# Radiative heat transfer exceeding the blackbody limit between macroscale planar surfaces separated by a nanosize vacuum gap

## Abstract

Using Rytov’s fluctuational electrodynamics framework, Polder and Van Hove predicted that radiative heat transfer between planar surfaces separated by a vacuum gap smaller than the thermal wavelength exceeds the blackbody limit due to tunnelling of evanescent modes. This finding has led to the conceptualization of systems capitalizing on evanescent modes such as thermophotovoltaic converters and thermal rectifiers. Their development is, however, limited by the lack of devices enabling radiative transfer between macroscale planar surfaces separated by a nanosize vacuum gap. Here we measure radiative heat transfer for large temperature differences (120 K) using a custom-fabricated device in which the gap separating two 5 × 5 mm2 intrinsic silicon planar surfaces is modulated from 3,500 to 150 nm. A substantial enhancement over the blackbody limit by a factor of 8.4 is reported for a 150-nm-thick gap. Our device paves the way for the establishment of novel evanescent wave-based systems.

## Introduction

Radiation heat transfer exceeding the blackbody limit at nanosize separation gaps has been experimentally confirmed in the scanning probe-surface1,2,3, scanning probe-film4, microsphere-surface5,6,7,8, microsphere-film9 and microsphere-nanostructured surface10 configurations. Although the accuracy of fluctuational electrodynamics at sub-10 nm gaps has been questioned in the experiments of Kittel et al.2, the validity of this framework has been confirmed both experimentally3 and theoretically11 down to separation gaps of 2 and 1 nm, respectively. Additional work involving micro/nanostructures has also experimentally demonstrated the enhancement of thermal radiation in the near field12,13,14,15,16. The micro/nanosize surfaces involved in the aforementioned experiments, however, limit the amount of radiation that can be exchanged, such that these configurations cannot be readily applied to engineering systems such as thermophotovoltaic power generators17,18,19,20,21,22,23,24,25,26 and thermal rectifiers27,28,29,30,31. While the development of evanescent wave-based devices typically requires macroscale surfaces separated by a nanosize vacuum gap, experimental research on near-field radiative heat transfer between macroscale surfaces has mainly focused on relatively large, microsize separation gaps at cryogenic32,33 and room34,35,36,37 temperatures. Recently, Ito et al.38 measured radiative heat transfer between millimetre-size fused quartz surfaces separated by pillars, also made of fused quartz, at a separation gap of 500 nm. The results were twice that of fluctuational electrodynamics predictions because of excessive heat conduction through the pillars, which prevent the application of this configuration to engineering systems. Lim et al.39 measured a radiative heat transfer enhancement of 2.91 relative to the blackbody limit between two microstrips of doped silicon (Si) separated by a 400-nm-thick gap. Yet, significant heat transfer and radiation enhancement necessitate larger surfaces and a smaller separation gap, respectively.

The difficulty associated with maintaining a nanosize vacuum gap between macroscale planar surfaces is the main bottleneck, currently preventing the application of near-field thermal radiation to engineering systems15.

Here we address this bottleneck by measuring radiative heat transfer via a custom-fabricated device consisting of two planar 5 × 5 mm2 intrinsic Si surfaces separated by a gap that can be modulated from 3,500 nm down to 150 nm via a compliant membrane and mechanical actuation. This device enables probing radiative heat transfer between macroscale surfaces for large temperature differences (ΔT120 K) in multiple regimes, including those dominated by either propagating or evanescent modes. An excellent agreement between experimental results and fluctuational electrodynamics predictions40,41 is obtained, and a radiative transfer enhancement of 8.4 relative to the blackbody limit is measured for a separation gap of 150 nm. The potential application of our device architecture to thermophotovoltaic power generation is also discussed.

