Coated silicon carbide (SiC) thin films can efficiently enhance near-field radiative heat transfer among metamaterials. In this study, the near-field heat transfer among graphene–SiC–metamaterial (GSM) multilayer structures was theoretically investigated. Graphene plasmons could be coupled both with electric surface plasmons supported by the metamaterial and with symmetric and anti-symmetric surface phonon polaritons (SPhPs) supported by SiC. The heat transfer among GSM structures was considerably improved compared to that among SiC-coated metamaterials when the chemical potential of graphene was not very high. In addition, the near-field heat transfer was enhanced among SiC–graphene–metamaterial multilayer structures, though the heat transfer among these structures was less than that among GSMs owing to the absence of coupling between symmetric SPhPs and graphene plasmons. Hence, heat transfer could be flexibly tuned by modifying the chemical potential of graphene in both configurations. These results provide a basis for active control of the near-field radiative heat transfer in the far-infrared region.
I. INTRODUCTION
Polder and van Hove1 first predicted that the heat transfer between two bodies at small distances could be greater than blackbody radiation, and since this pioneering work near-field radiative heat transfer has gained increasing research interest both theoretically2–9 and experimentally.10–13 Heat transfer among closely-spaced bodies can drastically surpass the well-known Stefan–Boltzmann law by orders of magnitude. When two objects are placed in close proximity, the contributions from evanescent waves cannot be ignored owing to the presence of near-field effects, including photon tunneling14 and excitation of surface polaritons.15,16 Therefore, materials that can support surface modes are often used to enhance near-field heat transfer.17–20 Similar enhancement effects have been predicted and observed for metals and doped semiconductors because of the presence of surface plasmons (SPs).21–26 This effect can be employed to improve the performance of near-field photovoltaic devices16,27–29 and near-field scanning thermal microscopy.30–33
Several studies have predicted that hyperbolic metamaterials can behave as broadband super-Planckian thermal emitters.3,6,34,35 In particular, surface waves of metamaterials can be excited for both s- and p-polarizations owing to negative permeability and permittivity; consequently, new channels can be opened to enhance heat transfer in the near-field regime. Joulain et al.4 have reported that radiative heat transfer increases in the near-field region between metamaterials and can support surface waves. Basu et al.8 have investigated near-field radiative transfer among metamaterials coated with SiC thin films where, for such structures, electric and magnetic SPs can be excited by the metamaterials and surface phonon polaritons (SPhPs) can be excited by SiC in different spectral regions. The results have demonstrated that near-field heat transfer between two bulk metamaterials can be enhanced through excitation of different types of surface waves by adding a thin SiC film on top of both of the semi-infinite metamaterials.
Graphene, a two-dimensional monolayer of carbon atoms, is a promising material with unusual electrical and optical properties.36–39 Graphene can support SPs that exhibit the same characteristics as those of noble metals. The SPs supported by graphene can cover the frequency range from terahertz to the near-infrared regions, and the frequency of the SPs can be easily tuned by altering the graphene chemical potential. Considering these novel characteristics, scholars have studied near-field heat transfer among graphene-based systems.5,7,40–42
In this paper, we theoretically investigated near-field heat transfer among graphene–SiC–metamaterial (GSM) multilayer structures. The heat transfer of the GSM structures could be improved over that of a SiC-coated metamaterial because the SPs supported by graphene could couple with electric SPs (ESPs) supported by the metamaterials and with the SPhPs from SiC. The heat transfer among GSM structures could be flexibly tuned by changing the chemical potential of graphene. For comparison, heat transfer among SiC–graphene–metamaterials (SGM) structures was also studied. The heat transfer among the SGM structures was less than that among GSM structures under the same conditions, which can be attributed to the absence of coupling between graphene plasmons and symmetric SPhPs.
II. THEORETICAL MODEL
Figure 1 shows a schematic of the GSM structures separated by a vacuum gap d. The thickness of both SiC films are denoted as t1, and the temperature is set at 300 K. The relative permittivity, εM, and permeability, μM, of the metamaterials can be described by4,8
where F = 0.56 is the split filling factor, ωp = 1014 rad−1 is the equivalent plasma frequency, ω0 = 0.4ωp, and γe = γm = γ = 0.01ωp. The permittivity of SiC is given by the Lorentz model43
where ε∞ = 6.7, ωL = 1.827 × 1014 rad−1, ωT = 1.495 × 1014 rad−1, and Γ = 0.9 × 1012 rad−1.
Schematic of the near-field radiative heat transfer between two GSM structures, showing the SiC thickness, t1, and the vacuum gap, d.
