Two flat plates at temperatures T₁ > T₂, separated by vacuum, exchange heat by thermal radiation. In the far field (gap d larger than the thermal wavelength λT ≈ ħc / kBT ≈ 7–10 μm at room temperature) the exchanged flux is capped by the Stefan–Boltzmann blackbody limit:
q_bb = σ (T₁⁴ − T₂⁴)
σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴
Once the gap shrinks well below λT — into the nanoscale — evanescent electromagnetic modes that would normally decay before reaching the other surface can instead tunnel across the gap, coupling to matching modes on the opposite plate. This near-field channel is forbidden in the far field but dominates at nanometer separations, and in the electrostatic (proximity) limit its contribution grows approximately as 1/d²:
q_nf(d) ≈ q_bb · [ 1 + A · (d_ref / d)² ] (illustrative scaling)
The prefactor A depends on how well the material's surface electromagnetic modes match the thermal spectrum. Polar dielectrics like SiO₂ support surface phonon-polaritons resonantly tuned to thermal-infrared frequencies, so their near-field enhancement is large — laboratory measurements between SiO₂ surfaces at gaps of a few tens of nanometers have reported flux 2–3 orders of magnitude above the blackbody limit (Shen, Narayanaswamy & Chen 2009; Rousseau et al. 2009). Metals like gold instead support surface plasmon-polaritons at much higher (visible/UV) frequencies that couple poorly to thermal-IR radiation, so their measured enhancement at the same gaps is far more modest.
- Gap slider — sets the plate separation on a log scale from 10 nm to 10 μm; the flux readouts and the tunneling-particle density update live.
- SiO₂ / Gold — switches the enhancement prefactor A to the strong-resonance dielectric case or the weak-coupling metal case.
- T₁ / T₂ sliders — set the hot and cold plate temperatures (T₁ is always kept above T₂); both the blackbody and near-field flux scale with T₁⁴ − T₂⁴.
- The dots crossing the gap are a visual analogy for photon tunneling density, not a literal ray trace — their rate and how tightly they stay collimated scale with the computed enhancement factor.
Real-world relevance: this effect underlies near-field thermophotovoltaics, nanoscale heat-assisted magnetic recording, scanning thermal microscopy, and proposed nanogap thermal management in dense electronics.