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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.

Version
v1 · 2026-09-28 · History
Domain-specific #
10301
Domain group
Natural Sciences
Origin domain
Physics
Subdomains
Mesoscopic Physics, Quantum Transport → Physics

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. Reservoirs set incoming occupations and the conductor's scattering matrix supplies the transmission values. Finite temperature, finite bias, energy dependence, inelastic processes, and multiple terminals require the corresponding energy integrals or Landauer–Büttiker generalization.

Structural Signature

Sig role-phrases:

  • phase-coherent conductor. Maintains quantum phase across the mesoscopic scattering region. Constitutive physical regime. If altered: Strong incoherent transport needs a different treatment or segmentation.
  • reservoir terminals. Set chemical potentials and supply incoming channel populations. Constitutive boundary conditions. If altered: An isolated finite system has no Landauer current relation.
  • transport channels. Provide propagating modes at the relevant energy. Constitutive decomposition. If altered: Counting unavailable modes overstates conductance.
  • transmission eigenvalues. Measure each channel's probability of traversing the scatterer. Identity-bearing scattering data. If altered: Density of states alone is insufficient.
  • conductance sum. Weights channel transmissions by the conductance quantum under stated degeneracy conventions. Constitutive output. If altered: Finite-bias or energy-dependent cases require integration/generalization.

What It Is Not

  • Ohm's law. Is bulk resistivity or coherent transmission primary?
  • Landauer–Büttiker formula. Are multiple terminals involved?
  • Conductance quantum. Is the unit being confused with the full relation?
  • Ballistic transport. Are all transmissions actually unity?

Scope of Application

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.

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.

Abstract Reasoning

  1. Define coherent region and reservoir terminals.
  2. Fix temperature, chemical potentials, and linear-response conditions.
  3. Find propagating transport channels at the relevant energy.
  4. Compute transmission eigenvalues from the scattering problem.
  5. 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.

Examples

Canonical

A quantum point contact has two channels with transmissions 1 and 0.4 at the Fermi energy, so its linear conductance is G0 times 1.4 under the stated convention.

Mapped back: phase-coherent conductor → point contact; reservoir terminals → left and right leads; transport channels → two modes; transmission eigenvalues → 1 and 0.4; conductance sum → 1.4 G0.

Applied / In Practice

A macroscopic hot resistor with strong inelastic scattering is fit by bulk resistivity and geometry; applying one coherent scattering region without dephasing treatment is not the simple Landauer case.

Mapped back: phase-coherent conductor → fails macroscopically; reservoir terminals → contacts present; transport channels → not coherently resolved; transmission eigenvalues → simple set invalid; conductance sum → bulk Ohm relation.

Structural Tensions

T1: channel idealization vs. inelastic reality. Transmission eigenchannels clarify coherent transport while real devices dephase and heat. Diagnostic: Where does coherence end?

T2: conductor resistance vs. contact contribution. Measured resistance includes reservoirs and channel injection, complicating a bulk-only picture. Diagnostic: Which boundaries define the scattering region?

Structural–Framed Character

Description turns on phase-coherent conductor, reservoir terminals, transport channels, transmission eigenvalues, conductance sum. Skeletal core. Independent transmission modes contribute additively to throughput according to passage probability. Domain-bound accent. Electrons, reservoirs, phase coherence, scattering matrices, conductance quanta, and chemical potentials define Landauer transport. Transfer remains bounded because Why not prime. Modal throughput is portable; this is a quantum electrical relation. The negative boundary is concrete: Any Ohm's law, conductance quantum, scattering matrix, ballistic transport, resistance measurement, quantum Hall plateau, or multi-terminal current equation is not automatically the simple Landauer formula. The Landauer formula is structural-formal within its physical assumptions, while transmissions and coherence are empirical/model-dependent. Its character: mesoscopic conductance as summed quantum transmission.

Structural Core vs. Domain Accent

Skeletal core. Independent transmission modes contribute additively to throughput according to passage probability.

Domain-bound accent. Electrons, reservoirs, phase coherence, scattering matrices, conductance quanta, and chemical potentials define Landauer transport.

Why not prime. Modal throughput is portable; this is a quantum electrical relation.

  • Transmission. Channels carry only their passed fraction.
  • Aggregation. Conductance sums channel contributions.
  • No strict parent is asserted.

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

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With

  • Ohm's law. Tell: Is bulk resistivity or coherent transmission primary?
  • Landauer–Büttiker formula. Tell: Are multiple terminals involved?
  • Conductance quantum. Tell: Is the unit being confused with the full relation?
  • Ballistic transport. Tell: Are all transmissions actually unity?

References

  • Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Landauer_formula (revision 1364161401).
  • Preserved source candidate: https://www.cambridge.org/core/books/mesoscopic-physics-of-electrons-and-photons/introduction-mesoscopic-physics/026F8AD6B95AC613676FB324677C8074

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.