Determining the Lorenz number for thermoelectric figure-of-merit assessment when the Wiedemann–Franz law fails

The dimensionless figure of merit $zT = S^2\sigma T/\kappa$ that rates every thermoelectric material depends on separating the measured total thermal conductivity $\kappa$ into an electronic part $\kappa_e$ and a lattice part $\kappa_L$, which in practice is done through the Wiedemann–Franz relation $\kappa_e = L\sigma T$ with an assumed Lorenz number $L$. In real thermoelectrics the assumption underlying this separation — that charge and heat are carried by the same transport mechanism — fails in several identifiable regimes (non-parabolic and multiband electronic structure, inelastic or strongly energy-dependent scattering, strong electron–phonon coupling, phonon drag, bipolar conduction), and no prescription for $L$ in those regimes is validated. The problem is to supply a measurable or first-principles prescription that yields $L$ for real thermoelectric materials in Wiedemann–Franz-violating regimes.

Background

A thermoelectric generator or cooler converts heat directly into electrical power (or vice versa) through the Seebeck effect, in which a temperature gradient across a material drives a voltage. The conversion efficiency of a material at temperature $T$ is governed by the dimensionless figure of merit $zT = S^2\sigma T/\kappa$, where $S$ is the Seebeck coefficient (the voltage generated per unit temperature difference), $\sigma$ is the electrical conductivity, and $\kappa$ is the total thermal conductivity. Because heat is carried both by lattice vibrations (phonons, contributing the lattice thermal conductivity $\kappa_L$) and by the mobile charge carriers (contributing the electronic thermal conductivity $\kappa_e$), any analysis of why a material performs well or badly requires splitting $\kappa = \kappa_e + \kappa_L$ into the two parts. This separation is almost never done by direct measurement: instead $\kappa_e$ is inferred from $\sigma$ using the Wiedemann–Franz law, an empirical relation dating to 1853, in the form $\kappa_e = L\sigma T$, where $L$ is the Lorenz number. In simple metals at low temperature, $L$ takes the universal Sommerfeld value $L_0 = (\pi^2/3)(k_B/e)^2 \approx 2.44\times10^{-8}\ \mathrm{W\Omega K^{-2}}$ ($k_B$ is Boltzmann's constant and $e$ the elementary charge). Thermoelectric semiconductors, however, are doped near the boundary between degenerate carrier statistics (Fermi level inside a band, metal-like) and non-degenerate statistics (Fermi level in the gap, classical Boltzmann occupation), where kinetic theory gives different values (e.g. $L = 2(k_B/e)^2$ for acoustic-phonon scattering with a constant mean free path), and $L$ then depends on temperature, doping, band structure, and the scattering mechanism. The prevailing practical prescription is an empirical fit by Kim, Gibbs, Tang, Wang, and Snyder (APL Materials 3, 041506, 2015, DOI 10.1063/1.4908244) that estimates $L$ from the measured Seebeck coefficient, $L \approx 1.5 + \exp(-|S|/116)$ with $S$ in $\mu\mathrm{V/K}$ and $L$ in $10^{-8}\ \mathrm{W\Omega K^{-2}}$, derived from a single parabolic band model (a single quadratic energy band) with acoustic-phonon scattering; its authors state accuracy of about 5% under those model assumptions but deviations up to about 20% for real PbTe-family and SiGe alloys. First-principles work (X. Wang, Askarpour, Maassen, and Lundstrom, J. Appl. Phys. 123, 055104, 2018, DOI 10.1063/1.5009939) shows $L$ in complex multiband thermoelectrics is strongly scattering-dependent, and ab initio electron–phonon calculations (e.g. Chen, Ma, and Li for tungsten, Phys. Rev. B 99, 020305(R), 2019, DOI 10.1103/PhysRevB.99.020305) have computed $L$ only for limited cases. Wang, Palma, Shang, Chen, and Liu (arXiv:2010.00664, published in Computational Materials Science 249, 113647, 2025, DOI 10.1016/j.commatsci.2024.113647) proposed a new route: $L$ is the ratio of two thermodynamic quantities, the electronic heat capacity $c_{el}$ and the electrochemical capacitance $c_N$ (the change in stored charge with electrochemical potential), via $LT = c_{el}/c_N$, so that $L$ can be computed from the electronic density of states alone, without solving the Boltzmann transport equation — the standard semiclassical kinetic equation for the carrier distribution under fields and gradients, which requires detailed knowledge of the scattering. The derivation assumes that electronic heat conduction and electrical conduction share the same transport mechanism ($\sigma = c_N\alpha_{el}$, with $\alpha_{el}$ the electronic thermal diffusivity); in the non-degenerate limit the prescription corresponds to a scattering energy-dependence $\tau_m(\varepsilon) \propto \varepsilon^{-1}$ (a 'constant phase space volume' picture, with $\tau_m$ the carrier relaxation time and $\varepsilon$ the carrier energy), and the method yields only the electronic part of $zT$, leaving $\kappa_L$ to be obtained separately. The route is validated in the paper against model limits, twelve simple metals, and Si$_{0.8}$Ge$_{0.2}$, and its conclusion claims a simplification over conventional Boltzmann transport theory; validity for materials with complex, inelastic, or non-Fermi-liquid scattering is not demonstrated there. Meanwhile, experiments keep uncovering regimes where the Wiedemann–Franz separation demonstrably fails: pronounced downward or upward deviations of $L$ from $L_0$ in semimetals, topological and correlated materials (e.g. WP2, WTe2, NdAlSi), hopping-type organic thermoelectrics (where charge moves by incoherent tunneling between localized molecular states), and the heavily doped alloy Bi$_{0.4}$Sb$_{1.6}$Te$_3$, where a 2025 direct field-dependent measurement revealed an anomalously low Lorenz number shaped by inelastic (energy-exchanging) electron–electron and electron–optical-phonon scattering that conventional transport models miss. In quantizing magnetic fields (fields strong enough that the carriers' transverse motion is frozen into discrete Landau levels), where Skinner and Fu (Sci. Adv. 4, eaat2621, 2018, DOI 10.1126/sciadv.aat2621) predicted large nonsaturating thermopower in Dirac and Weyl semimetals (materials with linear electron dispersion), transport is tensorial and the figure of merit is instead evaluated from the total measured thermal conductivity, outside the scalar Wiedemann–Franz separation considered here. Because essentially every reported $\kappa_L$ and $zT$ value in the thermoelectric literature passes through an assumed $L$, the unresolved question — what prescription actually determines $L$ in these regimes — bears on the reliability of the field's central performance metric.

