Radiative heat transfer has aroused a great deal of interest in the past few decades; many studies focus on the near-field radiative heat transfer due to its potential applications in thermal imaging,^[1–4] radiative cooling,^[5] near-field thermophotovoltaics,^[6,7] and thermal energy conversion.^[8] The near-field radiative heat transfer that can exceed the blackbody limit has been extensively studied in various systems.^[6–12] In the presence of resonant surface modes, such as surface phonon polaritons (SPhPs) and surface plasmon polaritons (SPPs), the heat transfer between two bodies in the near-field regime can be enhanced significantly. The materials that can support SPhPs^[11] or SPPs^[12] were employed to control the heat transfer in several studies. In addition, the hyperbolic mode (HM) excited by hyperbolic metamaterial (HMM) often covers a wider frequency range than SPhP or SPP, which is limited to a small frequency band around its resonance frequency. Therefore, the near-field radiative heat transfer between hyperbolic materials has also been investigated by many researchers.^[9,10]

Terahertz (THz) wave, a type of electromagnetic wave with a frequency in a range from 0.1 THz to 10 THz, has received increasing attention because of its promising applications in spectroscopy,^[13,14] imaging,^[15] bio-sensing,^[16] communication,^[17] etc. In the past few years, many attempts have been made to improve the absorption of THz waves, which facilitates the development of THz detectors,^[18,19] thermal emitters, and sensors.^[20,21] In addition, the THz technology has potential applications in near-field microscopy and thermal management.^[22] Furthermore, the contribution of THz waves to near-field radiation transfer is also an interesting subject of investigation.^[23–25] However, there are few natural materials except graphene^[26,27] capable of supporting THz SPhP or SPP, which plays an important role in enhancing radiative heat transfer. Graphene, a newly emerged two-dimensional (2D) mono-layered carbon-atom-arranged material, has received a great deal of attention in the field of nanophotonics due to its excellent optical properties.^[28] Graphene can support SPPs in a frequency range from THz to mid-infrared frequencies due to its unique Dirac band structure. Furthermore, when graphene is connected to other dielectric materials,^[29] the hyperbolic modes can appear at THz or mid-infrared frequencies by tuning its chemical potential. Therefore, the HMM with a hyperbolic band can be constructed to enhance the radiation transfer in THz and near-field frequency regimes.

In this paper, we theoretically investigate the near-field radiative heat transfer between HMMs each of which is periodically stacked by graphene and silicon (Si) in the THz region. On the one hand, we make full use of the advantage that the hyperbolic mode covers a wider frequency range than the surface resonance mode. The proposed structure can enhance the near-field radiative heat transfer in a wider frequency range. On the other hand, when the suitable width of Si and chemical potential of graphene are selected, the radiative heat transfer of THz waves between graphene-based HMMs can be enhanced remarkably compared with that between graphene-covered Si bulks. Furthermore, the near-field radiative heat transfer in the THz frequency regime between graphene-based hyperbolic metamaterials can be tuned actively by controlling the chemical potential of graphene.

As have been studied in Refs. [30] and [31], the HMM is a special class of uniaxial anisotropic material in which the dispersion relation of the electromagnetic wave is given by the following equation:^[32]

A periodic multilayer structure composed of graphene and Si as shown in Fig. 1(a) is considered. The permittivities in the directions perpendicular and parallel to the graphene surface can be described by the effective medium theory (EMT). The graphene-based periodic structure in the long wavelength regime can be modeled with uniaxial effective permittivity by the following expressions:^[33]

Fig. 1.

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Fig. 1. (color online) (a) Schematic of the THz wave transfer between two graphene-based HMMs separated by a vacuum gap D. (b) The real part of the effective permittivities parallel (ε‖) and perpendicular (ε⊥) to the surface of graphene as a function of frequency. The width of Si is d = 50 nm, and the chemical potential of graphene is μ = 0.1 eV. (c) Variation of the real part of ε‖ with frequency for different chemical potentials with d = 50 nm. (d) Variation of the real part of ε⊥ with frequency for different thickness with μ = 0.1 eV.

The effective permittivities of the graphene-based HMMs are shown in Fig. 1(b). This figure shows a hyperbolic band with ε_‖ < 0 and ε_⊥ > 0 for a p-polarized wave when the frequency is lower than 7.62 THz. The frequency for ε_‖ = 0 is defined as the critical frequency. The real part of ε_‖ is always negative in the hyperbolic region, while at higher frequencies, the real part of ε_⊥ is positive and tends to coincide with. Moreover, the location of the critical frequency can be actively controlled by modulating the chemical potential as shown in Fig. 1(c). This figure shows that the critical frequency shifts toward higher frequencies with the chemical potential increasing. Therefore, the hyperbolic band can be tuned flexibly. In addition, the effective permittivity can also be tuned by changing the thickness of the dielectric layer. The variations of the real part of ε_‖ with frequency for different dielectric thickness are shown in Fig. 1(d) at a fixed chemical potential of μ = 0.1 eV. The critical frequency moves to lower frequencies when the thickness of the dielectric layer increases, resulting in a narrow hyperbolic band.

