1Tongji University, School of Physics Science and Engineering, Center for Phononics and Thermal Energy Science, China-EU Joint Lab on Nanophononics, Shanghai, China
2Guangdong University of Technology, School of Physics and Optoelectronic Engineering, Guangzhou, China
Wen-Hao Mao, Yuanyang Du, Jiebin Peng, Jie Ren, "Material-engineered near-field heating and cooling with drifted plasmon–phonon polaritons," Adv. Photon. 7, 036006 (2025)
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【AIGC One Sentence Reading】:We reveal heating-cooling transition in near-field radiative heat transfer via graphene & metamaterial, aiding efficient thermal management.
【AIGC Short Abstract】:Our study reveals an unconventional heating-cooling transition in near-field radiative heat transfer between a graphene metasurface and a hyperbolic metamaterial, driven by competition between high- and low-frequency modes. This finding suggests efficient modulation of heat transfer for thermal management applications.
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Abstract
We investigate near-field radiative heat transfer between a current-driven graphene metasurface and an anisotropic magneto-dielectric hyperbolic metamaterial covered with a graphene metasurface according to fluctuational electrodynamics theory. Remarkably, we discover an unconventional radiative cooling flux accompanied by a heating–cooling transition. This phenomenon results from the competition between the high-frequency heating modes and low-frequency cooling modes. Our findings demonstrate a characteristic modulation of radiative heat transfer with implications for efficient thermal management applications.
The maximum possible radiative heat flux in the far-field region is limited by the classical Stefan–Boltzmann law. However, when the distance between two objects becomes smaller than the characteristic wavelength of thermal radiation, near-field radiative heat transfer (NFRHT) can be many orders of magnitude larger than this limitation.1–4 This phenomenon has been observed in various experiments using different materials, geometrical shapes, and gaps ranging from micrometers to a few nanometers.5 These studies have led people to believe that NFRHT could have a significant impact on various technologies, including scanning thermal microscopy,6–8 active noncontact thermal management,9–11 thermophotovoltaics,12,13 and other energy conversion devices.14
Currently, a significant research direction in the radiative heat transfer field is to discover materials that can greatly enhance NFRHT. So far, the largest NFRHT enhancements have been reported in polar dielectrics, where surface phonon polaritons (SPhPs) dominate NFRHT.15,16 Similar enhancements have been predicted and observed in doped semiconductors due to surface plasmon polaritons (SPPs)17,18 and magnetic media in which surface magnon polaritons (SMPs) can be excited.19,20 Another key issue in this field is the active control and modulation of NFRHT. Several strategies have been proposed, such as applying an electric field to ferroelectric materials,21 a magnetic field to magneto-optical materials, 22,23 or magnetic Weyl semimetals24–26 and utilizing current drift.27
The commonly used control methods for near-field thermal radiation are related to breaking the time-reversal symmetry, such as applying a bias to one of the radiation layers. However, there are fewer means to control spatial symmetry. Benefiting from the progress of nanomaterial manufacturing technology, it is possible to manufacture complex and sophisticated metamaterials. Materials engineering is an important means of controlling and breaking spatial symmetry. The concept of materials engineering is shown in Fig. 1(a).
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Figure 1.Concept and implementation for NFRHT. (a) Material-engineered near-field heating and cooling under nonequilibrium drift. (b) Schematic setup for radiative heat transfer. The top layer is a graphene metasurface at temperature , which is electrically biased along direction, and the electron drifting velocity is . The bottom layer is AMDHM covered with a graphene metasurface at temperature . The AMDHM is realized by embedding SiC nanowires into magnetic host hyperbolic metamaterials. is volume filling factor of SiC NWs. In the calculation, the parameters are as follows: , , , , , . Here, is the Fermi velocity of graphene.
