Landaura

Landaura, a canyon-shaper titaness of The SI Guard, widens a gorge one lane at a time, and the current climbs in equal steps.

The Physics

In a narrow ballistic conductor, such as a quantum point contact in a two-dimensional electron gas, electrons travel in a few transverse modes, like lanes on a road. Each fully open lane carries one conductance quantum, counting both spin states. Widening the constriction opens one lane at a time, so the conductance rises in a staircase. This was first seen in 1988 in GaAs point contacts (van Wees et al.; Wharam et al.).

G = (2e²/h) Σ_n T_n, G₀ = 2e²/h ≈ 7.748 091 729 × 10⁻⁵ S

• G: conductance (siemens, S)
• e: elementary charge, exact in the 2019 SI
• h: Planck constant, exact in the 2019 SI
• T_n: transmission probability of mode n (0 to 1)
• 2: spin degeneracy, two spin states per lane
• G₀: conductance quantum, exact (CODATA 2022)

This is the Landauer formula, after Rolf Landauer. Because e and h are fixed, G₀ is exact: 1/G₀ ≈ 12 906.4 Ω, which is half the von Klitzing constant R_K = h/e².

Honest caveat: The resistance of a perfect lane is not caused by collisions in the channel; it arises where the narrow channel meets the wide contacts. Clean plateaus need low temperature and a ballistic channel; real point-contact steps are accurate only to about 1%, unlike the quantum Hall effect, so G₀ is not used as a resistance standard. The factor 2 disappears if a magnetic field lifts the spin degeneracy.

On the sticker

Landaura casts a green beam down the canyon of gate-rock as she pushes the walls apart. Printed text: LANDAURA with the line G = 2e²/h. Pixel-art kiss-cut vinyl sticker, white, in 2″, 3″, 4″ and 6″ sizes.