Inherent figure of merit of thermoelectric composites and under-load figure-of-merit limits of segmented thermoelectric legs

For a homogeneous thermoelectric material the dimensionless figure of merit $ZT=\sigma\alpha^2T/\kappa$ uniquely fixes the efficiency of energy conversion, but for macroscopically inhomogeneous thermoelectrics this uniqueness is lost and the achievable figure of merit is set by how inhomogeneity and nonlinear coupled transport renormalize the effective coefficients. This problem asks for a quantitative theory of those limits in two linked settings. (i) For two-phase composites: determine the concentration dependence $\tilde Z(p)$ of the inherent figure of merit — the dimensionless parameter, known exactly only for two-dimensional self-dual media at the percolation threshold, that measures thermal-to-electric conversion internal to the composite and renormalizes the effective electrical and thermal conductivities — for general phase concentrations and property contrasts, extend it to three dimensions, and verify it beyond the self-consistent-field approximation. (ii) For segmented legs (inhomogeneous on the device scale): generalize the conditions under which a generator under operating load performs better than predicted by its effective transport coefficients beyond the double-segmented, constant-property case in which they are known. Both cores remain open: the composite thread has advanced only within mean-field approximations, and the segmented under-load conditions have not been generalized or tested since 2017.

Background

A thermoelectric converter exploits the Seebeck effect: a temperature difference across a semiconductor or metal generates an electromotive force proportional to the Seebeck coefficient $\alpha$. The local transport of charge and heat in a thermoelectric is described by the Onsager–de Groot–Callen theory, in which electric current and heat current are coupled linear forms in the electric field and the temperature gradient; the coupling produces Peltier heat (liberated or absorbed where current crosses a junction between materials with different $\alpha$) and Joule heat. For a homogeneous material with electrical conductivity $\sigma$, thermal conductivity $\kappa$, and Seebeck coefficient $\alpha$ at temperature $T$, essentially all performance figures — generator efficiency, maximum power, cooling characteristics — are governed by the single dimensionless combination $ZT=\sigma\alpha^2T/\kappa$, Ioffe's figure of merit.

Modern thermoelectric materials are rarely homogeneous: they are composites of two or more phases (e.g. nanoinclusions in a matrix) or segmented legs in which different materials are stacked along the temperature gradient to match each material to its optimal temperature window. For a two-phase, macroscopically inhomogeneous medium the volume-averaged response is described by effective kinetic coefficients $\sigma_e$, $\kappa_e$, $\alpha_e$, which are nonlinear functionals of the phase constants and geometry once thermoelectric coupling is included: the Peltier and Joule heat released inside the material feeds back on the temperature field and thereby on the measured effective conductivities. In linear response (neglecting this feedback), the composite theory of Bergman and Levy (1991) and Bergman and Fel (1999) shows that the effective figure of merit of a composite cannot exceed the largest figure of merit of its constituents, even though the power factor $\sigma\alpha^2$ can be enhanced — the conventional bound referenced throughout this problem.

Snarskii, Adzhigai, and Bezsudnov (2005/2006) identified a further composite-specific parameter that has no homogeneous analogue. For two-dimensional self-dual media — media invariant under interchange of the two phases, a class that includes a two-dimensional random medium exactly at its percolation threshold $p_c=1/2$, where percolation denotes the emergence of a connected path of one phase — the exact effective kinetic coefficients are known at arbitrary contrast by Dykhne duality (an exact two-dimensional symmetry that relates any two-phase medium to the medium with its two phases interchanged, and thereby fixes the effective coefficients of statistically phase-symmetric media in closed form), and they factorize as $\sigma_e^{nlin}=\sigma_e^{lin}/(1+\tilde Z T)$ and $\kappa_e^{nlin}=\kappa_e^{lin}(1+\tilde Z T)$, where the superscripts distinguish coefficients computed with and without thermoelectric feedback. The dimensionless $\tilde Z T$, built from the differences of the phase constants and proportional to $(\alpha_1-\alpha_2)^2$, was named the inherent figure of merit: it quantifies conversion of heat into electricity dissipated inside the composite, rather than delivered to an external load, and its growth lowers the effective figure of merit $Z_e$. Away from the self-dual point, effective coefficients are only accessible through the self-consistent field approximation (the Bruggeman-type effective-medium approximation, which treats each inclusion as embedded in the averaged medium); the resulting nonlinear self-consistent equations cannot be solved in closed form, and when the phases obey the Wiedemann–Franz relation $\sigma_1/\kappa_1=\sigma_2/\kappa_2$ (equal ratios of electrical to thermal conductivity) the extraction of $\tilde Z(p)$ is ambiguous up to the choice of an averaging function $M(a,b)$ satisfying Kolmogorov's axioms for a mean. The source treats the verification of $\tilde Z$ beyond the self-consistent-field approximation as open: $\tilde Z$'s predictions have not been checked against fully resolved numerical simulations of the nonlinear thermoelectric transport equations in inhomogeneous media, and the source reserves that verification for later work.