## Results

### Experimental procedure

The near-field radiative heat transfer device, shown in Fig. 1, was manufactured using standard microfabrication techniques, as detailed in Supplementary Note 1 and Supplementary Fig. 1. The device consists of two 2.2 × 2.2 cm2 Si substrates separated by four rigid, 3.5-μm-tall SU-8 posts with a diameter of 250 μm. The bottom substrate was fabricated from a 525-μm-thick intrinsic Si wafer, while the top substrate was manufactured from a 521-μm-thick Si-on-insulator wafer with a 1-μm-thick buried silicon dioxide (SiO2) layer. The surface roughness of the Si substrates was less than 1.2 nm as measured with a Zygo NewView optical profilometer. A 501-μm-deep, 3.5-mm-wide trench was etched on the backside of the top Si substrate using deep-reactive ion etching and a buffered oxide etch solution. This resulted in a 20-μm-thick compliant Si membrane, allowing the 5 × 5 mm2 emitter to move relative to the receiver under an applied force (the emitter–receiver portion of the device is identified by a dashed box in Fig. 1a,b). To avoid contact between the emitter and the receiver, four SiO2 stoppers with a diameter of 5 μm and height of 150 nm were fabricated on the lower Si substrate, thus fixing the minimum separation gap d to 150 nm. The two Si substrates were precisely aligned and bonded in an EVG 520 IS wafer bonder. Testing of the device was conducted in a vacuum chamber (P≈10−4 Pa) located in a class 1,000 clean room tent.

### Heat transfer measurements

The heat rate supplied by the TE heat pump as a function of the measured temperature difference ΔT between the emitter and receiver was compared against numerical predictions. Unprocessed experimental heat rates Q that include radiation heat transfer between the emitter and receiver Qe–r as well as the background heat transfer Qback are shown in Fig. 2a for temperature differences ΔT (=TeTr) up to 120 K. The separation gap d between the emitter and receiver was modulated by using calibrated masses ranging from 0.9 to 5 g, as detailed in Supplementary Note 3. The numerical predictions were obtained via a coupled fluctuational electrodynamics-COMSOL Multiphysics comprehensive heat transfer model of the device taking into account radiation between the emitter and receiver, heat transfer by radiation outside the emitter–receiver region and conduction through the SiO2 stoppers and SU-8 posts. The details of the comprehensive model are provided in the Methods Section. Figure 2b shows the temperature distribution in the device obtained from the model for a heat rate Q of 0.92 W and a separation gap d of 150 nm. The agreement between experimental data and numerical predictions when the device is in the open (d=3,500±22 nm) and closed (d=150±5 nm) position is remarkable. The uncertainty associated with these gap sizes, identified as coloured bands in Fig. 2a, was determined experimentally by measuring the variation of the height of the SiO2 and SU-8 layers used to create the stoppers and the posts, respectively. Using nominal gap values of 3,500 and 150 nm in the numerical simulations, a maximum relative difference between experiments and predictions of 9.1% is obtained for a 3,500-nm-thick gap and ΔT=15.5 K, while a minimum relative difference of less than 0.1% is achieved for a 150-nm-thick gap and ΔT=84.2 K. In addition, for all cases presented in Fig. 2a, radiation largely dominates heat transfer through the device. According to the model, the portion of the heat rate because of conduction reaches a maximum of 11.7% when the gap size is 3,500 nm and the temperature difference is 3.8 K. For the case when the device is in the closed position (d=150 nm) and the temperature difference is 115.6 K, the portion of the total heat rate due to conduction is at a minimum of 6.6%. As explained in Supplementary Note 3 and Supplementary Fig. 4, it was not possible to determine exactly the intermediate gap sizes between the open and closed positions. As the force applied on the device was increased, it was observed that the heat rate Q increased, because of a larger proportion of evanescent modes contributing to heat transfer, until the device was in the closed position. Intermediate gap sizes, shown in Fig. 2a, were estimated using the comprehensive model in combination with the measured heat rates and temperatures (see Methods section).

The radiative heat flux between the 5 × 5 mm2 emitter and receiver without the background heat rate is shown in Fig. 3a as a function of the temperature difference. The agreement between experimental data and fluctuational electrodynamics predictions in the open and closed positions, for which the gap sizes were measured experimentally, is excellent. A maximum radiation transfer enhancement over the blackbody limit by a factor of 8.4 was measured for a gap size of 150 nm and a temperature difference of 115.6 K. This constitutes the largest value recorded between two macroscale planar surfaces at non-cryogenic temperatures. The mechanism responsible for this enhancement can be understood by inspecting the dispersion relations shown in Fig. 3b, where the heat flux is plotted as a function of the angular frequency ω and parallel wavevector kρ for gap sizes of 3,500, 1,000, 500 and 150 nm, and a temperature difference of 120 K. Modes that are propagating in both Si and vacuum are contained within the region kρ<k0 (=ω/c0), where k0 is the magnitude of the wavevector in vacuum. Planck’s theory of heat radiation solely accounts for these modes. Frustrated modes, propagating in Si and evanescent in vacuum, are characterized by parallel wavevectors k0<kρ<Re(n)k0, where n is the refractive index of Si. Surface modes are evanescent in both Si and vacuum and are described by kρ>Re(n)k0. The dispersion relations show clearly that the enhancement of radiative heat transfer is solely due to the additional contribution of frustrated modes in the near field, as intrinsic Si does not support surface modes such as surface phonon–polaritons or surface plasmon–polaritons. Here with a single device, we measured radiative heat transfer in various regimes, including those dominated by either propagating or evanescent modes. Indeed, although heat transfer via frustrated modes occurs at a gap of 3,500 nm, their contribution is insufficient to exceed the blackbody predictions (68% of the heat flux is due to propagating modes). Conversely, radiation heat transfer is largely dominated by frustrated modes for a 150-nm-thick gap and accounts for 88% of the heat flux between the emitter and the receiver.