In this study, we investigate the NFRHT between a current-driven graphene metasurface and an anisotropic magneto-dielectric hyperbolic metamaterial (AMDHM) covered with a graphene metasurface based on fluctuational electrodynamics.3,28–31 We find that in the presence of current flowing, a net radiative heat flux occurs although graphene metasurface and AMDHM are at the same temperature. This radiative heat flux can adjusted by varying the composition of the AMDHM. The AMDHM is realized by embedding SiC nanowires (NWs) into magnetic host hyperbolic metamaterials (HMs). As the volume filling factor of SiC NWs increases, the radiation heat flux exhibits a nonmonotonic change, first showing heating and then turning into cooling, experiencing a heating–cooling transition. Decomposing the radiation heat flux into frequency and momentum space, we find that the transition is determined by the competition between high-frequency heating modes and low-frequency cooling modes supported by nonreciprocal surface plasmon–phonon polaritons.
2 Methods
From fluctuational electrodynamics, the radiative heat flux between two parallel layers can be calculated as where is the photon frequency, is the photon wave vector parallel to the surface, is the photon distribution function in the two layers, and is the energy transmission coefficient.
When the layer is in equilibrium, the photon distribution satisfies Bose–Einstein statistics, , where is the Boltzmann constant and is the temperature. In the presence of electric current, the distribution is modified as where is the electron drifting velocity. The term represents the Doppler frequency shift of the photon due to current flowing.32–34 This method is different from tuning the chemical potential of photons. A finite photon chemical potential is achieved by applying voltage in a biased semiconductor junction owing to interactions of photons with electrons and holes. The Dopper model primarily focuses on the fluid properties of Dirac electrons. Although they have a similar form, their physical essence is different. Compared with the voltage-controlled photonic chemical potential, the frequency shift in the Doppler model is related to the wave vector, which can lead to more band engineering methods in the Doppler model.
The energy transmission coefficient can be obtained by35where is the magnitude of photon wave vector in vacuum, is the magnitude of photon wave vector perpendicular to the layer plane, is the photon wave vector perpendicular to the layer plane, and is the separation distance between the two layers. The Fabry–Pérot-like matrix describes photons multiple scattering processes between the two interfaces. is the reflection coefficient matrix for interface . The first and second lines in Eq. (3) represent the contributions of the propagating wave and evanescent wave, respectively.
We choose a two-layer configuration to investigate NFRHT, as shown in Fig. 1(b). The top layer is a graphene metasurface with electric current flowing along graphene nanoribbons direction (). The bottom layer is a semi-infinite AMDHM covered with a graphene metasurface. On the one hand, the graphene metasurface on the bottom layer provides SMPs, which can easily resonate with SMPs in the current-driving layer. On the other hand, it plays a role in the formation of hybrid mode with AMDHMs, which selectively enhances the density of states of surface modes at different frequencies. The graphene metasurfaces consist of graphene nanoribbons and air gaps. The strip periodicity of the graphene metasurface is expressed as , and the width of graphene nanoribbons is expressed as . The distance between the two layers is expressed as , which ensures that radiative heat transfer is in the near-field region. The two layers are kept at the same temperature (300 K). For the two graphene metasurfaces, chemical potential is set at 0.1 eV above the Dirac degeneracy point, which corresponds to -type doping. Contrary to small drift velocity in conventional metals, the electron drift velocity in graphene is orders of magnitude larger and can be comparable with the Fermi velocity of graphene.36–38 In this work, we set with .
For the graphene metasurface, the strip periodicity is chosen to satisfy the deep subwavelength periodicity assumption, i.e., (where is the thermal wavelength at room temperature). Therefore, the optical conductivity tensor39 of the graphene metasurface can be characterized using the effective medium theory (EMT), where is the nonlocal correction parameter taking into account the near-field coupling of adjacent graphene nanoribbons. is the nonlocal optical conductivity tensor of graphene.40 Furthermore, we use a Dopper shift model to41 obtain the drift effects of the metasurface in direction,
The optical properties of graphene are anisotropic, and the corresponding SPPs become nonreciprocal due to current drift.
