Why does a resistance stop changing?
Send current along a thin conducting layer in a perpendicular magnetic field, and a sideways voltage appears: the Hall voltage. It usually changes with field or electron density. The surprise in the integer quantum Hall effect is an interval over which this voltage divided by current barely moves. A flat stretch in the transverse resistance appears at the same time as a deep minimum in resistance along the current. The two observations belong together: one identifies the special transverse response, and the other tests whether little energy is being lost to heat along the layer.
In 1980 Klaus von Klitzing, Gerhard Dorda and Michael Pepper reported this behavior in electrons confined near a silicon interface. Changing the transistor gate voltage changed their surface density. The resistance settled near values determined by Planck’s constant, the elementary charge and an integer, with an accuracy that made the device interesting as an electrical reference. Why should a fabricated piece of silicon remember a universal number?
Sources: [2]
The two voltages come from different pairs of contacts
The electrons occupy a thin conducting layer, so their low-energy motion has two spatial directions. A magnetic field passes perpendicular to this plane. Current enters at one end of a Hall bar and leaves at the other. Small side contacts measure a voltage across the width; another pair measures the voltage between two positions along the bar. Voltage probes draw very little current, which separates their purpose from the current-carrying contacts.
The transverse voltage divided by current is the Hall resistance. The longitudinal voltage tests energy loss along the sample. A misplaced transverse probe can also collect part of the longitudinal voltage. Reversing field or current, comparing contact pairs and measuring contact resistance help expose that contamination. These procedures matter especially when a plateau is used as a standard: a plausible staircase on an oscilloscope is much easier to obtain than a resistance value with a defensible uncertainty.
Sources: [5]
Landau levels provide the energy scale
A perpendicular field quantizes the orbital motion into Landau levels. Each level has a large number of available states proportional to magnetic field and sample area. Dividing electron surface density by the number of states per area gives the filling factor. That single-level capacity is elementary charge times magnetic field divided by Planck’s constant. Density is in inverse square metres, field in tesla, and elementary charge is a positive magnitude.
An integer filling is a useful starting picture: a whole number of levels is occupied. But perfectly sharp levels alone would pick out isolated densities or fields, not broad flat intervals. Real levels broaden in an imperfect sample. Explaining why many of those added states can accept electrons without changing the measured Hall response is the next step. Spin and valley splittings can further divide the levels, so an actual silicon sequence requires more detail than a spinless orbital diagram.
For example, at five tesla one resolved Landau level supplies about 1.21 quadrillion states per square metre. An ideal two-level filling would therefore require about 2.42 quadrillion electrons per square metre. These numbers follow from flux counting, not a fitted material constant. Sweeping density through that value gives a useful way to distinguish a sharp ideal filling from the finite experimental plateau around it.
Sources: [1]
Localized states let the density change
Disorder makes some electronic states concentrate in restricted regions. Electrons in such states can be added as the gate voltage changes, yet they do not carry charge across a large sample. States near the centre of a broadened Landau level can remain extended and participate in the transition between plateaus. Thus the same integer plateau can persist over a range of density, even though the filling factor is not exactly that integer at every point.
This explains a seemingly contradictory role for disorder. Some disorder helps produce a visible plateau by supplying localized states. Too much disorder can erase the energy separation needed to resolve quantum Hall transport. Localization also has a length scale: a state that appears confined in a small calculation may still communicate across a larger device. Low temperature suppresses thermally activated access to conducting states, which is why a resistance minimum becomes more convincing as it deepens on cooling.
Sources: [1][4]
Resistance and conductance must be converted together
An electric field can drive current both along and across its direction. Resistance and conductance therefore need a two-direction conversion. For an isotropic sheet, the magnitude of Hall conductivity equals the magnitude of transverse sheet resistivity divided by the sum of the squared longitudinal and transverse sheet resistivities. Sheet resistivity is measured in ohms and sheet conductivity in siemens. Their signs also depend on the field, charge and coordinate directions.
Only when longitudinal resistivity is sufficiently small does the Hall conductivity become the reciprocal of Hall resistivity. In the ideal integer limit its magnitude is an integer times the conductance quantum. A two-terminal resistance can include contacts and leads, and longitudinal resistance also depends on bar geometry. Quoting a quantized conductivity from one raw voltage therefore requires the measurement geometry and the longitudinal check. This distinction prevents a common error: a resistance plateau at one half of the von Klitzing resistance corresponds to twice the conductance quantum.
Sources: [1][6]
From a stable plateau to a calibrated resistor
A 1985 experiment by Marvin Cage, Ronald Dziuba and Bruce Field tested the effect as a resistance reference. They compared a room-temperature resistor with two gallium-arsenide Hall devices over twelve months, using different voltage probes and both magnetic-field directions. Their published high-resolution plateau was flat within about 0.01 parts per million over a field interval. They still corrected offsets and temperature dependence.
That work distinguishes repeatability from absolute accuracy. Agreement between two devices and over time tests stability; connecting the result to an existing unit requires a calibrated comparison chain and its uncertainty. Current reversal suppressed thermal voltages and slow drift. Excess current or temperature could produce longitudinal dissipation and shift the apparent plateau. Their comparison procedure repeatedly recovered the quantum value within measured operating conditions and uncertainty.
Sources: [5]
The number is exact; a realization still has uncertainty
In the formula, Hall resistance is in ohms; Planck’s constant and the positive elementary charge set the scale, and the positive integer labels the plateau. The scale is easy to estimate. The von Klitzing resistance is about 25,812.807 ohms, so the second integer plateau is about 12,906.404 ohms. A current of one microampere then gives a transverse voltage of about 12.906 millivolts. The integer labels a collective response of the electron system; it does not count individual electrons passing one at a time through a voltage probe.
Since the 2019 revision of the SI, Planck’s constant and the elementary charge have fixed exact values. Their ratio defines an exact von Klitzing constant. A physical device still has finite measurement uncertainty because cooling, contacts, leakage and comparison instruments are physical operations. This separation between an exact definition and its experimental realization is central to modern metrology. Improving materials and apparatus makes the same quantum value easier to recover with smaller uncertainty.
Sources: [6]
Why the flat interval mattered
A flat interval reveals more than one agreement with a predicted number. Electrons continue to enter the layer, yet the states accepting them can remain localized while the integer transverse response persists. Cooling, changing current and comparing contacts then test whether the device stays in that condition. The connection between a microscopic quantum state and a reproducible macroscopic voltage is what made this discovery a foundation of electrical metrology.
Sources: [3]