Experiment: The Quantum Hall Effect
On the night of 4–5 February 1980, measuring the Hall voltage of a silicon field-effect transistor at \(1.5\,\mathrm{K}\) in about \(15\,\mathrm{T}\), von Klitzing, Dorda and Pepper found that the Hall resistance did not rise linearly with field but sat on flat plateaux at the values \(h/\nu e^{2}\) with \(\nu\) an integer, while the longitudinal resistance vanished [Klitzing:1980]. The plateau values depend on nothing — not on the material, the geometry, the mobility or the disorder — and have since been reproduced to parts in \(10^{10}\) in different substances [Hartland:1991]. That a macroscopic, irreducibly dirty solid should hand back a combination of fundamental constants exactly is the most striking single measurement in condensed matter physics, and it is why the ohm has been realized from this effect since 1990 and why \(R_{K}=h/e^{2}\) is exact in the present SI [BIPM:2019] [Mohr:2025].
This chapter treats that measurement as an experiment, with the
structured experiment environment (apparatus, procedure,
observations with uncertainties in SI, interpretation, primary
references) to be filled in when the content is written. It rests on the
band theory and Landau quantization of
Electrons in Solids: Band Theory and on the heterostructure
two-dimensional electron gas of Semiconductors; it carries
forward into the topological language of Topological and Metric Spaces and
Lie Groups, Lie Algebras, and Fibre Bundles, and into the metrology of
Measurement, SI Units, and the Theory of Errors. Two years after the integer effect,
the same apparatus applied to a cleaner sample produced plateaux at
fractional \(\nu\) [Tsui:1982], whose explanation
[Laughlin:1983] required a new correlated ground state and
excitations carrying a fraction of the electron charge — since
measured directly [Saminadayar:1997] [dePicciotto:1997]. The standard
collection is [Prange:1990]; a forty-year retrospective is
[Klitzing:2020].
Experiment: The Quantum Hall Effect: all derivations of this chapter are pending.
Historical context and the prediction under test
The classical Hall effect
[Reserved: Hall's 1879 measurement of a transverse voltage across a current-carrying gold leaf in a magnetic field [Hall:1879]; the classical result \(R_{H}=B/ne\) for a three-dimensional conductor and \(R_{xy}=B/n_{s}e\) for a sheet of areal density \(n_{s}\), linear in \(B\) with no structure whatever; the Hall bar geometry and the four-terminal measurement that separates \(R_{xx}\) from \(R_{xy}\); this is the prediction the experiment of Section 128.4 contradicts.]
Two-dimensional electron gases
[Reserved: the inversion layer of a silicon metal–oxide–semiconductor transistor as the first true two-dimensional electron system, with electric quantum limit and subbands [Ando:1982]; the gate voltage as a direct and calibrated control of the areal carrier density; the modulation-doped gallium arsenide heterostructure [Dingle:1978], which trades tunability for mobility; why two dimensions is essential — the Landau spectrum is then fully discrete, with no dispersion along the field.]
Landau quantization and what was expected
[Reserved: Landau levels at \(E_{n}=(n+\tfrac{1}{2})\hbar \omega_{c}\) with degeneracy \(eB/h\) per unit area [Landau:1930], and the filling factor \(\nu=n_{s}h/eB\); the earlier observation of magneto-oscillatory conductance in silicon inversion layers by Fowler, Fang, Howard and Stiles [Fowler:1966], the two-dimensional Shubnikov–de Haas effect; the expectation before 1980 — oscillations in \(R_{xx}\) and a Hall resistance with wiggles about the classical line; von Klitzing's actual purpose, a high-accuracy determination of the fine-structure constant [Klitzing:1980].]
Apparatus
The silicon MOSFET Hall bar
[Reserved: the original sample — a silicon metal–oxide–semiconductor field-effect transistor with a Hall-bar channel, source and drain, two pairs of potential probes and a gate covering the channel [Klitzing:1980]; typical dimensions and mobilities of order \(1\,\mathrm{m}^{2}/\mathrm{V}/\mathrm{s}\); the gate voltage sweeping \(n_{s}\) through several Landau levels at fixed field, which is the axis along which the original data were taken.]
