Applications
The quantization of the Hall conductance has the important property of being incredibly precise. Actual measurements of the Hall conductance have been found to be integer or fractional multiples of e2/h to nearly one part in a billion. This phenomenon, referred to as "exact quantization", has been shown to be a subtle manifestation of the principle of gauge invariance. It has allowed for the definition of a new practical standard for electrical resistance, based on the resistance quantum given by the von Klitzing constant RK = h/e2 = 25812.807557(18) Ω. This is named after Klaus von Klitzing, the discoverer of exact quantization. Since 1990, a fixed conventional value RK-90 is used in resistance calibrations worldwide. The quantum Hall effect also provides an extremely precise independent determination of the fine structure constant, a quantity of fundamental importance in quantum electrodynamics.
History
The integer quantization of the Hall conductance was originally predicted by Ando, Matsumoto, and Uemura in 1975, on the basis of an approximate calculation which they themselves did not believe to be true. Several workers subsequently observed the effect in experiments carried out on the inversion layer of MOSFETs. It was only in 1980 that Klaus von Klitzing, working with samples developed by Michael Pepper and Gerhard Dorda, made the unexpected discovery that the Hall conductivity was exactly quantized. For this finding, von Klitzing was awarded the 1985 Nobel Prize in Physics. The link between exact quantization and gauge invariance was subsequently found by Robert Laughlin. Most integer quantum Hall experiments are now performed on gallium arsenide heterostructures, although many other semiconductor materials can be used. The integer quantum Hall effect has also been found in graphene at temperatures as high as room temperature.
Integer Quantum Hall Effect - Landau Levels
In two dimensions, when classical electrons are subjected to a magnetic field they follow circular cyclotron orbits. When the system is treated quantum mechanically, these orbits are quantized. The energy levels of these quantized orbitals take on discrete values: 
, where ωc = eB/m is the cyclotron frequency. These orbitals are known as Landau levels, and at weak magnetic fields, their existence gives rise to many interesting "quantum oscillations" such as the Shubnikov-de Haas oscillations and the de Haas-van Alphen effect (which is often used to map the Fermi surface of metals). For strong magnetic fields, each Landau level is highly degenerate (i.e. there are many single particle states which have the same energy En). Specifically, for a sample of area A, in magnetic field B, the degeneracy of each Landau level is N = gsBA / φ0 (where gs represents a factor of 2 for spin degeneracy, and φ0 is the magnetic flux quantum). For sufficiently strong B-fields, each Landau level may have so many states that all of the free electrons in the system sit in only a few Landau levels; it is in this regime where one observes the quantum Hall effect.
Mathematics
The integers that appear in the Hall effect are examples of topological quantum numbers. They are known in mathematics as the first Chern numbers and are closely related to Berry's phase. A striking model of much interest in this context is the Azbel-Harper-Hofstadter model whose quantum phase diagram is the Hofstadter's butterfly shown in the figure. The vertical axis is the strength of the magnetic field and the horizontal axis is the chemical potential, which fixes the electron density. The colors represent the integer Hall conductances. Warm colors represent positive integers and cold colors negative integers. The phase diagram is fractal and has structure on all scales. In the figure there is an obvious self-similarity.
Concerning physical mechanisms, impurities and/or particular states (e.g., edge currents) seem to be important for the 'integer' effect, whereas in the fractional quantum Hall effect the Coulomb interaction is considered as the main reason. Finally, concerning the observed strong similarities between integer and fractional quantum Hall effect, the apparent tendency of electrons, to form bound states of an odd number with a magnetic flux quantum, i.e. composite fermions, is considered.
Hofstadter's Butterfly
Fuente: http://en.wikipedia.org/wiki/Quantum_Hall_effect
Nombre: Rodriguez B. Joiver I.
