Wait, what? Make a wire shorter and cleaner, and its resistance does not necessarily keep behaving like the resistance of an ordinary macroscopic wire. At small enough scales, electrons can cross a device without repeatedly scattering inside it, and conductance begins to look like a problem about which quantum channels transmit.
This is the learning job of the Landauer picture. Instead of starting with a local resistivity and asking how much energy electrons lose per unit length, Landauer transport starts with reservoirs, incoming quantum states, a scattering region and transmission probabilities.
reservoirs → occupied incoming states → transmission/reflection → outgoing states → current
Quick Answer
For a simple two-terminal conductor in the linear-response, low-temperature, non-interacting picture, the electrical conductance can be written
G = (2e²/h) ΣnTn,
where Tn is the transmission probability of quantum channel n. If one spin-degenerate channel transmits perfectly, it contributes the conductance quantum G₀ = 2e²/h. The important idea is not that every small conductor must have conductance equal to an integer multiple of G₀. It is that conductance is built from the transmission of available quantum modes.
Learning Ladder: Beginner to Professional
| Stage | What the learner should be able to do |
|---|---|
| Beginner | Distinguish a particle passing through a channel from a particle being scattered back. |
| Secondary / O-Level | Use current, voltage, resistance and wave ideas without assuming every conductor is ohmic at every scale. |
| JC / A-Level | Connect electron waves, energy levels, Fermi occupation and potential barriers to transmission. |
| Undergraduate | Use the Landauer formula, conductance quantum, quantum point contacts, reservoirs and ballistic mean-free-path conditions. |
| Advanced / Professional | Distinguish two-terminal from four-terminal resistance, finite-temperature transport, interaction effects, dephasing, shot noise and limits of the single-particle picture. |
1. Begin With the Classical Picture—and Identify Its Assumption
In an ordinary metal wire, a familiar model is R = ρL/A. Resistance increases with length because electrons repeatedly scatter from phonons, disorder, defects and other electrons while moving through the material. That bulk relation is powerful when the conductor is much longer than the relevant scattering lengths.
But if the device length becomes shorter than the distance over which electrons typically scatter, the assumption of many internal collisions stops being appropriate. Transport can become ballistic.
2. Ballistic Does Not Mean “Electrons Feel Nothing”
A ballistic electron can still be guided by electrostatic potentials, magnetic fields and device boundaries. “Ballistic” means that scattering inside the active region is sufficiently weak that the electron can traverse it while retaining phase and momentum information to a useful approximation.
A common practical condition is that the device length L is shorter than an appropriate elastic mean free path. Phase coherence may require a second, often longer or shorter, scale: the phase-coherence length.
3. Replace the Long Wire With Reservoirs and a Scattering Region
The Landauer model imagines a small conductor connected to large electron reservoirs. The reservoirs establish occupation distributions characterised by their electrochemical potentials and temperature. The central device does not need to be assigned a local equilibrium at every point. Instead, incoming quantum states arrive from the leads, scatter, and either transmit or reflect.
This changes the question from “How much resistivity is inside each tiny slice?” to “What fraction of each available mode reaches the other reservoir?”
4. Confinement Turns a Wide Conductor Into Discrete Transverse Modes
In a narrow channel, the electron wave cannot take arbitrary transverse shapes. Boundary conditions produce discrete transverse modes, much as a string or waveguide supports discrete standing-wave patterns. Each propagating mode becomes a possible transport channel.
As a constriction widens or its electrostatic potential is changed, additional transverse modes fall below the Fermi energy and begin to carry current.
5. Why a Perfect Channel Contributes a Fixed Conductance
For a one-dimensional channel, increasing the electron velocity increases the current carried by each occupied state—but it simultaneously decreases the one-dimensional density of states per unit energy. Those factors cancel in a way that leaves a universal conductance scale set by the elementary charge e and Planck’s constant h.
With two spin states treated as degenerate, a perfectly transmitting channel contributes
G₀ = 2e²/h ≈ 7.748 × 10−5 S.
The factor of two is not a mysterious extra constant. It reflects spin degeneracy in the simple case. If degeneracy is lifted, the natural channel accounting can involve e²/h per spin-resolved channel.
6. Transmission Probability Is the Central Variable
A channel need not transmit perfectly. Disorder, barriers, geometric mismatch and interference can reduce the transmission probability T from 1 toward 0. For independent channels, the low-temperature linear conductance is
G = (2e²/h) ΣTn.
This immediately explains why the conductance quantum is a scale, not a rule that every measured conductance must be an exact integer multiple of it.
