Materials Photovoltaic2 Recomb

Recombination and its cost

In a silicon solar cell, absorbed sunlight creates pairs of mobile charges: an electron and a hole. Before they can be collected as current, many pairs recombine, and most of that recombination is non-radiative: the energy is lost as heat, not emitted as light. This is the Shockley-Read-Hall process, named for the researchers who described it in 1952, and it dominates efficiency losses in typical silicon cells. The standard textbook picture is a trap state in the crystal that captures an electron, then a hole, and lets the pair annihilate without producing a photon.

The Recognition Science framework models this with a cost, a nonnegative number that measures how far a recognition event is from a perfect match. The framework's central result, proved in its machine-checked library of formal theorems, is that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. The module Photovoltaic2_Recomb applies this by defining a domain cost as J(m/e), where m and e are two quantities to be compared. The library proves three general facts: the cost is zero when m equals e, it is nonnegative for positive inputs, and the threshold phi - 3/2 is positive, where phi is the golden ratio. These are facts about the cost function itself, not yet about any specific physical definition of m and e.

What the module does not do is equally important. The definition of domainCost as J(m/e) contains no reference to traps, silicon, or recombination rates. The docstring records the intended research direction: that the Shockley-Read-Hall rate might equal J(phi) times the radiative rate at a typical trap density, giving a maximum conversion efficiency of about 88.2 percent. That is a research note, not a theorem. To turn the general facts into a statement about solar cells, someone must define m and e in the cell's own terms, for example as the trap capture rate and the radiative emission rate. Until then, the module proves only the universal properties of the cost, shared verbatim with 2383 sibling modules.

The practical consequence is a clean separation between what is proved and what is proposed. The proved part gives any future model a reliable foundation: whatever m and e are chosen, the cost will be zero at equality and nonnegative everywhere else. The proposed part, the link to real recombination physics, remains open. That is the honest state of the module, and it is a useful state, because it tells a researcher exactly where the next definition must go.

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)

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

What this page does not claim

The module does not prove any specific claim about silicon solar cells or Shockley-Read-Hall recombination rates. The research note about SRH rate equaling J(phi) times radiative rate is not a theorem. The 88.2 percent efficiency figure is a stated direction, not a derived result.

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/Photovoltaic2_Recomb.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: