Marginal channel model
The occurrence of the quantum Hall effect and the Schubnikow-de-Haas effect in two-dimensional electron gases can be explained with the marginal channel model. By taking edge effects into account, edge channels are formed which can explain the anomalies of the above effects.
Quantum mechanical interpretation
When an external magnetic field is applied, the density of states of the electrons changes. Landa levels develop as a result . These are discrete energy levels spin-started in small fields . The description by the marginal canal model assumes that the Fermi level is between two landing levels and the system is in the ground state . All levels below the Fermi level are therefore fully occupied.
At the edges of a sample, the otherwise negligible edge potential must be included in the Schrödinger equation. (This increase can make plausible, considering the This causes the Landau levels are turned up work function - are fed to the hurdle "edge" overcome energy must taking as analogue). This increase in levels results in intersections of the energy levels with the Fermi energy . Conditions arise at the Fermi edge , which are referred to as marginal canals . This enables the load carriers to move freely.
Classic interpretation
Classically one can describe the edge canals by so-called "skipping orbits" ("hopping orbits"). Because of Lenz's rule, electrons are forced into a curved path by a magnetic field , the cyclotron orbit . If there is an electron inside the sample, the circular path can be traversed without any further restriction ( scattering is not considered). This corresponds to the ideal view without edge effects.
If you consider electrons at a distance smaller than the cyclotron radius to the edge of the sample, it becomes clear from geometrical considerations that they can no longer run through undisturbed circular paths. They hit the edge within one cycle and are reflected there. The term "skipping orbits" is derived from this movement. This results in a net movement of the charge carriers along the boundary, which enables current to flow. The current flow is therefore restricted to the edges of the sample.
The result is a very effective spatial separation of charge carriers that move in different directions. The electronic wave function drops off quickly, so that the overlap of the states of different edges becomes very small. Furthermore, they cannot scatter to the other side of the sample, since the Fermi energy is located between two landing levels and there are no free states into which scattering could occur. The probability of scattering between charge carriers is thus effectively suppressed. As a result, the probability of backscattering also tends to zero and the line is thus free of resistance . This effect is called the Schubnikow-de-Haas effect after its discoverer.
This means that electrons that enter an edge channel at one point (e.g. a contact) have to move until the next contact. Even after they have been scattered, they are forced further in this direction. The rightly raised question of how the Fermi level can be between an occupied and an unoccupied landing level is explained by impurities and foreign atoms. Without them, the Fermi level could not be in between. This is also confirmed experimentally. In the case of extremely pure samples, the measured oscillations become weaker again; if the density of impurities is too high, the effect is suppressed by the high probability of scattering. In this case, the charge carriers can no longer run through a complete cyclotron path and interfere with themselves.
literature
- S. Datta: Electronic transport in mesoscopic systems . Cambridge University Press, Cambridge 1995, ISBN 978-0-521-59943-6 (English).
- JH Davies: The physics of low-dimensional semiconductors: An introduction . Cambridge University Press, Cambridge 1998, ISBN 978-0-521-48491-6 (English).
- D. Yoshioka: The Quantum Hall Effect . Springer Verlag, Berlin 2002, ISBN 978-3-540-43115-2 (English).
- G. Czycholl: Theoretical solid state physics . Springer Verlag, Berlin 2004, ISBN 3-540-20824-0 .
Web links
- D. Tong: Lectures on the Quantum Hall Effect . 2016, p. 46ff , arxiv : 1606.06687 (English).
- Jürgen Smoliner: Lecture notes on VO Semiconductor Electronics at the Institute for Solid State Electronics, Vienna University of Technology . 2017, p. 357ff ( tuwien.ac.at [PDF]).
- Advanced internship: The quantum Hall effect. Giessen University, I. Institute of Physics, Dept. of Micro and Nanostructuring, Prof. Dr. Peter J. Sure.