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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 19 — Sep. 23, 2013
  • pp: 22173–22185
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Near-field thermal radiation between graphene-covered doped silicon plates

Mikyung Lim, Seung S. Lee, and Bong Jae Lee  »View Author Affiliations


Optics Express, Vol. 21, Issue 19, pp. 22173-22185 (2013)
http://dx.doi.org/10.1364/OE.21.022173


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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

It is well known that radiative heat transfer between two objects can exceed the maximum governed by Planck’s law of blackbody radiation if the distance between two objects is smaller than the characteristic wavelength of thermal radiation determined by Wien’s displacement law [1

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

3

3. Z. M. Zhang, Nano/Microscale Heat Transfer (McGraw-Hill, 2007).

]. In this near-field regime, evanescent waves play an essential role in heat transfer through photon tunneling across the vacuum gap. If materials support the surface plasmon polariton (SPP), photon tunneling can be further enhanced because of amplified evanescent waves at the resonance condition [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]

, 3

3. Z. M. Zhang, Nano/Microscale Heat Transfer (McGraw-Hill, 2007).

]. A plethora of theoretical and experimental works has demonstrated the enhanced near-field thermal radiation between various materials, such as polar materials and metals [1

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

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

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

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]

]. In particular, in accordance with current boom of miniaturized electronic devices with state-of-the-art MEMS/NEMS technology using doped Si, theoretical studies on the near-field thermal radiation between doped Si plates have also been reported [10

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

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]

]. Because optical constants (i.e., refractive index and extinction coefficient) of doped Si highly depend on both doping concentration and temperature [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]

], there may exist mismatch in SPP resonance frequencies of doped Si plates at different doping concentrations and temperatures, which in turn can impede the enhancement of thermal radiation. In order to minimize the mismatch between SPP resonance frequencies of such an asymmetric layered system, we propose to employ a monolayer of graphene with its chemical potential as a tuning parameter.

2. Theoretical modeling

Let us consider two semi-infinite, 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 1017 to 1021 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]

]. For simplicity, temperature of the source and receiver is set to be 400 and 300 K, respectively.

Fig. 1 Schematic of the near-field thermal radiation between two doped Si plates with graphene separated by vacuum gap d (a) in three-dimensional view and (b) in cylindrical coordinate. A monolayer of graphene is modelled as surface conductivity σ.

Formulations of near-field thermal radiation start with two semi-infinite bulk media separated by a vacuum gap 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

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]

]:
E(x,ω)=iωμ0Vsd3xG¯¯(x,x,ω)j(x,ω)
(1)
where ω is the angular frequency, μ0 is the magnetic permeability of vacuum, and Vs is volume of the source. As shown in Fig. 1, cylindrical coordinate is used such that space variable x = r + z, where and 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]

]
G¯¯(x,x,ω)=i4π0βdβ1k1z(s^t12ss^+p^2t12pp^1)eiβr^(rr)ei{k2z(zd)k1zz}
(2)
where ŝ = × and i =(β/ki)−(kiz/ki) are polarization vectors. In Eq. (2), t12s and t12p represent the transmission coefficients from the source to the receiver for a given polarization, where β is the parallel wavevector component, and kiz=ω2/c02εiβ2 is the normal wavevector component with the speed of light in vacuum c0 (i.e., ki2=β2+kiz2). With the magnetic field H(x, ω) obtained using Maxwell’s equation, the spectral energy flux can be expressed by the ensemble average of the Poynting vector, S=12ReE×H*, where Re() takes the real part of a complex quantity and * denotes the complex conjugate. Finally, the spectral heat flux from the source (i.e., body 1) to the receiver (i.e., body 2) can be calculated based on the z-component of the Poynting vector at z = d; that is, 〈Sz〉, which can then be expressed as
qω,12=Sz=0S(β,ω)dβ=γ=p,s0Sγ(β,ω)dβ
(3)
where superscript γ indicates a polarization index. Expression of Sγ (β, ω) is different for propagating (i.e., β < ω/c0) and for evanescent (i.e., β > ω/c0) 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]

],
Spropγ(β,ω)=Θ(ω,T1)π2×β(1|r01γ|2)(1|r02γ|2)4|1r01γr02γei2k0zd|2Sevanγ(β,ω)=Θ(ω,T1)π2×βIm(r01γ)Im(r02γ)e2Im(k0z)d|1r01γr02γei2k0zd|2
(4)
where Im() takes the imaginary part of a complex quantity, and rijγ is the Fresnel reflection coefficient at the ij interface for a given polarization state. In the above equation, T1 is the temperature of body 1 and Θ(ω,T1)=h¯ωexp{h¯ω/(kBT1)}1 is the mean energy of Planck oscillator, where is the Planck constant divided by 2π and kB is the Boltzmann constant. The net heat flux q″net between body 1 and body 2 can be calculated as qnet=0dωqω,net=0dω[qω,12qω,21].