To describe the near-field radiative heat flux, the radiative transfer coefficient (RTC) h(D) of THz wave between HMMs, which are assumed to be at a local thermal equilibrium, is calculated from the following equation:^[36]

Figure 2 shows the contours of exchange function s(ω,q) for graphene-covered silicon bulk and HMM in the left and right panels, respectively. The chemical potentials of graphene are μ = 0.05, μ = 0.1, and μ = 0.15 eV and eV from top to bottom panels. The frequency ranges are limited in the THz region. The width of Si is still set to be d = 50 nm. Although each exchange function exhibits a bright region, the characteristics of the modes in the left and right panels are different. For the graphene-covered structure, the contribution to the exchange function is from graphene-supported surface plasmons (SPs), whereas for HMM, this contribution is from HM. The SPs and HM appear in different frequency regions even if the chemical potentials are the same for both configurations, which is due to the fact that the HM excited by the HMM often covers a wider frequency range than SPhPs or SPPs whose frequency is limited to its resonance frequency. In addition, the chemical potential has a distinct effect on the SPs and HM, leading to a shift toward higher frequencies when the chemical potential increases. The exchange function can represents the electromagnetic wave tunneling probability between two HMMs (or graphene-covered Si bulks). It depends on both the frequency and the dimensionless wave vector component along the x direction. However, to further compare the advantages of different configurations, we display the STF for two different configurations in the following study.

Fig. 2.

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	Fig. 2. (color online) Exchange function s(ω,q) for different chemical potentials of (a)–(c) graphene-covered semi-infinite structure and (d)–(f) graphene-based HMM. Chemical potential is μ = 0.05 eV ((a) and (d)), 0.1 eV ((b) and (e)), and 0.15 eV ((c) and (f)). The vacuum gap is fixed to be D = 150 nm, and the thickness of Si is set to be d = 50 nm.

In Fig. 3, the STFs are shown as a function of frequency for different chemical potentials. To compare the STF of HMM with that of graphene-covered Si bulk, we define a dimensionless parameter Γ = H(ω)/H_g−Si (ω), where H(ω) and H_g−Si (ω) are the STFs of HMMs and graphene-covered Si, respectively. In our calculation, the vacuum gap is fixed to be 150 nm, and the thickness of Si is maintained at 50 nm to satisfy the validity of EMT. The black solid, red dashed, and blue dotted lines correspond to the cases with chemical potentials of 0.05 eV, 0.1 eV, and 0.15 eV, respectively. For each chemical potential selected, a peak appears in the STF curve as shown in Fig. 3. Each value of Γ is smaller than 1.0 at lower frequency; thus, the radiative transfer between graphene-covered Si bulks is larger than that between HMMs in such a frequency range. However, when the frequency increases, the value of Γ increases very rapidly. The maximum value of Γ can reach 3.5 when μ = 0.15 eV. Therefore, the radiative heat transfer can be significantly enhanced between HMMs than that between graphene-covered Si bulks. However, when the frequency is beyond the THz range, the difference in curve among the three different chemical potentials almost disappears. In addition, as the chemical potential increases, the peak moves toward higher frequencies with an increasing value. Therefore, the radiative transfer of THz wave might be tuned by the chemical potential. The STF does not drop to zero over the THz band, which is because of the weak contribution of other surface waves.

Fig. 3.

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	Fig. 3. (color online) Plots of ratio between STFs of HMM and graphene-covered Si bulk versus frequency for different chemical potentials. The vacuum gap is set to be 150 nm, and the thickness of Si is 50 nm in HMM.

We have analyzed qualitatively the radiative transfer between HMMs using STF. In addition, we find that the radiative transfer can be larger or less for the HMMs than that for the graphene-covered Si bulk in different frequency ranges. The properties of graphene are determined by its chemical potential; thus, the chemical potential significantly affects the property of graphene-based HMM. Here, we study the RTC through the total THz frequency range (from 0.1 THz to 10 THz). Similarly, we define a dimensionless parameter γ_μ, which is the ratio of the RTC of HMMs to the RTC of graphene-covered Si bulk, that is γ_μ = h(μ)/h_g−Si(μ). The thickness of Si in HMM is 50 nm in Eq. (4). Figure 4 shows the γ_μ as a function of chemical potential for different vacuum gaps. The values of γ_μ for vacuum gaps D = 300 and D = 1000 nm are displayed with black solid and red dashed curves, respectively. The value of γ_μ first increases and then decreases as the chemical potential increases for each case. The values of γ_μ reach the maximum values at 0.05 eV and 0.075 eV for vacuum gaps of 300 and 1000 nm, respectively. The value of γ_μ is larger for vacuum gap 300 nm than that for 1000 nm when the chemical potential is less than 0.085 eV; after which, the value of γ_μ is smaller for D = 300 nm than that for D = 1000 nm. For most of the chemical potential values in our study, γ_μ is larger than 1.0. Therefore, the radiative transfer between HMMs is stronger than that between graphene-Si structures. However, when the chemical potential is larger than 0.134 eV, the radiative transfer of THz wave between HMMs cannot catch up with that between graphene-covered Si structures when the vacuum gap is 300 nm. For vacuum gap D = 1000 nm, the value of γ_μ is smaller than 1.0 when the chemical potential is larger than 0.148 eV. Therefore, HMMs can enhance the near-field radiative transfer of THz wave by selecting a suitable chemical potential. Furthermore, we can actively control the radiative transfer by tuning the chemical potential.