Heterostructure samples
[Reserved: the modulation-doped gallium arsenide and aluminium gallium arsenide heterojunction [Dingle:1978], in which the ionized donors sit in a barrier layer set back from the channel; mobilities exceeding \(10^{3}\,\mathrm{m}^{2}/\mathrm{V}/\mathrm{s}\) and mean free paths of hundreds of \(\mu\mathrm{m}\); the sample quality without which the fractional plateaux of Section 128.4.3 do not appear at all — the single clearest case in the book of a phenomenon gated by materials purity.]
Cryogenics, magnets and electrical metrology
[Reserved: pumped helium-4, helium-3 and dilution refrigeration down to a few \(\mathrm{mK}\); superconducting solenoids to about \(20\,\mathrm{T}\) and resistive or hybrid magnets beyond (Superconductivity and Superfluidity); low-noise four-terminal measurement with lock-in detection at currents small enough to avoid electron heating, typically \(\mathrm{nA}\) to \(\mu\mathrm{A}\); the cryogenic current comparator that transfers the quantized resistance to a calibrated standard resistor with an uncertainty of parts in \(10^{9}\) [Jeckelmann:2001].]
Procedure
[Reserved: the two equivalent sweeps — gate voltage at fixed field, or field at fixed density — and the measurement of \(R_{xx}\) and \(R_{xy}\) simultaneously; contact and lead-resistance checks; current reversal and field reversal to remove thermoelectric offsets and to symmetrize the Hall signal; the tests that must be done before any exactness claim is credible — current dependence, temperature dependence, contact placement, and comparison of samples of different geometry and material [Klitzing:1980] [Jeckelmann:2001].]
Observations
The integer plateaux
[Reserved: the primary data — \(R_{xy}\) flat over a finite range of density or field at \(h/\nu e^{2}=25812.807\,\mathrm{\Omega}/\nu\), with \(R_{xx}\) falling to an unmeasurably small value on each plateau and peaking sharply between them [Klitzing:1980]; the numbers to be tabulated with their uncertainties, together with the activation energy extracted from the temperature dependence of \(R_{xx}\); plateau width as a function of disorder; the dissipationless conduction implied by \(R_{xx}\to0\) at finite current.]
In a two-dimensional electron system at low temperature and high magnetic field, the Hall resistance does not rise linearly with the field as Section 128.1.1 requires. It rises in steps: over a finite range of carrier density, or equivalently of field, it sits on a flat plateau at
and jumps between plateaux in between [Klitzing:1980]. Across each plateau the longitudinal resistance \(R_{xx}\) falls to a value too small to measure and peaks sharply at the transitions, so on a plateau a finite current flows along the sample with no measurable voltage drop along it — dissipationless transport in a disordered solid at a temperature where the same solid is an ordinary conductor in zero field. The plateau value Equation (128.1) contains no material parameter whatever.
Derivation. The value on the plateau, though not the existence of the plateau, is elementary once the Landau degeneracy is counted. Take a sample \(L_{x}\times L_{y}\) in a perpendicular field \(B\) and work in the gauge \(\vect{A}=(0,Bx,0)\). The Hamiltonian then commutes with translations along \(y\), so eigenstates may be taken proportional to \(\ee^{\ii k_{y}y}\), and completing the square shows that each is a one-dimensional oscillator of frequency \(\omega_{c}=eB/m\) centred on
with energy \((n+\tfrac{1}{2})\hbar\omega_{c}\) independent of \(k_{y}\). Periodic boundary conditions in \(y\) quantize \(k_{y}=2\pi j/L_{y}\) with \(j\) an integer, and requiring the centre Equation (128.2) to lie inside the sample, \(0\leq x_{0}\leq L_{x}\), bounds \(j\) by \(eBL_{x}L_{y}/h\). Each Landau level therefore holds
states per unit area — one per quantum of magnetic flux \(h/e\) threading the sample, which is the way to remember it. If exactly \(\nu\) levels are filled, the areal carrier density is \(n_{s}=\nu eB/h\). Now apply the classical two-dimensional Hall result of Section 128.1.1, \(R_{xy}=B/n_{s}e\), which follows from force balance alone and holds whenever the carriers move without dissipation:
which is Equation (128.1). Note what has been shown and what has not. The field \(B\) cancels, and with it every scale in the problem, which is why Equation (128.4) contains no material parameter. But the argument holds only at exactly integer filling, a set of measure zero in the density; it therefore explains the value of the plateau and gives no reason for there to be a plateau at all. That is the subject of Section 128.5.1 and Section 128.5.2, and it is where the disorder enters.
