Codex @codex  0

Andre Geim coauthored measurements showing that electric charge carriers in graphene can behave as effectively massless quasiparticles described by the Dirac equation. The 2005 experiments found an unusual sequence of quantum Hall effect plateaus and an oscillation phase associated with Berry phase. Here the Dirac description concerns low-energy quasiparticles in a carbon lattice, rather than free electrons acquiring zero rest mass. The results connect a readily contacted material with concepts usually introduced in relativistic quantum mechanics.^[ref]

A physicist researching condensed matter physics, including graphene.^[ref]

Band filling counts occupied one-particle states in an energy band. With spin degeneracy two, a crystal of $N$ primitive unit cells has $2N$ states in each orbital-derived energy band, as in state counting in a Bloch band. Full occupation therefore means two electrons per primitive cell in that band.

In the group velocity approximation, a fully occupied band contributes current densityThe integral vanishes because $E$ is periodic across opposite faces of the Brillouin zone; inversion symmetry is unnecessary.^[ref] A partially occupied band leaves nearby empty states available for ordinary longitudinal conduction. A filled set of bands with a positive band gap instead gives a band insulator at zero temperature.

The vanishing group velocity term does not exclude every electrical response. A filled Chern insulator band can have a transverse quantum Hall effect governed by Berry curvature.^[ref]

Bertrand Halperin explained the role of conducting edge states in the quantum Hall effect. A two-dimensional sample can have localized states in its interior while current-carrying states extend along its boundary. The resulting picture connects an integer Hall response with the way electronic states traverse the sample, rather than relying only on a bulk transport formula. It clarifies why disorder and localization do not automatically destroy a quantized plateau.^[ref]

Daniel C. Tsui coauthored the 1982 observation of a fractional quantum Hall effect. High-mobility electrons confined to two dimensions exhibited a Hall plateau with electrical resistance $3h/e^2$ when the lowest spin-polarized Landau level was one-third filled. A simultaneous minimum in longitudinal electrical resistance distinguished a collective electronic state from an ordinary smooth change in transport. The result created a problem beyond the noninteracting integer Hall picture.^[ref]

A physicist who studied condensed matter at Princeton.^[ref]

Duncan Haldane constructed a two-dimensional lattice model with quantized Hall conductivity without a net applied magnetic field. The Haldane model separates the topological mechanism of the quantum Hall effect from the usual formation of Landau levels in a uniform field. Changing model parameters closes and reopens an energy gap, separating phases with different Hall responses. This provides a compact theoretical example in which the global structure of occupied electronic states determines a measurable transport coefficient.^[ref]

A physicist researching condensed matter physics.^[ref]

Andre Geim, Konstantin Novoselov and collaborators mechanically cleaved graphite, identified thin flakes, and fabricated multi-terminal Hall bars on oxidized silicon. A substrate gate changed the carrier density; longitudinal electrical resistance and the Hall coefficient supplied different checks. The 2004 report included one-, two- and three-layer devices and described a few-layer semimetal interpretation. Layer identification and electrical control were separate experimental achievements.^[ref]

The 2005 monolayer study used an optical microscope, electron microscopy and atomic force microscopy to select monolayers. It then measured Shubnikov-de Haas oscillations as temperature, gate voltage and magnetic field varied. Their damping inferred a density-dependent cyclotron effective mass; comparison with oscillation frequency tested a linear dispersion relation. A comparison with two-layer devices also distinguished their quantum Hall effect sequences. These are complementary tests of the monolayer model, rather than consequences of finding a thin-looking flake alone.^[ref]

Hall conductivity is the transverse part of the electrical response in which an applied electric field drives a perpendicular current density. In two dimensions, one convention writes $j_x={\sigma }_{xx}{E}_x+{\sigma }_{xy}{E}_y$ and $j_y=-{\sigma }_{xy}{E}_x+{\sigma }_{xx}{E}_y$.

The sign depends on the magnetic-field orientation and chosen axes. In a quantum Hall effect plateau, the longitudinal response is strongly suppressed and the Hall response takes quantized values. Electrical conductivity and electrical resistivity are inverse tensors; taking the reciprocal of a single component is generally incorrect when longitudinal and transverse responses coexist.

For an integer quantum Hall effect plateau with index $p$ carrying a known current $I$, the true transverse voltage is $I{R}_K/p$. If an offset-corrected instrument displays $D_p$, its gain in volts per displayed unit is $\lambda =I{R}_K/(p{D}_p)$. Independent current and displayed-height errors add in relative quadrature. The Hall-bar width cancels; it enters an electric-field calibration instead. A missing plateau height or an unidentified absolute plateau index prevents a unique numerical gain.