7. Quantum Point Contacts Make Conductance Steps Visible
A quantum point contact is a narrow constriction whose width can be controlled electrostatically. In a clean two-dimensional electron gas at low temperature, opening the constriction allows transverse modes to enter one by one. If each new mode is nearly fully transmitted, the conductance rises in steps close to integer multiples of 2e²/h.
The celebrated 1988 quantum-point-contact experiments made this quantisation visible and helped establish mesoscopic transport as a field where wave mechanics appears directly in electrical conductance.
8. The Conductance Quantum Is Not the Same as the Quantum Hall Effect
Both involve fundamental constants and quantised conductance, but the physical mechanisms differ. A quantum point contact can show channel quantisation from transverse confinement at zero magnetic field. The quantum Hall effect involves Landau quantisation and chiral edge transport in a strong magnetic field. Landauer–Büttiker reasoning is useful for both, but the states that carry current are different.
9. Where Is the Resistance if the Channel Is Ballistic?
This is one of the most important conceptual questions. A perfectly transmitting ballistic channel can still show a finite two-terminal resistance. The measured voltage difference refers to macroscopic reservoirs and contacts where many modes, occupations and irreversible relaxation are established. The resistance of the complete measurement is therefore not the same thing as dissipation uniformly distributed inside a tiny perfect channel.
The Landauer picture forces us to include the contacts as part of the physical measurement system.
10. Two-Terminal and Four-Terminal Resistance Need Not Mean the Same Thing
In a two-terminal measurement, the same leads carry current and define the voltage, so contact and interface effects are included. A four-terminal arrangement can attempt to measure the voltage of additional probes while drawing negligible current through them. In mesoscopic systems the probes themselves alter boundary conditions and occupations, so “remove contact resistance” is not always as simple as it sounds in a macroscopic circuit.
11. Finite Temperature Smears the Simple Step Picture
At zero temperature, the Fermi distribution changes sharply at the Fermi energy. At finite temperature, occupations are smeared over an energy window of order kBT. If the transmission T(E) changes significantly across that window, conductance becomes an energy-weighted integral rather than a single value of T evaluated exactly at one energy.
The general Landauer current is built from the difference between the left and right reservoir Fermi functions multiplied by energy-dependent transmission.
12. Ballistic Is Not the Same as Phase-Coherent
An electron may cross a device without losing much momentum yet still lose phase coherence through interactions with its environment. Conversely, coherent interference can survive multiple elastic scattering events if those events do not randomise phase. Mean free path and phase-coherence length therefore answer different questions.
Professional quantum transport keeps momentum relaxation, energy relaxation and dephasing separate.
13. Shot Noise Reveals More Than Average Conductance
Conductance depends on ΣTn. Shot-noise power in the Landauer–Büttiker framework contains factors Tn(1−Tn). A perfectly open channel transmits deterministically and contributes little partition noise, while a partially transmitting channel produces fluctuations because each carrier is probabilistically transmitted or reflected.
This is a deep measurement lesson: two devices can have the same average conductance but different sets of transmission eigenvalues, and noise measurements can help distinguish them.
14. The 0.7 Anomaly Shows Where the Simplest Picture Breaks
Real quantum point contacts commonly show an extra shoulder near about 0.7 times 2e²/h before the first full plateau. The precise microscopic interpretation has been debated for decades and involves electron–electron interactions, spin physics and the detailed potential landscape.
The 0.7 anomaly is pedagogically valuable because it marks the boundary of the clean non-interacting channel model. A model can explain the main staircase and still fail on a reproducible many-body feature.
15. Landauer Transport Is a Scattering Theory, Not a Universal Theory of Matter
The simplest Landauer formula treats transport in terms of effectively independent quasiparticles and transmission probabilities. Strong interactions, inelastic scattering, time-dependent driving and non-equilibrium many-body states can require more general techniques such as non-equilibrium Green’s functions or kinetic approaches.
That does not make Landauer “wrong”. It makes its assumptions visible.
16. Atomic Contacts Make Channel Transmission Almost Tangible
When metallic contacts narrow to only a few atoms, individual electronic channels can dominate transport. Experiments on atomic-scale gold contacts have observed conductance near integer multiples of the conductance quantum under suitable conditions. Structure, orbital character and transmission determine whether those plateaus are exact or approximate.
17. Graphene and Topological Materials Add New Channel Physics
In graphene, topological insulators and other low-dimensional materials, degeneracy, edge states, valley structure, spin–orbit coupling and unusual dispersion can alter the effective channel count and transmission. The Landauer framework remains useful, but the simple “one ordinary parabolic subband equals 2e²/h” picture must be adapted to the actual band structure and symmetries.
18. Why the SI Makes the Conductance Quantum Especially Clean
Since the 2019 SI redefinition, the numerical values of e and h are fixed exactly in SI units. Relations built directly from these constants therefore play a central role in quantum electrical metrology. The conductance quantum and the von Klitzing resistance are linked by fundamental constants rather than a material artefact.
