Landauer formula¶
A mesoscopic-transport relation expressing coherent electrical conductance as the conductance quantum multiplied by the sum of transmission probabilities of the conductor's available quantum channels.
Core Idea¶
The Landauer formula describes mesoscopic conductance through transmission. For a phase-coherent two-terminal conductor in linear response, each available quantum channel contributes according to its transmission eigenvalue, giving G = G0 Σn Tn under the chosen spin convention. A fully transmitted channel contributes one conductance quantum; partial scattering reduces its contribution. A fully transmitted channel contributes one conductance quantum; partial scattering reduces its contribution.
Scope of Application¶
Use the formula for coherent mesoscopic conductors after stating terminals, response regime, degeneracy convention, and transmission calculation. Use the formula for coherent mesoscopic conductors after stating terminals, response regime, degeneracy convention, and transmission calculation.
- Quantum point contacts. Counts partially open channels.
- Nanoscale wires. Relates scattering to conductance.
- Molecular electronics. Models coherent junction transmission.
- Ballistic devices. Approaches quantized channel conductance.
- Multiterminal transport. Uses the Büttiker extension.
Clarity¶
Conductance is not determined only by material resistivity at this scale. Contacts, modes, and transmission across the whole scattering region enter explicitly. The closest near miss sets the boundary: The Landauer–Büttiker relation is closest: it generalizes the same scattering framework to multiple terminals and currents. A positive case must satisfy this test: The Landauer formula applies when coherent transport between reservoirs can be represented by channel transmission probabilities under the stated response regime.
Manages Complexity¶
The formula compresses a scattering matrix into transmission eigenvalues, preserving current-relevant information while discarding phases not needed for two-terminal linear conductance. The central channel idealization–inelastic reality tradeoff is this: Transmission eigenchannels clarify coherent transport while real devices dephase and heat. A second conductor resistance–contact contribution tension matters because Measured resistance includes reservoirs and channel injection, complicating a bulk-only picture.
Abstract Reasoning¶
Use three linked moves: define coherent region and reservoir terminals; fix temperature, chemical potentials, and linear-response conditions; find propagating transport channels at the relevant energy. As a collapse test, the simple case exits when bias is not small, scattering varies materially across the transport window, inelastic effects dominate, or terminals exceed the two-reservoir setup. A fourth check is to compute transmission eigenvalues from the scattering problem. A final check is to apply the correct quantum and generalization for degeneracy, energy window, and terminal count.
Knowledge Transfer¶
Channel-throughput summation transfers structurally to waves and networks, but fermionic reservoirs, quantum coherence, and electrical conductance delimit the formula. The nearest stopping boundary is explicit: The Landauer–Büttiker relation is closest: it generalizes the same scattering framework to multiple terminals and currents. The inclusion test remains: The Landauer formula applies when coherent transport between reservoirs can be represented by channel transmission probabilities under the stated response regime. The structure no longer applies when the simple case exits when bias is not small, scattering varies materially across the transport window, inelastic effects dominate, or terminals exceed the two-reservoir setup. No canonical parent prime is currently asserted; broader structural comparisons remain related-prime analogies until separately adjudicated in the DAG. Channels carry only their passed fraction. Conductance sums channel contributions.
Neighborhood in Abstraction Space¶
Landauer formula sits in a moderately populated region (52nd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- Quantum Point Contact — 0.89
- Aharonov–Casher effect — 0.86
- Elliott formula — 0.86
- Lieb–Liniger model — 0.85
- Lattice Boltzmann Methods — 0.85
Computed from structural-signature embeddings · 2026-10-08