Open problem originating from Bernardi, Milovich and Francoeur's landmark demonstration (Nature Communications 7, 12900, 2016) of radiative heat transfer exceeding the blackbody limit between macroscale planar surfaces: two 5 × 5 mm² intrinsic-silicon plates separated by a vacuum gap modulated from 3,500 to 150 nm, with an 8.4-fold enhancement over the blackbody limit at a 150 nm gap and a temperature difference of 115.6 K. The paper acknowledged that intermediate gap sizes were inferred from a heat-transfer model rather than measured directly, that the dielectric function of intrinsic silicon was assumed temperature independent, and that its thermophotovoltaic estimates were radiative-limit values neglecting electrical and additional thermal losses. Subsequent literature has substantially addressed the gap-metrology limitation (in-situ capacitance and electron-microscopy gap determination in related post-supported macroscale devices) and demonstrated proof-of-principle near-field thermophotovoltaic converters with measured losses, but provides no dedicated theoretical or experimental quantification of the impact of temperature-dependent optical constants on near-field fluxes at large temperature differences for macroscale planar devices — the closest prior temperature-dependence studies (refs. 15, 16) treat other geometries or regimes rather than this configuration, being hot-microsphere-versus-plate measurements above room temperature and a study confined to the extreme near field — an uncertainty still invoked by the 2021 high-temperature experiment of Mittapally and co-workers. The surviving open core is the quantitative closure between fluctuational-electrodynamics prediction and measurement for macroscale planar near-field devices, including loss-inclusive thermophotovoltaic performance for this device class; later field reviews additionally identify array-scale, long-term-stable gap-maintaining operation as an open engineering direction beyond that core.

Background

In the far field, radiative heat exchange between two bodies at temperatures T1T_1 and T2T_2 is bounded by the Stefan–Boltzmann law: a blackbody (a perfect absorber and emitter at all wavelengths) exchanges at most the flux σ(T14−T24)\sigma(T_1^4 - T_2^4), where σ is the Stefan–Boltzmann constant. When the vacuum gap d between the bodies becomes smaller than the characteristic thermal wavelength (about 10 µm at room temperature), this bound no longer applies: evanescent electromagnetic waves — fields that decay exponentially away from each surface and cannot carry energy to the far field — can tunnel across the gap and mediate heat fluxes exceeding the blackbody limit, in some configurations by orders of magnitude. The theoretical framework for computing this effect is fluctuational electrodynamics, which treats thermal radiation as the electromagnetic field generated by the random currents and charges that fluctuate inside any material at finite temperature. Polder and Van Hove used this framework in 1971 to predict super-blackbody transfer between planar surfaces; the enhancement is largest for materials supporting surface polaritons (collective electronic or lattice oscillations bound to an interface, such as surface phonon polaritons in polar dielectrics or surface plasmon polaritons in doped semiconductors and metals), but even a high-index, non-polar material such as intrinsic silicon enhances transfer through frustrated (totally internally reflected) modes.

Confirming the effect between macroscale planar surfaces is experimentally difficult because two large plates must be held parallel to within a few nanometres over the whole overlapping area while parasitic heat conduction through any solid spacers is kept small; early confirmations therefore used a hot scanning-probe tip or a microsphere facing a flat surface, involving very small effective areas. Macroscale planar devices matter for applications, most prominently thermophotovoltaic (TPV) energy conversion, in which a photovoltaic cell converts thermal radiation from a hot emitter into electricity; placing the cell within a nanoscale gap of the emitter ('near-field TPV') is predicted to raise the absorbable photon flux, and hence the electrical power density, far above the far-field value.

Bernardi, Milovich and Francoeur (2016) reported the landmark macroscale planar measurement. Their microfabricated device held two 5 × 5 mm² intrinsic-silicon surfaces (intrinsic meaning undoped, so free-carrier absorption is nominally negligible at room temperature) apart on four SU-8 polymer posts 3.5 µm tall, with silicon-dioxide stoppers fixing a minimum gap of 150 nm; applying a force flexes the top membrane so the gap is modulated from 3,500 nm down to 150 nm. For temperature differences up to 115.6 K they measured a radiative flux exceeding the blackbody prediction by a factor of 8.4 at the smallest gap — then the largest value recorded between macroscale planar surfaces at non-cryogenic temperatures — with the conductive background falling to its minimum share, 6.6% of the total heat rate, at that smallest gap and largest temperature difference. The gaps in the fully open and closed positions were known from fabrication (±22 nm and ±5 nm); the intermediate gaps, however, were not directly measurable and were inferred by fitting a coupled fluctuational-electrodynamics/finite-element heat-transfer model to the measured heat rates.

