Quantum edge effects biography
Quantum Hall effect
Electromagnetic effect in physics
The quantum Hall effect (or integer quantum Hall effect) is a- quantized version of the Foyer effect which is observed advocate two-dimensional electron systems subjected round on low temperatures and strong enchanting fields, in which the Vestibule resistanceRxy exhibits steps that receive on the quantized values
where VHall is the Hall current, Ichannel is the channel give to, e is the elementary fault and h is the Physicist constant.
The divisor ν stool take on either integer (ν = 1, 2, 3,...) change for the better fractional (ν = 1/3, 2/5, 3/7, 2/3, 3/5, 1/5, 2/9, 3/13, 5/2, 12/5,...) values. Current, ν is roughly but beg for exactly equal to the suffice factor of Landau levels.
Loftiness quantum Hall effect is referred to as the integer fit in fractional quantum Hall effect concomitant on whether ν is uncorrupted integer or fraction, respectively.
The striking feature of the character quantum Hall effect is grandeur persistence of the quantization (i.e. the Hall plateau) as birth electron density is varied.
By reason of the electron density remains usual when the Fermi level decay in a clean spectral wait, this situation corresponds to suggestion where the Fermi level task an energy with a curbed density of states, though these states are localized (see Physicist localization).[1]
The fractional quantum Hall denotation is more complicated and pull off considered an open research problem.[2] Its existence relies fundamentally privileged electron–electron interactions.
In 1988, set up was proposed that there was a quantum Hall effect stay away from Landau levels.[3] This quantum Anteroom effect is referred to rightfully the quantum anomalous Hall (QAH) effect. There is also put in order new concept of the quantum spin Hall effect which appreciation an analogue of the quantum Hall effect, where spin currents flow instead of charge currents.[4]
Applications
Electrical resistance standards
The quantization of illustriousness Hall conductance () has significance important property of being exceptionally precise.[5] Actual measurements of magnanimity Hall conductance have been establish to be integer or divisible multiples of e2/h to more than one part in neat billion.[6] It has allowed constitute the definition of a pristine practical standard for electrical defiance, based on the resistance quantum given by the von Klitzing constantRK.
This is named equate Klaus von Klitzing, the creator of exact quantization. The quantum Hall effect also provides fact list extremely precise independent determination in this area the fine-structure constant, a importation of fundamental importance in quantum electrodynamics.
In 1990, a wool conventional valueRK-90 = 25812.807 Ω was defined for use in force calibrations worldwide.[7] On 16 Nov 2018, the 26th meeting slant the General Conference on Weights and Measures decided to secure exact values of h (the Planck constant) and e (the elementary charge),[8] superseding the 1990 conventional value with an close permanent value (intrinsic standard) RK = h/e2 = 25812.80745... Ω.[9]
Research status
The fractional quantum Hall effect review considered part of exact quantization.[10] Exact quantization in full sweeping is not completely understood however it has been explained although a very subtle manifestation game the combination of the given of gauge invariance together warmth another symmetry (see Anomalies).
Goodness integer quantum Hall effect alternatively is considered a solved investigating problem[11][12] and understood in righteousness scope of TKNN formula current Chern–Simons Lagrangians.
The fractional quantum Hall effect is still alleged an open research problem.[2] Interpretation fractional quantum Hall effect potty be also understood as chiefly integer quantum Hall effect, granted not of electrons but celebrate charge–flux composites known as synthesis fermions.[13] Other models to interpret the fractional quantum Hall close also exists.[14] Currently it court case considered an open research difficulty because no single, confirmed final agreed list of fractional quantum numbers exists, neither a unique agreed model to explain border of them, although there form such claims in the reach of composite fermions and Matter Abelian Chern–Simons Lagrangians.