To achieve the realization of the AMDHM, one can embed SiC nanowire arrays into magnetic HMs. As a result, the two-order uniaxial dielectric function tensor and permeability tensor of the AMDHM can be expressed as follows:42where the subscripts and indicate the direction perpendicular and parallel to the optical axis ( axis in this study), respectively. Due to the fact that the characteristic size of the AMDHM is much smaller than the characteristic wavelength of thermal radiation, the EMT can be applied to describe the properties of the considered AMDHM. To ensure the feasibility of EMT, the gap distance between the two emitting layers should satisfy the condition , where is the period of the AMDHM.
According to EMT theory,35 the components of is given by43where the subscript denotes the host metamaterials, refers to the inserted SiC NWs, and is the volume filling factor of SiC NWs. Moreover, and are obtained from and similarly. For more details on the host HM and the inserted nanowires, see the Supplementary Material.
The AMDHM exhibits hyperbolic intervals in their dielectric function and magnetic permeability. This allows for the coexistence of surface plasmon–phonon polaritons (SPPhPs) and surface plasmon-magnon polaritons (SPMPs) in both graphene metasurfaces and AMDHMs systems. The SPPhPs manifest in the -polarization mode, whereas the SPMPs manifest in the -polarization mode. We calculate the contribution of -polarized waves and -polarized waves to the heat flow separately. The results show that the contribution of -polarized waves is three orders of magnitude smaller than that of the -polarized waves. Thus, -polarized waves dominate the radiative heat transfer and we can safely ignore the contribution of -waves. As a result, we will not consider the influence of the SPMPs in the following discussion.
For the convenience of data analysis, we define the energy transmission function and -integrated energy transmission function by where the decomposition of the two-dimensional wave vector is in the Cartesian coordinate system.
Finally, the dispersion relation for the nonreciprocal SPPhPs in the cavity formed by a graphene metasurface and the AMDHM covered with a graphene metasurface can be written as the singularity of the Fabry–Pérot-like denominator matrix,
3 Results and Discussion
We have discovered that changing the volume filling factor of SiC NWs can effectively modulate the dielectric function and magnetic permeability of the AMDHM, which in turn affects the hyperbolic frequency range. To investigate this, we calculated the heat flow at different volume filling factors of SiC NWs. Figure 2(a) shows the results when the top radiator releases heat to the bottom radiator, with positive heat flow. When the volume filling factor , the graphene metasurface driven by drift current releases heat. As the volume filling factor increases, the heat flow gradually increases, reaching a maximum value of . After reaching the maximum value, the heat flow gradually decreases with the increase of the filling factor. When the volume filling factor is , there is almost no heat exchange between the radiators. When the volume filling factor is greater than 0.5, the heat flow first decreases and then increases with the increase of the filling factor. At a volume filling factor of , the graphene metasurface driven by drift current has the strongest heat absorption effect, and can reach . The heat flow exhibits a nonmonotonic change with the increase of the volume filling factor, and the current-driven graphene metasurface can switch from heat release to heat absorption.
Figure 2.(a) Dependence of radiative heat flux on volume filling factor . (b) Contour plot of heat flux spectral function as a function of volume filling factor and frequency . Three vertical slash lines indicate . Note that we use arbitrary unit for .
We conducted further investigation on the change of heat flow with volume filling factor by calculating the NFRHT spectrum, , which is presented in Fig. 2(b). The high-frequency region contributes positively to the heat flow, and the peak value initially increases and then decreases with the increase of the volume filling factor . The low-frequency region contributes negatively to the heat flow, and the absolute value of the peak value initially increases and then decreases with the increase of the volume filling factor . We find that the transition line between heating and cooling modes remains consistent with zero point of , i.e., the blue line. Below that point, AMDHM supports hyperbolic SPhPs () and the wavevector in -direction () is high-confined. This will make it easy to match SPhPs in AMDHMs with graphene’s SPPs, creating a hybrid SPPhPs channel for heat transfer. However, above that point, AMDHM supports elliptical SPhPs () making it difficult to match SPhPs with graphene’s SPPs until the frequency value reaches the second hyperbolic interval, i.e., above the orange line. In that region, the AMDHM also supports hyperbolic SPhPs due to anisotropic properties and another SPPhPs channel for heat modes can be created.
