Between a gold-coated scanning-thermal-microscope tip and a planar gold surface separated by 0.2–7 nm in ultra-high vacuum, Kloppstech et al. (2017) measured a heat flux about five orders of magnitude above the blackbody limit and four orders of magnitude above the prediction of fluctuational electrodynamics, decaying approximately linearly with distance over the first ~2 nm — a distance dependence that existing phonon-tunnelling models, which predict algebraic decay, did not reproduce. Later literature has sharpened but not settled the problem: anomalously large thermal conductances of metallic nanogaps have been shown to arise from adsorbed water or surface contaminants bridging the gap (Cui et al. 2017; López-Nebreda et al. 2025), yet a 2025 measurement by the same group as the source still finds a roughly 100-fold excess over fluctuational electrodynamics in the extreme near field (Geesmann et al. 2025), and no microscopic theory reproduces both the amplitude and the distance dependence of the original data for realistic geometries. The surviving open core is to determine, experimentally and theoretically, the mechanism(s) — intrinsic or extrinsic — that mediate heat transfer between metallic bodies across sub-10-nm vacuum gaps — the crossover between near-field thermal radiation and contact conduction — and to delineate the gap range in which macroscopic, local-response fluctuational electrodynamics remains quantitatively valid.

Background

When two bodies at different temperatures face each other across a vacuum gap, heat flows by thermal electromagnetic radiation. At separations below the thermal wavelength (about 10 μm at room temperature), evanescent electromagnetic waves concentrated near each surface dominate the exchange; this regime is called near-field radiative heat transfer (NFRHT). The standard macroscopic description is fluctuational electrodynamics (FE), formulated by Polder and van Hove in 1971: thermally fluctuating currents inside each body, treated via the fluctuation–dissipation theorem and a local, frequency-dependent dielectric response, radiate into the other body. FE predicts that the heat flux between two surfaces grows roughly as the inverse square of the gap distance as the gap becomes small, and it accurately describes a large body of experiments down to gaps of a few tens of nanometres. As the gap shrinks to a few nanometres — comparable to interatomic spacings — additional channels beyond FE are expected to switch on and to cross over to ordinary contact heat conduction at zero gap: phonon tunnelling, in which lattice vibrations of one body couple to those of the other across the gap through short-range electrostatic and atom–atom interactions (modelled, for example, by Pendry, Sasihithlu and Craster), and electron tunnelling, in which electrons quantum-mechanically traverse the vacuum barrier between the two metals, carrying charge and heat.

Kloppstech, Könne, Biehs, Rodriguez, Worbes, Hellmann and Kittel (Nature Communications 8, 14475, 2017) measured heat transfer in precisely this crossover regime. They used a near-field scanning thermal microscope (NSThM) — a scanning tunnelling microscope whose gold-coated thermocouple probe serves simultaneously as electrode and as heat-flux sensor — against a planar gold sample, in ultra-high vacuum (about 10−1010^{-10} mbar), at tip–sample distances from 0.2 to 7 nm. The measured near-field heat flux exceeded the blackbody value by a factor of about 5×1055\times10^{5} and exceeded the FE prediction — computed exactly with a boundary-element method (a numerical solver that discretizes only the surfaces to solve the electromagnetic scattering problem for the realistic tip geometry: a roughly 30 nm radius apex on a conical shank) — by about four orders of magnitude. The flux was strongly localized, decaying approximately linearly with distance over the first ~2 nm. The authors showed that nonlocal corrections to the dielectric response (a Lindhard–Mermin model, in which the metal's response depends on wavevector as well as frequency) are too weak to close the gap, that the proximity approximation — the common shortcut that treats a curved surface as a mosaic of small mutually parallel planar patches, each described by the two-flat-surfaces formula — fails badly for such a sharp tip, and that available phonon-tunnelling theories, while able to explain fluxes much larger than FE, predict an algebraic distance decay inconsistent with the measured near-linear one. They concluded that whether the carriers are phonons, photons beyond local-response FE, electrons, or an as-yet unmodelled channel remains open, asked at what distances macroscopic FE ceases to be valid for probes with nanometre radii of curvature, and noted that a single experiment spanning sub-nanometre to hundreds-of-nanometre gaps (for example using a cryogenic transition-edge sensor, an ultra-sensitive resistance-thermometry heat detector) was still lacking. Subsequent work has made the question sharper rather than settled: independent experiments found that anomalously large conductances of Ångström-to-nanometre Au–Au gaps collapse below the detection limit once the surfaces are rigorously cleaned, while a 2025 measurement by the same group as the source still observes a roughly 100-fold excess over FE just before contact. The problem is therefore now one of disentangling genuine vacuum-gap transport from extrinsic artifacts and of building a theory that spans the full crossover.

Problem Statement

Determine, with quantitative experimental and theoretical support, the physical mechanism(s) that mediate heat transfer between metallic bodies across vacuum gaps from sub-nanometre to about 10 nm — the crossover regime between near-field thermal radiation and contact heat conduction — taking the Au-tip/planar-Au geometry of Kloppstech et al. (2017) as the defining anomaly. The open problem is to resolve: (i) the carrier identity question — whether the measured giant fluxes (exceeding fluctuational electrodynamics by factors of roughly 10210^{2} to 10410^{4} in the extreme near field) are intrinsic vacuum-gap transport via phonon tunnelling, electron tunnelling, or photon-mediated transfer beyond local-response fluctuational electrodynamics, or are extrinsic, mediated by adsorbed water or contamination bridging the gap; (ii) the theoretical description — a model spanning the radiation-to-conduction crossover for realistic tip or sphere geometries (nanometre radii of curvature, roughness, actual material response, possibly nonlocal and many-body effects) that reproduces both the amplitude and the distance dependence of the original Kloppstech et al. (2017) data insofar as those data are intrinsic (otherwise accounting for them as extrinsic artifacts), reduces to fluctuational electrodynamics at larger gaps, and to contact conduction at zero gap; and (iii) the validity boundary — the gap distance and probe-curvature range within which macroscopic local-response fluctuational electrodynamics remains quantitatively accurate for metallic bodies. These three facets are aspects of one objective — an experimentally verified, physically identified mechanism for nanometre-gap heat transfer.

References

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