Illustrated Physics
physics, drawn
Quantum

The heat that crosses a gap too narrow for light

Stefan and Boltzmann's law is usually read as a ceiling: no surface can radiate more heat than a perfect black body at the same temperature, 6.1 watts per square metre for every degree of difference at room temperature. It is a ceiling only for surfaces that are far apart. Bring two of them closer than the wavelength of their thermal glow and they begin to exchange waves that never leave either surface — waves that cling to it and die away within a wavelength — and those waves tunnel across the gap. Two plates of silicon carbide ten nanometres apart exchange fifteen hundred times what two black bodies can, almost all of it in a single colour, and the colour is the one in which silicon carbide is too shiny to glow at all.
15 min read 5 figures Where the model stopsWhat happens at the edgeApproximations that lie

Assumes: The curve that would not come down · The glow that says nothing about the surface

The curve that would not come down derived Planck’s law by counting the ways light can fit into a box and giving each an energy that falls off at high frequency. The gas that nobody counted treated that radiation as a gas of photons, and the glow that says nothing about the surface found Kirchhoff’s result, that the radiation inside a cavity depends on its temperature alone and nothing about its walls — a surface that absorbs poorly at some wavelength emits poorly there in exactly the same proportion, so no surface can outshine a black body. The height a planet is seen from used the same law on a whole atmosphere.

From those comes the Stefan–Boltzmann law, σT4\sigma T^4, and its consequence for heat flow between two surfaces: at room temperature two perfect black bodies exchange 6.1 watts per square metre for each degree of temperature difference between them, and nothing real exchanges more. That is written into the design of vacuum flasks, spacecraft radiators and cryostats. It is also, strictly, a statement about surfaces far apart. This essay follows what happens when they are brought closer than the wavelength of their own glow, and finds that the ceiling disappears — that two surfaces a few nanometres apart can exchange heat by radiation hundreds of thousands of times faster than black bodies, in a way that depends on almost nothing but the surfaces’ own material.

What Planck’s count leaves out

Planck counted the waves that travel. In the language of a flat surface, a travelling wave in the gap has a wavenumber along the surface, kk, no larger than ω/c\omega/c; anything with a larger lateral wavenumber cannot satisfy the wave equation in vacuum as a travelling wave, and the equation offers it instead as an evanescent wave, which ripples along the surface and dies away exponentially with distance from it, within about 1/k1/k.

Every hot surface is covered in such waves. Its charges jiggle thermally, at every frequency and with every lateral wavelength, and the fields they make include components of every lateral wavenumber. Those with k<ω/ck < \omega/c leave the surface as light and are what Planck counted. Those with k>ω/ck > \omega/c cannot leave. They cling to the surface within a fraction of a wavelength of it, carry energy along it but not away from it, and for a surface on its own contribute nothing to the radiation that reaches anything far away.

Put a second surface within that clinging layer and the evanescent field reaches it. The reflection that happens where the glass is not found the optical version: light totally reflected inside one prism can be made to cross into a second, brought within a wavelength, by frustrating the reflection, exactly as a particle tunnels through a wall. Thermal evanescent waves tunnel the same way. They carry energy from the hotter surface to the colder across a gap they could never cross as light, and they do it through channels that Planck’s count did not include because in a cavity of any ordinary size they do not reach.

How much more

The calculation of how much heat crosses is fluctuational electrodynamics, worked out by Sergei Rytov in the 1950s and applied to two plates by Dirk Polder and Michel Van Hove in 1971. The fluctuating currents in each body are fixed by its temperature and its absorption through the fluctuation–dissipation theorem — the same theorem half a kT in a piece of wire used to find the thermal noise of a resistor — and the heat that crosses is a sum over every frequency and every lateral wavenumber of the probability that a wave of that kind gets from one body to the other.