By the definition in Eq. (8), the energy transmission function characterizes the strength of radiative heat transfer at different frequencies and wave vectors (along -direction). Combined with the dispersion relation of SPPhPs, we can look into the details on thermal excitation. For representative volume fill factor , the results are shown in Fig. 3.
Figure 3.Contour plot of -integrated energy transmission function as a function of and . Spectral function at fixed is plotted on the side. (a) . (b) . (c) . Black dashed lines in contour plots represent dispersion relation of surface plasmon–phonon polaritons obtained at . Heating and cooling peaks in the curve of are marked as and , respectively. Note that we use arbitrary unit for and .
Figure 3(a) shows the -integrated energy transmission function for a volume filling factor of . The dispersion relations of SPPhPs (SPhPs in AMDHM) are represented by the black (green) dashed line on the background of in plane, and the dispersion relations of SMPs in graphene are similar to the black dashed line. The spectral function is just the integration of over , and the resulting curve is plotted on the side. It is evident that the SPPhP exhibit strong nonreciprocity in the high-frequency region. The high-frequency heating mode spectrum ranges align with the AMDHM’s hyperbolic range, and the frequency range of the peak values in heating closely aligns with the resonance frequency of the AMDHM’s SPhPs. The dielectric constant of AMDHM is close to plus or minus zero in the high-frequency hyperbolic range (, ). That means there is an -near-zero hyperbolic surface mode, and it enhances the density of states for high-frequency heating modes in graphene metasurfaces by forming hybrid modes. This is demonstrated by the high peak at frequency in the curve. On the contrary, due to the absence of -near-zero modes in the low-frequency range, the hyperbolic enhancement effect is less effective than the high-frequency heating mode, leading to that the cooling effect is relatively weak in the low-frequency region, indicated by the small peak at frequency in the curve. The positive heat flow contribution in the region is significantly larger than the negative heat flow contribution in the region, resulting in a strong overall heating effect. Thus, the system displays a robust heating effect.
The results for the volume filling factor are shown in Fig. 3(b). We can see that the SPPhP exhibit strong nonreciprocity in both the high- and low-frequency regions. In the low-frequency region, we find the frequency of is equal to , which is consistent with the peak position in the spectrum. That means the -near-zero hyperbolic mode in AMDHM enhances the density state of cooling modes in graphene metasurfaces, and the system displays a strong cooling effect. This is demonstrated by the sharp peak at . Conversely, in the high-frequency region, the positive heat flow contribution in the region is stronger than the negative heat flow contribution in the region, resulting in a strong heating effect. This is indicated by the relatively smooth peak at . However, the net heating effect in high-frequency region and the net cooling effect almost cancel each other. Thus, the system exhibits minimal radiative heat transfer overall at this volume filling factor. A similar low-frequency enhancement is also found in Fig. 3(c) with . It demonstrates that the SPPhP exhibits significant nonreciprocity at low frequencies. In the low-frequency region, the competition between heating modes and cooling modes leads to a strong cooling effect. By contrast, the high-frequency region experiences minimal thermal excitation. This results in a pronounced cooling effect overall.
From the previous analysis, we find that with the drifted current along -direction, the SPPhP shows nonreciprocity along direction. The modes with positive contribute to heating and those with negative contribute to cooling. Overall, if the proportion of thermal excitation in the wave vector space of is greater than that in , it exhibits a heating effect. Otherwise, it shows a cooling phenomenon. Next, we fix the frequency at peak frequencies and analyze the momentum-resolved energy transmission function for representative volume filling factor.
