Materials Thermoelectric Figure Merit

The figure of merit and its limits

Thermoelectric materials convert temperature differences into voltage, and vice versa. The standard measure of how well a material does this is the dimensionless figure of merit, ZT = S²σT/κ, where S is the Seebeck coefficient, σ the electrical conductivity, T the absolute temperature, and κ the thermal conductivity. A good thermoelectric needs high electrical output and low heat leakage, which is why the best known materials reach ZT values around 2 to 3.

The figure of merit was introduced by Edmund Altenkirch in 1909, and it remains the central design target in thermoelectric research. The trade-off is sharp: raising S usually lowers σ, and lowering κ often degrades σ. Most engineering effort goes into nanostructuring and alloying to decouple these properties, with bismuth telluride and lead telluride as the classic workhorse compounds.

In Recognition Science, a framework that treats physical structure as forced by the cost of recognition events, the figure of merit takes a specific structural form. The module defines a cost function domainCost(m, e) = Jcost(m/e), where Jcost is the recognition cost function and m and e are positive real parameters. The three proved facts are that this cost vanishes when m equals e, that it is nonnegative for all positive inputs, and that the threshold phi - 3/2 is positive. These are general properties of the cost function, not results about any particular thermoelectric material.

The research note attached to the module records an intended target: a structural value for ZT at optimal doping, phi - 1 ≈ 0.618, derived from the golden ratio phi. That value is not proved in the module. The module itself contains no definition of m and e in thermoelectric terms, so it establishes nothing specific to thermoelectrics. What it does establish is a template: if one could define m and e as, say, charge and energy scales in a thermoelectric, the cost function's properties would apply.

The practical consequence is a boundary on what the framework can currently claim. The three theorems are real and machine-checked, but they are universal facts about Jcost, shared verbatim with 2383 sibling modules. The thermoelectric prediction, ZT ≈ 0.618, remains a research hypothesis awaiting a physical definition of its parameters.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The module does not prove any thermoelectric-specific result, including the ZT value of 0.618. This page does not claim that the recognition cost function has been measured in any material. The framework does not derive the figure of merit from first principles of transport physics.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Materials/ThermoelectricFigure_Merit.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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