Fig. 4.

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	Fig. 4. (color online) Plots of normalized RTC with respect to chemical potential μ for vacuum gap D = 300 nm and 1000 nm. RTC is normalized to graphene-covered Si bulk.

Figure 4 shows that radiation transfer depends on the chemical potential and vacuum gap. Figure 5(a) shows the relationship between RTC and vacuum gap for different chemical potentials. The black solid, red dashed, and blue dotted curves correspond to RTC with chemical potentials of 0.05 eV, 0.075 eV, and 0.1 eV, respectively. The RTC is also normalized to that of graphene-covered Si bulk, and γ_D = h(D)/h_g−Si(D) is defined. For a given chemical potential, the value of γ_D first increases rapidly, and then decreases slowly with the increase of vacuum gap. The radiation transfer between graphene-covered Si bulks is dominated by SPs, and that between HMMs is dominated by HM. The variation behavior of the RTC curve shows that evanescent SPs play an important role in radiation transfer for smaller vacuum gaps, while HM contributes to the RTC importantly even when the vacuum gap is large. As the vacuum gap continues to increase, γ_D decreases. However, the radiation transfer between HMMs is still stronger than that between graphene-covered Si bulks. The values of RTC decline and the peak values shift toward the higher vacuum gaps when the chemical potential increases. Finally, we investigate the effect of the thickness of Si in HMM on RTC. Figure 5(b) shows the relationships between normalized RTC (γ_d = h(d)/h_g−Si(d)) and thickness d for different chemical potentials. The distance between two HMMs is set to be D = 300 nm, and the calculation range of d is from 50 nm to 2000 nm. We use the transmission matrix method^[37] rather than the EMT to obtain the reflection coefficient of HMMs because the EMT is valid only when D > d. Figure 5(b) shows that when the chemical potential is not extremely large, the radiation transfer between HMMs is always larger than that between graphene-covered Si bulks. Moreover, γ_d decreases as d increases. The value becomes 1.0 when d is sufficiently thick; thus, such an HMM structure is equivalent to a graphene-covered Si structure.

Fig. 5.

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	Fig. 5. (color online) Plots of normalized RTC as a function of (a) vacuum gap with μ = 0.05 eV (black solid line), μ = 0.075 eV (red dashed line), and μ = 0.1 eV (blue dotted line). (b) Plots of RTC sets varying with dielectric thickness, where D = 300 nm.

Finally, we show the RTCs for HMMs (black dashed curve) and graphene-covered Si bulks (blue dotted line) as a function of temperature in Fig. 6. In both HMM and graphene-covered Si configurations, the chemical potentials and vacuum gaps are set to be 0.06 eV and 300 nm, respectively. It is shown that the RTCs of both configurations increase as the temperature increases. Although the difference in RTC between HMM and graphene-covered Si bulk is very small in a low temperature range, the RTC between HMMs is larger than that between graphene-covered Si bulks in our studied temperature range. In addition, we also display the normalized RTC (red solid curve) as a function of temperature. We can see that the normalized RTC increases rapidly as the temperature increases at a lower temperature regime (<150 K), while at higher temperature (> 150 K), the normalized RTC increases slowly and almost keeps constant. This means that the temperature has a more important effect on the RTC of the HMM configuration than that on the RTC of the graphene-covered Si configuration at lower temperature, which can also be seen from the comparison between the black dashed line and blue dotted line. Therefore, we can see that at room temperature HMM is better than graphene-covered Si bulk in enhancing near-field radiative heat transfer in the THz frequency range.

Fig. 6.

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	Fig. 6. (color online) RTCs for HMMs and graphene-covered Si bulks as well as normalized RTC as a function of temperature, with the chemical potential μ and vacuum gap D being 0.06 eV and 300 nm, respectively.

We propose graphene-based hyperbolic metamaterials to investigate the contribution of THz wave on the near-field radiative heat transfer. We find that due to the existence of the hyperbolic mode in the HMM, the THz radiative heat transfer in some frequency ranges can be enhanced remarkably compared with that between graphene-covered Si bulks. Even in the whole THz frequency range, the radiative transfer coefficient can also be enhanced. More importantly, the near-field radiative heat transfer in the THz band can be actively controlled by changing the chemical potential of graphene. We also show that when the width of Si is sufficiently large, the radiation transfer coefficient of HMM is equal to that of graphene-covered Si bulk. We believe that the results obtained in this study are useful for actively controlling the energy transfer in the THz frequency regime.