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Stairohm

Quantum Hall Effect / Resistance Plateaus Physics

\[\begin{gathered} R_{xy} = \frac{h}{\nu e^2} = \frac{R_K}{\nu} \\ \nu = 1, 2, 3, \ldots \end{gathered}\]
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Stairohm is a Pixelated Physics law character, one of our quantum designs. Integer quantum Hall effect: Hall resistance locks onto exact plateaus at h/ne², and some disorder helps make the steps flat. This is a 3" × 3" pixel-art sticker. The physics panel below gives the equation, what each symbol means and the sources.

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Size3″ × 3″

3″ × 3″ kiss-cut sticker.

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The physics

In a thin conductor at low temperature and strong magnetic field, the Hall resistance locks onto exact plateaus at h/e² divided by a whole number. These plateaus are so precise and reproducible that they now serve as the standard of electrical resistance.

Stairohm manages a staircase where every step is exact.

Push a current along a thin strip of conductor in a magnetic field and the charges are pushed sideways, building a voltage across the strip. That is the Hall effect. The ratio of the sideways voltage to the current, the Hall resistance, normally rises smoothly as the field grows.

In the spring of 1980, at the High Magnetic Field Laboratory in Grenoble, Klaus von Klitzing measured specially designed transistors called MOSFETs a few degrees above absolute zero in extremely strong fields, with the electrons forced to move in an extremely thin surface layer. The Hall resistance did not ramp. It climbed in flat steps, and on each landing it sat at h/e² divided by a whole number, the first line on this page, to better than one part in ten million. With Gerhard Dorda and Michael Pepper he published it that year as a new way to measure the fine-structure constant. He received the 1985 Nobel Prize.

The landings are where the trick lives. In a strong field the electrons’ allowed energies bunch into Landau levels. Some disorder in the sample traps states between those levels, so as the field rises the Hall resistance stays put while the trapped states fill. Too much disorder washes the steps out. In 1981 Robert Laughlin explained why the step values are exact. And only while you stand on a landing does the resistance along the strip, \(R_{xx}\), drop to zero.

The steps are so reliable that h/e² now anchors the ohm. Since 20 May 2019, with h and e fixed exactly in the SI, the von Klitzing constant is exact too: 25 812.807 45… Ω, the second line.

Exact steps. Dirt helps.

Equations

\[\begin{gathered} R_{xy} = \frac{h}{\nu e^2} = \frac{R_K}{\nu} \\ \nu = 1, 2, 3, \ldots \end{gathered}\]
\[\begin{aligned} R_K &= \frac{h}{e^2} \\ &= 25\,812.807\,45\ldots\ \Omega \end{aligned}\]
\[R_{xx} = 0\ \text{on a plateau}\]
SymbolMeaningUnit
\(R_{xy}\) Hall resistance: transverse voltage divided by current \(\mathrm{Ω}\)
\(h\) Planck constant (exact in the SI since 2019) \(\mathrm{J·s}\)
\(e\) elementary charge (exact in the SI since 2019) \(\mathrm{C}\)
\(\nu\) filling factor: an integer on each plateau \(\mathrm{1}\)
\(R_K\) von Klitzing constant, h/e² \(\mathrm{Ω}\)
\(R_{xx}\) longitudinal resistance along the sample \(\mathrm{Ω}\)

Sources

  1. von Klitzing, Dorda & Pepper (1980) New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance, Phys. Rev. Lett. 45, 494
  2. Laughlin (1981) Quantized Hall conductivity in two dimensions, Phys. Rev. B 23, 5632
  3. NobelPrize.org — The Nobel Prize in Physics 1985 (Klaus von Klitzing, for the discovery of the quantized Hall effect)
  4. NIST CODATA — von Klitzing constant R_K (exact)
  5. BIPM — The International System of Units (SI Brochure, 9th ed., 2019)
  6. NobelPrize.org — Press release, The 1985 Nobel Prize in Physics (crucial experiment in spring 1980 at the high-field magnet laboratory in Grenoble; MOSFET samples)
  7. NobelPrize.org — Klaus von Klitzing, Biographical (High Magnetic Field Laboratory, Grenoble, 1979–80)

Source Klitzing, K. v.; Dorda, G.; Pepper, M. (1980). New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance. Physical Review Letters 45, 494–497.doi.org/10.1103/PhysRevLett.45.494

Field notes

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