An engine with one number in it

The three-reservoir refrigerator removes the work from a refrigerator. This one removes everything else: no pistons, no valves, no working fluid, no circulation of any kind. A thermoelectric generator is two pieces of doped semiconductor joined at one end — one gradient driving the other thing, which is the cross-coupling this one is built on — hot at that end and cold at the other, with a voltage across the pair.

Its efficiency has the same Carnot ceiling as a steam turbine and reaches a fraction of it that is a single expression:

$$\eta = \frac{T_h - T_c}{T_h}\cdot\frac{\sqrt{1 + Z\bar{T}} - 1}{\sqrt{1 + Z\bar{T}} + T_c/T_h},$$

where $Z\bar{T}$ is one dimensionless combination of the material’s properties and nothing else about the material appears anywhere.

What the group is, and why it has that shape #

The group is

$$ZT = \frac{S^2\sigma T}{\kappa},$$

with $S$ the Seebeck coefficient — volts produced per kelvin of temperature difference — $\sigma$ the electrical conductivity, and $\kappa$ the thermal conductivity. Each appears for a reason that can be said in a clause.

$S$ appears squared because the device is a voltage source with an internal resistance: the voltage it produces is $S\Delta T$, and the power delivered into a matched load goes as the square of the voltage.

$\sigma$ appears once because the internal resistance is what the current has to fight, and a higher conductivity means less power dissipated inside the device on the way out.

$\kappa$ appears in the denominator because heat leaking straight from the hot end to the cold end through the material does nothing useful at all. It is the direct analogue of a short circuit across a battery, and it is what stops a metal being a good thermoelectric however well it conducts.

And $T$ appears because the group has to be dimensionless, and because a device is judged over the temperature range it works in — which is why the quantity quoted is $ZT$ at a stated temperature and not $Z$.

The expression’s two limits are what make it a figure of merit rather than a fitted constant. As the group goes to zero the efficiency goes to zero. As it goes to infinity the second factor goes to one and the efficiency goes to Carnot. Every material sits somewhere between, and where it sits is one number.

The three ingredients that will not cooperate #

If the three quantities could be chosen independently, a good thermoelectric would be easy: pick a large Seebeck coefficient, a metal’s conductivity and a glass’s thermal conductivity. They cannot be chosen independently.

The conflict is not an empirical nuisance; it comes out of one model of a doped semiconductor with one adjustable parameter, which is where the Fermi level sits relative to the band edge.

With few carriers, each one that moves from the hot end to the cold end carries a large amount of entropy relative to the number available, so the Seebeck coefficient is large — and there are few of them, so the conductivity is small.

With many carriers, the conductivity is large and the Seebeck coefficient is small, because the Fermi level is deep inside the band and a carrier at the hot end is barely different from one at the cold end.

The product peaks in between, and the peak is at a carrier concentration of around $10^{20}$ per cubic centimetre — which is a thousand times more than a doped silicon transistor and a thousand times less than a metal. Every thermoelectric material in industrial use is in that window, and the doping level is the first parameter optimised and the one with the least left in it.

Where the Seebeck coefficient comes from #

It is worth saying what the coefficient actually measures, because “volts per kelvin” is a specification rather than an account.

A conductor with one end hot has more energetic carriers at that end. They diffuse toward the cold end faster than the cold end’s carriers diffuse back, charge builds up, and the resulting electric field grows until it stops the net flow. The voltage at that point, divided by the temperature difference, is the Seebeck coefficient.

What decides its size is how much the carriers’ behaviour differs between the two ends — how strongly their number and their mobility depend on energy. In a metal that dependence is weak, because only carriers within a thermal energy of the Fermi level participate at all and the Fermi level is far above it: metals have Seebeck coefficients of a few microvolts per kelvin. In a lightly doped semiconductor the dependence is strong, and the coefficients are hundreds of microvolts per kelvin.

Written properly, the coefficient is the entropy carried per unit charge. That is the deepest form of the statement and it explains both the sign and the size: a carrier moving from hot to cold carries its own entropy with it, the voltage is the electrical work that entropy is worth, and a material with few carriers has a large entropy per carrier for the same counting reason a dilute gas has a large entropy per particle.

It also explains the conflict. Entropy per carrier falls as the carriers are packed in, for exactly the reason entropy is a count and the count of available arrangements falls as the states fill.