Schematic of the near-field radiative heat transfer between two GSM structures, showing the SiC thickness, t1, and the vacuum gap, d.
The near-field radiative heat flux can be obtained from fluctuational electrodynamics.1 The expression for the radiative heat transfer coefficient (HTC) between two media is given as
where is the mean energy of a Plank oscillator at thermal equilibrium temperature T; is the reduced Planck constant; kB is the Boltzmann constant; and is the wave vector component in vacuum along the z direction, c is the speed of light in vacuum, and q is the wave vector component along x direction. Only the contributions from evanescent waves (q > ω/c) are dominant in the near-field regime, whereas those from propagating waves (q < ω/c) are negligible. To describe the physical process, we define the spectral heat transfer coefficient (SHTC), H(ω,d), as
where s(ω, q) is the exchange function used to describe the frequency correlation of heat transfer. The exchange function for j-polarization (j=s, p) is given by the equation
where the reflection coefficients (i = 1,2) of a multilayer structure can be obtained using thin-film optics for both the s- and p-polarizations44,45
where , and κ = qc/ω. Using the electromagnetic field boundary condition, the reflection coefficients (i = 1,2) for s- and p-polarized electromagnetic waves from the surface of a general medium–graphene–medium sandwich structure can be derived as
where μ0 is the permeability of vacuum, , ; and and (εr2 and μr2) are the relative permittivity and permeability of medium 1 (medium 2), respectively. The relative permeability for the nonmagnetic medium is μr = 1.0, and that for the metamaterials is given by Eq. (2). The optical conductivity σ(ω) of graphene can be decomposed into intraband (i.e., Drude), σD, and interband, σI, contributions,37,38 such that
where G(ξ) = sinh(ξ/kBT)/[cosh(μ/kBT) + cosh(ξ/kBT)], and μ is the chemical potential of graphene. The chemical potential is determined by the concentration of current carrier , where νF is the Fermi velocity. The chemical potential can also be tuned freely because n0 can be easily controlled by the grid voltage. The relaxation time τ is caused by electronic impurities, electronic defects, and electron-phonon scattering; and is given by , where μdc is the electronic mobility.
III. RADIATIVE HEAT TRANSFER BETWEEN TWO GSM STRUCTURES
First, we investigated the radiative heat transfer between two GSM structures. The thicknesses of the SiC films were both t1 = 10 nm, while the vacuum gap was set at 10 nm. The exchange function between two GSM structures for different chemical potentials is shown in Fig. 2. For comparison, the exchange function for the SiC-coated metamaterials (GSM without graphene) is shown in Fig. 2(a). A metamaterial is a magnetodielectric material, which can simultaneously support magnetic SPs (MSPs) and ESPs. As shown in Fig. 2(a), two modes at around 3 and 4.5 × 1013 rad/s respectively correspond to the electric and magnetic resonance for emission from the metamaterial substrates. In addition, another resonance mode appears near the frequency 178 THz that corresponds to the SPhPs excited by SiC. With the strong coupling of the surface waves generated at the vacuum–film interface within and between SiC thin films, we can clearly see the splitting of the near-field flux into symmetric and anti-symmetric resonance modes. This result has been previously discussed in detail in Refs. 8 and 46. Furthermore, when the SiC film is covered with a graphene sheet, the ESPs excited by the metamaterial can not only couple with the SPs supported by graphene, but the SPhPs supported by SiC can also strongly couple with the graphene plasmons. From Figs. 2(b), 2(c) and 2(d) we can clearly see that the coupled surface modes acutely depend on the chemical potential. As the chemical potential increases, the corresponding ESPs and anti-symmetric resonance modes move to higher frequencies and their spectra become broader. However, the graphene has no effect on the MSPs, which is owing to the absence of coupling between MSPs and p-polarized SPs excited by graphene. Additionally, coupling of the SPhPs generates a symmetric resonance mode at the vacuum–film interface. In contrast, in Fig. 2(a) the positions of the symmetric resonance mode are independent of the chemical potential, while the amplitude decreases as the chemical potential increases.
Exchange function s(ω, q) for the GSM structure as a function of ω and κ for structures (a) without graphene, and with graphene possessing a chemical potential (μ) of (b) 0.1, (c) 0.2 and (d) 0.3 eV. The vacuum gap is 10 nm and the SiC layer is 10 nm thick. Labeled are the electric surface plasmon (ESP), magnetic surface plasmon (MSP), and symmetric (S-SPhPs) and anti-symmetric (As-SPhPs) surface phonon polaritons.