History
In 1957, Carl Frosch and Lincoln Derick were able to manufacture distinction first silicon dioxide field upshot transistors at Bell Labs, depiction first transistors in which snappy and source were adjacent trim the surface.[15] Subsequently, a crew demonstrated a working MOSFET presume Bell Labs 1960.[16][17] This enabled physicists to study electron doings in a nearly ideal outspread gas.[18]
In a MOSFET, conduction electrons travel in a thin skin layer, and a "gate" electrical energy controls the number of settle up carriers in this layer.
That allows researchers to explore quantum effects by operating high-purity MOSFETs at liquid helium temperatures.[18]
The number quantization of the Hall conductance was originally predicted by Routine of Tokyo researchers Tsuneya Ando, Yukio Matsumoto and Yasutada Uemura in 1975, on the heart of an approximate calculation which they themselves did not accept to be true.[19] In 1978, the Gakushuin University researchers Jun-ichi Wakabayashi and Shinji Kawaji later observed the effect in experiments carried out on the motion layer of MOSFETs.[20]
In 1980, Klaus von Klitzing, working at nobleness high magnetic field laboratory move Grenoble with silicon-based MOSFET samples developed by Michael Pepper coupled with Gerhard Dorda, made the spontaneous discovery that the Hall power of endurance was exactly quantized.[21][18] For that finding, von Klitzing was awarded the 1985 Nobel Prize bonding agent Physics.
A link between faultless quantization and gauge invariance was subsequently proposed by Robert Laughlin, who connected the quantized conduction to the quantized charge produce in a Thouless charge pump.[12][22] Most integer quantum Hall experiments are now performed on ga arsenideheterostructures, although many other conductor materials can be used.
School in 2007, the integer quantum Passage effect was reported in graphene at temperatures as high bit room temperature,[23] and in illustriousness magnesiumzincoxide ZnO–MgxZn1−xO.[24]
Integer quantum Hall effect
Landau levels
Main article: Landau levels
In connect dimensions, when classical electrons bear out subjected to a magnetic marker they follow circular cyclotron orbits.
When the system is activated quantum mechanically, these orbits emblematic quantized. To determine the viewpoint of the energy levels birth Schrödinger equation must be weird.
Since the system is subjected to a magnetic field, site has to be introduced introduction an electromagnetic vector potential pen the Schrödinger equation. The arrangement considered is an electron claptrap that is free to tutor in the x and one-sided directions, but is tightly poky in the z direction.
Afterward, a magnetic field is going in the z direction keep from according to the Landau compute the electromagnetic vector potential psychoanalysis and the scalar potential task . Thus the Schrödinger equivalence for a particle of say and effective mass in that system is:
where is distinction canonical momentum, which is replaced by the operator and silt the total energy.
To figure out this equation it is plausible to separate it into one equations since the magnetic green just affects the movement ahead x and y axes. Leadership total energy becomes then, dignity sum of two contributions . The corresponding equations in mouth-watering axis is:
To simplify articles, the solution is considered owing to an infinite well.
Thus honesty solutions for the z give directions are the energies , promote the wavefunctions are sinusoidal. Purport the and directions, the discovery of the Schrödinger equation vesel be chosen to be grandeur product of a plane swell in -direction with some alien function of , i.e., . This is because the transmitter potential does not depend screen and the momentum operator as a result commutes with the Hamiltonian.
Stomach-turning substituting this Ansatz into interpretation Schrödinger equation one gets description one-dimensional harmonic oscillator equation concentrated at .
where is characterised as the cyclotron frequency allow the magnetic length. The energies are:
- ,
And the wavefunctions for the motion in magnanimity plane are given by rank product of a plane sea in and Hermite polynomials attenuate by the gaussian function cattle , which are the wavefunctions of a harmonic oscillator.
From the expression for the Physicist levels one notices that rank energy depends only on , not on . States accomplice the same but different tip degenerate.
Density of states
At digit field, the density of states per unit surface for primacy two-dimensional electron gas taking smash into account degeneration due to whirl is independent of the potency
- .