Figures 4(a) and 4(b) show in plane when the volume filling factor . -integrated results, , are plotted on the side. Figures 4(a) and 4(b) correspond to the peak and peak in Fig. 3(a), respectively. As the system is equivalent in the direction, is symmetric in the direction. The SPPhP exhibits nonreciprocity along the direction. At frequency , the degree of the nonreciprocity is low. This manifests in the curve of versus . The positive and the negative peaks are close in magnitude. Thus, a weak cooling effect is observed at this frequency. At frequency , the SPPhP exhibits strong nonreciprocity between and . This can be seen from the two-dimensional result of and the curve of . The contribution of heating from modes greatly exceeds the contribution of cooling from modes, resulting in an overall heat effect at this frequency. Combining the results of and , we can qualitatively determine that the system exhibits a very strong overall heating effect for this volume filling factor, which is consistent with the results in Fig. 2.
Figure 4.Contour plot of energy transmission function as a function of at fixed and . -integrated energy transmission function is plotted on the side. (a), (b) . (c), (d) . (e), (f) . and correspond to the heating and cooling peaks in Fig. 3, respectively. Note that we use arbitrary unit for and .
A similar analysis can be carried out for the volume filling coefficient and . The results for are shown in Figs. 4(c) and 4(d). The SPPhP displays the same large degree of nonreciprocity at frequency and . This means that the cooling effect at and the heating effect at almost balance with each other, making the overall heat transfer negligible. The results for are shown in Figs. 4(e) and 4(f). At this volume filling factor, thermal excitation is concentrated in the low-frequency region. At frequency , the SPPhP displays strong nonreciprocity along direction. The cooling contribution dominates over the heating contribution. This leads to an overall cooling effect.
4 Conclusion
In this work, we have explored the NFRHT between a current-driven graphene metasurface and an AMDHM covered with a graphene metasurface. By applying the fluctuational electrodynamics theory, we have uncovered an atypical radiative cooling flux that arises from the volume filling factor of SiC NWs, leading to a transition from heating to cooling. As the volume filling factor has a strong regulatory effect on the excitation of the SPPhPs in the system, the radiative heat transfer exhibits nonlinear changes with the volume filling factor increasing. In this article, we report a maximum heating flux and a maximum cooling flux . These numeral results are two orders of magnitude larger than those in experiments. In Ref. 34, the authors reported a maximum heating/cooling flux between an unbiased photodiode and a planar surface. The heating flux also exceeds other theoretical calculations44 in the same vacuum separation due to the enhancement effect of the hyperbolic modes in the graphene metasurface, i.e., to at 100 nm. This discrepancy arises from the enhancement of the hybrid hyperbolic mode between the graphene metasurface and AMDHM. By investigating the energy transmission function at different frequencies and wave vectors, we have found that the heating or cooling is determined by the competition between high-frequency heating modes and low-frequency cooling modes that are supported by nonreciprocal SPPhPs. These discoveries have significant implications for the advancement of efficient thermal management applications.
Wen-Hao Mao received his PhD in physics from Huazhong University of Science and Technology in 2023 and was a postdoctoral fellow at Tongji University. His research interests include electron-phonon coupling, non-Hermitian physics, chiral phonons, and near-field thermal radiation.
Yuanyang Du graduated with a BS degree in physics from Zhejiang Normal University in 2020 and an MS degree in physics from Tongji University in 2023. His research specializes in near-field thermal radiation, investigating nanoscale heat transfer, radiative energy optimization, and coherent thermal effects.
Jiebin Peng received his PhD from Department of Physics, National University of Singapore in 2017 and was a postdoctoral fellow at Tongji University. Currently, he is a lecturer of Guangdong University of Technology. His current research interests include thermal photonics, nonequilibrium transport and nonequilibrium Green’s function.
Jie Ren is a full professor in the School of Physics Science and Engineering at Tongji University since 2015. He obtained his BS degree from University of Science and Technology of China in 2006, then worked as a research assistant at University of Fribourg, Switzerland. After that, he obtained PhD from National University of Singapore in 2012. From 2012 to 2015, he worked as the director-funded fellow at Los Alamos National Laboratory, and a post-doctorate researcher at Massachusetts Institute of Technology. His current research interests include quantum phononics, near-field photonics and acoustic, nonequilibrium statistical physics, topological metamaterials, and AI physics.