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

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.

setup used to calibrate the measurement system for the case of conduction. It consists of a 1.1-mm-thick layer of borosilicate glass sandwiched between two copper (Cu) heat spreaders. The temperature difference is maintained by a thermoelectric (TE) heat pump and a TE cooler. The power supplied to the TE heat pump, P HP , is approximately equal to the heat rate through the glass, Q. The temperatures on either side of the glass layer, T 1 and T 2 , are measured using thermistors embedded in the Cu heat spreaders. b, Heat rate, Q, as a function of temperature difference, ΔT.
The symbols indicate unprocessed experimental results while the dashed line correspond to Fourier's law using a thermal conductivity κ = 1.0 W m -1 K -1 for borosilicate glass. setup by measuring radiation between Si surfaces separated by nanosize polystyrene spherical particles. a, Schematic of the experimental setup used to calibrate the measurement system for the case of near-field radiation. It consists of two silicon (Si) substrates separated by vacuum gap sizes of 500 nm and 200 nm using polystyrene spherical particles. The temperature difference between the emitter and receiver is maintained by a thermoelectric (TE) heat pump and a TE cooler. The power supplied to the TE heat pump, P HP , is approximately equal to the heat rate, Q. The temperatures T e,o and T r,o are measured by thermistors and are approximately equal to T e and T r , respectively. b, Heat rate, Q, as a function of temperature difference between the emitter and receiver, ΔT. The symbols show unprocessed experimental measurements, while the colored bands are numerical simulations obtained from fluctuational electrodynamics (FE). Experimental results at a gap size of 500 nm exceed blackbody predictions by a factor of 4.6 at a temperature difference of 119.0 K. At a gap size of 200 nm, blackbody predictions are exceeded by a factor of 8.1 at a temperature difference of 108.9 K. a b Supplementary Figure 4 | Modeling of the membrane. a, Stress distribution in the membrane when a force, F, of 4 mN is applied to the emitter. The maximum stress, σ max , of 6.13 MPa occurring at the corners of the membrane is significantly smaller than the Si yield strength of 7 GPa. The model shows that the displacement, ɗ, of the emitter relative to the receiver under an applied force of 4 mN is enough to bring the device in the closed position. b, Numerical predictions of the emitter displacement, ɗ, and membrane maximum stress, σ max , as a function of the applied force, F, for the 20-µm-thick, 3.5-mm-wide membrane. The resulting spring coefficient, ƙ, is predicted to be 1282 N m -1 .