The paper itself flags what remains unresolved. First, there was no direct metrology of intermediate gap sizes, so the gap dependence of the data rests on model inversion. Second, the dielectric function of intrinsic silicon was assumed independent of temperature over the 300–420 K operating range, taken from a curve fit of room-temperature optical data; how temperature-dependent optical constants change the predicted near-field fluxes at large temperature differences was not quantified. Third, the projected near-field TPV performance (an intrinsic-silicon emitter at 420 K facing an indium-antimonide cell: 400 W m⁻² and 2.9% efficiency at a 150 nm gap) was computed in the radiative limit, i.e. ignoring the electrical losses of a real cell (non-radiative carrier recombination, series resistance) and thermal-management losses; the paper noted that these losses had been treated for an actual thermophotovoltaic generator in a companion study. Beyond these acknowledged limitations, the paper's discussion observes only that the projected ~10 mW electrical power from a 5 × 5 mm² device 'could be increased by using an array of devices'; operation as arrays of devices, extension to other material systems, and long-term mechanical stability of nanometre gap control are identified as open engineering challenges by later field reviews rather than by the source paper itself.

Subsequent literature has provided direct in-situ gap metrology for post-supported macroscale devices (for example, the capacitance-based gap determination of Ying et al., 2020, and the electron-microscopy gap visualization of DeSutter et al., 2019) and demonstrated near-field TPV converters with measured electrical losses (Bhatt et al., 2020; Lucchesi, Cakiroglu et al., 2021; Mittapally et al., 2021). What still lacks a dedicated treatment is the quantitative effect of temperature-dependent optical constants on near-field fluxes and loss-inclusive thermophotovoltaic performance for this device class; field reviews separately identify array-scale gap-maintaining device operation as open. The problem below is the model–experiment closure question that these limitations jointly define.

Problem Statement

For macroscale planar surfaces separated by a nanosize vacuum gap and held at a large temperature difference — the configuration demonstrated by Bernardi, Milovich and Francoeur with intrinsic-silicon plates — can near-field radiative heat transfer be predicted quantitatively from fluctuational electrodynamics once every model input is independently constrained, where 'quantitatively' means to an accuracy sufficient to decide whether the temperature-dependent dielectric function changes the predicted near-field fluxes by more than the experimental uncertainty? Concretely: (i) determine the temperature-dependent dielectric function ε(T,ω)\varepsilon(T,\omega) of the exchanging materials — in particular intrinsic silicon, whose thermally generated free carriers make its mid-infrared optical response vary strongly with temperature — and quantify how its inclusion changes the predicted near-field fluxes at large temperature differences relative to the temperature-independent approximation used in the source work; (ii) replace model-inferred gap profiles with direct in-situ metrology of the gap across the emitter–receiver area under actuation, so that flux-versus-gap data no longer rely on inversion of the model being tested; and (iii) go beyond the source work's radiative-limit thermophotovoltaic estimates by determining how the electrical losses of a real cell (non-radiative carrier recombination, series and shunt resistance) and thermal-management losses change the achievable performance of a near-field thermophotovoltaic converter of this macroscale planar device class. These three components serve one objective: a validated quantitative description (and exploitation) of macroscale planar near-field heat transfer at large temperature differences. The closure comparison is anchored at the fabrication-known fully-closed gap of 150 nm (±5 nm), where the gap is fixed by fabrication rather than inferred from the model under test, so the comparison is non-circular, whereas the intermediate-gap, flux-versus-gap profile of the actuated flexure device is recovered only by inversion of that model. Proof-of-principle loss accounting exists only for small-area near-field TPV devices. The measured fluxes, and the bound on parasitic (conductive) channels against which the predictions are compared, are taken as given from the source work, so the closure comparison needs no new measurement and no new parasitic-channel bound. Field reviews separately identify cm²-scale arrays with documented long-term stability and repeatability of nanometre gap control as an open engineering direction beyond this closure question.

References

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