Score quantity Hall effect

Introduction

Before the three of them newly discovered, physicists believe that in addition to a class of quarks, elementary particles in the universe brought charges are all carried by an electron charge -e (e = 1.6 × 10-19 coulombs) is an integer multiple. And according to their category may quark with ± 1e / 3 or ± 2e / 3 charges. Under normal conditions quark, only present in the nucleus, they are not free flowing electrons. Therefore, in general do not expect physicist condensate system can be seen as quark fraction having particles like electrons or excited state charge.

The idea was challenged in 1982, Cui Qi and Stemmer in two-dimensional electron systems, found that the fractional quantum Hall effect. Houlaokelin year proposed a novel theory that the two dimensional electron gas system due to the Coulomb interaction between electrons of power, may form a quantum incompressible liquid (incompressiblequantumfluid) in a strong magnetic field, exhibit a fractional charge . There are no fractional charge can be said to be very mysterious, and surprising, in fact, it can be derived from the known quantum rules.

Laughlin had also wanted to use his theory to explain why quarks mixed fractions electron charge, although this idea has not been successful. After Laughlin's theory appears, immediately the other a theoretical physicist determination is the right idea. But for many people, his theory is still difficult to understand. After that five to six years, a number of important papers start to appear, the more obscure the Laughlin theory explain the concept more clearly, but also further promote his theory to a number of different physical conditions, so that the whole theory more complete. The following is a brief description of what the fractional quantum Hall effect, and its theoretical explanation. Object

Hall conductance

Our study is a two-dimensional electron gas. Assumed that the electronic activity only on the x-y plane, while a uniform magnetic field in the z-axis direction B. Hall effect is the x-axis direction when a current I, in the y-axis direction will be a potential difference V _H .

We measured the I and V _H Hall conductance can be obtained R _H ,

E _{H < / sub> is the y-axis direction electric field strength, J is the current density. Let's start with the angle of view of classical electromagnetism R H is.}

v proceeds when the electron in the negative x-axis direction at a speed, it will be along the negative power eE y-axis direction of the Lorentz force, the size of evB / c, y-axis direction will also be < sub> H . These two forces must be equal to the electronic offset in the x-axis without moving. H is proportional to the relationship between the magnetic field B

Therefore, in the classical theory, we would expect the measured R _{. However, Cui Qi and Stemmer The amount to R H . We see R H and B is not a simple proportional relationship, but when R will remain H up to the vicinity of the special magnetic field (indicated by arrow) of change, we can see that as the platform-like area appears. Then R H and then increased to the next platform, in particular if the two-dimensional electron system in the vicinity of the magnetic field that have particular stability. In R H while in the platform, parallel to the current direction and landed potential difference V is zero, meaning that a certain time into the two-dimensional electron superfluid state, the current I is not required by the potentiometer V poor to drive.}

quantum Hall effect

experimental data closer look will find, on the platform R _{H value} is h / e ² (where h is Planck's constant) multiplied by 1 / ν. ν may be an integer 1,2,3 ... etc., or 1 / 3,2 / 3,2 / 5, ...... like score. When ν is an integer, the integer quantum Hall effect we call; if ν is a fraction, we call it fractional quantum Hall effect .

Why it is a "quantum" effect it? Because Planck's constant appeared. This is from the classic formula is not apparent. In fact, the integer quantum Hall effect, the German physicist von Klitzing (vonKlitzing) discovered in 1980, he also won the Nobel Prize in physics in 1985. Cui Qi Stemmer and further lower the high magnetic field and temperature conditions found that the fractional quantum Hall effect. Next, a brief introduction to how quantum mechanics from the point of view of the Hall effect, and explain the meaning of ν.

In quantum mechanics, the electron wave is considered, it follows the movement of the Schrödinger equation, the solution is necessary to understand the electronic behavior of the Schrodinger equation. When a large number of electrons, and when the strong interaction between electrons can not be ignored, for the Schrodinger equation we are almost impossible to get a complete and accurate solution. Laughlin's contribution is that he can write a wave function, to express the important physical properties of two-dimensional electron systems. To understand this theory, you must first know if you ignore the Coulomb interaction between the electrons, the behavior of a single electron in the magnetic field of. This issue has been well-known Russian physicist Landau (LevLandau) in 1930 solved. He found that in some two-dimensional electron only electronic states, where n and i are marked quantum state quantum number.

quantum state having an energy En. This is called En Landau level. E is important regardless of the quantum number i, i.e., there are many different quantum states have the same energy, i.e. degenerate state (degeneratestates) occurs. How many different quantum states are on the same level it? Landau from solutions we can calculate the total of N _d a, and it can be seen even at different energy levels, which are the same number of degenerate states,

where A is an electron in the active area of ​​the region on a two-dimensional plane, BA is the magnetic flux. ψ ₀ is defined as a flux quantum (fluxquantum) units. Roughly speaking, it corresponds to a magnetic flux quantum a quantum state. It appeared earlier that the parameter is defined as

where N _e as the number of electrons. For example, when ν = 1, and the same number of electrons to the lowest energy level states degenerate number.

because electrons are fermions, two electrons can not fall on the same quantum state, so N _e Ne electronic quantum states requires, when it is just 1 ν = Ne electrons occupy the lowest Landau level. If ν = 1/3 The number of the electrons only N _d 1/3, i.e., only every three quantum states "assigned" to an electron. From the physical meaning of the parameters ν can see why people would call it Fill Factor (fillingfactor). ν = 1 represents the electron completely fill the lowest energy, ν = 1/3 is the lowest energy level of the quantum state is only one third of the electrons fill up. Next we calculate the fill factor when the system is ν, RH have much?

