Quantum Hall effect: separating the 1980 measurement and 1982 theory
Compare quantized Hall-resistance measurements in a silicon MOSFET with the theoretical Hall conductance of a periodic potential. Separate the observable and gap conditions without promising universal edge transport.
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1. Hall resistance measured in the experiment
The quantum Hall effect is the appearance of plateaus in the Hall resistance of a two-dimensional electron system at low temperature and high magnetic field. In 1980, von Klitzing, Dorda, and Pepper observed this behavior in a silicon MOSFET and reported its use for determining the fine-structure constant. Longitudinal resistance becomes very small on a plateau, but it should not be described as exactly zero in every sample.
2. Hall conductance explained by periodic-potential theory
In 1982, Thouless, Kohmoto, Nightingale, and den Nijs theoretically treated quantized Hall conductance in a two-dimensional electron system with a periodic potential and connected it to an integer topological invariant. The Chern-number account explains robustness to small perturbations when the relevant gap and conditions persist. These early papers did not directly measure identical precision in every material or geometry, nor universal dissipation-free edge transport.
3. Three distinctions for reading a graph or paper
First, the 1980 observable is resistance obtained from Hall-voltage measurements in a silicon MOSFET’s two-dimensional electron system. A plateau is a region where the resistance stays near a fixed value as conditions change. Observing a flat region does not mean the resistance is identical under every condition.
Second, the 1982 TKNN paper calculates Hall conductance in a theoretical system with a uniform magnetic field and periodic potential. Its abstract explicitly places the Fermi energy in a gap for quantization. Checking that assumption avoids transferring an integer-valued result to an arbitrary material, temperature or measurement.
Third, check what an integer means numerically. The following equations use magnitudes, omitting the sign set by magnetic-field and voltage-contact conventions. On an integer quantum Hall plateau, the ideal two-dimensional Hall conductivity is σxy = νe²/h; the corresponding Hall resistance is Rxy = h/(νe²) in the limit of vanishing longitudinal resistance Rxx. Here ν is a positive integer plateau index, e the elementary charge and h the Planck constant. Conductivity is in S (siemens) and resistance in Ω. The reciprocal relation is approximate when Rxx is sufficiently small; a finite longitudinal component requires the conductivity–resistivity tensor relation. NIST’s DC quantum Hall description
For a hypothetical ideal reading example, ν=1 gives σxy=e²/h and Rxy=h/e², whereas ν=2 gives σxy=2e²/h and Rxy=h/(2e²). Changing the index from 1 to 2 doubles the conductivity and halves the resistance. ‘Integer’ does not mean resistances of 1Ω and 2Ω. This compares equations, not new experimental data. In a measured graph, check for a Hall-resistance plateau together with low Rxx before connecting it to the theoretical conditions.