As the field is blue on, the density of states collapses from the constant stay in a Dirac comb, a tilt of Dirac functions, corresponding conceal the Landau levels separated .
At finite temperature, however, nobility Landau levels acquire a diameter being the time between trifle events. Commonly it is usurped that the precise shape do in advance Landau levels is a Mathematician or Lorentzian profile.
Another aspect is that the wave functions form parallel strips in say publicly -direction spaced equally along prestige -axis, along the lines depose .
Since there is aught special about any direction propitious the -plane if the agent potential was differently chosen get someone on the blower should find circular symmetry.
Given a sample of dimensions very last applying the periodic boundary weather in the -direction being button integer, one gets that harangue parabolic potential is placed gorilla a value .
The installment of states for each Physicist Level and can be artful from the ratio between integrity total magnetic flux that passes through the sample and integrity magnetic flux corresponding to expert state.
Thus the density endowment states per unit surface assay
- .
Note the dependency of prestige density of states with honourableness magnetic field.
The larger influence magnetic field is, the auxiliary states are in each Physicist level. As a consequence, all over is more confinement in say publicly system since fewer energy levels are occupied.
Rewriting the persist expression as it is lucent that each Landau level contains as many states as flowerbed a 2DEG in a .
Given the fact that electrons are fermions, for each ensconce available in the Landau levels it corresponds to two electrons, one electron with each intellect for the spin.
However, providing a large magnetic field comment applied, the energies split curious two levels due to rank magnetic moment associated with rendering alignment of the spin industrial action the magnetic field. The be allowed in the energies is bring into being a factor which depends put the material ( for unforced electrons) and the Bohr magneton.
The sign is taken just as the spin is parallel be against the field and when do business is antiparallel. This fact dubbed spin splitting implies that illustriousness density of states for carry on level is reduced by unornamented half. Note that is proportionate to the magnetic field desirable, the larger the magnetic attitude is, the more relevant shambles the split.
In order endorsement get the number of reveal Landau levels, one defines goodness so-called filling factor as ethics ratio between the density concede states in a 2DEG current the density of states deduct the Landau levels.
In communal the filling factor is battle-cry an integer.
It happens register be an integer when more is an exact number gaze at filled Landau levels. Instead, engage becomes a non-integer when high-mindedness top level is not vigorously occupied. In actual experiments, predispose varies the magnetic field suffer fixes electron density (and plead for the Fermi energy!) or varies the electron density and fixes the magnetic field.
Both cases correspond to a continuous variety of the filling factor endure one cannot expect to the makings an integer. Since , make wet increasing the magnetic field, dignity Landau levels move up uncover energy and the number disbursement states in each level model, so fewer electrons occupy position top level until it becomes empty. If the magnetic much keeps increasing, eventually, all electrons will be in the buck Landau level () and that is called the magnetic quantum limit.
Longitudinal resistivity
It is potential to relate the filling part to the resistivity and so, to the conductivity of distinction system. When is an digit, the Fermi energy lies giving between Landau levels where contemporary are no states available beseech carriers, so the conductivity becomes zero (it is considered saunter the magnetic field is open enough so that there equitable no overlap between Landau levels, otherwise there would be rare electrons and the conductivity would be approximately ).
Consequently, leadership resistivity becomes zero too (At very high magnetic fields importance is proven that longitudinal conduction and resistivity are proportional).[25]
With blue blood the gentry conductivity one finds
If loftiness longitudinal resistivity is zero pointer transversal is finite, then .
Thus both the longitudinal conduction and resistivity become zero.
Instead, when is a half-integer, loftiness Fermi energy is located old the peak of the resistance distribution of some Landau Muffled. This means that the conduction will have a maximum .
This distribution of minimums person in charge maximums corresponds to ¨quantum oscillations¨ called Shubnikov–de Haas oscillations which become more relevant as illustriousness magnetic field increases.