We can understand why ν = 1, (or ν = 2,3, ...) the system will be "stability":

Because electronic completely fill the lowest energy level ( or the lowest energy levels 2, 3, ...). (In fact, to fully explain the existence of the platform, we have to consider the effect of impurities, but that is not the scope of this article to discuss.) However, at ν = 1/3, electronic behavior, what is special about it? Before explaining the first description of a novel concept: any child (anyon). We know that in the microscopic world of particles into fermions (fermion) and bosons (boson). In the case of two particles, for example, if two particles of quantum states, this wave function, i.e., a particle located in the probability amplitude.

when the particle Fermi neutrons, θ = ± π, ± 3π, ......,

When the Boson particles, θ = 0, ± 2π, ± 4π, .......

So for fermions, it represents two fermions can not be on the same point (quantum states). However, the above classification applies only on the condition of three-dimensional space. If the particles can only act in two dimensions, it may be any real number, not limited to θ = 0, ± π, ± 2π, ....... This category is neither a fermion nor boson particle is called any child. This is the first to propose the concept of two Norwegian academics Laina Si (J.M.Leinaas) and Michael Han (J.Myrheim). Later, American scholar Wei Qieke (F.Wilczek) in the following manner to clarify any child.

to imagine with a charged particle with a (ideally) zero radius beam flux (magneticfluxtube), assuming that the charge is e, the magnetic flux of φ. Such particles, if two switching positions.

We fixed to the second particles, the first particles from the second particles to the bypass, and then pan left to move so that, at the same time move. This process we can calculate the probability amplitude of quantum mechanics, the beam flux particles are found than when the amplitude of a phase without multiple factor (phasefactor).

This extra phase factor is substantially due to the charge of a particle is affected by the magnetic flux from the second beam of particles, this is purely quantum effect.

In classical physics, and no magnetic field on the first particle through the road, so the first particle is not affected by the second magnetic flux particle beam. The quantum effect is peculiar when the Aha Ivanov (Y.Aharonov) and Bohm (D.Bohm) discovered in 1959, so this factor is called A-B phase factor.

due to the presence of the A-B phase, bosons fermions transformed into a! [Strictly speaking, for any child, the wave function is not a good idea. Careful readers may have found that if counter-clockwise direction bypassing the A-B phase factor eiθ becomes eiθ. In other words, the wave function of the particle motion will be subject to the process varies, so the number of Han and the general wave different meaning. Feynman (R.P.Feynman) path integral (pathintegral) Here is a better expression language. ] Since electrons are fermions, so we can think of it as one kind of bosons (referred to as Bose electrons), which is the same electric charge and electrons, and are 2n + 1 accompanied with the flux quantum.

Statistics field

emphasized here differs from the magnetic flux quantum in a uniform magnetic field actually penetrates the experimental system B. This change statistical behavior of particles (the fermions into bosons or any other sub) field, we can be called a virtual statistical magnetic field or magnetic field. In fact, statistics magnetic field situation is not necessarily toward the positive z-axis direction; it can also point to the negative z-axis direction, but there will still turn into a Bose-electronic electronic functions.

and we come back to see the electronic system fill factor ν = 1/3. Then Bose electronics will also feel the effect of the applied magnetic field B and magnetic field of statistics. If the selected statistical magnetic field toward the negative z-axis direction, then the average magnetic field B and the statistical magnetic field cancel each other: Because of (7) shows that ν = 1/3, an electronic "Assignment" has fluxon three external magnetic field B its size is exactly equal to the three flux quantum statistics; therefore Bose electronics will think it is not subjected to a magnetic field. The following description ν = 1/3. By the absence of magnetic field, electrons via Porth Bose - Einstein polymerization (Bose-Einsteincondensation) effect aggregated together into a superfluid state, which indicates the stability of ν = 1/3 in. As for the other platforms fill factor

ν = 1 / 5,2 / 3,2 / 5,3 / 5,1 / 7, ......

and the like may be used glass color electronic point of view to understand.

interaction between the electrons is very important, because the relationship between mutually exclusive, and will not spread like electronic move closer together, so as the statistics of the magnetic field the electrons move to be able to offset evenly applied magnetic field . In other words, since the Coulomb repulsion, the formation of a new form of two-dimensional electron incompressible fluid, the above idea can only be achieved.

In fact, the idea was not initially Laughlin Bose electrons to explain the experimental data, but a guess first mentioned earlier two-dimensional electron ground state wave function for ν = 1/3-state, and calculates this wave function corresponding to the electron density is exactly the electron density ν = 1/3 states. He also proved that when the small number of electrons, the wave function and numerical solutions are very close, almost unanimously. He wave functions may be applied to the adjusted parameters ν = 1 / 5,1 / 7, ...... state.

The first function Lauck Limpo and Bose electronics to link up the idea of ​​the American physicist Wen Ge (S.Girvin) and McDonnell (A.H.MacDonald). Laughlin write further excited state wave function ν = 1/3, wherein the charge indicated (quasi-) particles is ± 1 / 3e. In general, we can see: 1 because when ν / = 3, with every three electron flux quantum, if the system is a multi-flux does not match with the electron beam, with respect to the uniform charge distribution, this flux with a beam like electron charge 1/3. Be careful to speak the truth also requires more math, I do not say here.

quantum Hall effect is one of the most important advances in condensed matter physics research over the past two decades, the experimental and theoretical aspects of surprising discoveries and ideas. So once again the Nobel Prize in recognition of achievements in this field over the years, is very appropriate.