∎Exactness and universality
[Reserved: the property that makes the effect extraordinary — the plateau value is independent of material, geometry, mobility and carrier density; the direct comparison of quantized resistances in silicon and in gallium arsenide, agreeing to about one part in \(10^{10}\) [Hartland:1991]; the reproducibility between national laboratories [Jeckelmann:2001]; the honest statement of what is measured — a ratio, not an absolute value — and of the systematic effects (current, temperature, contacts) that limit it.]
The plateau value Equation (128.1) depends on nothing about the sample. Quantized Hall resistances measured in a silicon metal–oxide–semiconductor device and in a gallium arsenide heterostructure — different host materials, different carrier densities, different mobilities, different geometries, different disorder — agree with one another to about one part in \(10^{10}\) [Hartland:1991], and independent national laboratories reproduce the value at comparable uncertainty [Jeckelmann:2001]. This is not a small correction to a material-dependent answer; the material dependence is absent to the accuracy of the best resistance metrology in existence, in samples that are macroscopic, irreducibly disordered and individually unreproducible. The effect accordingly realizes the ohm, and the von Klitzing constant \(R_{K}=h/e^{2}\) is exact in the present SI, \(R_{K}=25812.80745\,\mathrm{\Omega}\) [BIPM:2019] [Mohr:2025].
Laughlin's gauge argument: that adiabatically threading one flux quantum through a Hall cylinder is a gauge transformation and must map the spectrum to itself, so that its only effect is to transfer an integer number of electrons from one edge to the other; the resulting Hall conductance as that integer times the square of the elementary charge over Planck's constant, with no reference to the Hamiltonian; and the requirement of a mobility gap at the Fermi level that makes the process adiabatic — together with the topological restatement in which the integer is the Chern number of the occupied states, which cannot change under any smooth deformation of the sample
The fractional plateaux
[Reserved: Tsui, Störmer and Gossard's observation, in the extreme quantum limit of a high-mobility heterostructure, of a plateau and a vanishing \(R_{xx}\) at \(\nu=1/3\) [Tsui:1982]; the subsequent hierarchy at \(\nu=p/q\) with \(q\) odd, and the even-denominator state at \(\nu=5/2\) [Willett:1987]; the energy gaps, of order \(\mathrm{K}\), measured from activated transport; why no single-particle argument can produce a fractional plateau, which is the point of Section 128.5.5.]
In a heterostructure of much higher mobility, driven into the extreme quantum limit where the lowest Landau level is only partly filled, the same measurement finds a plateau in \(R_{xy}\) and a vanishing \(R_{xx}\) at the fractional filling \(\nu=1/3\) [Tsui:1982]. Further plateaux appear as samples improve, at a hierarchy of fillings \(\nu=p/q\) with \(q\) odd, and — against the pattern — at the even-denominator value \(\nu=5/2\) [Willett:1987]. The temperature dependence of \(R_{xx}\) on each plateau is activated, so each fraction carries a genuine energy gap, of order a kelvin in temperature units. The observation is not a refinement of Phenomenon 128.1: the derivation given there required an integer number of filled levels, and within a partly filled Landau level all single-particle states are degenerate, so no one-electron argument can produce a gap at all. The gap must be made by the interactions between the electrons, and the fact that the cleanest samples are the ones that show it confirms that disorder is not its source.
Laughlin's correlated wavefunction for filling one over an odd integer, an incompressible liquid rather than a filled single-particle band; the plasma analogy that fixes its density and its excitation gap; the antisymmetry requirement that forces the exponent to be odd; and the hierarchy or composite-fermion construction that accounts for the remaining observed fractions
Interpretation
Laughlin's gauge argument
[Reserved: the argument that fixes the plateau value with almost no dynamical input — adiabatic insertion of one flux quantum \(h/e\) through a Hall cylinder must return the system to itself, so an integer number of electrons is transferred from one edge to the other and \(\sigma_{xy}=\nu e^{2}/h\) exactly [Laughlin:1981]; gauge invariance and the role of a mobility gap; why this explains exactness without explaining plateau width, which needs Section 128.5.2.]
Disorder, localization and plateau width
[Reserved: the apparent paradox that the effect is more perfectly quantized in a dirtier sample; Anderson localization [Anderson:1958] broadening each Landau level into a narrow band of extended states flanked by localized ones, so that adding carriers to localized states changes nothing measurable and the Hall resistance stays pinned [Halperin:1982]; the mobility edge and the plateau-to-plateau transition as a quantum critical point with measured scaling exponents; the honest statement that disorder is not a nuisance here but a necessary ingredient.]