Exchange function s(ω, q) for the GSM structure as a function of ω and κ for structures (a) without graphene, and with graphene possessing a chemical potential (μ) of (b) 0.1, (c) 0.2 and (d) 0.3 eV. The vacuum gap is 10 nm and the SiC layer is 10 nm thick. Labeled are the electric surface plasmon (ESP), magnetic surface plasmon (MSP), and symmetric (S-SPhPs) and anti-symmetric (As-SPhPs) surface phonon polaritons.
The contribution of graphene to the near-field heat transfer can be further understood from the spectral plot, as shown in Fig. 3 for different chemical potentials. The SHTC for the SiC-coated metamaterials are shown (black solid lines in Fig. 3) as a comparison. In our calculation, the vacuum gaps d are set at 10 and 50 nm in Figs. 3(a) and 3(b), respectively, and the SHTC H(ω, d) is normalized to the blackbody result HBB = ω2/2πc2. In Fig. 3(a), we can see the appearance of four peaks in the SHTC curve that correspond to the MSPs, the ESPs from metamaterials, and the symmetric and anti-symmetric resonance modes of SPhPs from SiC, respectively. As the chemical potential of graphene increases, the position of the MSPs remains constant while the ESPs shift toward higher frequencies after coupling with the graphene plasmon. The anti-symmetric resonance modes of SPhPs also shifts to higher frequencies, in a manner similar to that of the ESPs, and its peak broadens. The position of the symmetric resonance modes of SPhPs is also constant, while the amplitudes of the symmetric resonance modes becomes suppressed. We also plot the SHTC between the GSM structures for a vacuum gap of 50 nm in Fig. 3(b), where the spectra exhibit a shifting trend similar to that of Fig. 3(a) and where the amplitude of the SHTC obviously decreases. In addition, the peak of the coupled SPhPs has a smaller shift for the vacuum gap of 50 nm compared to the peak for the vacuum gap of 10 nm, which means that the SPhPs are more easily modulated by graphene in a smaller vacuum gap than in a larger vacuum gap. By considering the area under the SHTC, we can predict that the radiative heat transfer is larger for the GSM structure than for the SiC-coated metamaterials. The exact results of the HTC can be obtained using Eq. (4).
Spectral heat transfer coefficients between two GSM structures as a function of ω for different chemical potentials and vacuum gaps (d) of (a) 10 and (b) 50 nm. Spectral heat flux is normalized to the blackbody result HBB = ω2/2πc2.
Spectral heat transfer coefficients between two GSM structures as a function of ω for different chemical potentials and vacuum gaps (d) of (a) 10 and (b) 50 nm. Spectral heat flux is normalized to the blackbody result HBB = ω2/2πc2.
The dependence of the HTC on the chemical potential is shown in Figs. 4(a) and 4(b) for 10 and 50 nm-wide vacuum gaps, respectively, where the HTC has been normalized to the blackbody result. In Fig. 4(a), the maximum HTC between the GSM structures is 1000 times higher than blackbody radiation when the low chemical potential value is much larger than that between the SiC-coated metamaterials. Compared to the SiC-coated metamaterials case, graphene is important for enhancing the near-field radiative heat transfer. The HTC between GSM structures strongly depends on the chemical potential and, further, the HTC does not exhibit a monotonic trend, but increases to its maximum value at a chemical potential of about 0.1 eV and subsequently decreases as the chemical potential further increases. Interestingly, when the chemical potential exceeds 0.25 eV, the HTC between GSM is even less than the value between SiC-coated metamaterials. The theoretical limit of heat transfer between hyperbolic metamaterials is given by3 , which is also plotted in Fig. 4(a), and we can see that our result is still far below the theoretical limit. The HTCs for both GSM and SiC-coated metamaterials exhibited an obvious decrease for a 50 nm vacuum gap compared to that for a 10 nm vacuum gap. The HTC for GSM still exhibits the identical trend of initially increasing to a maximum and subsequently decreasing with increasing chemical potential, though with a difference in the value of the chemical potential that corresponds to the maximum HTC changes. However, the HTC for GSM is always larger than the HTC of the SiC-coated metamaterials at the chemical potential range of 0.08–0.4 eV. The dependence of HTC on the vacuum gap for different chemical potentials is displayed in Fig. 4(c), where the HTC decreases rapidly as the vacuum gap increases for all the cases. When the vacuum gap widens, the differences of HTC between different cases become smaller because all of the surface waves are dominant in the near-field region. When the vacuum gap widens, the effect of the evanescent surface waves on the HTC significantly weakens. The contribution from graphene possessing even a small chemical potential is particularly remarkable in enhancing the radiative heat transfer for small vacuum gaps.
