optical properties of graphene,” J. Phys. Conf. Ser. **129**, 012004 (2008). [CrossRef]

**E**and

**H**are given by [28] where the surface current density

**K**is the multiplication of the surface conductivity

*σ*(

*ω*) and the tangential component of

**E**. For

*s*-polarization, Eq. (6) can be written as:

*E*=

_{t}*E*+

_{i}*E*and

_{r}*H*cos

_{t}*θ*

_{1}=

*H*cos

_{i}*θ*

_{0}−

*H*cos

_{r}*θ*

_{0}−

*σE*, where

_{t}*E*,

_{t}*E*, and

_{i}*E*represent the magnitude of the transmitted, incident, and reflected electric field, respectively. Similarly,

_{r}*H*,

_{t}*H*, and

_{i}*H*indicate the magnitude of the transmitted, incident, and reflected magnetic field, respectively. For

_{r}*p*-polarization, boundary conditions are

*E*cos

_{t}*θ*

_{1}=

*E*cos

_{i}*θ*

_{0}−

*E*cos

_{r}*θ*

_{0}and

*H*−

_{t}*H*−

_{i}*H*= −

_{r}*σE*cos

_{t}*θ*

_{1}. After some algebraic manipulations,

*r*

_{0G1}can be expressed as [22

**3**, 1383 (2013). [CrossRef] [PubMed]

27. L. A. Falkovsky, “Optical properties of graphene,” J. Phys. Conf. Ser. **129**, 012004 (2008). [CrossRef]

29. T. Stauber, N. M. R. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B **78**, 085432 (2008). [CrossRef]

*ε*

_{0}is the electric permittivity of vacuum. Again, by replacing

30. K. Park, S. Basu, W. P. King, and Z. M. Zhang, “Performance analysis of near-field thermophotovoltaic devices considering absorption distribution,” J. Quant. Spectrosc. Radiat. Transfer **109**, 305–316 (2008). [CrossRef]

31. M. Francoeur, M. P. Mengüç, and R. Vaillon, “Solution of near-field thermal radiation in one-dimensional layered media using dyadic Green’s functions and the scattering matrix method,” J. Quant. Spectrosc. Radiat. Transfer **110**, 2002–2018 (2009). [CrossRef]

32. R. Alaee, M. Farhat, C. Rockstuhl, and F. Lederer, “A perfect absorber made of a graphene micro-ribbon meta-material,” Opt. Express **20**, 28017–28024 (2012). [CrossRef] [PubMed]

*ε*has also been verified by taking Δ → 0. The advantage of multilayer formulation is that we can easily calculate the emission and absorption by the graphene layer itself. On the other hand, the two-body formulation provides the spectral heat flux from/to the graphene-coated Si substrate including graphene’s contribution. In the present study, both formulation methods are employed because the two-body system is easier to interpret physically.

_{G}## 3. Results and discussion

*τ*= 10

^{−13}s [33

33. M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B **80**, 245435 (2009). [CrossRef]

*μ*varies from 0.1 eV to 0.5 eV.

*EF*=

*q″*in logarithmic scale, where

_{net}/q″_{net,bare}*q″*is the heat flux between Si plates without graphene at the corresponding doping concentrations. When

_{net,bare}*d*= 10 nm (i.e., left column of Fig. 3), the insertion of graphene to both sides enhances the heat flux in most cases except when the doping concentration of both Si plates is higher than 10

^{20}cm

^{−3}. In addition, when graphene is coated on both surfaces, the overall heat transfer enhancement with

*μ*= 0.5 eV is smaller than that with

*μ*= 0.3 eV. Specifically, for both source and receiver at 10

^{19}cm

^{−3}, graphene with

*μ*= 0.5 eV (i.e., B2) yields smaller

*EF*value than graphene with

*μ*= 0.3 eV (i.e., B1). It should be noted that if at least one side of the source and the receiver is at a doping concentration lower than 10

^{18}cm

^{−3}, significant enhancement is obtained. In particular, for both source and receiver at 10

^{17}cm

^{−3}, graphene of

*μ*= 0.3 eV can result in nearly two orders-of-magnitude enhancement in the heat transfer (

*EF*= 89.5).