Supplementary Note 1. Device fabrication.
Fabrication of the bottom silicon (Si) substrate. The main steps required in fabricating the bottom Si substrate are shown in Supplementary Fig. 1a. Starting with a 525-µm-thick, 10-cmdiameter intrinsic Si wafer, a 150-nm-thick layer of silicon dioxide (SiO 2 ) was grown using thermal oxidation. Most of the SiO 2 layer was etched away using UV lithography and a buffered oxide etch (BOE) solution consisting of 25% HF:40% NH 4 F leaving a set of four 5-µm-diameter stoppers per device. The purpose of these stoppers was to prevent the emitter from making contact with the receiver when the device was in the closed position. Next, a 3.5-µm-thick layer of SU-8 3005 negative photoresist was spun onto the wafer. A spin speed of 4500 rpm lasting 30 seconds was required to achieve the desired thickness. Using UV lithography, 250-µm-diameter areas were exposed and developed in order The near-field radiative heat transfer device was replaced with a 5 × 5 mm 2 layer of borosilicate glass, and its thermal conductivity κ was retrieved by measuring the heat rate Q as a function of the temperature difference ΔT. Note that the cold side of the borosilicate layer, measured by a thermistor, was maintained at a constant temperature T 2 of 300 K in all experiments. Using Fourier's law and assuming one-dimensional conduction, a thermal conductivity κ of 1.0 W m -1 K -1 was experimentally determined. This value is in good agreement with published data for borosilicate glass 2 .
Radiation heat transfer between Si surfaces separated by nanosize polystyrene spherical particles. In order to calibrate the measurement method for the case of radiation, the near-field radiative heat transfer device was replaced by two 5 × 5 mm 2 , 525-µm-thick layers of intrinsic Si separated by vacuum gaps of 500 nm and 200 nm maintained by polystyrene spherical particles. This technique was used by Hu et al. 3 for measuring near-field radiative heat transfer between SiO 2 plates separated by a 1.6-µm-thick vacuum gap, since the polystyrene particles have a low thermal conductivity κ p of 0.18 W m -1 K -1 and are essentially transparent in the infrared spectral band. A schematic of the setup and associated results are provided in Supplementary Fig. 3.
The Si emitter and receiver were cleaned in a UV/ozone chamber and were then rinsed in acetone, isopropanol and deionized water prior to depositing the polystyrene particles. The 500nm-diameter particles were in a suspension of deionized water at a concentration of 1.15×10 11 particles/mL. The suspension was sonicated with an Elmasonic Bath Sonicator to ensure a uniform distribution of particles. Since the particles were in a fairly high initial concentration, it was necessary to dilute them to ensure that only a monolayer of particles was deposited on the Si surface and to minimize conduction heat transfer between the emitter and receiver. The dilution was performed in two steps. First, 0.05 mL of the particle suspension was diluted with 100 mL of deionized water; second, 1 mL of the intermediate suspension from the first step was diluted with 8 115 mL of deionized water. The resulting concentration was 5×10 6 particles/mL. Using a syringe, 0.02 mL of the suspension was deposited on the Si receiver leaving approximately 10 4 polystyrene particles on the surface. The Si receiver was then dried on a hotplate before being aligned with the Si emitter and placed in the vacuum chamber. Preparation of the samples separated by 200-nm-diameter polystyrene particles was accomplished in a similar fashion.
These particles were initially suspended in deionized water at a concentration of 1.8×10 12 particles/mL and were diluted to a concentration of 7.8×10 6 particles/mL in two steps using sonication. A syringe was used to deposit 0.02 mL of the suspension on the surface of the Si receiver resulting in approximately 1.6×10 5 particles. According to the manufacturer specifications, the standard deviation in particle sizes was ±74 nm and ±30 nm for the 500 nm and 200 nm particles, respectively. This standard deviation was taken into account in the fluctuational electrodynamics simulations and is shown as colored bands in Supplementary Fig.   3b.
The heat flow through the sample Q (≈ P HP ) supplied by the thermoelectric (TE) heat pump is split into two contributions, namely the heat rate by radiation between the emitter and receiver Q e-r , and the background heat rate Q back (see Fig. 1c). Here, the background heat rate is solely due to conduction through the polystyrene particles. This background heat rate was determined by estimating the contact area between the particles and the Si surfaces via a Hertz model.
Taking into account the force exerted by the masses of the TE heat pump, Si emitter, thermistor and copper (Cu) heat spreader, the contact area A between the 500 nm polystyrene particles and Si was determined to be 2241 nm 2 . The background heat rate was thus estimated using Fourier's law: where N is the number of particles while D is the particle diameter which is the same as the gap size d separating the emitter and receiver. For a fixed receiver temperature of 300 K, the background heat rate was estimated to be 8.1×10 -5 W and 9.7×10 -4 W for temperature differences of 1 K and 120 K, respectively. Since the background heat transfer due to conduction through the polystyrene particles was always less than 1% of the heat rate by radiation, it was assumed that Q ≈ Q e-r . The contact area between the 200 nm particles and the Si surfaces was estimated to be 194 nm 2 using the same process as described earlier, thus resulting in background heat rates of 2.7×10 -5 W to 3.4×10 -3 W for temperature differences of 1 K and 120 K, respectively. Although this heat rate was slightly larger than for the case of 500-nm-diameter particles, it was always less than 1.7% of the heat rate by radiation such that Q ≈ Q e-r was again assumed.
In general, there is a good agreement between experimental results and fluctuational electrodynamics predictions. For the case of 500 nm particles, the measured heat rate is 148.2×10 -3 W at a temperature difference of 119.0 K. This differs from the predicted value of 143.5×10 -3 W by 3.2%. At the lower end of the temperature difference, the experimentally measured heat rate is 5.3×10 -3 W for a temperature difference of 4.6 K, which differs from the prediction of 3.7×10 -3 W by 45.4%. However, the absolute value of the difference between the measured and experimental heat rates is 1.7×10 -3 W, which is actually smaller than the difference at the high end of the temperature range of 4.6×10 -3 W. For 200 nm particles, the measured heat rate is 227.7×10 -3 W at a temperature difference of 108.9 K. This differs from the prediction of 205.1×10 -3 W by 11.0%. At the lower end of the temperature difference, the experimentally measured heat rate is 24.5×10 -3 W for a temperature difference of 14.7 K. This differs from the prediction of 18.7×10 -3 W by 31.3%. However, much like the case with 500 nm particles, the absolute value of the difference between the measured and experimental heat rates is 5.9×10 -3 W which is less than the difference at the high end of the temperature range of 22.6×10 -3 W. These discrepancies may be attributed to the difficulty in aligning the Si emitter and receiver. Unlike the near-field radiative heat transfer device, the Si emitter and receiver were aligned manually.
Also, the imperfect dispersion of particles on the receiver and possible perturbations of the thermal near field by the polystyrene particles could have contributed to the discrepancies.

Supplementary Note 3. Membrane design.
As shown in Supplementary Fig. 1b, the device layer of an SOI wafer constitutes the membrane allowing the emitter to move relative to the receiver. The thickness of the device layer thus determined the thickness of the membrane. When designing the membrane, compliance and strength were considered. Since SOI wafers can be purchased with a variety of device layer thicknesses, a COMSOL model was created in order to determine an appropriate membrane thickness and width. Supplementary Fig. 4 shows the results for a 3.5-mm-wide, 20-µm-thick membrane. According to the model, a force of 4 mN is required to displace the emitter by 3.39 µm. At this deflection, the peak stress is 6. applied load of 5 g. This may be due to the issue encountered during fabrication that was discussed in Supplementary Note 1. This resulted in the membrane being slightly thicker in certain areas causing it to be stiffer than designed.