It should be noted that 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]

], 1|r01γ|2 and 1|r02γ|2 represent the spectral emittance of the source and the spectral absorptance of the receiver, respectively, for propagating waves. Similarly, Im(r01γ) and Im(r02γ) can be regarded as a counterpart of the spectral emittance of the source and the spectral absorptance of the receiver, respectively, for evanescent waves. Therefore, extension of Eq. (4) to the near-field thermal radiation between thin-film-coated semi-infinite bulk media, as shown in Fig. 1, is straightforward. Simply, we only need to modify the Fresnel reflection coefficients at the interfaces with vacuum: that is, r01γr0F1γ and r02γr0F2γ, where subscript 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 r0F1γ at the vacuum-source interface, and the thin-film-coated semi-infinite receiver as another single body with r0F2γ at the vacuum-receiver interface (i.e., two-body system with modified Fresnel reflection coefficient at the vacuum interfaces). Such two-body formulation has been derived for the near-field thermal radiation between two thin films [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]

] as well as that between a semi-infinite body and a coated semi-infinite body [26

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]

], and has also been employed for graphene-coated media [19

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

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

].

In order to calculate the Fresnel reflection coefficient at the interface between a graphene-coated body and vacuum, the surface conductivity of graphene is modelled as σ(ω)= σI(ω)+ σD(ω); that is, a summation of interband and intraband (Drude) contributions of
σI=e24h¯[G(h¯ω2)+i4h¯ωπ0G(ξ)G(h¯ω/2)(h¯ω)24ξ2dξ]σD=iω+iτ2e2kBTπh¯2ln[2cosh(μ2kBT)]
(5)
where e is the electron charge, G(ξ)=sinh(ξkBT)/[cosh(μkBT)+cosh(ξkBT)], and 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]

]. With this surface conductivity of graphene, the Fresnel reflection coefficient r0G1γ at the interface between body 0 and body 1 separated by a monolayer of graphene can be obtained as follows. When the incident field in vacuum (i.e., body 0) partially transmits into body 1 or reflects into vacuum again, the boundary conditions for the tangential components of E and H are given by [28

28. J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

]
z^×(E1E0)=0z^×(H1H0)=K
(6)
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: Et = Ei + Er and Ht cos θ1 = Hi cos θ0Hr cos θ0σEt, where Et, Ei, and Er represent the magnitude of the transmitted, incident, and reflected electric field, respectively. Similarly, Ht, Hi, and Hr indicate the magnitude of the transmitted, incident, and reflected magnetic field, respectively. For p-polarization, boundary conditions are Et cos θ1 = Ei cos θ0Er cos θ0 and HtHiHr = −σEt cosθ1. After some algebraic manipulations, r0G1 can be expressed as [22

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

, 27

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

, 29

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]

]:
r0G1s=k0zk1zσμ0ωk0z+k1z+σμ0ωr0G1p=ε1k0zk1z+(σk0zk1zωε0)ε1k0z+k1z+(σk0zk1zωε0)
(7)
where ε0 is the electric permittivity of vacuum. Again, by replacing r01γr0G1γ and r02γr0G2γ in Eq. (4), the near-field heat transfer between graphene-coated Si plates can be calculated.

Alternatively, the near-field thermal radiation between doped Si plates coated with graphene layer in Fig. 1 can be modelled as a multilayer system. The dyadic Green’s function for multilayer structures can be expressed by considering forward and backward waves in each layer [30

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

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]

]. In order to consider the fluctuating current source in the graphene layer, we regard graphene sheet as a thin film with a finite thickness Δ = 0.5 nm and a dielectric function εG(ω)=1+iσ(ω)/Δωε0[32

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]

]. Figure 2 shows comparison of the spectral heat flux between the graphene-coated Si plates calculated by taking the configuration as a multilayer system and as a two-body system with modified Fresnel reflection coefficients at vacuum-source and vacuum-receiver interfaces. These two methods essentially provide the identical results. For the multilayer system, convergence of εG 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.

Fig. 2 Comparison of the spectral heat flux between the source (graphene-coated Si at 400 K) and the receiver (graphene-coated Si at 300 K) calculated by taking the configuration as multilayer system (symbols) and as two-body system with modified Fresnel reflection coefficients (lines).

3. Results and discussion

In the following, we consider the near-field thermal radiation of four configurations: (i) no graphene is coated on the source and receiver; (ii) graphene is coated on the source only; (iii) graphene is coated on the receiver only; and (iv) both source and receiver are coated with graphene. For graphene, the relaxation time is set to be 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]

], while chemical potential μ varies from 0.1 eV to 0.5 eV.

