The one-third plateau was a new problem
In 1982 Daniel Tsui, Horst Störmer and Arthur Gossard found a Hall plateau where the lowest spin-polarized Landau level was only one third filled. A Landau level is an energy step created by orbital motion in a magnetic field; spin polarization aligns the electrons’ intrinsic magnetic moments. An integer number of filled levels could no longer explain the result. Their high-mobility gallium-arsenide heterostructure also showed a longitudinal resistance minimum at that filling. A special electronic state had appeared within a partly filled level.
The fraction names the Hall conductance in units of the conductance quantum. Its Hall resistance is three times the von Klitzing resistance, about 77.4 kilohms, when longitudinal dissipation is negligible. Confusing those two quantities reverses the meaning of the observation. The original paper proposed a Wigner solid or a charge-density wave as possible interpretations. The now familiar liquid explanation came afterward. The discovery was a transport measurement; the identity of the state required further theory and experiments.
Sources: [2]
A magnetic field makes interactions decisive
A Landau level contains many states with the same orbital energy. At partial filling, rearranging electrons among those states costs little single-particle kinetic energy. Repulsion between electrons can therefore decide how they arrange their motion. The field has not created extra electric charge or removed the repulsion; it has changed the relative importance of the available energy scales.
Clean samples were essential because strong uncontrolled disorder could hide this collective state. A heterostructure confines electrons close to an interface and separates them from much of the ionized donor disorder. Low temperature limits the population of excitations across the many-electron energy gap. Some disorder can then pin the extra charged excitations and give a plateau finite width. This is a more demanding situation than an orbital-level gap in a simple integer model: the relevant gap must first be created by correlations within the level.
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Laughlin built the avoidance into a wavefunction
Robert Laughlin’s 1983 proposal described a liquid whose electrons strongly avoid close encounters. A many-electron wavefunction assigns an amplitude to an entire arrangement of positions. In Laughlin’s trial state, every pair contributes a factor that vanishes when those two positions coincide. An odd power also gives the required sign change when two electrons are exchanged. At one-third filling, the power is three.
For the one-third state, each pair contributes the cube of its position difference; a magnetic-field-dependent envelope completes the trial wavefunction. This cubic zero suppresses close approaches more strongly than the simple zero required when identical fermions exchange places. The odd power also keeps the required sign change under exchange. Laughlin compared the trial states with small-system calculations and estimated their energy against competing states. The construction was an approximation to the interacting semiconductor, but it joined the observed filling, a gap and the transport response in one physical description.
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Incompressible means that charged excitations cost energy
Calling the state a liquid does not mean that it has ordinary water-like compressibility. Near the selected density, changing the local charge creates excitations with a finite energy cost. That is the relevant meaning of an incompressible quantum Hall fluid. The electrons need not occupy a crystalline array, and the state can support low-energy motion at its boundary even when charged excitations in the interior are gapped.
A quasihole is a localized deficit of electronic charge relative to the uniform liquid. Laughlin’s construction predicts a deficit of one third of an elementary charge in the one-third state. The whole electron system rearranges around the disturbance. No free electron has been cut into three pieces that could be removed into vacuum. The fractional object exists as an excitation of that medium. The Hall plateau describes the average response of the whole liquid. To count the charge carried in individual transfer events, experimenters turned to current fluctuations.
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Shot noise turns current fluctuations into a charge measurement
A steady average current can conceal discrete transfer events. If weak tunnelling events occur independently, their random timing produces shot noise whose spectral density is proportional to event charge and average tunnelling current. Smaller charges at the same average current make more frequent, smaller events and a smaller noise level. The relevant current is the weak backscattered current, not the entire current flowing along an edge.
In 1997 Saminadayar, Glattli, Jin and Etienne measured this noise near one-third filling. Gates formed a narrow constriction, a quantum point contact, that weakly coupled opposite edges. A small part of the current could tunnel between them. In the low-temperature, weak-backscattering limit, the slope of noise against that transferred current agreed with one-third charge. In the expression, the noise is a one-sided low-frequency current spectral density in squared amperes per hertz, the charge is the amount transferred per event in coulombs, and the backscattered current is in amperes. Here the inferred event charge is one third of the elementary charge. Correlated transfers or strong tunnelling require a more detailed noise theory.
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Calibration and a second voltage scale strengthened the result
The charge inference depends on an absolute noise calibration. Thermal noise remains at zero applied bias, and the amplifier adds its own background. Measuring those contributions is part of the experiment. At small bias, thermal fluctuations dominate. At larger bias, the excess noise approaches the shot-noise form. The crossover voltage depends on charge as well as temperature, so its position gives another test beyond fitting a single high-bias slope.
The crossover scale is roughly twice Boltzmann’s constant times temperature divided by event charge. For an illustrative electron temperature of fifty millikelvin, it is about twenty-six microvolts for one-third charge. For electron charge it would be about nine microvolts. Heating changes the temperature and moves the crossover, so a fit must be consistent with independent thermometry and the measured current-voltage relation. Agreement across the curve is stronger evidence than one noisy point below an electron-charge prediction. These experiments measured fractional charge. The phase acquired when two quasiparticles exchange positions requires a different, phase-sensitive experiment.
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An independent group used the same physical question
De-Picciotto, Reznikov, Heiblum, Umansky, Bunin and Mahalu also reported one-third-charge noise in 1997. Their gallium-arsenide device used surface gates to form a point contact, a dilution refrigerator and a cryogenic preamplifier. They calibrated the gain with a known noise source and used equilibrium noise versus conductance to determine electron temperature. These details explain why their paper contains noise spectra and calibration plots alongside transport data.
The two experiments did not photograph fractional particles. They measured fluctuations of electrical current and tested the relation between those fluctuations and discrete charge transfer. Independent apparatus and analysis reduced the likelihood that one particular amplifier or sample artifact explained the result. Together with the fractional plateau and correlated-state theory, the charge measurements connected a striking macroscopic number to the elementary excitations of an interacting liquid.
The fractional Hall discovery changed what an electron system could contain. Repulsion organized a partly filled level into a liquid, and a local disturbance of that liquid carried a charge smaller than the electron’s. The sequence from plateau to a trial wavefunction and then to current noise shows how a new collective state can supply new elementary excitations. Their properties emerge from the arrangement of many ordinary electrons.
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