Quantum Hall Effect

The quantum Hall effect is a phenomenon in which a two-dimensional system of electrons, typically at low temperature and in a strong perpendicular magnetic field, exhibits precisely quantized transverse electrical conductance. Instead of changing continuously, the Hall response forms plateaus accompanied by very small longitudinal resistance. Its integer and fractional forms connect quantum mechanics, collective electron behavior, and topology, and provide a fundamental electrical resistance standard. (nist.gov)

Physical setting and observable quantities

In the ordinary Hall effect, a magnetic field deflects moving charge carriers through the Lorentz force. Charge accumulates across a conductor until a transverse electric field develops. The resulting Hall voltage is measured perpendicular to the applied electric current. For a simple two-dimensional system with one carrier type, classical theory predicts a Hall resistance whose magnitude is $B/(n_s e)$, where $B$ is the perpendicular field, $n_s$ is the carrier density per unit area, and $e$ is the elementary charge. (davidtong.org)

In the quantum regime, the characteristic plateau values are

where $h$ is the Planck constant and $\nu$ labels the plateau. The resistance expression applies when longitudinal resistivity vanishes; more generally, conductivity and resistivity are inverse tensors, not independently reciprocal components. Signs depend on carrier type and field orientation. The integer effect has integer plateau labels, whereas fractional states occur at particular rational values. (davidtong.org)

Experimental platforms include silicon metal–oxide–semiconductor field-effect transistors, gallium arsenide semiconductor heterostructures, and graphene. In these systems, carrier motion perpendicular to the conducting layer is restricted, while motion within the layer remains available. (tsapps.nist.gov)

Landau levels and filling factor

A perpendicular magnetic field quantizes electron orbital motion into Landau levels. For a conventional electron system with a parabolic dispersion, ignoring spin splitting,

where $N=0,1,2,\ldots$, $\hbar=h/(2\pi)$, and $m^\ast$ is the effective mass. Each level contains many distinct orbital states: its degeneracy per unit area, for each resolved internal component, is $eB/h$. The filling factor

therefore measures carrier density relative to this orbital degeneracy. Spin and other internal degrees of freedom influence the observed sequence of states. (davidtong.org)

Low temperature helps resolve the relevant energy gaps rather than thermally populating excitations across them. Graphene has a different Landau-level spectrum, including a zero-energy level and unequal spacing between levels, reflecting its approximately linear low-energy dispersion. Thus, the microscopic spectrum varies between materials even though quantized Hall resistance remains a common phenomenon. (nist.gov)

Integer quantization and topology

The integer quantum Hall effect can largely be understood using electrons occupying Landau levels together with disorder-induced localization. Disorder broadens the levels and localizes many bulk states. As magnetic field or density changes, localized states can gain or lose occupation without changing the quantized Hall conductance. This explains why plateaus occupy finite intervals rather than occurring only at isolated integer fillings. Longitudinal transport becomes appreciable during transitions between plateaus. (davidtong.org)

A complementary description uses edge states. At sample boundaries, the confining potential creates conducting channels with a preferred propagation direction. For a simple integer state, each channel contributes $e^2/h$ to conductance. An insulating bulk and conducting boundaries are therefore compatible, rather than contradictory. (davidtong.org)

The deeper quantization is expressed through a Chern number, an integer topological invariant associated with occupied electronic states. In an appropriate band description, $\sigma_{xy}=C e^2/h$, up to sign convention. This invariant cannot change continuously while the system remains in the same gapped phase. The topological formulation explains the resistance of quantization to many microscopic perturbations. (davidtong.org)

Fractional quantum Hall states

The fractional quantum Hall effect requires electron–electron interactions and cannot be explained simply by independently filling single-particle levels. At selected fractional fillings, electrons form correlated, incompressible quantum fluids. Examples include states at $\nu=1/3$ and $2/5$; not every rational filling produces such a state. (nist.gov)

For fillings $\nu=1/m$, with odd $m$, the Laughlin wave function describes an important family of electronic states. Their elementary quasiparticle excitations carry charge of magnitude $e/m$. These are collective disturbances of the fluid, not fragments into which an elementary electron has literally broken. (davidtong.org)

Such excitations can be anyons, whose exchange behavior differs from that of fermions and bosons. Fractional Hall phases also exhibit topological order, characterized by properties such as fractional excitations and ground-state degeneracy dependent on spatial topology. These features distinguish them from phases classified solely by conventional symmetry breaking. (davidtong.org)

Discovery and resistance metrology

Klaus von Klitzing discovered integer quantization in 1980 using silicon devices prepared by Gerhard Dorda and Michael Pepper, and received the 1985 Nobel Prize in Physics. Daniel Tsui and Horst Störmer discovered the fractional effect in 1982; Robert Laughlin subsequently supplied its foundational correlated-state explanation. Laughlin, Störmer, and Tsui shared the 1998 physics prize for the discovery of a quantum fluid with fractionally charged excitations. (davidtong.org)

In metrology, integer Hall plateaus realize resistance through the von Klitzing constant,

The 2019 revision of the International System of Units fixed exact values of $h$ and $e$, making $R_K$ exact, although any experimental realization retains measurement uncertainty. Graphene devices extend practical operating conditions and support arrays for realizing additional resistance values. (tsapps.nist.gov)