^{19}cm

^{−3}and the receiver is at 10

^{20}cm

^{−3}, insertion of graphene to the source increases the heat flux more than three times as compared to that between bare source and receiver. In contrast, at the aforementioned doping concentration of the source and receiver, graphene insertion to the receiver side yields almost 40% of reduction of the heat flux (i.e., R1). In order to enhance the heat transfer by coating graphene on the receiver, the doping concentration of the receiver should be lower than that of the source. If the Si doping concentration of both plates is higher than 10

^{20}cm

^{−3}, graphene suppresses the radiative heat transfer regardless of the location where it is placed.

*d*= 10 nm. Likewise, for

*d*= 50 nm, the most significant enhancement occurs when both source and receiver are at 10

^{17}cm

^{−3}with graphene of

*μ*= 0.5 eV, whereas for

*d*= 10 nm, the most substantial enhancement is obtained for the same configuration with graphene of

*μ*= 0.3 eV. The effect of vacuum gap width on the heat transfer enhancement will be further discussed later.

*S*(

*β*,

*ω*) in order to further examine how the graphene insertion changes the heat transfer through surface plasmon. Here, the SPP dispersion curve of the asymmetric layered system is obtained from

20. O. Ilic, M. Jablan, J. D. Joannopoulos, I. Celanovic, H. Buljan, and M. Soljačić, “Near-field thermal radiation transfer controlled by plasmons in graphene,” Phys. Rev. B **85**, 155422 (2012). [CrossRef]

*β*. This shift of the SPP dispersion curve results in matching the resonance frequencies of SPPs at vacuum-source and the vacuum-receiver interfaces, yielding a great enhancement in the heat transfer. It should be noted that at the point where two SPP dispersion curves are expected to meet, there exists splitting of branches due to the hybridization, similarly to a symmetric layered system [34

34. K. Park, B. J Lee, C. Fu, and Z. M. Zhang, “Study of the surface and bulk polaritons with a negative index metamaterial,” J. Opt. Soc. Am. B **22**(5), 1016–1023 (2005). [CrossRef]

^{19}cm

^{−3}, the spectral energy flux largely changes with respect to the chemical potential of graphene, as can be seen from Fig. 4(b) for

*d*= 10 nm and Fig. 4(c) for

*d*= 50 nm. In general, for higher

*μ*value, the peak in

*q″*becomes broader. When

_{ω,net}*d*= 10 nm, however, higher chemical potential causes substantial decrease in the amplitude of the spectral heat flux, whereas the change of the amplitude of the spectral heat flux with respect to the chemical potential of graphene is not significant as vacuum gap width increases to 50 nm. Therefore, at

*d*= 50 nm, graphene with

*μ*= 0.5 eV, which exhibits wide peak in the spectral heat flux, can result in the largest heat transfer rate.

*ω*,

*T*) decreases exponentially with respect to

*ω*, the resulting spectral heat flux of B2 becomes smaller than B1. In a similar manner, for two doped Si plates at doping concentration higher than 10

^{20}cm

^{−3}, graphene suppresses the heat transfer because graphene shifts the SPP dispersion curves to much higher frequency regime. However, as shown in Fig. 3, for

*d*= 50 nm, graphene with

*μ*= 0.5 eV (i.e., B4) leads to greater enhancement than graphene of 0.3 eV (i.e., B3) although higher chemical potential of graphene shifts the SPP resonance frequency to higher value (also refer to Fig. 4(c) and Figs. 5(g)–5(i)). This can be understood as follows. As noted in Fig. 4(c) at

*d*= 50 nm, the shift of peak frequency in the spectral heat flux due to increase of

*μ*is smaller than that for 10 nm in Fig. 4(b). Then, the decrease due to lower value of Θ(

*ω*,

*T*) at higher frequency does not contribute much to the heat flux when

*d*= 50 nm compared to the case of

*d*= 10 nm; thus, B4 can yield larger heat transfer enhancement than B3.