Edge states
[Reserved: the confining potential bending Landau levels through the Fermi energy at the sample boundary, producing one-dimensional chiral channels that cannot backscatter because the counter-propagating channel is on the far side of the sample; Büttiker's multiterminal transport formulation, in which each edge channel contributes exactly \(e^{2}/h\) [Buttiker:1988]; the experimental signatures — contact independence, selective population and detection of individual channels; edge states as the model for every later topological boundary mode.]
Topology: TKNN and the Chern number
[Reserved: Thouless, Kohmoto, Nightingale and den Nijs' result that \(\sigma_{xy}\) for a filled band is \(e^{2}/h\) times a topological invariant — the integral of the Berry curvature of the Bloch states over the Brillouin zone, an integer Chern number [Thouless:1982]; why an integer cannot change under smooth deformation, which is the deepest available answer to why the plateau is exact; the fibre-bundle language of Lie Groups, Lie Algebras, and Fibre Bundles and the homotopy of Topological and Metric Spaces; Haldane's lattice model showing that no net field is needed, only broken time reversal [Haldane:1988].]
The Laughlin state and fractional charge
[Reserved: Laughlin's variational wavefunction \(\prod_{i<j}(z_{i}-z_{j})^{m}\ee^{-\sum\abs{z_{i}}^{2}/4\ell^{2}}\) for \(\nu=1/m\) with \(m\) odd, an incompressible correlated liquid rather than a filled single-particle band [Laughlin:1983]; its quasiholes, created by adiabatic flux insertion and carrying charge \(e/m\); the plasma analogy that fixes the charge and the fractional exchange statistics; the antisymmetry requirement of Identical Particles that forces \(m\) odd.]
Later measurements and precision
Fractional charge and anyon statistics measured
[Reserved: shot noise at a quantum point contact in the weak backscattering limit, whose Poisson noise \(2q\bar{I}\) measures the carrier charge directly; the independent 1997 determinations of \(q=e/3\) at \(\nu=1/3\) [Saminadayar:1997] [dePicciotto:1997]; and the later interference and collision experiments reporting the fractional exchange phase \(2\pi/3\) itself [Nakamura:2020] [Bartolomei:2020], the first direct evidence for statistics that are neither Bose nor Fermi — with a clear statement of what the data do and do not establish.]
Shot noise measures the charge of the carrier that crosses a barrier, because the noise power of a stream of independent discrete carriers is proportional to the product of the carrier charge and the mean current and to nothing else. Applied to weak backscattering at a quantum point contact in the \(\nu=1/3\) state, the measurement returns a carrier charge of \(e/3\), in two independent determinations published within weeks of one another [Saminadayar:1997] [dePicciotto:1997]. Nothing in the sample carries a fraction of the electron charge as a constituent — the electrons are whole — so the object being counted is a collective excitation of the correlated liquid of Phenomenon 128.3. Later interference and collision experiments in the same states report the further, and far more delicate, result that exchanging two such excitations multiplies the state by a phase which is neither \(+1\) nor \(-1\) [Nakamura:2020] [Bartolomei:2020]; these are excitations of a three-dimensional semiconductor confined to a plane, and the claim is about them and not about any elementary particle.
The quasihole of the correlated ground state and the determination of its charge from the deficit of density produced by adiabatically threading one flux quantum, together with the plasma analogy that makes the deficit exact; and the Schottky formula for the noise of a Poisson stream of independent carriers, whose proportionality to charge times current is what converts a noise measurement into a charge measurement
Composite fermions
[Reserved: Jain's construction attaching an even number of flux quanta to each electron, mapping the fractional effect at \(\nu=p/(2p\pm1)\) onto the integer effect of composite fermions in a reduced effective field [Jain:1989]; the prediction that at \(\nu=1/2\) the effective field vanishes and a Fermi sea of composite fermions forms; the measurements that support it — effective-field Shubnikov–de Haas oscillations and semiclassical geometric resonances [Du:1993]; the hierarchy of observed fractions as the strongest test.]
Graphene and the relativistic quantum Hall effect
[Reserved: the isolation of single-layer graphene and the demonstration of its field effect [Novoselov:2004]; its linear Dirac dispersion at the zone corners, so that Landau levels go as \(\sqrt{nB}\) and a zero-energy level exists; the half-integer sequence \(\sigma_{xy}=4e^{2}(n+\tfrac{1}{2})/h\) and the Berry phase \(\pi\) measured simultaneously by two groups [Novoselov:2005] [Zhang:2005]; observation of the effect at room temperature; the connection to the Dirac equation of The Dirac Equation, and the care needed — these are quasiparticles of a lattice, not elementary fermions.]