Figure 3 plots the enhancement factor EF = q″net/q″net,bare in logarithmic scale, where q″net,bare is the heat flux between Si plates without graphene at the corresponding doping concentrations. When 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 1020 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 1019 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 1018 cm−3, significant enhancement is obtained. In particular, for both source and receiver at 1017 cm−3, graphene of μ = 0.3 eV can result in nearly two orders-of-magnitude enhancement in the heat transfer (EF = 89.5).

Fig. 3 Contour plot of the calculated radiative heat flux normalized by that between bare doped Si surfaces. In each panel of the figure, x-axis indicates the doping concentration of the source (1017 ∼ 1021 cm−3) and y-axis represents the doping concentration of the receiver (1017 ∼ 1021 cm−3). The left column corresponds to the case of d = 10 nm, and the right column is for d = 50 nm.

If graphene is coated on the source only, the heat flux is enhanced only when the doping concentration of the source is lower than that of the receiver. For S1 when the source is at 1019 cm−3 and the receiver is at 1020 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 1020 cm−3, graphene suppresses the radiative heat transfer regardless of the location where it is placed.

As the vacuum gap width increases to 50 nm (i.e., right column of Fig. 3), if graphene is coated on both Si plates, graphene with higher chemical potential (i.e., B4) results in larger heat transfer enhancement than graphene with lower chemical potential (i.e., B3), which is opposite to the case of B1 and B2 at d = 10 nm. Likewise, for d = 50 nm, the most significant enhancement occurs when both source and receiver are at 1017 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.

In order to elucidate the heat transfer enhancement mechanism associated with graphene insertion, the spectral energy flux is plotted in Fig. 4 for selected cases listed in Table 1. In Fig. 4(a), S1 shows higher and more broadened peak in the spectral heat flux than N1, whereas R1 exhibits the lowest peak. Since the radiative heat transfer in near-field regime is greatly affected by SPP, Fig. 5 plots the plasmon dispersion curves with contour of 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 1r01pr02pei2k0zd=0, where r01p and r02p represent the Fresnel reflection coefficients at vacuum-source and vacuum-receiver interfaces, respectively [20

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]

]. For N1 in Fig. 5(a), dielectric functions of the source and receiver are different as their temperatures and doping concentrations are different, leading to two separate SPP dispersion curves. When graphene is coated on the source as S1 in Fig. 5(b), the SPP dispersion at the vacuum-source interface, which is located at lower frequency than that at the vacuum-receiver interface, is shifted to the higher frequency region at a given 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]

]. If graphene is coated on the receiver (i.e., R1), the upper SPP branch at the vacuum-receiver interface is shifted to further higher frequency regime, resulting in a larger mismatch in the resonance frequencies of the SPP at vacuum-source and vacuum-receiver interfaces (refer to Fig. 5(c)). This in turn reduces the heat transfer as shown in Fig. 4(a). Consequently, if graphene is coated on the Si whose SPP resonance frequency is lower than that of the other side (i.e., whose doping concentration is lower than that of the other side), the radiative heat flux can be enhanced.

Fig. 4 Spectral energy flux between two doped Si plates: (a) d = 10 nm, Source (1019 cm−3), Receiver (1020 cm−3); (b) d = 10 nm, Source (1019 cm−3), Receiver (1019 cm−3); and (c) d = 50 nm, Source (1019 cm−3), Receiver (1019 cm−3).
Fig. 5 Contour plot of S(β, ω) with respect to the parallel wavevector component β normalized by plasma frequency (ωp = 2.90 × 1014 rad/s) of doped Si at 1019 cm−3 and 400 K. SPP dispersion curves are also overlaid. Source and receiver configurations for each case are listed in Table 1.

Table 1. Source and receiver configurations (N: no graphene; B: graphene on both sides; S: graphene on source only; R: graphene on receiver only).

table-icon
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When graphene is coated on both source and receiver, graphene can enhance the heat transfer also by broadening spectral heat flux. For both source and receiver at 1019 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″ω,net becomes broader. When 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.

It can be seen from SPP dispersion curves in Figs. 5(d)–5(f) that graphene does help SPPs at vacuum-source and vacuum-receiver interfaces occur at similar frequencies but at higher frequency regime. Shift of SPP dispersion curves to higher frequency regime becomes larger as the chemical potential of graphene increases. Since the mean Planck oscillator energy Θ(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 1020 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.

As seen from Figs. 5(a), 5(d), and 5(g), if there is no graphene, the maximum of 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

35. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

]. 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 Sp(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]

]. However, when graphene covers both the source and receiver (i.e., B1, B2, B3, and B4), the maximum of 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 Sp(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.