Heat transfer coefficients h(d) as a function of the chemical potential μ for vacuum gaps of (a) 10 and (b) 50 nm. (c) Heat transfer coefficients h(d) as a function of the vacuum gap. The heat transfer coefficient is normalized to the blackbody value hBB = 38.2852 Wm−2K−1. For comparison, in (a, b) the heat transfer coefficient for SiC-coated metamaterials is also plotted (black solid lines), as well as in (a) the theoretical limit of heat transfer between hyperbolic metamaterials (blue dotted line).
Heat transfer coefficients h(d) as a function of the chemical potential μ for vacuum gaps of (a) 10 and (b) 50 nm. (c) Heat transfer coefficients h(d) as a function of the vacuum gap. The heat transfer coefficient is normalized to the blackbody value hBB = 38.2852 Wm−2K−1. For comparison, in (a, b) the heat transfer coefficient for SiC-coated metamaterials is also plotted (black solid lines), as well as in (a) the theoretical limit of heat transfer between hyperbolic metamaterials (blue dotted line).
The HTC for different thicknesses of the SiC layer as a function of vacuum gap are shown in Fig. 5. We can see that the thickness of the SiC has an obvious impact on the HTC. For the graphene-covered metamaterials (i.e., SiC thickness is zero), the HTC (black solid line in Fig. 5) is smaller than that between the MSG structures with a 10 nm-thick layer of SiC (red dashed line in Fig. 5) in our studied vacuum gap range. This result can be easily understood when one considers the absence of a contribution to the HTC from the SPhPs excited by SiC. However, when the SiC thickness increases, the HTC exhibits an obvious decrease, as can be seen from the comparison of the data in Fig. 5. In particular, when the SiC is sufficiently thick (green dash-dotted line in Fig. 5), there is almost no difference between the HTC of the MSG structures and that of the graphene-covered SiC structures. This is because the SiC film behaves as a semi-infinite medium that will inhibit the contributions of the MSPs and ESPs excited by the metamaterials.
Heat transfer coefficients h(d) as a function of vacuum gap d for different thicknesses of the SiC layer.
Heat transfer coefficients h(d) as a function of vacuum gap d for different thicknesses of the SiC layer.
IV. RADIATIVE HEAT TRANSFER BETWEEN TWO SGM STRUCTURES
Based on the results above, coupling is not only allowable in ESPs, but symmetric and anti-symmetric couplings of SPhPs with graphene plasmons are also acceptable. To confirm this hypothesis, we investigated the radiative heat transfer when the positions of the SiC and graphene are exchanged; that is, we investigated the radiative heat transfer between two SGM structures. To enable comparison between these studies and those reported above, all of the parameters used were identical with those of the GSM structure. Figure 6 shows a contour plot of the exchange function s(ω, q) as a function of the angular frequency ω and of κ for different chemical potentials, where the vacuum gap and the SiC film thickness are both set at 10 nm. The contour of the SiC-coated metamaterials is shown in Fig. 6(a) as a comparison. In Fig. 6(b), the ESPs from the metamaterials and the anti-symmetric SPhPs from SiC can couple with graphene plasmons despite the changes in the position of the graphene. The coupled ESPs and SPhPs in the SGM structure shift to higher frequencies, and exhibit a similar shift behavior with that of the GSM structure. The MSPs, however, are independent of the graphene influence and, in addition, the position and amplitude for the symmetric SPhPs do not vary with the chemical potential, which is different than the results of the GSM structure case. In this case, the symmetric modes arise from the coupling of the surface waves at the vacuum–film interface. When graphene is sandwiched between the SiC and the metamaterials, the coupling of the symmetric SPhPs and graphene plasmons is absent, and the graphene therefore has no effect on the symmetric SPhPs.
Exchange function s(ω, q) for the SGM structure as a function of ω and κ for structures (a) without graphene, and with graphene possessing a chemical potential (μ) of (b) 0.1, (c) 0.2 and (d) 0.3 eV. The vacuum gap is 10 nm and the SiC layer is 10 nm thick. Labeled are the electric surface plasmon (ESP), magnetic surface plasmon (MSP), and symmetric (S-SPhPs) and anti-symmetric (As-SPhPs) surface phonon polaritons.
Exchange function s(ω, q) for the SGM structure as a function of ω and κ for structures (a) without graphene, and with graphene possessing a chemical potential (μ) of (b) 0.1, (c) 0.2 and (d) 0.3 eV. The vacuum gap is 10 nm and the SiC layer is 10 nm thick. Labeled are the electric surface plasmon (ESP), magnetic surface plasmon (MSP), and symmetric (S-SPhPs) and anti-symmetric (As-SPhPs) surface phonon polaritons.