Problem Statement

Determine whether a prescription exists — based either on independently measurable quantities or on a first-principles (e.g. electronic-structure plus scattering) calculation — that determines the Lorenz number $L$ entering the Wiedemann–Franz separation $\kappa_e = L\sigma T$, and through it the electronic contribution to the figure of merit $zT_{\mathrm{el}} = S^2/L$ and the lattice contribution $\kappa_L = \kappa - L\sigma T$, for real thermoelectric materials in regimes where the assumption that charge and heat transport share the same mechanism fails; if such a prescription exists, supply it, and if in a listed regime no prescription built from independently measurable or first-principles (electronic-structure plus scattering) inputs determines a scalar $L$ with $\kappa_e = L\sigma T$, state that explicitly and justify it. These regimes are: non-parabolic and multiband band structures (band edges that are not single quadratic bands); inelastic or strongly energy-dependent carrier scattering (electron–electron, electron–optical-phonon; scattering that exchanges energy between carriers and the scattering channel); strong electron–phonon coupling; phonon drag (transport augmented by nonequilibrium phonons dragging charge carriers); and bipolar conduction (simultaneous electron and hole transport, which adds an electron–hole pair contribution to heat flow). The existence question is posed separately for each of these five regimes: for each regime, either a prescription built from independently measurable or first-principles (electronic-structure plus scattering) inputs determines a scalar $L$ with $\kappa_e = L\sigma T$, or no such prescription exists. Observing or citing that the scalar Wiedemann–Franz form breaks down in a regime does not by itself resolve that regime, a per-regime verdict resolves that regime, and a single prescription need not span all five regimes.

References

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