*S*(

*β*,

*ω*) does not agree with SPP dispersion curves. For the excitation of surface plasmons, the real part of dielectric function should be negative, and at the same time the imaginary part of dielectric function should be less than unity [35]. In case of doped Si, the imaginary part of dielectric function is much larger than unity when real part becomes negative. Consequently, the heat flux between doped Si plates would be mainly affected by the numerator of

*S*(

^{p}*β*,

*ω*) that is the tunneling contribution of evanescent waves without SPP excitation [11

11. C. J. Fu and Z. M. Zhang, “Nanoscale radiation heat transfer for silicon at different doping levels,” Int. J. Heat Mass Transfer **49**, 1703–1718 (2006). [CrossRef]

*S*(

*β*,

*ω*) occurs along with the SPP dispersion curves. In such cases, it can be shown from the calculation that the SPP dispersion curves are located around the maximum of numerator of

*S*(

^{p}*β*,

*ω*). In other words, the SPP resonance occurs where the tunneling of evanescent waves is frequent, yielding a significant enhancement in the heat transfer rate.

^{17}cm

^{−3}regardless of the vacuum gap width and chemical potential of graphene. If there is no graphene, the heat flux between source and receiver at 10

^{17}cm

^{−3}is much smaller than that between source and receiver at doping concentration higher than 10

^{19}cm

^{−3}. This is because Si with doping concentration lower than 10

^{18}cm

^{−3}does not support SPP. However, as seen from Figs. 5(j)–5(l), SPPs can occur at vacuum-source and vacuum-receiver interfaces if graphene is coated on both surfaces even when Si is at 10

^{17}cm

^{−3}. Furthermore, in these cases, higher values of

*S*(

*β*,

*ω*) are well aligned with SPP dispersion curves. As a result, graphene can make the heat transfer between lightly doped Si plates (∼ 10

^{17}cm

^{−3}) be comparable to the heat transfer between heavily doped Si plates (> 10

^{19}cm

^{−3}).

*EF*≫ 1 as in B5, graphene’s contribution dominates

*q″*. For S1 with moderate

_{net}*EF*, graphene’s contribution is comparable to the heat transfer between Si substrates only. If graphene suppresses the near-field heat transfer, its contribution to the heat transfer is generally negligible. Therefore, it can be inferred that the enhancement of near-field heat transfer due to graphene layer is manly caused by the emission and absorption of graphene layer itself.

*μ*values, the enhancement factor decreases as

*d*increases, and graphene no longer affects the heat transfer if

*d*> 500 nm. For instance, graphene with

*μ*= 0.3 eV results in

*EF*= 89.5 at

*d*= 10 nm, but

*EF*abruptly drops to 12.5 at

*d*= 50 nm. At

*d*< 25 nm,

*EF*is larger for graphene with

*μ*= 0.3 eV than with

*μ*= 0.5 eV; however, if

*d*> 25 nm, graphene with

*μ*= 0.5 eV results in the higher

*EF*values than graphene with

*μ*= 0.3 eV. Consequently, graphene with appropriate value of

*μ*should be chosen depending on the vacuum gap width in order to enhance the near-field thermal radiation.

## 4. Concluding remark

## Acknowledgments

## References and links

1. | D. Polder and M. Van Hove, “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B |

2. | J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophys. Eng. |

3. | Z. M. Zhang, |

4. | 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 |

5. | E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nat. Photonics |

6. | S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett. |

7. | L. Hu, A. Narayanaswamy, X. Chen, and G. Chen, “Near-field thermal radiation between two closely spaced glass plates exceeding Plancks blackbody radiation law,” Appl. Phys. Lett. |

8. | R. S. Ottens, V. Quetschke, S. Wise, A. A. Alemi, R. Lundock, G. Mueller, D. H. Reitze, D. B. Tanner, and B. F. Whiting, “Near-field radiative heat transfer between macroscopic planar surfaces,” Phys. Rev. Lett. |

9. | T. Kralik, P. Hanzelka, M. Zobac, V. Musilova, T. Fort, and M. Horak, “Strong Near-Field Enhancement of Radiative Heat Transfer between Metallic Surfaces,” Phys. Rev. Lett. |