Asignatura: CRF
, where ωc = eB/m is the cyclotron frequency. These orbitals are known as Landau levels, and at weak magnetic fields, their existence gives rise to many interesting "quantum oscillations" such as the Shubnikov-de Haas oscillations and the de Haas-van Alphen effect (which is often used to map the Fermi surface of metals). For strong magnetic fields, each Landau level is highly degenerate (i.e. there are many single particle states which have the same energy En). Specifically, for a sample of area A, in magnetic field B, the degeneracy of each Landau level is N = gsBA / φ0 (where gs represents a factor of 2 for spin degeneracy, and φ0 is the magnetic flux quantum). For sufficiently strong B-fields, each Landau level may have so many states that all of the free electrons in the system sit in only a few Landau levels; it is in this regime where one observes the quantum Hall effect.Mathematics
The integers that appear in the Hall effect are examples of topological quantum numbers. They are known in mathematics as the first Chern numbers and are closely related to Berry's phase. A striking model of much interest in this context is the Azbel-Harper-Hofstadter model whose quantum phase diagram is the Hofstadter's butterfly shown in the figure. The vertical axis is the strength of the magnetic field and the horizontal axis is the chemical potential, which fixes the electron density. The colors represent the integer Hall conductances. Warm colors represent positive integers and cold colors negative integers. The phase diagram is fractal and has structure on all scales. In the figure there is an obvious self-similarity.
Concerning physical mechanisms, impurities and/or particular states (e.g., edge currents) seem to be important for the 'integer' effect, whereas in the fractional quantum Hall effect the Coulomb interaction is considered as the main reason. Finally, concerning the observed strong similarities between integer and fractional quantum Hall effect, the apparent tendency of electrons, to form bound states of an odd number with a magnetic flux quantum, i.e. composite fermions, is considered.
Hofstadter's Butterfly
Fuente: http://en.wikipedia.org/wiki/Quantum_Hall_effect
Nombre: Rodriguez B. Joiver I.
Asignatura: CRF
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The quantum Hall effect calculations deserve expansion, for more detailed views of the relativistic quantum effects involved in subatomic particle states and reactions. That progress depends on the data density of the atomic topological function applied. Recent advancements in quantum science have produced the picoyoctometric, 3D, interactive video atomic model imaging function, in terms of chronons and spacons for exact, quantized, relativistic animation. This format returns clear numerical data for a full spectrum of variables. The atom's RQT (relative quantum topological) data point imaging function is built by combination of the relativistic Einstein-Lorenz transform functions for time, mass, and energy with the workon quantized electromagnetic wave equations for frequency and wavelength.
ResponderEliminarThe atom labeled psi (Z) pulsates at the frequency {Nhu=e/h} by cycles of {e=m(c^2)} transformation of nuclear surface mass to forcons with joule values, followed by nuclear force absorption. This radiation process is limited only by spacetime boundaries of {Gravity-Time}, where gravity is the force binding space to psi, forming the GT integral atomic wavefunction. The expression is defined as the series expansion differential of nuclear output rates with quantum symmetry numbers assigned along the progression to give topology to the solutions.
Next, the correlation function for the manifold of internal heat capacity energy particle 3D functions is extracted by rearranging the total internal momentum function to the photon gain rule and integrating it for GT limits. This produces a series of 26 topological waveparticle functions of the five classes; {+Positron, Workon, Thermon, -Electromagneton, Magnemedon}, each the 3D data image of a type of energy intermedon of the 5/2 kT J internal energy cloud, accounting for all of them.
Those 26 energy data values intersect the sizes of the fundamental physical constants: h, h-bar, delta, nuclear magneton, beta magneton, k (series). They quantize atomic dynamics by acting as fulcrum particles. The result is the exact picoyoctometric, 3D, interactive video atomic model data point imaging function, responsive to software application keyboard input of virtual photon gain events by relativistic, quantized shifts of electron, force, and energy field states and positions. This system also gives a new equation for the magnetic flux variable B, which appears as a waveparticle of changeable frequency. Molecular modeling and chip design engineering application software developer features for programming flow are built-in.
Images of the h-bar magnetic energy waveparticle of ~175 picoyoctometers are available online at http://www.symmecon.com with the complete RQT atomic modeling manual titled The Crystalon Door, copyright TXu1-266-788. TCD conforms to the unopposed motion of disclosure in U.S. District (NM) Court of 04/02/2001 titled The Solution to the Equation of Schrodinger.