## 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 [1–3]. 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, 3]. A plethora of theoretical and experimental works has demonstrated the enhanced near-field thermal radiation between various materials, such as polar materials and metals [1,2,4–9]. 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 [10–12]. Because optical constants (i.e., refractive index and extinction coefficient) of doped Si highly depend on both doping concentration and temperature [13], 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.

Graphene, a two-dimensional (2-D) lattice of carbon atoms with exceptionally high crystal and electronic quality, draws enormous attention due to its broad potential applications in electronic and photonic devices [14, 15]. Because plasmon frequency of graphene can be tuned to near-infrared spectral region by modifying electron density [16], several recent studies also focus on the role of graphene on the near-field radiative heat transfer [17–20] and its application on near-field thermophotovoltaic devices [21, 22]. Although previous studies have investigated the effect of graphene on the near-field heat transfer, they mainly considered suspended graphene [17,18,20,21] or graphene on glass [19] and InSb [22]. In the present study, however, we consider a graphene-coated asymmetric doped Si plates of different optical constants (i.e., with different doping concentrations and temperatures) to investigate more general effects of graphene on modifying the near-field thermal radiation.

## 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 10^{17} to 10^{21} cm^{−3}, and the resulting dielectric function *ε* is obtained from the functional expressions in Basu *et al.* [13]. For simplicity, temperature of the source and receiver is set to be 400 and 300 K, respectively.

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, 23]:

*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]

**ŝ**=

**r̂**×

**ẑ**and

**p̂**

*=(*

_{i}*β/k*)

_{i}**ẑ**−(

*k*)

_{iz}/k_{i}**r̂**are polarization vectors. In Eq. (2), ${t}_{12}^{s}$ and ${t}_{12}^{p}$ represent the transmission coefficients from the source to the receiver for a given polarization, where

*β*is the parallel wavevector component, and ${k}_{iz}=\sqrt{{\omega}^{2}/{c}_{0}^{2}{\epsilon}_{i}-{\beta}^{2}}$ is the normal wavevector component with the speed of light in vacuum

*c*

_{0}(i.e., ${k}_{i}^{2}={\beta}^{2}+{k}_{iz}^{2}$). 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, $\u3008\mathbf{S}\u3009=\frac{1}{2}\text{Re}\u3008\mathbf{E}\times {\mathbf{H}}^{*}\u3009$, 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, 〈

*S*〉, which can then be expressed as

_{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],

*i*−

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

*T*

_{1}is the temperature of body 1 and $\mathrm{\Theta}\left(\omega ,{T}_{1}\right)=\frac{\overline{h}\omega}{\text{exp}\left\{\overline{h}\omega /\left({k}_{B}{T}_{1}\right)\right\}-1}$ is the mean energy of Planck oscillator, where

*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 ${q}_{\mathit{net}}^{\u2033}={\int}_{0}^{\infty}d\omega \hspace{0.17em}{q}_{\omega ,\mathit{net}}^{\u2033}={\int}_{0}^{\infty}d\omega \left[{q}_{\omega ,1\to 2}^{\u2033}-{q}_{\omega ,2\to 1}^{\u2033}\right]$.

_{net}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],
$1-{\left|{r}_{01}^{\gamma}\right|}^{2}$ and
$1-{\left|{r}_{02}^{\gamma}\right|}^{2}$ represent the spectral emittance of the source and the spectral absorptance of the receiver, respectively, for propagating waves. Similarly,
$\text{Im}\left({r}_{01}^{\gamma}\right)$ and
$\text{Im}\left({r}_{02}^{\gamma}\right)$ 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,
${r}_{01}^{\gamma}\to {r}_{0F1}^{\gamma}$ and
${r}_{02}^{\gamma}\to {r}_{0F2}^{\gamma}$, 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
${r}_{0F1}^{\gamma}$ at the vacuum-source interface, and the thin-film-coated semi-infinite receiver as another single body with
${r}_{0F2}^{\gamma}$ 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] as well as that between a semi-infinite body and a coated semi-infinite body [26], and has also been employed for graphene-coated media [19, 22].

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*)= *σ _{I}*(

*ω*)+

*σ*(

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

*e*is the electron charge, $G(\xi )=\text{sinh}\left(\frac{\xi}{{k}_{B}T}\right)/\left[\text{cosh}\left(\frac{\mu}{{k}_{B}T}\right)+\text{cosh}\left(\frac{\xi}{{k}_{B}T}\right)\right]$, and

*τ*and

*μ*represent the relaxation time and chemical potential of graphene, respectively [27]. With this surface conductivity of graphene, the Fresnel reflection coefficient ${r}_{0G1}^{\gamma}$ 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]

**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, 27, 29]:

*ε*

_{0}is the electric permittivity of vacuum. Again, by replacing ${r}_{01}^{\gamma}\to {r}_{0G1}^{\gamma}$ and ${r}_{02}^{\gamma}\to {r}_{0G2}^{\gamma}$ 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, 31]. 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
${\epsilon}_{G}(\omega )=1+i\frac{\sigma (\omega )/\mathrm{\Delta}}{\omega {\epsilon}_{0}}$[32]. 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 *ε _{G}* 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.

## 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], 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″*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).

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 10^{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.

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 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.

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
$1-{r}_{01}^{p}{r}_{02}^{p}{e}^{i2{k}_{0z}d}=0$, where
${r}_{01}^{p}$ and
${r}_{02}^{p}$ represent the Fresnel reflection coefficients at vacuum-source and vacuum-receiver interfaces, respectively [20]. 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]. 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.

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 10^{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″ _{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 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.

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]. 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]. 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

*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.

It should be noted that the most significant enhancement occurs when graphene is coated on both source and receiver at 10^{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}).

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.

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.

## 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).

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