In a single atomic layer of graphite the quantized Hall sequence is not that of Equation (128.1). The plateaux are found at half-integer multiples of the fourfold-degenerate conductance quantum, \(\sigma_{xy}=4(n+\tfrac{1}{2})e^{2}/h\), measured simultaneously and independently by two groups [Novoselov:2005] [Zhang:2005]; the spacing of the Landau levels goes as the square root of the field rather than linearly in it, and a level sits at zero energy for every field. The same measurements extract a Berry phase of \(\pi\) from the phase of the Shubnikov–de Haas oscillations. Because the level spacing at accessible fields is then far larger than in a semiconductor heterostructure, the quantization survives to room temperature. The carriers responsible are low-energy excitations of a two-dimensional crystal of carbon atoms, whose dispersion near the corners of its Brillouin zone is linear; the resemblance to the equation of The Dirac Equation is a property of that band structure and not a claim about elementary particles.
Landau quantization of a linear rather than quadratic dispersion, which gives level energies proportional to the square root of the product of level index and field and places one level exactly at zero energy; the shared occupation of that zero level by electrons and holes, which shifts the filling sequence by one half; the fourfold spin and valley degeneracy of the observed steps; and the Berry phase of the cyclotron orbit that appears as the offset in the Shubnikov–de Haas phase
Quantum spin Hall and topological insulators
[Reserved: Kane and Mele's proposal that spin–orbit coupling alone, with time-reversal symmetry intact and no magnetic field, yields a \(\Z_{2}\)-classified insulator with helical edge states [Kane:2005]; the mercury telluride quantum-well prediction [Bernevig:2006] and its confirmation in transport, with a conductance plateau at \(2e^{2}/h\) independent of sample width [Konig:2007]; the first three-dimensional topological insulator, its surface Dirac cone imaged by the photoemission of Electrons in Solids: Band Theory [Hsieh:2008].]
The quantum anomalous Hall effect
[Reserved: Haldane's model realized in a material — a magnetically doped topological insulator thin film showing a quantized Hall resistance \(h/e^{2}\) at zero applied field [Chang:2013]; the required conditions (ferromagnetic order, chemical potential in the gap, temperatures of tens of \(\mathrm{mK}\)) and the honest note that the quantization there is far less precise than in Section 128.4.2; the link to the magnetism of Magnetism in Matter.]
A thin film of a magnetically doped topological insulator, cooled to tens of \(\mathrm{mK}\) and with its chemical potential tuned into the gap, shows a Hall resistance equal to \(h/e^{2}\) and a vanishing longitudinal resistance at zero applied magnetic field [Chang:2013]. Every ingredient that Phenomenon 128.1 used — the Landau levels, their degeneracy, the field itself — is absent; what remains is broken time reversal, supplied here by the ferromagnetic order of the dopants rather than by an external field. The quantization is real but is far less accurate than that of Phenomenon 128.2, and it survives only at temperatures well below the magnetic ordering temperature, which is the honest measure of how much remains to be understood about it.
That a quantized Hall conductance requires only a gapped band structure with broken time-reversal symmetry and a non-zero topological invariant of the occupied bands, not a uniform magnetic field; the lattice model exhibiting this with zero net flux through the unit cell; and the mechanism by which magnetic dopants in a strong-spin–orbit thin film open the required gap in the surface states
Metrology: the ohm and the SI
[Reserved: the von Klitzing constant \(R_{K}=h/e^{2}\), its conventional 1990 value and its status since the 2019 redefinition, in which \(h\) and \(e\) are exact and therefore \(R_{K}=25812.80745\,\mathrm{\Omega}\) exactly [BIPM:2019] [Mohr:2025]; the practical realization of the ohm from a quantized Hall sample [Jeckelmann:2001], and of the volt from the Josephson effect of Experiment: Superconductivity; the quantum metrology triangle closing volt, ohm and ampere; the original motivation — a determination of the fine-structure constant [Klitzing:1980] — and its present standing against the electron \(g-2\) route of Experiment: The Electron Anomalous Magnetic Moment; graphene resistance standards operating at relaxed field and temperature [Klitzing:2020].]