The heat two surfaces exchange when they almost touch. The radiative heat-transfer coefficient between two parallel half-spaces at 300 K, in W/m²K on a logarithmic axis, against the width of the vacuum gap between them, also logarithmic, from fluctuational electrodynamics: silicon carbide (solid) and gold (dashed), with two perfect black bodies, 4σT³ = 6.12 W/m²K, for comparison (dotted). Far apart, silicon carbide exchanges 3.3 W/m²K — less than black — and gold, a good mirror, only 0.024. Closer than the thermal wavelength, about 17 µm, waves that cannot leave either surface begin to tunnel across the gap. At 100 nm silicon carbide exchanges 137 W/m²K, at 10 nm 9338, at 1 nm 9.3·10⁵ — rising as the inverse square of the gap, 1.5·10⁵ times the black-body value at 1 nm. Gold exceeds black only below a few hundred nanometres and then grows much more slowly: 1284 at 10 nm, 2561 at 1 nm.
Fig. 1 Radiative heat-transfer coefficient between two parallel half-spaces at 300 K against the gap, both logarithmic: silicon carbide (solid), gold (dashed), two black bodies, 6.12 W/m²K (dotted). Far apart, SiC exchanges 3.3 and gold 0.024. At 100 nm SiC exchanges 137 W/m²K, at 10 nm 9,338, at 1 nm 930,000 — as 1/d². Gold passes black below a few hundred nanometres and then grows slowly.

The figure shows the result for two materials. Far apart — at a hundred micrometres, several thermal wavelengths — silicon carbide exchanges 3.3 watts per square metre per kelvin, about half the black-body value, because it is not a perfect absorber, and gold, a good mirror, exchanges only 0.024. Kirchhoff’s law holds and the black body is the ceiling. Closer than the thermal wavelength, about seventeen micrometres at room temperature, both curves rise. Gold crosses the black-body line at a few hundred nanometres and continues rising slowly. Silicon carbide rises steeply, as the inverse square of the gap: 137 at a hundred nanometres, 9,300 at ten, 930,000 at one. At a nanometre two slabs of silicon carbide exchange heat by radiation a hundred and fifty thousand times faster than two black bodies could.

The effect was predicted in 1971 and measured convincingly only from 2009, with a small sphere held near a flat surface on the arm of an atomic-force microscope, because keeping two surfaces parallel, clean and nanometres apart without touching is extremely hard. Measurements now reach gaps of a few nanometres, and at the smallest gaps they are complicated by the question of whether radiation is still the right description when the gap is only a few atoms wide.

Almost all of it in one colour

The most surprising feature of the near-field heat is its spectrum.

The near field carries its heat in one colour. The heat-transfer coefficient per unit angular frequency between two silicon-carbide half-spaces 10 nm apart at 300 K (solid), and between two black bodies at any distance (dashed), in W/m²K per 10¹² rad/s on a logarithmic axis, against angular frequency, also logarithmic. The black bodies share their heat across the whole thermal spectrum. The narrow gap sends almost all of it through a line at 1.786·10¹⁴ rad/s — a wavelength of 10.55 µm — where silicon carbide's permittivity is −1 and its surface carries a resonant wave of charge and lattice vibration, the surface phonon polariton. At its peak the narrow gap carries 1.5·10⁵ times what black bodies exchange at the same frequency.
Fig. 2 Heat-transfer coefficient per unit angular frequency between two SiC half-spaces 10 nm apart at 300 K (solid) and between two black bodies (dashed), logarithmic. The black bodies share heat across the whole thermal spectrum; the narrow gap sends nearly all of it through a line at 1.786 × 10¹⁴ rad/s, 10.55 µm, where it is 1.5 × 10⁵ times the black-body value.

Black bodies share their heat across the whole thermal spectrum, in the broad hump of Planck’s curve. Two silicon-carbide surfaces ten nanometres apart send almost all of theirs through a single narrow line, at an angular frequency of 1.786×10141.786 \times 10^{14} per second — a wavelength of 10.55 micrometres — where the spectral heat flow is a hundred and fifty thousand times the black-body value.

That line is a surface resonance. Silicon carbide is a polar crystal: its silicon and carbon atoms carry opposite charges, and when the lattice vibrates they move in opposite directions and make an oscillating dipole. Between the frequencies of its transverse and longitudinal lattice vibrations its permittivity is negative, and at the frequency where the permittivity is exactly −1-1 the surface supports a wave in which the lattice vibration and an electromagnetic field travel along the surface together, bound to it: a surface phonon polariton. It is the lattice-vibration counterpart of the wave a surface is enough to hold, and like any surface wave it has lateral wavenumbers far larger than ω/c\omega/c, so it lives entirely in the evanescent channels. Two such surfaces facing each other across a narrow gap couple their resonances, and the heat pours through them.

