Between two metallic surfaces separated by a vacuum gap of a few nanometres, measured heat fluxes have been reported that exceed the predictions of fluctuational electrodynamics (the standard theory of near-field thermal radiation) by factors of 10210^2 to 10410^4, while other experiments of the same kind find agreement with the theory down to sub-nanometre gaps. The open problem is to determine the physical origin of this giant near-field heat flux: whether it is carried by a phonon-mediated tunnelling mechanism, by a photon-mediated process beyond macroscopic fluctuational electrodynamics, by electronic tunnelling, by surface contamination such as water menisci, or by a mechanism not yet identified — and to build quantitative models that predict absolute magnitudes and distance dependences for realistic metallic tip/sphere–sample geometries across the 0.1–10 nm radiative-to-contact crossover. The question was posed by Kloppstech et al. after ultrahigh-vacuum measurements between a gold scanning-thermal-microscope tip and an Au(111) sample; it remains actively contested, with 2025 experiments from two groups reaching opposite conclusions (persistent enhancement after in-situ cleaning versus water-meniscus artifacts), and with no existing theory reproducing both the magnitude and the distance dependence of the anomalous data.

Background

Radiative heat transfer between two bodies is limited in the far field by the Stefan–Boltzmann blackbody law. At surface separations smaller than the thermal wavelength (roughly 10 μm at room temperature), evanescent electromagnetic waves that decay exponentially away from each surface tunnel across the gap and enhance the exchange beyond this limit — near-field radiative heat transfer (NFRHT). The standard theory, fluctuational electrodynamics (FE), introduced by Rytov and co-workers and formulated for arbitrary geometries by Polder and van Hove, treats the thermal charge-current fluctuations inside each body as statistically independent random sources radiating through the classical electromagnetic Green's function of the whole system. FE with measured (local) dielectric functions has been confirmed experimentally for many material combinations down to gaps of a few tens of nanometres. At gaps of a few nanometres or less — the extreme near field, where the gap approaches interatomic distances and heat transport should cross over from radiative to conductive — macroscopic FE is expected to break down: it neglects the spatial dispersion (nonlocality) of the optical response, correlations between the fluctuating currents of the two facing bodies, and the possibility that matter-based carriers cross the vacuum gap. Candidate beyond-FE channels include phonon tunnelling, in which short-range forces (overlap of electronic wave functions, electrostatic coupling of surface atoms) let lattice vibrations carry heat across a gap without material contact, typically with a power-law distance dependence d−nd^{-n}; electronic heat tunnelling, in which electrons quantum-tunnel between the metals (with the Nottingham effect — net heating or cooling because tunnelling electrons enter and leave a metal at different average energies — appearing at finite bias voltage); and surface contamination, in which adsorbed molecules or water menisci physically bridge the gap and conduct heat.

The question was sharpened by experiments with a near-field scanning thermal microscope (NSThM), a scanning-tunnelling-microscope-type instrument whose gold-coated tip carries a calibrated thermocouple temperature sensor, so that the measured thermovoltage yields the absolute heat flux from tip to sample while the tunnelling current fixes the zero of the tip–sample distance. Kloppstech et al. measured, in ultrahigh vacuum (about 10−1010^{-10} mbar) between such a tip (apex radius about 30 nm) and an in-situ sputter-annealed Au(111) sample (a gold single crystal whose (111) surface was cleaned inside the vacuum by argon-ion bombardment followed by annealing) held at 120 K (temperature difference 160 K), the heat flux at gaps of 0.2–7 nm: it exceeded the blackbody value by more than five orders of magnitude and the FE prediction (computed with a boundary-element method — a numerical technique that solves the electromagnetic scattering problem by discretizing only the surfaces of the bodies — for the real tip geometry) by about four orders of magnitude, with an almost linear decay of the flux up to 2 nm that excludes the d−nd^{-n} scalings of the phonon-tunnelling models available at the time. Joule heating by the tunnelling electrons was bounded at about 6% of the signal. In contrast, contemporary sphere–plane experiments by the Reddy group (Kim et al. 2015; Cui et al. 2017) found, after tip cleaning, no contradiction with FE down to gaps of about 0.2 nm. The experimental record is thus itself split, and the source paper closes by stating that whether the measured flux can be explained by phonon or photon tunnelling, or whether there is yet another unknown mechanism at play, remains open, with quantitative models spanning the 0.1–10 nm crossover for realistic geometries explicitly lacking.

Problem Statement

Determine the physical mechanism responsible for the measured heat flux between metallic (gold) surfaces separated by nanometre-scale vacuum gaps — the flux exceeding the predictions of macroscopic fluctuational electrodynamics by roughly two to four orders of magnitude in ultrahigh-vacuum measurements — and thereby establish a quantitative, experimentally testable description of heat transfer across the radiative-to-contact crossover between 0.1 and 10 nm. The open question, to be settled with evidence sufficient to reconcile the conflicting experiments, is: (i) whether the excess flux between in-situ-cleaned gold surfaces in ultrahigh vacuum — the roughly 10410^4-fold enhancement over fluctuational electrodynamics at 0.2–7 nm with an almost linear decay up to about 2 nm reported by Kloppstech et al., and the roughly 10210^2-fold enhancement below about 18 nm persisting after in-situ argon-ion sputtering reported by Geesmann et al. — is a genuine heat-transfer channel or an experimental artifact such as contamination or water menisci; and (ii) if genuine, which channel dominates in which distance range — phonon-mediated tunnelling, a photon-mediated process beyond macroscopic fluctuational electrodynamics (nonlocal response, fluctuation cross-correlations, or other quantum corrections), or electronic tunnelling including the Nottingham effect — with a model that predicts both the absolute magnitude and the distance dependence for realistic tip/sphere–sample geometries and can be compared quantitatively with calibrated experimental data.

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