## Discussion

Near-field thermal radiation research is motivated by potential applications to energy conversion, heat flow management, imaging and micro/nanomanufacturing17,21,31,42,43,44,45,46,47. Among those, thermophotovoltaic power generation capitalizing on evanescent modes is a promising application of our device, as this technology requires both large surfaces and considerable near-field enhancement. For instance, when considering both the near-field enhancement and the size of the surfaces, our device provides a radiative heat rate more than one order of magnitude larger than in ref. 39. In addition, conductive heat losses in our device, negatively having an impact on thermophotovoltaic power generation22,25,48, are minimized to 6.6% of the total heat rate when the near-field enhancement is maximum. For an emitter temperature of 420 K, the photon energy at which thermal emission is maximum is 0.18 eV, such that our device could operate as a thermophotovoltaic power generator by replacing the receiver by a cell made of indium antimonide (InSb) having an absorption bandgap energy of 0.17 eV (ref. 26). In the radiative limit23, we estimated that the output power density and conversion efficiency of a thermophotovoltaic power generator made of intrinsic Si and InSb maintained at 420 and 300 K, respectively, were 400 W m−2 and 2.9% for a 150-nm-thick gap. This implies that a 5 × 5 mm2 surface area would produce an electrical power of 10 mW, a value that could be increased by using an array of devices. In comparison, a similar thermophotovoltaic system operating in the far-field regime with a blackbody source would lead to a significantly lower output power density of 33.4 W m−2 and the same conversion efficiency of 2.9%. A thermophotovoltaic device operating with an emitter temperature of 420 K could potentially be used for recycling waste heat in electronic devices such as solar cells, where output power density is more important than conversion efficiency48. In an actual thermophotovoltaic power generator, one must also consider the impacts of electrical and thermal losses on system performance. This was done by Bernardi et al.25, where the results showed that, despite its broadband near-field enhancement, an emitter such as intrinsic Si supporting strictly propagating and frustrated modes is more beneficial to the performance of evanescent wave-based thermophotovoltaic power generators than a radiatively optimized emitter supporting surface modes.

In summary, using a custom-fabricated device made of two 5 × 5 mm2 intrinsic Si surfaces separated by a tunable vacuum gap, we measured a maximum radiation transfer enhancement over the blackbody limit by a factor of 8.4 for a 150-nm-thick gap and a temperature difference of 115.6 K while minimizing heat conduction. Our near-field radiative heat transfer device, capable of delivering considerable radiation heat rates because of the large near-field enhancement and surface size, paves the way to the development of thermophotovoltaic systems converting evanescent modes into electrical power.

## Methods

### Experimental uncertainty analysis

The uncertainty associated with the experimental data is because of the temperature and heat rate measurements. For the temperature, the uncertainty stems from the ohmmeter (BK Precision, 889B) used to measure the resistance of the thermistors as well as the thermistors themselves. The error in the resistance measurement is given by the manufacturer specifications as ±(0.2%+0.1 Ω) within the range from 100 to 1,000 Ω, and ±(0.1%+1.0 Ω) for the range of 1–10 kΩ. The uncertainty introduced by the thermistors is a function of temperature and the thermistors’ change of resistance with temperature. A resistance reading of 9,225 Ω corresponds to a temperature of 300±0.46 K while a resistance reading of 185.8 Ω corresponds to a temperature of 420±0.85 K. Combining the uncertainties introduced by the ohmmeter and the thermistors results in an overall uncertainty of ±0.48 K at 300 K and ±0.89 K at 420 K.