The colour it cannot shine in

That line has a strange relation to the surface’s ordinary glow.

The colour silicon carbide cannot shine in is the one it passes most. The emissivity of a flat silicon-carbide surface seen straight on, one minus its reflectance, against wavelength in micrometres. Between 10.3 and 12.6 µm the crystal's lattice vibrations make its permittivity negative and the surface reflects almost everything — the reststrahlen band — so there it glows hardly at all: 4.8 per cent at 11 µm. The near-field heat between two such surfaces is carried at 10.55 µm (dashed), inside that band, by surface waves that exist only because the permittivity there is negative. The same property that keeps the surface from radiating into space makes it an exceptionally strong exchanger of heat with a surface a few nanometres away.
Fig. 3 Emissivity of a flat SiC surface straight on, against wavelength. Between 10.3 and 12.6 µm (shaded) the permittivity is negative and the surface reflects almost everything, so it barely glows — 4.8 per cent at 11 µm. The near-field heat is carried at 10.55 µm (dashed), inside that band.

Where the permittivity is negative a surface reflects almost everything that falls on it, just as a metal does; for silicon carbide that is the band from 10.3 to 12.6 micrometres, called the reststrahlen band, from the German for “residual rays”, because early infrared experimenters isolated these wavelengths by bouncing light off several crystals in succession. By Kirchhoff’s law a surface that reflects almost everything at some wavelength emits almost nothing there. Silicon carbide’s emissivity at 11 micrometres is under five per cent. Seen from far away, the crystal is nearly dark in exactly the band where it is brightest from a few nanometres.

The two facts are the same fact. The negative permittivity that makes the surface a mirror for waves arriving from far away is what allows a wave to be bound to it, and the bound wave cannot leave: it is the energy that would have been radiated, held at the surface instead. Kirchhoff’s result, that no surface can outshine a black body, is a statement about what can escape to infinity, and it remains true. The near field is not a violation of it but a different question — what can be exchanged with something that has not been placed at infinity — and the answer to that question depends on the surface’s material in exactly the way the glow that says nothing about the surface showed the far field does not.

Where the channels come from

The inverse square of the gap has a simple origin, visible in the probability that each wave crosses.

The waves that tunnel, and how many of them there are. The probability that a wave of p polarisation at the surface resonance crosses a gap between two silicon-carbide surfaces, against its wavenumber along the surface as a multiple of ω/c, on a logarithmic axis, for gaps of 10 nm and 100 nm. Left of 1 the waves propagate, and the surfaces, which reflect strongly at this frequency, pass few of them — the far field. Right of 1 the waves are evanescent: they decay away from each surface, but across a narrow gap they tunnel, and near the resonance they cross with probability close to one. For the 100 nm gap the open channels reach 59 times ω/c, for the 10 nm gap 573 times — about 3.5/d in both cases. The number of open channels per unit area grows as the square of that reach, which is where the 1/d² comes from.
Fig. 4 Probability that a p-polarised wave at the surface resonance crosses a gap between two SiC surfaces, against its lateral wavenumber as a multiple of ω/c, for gaps of 10 nm and 100 nm. Left of 1 (shaded) the waves propagate and few cross; right of 1 they tunnel, nearly all of them, up to about 3.5/d: 59 times ω/c for 100 nm, 573 times for 10 nm.

At the resonance frequency the propagating waves, left of 1 in the figure, barely cross: the surfaces reflect them. The evanescent waves to the right cross with a probability close to one, up to a lateral wavenumber of about 3.5/d3.5/d, beyond which their decay across the gap, as e−2kde^{-2kd}, shuts them off. Each lateral wavenumber is a channel, and the number of channels per unit area of surface within a disc of radius KK in wavenumber is proportional to K2K^2. The open channels reach to K≈3.5/dK \approx 3.5/d, so their number grows as 1/d21/d^2, and since each carries at most its thermal share of energy, the heat grows as 1/d21/d^2.

That is the same counting Planck did, with a different radius. Planck’s channels fill a disc of radius ω/c\omega/c, set by the frequency and hence by the temperature, and that is why the black-body flux is fixed by the temperature alone. The near-field channels fill a disc of radius 1/d1/d, set by the gap. There is no black-body limit on the near field because the limit came entirely from the size of that disc.