The segmented counterpart of the question concerns legs that are inhomogeneous on the device scale. The standard design tool for segmentation is the compatibility framework of Snyder and Ursell (2003), which judges which materials combine efficiently under the constant property model (CPM, all material constants uniform within each segment). It is formulated essentially at open circuit. Under realistic operating load (finite current), Angst and Wolf (2017) showed within the Onsager–de Groot–Callen theory that a double-segmented generator usually performs worse than predicted by its effective transport coefficients, because the current and the interface Peltier effect self-organize the temperature profile so as to shrink the temperature drop across the segment with the larger $|\alpha|$; they nevertheless derived conditions, with Joule heating included, under which the opposite — enhanced power — occurs. The conditions under which an inhomogeneous leg under load performs better than predicted by its effective transport coefficients are thereby established only for the double-segmented, constant-property, one-dimensional case.

Problem Statement

Determine the limits that macroscopic inhomogeneity and nonlinear coupled thermoelectric transport impose on the achievable figure of merit of thermoelectric structures beyond the homogeneous-material $ZT$. The problem has two linked components sharing one objective — a quantitative theory of the effective/inherent figure of merit of inhomogeneous thermoelectric media — and a completion boundary set by the open questions stated in the source literature.

(1) Composite half (Snarskii, Adzhigai, Bezsudnov, 2005/2006): for a two-phase thermoelectric composite, determine the concentration dependence $\tilde Z(p)$ of the inherent figure of merit — the dimensionless parameter that quantifies thermal-to-electric energy conversion internal to the composite and renormalizes the effective electrical and thermal conductivities via relations of the form $\sigma_e^{nlin}=\sigma_e^{lin}/(1+\tilde Z T)$, $\kappa_e^{nlin}=\kappa_e^{lin}(1+\tilde Z T)$ at the two-dimensional self-dual point — for general phase concentrations $p\neq 1/2$ and general contrasts of the local thermoelectric constants (beyond the Wiedemann-Franz-constrained case), resolving or bounding the ambiguity introduced by the choice of averaging function $M(a,b)$; establish whether a well-defined three-dimensional analogue of $\tilde Z$ exists and determine it, given that no self-dual exact results are available in three dimensions; and verify and quantify $\tilde Z$ beyond the self-consistent-field approximation.

(2) Segmented half (Angst and Wolf, 2017): for thermoelectric legs that are inhomogeneous on the device scale, generalize the conditions under which a generator under operating load (finite current, Joule and interface Peltier heating included) performs better than predicted by its effective transport coefficients — the enhanced-power regime derived so far only for double-segmented, constant-property, one-dimensional generators, which otherwise usually perform worse than that prediction — to arbitrary segmentation or continuously graded legs, temperature-dependent material properties, and non-one-dimensional geometries.

The two components and the parts enumerated above are independent: no part is a prerequisite for any other. Any general treatment of these limits must reduce to the known special cases — the exact two-dimensional self-dual results at $p_c=1/2$ and the double-segmented constant-property conditions.

References

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