The uncertainty associated with the heat rate is introduced by the power supply (BK Precision, 9121 A) connected to the TE heat pump. The uncertainty in the supplied current is ±(0.05%+2 mA) and the uncertainty in the supplied voltage is ±(0.02%+3 mV). In addition, the resistance associated with the wires needs to be accounted for since it induces a small amount of power dissipation. A four-wire sensing technique was employed to account for the resistance of the wires leading up to the device. The resistance associated with the TE leads (see Fig. 1d) was measured to be 0.08±0.9 mΩ. For example, when 1.2 A±2.6 mA and 0.84 V±3.2 mV is provided by the power supply (1.01 W±6.0 mW), the power dissipated in the TE leads is 115.2±1.8 mW. The power supplied to the TE heat pump, which is equivalent to the total heat rate, is then determined to be 894.8±7.8 mW. These uncertainties are plotted as error bars in Fig. 2a.

### Computational model

Near-field radiative heat transfer was modelled using fluctuational electrodynamics40. The net radiative heat flux because of propagating and evanescent waves in the emitter–receiver portion of the device was calculated as follows41:

where the subscripts 0, 1 and 2, respectively, refer to the vacuum, the top Si substrate and bottom Si substrate, TE and TM refer to transverse electric and transverse magnetic polarizations, Θ(ω,T) is the mean energy of an electromagnetic state calculated as , kρ and kz are the components of the wavevector parallel and perpendicular to the surface of the layers, while is the Fresnel reflection coefficient at the interface of media i and j in polarization state γ. Outside the emitter–receiver portion of the device, equations (1) and (2) were modified to account for the fact that the 20-μm-thick membrane was optically thin. As such, the background-radiative heat flux due to propagating and evanescent waves was calculated as49:

where and are the reflection and transmission coefficients of layer 1 in polarization state γ. These coefficients are calculated as and , where t is the thickness of layer 1, while is the Fresnel transmission coefficient at the interface of media i and j in polarization state γ. The net radiative heat flux used for producing the numerical results shown in Fig. 2a,b was obtained by summing equations (1) and (2) in the emitter–receiver region, and by summing equations (3) and (4) outside the emitter–receiver region. The net radiative heat flux in the emitter–receiver portion of the device shown in Fig. 3a was calculated by summing equations (1) and (2). The dispersion relations in Fig. 3b were generated by solving equations (1) and (2) per unit angular frequency ω and per unit parallel wavevector kρ. Note that the dielectric function of intrinsic Si was assumed to be independent of temperature in the range of operation and was obtained by curve-fitting the experimental data from ref. 50.

Theoretical curves of heat rate Q as a function of the temperature difference ΔT between the emitter and receiver for a specific separation gap d were calculated using a coupled fluctuational electrodynamics-COMSOL Multiphysics comprehensive model to account for radiation transfer in the emitter–receiver region Qe–r as well as the background heat transfer Qback. Near-field radiative heat transfer was included in COMSOL by defining a fictional material, in place of the vacuum gap, characterized by a local, temperature-dependent effective thermal conductivity. In the emitter–receiver portion of the device, the effective thermal conductivity was calculated as keff=qdT, where q is the sum of equations (1) and (2). The effective thermal conductivity outside the emitter–receiver region was derived using keff=T, where q is the sum of equations (3) and (4), and by using the Derjaguin approximation7 to account for the variations of the separation distance δ, assumed to be linear, when the device was not in the open position. Heat conduction through the SU-8 posts and SiO2 stoppers was calculated using temperature-independent thermal conductivities of 0.2 and 1.3 W m−1 K−1 listed in refs 51, 52, respectively, while the temperature-dependent thermal conductivity provided in ref. 53 was used for intrinsic Si. A typical value for contact resistance of 2.5 × 10−5 K m2 W−1 was imposed between the Si and SiO2 stoppers as well as between the Si and SU-8 posts54. For a given gap d separating the emitter from the receiver, heat transfer simulations in the device were initiated by imposing the heat rate supplied by the TE heat pump PHP (≈Q), while the TE cooler was modelled as a constant temperature boundary condition (300 K) at the bottom face of the receiver. For these conditions, the temperature distribution in the device, and thus the temperature difference ΔT between the emitter and receiver, was determined using an iterative method where the effective thermal conductivity of the vacuum gap and the thermal conductivity of Si were calculated at the updated temperature. Iterative computations were repeated until a maximum absolute temperature difference between two successive iterations less than 0.001 K was obtained everywhere in the spatial grid. These simulations were repeated for a series of heat inputs Q ranging from 0 to 1.4 W in increments of 0.04 W, thus allowing the generation of a theoretical heat rate Q as a function of the temperature difference ΔT between the emitter and receiver for a specific separation gap d. For instance, Fig. 2b shows the temperature distribution in the device for a heat input Q of 0.92 W and a separation gap d of 150 nm. For these conditions, the emitter reaches a uniform temperature of 420 K, but heat is also dissipated outside the emitter region in the top Si substrate. It can also be seen that the bottom Si substrate has a nearly uniform temperature of 300 K.