The heat crosses in ripples the size of the gap. The fraction of the heat exchanged between two silicon-carbide surfaces at 300 K that is carried by waves whose wavenumber along the surface is below k, against k times the gap d, on a logarithmic axis, for gaps of 10 nm and 100 nm. For the 10 nm gap half the heat is carried by waves with k d below 2.15, and for the 100 nm gap by k d below 1.62: in both, the ripples that carry it have a wavelength along the surface of a few times the gap, whatever the gap. Heat radiated into space is carried by waves at least as long as the thermal wavelength; heat exchanged across a narrow gap is carried by ripples set by the gap itself, which is why it grows without any limit Planck's law would recognise as the gap closes.
Fig. 5 Share of the heat between two SiC surfaces at 300 K carried by waves with lateral wavenumber below k, against k times the gap, for gaps of 10 nm and 100 nm. Half the heat is carried below kd = 2.15 for 10 nm and 1.62 for 100 nm: ripples with a lateral wavelength of a few times the gap, whatever the gap.

The last figure makes the scaling explicit: plotted against wavenumber times gap, the heat carried by each range of ripples looks nearly the same for a ten- and a hundred-nanometre gap. Half of it is carried by ripples with a lateral wavelength of about three times the gap or longer. A gap ten times smaller moves the heat into ripples ten times finer, and there are a hundred times as many of them per unit area.

Why a ceramic beats a metal

The first figure has a result that runs against intuition: gold, an excellent conductor of heat and electricity, exchanges far less heat across a narrow gap than silicon carbide, an insulating ceramic. The reason is the thermal factor that multiplies every channel.

A channel carries energy only if the modes feeding it are thermally excited, and a mode of angular frequency ω\omega holds, on average, the energy ℏω/(eℏω/kT−1)\hbar\omega/(e^{\hbar\omega/kT} - 1). At room temperature kT/ℏkT/\hbar is about 4×10134 \times 10^{13} per second, and modes much above a few times that are nearly empty. Silicon carbide’s surface resonance sits at 1.79×10141.79 \times 10^{14} per second, where ℏω/kT\hbar\omega/kT is about 4.5 — on the tail of the thermal distribution, but not so far that nothing is there. Gold has a surface resonance too, the surface plasmon, but it sits in the ultraviolet, at about 101610^{16} per second, where ℏω/kT\hbar\omega/kT is over two hundred and the thermal occupation is e−200e^{-200}. Gold’s surface waves exist and would carry heat superbly if they were occupied; at room temperature they are empty. What gold exchanges across a narrow gap is carried instead by the other polarisation — by thermally fluctuating eddy currents in the metal, which induce currents in the other surface through their magnetic fields — and that mechanism saturates rather than growing as 1/d21/d^2.

So the best materials for near-field heat transfer are those with a surface resonance near the peak of the thermal spectrum at the temperature of interest: polar dielectrics such as silica, silicon carbide and hexagonal boron nitride at room temperature, doped semiconductors whose plasma frequency can be tuned into the infrared, and, for higher temperatures, materials whose resonances sit correspondingly higher. The far-field rule of thumb — good absorbers are good emitters, black is best — is replaced by a rule about resonances, and the black body is no longer the benchmark it was.

Ten nanometres of vacuum, and a tenth of a millimetre of glass

The size of the effect is easier to feel against ordinary conduction. A heat-transfer coefficient is a conductance per unit area, and the same number describes a slab of solid: a sheet of window glass, whose thermal conductivity is about one watt per metre per kelvin, has a coefficient of one over its thickness. A millimetre of glass passes a thousand watts per square metre per kelvin. The ten-nanometre vacuum gap between silicon-carbide plates passes 9,300 — as much as a tenth of a millimetre of glass. At one nanometre the vacuum conducts like a glass sheet a micrometre thick.