It should be noted that the most significant enhancement occurs when graphene is coated on both source and receiver at 1017 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 1017 cm−3 is much smaller than that between source and receiver at doping concentration higher than 1019 cm−3. This is because Si with doping concentration lower than 1018 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 1017 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 (∼ 1017 cm−3) be comparable to the heat transfer between heavily doped Si plates (> 1019 cm−3).

Figure 6 shows the contribution of graphene to the near-field thermal radiation in B5, S1, and R1. For a given configuration, the net heat flux between Si substrates only can be easily calculated by using the multilayer formulation. Graphene’s contribution to the near-field thermal radiation is estimated from the difference between the total net heat flux and the net heat flux between Si substrates only, for the corresponding configuration. When EF ≫ 1 as in B5, graphene’s contribution dominates q″net. For S1 with moderate 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.

Fig. 6 Contribution of graphene to the net heat transfer.

Distance dependence of the near-field thermal radiation between graphene-coated Si plates is illustrated in Fig. 7. For all 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.

Fig. 7 Net heat transfer between graphene-coated Si plates at 1017 cm−3 with respect to the vacuum gap width.

4. Concluding remark

We have systemically investigated the effect of graphene on the near-field thermal radiation of asymmetric layered system with doped Si at different doping concentrations. It was found that the radiative heat flux between doped Si plates could be either enhanced or suppressed by introducing graphene layer, depending on the Si doping concentration and chemical potential of graphene. Graphene could enhance the heat flux if it matches resonance frequencies of surface plasmon at vacuum-source and vacuum-receiver interfaces. In particular, significant enhancement was achieved when graphene is coated on both surfaces that originally does not support the surface plasmon resonance. On the other hand, graphene barely induced the heat transfer enhancement if the vacuum gap width becomes greater than 500 nm.

Acknowledgments

This research was supported by Basic Science Research Program through the National Science Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning ( NRF-2012RA1A1006186 and NRF-20100027050).

References and links

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]

3.

Z. M. Zhang, Nano/Microscale Heat Transfer (McGraw-Hill, 2007).

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]

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 3, 514–517 (2009). [CrossRef]

6.

S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett. 9, 2909–2913 (2009). [CrossRef] [PubMed]

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. 92, 133106 (2008). [CrossRef]

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. 107, 014301 (2011). [CrossRef] [PubMed]

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]

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]

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 SiO2,” 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 SiO2,” Phys. Rev. B 83, 241407 (2011). [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]

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]

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]

24.

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

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]

27.

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

28.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

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]

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]

33.

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

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]

35.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

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

  1. D. Polder and M. Van Hove, “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B4, 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]
  3. Z. M. Zhang, Nano/Microscale Heat Transfer (McGraw-Hill, 2007).
  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. B77, 035431 (2008). [CrossRef]
  5. E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nat. Photonics3, 514–517 (2009). [CrossRef]
  6. S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett.9, 2909–2913 (2009). [CrossRef] [PubMed]
  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.92, 133106 (2008). [CrossRef]
  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.107, 014301 (2011). [CrossRef] [PubMed]
  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]
  11. C. J. Fu and Z. M. Zhang, “Nanoscale radiation heat transfer for silicon at different doping levels,” Int. J. Heat Mass Transfer49, 1703–1718 (2006). [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 Transfer132, 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 Transfer132, 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 SiO2,” J. Phys. Condens. Matter22, 462201 (2010). [CrossRef]
  18. A. I. Volokitin and B. N. J. Persson, “Near-field radiative heat transfer between closely spaced graphene and amorphous SiO2,” Phys. Rev. B83, 241407 (2011). [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. B85, 155418 (2012). [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. B85, 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. Express20, A366–A384 (2012). [CrossRef] [PubMed]
  22. R. Messina and P. Ben-Abdallah, “Graphene-based photovoltaic cells for near-field thermal energy conversion,” Scientific Reports3, 1383 (2013). [CrossRef] [PubMed]
  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]
  24. J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B4, 481–489 (1987). [CrossRef]
  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. B58, 423–431 (2007). [CrossRef]
  27. L. A. Falkovsky, “Optical properties of graphene,” J. Phys. Conf. Ser.129, 012004 (2008). [CrossRef]
  28. J. D. Jackson, Classical Electrodynamics (Wiley, 1999).
  29. T. Stauber, N. M. R. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B78, 085432 (2008). [CrossRef]
  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. Transfer109, 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. Transfer110, 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. Express20, 28017–28024 (2012). [CrossRef] [PubMed]
  33. M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B80, 245435 (2009). [CrossRef]
  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. B22(5), 1016–1023 (2005). [CrossRef]
  35. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

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