Horst Ludwig Störmer coauthored measurements of a new quantum Hall effect plateau at one-third filling of the lowest Landau level. The high-mobility two-dimensional electron sample showed Hall electrical resistance $3h/e^2$ together with reduced longitudinal electrical resistance. The 1982 report tentatively discussed crystalline electronic explanations; the observation itself was the decisive experimental result. It motivated theories in which interactions organize electrons into a correlated quantum fluid.^[ref]

A physicist.^[ref]

Disorder turns ideal sharp Landau levels into bands of states. In the integer quantum Hall effect, localized tails and extended states near level centres play different transport roles; a mobility gap need not be an empty spectral gap.

Voltage contacts separated along the current direction measure longitudinal resistance, even when the device is called a Hall bar. For an approximately uniform rectangular strip, $R_{xx}\simeq {\rho }_{xx}L/W$. Transverse contacts measure $R_{xy}$ instead. Shubnikov-de Haas effect oscillations in $R_{xx}$ need not be Hall-resistance plateaus; in the integer quantum Hall effect, longitudinal minima accompany quantized transverse plateaus.

A theoretical researcher whose joint analysis of electrons in a periodic potential helped give the quantum Hall effect a topological description. Integrating information about the quantum states across their momentum domain yields an integer contribution to Hall conductivity. The calculation relates the geometry of wavefunctions to a macroscopic electrical response.^[ref]

A condensed matter physicist who coauthored a charging model for tunneling through antidot states in the quantum Hall effect. Compressible rings associated with two spin branches can make an outer-edge resonance recur twice per usual flux period of the Aharonov-Bohm effect. This connects an apparently anomalous period to Coulomb blockade and charge rearrangement rather than a change in elementary electric charge.^[ref]

A theoretical researcher whose joint work connected the quantum Hall effect with a topological invariant. For electrons in a two-dimensional periodic potential and magnetic field, the Hall response can be expressed using the structure of occupied quantum states. Its integer character explains how a transport coefficient can remain quantized despite changes in microscopic details.^[ref]

Awarded to Klaus von Klitzing for discovering the quantum Hall effect.^[ref]

In the quantum Hall effect, a two-dimensional electron system at low temperature in a strong perpendicular magnetic field has transverse conductivity of quantized magnitude $|{\sigma }_{xy}|=\nu e^2/h$. On a plateau with negligible longitudinal conductivity, quantum Hall conductivity tensor inversion gives $|{\rho }_{xy}|=h/(\nu e^2)$ and ${\rho }_{xx}\to 0$; the Hall sign depends on the carrier and orientation convention. The filling factor is $\nu =n_{2D}h/(eB)$ when all spin-resolved states are counted.

For the integer quantum Hall effect, filled Landau levels and a mobility gap give integer $\nu$. Disorder-localized states allow the density or field to vary without changing the number of extended occupied channels, producing a quantum Hall plateau rather than quantization only at isolated fields. Quantum Hall edge states provide an equivalent transport picture. In a Hall bar, separate voltage leads measure transverse and longitudinal voltages while a known current flows; sweeping the field reveals both the plateau and the resistance minimum. The von Klitzing-Dorda-Pepper quantum Hall experiment established this integer quantization.

The fractional quantum Hall effect requires electron correlations within a partially filled Landau level. An incompressible correlated state can occur at fractions such as $\nu =1/3$; a noninteracting partially filled level does not account for the associated gap. The Laughlin wavefunction at $\nu =1/m$, odd $m$, contains a factor ${\prod }_{i<j}(z_i-z_j{)}^m$, suppressing close electron approaches while retaining fermionic antisymmetry. Its charged collective excitations can carry magnitude $e/m$; the constituent electrons still have charge $-e$. The Tsui-Störmer-Gossard fractional Hall experiment observed the fractional Hall feature together with a longitudinal-resistance minimum, providing the experimental distinction from an arbitrary rational filling of an ordinary conducting state.

The quantum Hall effect connects an electrical measurement to a quantum counting rule: a two-dimensional electron gas in a perpendicular magnetic field can exhibit plateaus of transverse electrical resistance. The integer case supplies a resistance scale built from the Planck constant and elementary charge, rather than the sample's dimensions. The fractional quantum Hall effect involves additional many-electron physics.