Vacuum is the standard insulator of thermal engineering, because nothing in it can conduct, and it remains one at ordinary scales. At the scale of modern devices it does not. The read head of a hard disk flies a few nanometres above the spinning platter, and the heat that crosses that gap by radiation is a measurable part of the head’s thermal budget. Components separated by vacuum gaps in microscopic machines and in the probes of scanning microscopes exchange heat through their near fields, and their designers have to include it. Across a gap of a few nanometres, “thermally isolated by vacuum” is a statement about far-field radiation that no longer describes where the heat goes.

Where the counting stops

The calculation treats the surfaces as smooth, flat, semi-infinite and described by the permittivity measured for bulk material. Each assumption fails at some gap. The inverse-square growth requires ripples of lateral wavelength a few times the gap, and when the gap approaches the spacing of the atoms the continuum description of the surface has nothing to say about such fine ripples; the permittivity itself becomes a function of wavenumber as well as frequency, and the growth is expected to level off at a gap of about a nanometre. Below that, electrons and lattice vibrations can cross the gap directly, and the distinction between radiation and conduction dissolves. Real surfaces are rough on the scale of nanometres, so the local gap varies and the average heat flow is dominated by the closest points.

For metals the story is different in detail. The evanescent heat between two gold surfaces is carried mainly by the other polarisation, by fluctuating eddy currents rather than by a surface resonance, and it grows more slowly, which is why the gold curve in the first figure bends over rather than continuing as 1/d21/d^2.

What the pictures cannot show

The figures compute heat exchanged between two infinite parallel plates at slightly different temperatures, which is the textbook geometry and not the experimental one. Measurements use a sphere or a sharp tip near a flat surface, and the heat is then dominated by the small region of closest approach; converting between the two geometries requires an approximation that is itself tested only over part of the range. The figures show the exchange coefficient and not the absolute flows, which for large temperature differences are no longer linear in the difference. And they cannot show that the fluctuating fields responsible for the heat also produce a force: the same fields, in the same gap, give the attraction between neutral surfaces that the attraction that weakens when light is too slow met. Heat and force are two readings of one spectrum of fluctuations, one weighted by the difference of temperatures and the other by the sum.

Still open: whether near-field heat can be made useful

The near field is already used in heat-assisted magnetic recording, where a tiny antenna focuses energy onto a spot smaller than any lens could, and in scanning thermal microscopy. The larger prize is energy conversion. A thermophotovoltaic cell turns the thermal radiation of a hot emitter into electricity, and its output is limited by how much radiation crosses from emitter to cell; bring the two within a hundred nanometres and the power can in principle rise tenfold or more while still passing through the cell’s band gap. Devices doing this over areas of square millimetres, with gaps held stable against thermal expansion and without the emitter touching the cell, have been demonstrated at small scale, and whether the approach can be built over the areas and temperature differences that would make it worthwhile is an engineering question still being worked on. A related question concerns heat flow at the smallest gaps, below a few nanometres, where several experiments have reported heat transfer larger than fluctuational electrodynamics predicts and others have not, and what carries it there is disputed.

The habit worth carrying away is to ask what a limit is counting. The black-body ceiling counts only the waves that can travel to infinity, whose number per unit area is fixed by the temperature; two surfaces closer than their thermal wavelength also exchange evanescent waves, whose number is fixed by the gap, and across ten nanometres of vacuum silicon carbide passes fifteen hundred times the black-body heat, almost all of it in the one colour it is too shiny to radiate. Kirchhoff’s law survives untouched, because it was always a statement about what reaches a distant absorber.

Part 6 of 6

This essay is one argument about Blackbody. The others:

Each wavelength sees from a different height. The weighting functions of 7 infrared channels in a carbon-dioxide band — how much each height contributes to the radiance a satellite measures in that channel — for absorption strengths chosen so that the peaks sit at 3, 8, 14, 20, 27, 35, 43 km. Each is scaled to its own maximum. A channel's peak is where the carbon dioxide above that height amounts to one optical depth: deeper, its emission is absorbed on the way out; higher, there is too little gas to emit. Each function is about 17 km wide at half its peak, the same for every channel, because the absorber's density falls with one scale height, 7 km. The dotted curve is the temperature profile of the standard atmosphere, on its own scale from 180 to 300 K, whose cold tropopause and warm stratosphere the channels between them straddle. The spectrum that is a thermometer at every height Part 5 — A satellite looking down at the Earth in the infrared sees, at each wavelength, the glow of whatever layer of air that wavelength escapes from. In the wings of carbon dioxide's great absorption band the air is transparent and the satellite sees the ground; towards the band's centre the gas grows opaque and each wavelength escapes from higher up. A single spectrum is therefore a stack of thermometers at different heights, and weather forecasts are built on reading it. What cannot be read from it is anything much thinner than the ten to twenty kilometres of air each of those thermometers averages over.

