Thermoelectric figure of merit ($ZT$)

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.

For a constant-property thermoelectric module consisting of two material legs, let ${\alpha }_{np}=S_p-S_n$ be their Seebeck coefficient difference, $R$ their total series electrical resistance, and $K$ their total parallel thermal conductance. The couple's thermoelectric figure of merit isIts units are ${\mathrm{K}}^{-1}$; multiply by an absolute temperature to form a dimensionless figure. This expression assumes that contact losses are absent or consistently included in $R$ and $K$.

For equal leg lengths $l$ and areas $A$, with equal electrical resistivity $\rho$ and thermal conductivity $\kappa$, one has $R=2\rho l/A$ and $K=2\kappa A/l$. Consequently $Z_{\mathrm{pair}}={\alpha }_{np}^2/(4\rho \kappa )$. The factor four follows from combining two series resistances and two parallel conductances; it disappears only if the symbols have already been defined as pair properties. With opposite equal Seebeck coefficients $S_p=-S_n=S$, the result reduces to $S^2/(\rho \kappa )$.

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 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]