Due to their all-solid-state operation, thermoelectric coolers based on the Peltier effect have many positive attributes: they require no regular maintenance, produce no vibrations, emit no greenhouse gasses, offer a high specific cooling capacity, and scale well to low power and small size. However, their efficiency is competitive only in low-power applications [1], which prevents their large-scale adoption. Thus, their use is restricted to cases where reliability and quiet operation are prioritized over energy efficiency.
Assuming optimal heat exchangers, the cooling performance of Peltier modules is limited only by the quality of the thermoelectric materials used to make them. This quality is quantified by a dimensionless figure of merit ZT = α2σT/κ, with thermal conductivity κ, electrical conductivity σ, and thermopower α. Conventional Peltier modules are made of tetradymite [2] materials (alloys of general formula (Bi,Sb)2(Te,Se)3 that crystallize in the \( \overline{3}m \) crystallographic point group) with thermal conductivity ~ 1(W/m/K) and ZT of order 1.
In order to improve the ZT of the materials, research in tetradymites has focused on reducing the lattice contributions to thermal conductivity by introducing nanostructures. Nanocomposites based on tetradymite alloys result in large ZT improvements, which is credited to phonon scattering [3]. Similar approaches based on phonon scattering have been effective in other materials, resulting in maximum ZT ~ 1.2 in n-type materials [4] and ZT ~ 1.4 in p-type materials [5]. The various mechanisms that lead to the remarkable thermoelectric performance of these alloys are reviewed in Ref. [2].
In this work, we take a different approach: we instead consider composites with micro-scale inclusions that interact with the host matrix as a general method to improve functionality of conventional materials. Some previous work has provided empirical evidence of improvements in the thermoelectric performance of composites over uniform materials [6]. However, we are not aware of any that provide a physical mechanism for the improvement, as we do here. Instead, the work has simply recounted the observation of a reduction in thermal conductivity that is not accompanied by an equivalent reduction in electrical conductivity, but they do not hypothesize or prove why that observation holds. Here, we make the explicit diagnostic: the reduction in thermal conductivity absent an equivalent reduction in electrical conductivity results from local doping of the matrix around an inserted phase (here, beads). We prove mathematically, using the effective medium theory, that this is possible. Then we proceed to prove it experimentally by contrasting the behavior of the composite with Ag-coated beads, which dope the sample uniformly, with that of the composite with Pb-doped beads, which dope it locally. We apply this new approach to p-type tetradymite alloys already optimized as described above, and show how this new approach yields an increase in ZT greater than 10%.
Composites of non-interacting thermoelectric materials with high electrical conductivity and low thermal conductivity have been tried, but have not produced increases in ZT. By non-interacting composites, we mean composites that contain inclusions of a component B in a matrix A where the physical properties (electrical and thermal conductivity, thermopower) of A are not modified by the presence of B, or vice-versa. The effective medium theory explains this result. A formal mathematical demonstration by D. Bergman and collaborators shows that for any combination of non-interacting materials, ZT cannot exceed the greatest value of any single constituent [7]. While the power factor α2σ can be enhanced, an accompanying increase in thermal conductivity will always limit the figure of merit [8]. The demonstration assumes that transport coefficients describing a composite are equal to the mean transport coefficients of each constituent weighted by their volume fraction φ. For example, Fig. 1 shows the application of this effective medium theory to three hypothetical cases based on a host material (A) that is a p-type tetradymite host with a thermopower α = 240 μV/K, an electrical conductivity σ = 60,000 S/m and a thermal conductivity κ = 1.2 W/m K. In case (a), perfectly electrically and thermally insulating spherical inclusions (component B) are added (Fig. 1a). The calculated relative change of the effective α,σ, and κ of the mixture as a function of φ are shown: at values of φ < 0.3, below the percolation threshold, α(φ) barely changes, while the thermal and electrical conductivity decrease in such a way that they offset each other and ZT is unaffected. Similarly, Fig. 1b shows how adding inclusions of a metal (component B in this case) with α = 0 μV/K and 100 times higher electrical and thermal conductivity than the host material results in values for the effective medium α(φ) that also barely changes, and once again, σ(φ) and κ(φ) that compensate each other. In this case the model predicts that the power factor will rise, but the thermal conductivity will rise congruently, again resulting in a ZT that is unaffected.
The only cases where the limits of the effective medium theory are exceeded are those where a guest material B alters the properties of the host A, e.g., by introducing new scattering mechanisms [9]. This is what we call interacting composites. Here, we experimentally explore the theoretical solution offered in Fig. 1c, describing the results of the effective medium theory applied to a composite based on insulating beads, but with a variation of the electrical conductivity at interfaces. The model is based on the same formulas but considers three constituents: a thermoelectric host, insulating inclusions, and narrow, electrically conductive channels where they meet. Assuming voids that are much more thermally and electrically insulating than the host but surrounded by channels that are much more electrically conductive than the host, the model in Fig. 1c predicts an increase in the figure of merit. The result in Fig. 1c shows how the power factor is expected to decrease slightly, but that effect is accompanied by an even larger loss in thermal conductivity.