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So, you are in the world where electrons move. Actually, you've always been. Huge - so huge that we can't give you even its order of magnitude - number of electrons are under permanent motion in our bodies, in your computer you are using to access this page, inside the Sun, on its surface, even in the interstellar vacuum - in other words, absolutely everywhere. If they stop moving, mankind - Universe in general - will cease to exist. Hence, understanding this motion under miscellaneous circumstances represents the problem of paramount importance. For centuries the best human brains have been attacking Mother Nature' mysteries. Nowadays, the achievements in stealing from the God his (her?) secrets are evaluated by awarding the most prominent scientists with Nobel Prize . Here we want to tell (and mainly, to show) you about one of the most recent such award in physics - for the year 1998 which came to Prof. Horst L.Stormer and Prof. Daniel C. Tsui for the experimental discovery and Prof. Robert B. Laughlin for the theoretical explanation of the Fractional Quantum Hall Effect, or, as the Nobel foundation's press release says: "For their discovery of a new form of quantum fluid with fractionally charged excitations".
"Wait,- you will stop us here, - if there is the Fractional Quantum Hall Effect, then the Integer one also should exist?"
And, surely, you will be right. Indeed, the Integer Quantum Hall Effect also exists - it was experimentally discovered in 1980. And, of course, the guy who discovered it - Prof. Klaus von Klitzing - was awarded by the Nobel Prize for this as well (in 1985).
Well, in both cases we will not annoy you with rigid quantum mechanical treatments of these beautiful phenomena - some more or less popular explanations with more links can be found on the corresponding places on the Internet - presenting instead some animation pictures based on classical mechanics and electrodynamics which will help you to feel the majesty of our world and smartness of the human brain attacking its mysteries.
You can click on each static picture below in order to force its content to move. Before continuing, we want to warn you that even though we made our best to make uploading and showing the images as fast as possible, it might take some time to do it - especially if your connection speed is low and your computer is not fast. Please, be patient while waiting. Your time will be reimbursed by contemplation of beautiful images in motion!
So, both Fractional and Integer Quantum Hall effects deal with magnetic field influence on charged carriers - electrons. As is well known, in the uniform magnetic field (we assume that it pierces your computer screen at the right angle) electron moves along the circle, something like that:
The charged particle moves along the orbit - it means that it carries electric current. In the cases like this one shown above physicists say that it is an azimuthal current. Its value is proportional to the magnetic field and to the charge/mass ratio of the electron.
On the other hand, sole uniform electric field (say, directed vertically to the floor) will accelerate electron upward - what is pretty obvious. In this case we have longitudinal current (from your floor to your ceiling - not along the circle). What will happen when we apply both crossed magnetic and electric fields? On the first sight, the answer looks quite unexpected: the trajectory is shown below (click on it if you want to see the animation):
It appears that on average the electron moves (and, accordingly, the longitudinal current flows) in the direction perpendicular to the both fields. How can we explain this? Well, let us substitute the electric field by the absolutely hard infinite horizontal plate (with the electron above it) - leaving the magnetic field intact, of course. Is this change justifiable? In many respects - yes. The most important similarity between the two models lies in the fact that both electric field and the plate while interacting with electron push it upward banning moving to infinite depth. And let us consider this second model when electron incidents on the plate at the right angle (for the other angles reasoning is the same). Since the plate is absolutely hard, it reflects the electron with the same magnitude of velocity - only its direction has changed: from downward before reflection to upward after. What will happen after the reflection? Yes, of course, magnetic field forces the electron to move along the circular orbit. As a result, it "makes" half-circle loop and hits the plate in the other point, and the whole picture is repeated again.
As you see, even this simplest situation brings some non-trivial results and requires non-traditional thinking. Complete theory of the Integer Effect is based on advanced quantum mechanics with statistical physics involved and still is under development.
Even more dramatic events take place in the case of the Fractional Quantum Hall Effect. Here another crucial actor appears on the scene - Coulomb interaction between electrons. Of course, you know that two particles of the same sign of the electric charge repulse each other, and with the opposite signs - attract. In the semiconductor samples and at the magnetic fields values which were used for the Integer Quantum Hall Effect, Coulomb repulsion between electrons was not the major factor and could be - for the most conditions - safely neglected. However, for the samples used by Stormer and Tsui and for the larger strengths of magnetic fields effects of electron-electron interaction start to play the dominant role. So, the Fractional Quantum Hall Effect is essentially a many-body phenomenon.
Since the smallest number of electrons bigger than one is two, let us consider this situation: two interacting via Coulomb force electrons are subjected to the uniform magnetic field. Even this simplest situation carries some the most characteristic features of the systems with the bigger number of the particles. Probably, this is why Laughlin used this model (strictly speaking - its quantum mechanical case) in one of his original papers - which brought him 15 years later the most prestigious scientific award.
In the absence of the repulsion between electrons situation is clear: each of the buddies moves along circular orbit (see above), and the total circular current of the system is the sum of the two currents corresponding to each electron - so, it is twice as big as for the one electron. Let us "turn on" Coulomb interaction. It can be easily shown that in this case total motion of the system is separated into two independent motions: the centre-of-mass motion -which reduces to the usual circular motion in the uniform field , and the relative motion - the motion of, say, the second electron seen by the first electron (or vice versa). This holds true for both quantum as well as classical mechanics case. It appears that the centre-of-mass motion is the same as for the one particle in the uniform magnetic field. Introduction of Coulomb interaction between electrons essentially influences the relative motion only. If without repulsion between particles it was - similar to the centre-of-mass motion - simply circular motion, in the case of the interaction it can take very whimsical and unusual forms. We show below two of the infinite set of trajectories (As usual, you can click on the image to make it animating). Few comments are in order. First, depending on the velocity and mutual distance correlation between electrons, relative motion might be closed trajectory (as the lower example shows) - what means that the path after some time repeats itself, as well as open trajectory. In this last case - which is shown in the upper picture and animation, - "relative" electron never repeats its motion filling instead by its path the whole area inside some ring with inner and outer radii. Mathematicians say that the path "densely" fills in the ring. Second, in both cases of closed and open trajectories the path never crosses the origin - even though it can pass it very closely. This can be readily explained: passing through the origin of the relative motion means that the distance between electrons is infitely small. As a result, the repulsion force between these two charged particles is infinitely high and does not allow the interacting charges to draw together. Third, the path can't go to infinity either - it is forbidden by magnetic field.
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