Where this goes next

What a thermal camera reads off a shiny surface. The temperature a camera calibrated for a black body reports, against the emissivity of the surface it is pointed at, for surfaces truly at 323 K, 373 K, 473 K in a room at 293 K. Every curve begins at the true temperature when the emissivity is one and ends at the room's temperature when it is zero, because a surface that emits nothing reflects everything and the camera is then looking at the room. polished aluminium at 0.05 reads 312 K; stainless steel, oxidised at 0.8 reads 451 K; matt black paint at 0.95 reads 468 K; human skin at 0.98 reads 471 K for a surface truly at 473 K. That is not a fault in the instrument. It follows from Kirchhoff's law — a poor emitter is a poor absorber and therefore a good reflector — so the shortfall in a shiny surface's own glow is made up almost exactly by whatever is reflected in it, and no measurement of the light leaving a surface can separate the two without knowing the emissivity in advance. The glow that says nothing about the surface A thermal camera pointed at a saucepan of boiling water reads a hundred degrees. Pointed at a polished aluminium block at the same temperature it reads about twenty-five, and the instrument is working perfectly. What it is measuring is emissivity as much as temperature — and a surface's emissivity is forced to equal its absorptivity, at every wavelength and every angle, by an argument with no physics of matter in it at all. The reflected beam does not leave from where it arrived. The lateral displacement of a totally reflected beam along the surface, for the two polarisations, at 550 nm from n = 1.5 into n = 1. Geometrical optics puts the outgoing ray at the point where the incoming one struck. It is not there: it is displaced forward along the surface by a distance comparable with a wavelength, which was measured by Goos and Hänchen in 1947 by reflecting a beam many times and looking at the accumulated offset. The displacement is different for the two polarisations — at 42°, 1361 nm for s and 2991 nm for p, at 45°, 350 nm for s and 560 nm for p, at 50°, 237 nm for s and 261 nm for p, at 60°, 183 nm for s and 127 nm for p, at 75°, 161 nm for s and 79 nm for p — which is why an unpolarised beam comes back slightly split. A displacement is only possible if the light spent time on the far side of a boundary it never crossed, and it is the most direct evidence there is that the evanescent field is a real field rather than a bookkeeping term. The reflection that happens where the glass is not Total internal reflection sends back every photon, which is why it is called total. It does not send them back from where they arrived — the beam re-emerges displaced along the surface, by a fraction of a wavelength, and a displacement is only possible if the light spent time on the far side of a boundary it never crossed. London's attraction, and the distance where it gives out. The dispersion energy between two identical atoms times the sixth power of their separation, relative to London's constant, against the separation in units of c/ω₀ — the reduced wavelength of each atom's strongest transition — on logarithmic axes, from the Casimir–Polder integral with a one-resonance polarisability. Close in, the product is one: London's inverse sixth power. It has fallen to a half by 3.82 c/ω₀, and far out it falls as one over the distance, so the energy goes as the inverse seventh power, reaching 23.2 per cent of London's value at ten reduced wavelengths and 2.44 per cent at a hundred. For two helium atoms, whose strongest line is at 58 nm, c/ω₀ is about 9 nm; for two sodium atoms, 94 nm. The attraction that weakens when light is too slow Two neutral atoms attract because each one's fluctuating electrons polarise the other, and the two fluctuations fall into step. London's calculation of the force, as the inverse sixth power of the distance, assumes the field from one atom reaches the other instantly. It does not. Beyond a distance set by the wavelength of the atoms' own strongest transition — nine nanometres for helium, about a hundred for sodium — the field returns too late to find the fluctuation that made it. The attraction then falls one power faster. Casimir and Polder worked this out in 1948, and it was the starting point for the Casimir force between plates.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

BlackbodyEmissivityEvanescent waveFluctuation dissipationHeat transferNear fieldSurface phonon polaritonTunnelling