FR32 · Heat across a vacuum gap

Near-Field Heat Transfer: Beating the Blackbody Limit

How can two plates exchange more heat than two black bodies? Through waves that tunnel across the gap. The blackbody limit counts only waves that travel freely between the plates, but each surface also holds bound waves that die off within about a wavelength. When the gap is far smaller than that, the bound waves reach the other plate and carry heat too. For silicon carbide plates 10 nm apart, the model gives over a thousand times the blackbody flux.

Close the vacuum gap between two hot plates and watch tunnelling waves carry up to a thousand times the blackbody heat flux.

Vacuum gap $1\,\mu\mathrm{m}$

Net heat flux $2.18\times10^{3}\,\mathrm{W\,m^{-2}}$

Over two black plates $2.20\times$

Carried by tunnelling $70.6\%$

Two plates, one vacuum gap

Vacuum gap

1\,\mu\mathrm{m}

Hot plate

400\,\mathrm{K}

Cold plate

300\,\mathrm{K}

Plate materials

Drag the edge of the cold plate, or press anywhere above the hot plate.

Advanced: material damping

Damping multiplier

1\times

Scales the phonon damping of SiC or the electron damping of gold in both plates. The small-gap flux of matched SiC plates depends only weakly on it, because a sharper line is also narrower.

Model, units and sources

$q=\int\frac{d\omega}{2\pi}\,[\Theta(\omega,T_1)-\Theta(\omega,T_2)]\int\frac{\beta\,d\beta}{2\pi}\sum_{j}\tau_j(\omega,\beta,d)$

Fluxes are in $\mathrm{W\,m^{-2}}$ ; the black-plate reference is $\sigma(T_1^{4}-T_2^{4})$ . When the temperatures are equal the Lab reports the linear-response conductance instead of a ratio of zero fluxes.

Joulain et al. (2005), Surf. Sci. Rep. 57, 59, Sec. 7 · Shchegrov et al. (2000), SiC parameters · Lambrecht & Reynaud (2000), gold parameters

Work through the heat-flux formula in the course

Model assumptions

Two flat, semi-infinite plates at uniform temperatures face each other across a vacuum gap. Each is described by a local dielectric function: SiC as a single Lorentz oscillator with the parameters of Shchegrov et al. (2000), gold as a Drude metal. The net flux is the Landauer-type sum of Joulain et al. (2005), Sec. 7, over travelling and tunnelling waves of both polarisations. Plates are infinite, so edges, roughness, finite thickness and heat conduction inside the plates are ignored. Below about 10 nm a local continuum model is no longer reliable.

Where in the spectrum the heat flows

Heat transfer coefficient against gap

Which channels are open

More numbers and how they are computed

Heat transfer coefficient: $0$
Two black plates, same temperatures: $0$
Share carried by travelling waves: $0$
Share carried by tunnelling s waves: $0$
Share carried by tunnelling p waves: $0$
Coefficient times gap squared: $0$
Gaps computed so far for the conductance chart: $0$

Channel transmissions are integrated over wavevector with adaptive Gauss–Kronrod quadrature at fixed frequencies, three Gauss–Legendre panels per decade of order six, then weighted by the two Planck occupations. Gaps between the 34 precomputed scan points are interpolated in log scale. Model version fr32-near-field-heat-transfer-v1.

Physics tutorial

Heat that tunnels across a vacuum gap

BackgroundTwo black plates facing each other exchange heat at the Stefan–Boltzmann rate, whatever the distance between them. That rule assumes the only carriers are waves that travel freely through the gap. When the gap is smaller than the thermal wavelength, about 10 µm at room temperature, evanescent waves bound to one surface reach the other and carry heat too.

Why it mattersNear-field heat transfer limits how close hot read heads and probe tips can work, and it is the reason near-field thermophotovoltaic converters are studied. It is also a clean test of fluctuational electrodynamics, because the same reflection coefficients predict both the far-field and the near-field results.

Start with the essentials

Focus question: How can two plates exchange more heat than two black bodies?
One-sentence intuition: The blackbody limit counts only the waves that can leave a surface. A plate also has many more bound modes with large in-plane wavevectors. Across a small gap they tunnel, and each one can carry heat almost as well as a travelling wave. Their number grows as one over the gap squared, so the flux does too, and for SiC almost all of it flows at the surface-mode frequency.