The lever that is actually free #

The conductivity and the Seebeck coefficient are tied to each other. The thermal conductivity is only partly tied to either.

Heat in a solid is carried by two things: the charge carriers, and the lattice vibrations. The carrier part is fixed by the electrical conductivity, through a ratio that is nearly the same for every conductor — a relation that ties the two conductivities together with a constant containing only Boltzmann’s constant and the electron charge.

The lattice part is not tied to anything electronic. In principle a material can conduct electricity like a metal and heat like a glass, and that combination — sometimes called a phonon glass and an electron crystal — is the whole target of the modern subject.

The strategies are all ways of scattering phonons without scattering electrons, and they work because the two have different wavelengths. Alloying introduces mass disorder, which scatters short-wavelength phonons and barely affects electrons. Nanostructuring introduces boundaries at tens of nanometres, which scatter mid-wavelength phonons and are transparent to electrons with a mean free path of a few nanometres. Filling cages in a crystal with loosely bound heavy atoms gives a low-frequency resonance that absorbs long-wavelength phonons. Between them these have taken the best materials’ figure of merit from about 0.8 in 1960 to about 2 today, almost entirely by reducing $\kappa$.

There is a ceiling even so. With the lattice contribution removed entirely, the thermal conductivity is the electronic part alone, and it cancels against the electrical conductivity in the group — leaving the figure of merit equal to the Seebeck coefficient squared divided by the constant that ties them. That is a bound with no conductivity in it at all, and raising it requires a larger Seebeck coefficient, which is the thing doping fights against.

Where these are used, and why the list is short #

A device with an efficiency of ten per cent and no moving parts occupies a particular niche, and the niche is defined by the alternatives being unavailable rather than by the efficiency being good.

Spacecraft beyond Mars, where sunlight is too weak for panels and nothing can be maintained, run on the heat of a decaying isotope through thermoelectric couples — a nucleus with no clock supplying a power source that cannot be switched off and cannot fail. The Voyagers have run continuously since 1977 and their power has fallen by rather less than half, most of that from the isotope’s decay rather than from the couples.

Small refrigerators, run backwards as heat pumps, cool camera sensors and laboratory instruments where a compressor’s vibration would be intolerable and the heat load is a few watts.

Waste heat from vehicles and industry is where the large numbers are and where the technology has not established itself. A car’s exhaust carries roughly a third of the fuel’s energy at 700 to 900 kelvin, and a generator recovering a tenth of it would be worth several per cent of the fuel — which has been demonstrated repeatedly in prototypes and has not reached production, because the cost of the material per watt is high and the alternative, which is a smaller engine with a battery, improved faster.

That last comparison is the same shape as the one in the essay before it. The thermodynamic ceiling has not moved and the alternatives have.

The second law, read off the same expression #

The efficiency expression has a feature worth extracting, because it says something about irreversibility that a Carnot argument alone does not.

Carnot’s ceiling is approached by a machine that runs infinitely slowly, and the reason a real engine falls short is that it runs at a finite rate. A thermoelectric generator is different: it has no rate to slow down. There is no cycle, no piston, nothing that could be made to take longer — and it still falls short of Carnot by a factor that is nothing to do with speed.

What it falls short by is the heat that leaks straight through. A thermoelectric leg conducts heat from the hot end to the cold end whether or not any current flows, and that leakage is an irreversible flow down a temperature gradient — the plainest entropy-generating process there is. The figure of merit is a ratio of the useful power to that unavoidable leakage, and the second factor in the efficiency is what it costs.

So the device fails to be reversible for a reason that is structural rather than temporal. The same material that carries the charge carries the heat, and the second job cannot be switched off. Making the group infinite means making the leakage zero, which means the thermal conductivity zero at finite electrical conductivity, and the limit is approached and never reached.

That is a cleaner instance of a general point than any engine cycle provides. Irreversibility is usually introduced through finite rates, and it is really about processes that cannot be undone — here, heat crossing a temperature difference with nothing extracted. The thermoelectric has no rate at all and is irreversible anyway.

One parabolic band, constant properties, and no parasitic path #

The model of the material is a single parabolic band. Real thermoelectrics have several bands, non-parabolic ones, and scattering that is not by acoustic phonons alone. The model reproduces the conflict between the ingredients, the location of the optimum doping to within a factor of a few, and the magnitudes to within tens of per cent — which is enough for the argument and not enough for a design.

The properties are treated as constants. Every one of them depends on temperature, often strongly, and a device with 300 kelvin across it operates over a range in which the figure of merit may vary twofold. The expression quoted uses an average, and a serious calculation integrates the local properties along the leg — which is why segmented legs, made of different materials in series, outperform single ones.

The efficiency assumes the load is matched and that the only heat path is through the legs. A real module has parasitic conduction through the air and the housing, and contact resistances at the four junctions that can dissipate a substantial fraction of the power in a short leg. Module efficiencies are consistently below what the material’s figure of merit predicts for that reason.

And the Carnot ceiling assumes reservoirs. A thermoelectric generator on an exhaust pipe cools the exhaust as it works, so the hot side’s temperature falls along the device, and the finite-reservoir arithmetic applies.

How a module is built out of the material #

Between a good material and a working generator there is a construction, and the arithmetic of it is short enough to give and is where half the performance is lost.

A single couple is two legs — one doped so the carriers are electrons and one so they are holes — joined by a metal strap at the hot end and connected to the circuit at the cold end. The two dopings matter: their Seebeck coefficients have opposite signs, so the voltages add round the loop rather than cancelling.

One couple produces a few tens of millivolts across a few hundred kelvin, which is useless, so a module is a hundred or two couples electrically in series and thermally in parallel, sandwiched between two ceramic plates. Series for the voltage, parallel for the heat: every couple sees the same temperature difference and the voltages add.

Three things are then traded against one another.

Leg length. A long leg has a large temperature difference across it and a large resistance; a short one has a small difference and small resistance. The optimum depends on how well heat can be got into and out of the faces, and for a module bolted to a hot surface with a fan on the other side it is usually a few millimetres.

Fill factor. The fraction of the module’s area that is semiconductor rather than gap. A high fill conducts more heat through and produces more power; a low one conducts less and produces less. What decides it is whether heat is plentiful — recovering exhaust heat wants a low fill, because the heat is free and the material is not.

And contacts. Four metal-to-semiconductor junctions per couple, each with a resistance in series with the leg. In a short leg the contact resistance can be comparable to the leg’s own, which halves the power, and making a low-resistance contact that survives thermal cycling at 800 kelvin for years is the single hardest manufacturing problem in the subject.

None of those appears in the figure of merit, and between them they account for most of the difference between a material’s promise and a module’s datasheet.

A material’s figure of merit is not a device’s efficiency #

They cannot show that the figure of merit is a property of a material and the efficiency is a property of a device. Two modules of the same material differ by a factor of two in output depending on leg geometry, contact quality and how well the heat is delivered to and removed from the faces, none of which appears in the group.

Nor can they show the cost. The materials with the best figures of merit contain tellurium, which is among the rarest stable elements in the crust, or lead, or germanium. The figure of merit says what a watt costs in efficiency and not what it costs in money, and for every application above a few watts the second has decided the outcome.

And they cannot show the Peltier side. The same couple run backwards pumps heat rather than producing power, and the same group governs it — the coefficient of performance of a thermoelectric cooler has the same square root in it. That the two are one device is the deepest statement in the subject, and nothing in an efficiency curve says so.

Still open: whether the figure of merit can be doubled again #

The best reported values have roughly doubled since 1990, from about one to about two, essentially all of it by suppressing lattice conduction. That route has a floor: the lattice conductivity cannot fall below the value a completely disordered solid has, which several of the best materials are already near.

Beyond it, the options require raising the electronic part. Proposals include band structures engineered to have many degenerate valleys, which raises the conductivity without lowering the Seebeck coefficient in the usual way; sharp features in the density of states near the Fermi level, which raise the Seebeck coefficient by making carriers above and below more different from one another; and low-dimensional systems, where the density of states can be shaped deliberately.

Each has produced encouraging measurements and none has produced a material that is reproducibly better in a module. The gap between a record figure of merit measured on a small sample in one direction at one temperature and a device that runs for years is where most of the subject’s disappointments have been, and it is a gap about materials engineering rather than about the physics in the expression.

The habit worth carrying away is about figures of merit. A single dimensionless group that governs a device’s performance is worth finding, and its usefulness is exactly proportional to how independent its ingredients are. This one is famous because it is genuinely the only material quantity in the efficiency, and it is difficult because its three ingredients share a carrier concentration — so the group is not three knobs but one, plus a lattice conductivity that is free and has a floor.