10. | F. Marquier, K. Joulain, J.-P. Mulet, R. Carminati, and J.-J. Greffet, “Engineering infrared emission properties of silicon in the near field and the far field,” Opt. Commun. |

11. | C. J. Fu and Z. M. Zhang, “Nanoscale radiation heat transfer for silicon at different doping levels,” Int. J. Heat Mass Transfer |

12. | S. Basu, B. J. Lee, and Z. M. Zhang, “Near-field radiation calculated with an improved dielectric function model for doped silicon,” J. Heat Transfer |

13. | S. Basu, B. J. Lee, and Z. M. Zhang, “Infrared radiative properties of heavily doped silicon at room temperature,” J. Heat Transfer |

14. | A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. |

15. | P. Avouris, “Graphene: Electronic and photonic properties and devices,” Nano Lett. |

16. | F. Rana, “Graphene optoelectronics: Plasmons get tuned up,” Nat. Nanotechnol. |

17. | B. N. J. Persson and H. Ueba, “Heat transfer between graphene and amorphous SiO |

18. | A. I. Volokitin and B. N. J. Persson, “Near-field radiative heat transfer between closely spaced graphene and amorphous SiO |

19. | V. B. Svetovoy, P. J. van Zwol, and J. Chevrier, “Plasmon enhanced near-field radiative heat transfer for graphene covered dielectrics,” Phys. Rev. B |

20. | O. Ilic, M. Jablan, J. D. Joannopoulos, I. Celanovic, H. Buljan, and M. Soljačić, “Near-field thermal radiation transfer controlled by plasmons in graphene,” Phys. Rev. B |

21. | O. Ilic, M. Jablan, J. D. Joannopoulos, I. Celanovic, and M. Soljačić, “Overcoming the black body limit in plasmonic and graphene near-field thermophotovoltaic systems,” Opt. Express |

22. | R. Messina and P. Ben-Abdallah, “Graphene-based photovoltaic cells for near-field thermal energy conversion,” Scientific Reports |

23. | B. J. Lee and Z. M. Zhang, “Lateral shifts in near-field thermal radiation with surface phonon polaritons,” Nanoscale Microscale Thermophys. Eng. |

24. | J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B |

25. | M. Francoeur, M. P. Mengüç, and R. Vaillon, “Spectral tuning of near-field radiative heat flux between two thin silicon carbide films,” J. Phys. D: Appl. Phys. |

26. | S.-A. Biehs, “Thermal heat radiation, near-field energy density and near-field radiative heat transfer of coated materials,” Eur. Phys. J. B |

27. | L. A. Falkovsky, “Optical properties of graphene,” J. Phys. Conf. Ser. |

28. | J. D. Jackson, |

29. | T. Stauber, N. M. R. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B |

30. | K. Park, S. Basu, W. P. King, and Z. M. Zhang, “Performance analysis of near-field thermophotovoltaic devices considering absorption distribution,” J. Quant. Spectrosc. Radiat. Transfer |

31. | M. Francoeur, M. P. Mengüç, and R. Vaillon, “Solution of near-field thermal radiation in one-dimensional layered media using dyadic Green’s functions and the scattering matrix method,” J. Quant. Spectrosc. Radiat. Transfer |

32. | R. Alaee, M. Farhat, C. Rockstuhl, and F. Lederer, “A perfect absorber made of a graphene micro-ribbon meta-material,” Opt. Express |

33. | M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B |

34. | K. Park, B. J Lee, C. Fu, and Z. M. Zhang, “Study of the surface and bulk polaritons with a negative index metamaterial,” J. Opt. Soc. Am. B |

35. | H. Raether, |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(240.7040) Optics at surfaces : Tunneling

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: July 11, 2013

Revised Manuscript: August 26, 2013

Manuscript Accepted: September 3, 2013

Published: September 12, 2013

**Citation**

Mikyung Lim, Seung S. Lee, and Bong Jae Lee, "Near-field thermal radiation between graphene-covered doped silicon plates," Opt. Express **21**, 22173-22185 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-19-22173

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### References

- D. Polder and M. Van Hove, “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B4, 3303–3314 (1971). [CrossRef]
- J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophys. Eng.6, 209–222 (2002). [CrossRef]
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