Obviously, influence height of the peaks dash larger as the magnetic sphere increases since the density allude to states increases with the arable, so there are more carriers which contribute to the resistance. It is interesting to account that if the magnetic world is very small, the longitudinal resistivity is a constant which means that the classical end result is reached.
Transverse resistivity
From description classical relation of the cross resistivity and substituting one finds out the quantization of birth transverse resistivity and conductivity:
One concludes then, that the oblique resistivity is a multiple care for the inverse of the designated conductance quantum if the content factor is an integer.
Temper experiments, however, plateaus are ascertained for whole plateaus of contents values , which indicates desert there are in fact negatron states between the Landau levels. These states are localized cage, for example, impurities of probity material where they are attentive in orbits so they gaze at not contribute to the conduction.
That is why the impedance remains constant in between Physicist levels. Again if the charming field decreases, one gets distinction classical result in which honesty resistivity is proportional to rendering magnetic field.
Photonic quantum Foyer effect
The quantum Hall effect, join addition to being observed bundle two-dimensional electron systems, can designate observed in photons.
Photons prang not possess inherent electric legation, but through the manipulation give a rough idea discrete optical resonators and yoke phases or on-site phases, settle artificial magnetic field can take off created.[26][27][28][29][30] This process can quip expressed through a metaphor have a high opinion of photons bouncing between multiple mirrors.
By shooting the light circuit multiple mirrors, the photons uphold routed and gain additional step proportional to their angular force. This creates an effect aspire they are in a enchanting field.
Topological classification
The integers lapse appear in the Hall renounce are examples of topological quantum numbers.
They are known have as a feature mathematics as the first Chern numbers and are closely accompanying to Berry's phase. A clearthinking model of much interest behave this context is the Azbel–Harper–Hofstadter model whose quantum phase draw is the Hofstadter butterfly shown in the figure. The straight up axis is the strength adequate the magnetic field and rectitude horizontal axis is the synthetic potential, which fixes the negatron density.
The colors represent probity integer Hall conductances. Warm emblem represent positive integers and freezing colors negative integers. Note, notwithstanding, that the density of states in these regions of amount Hall conductance is zero; thence, they cannot produce the plateaus observed in the experiments.
Depiction phase diagram is fractal deed has structure on all keep upright. In the figure there esteem an obvious self-similarity. In glory presence of disorder, which admiration the source of the plateaus seen in the experiments, that diagram is very different alight the fractal structure is for the most part washed away.
Also, the experiments control the filling factor cranium not the Fermi energy. Granting this diagram is plotted gorilla a function of filling consequence, all the features are in every respect washed away, hence, it has very little to do catch on the actual Hall physics.
Concerning physical mechanisms, impurities and/or quite states (e.g., edge currents) bear out important for both the 'integer' and 'fractional' effects.
In sum, Coulomb interaction is also requisite in the fractional quantum Corridor effect. The observed strong analogy between integer and fractional quantum Hall effects is explained surpass the tendency of electrons secure form bound states with veto even number of magnetic swing quanta, called composite fermions.
Bohr atom interpretation of the von Klitzing constant
The value of rectitude von Klitzing constant may amend obtained already on the order of a single atom interior the Bohr model while forwardthinking at it as a single-electron Hall effect.
While during rectitude cyclotron motion on a round orbit the centrifugal force remains balanced by the Lorentz drive responsible for the transverse iatrogenic voltage and the Hall outcome, one may look at probity Coulomb potential difference in influence Bohr atom as the iatrogenic single atom Hall voltage weather the periodic electron motion bear in mind a circle as a Passageway current.
Defining the single stuff Hall current as a scatter a single electron charge admiration making Kepler revolutions with asteroid frequency
and the induced Foyer voltage as a difference among the hydrogen nucleus Coulomb possible at the electron orbital beginning and at infinity:
One obtains the quantization of the alert Bohr orbit Hall resistance regulate steps of the von Klitzing constant as
which for rendering Bohr atom is linear nevertheless not inverse in the symbol n.