Evidence: What Proves What?
- Conductance versus gate voltage: can reveal quantised plateaus as transverse modes open.
- Temperature dependence: tests thermal smearing and interaction-driven anomalies.
- Magnetic-field dependence: lifts degeneracies and reveals spin, edge-state or orbital effects.
- Source–drain spectroscopy: maps subband energy scales and non-linear transport.
- Shot-noise measurement: constrains transmission probabilities beyond their sum.
- Scanning-gate or local-probe experiments: can image or perturb electron trajectories and interference.
- Device-length comparison: helps distinguish ballistic from diffusive regimes.
- Multi-terminal transport: tests reciprocity, non-local response and contact/probe effects.
Misconceptions Worth Hunting
- “Ballistic means zero resistance in every measurement.” A finite two-terminal resistance remains because reservoirs and contacts matter.
- “Every nanoscale conductor must have conductance equal to an integer times 2e²/h.” Partial transmissions and lifted degeneracies give other values.
- “2e²/h comes from two wires.” The factor two normally comes from spin degeneracy in the simple case.
- “Landauer conductance is just Ohm’s law rewritten.” It is a transmission-based microscopic framework with different assumptions.
- “Mean free path and coherence length are the same.” They quantify different loss processes.
- “A conductance plateau proves there are no interactions.” The 0.7 anomaly shows many-body effects can coexist with the staircase.
- “The channel dissipates energy uniformly because it has a measured resistance.” Dissipation and reservoir relaxation must be distinguished from the ballistic region itself.
- “A mathematical fit to T(E) uniquely identifies microscopic scattering.” Different physical models can sometimes produce similar transmission curves.
Transfer Checks
- A channel has T = 0.5 and is spin-degenerate. What is its simple Landauer conductance? (2e²/h) × 0.5 = e²/h.
- Two fully transmitting spin-degenerate modes are open. What conductance is expected? 4e²/h.
- A device is shorter than the elastic mean free path but much longer than the phase-coherence length. Is it automatically fully coherent? No.
- The same conductance is measured in two devices, but one has many weak channels and the other one nearly perfect channel. What extra observable could help distinguish them? Shot noise.
- A plateau deviates from integer quantisation only when temperature rises. Name at least two possibilities. Thermal broadening and temperature-dependent scattering/interactions.
- A 0.7 shoulder persists in a clean QPC. Does the ideal non-interacting Landauer staircase fully explain it? No; interaction physics becomes important.
How We Know the Learning Has Held
A learner should be able to explain why a short conductor requires a transmission model, derive the physical meaning of the conductance quantum, distinguish spin-resolved from spin-degenerate channels, explain quantum point-contact plateaus, identify contact resistance as a measurement-system issue, move from zero-temperature G = (2e²/h)ΣTn to an energy-integral picture at finite temperature, and state where interactions or dephasing force a richer model.
Model Limits
The simplest Landauer equation assumes a steady state, well-defined reservoirs and effectively elastic single-particle scattering through the central region. Strong electron correlations, superconductivity, time-dependent fields and inelastic processes require extensions. “Ballistic” is scale-dependent, not an intrinsic yes/no label for a material. Quantised steps are broadened by temperature, imperfect transmission and experimental resolution. Finally, assigning a microscopic mechanism from conductance alone is often underdetermined; complementary spectroscopy, noise and magnetic-field measurements are needed.
Research Foundations and Freshness Check
- NIST — Landauer’s formula and its transport assumptions.
- NIST — Quantum Conductance, current project page highlighting quantum electrical metrology.
- NIST — Quantum Metrology Triangle and the 2019 SI, updated 13 August 2025.
- APS Physics — Kondo Physics in a Quantum Channel.
- Nature — Microscopic origin of the 0.7 anomaly in quantum point contacts.
- Physical Review Research 2025 — noninteger conductance steps in HgTe quantum point contacts.
Connect This to the eduKate Science Estate
This article owns the narrow Physics job of how transmission channels determine ballistic mesoscopic conductance. For the broader electronic-structure route, continue to How to Learn Semiconductors and Transistors. For quantum-state reasoning, continue to How to Learn Quantum Measurement. The wider corridor remains Physics: Energy, Forces, Electricity, Light, Sound and Waves.
The Quiet Ending
The beginner asks, “Why does a narrow wire have resistance?” The developing physicist asks, “Which quantum modes are open?” The advanced learner asks, “What are their transmission probabilities and coherence conditions?”
The professional asks: which part of the measured resistance belongs to transmission through the device, which part belongs to contacts and reservoirs, and what evidence tells us that the Landauer assumptions are still valid?
