quantum Hall effect

quantum Hall effect
Name	Quantum Hall Effect
Caption	A schematic of the Hall effect, the classical precursor to the quantum phenomenon.
Discovered by	Klaus von Klitzing
Discovery date	1980
Related concepts	Hall effect, Landau quantization, Topological insulator

quantum Hall effect. The quantum Hall effect is a quantum-mechanical version of the Hall effect, observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields. It is characterized by the quantization of the Hall conductance into exact integer or fractional multiples of fundamental constants. This remarkable precision has made it a cornerstone of modern condensed matter physics and metrology, leading to Nobel Prizes and deep insights into topological order.

Discovery and history

The effect was first discovered in 1980 by Klaus von Klitzing while conducting experiments on the silicon MOSFET at the Grenoble High Magnetic Field Laboratory. His observation of precisely quantized Hall resistance in the integer quantum Hall effect was so unexpected and precise that it earned him the Nobel Prize in Physics in 1985. Just two years later, in 1982, Daniel Tsui and Horst Störmer, working at Bell Labs, discovered the fractional quantum Hall effect in ultra-high-mobility gallium arsenide heterostructures. This discovery, for which Tsui, Störmer, and theorist Robert Laughlin shared the 1998 Nobel Prize, revealed entirely new states of matter governed by many-body physics.

Basic principles and theory

The phenomenon occurs when a two-dimensional electron gas, such as that found in a MOSFET or a GaAs/AlGaAs heterostructure, is cooled to near absolute zero and placed in a powerful perpendicular magnetic field. The magnetic field quantizes the kinetic energy of the electrons into discrete Landau levels, a consequence of Landau quantization. When the Fermi level lies in the gap between these levels, the longitudinal resistivity vanishes, and the Hall conductance becomes quantized. The theoretical foundation relies heavily on concepts from gauge invariance and topological invariance, with the work of David Thouless and others on the TKNN invariant providing a formal explanation for the integer effect's robustness.

Integer quantum Hall effect

In the integer quantum Hall effect, the Hall conductance is quantized in integer multiples of the fundamental constant \(e^2/h\), where \(e\) is the elementary charge and \(h\) is Planck's constant. This quantization is extraordinarily precise, independent of the sample's material details or geometry, making it a primary standard for electrical resistance. The effect is explained by the existence of localized states in the tails of the Landau levels and extended states at their centers, with the Kubo formula and the concept of Chern number playing key theoretical roles. The precision of this effect led to the adoption of the von Klitzing constant in metrology.

Fractional quantum Hall effect

The fractional quantum Hall effect is a more profound many-body phenomenon where the Hall conductance is quantized in fractions of \(e^2/h\), such as 1/3 or 2/5. This indicates the formation of new quantum states where electrons, interacting strongly via Coulomb interaction, bind with magnetic flux quanta to form composite particles known as composite fermions. Robert Laughlin's groundbreaking wavefunction for the 1/3 state successfully described these incompressible quantum fluids. Subsequent theories, including the concept of anyon statistics proposed by Frank Wilczek, suggest the excitations in these states are neither fermions nor bosons but possess fractional charge and statistics.

Experimental methods and observations

Key experiments are performed using dilution refrigerators capable of reaching temperatures below 100 millikelvin and superconducting magnets producing fields exceeding 10 tesla. High-mobility samples, typically GaAs/AlGaAs heterostructures grown by molecular beam epitaxy, are patterned into a Hall bar geometry. Measurements of the longitudinal resistivity and Hall voltage reveal characteristic plateaus in the Hall resistance coinciding with zero longitudinal resistance. Advanced techniques like scanning probe microscopy and transport in graphene or other two-dimensional materials like transition metal dichalcogenides have further expanded the study of these quantum phases.

Applications and technological impact

The most direct application is in metrology, where the integer quantum Hall effect provides the resistance standard, defining the ohm in the International System of Units. The effect's precision is utilized by national metrology institutes like the National Institute of Standards and Technology and the Physikalisch-Technische Bundesanstalt. Furthermore, the study of topological states in the quantum Hall regime has profoundly influenced the field of topological insulators, with potential applications in fault-tolerant quantum computing via topological quantum computation. Research into these phenomena continues to drive advancements in nanotechnology and the understanding of exotic quantum matter.

Category:Condensed matter physics Category:Quantum mechanics Category:Electrical phenomena Category:Nobel Prize in Physics