Electric potential difference → Electrical work

Matrix context

D = disequilibrium row · C = coupling family column. A cell reads the state of the atlas for that pair.

D.06 × C.01 · Electric potential difference × Thermoelectricfamily: Thermoelectric

Typed bounds on this route

Every limit the atlas holds for this route, as constraint records: reached through bounded_by claims on the route's own entities. Hard bounds (an upper bound or a formula) decide the thermodynamic-bound check above; benchmarks, constitutive relations and resource limits are listed and never decisive. No prose limit exists anywhere (pass 42).

• Carnot limit · formula-bound · hard, applies to this route
  η ≤ 1 − T_c/T_h for any heat engine between reservoirs at T_h and T_c

Steps

1. 01
   Electric potential difference —drives→ Ettingshausen effectestablishedclaim record

   electrical → thermal

   • a current in a conductor with a perpendicular magnetic field

   solid · conductor required · magnetic field required

   refs [1] [2]
2. 02
   Ettingshausen effect —produces→ Temperature gradientestablishedclaim record

   thermal → thermal

   refs [2] [3]
3. 03
   Temperature gradient —drives→ Heat conductionestablishedclaim record

   thermal → thermal

   q = −κ · ∇T · thermal conductivity κ [W/(m K)]

   refs [4]
4. 04
   Heat conduction —produces→ Phononsestablishedclaim record

   thermal → phonon

   • in an electrically insulating or semiconducting solid the heat current is carried by phonons

   solid

   refs [5]
5. 05
   Phonons —drives→ Phonon dragestablishedclaim record

   phonon → electrical · loss to thermal

   • a phonon heat current in a semiconductor at low temperature

   solid · semiconductor required · cryogenic

   Phonon drag is a component of the same conductor's Seebeck coefficient, not a second converter in series with it: Yalamarthy et al. 2019 attribute up to 32 % (300 K) and 88 % (50 K) of the Seebeck coefficient of a thick AlGaN/GaN 2DEG to drag. The compiler therefore treats the heat-conduction → phonon-drag spelling as representation-equivalent to the thermoelectric generator.

   refs [6] [3] [7]
6. 06
   Phonon drag —produces→ Charge carriersestablishedclaim record

   electrical → electrical

   refs [6] [3]
7. 07
   Charge carriers —converts into→ Electrical workestablishedclaim record

   electrical → electrical · loss to thermal

   • carriers driven through an external load at the potential difference the source maintains

   refs [8]

Evidence for the complete composition · 0

No source on record for this exact composition. The sources below support the individual steps, not the route as a whole.

Evidence for the constituent relations · 8

1. [1]
   A. von Ettingshausen, W. Nernst. Ueber das Auftreten electromotorischer Kräfte in Metallplatten, welche von einem Wärmestrome durchflossen werden und sich im magnetischen Felde befinden (1886)

   Annalen der Physik 265, 343–347 · doi:10.1002/andp.18862651010 · crossref ✓ 2026-09-19 · record
2. [2]
   H. B. Callen. The application of Onsager's reciprocal relations to thermoelectric, thermomagnetic, and galvanomagnetic effects (1948)

   Physical Review 73, 1349 · doi:10.1103/PhysRev.73.1349 · crossref ✓ 2026-09-19 · record
3. [3]
   H. J. Goldsmid. Introduction to Thermoelectricity (2nd ed.) (2016)

   Springer Series in Materials Science 121 · doi:10.1007/978-3-662-49256-7 · crossref ✓ 2026-09-19 · record
4. [4]
   A. Bejan. Advanced Engineering Thermodynamics (4th ed.) (2016)

   Wiley · doi:10.1002/9781119245964 · crossref ✓ 2026-09-19 · record
5. [5]
   C. Kittel. Introduction to Solid State Physics (8th ed.) (2005)

   Wiley · book, no DOI · record
6. [6]
   C. Herring. Theory of the thermoelectric power of semiconductors (1954)

   Physical Review 96, 1163 · doi:10.1103/PhysRev.96.1163 · crossref ✓ 2026-09-19 · record
7. [7]
   A. S. Yalamarthy et al.. Significant Phonon Drag Enables High Power Factor in the AlGaN/GaN Two-Dimensional Electron Gas (2019)

   Nano Letters 19, 3770–3776 · doi:10.1021/acs.nanolett.9b00901 · record
8. [8]
   N. W. Ashcroft, N. D. Mermin. Solid State Physics (1976)

   Holt, Rinehart and Winston · book, no DOI · record