Between sputter-cleaned gold surfaces separated by vacuum gaps below roughly 18 nm, the measured heat flux exceeds the prediction of fluctuational electrodynamics by up to two orders of magnitude, and the excess survives in-situ argon-ion sputtering in ultra-high vacuum (Geesmann et al., Phys. Rev. Lett. 135, 166202 (2025)). Neither phonon tunneling, electron tunneling, nor any existing extension of the theory reproduces its magnitude and distance dependence, while other experiments find no excess after aggressive cleaning or attribute anomalous nanogap signals to water menisci in standard high vacuum. The open problem is to identify the microscopic channel, correction, or artifact responsible for the sputtering-resistant excess, determine its absolute magnitude and distance dependence in a geometry with independently characterized nanoscale contact area, and thereby establish a quantitative, experimentally validated description of heat transfer across the near-field to extreme-near-field (radiation-to-conduction) crossover in vacuum.

Background

When two bodies at different temperatures are separated by a vacuum gap, they exchange heat by thermal radiation. At gaps comparable to or smaller than the thermal wavelength (about 10 micrometers at room temperature), the exchange is dominated by evanescent electromagnetic waves that exist only near surfaces; this regime is called near-field radiative heat transfer (NFRHT). The standard theory, fluctuational electrodynamics (FE), formulated by Rytov and co-workers, treats the thermal electromagnetic fluctuations inside each body as statistically independent random sources characterized by the body's macroscopic, local (neglecting spatial dispersion, i.e., the dependence of the response on wavevector) dielectric function. FE correctly predicts the orders-of-magnitude enhancement of heat flux over the blackbody limit observed at nanometre gaps, and for polar dielectrics the predicted conductance grows roughly as the inverse square of the gap dd. Below about 10 nm — the extreme near field — several assumptions of FE become questionable: the local dielectric response, the neglect of atomistic transport channels, and the statistical independence of the fluctuations in the two facing bodies. Candidate microscopic channels discussed in the literature include phonon tunneling, i.e. heat carried by the electrostatic coupling of surface-atom vibrations across the vacuum gap, and electron tunneling, i.e. heat carried by the overlap of electronic wave functions across the gap without net charge transport; both are predicted to become substantial mainly below about 1 nm, far below the few-nanometre gaps at which the persistent anomalous excesses are reported.

Experiments in this regime disagree sharply. Kim et al. (Nature 528, 387 (2015)) found FE accurate for gaps down to about 2 nm between dielectric and metal surfaces. Cui et al. (Nat. Commun. 8, 14479 (2017)) measured Au gaps from a few Angstroms to 5 nm using a probe with embedded nanoscale thermocouples and, after aggressive cleaning, found signals below their detection limit down to 0.2 nm, suggesting that reported giant excesses arise from contaminants bridging the gap. In contrast, Kloppstech et al. (Nat. Commun. 8, 14475 (2017)), using a near-field scanning thermal microscope (NSThM — a scanning tunneling microscope whose thermocouple tip simultaneously serves as a heat-flux sensor), measured heat fluxes about four orders of magnitude above FE between gold surfaces from roughly 7 nm down to 2 Angstroms, with an approximately linear dependence on gap that no theoretical model reproduces; Jarzembski et al. (Phys. Rev. B 106, 205418 (2022)) later reported elevated fluxes in the near-field to asperity-contact range, supporting the existence of an excess beyond the NSThM technique.

The present problem originates with the experiment of Geesmann et al. (Phys. Rev. Lett. 135, 166202 (2025)). In ultra-high vacuum (2×10−102\times10^{-10} hPa), between a gold-coated silica sphere of 20 µm diameter held near room temperature and a cooled (about 150 K) atomically flat gold film cleaned by in-situ argon-ion sputtering and annealing, FE matches the measured heat flux for gaps above roughly 18–40 nm. Below about 18 nm the flux rises rapidly to roughly 100 times the FE prediction, and this excess survives in-situ sputtering of the sphere. A tunneling current appears only below 1 nm, and a metallic contact would require electrical currents hundreds of times larger than observed, so ordinary electron conduction through a contact is excluded; molecular contamination bridging the gap is implausible after sputtering at this pressure. At the smallest gaps the measured response is dominated by an unknown nanoscale protrusion on the sphere whose geometry cannot be specified. The authors conclude that the heat flux in the extreme near field is still not well understood and call for sharp-tip measurements in controlled geometry and for theory that couples microscopic tunneling channels and atomic-scale geometry to the radiative and conductive channels.

Problem Statement

Identify the physical origin of the reproducible, cleaning-resistant excess of heat transfer between clean gold surfaces in vacuum at gaps below roughly 20 nm — reaching about two orders of magnitude above the prediction of fluctuational electrodynamics in the sphere-plane experiment of Geesmann et al. — and establish a quantitative description of heat transfer across the radiation-to-conduction crossover in vacuum. The open problem requires determining: (a) which microscopic channel(s), theory correction, or demonstrable experimental artifact carries the excess (phonon tunneling, electron tunneling, nonlocal or correlated-fluctuation corrections to fluctuational electrodynamics, geometry-induced near-field concentration at protrusions, or another mechanism); (b) the absolute magnitude and distance dependence of that contribution, predicted for a geometry whose nanoscale contact area and gap are independently characterized; and (c) whether the resulting description quantitatively reproduces the measured heat flux, within experimental uncertainties, from the near-field regime down to contact and reconciles the contradictory prior reports (ultra-high-vacuum sputtered excess versus high-vacuum water-meniscus signals versus cleaned-probe agreement with fluctuational electrodynamics). These three components serve one objective — a validated microscopic account of extreme near-field heat flow between well-characterized clean metal surfaces. The excess may be accounted for either by a theoretical or computational account that reproduces the measured magnitude and distance dependence of that heat flow within experimental uncertainties, or by tracing the excess, with direct and quantitative evidence, to a specific identified mechanism or artifact that accounts for the published data. The account must quantify the contribution and be validated against the measured heat flux, which the account need not itself measure.

References

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