Core mathematical model

Net flux as a sum over channels

q=\int_{0}^{\infty}\frac{d\omega}{2\pi}\,\bigl[\Theta(\omega,T_1)-\Theta(\omega,T_2)\bigr]\int_{0}^{\infty}\frac{\beta\,d\beta}{2\pi}\sum_{j=s,p}\tau_j(\omega,\beta,d)

Each in-plane wavevector $\beta$ and polarisation is a channel with transmission $0\le\tau_j\le1$ , and $\Theta$ is the mean energy of a mode. This is Eqs. 90 and 91 of Joulain et al. (2005), rewritten per channel.

Travelling and tunnelling channels

\tau_j=\begin{cases}\dfrac{(1-|r_{1}|^{2})(1-|r_{2}|^{2})}{|1-r_{1}r_{2}e^{2i\gamma d}|^{2}}, & \beta<k_0\\[2ex]\dfrac{4\operatorname{Im}r_{1}\operatorname{Im}r_{2}\,e^{-2|\gamma|d}}{|1-r_{1}r_{2}e^{-2|\gamma|d}|^{2}}, & \beta>k_0\end{cases}

Here $r_1,r_2$ are the Fresnel coefficients of the two plates for polarisation $j$ and $\gamma=\sqrt{k_0^{2}-\beta^{2}}$ . For black plates the first line is 1 and the second is 0, which gives $\sigma(T_1^{4}-T_2^{4})$ .

Why the small-gap flux grows as one over the gap squared

\tau_p\approx\frac{4\operatorname{Im}r_{1}\operatorname{Im}r_{2}\,e^{-2\beta d}}{|1-r_{1}r_{2}e^{-2\beta d}|^{2}}\ \Rightarrow\ \int\beta\,d\beta\,\tau_p\propto\frac{1}{d^{2}}

When $d\ll\lambda$ , the p-polarised coefficients no longer depend on $\beta$ and the integral is cut off at $\beta\sim1/d$ . For SiC at 300 K, $h\,d^{2}$ stays near $9.4\times10^{-13}\,\mathrm{W\,K^{-1}}$ between 10 and 20 nm.

Common difficulties

The Stefan–Boltzmann law is a hard ceiling

Typical misconceptionNo pair of surfaces can exchange more than two black bodies.

Better mental modelThe law is a ceiling only for travelling waves. Tunnelling channels are extra, and at a 10 nm gap SiC plates exchange more than a thousand times the black-plate flux in this model.

Any material benefits equally

Typical misconceptionBringing any two hot plates close gives the same enhancement.

Better mental modelThe tunnelling channels open only where both plates absorb at the same frequency. SiC facing gold transfers less than two black plates at 100 nm, because gold has no surface mode at the SiC line.

Run the experiment

01
Start in the blackbody regime
Choose Blackbody regime and read the ratio to two black plates.
What to observe: At 20 µm the ratio is about 0.57: SiC reflects strongly in its band, so it does worse than a black body.
02
Predict, then close the gap
Guess how the flux changes when the gap shrinks tenfold below 1 µm, then drag the cold plate down.
What to observe: The flux grows about ninefold from 1 µm to 100 nm and about seventyfold from 100 nm to 10 nm, approaching the hundredfold of a one-over-gap-squared law. At 10 nm the ratio exceeds a thousand and the spectrum is one narrow peak.
03
Explain with mismatched plates
Compare Gold plates and Mismatched plates at 100 nm with the SiC pair.
What to observe: Gold pairs gain a factor of about eight, almost all in s polarisation. SiC facing gold falls to about a fifth of the black-plate flux.

Near-Field Heat Transfer: Beating the Blackbody Limit

Heat that tunnels across a vacuum gap

Core mathematical model

Net flux as a sum over channels

Travelling and tunnelling channels

Why the small-gap flux grows as one over the gap squared

Common difficulties

The Stefan–Boltzmann law is a hard ceiling

Any material benefits equally

Run the experiment

Start in the blackbody regime

Predict, then close the gap

Explain with mismatched plates

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