Validation of the coupled fluctuational electrodynamics-COMSOL Multiphysics model was performed by comparing numerical predictions against unprocessed experimental data measured when the device was in the open (d=3,500±22 nm) and closed (d=150±5 nm) positions. The uncertainty associated with these two gap sizes was used to calculate a theoretical band of heat rate Q as a function of the temperature difference ΔT. These bands, however, are fairly small because of the small uncertainty associated with these gap sizes, and are thus hardly visible in Fig. 2a.

### Estimation of intermediate separation gap sizes

There was no direct method for measuring the separation gap between the emitter and receiver, except when the device was in the open and closed positions. Since the coupled fluctuational electrodynamics-COMSOL Multiphysics model was in excellent agreement with the measured heat rate as a function of the temperature difference in both the open and closed positions (see Fig. 2a), intermediate separation gap sizes were estimated using the aforementioned model. Specifically, a nominal gap size d was determined by best fitting experimental data of nominal heat rate Q versus temperature difference ΔT with numerical predictions. The uncertainty associated with the estimated gaps, shown in coloured bands in Figs 2a and 3a, was derived by best fitting experimental data of maximum Q versus minimum ΔT, and minimum Q versus maximum ΔT with numerical predictions. These maximum and minimum values were determined from the uncertainty associated with the measured temperatures and heat rates.

Even if intermediate gap sizes were not determined from an independent measurement, the experimental results show clearly that heat transfer increases as the separation gap decreases because of an increasing contribution of evanescent modes. Heat transfer reaches saturation when the emitter comes into contact with the SiO2 stoppers. These observations are consistent with fluctuational electrodynamics predictions.

### Data availability

The data that support the findings of this study are available from the corresponding author upon request.

How to cite this article: Bernardi, M. P. et al. Radiative heat transfer exceeding the blackbody limit between macroscale planar surfaces separated by a nanosize vacuum gap. Nat. Commun. 7:12900 doi: 10.1038/ncomms12900 (2016).

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## Acknowledgements

M.F. and M.P.B. acknowledge financial support from the National Science Foundation (Grant No. CBET-1253577). Financial support for D.M. was provided by the Undergraduate Research Opportunities Program (UROP) at the University of Utah. We also thank Rodolphe Vaillon for making a critical reading of this paper and John DeSutter for providing thermophotovoltaic performance predictions. This work was performed in part at the Utah Nanofab sponsored by the College of Engineering, Office of the Vice President for Research and the Utah Science Technology and Research (USTAR) initiative of the State of Utah. We appreciate the support of the staff and facilities that made this work possible.

## Author information

Authors

### Contributions

This work was conceived by M.P.B. and M.F. Design, fabrication and testing of the device, as well as the associated numerical simulations, were performed by M.P.B. under the supervision of M.F. Validation of the experimental system via heat conduction and thermal radiation measurements was done by D.M. under the supervision of M.P.B. and M.F. The manuscript was written by M.P.B. and M.F. with comments from D.M.

### Corresponding author

Correspondence to Mathieu Francoeur.

## Ethics declarations

### Competing interests

The authors declare no competing financial interests.

## Supplementary information

### Supplementary Information

Supplementary Figures 1-4, Supplementary Notes 1-3 and Supplementary References (PDF 7969 kb)

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Bernardi, M., Milovich, D. & Francoeur, M. Radiative heat transfer exceeding the blackbody limit between macroscale planar surfaces separated by a nanosize vacuum gap. Nat Commun 7, 12900 (2016). https://doi.org/10.1038/ncomms12900

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