In Fig. 7, the SHTC H(ω, d) is plotted as a function of angular frequency for SiC-coated metamaterials, as well as the SGM structure with chemical potentials of μ = 0.1, 0.2 and 0.3 eV. All of the parameters in the calculations are the same as those used in Fig. 6. Four peaks can be seen in the spectra of the SHTC for the SGM structure in Fig. 7, which is similar to that seen for the GSM structure. The peaks of ESPs and SPhPs shift toward higher frequencies as the chemical potential increases, though the shift range of the SGM spectral peaks is less than that of the GSM spectral peaks. Therefore, the effect of graphene upon the SHTC is relatively weaker when embedded between SiC and metamaterials. The positions and the amplitudes of the MSP and symmetric SPhP peaks do not vary with chemical potential, which exhibits the same result as the contour analysis of s(ω, q) and, in addition, the shift range of the anti-symmetric SPhP peaks is smaller than that of the GSM structure.
Spectral heat transfer coefficient for the SGM structure as a function of ω for different chemical potentials. The vacuum gap is 10 nm and the spectral heat flux is normalized to the blackbody result HBB = ω2/2πc2.
Spectral heat transfer coefficient for the SGM structure as a function of ω for different chemical potentials. The vacuum gap is 10 nm and the spectral heat flux is normalized to the blackbody result HBB = ω2/2πc2.
In Figs. 8(a) and 8(b), the normalization of HTC to the blackbody result is shown for these SGM structures with vacuum gaps of 10 and 50 nm, respectively. Compared to the HTC values between GSM structures, the HTC values between SGM structures obviously decrease for both vacuum gap values, indicating the weakened contributions of the graphene plasmon that arises from the absence of coupling between the graphene plasmons and symmetric SPhPs. For the vacuum gap d = 10 nm, the HTC is seen to continuously decrease as the chemical potential increases and, for a chemical potential greater than 0.23 eV, the HTC of SGM is even lower than that of SiC-coated metamaterials. For the vacuum gap of 50 nm, the HTC follows a non-monotonic trend of initially increasing to a maximum value and then decreasing as the chemical potential increases. For this vacuum gap value, however, the HTC for the SGM is not less than that for SiC-coated metamaterials until the chemical potential reaches 0.27 eV. In Fig. 8(c), the HTC is shown as a function of vacuum gap for different chemical potentials. The differences of the HTC between different configurations are larger at small vacuum gaps, while the differences of the HTC are very small for larger vacuum gaps by virtue of the near-field effect.
Heat transfer coefficients h(d) as a function of the chemical potential μ for a vacuum gap (d) of (a) 10 and (b) 50nm. For comparison, in (a, b) the heat transfer coefficient for SiC-coated metamaterials is also plotted (black solid lines). (c) Heat transfer coefficients h(d) as a function of the vacuum gap. The heat transfer coefficient is normalized to the blackbody value hBB = 38.2852 Wm−2K−1.
Heat transfer coefficients h(d) as a function of the chemical potential μ for a vacuum gap (d) of (a) 10 and (b) 50nm. For comparison, in (a, b) the heat transfer coefficient for SiC-coated metamaterials is also plotted (black solid lines). (c) Heat transfer coefficients h(d) as a function of the vacuum gap. The heat transfer coefficient is normalized to the blackbody value hBB = 38.2852 Wm−2K−1.
From the results above, we can ascertain that the chemical potential significantly influences the HTC. For small vacuum gaps, graphene possessing a lower chemical potential can enhance the radiative heat transfer efficiently for both GSM and SGM structures, while graphene possessing a higher chemical potential can hinder the near-field radiative heat transfer. Therefore, using graphene can actively control the near-field radiative heat transfer without adding any more volume.
V. CONCLUSIONS
We studied the near-field radiative heat transfer between GSM structures and found that metamaterials not only support ESPs but also the symmetric and anti-symmetric SPhPs coupling of SiC with graphene plasmons. This strong coupling resulted in the enhancement of the heat transfer under a low chemical potential of graphene. However, graphene at a high chemical potential may inhibit the heat transfer. In addition, we investigated the near-field radiative heat transfer between SGM structures. Because the coupling between symmetric SPhPs and graphene plasmons was absent, the heat transfer was not as high as that brought by the GSM structures. We therefore demonstrated that the heat transfer is flexibly tunable by graphene for both configurations, and the results obtained herein pave the way for active measurement of the near-field heat flux.
ACKNOWLEDGMENTS
This work is supported by the National Natural Science Foundation of China (Nos. 61367006, 11364033, 11664024, 11264029) and the Natural Science Foundation of Jiangxi Province (No. 20151BAB202017).