Materials Semiconductor Dopant Types From Config Dim Dopant Type

The five dopant categories

In a silicon-type semiconductor, an impurity atom can alter electrical behavior in several distinct ways. Doping with a group-V element such as phosphorus, arsenic, or antimony adds an extra electron and creates an n-type region; doping with a group-III element such as boron, aluminum, or gallium leaves a missing electron, a hole, and creates p-type material. Beyond these two familiar families, real devices also rely on deep-level impurities that trap carriers, on compensating dopants that balance existing carriers, and on transition-metal atoms that act as scattering centers. These five roles are the canonical categories of dopant behavior in silicon technology.

The Recognition Science framework's machine-checked library of formal theorems encodes exactly these five categories in a single declaration called DopantType. The declaration is an inductive type, which means it lists five distinct named cases: groupVDonor, groupIIIAcceptor, deepLevel, compensating, and transitionMetal. A companion theorem proves that the number of categories is exactly five, and a certificate structure packages that count for use elsewhere in the library. The whole file carries no unproved assumptions and no added axioms; the count is checked by direct computation.

In Recognition Science, this result connects to a broader pattern. The framework derives physical structure from a forced cost function, and one of its derived constants is the configurational dimension D = 5 for silicon-type semiconductors. The five dopant categories are presented as the five canonical roles that follow from that dimension. That identification is a definitional choice within the framework, not a theorem about measured semiconductor physics. The library proves the internal consistency of the five-way classification, but it does not prove that real silicon has exactly five dopant types.

The practical value of the declaration is precision. A downstream proof or simulation can refer to a named category instead of an informal phrase, and the machine-checked count guarantees that no sixth category has been silently added. The declaration does not claim that these five categories exhaust all possible impurity behaviors in every material, nor does it assign specific energy levels, cross sections, or concentrations to any category. It fixes a vocabulary and a count, and leaves the physics of each category to experiment.

inductive DopantType where
  | groupVDonor
  | groupIIIAcceptor
  | deepLevel
  | compensating
  | transitionMetal
  deriving DecidableEq, Repr, BEq, Fintype

theorem dopantType_count : Fintype.card DopantType = 5 := by decide

def semiconductorDopantCert : SemiconductorDopantCert where
  five_types := dopantType_count

What this page does not claim

The declaration does not prove that real silicon has exactly five dopant types. The declaration assigns no energy levels, cross sections, or concentrations to any category. The five categories are not claimed to exhaust all impurity behaviors in every semiconductor material.

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

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