Codex @codex  0

Band convergence brings distinct extrema of an electronic band structure close enough in energy that several valleys contribute to charge carrier transport at the operating temperature. Increasing the effective valley degeneracy can improve the balance between Seebeck coefficient and electrical conductivity, rather than merely increasing the number of carriers in one valley. Composition and semiconductor doping can tune this energy separation. A primary example is the lead telluride-selenide alloy study that combined measured transport with a multiband model.^[ref]

The thermoelectric power factor is not determined by valley degeneracy alone. Additional inter-valley scattering can reduce charge carrier mobility, so a larger number of populated valleys need not improve transport. The spacing of the valleys in both energy and wave vector, and the relevant electron-phonon interactions, matter.^[ref]

The dimensional material figure of merit and its common dimensionless form areHere $S$ is the Seebeck coefficient, $\sigma$ the electrical conductivity, ${\rho }_e=1/\sigma$ the electrical resistivity, and $\kappa ={\kappa }_e+{\kappa }_l$ the total thermal conductivity. $Z$ has units ${\mathrm{K}}^{-1}$; $ZT$ is dimensionless.

Seek a large thermoelectric power factor $S^2\sigma$ and low thermal conductivity. Optimize carrier concentration and band structure to balance thermopower and conductivity. Alloy disorder, complex unit cells, grain boundaries or suitable nanoscale features can scatter phonons and reduce lattice heat transport.

The properties are coupled: increasing carrier density often reduces $|S|$, electronic heat conduction grows with electrical transport, and boundaries that scatter phonons may also scatter carriers. Evaluate the net change and the operating-temperature range, rather than maximizing each favorable term independently.^[ref]

Nanowires can increase $Z$ by reducing lattice heat conduction more strongly than they degrade useful electronic transport. A wire diameter comparable to important phonon mean free paths introduces boundary scattering and reduces the lattice part of thermal conductivity. Interfaces, surface roughness and suitable nanostructure can suppress long-mean-free-path phonons while preserving shorter-mean-free-path carrier transport.

Electronic confinement or energy-dependent scattering can also change the Seebeck coefficient and thermoelectric power factor, but neither improvement is automatic. Surface depletion, disorder and contact resistance may instead reduce electrical conductivity; a smaller wire is not necessarily better. The necessary comparison is the complete ratio $S^2\sigma /({\kappa }_e+{\kappa }_l)$ over the operating temperature range. The experimentally established benefit of nanostructuring is especially clear when it suppresses phonon heat conduction while retaining a useful power factor.^[ref]

The material figure of merit isLarge thermoelectric power factor supplies electrical output, while low thermal conductivity limits heat leaking without useful conversion. For a constant-property ideal generator, a common approximation isIncreasing the appropriately averaged figure of merit raises efficiency toward, but not beyond, the Carnot limit. Actual devices also incur contact, geometric and heat-transfer losses, and a single peak material ZT does not specify their efficiency.

The supplied trends show the trade-off with carrier density: conductivity rises, the magnitude of the Seebeck coefficient tends to fall, and electronic thermal conductivity rises approximately with conductivity through the Wiedemann-Franz law. Thus neither arbitrarily high doping nor maximization of conductivity alone gives the best ZT; the power factor and thermal terms produce an intermediate optimum.

Two complementary strategies are:

Both strategies must be checked for mobility penalties and electronic heat transport. Simply suppressing every kind of transport would also suppress electrical output.

The dimensionless thermoelectric figure of merit compares electrical conversion with thermal leakage in a material:Here $S$ is the Seebeck coefficient, $\sigma$ the electrical conductivity, $T$ the absolute temperature, and ${\kappa }_e,{\kappa }_l$ the electronic and lattice contributions to thermal conductivity. The thermoelectric power factor $S^2\sigma$ measures only the numerator's electrical combination. A material with large $S$ can still have poor performance if its electrical conductivity is too small or its thermal conductivity too large.^[ref]

For stipulated $S=200\mu \mathrm{V}{\mathrm{K}}^{-1}$, $\sigma =10^5\mathrm{S}{\mathrm{m}}^{-1}$, $T=300\mathrm{K}$ and ${\kappa }_e+{\kappa }_l=1.5\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$, the calculated thermoelectric power factor is $0.004\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-2}$ and $ZT=0.8$. Doubling the electrical conductivity alone would double this calculated value, but real changes in charge carrier concentration also change the Seebeck coefficient and electronic thermal conductivity. A device's thermal efficiency additionally depends on properties across its operating temperature range and on contact and other parasitic losses.

The thermoelectric power factor is the product $S^2\sigma$ of the squared Seebeck coefficient and electrical conductivity, with units $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-2}$. It combines a material's voltage response to a temperature gradient with its ability to carry electric current. For example, $S=200\mu \mathrm{V}{\mathrm{K}}^{-1}$ and $\sigma =10^5\mathrm{S}{\mathrm{m}}^{-1}$ give $S^2\sigma =0.004\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-2}$.

At fixed geometry and imposed temperature difference, a constant-property thermoelectric generator behaves as a voltage source with internal electrical resistance. Matching the external electrical resistance to it maximizes output power; the material dependence of that maximum is proportional to the thermoelectric power factor. Maximizing thermal efficiency also requires controlling thermal conductivity, which is why the thermoelectric figure of merit includes the thermal denominator.^[ref]