...gnetic field instead of the monotonic increase. hat is more, these steps occur at incredibly precise values of resistance hich are the same no matter hat sample is investigated. The resistance is quantised in units of he2 divided by an integer. This is the QUANTUM HALL EFFECT.The figure shos the integer quantum Hall effect in a GaAs-GaAlAs heterojunction, recorded at 30mK. The QHE can be seen at liquid helium temperatures, but in the millikelvin regime the plateaux are much ider. Also included is the diagonal component of resistivity, hich shos regions of zero resistance corresponding to each QHE plateau. In this figure the plateau index is, from top right, 1, 2, 3, 4, 6, 8.... Odd integers correspond to the Fermi energy being in a spin gap and even integers to an orbital LL gap. As the spin splitting is small compared to LL gaps, the odd integer plateaux are only seen at the highest magnetic fields. Important points to note areThe value of resistance only depends on the fundamental constants of physics e the electric charge and h Planks constant. It is accurate to 1 part in 100,000,000. The QHE can be used as primary a resistance standard, although 1 klitzing is a little large at 25,813 ohm!Explanation of the Quantum Hall EffectThe zeros and plateaux in the to components of the resistivity tensor are intimately connected and both can be understood in terms of the Landau levels LLs formed in a magnetic field.In the absence of magnetic field the density of states in 2D is constant as a function of energy, but in field the available states clump into Landau levels separated by the cyclotron energy, ith regions of energy beteen the LLs here there are no alloed states. As the magnetic field is sept the LLs move relative to the Fermi energy. hen the Fermi energy lies in a gap beteen LLs electrons can not move to ne states and so there is no scattering. Thus the transport is dissipationless and the resistance falls to zero. The classical Hall resistance as just given by BNe. Hoever, the number of current carrying states in each LL is eBh, so hen there are i LLs at energies belo the Fermi energy completely filled ith ieBh electrons, the Hall resistance is hie2. At integer filling factor this is exactly the same as the classical case. The difference in the QHE is that the Hall resistance can not change from the quantised value for the hole time the Fermi energy is in a gap, i.e beteen the fields a and b in the diagram, and so a plateau results. Only hen case c is reached, ith the Fermi energy in the Landau level, can the Hall voltage change and a finite value of resistance appear.This picture has assumed a fixed Fermi energy, i.e fixed carrier density, and a changing magnetic field. The QHE can also be observed by fixing the magnetic field and varying the carrier density, for instance by seeping a surface gate.Dirt and disorderAlthough it might be thought that a perfect crystal ould give the strongest effect, the QHE actually relies on the presence of dirt in the samples. The effect of dirt and disorder can best be though of as creating a background potential landscape, ith hills and valleys, in hich the electrons move. At lo temperature each electron trajectory can be dran as a contour in the landscape. Most of these contours encircle hills or valleys so do not transfer an electron from one side of the sample to another, they are localised states. A fe states just one at T0 in the middle of each LL ill be extented across the sample and carry the current. At higher temperatures the electrons have more energy so more states become delocalised and the idth of extended states increases.The gap in the density of states that gives rise to QHE plateaux is the gap beteen extended states. Thus at loer temperatures and in dirtier samples the plateaus are ider. In the highest mobility semiconductor heterojunctions the plateaux are much narroer.Some Interesting Variants on the QHEIn very high mobility samples extra plateaux appear beteen the regular quantum Hall plateaux, at resistances given by he2 divided by a rational fraction pq instead of an integer. This is the fractional quantum Hall effect FQHE. Early observations found that q as alays an odd number and that certain fractions gave rise to much stronger features than others. The FQHE is much more difficult to explain since it originates from many electron correlations, but for this reason has been of great interst to theoreticians and experimentallists alike.In some materials there are more than one species of charge carrier. These may be elecrons in different conduction band minima, different spatially confined subbands or electrons and holes simultaneously present. The numbers and mobilities of all the species have to be considered to find the transport coefficients.If there are electrons and holes the total filling factor is the difference beteen the filling factors for electrons and holes. At certain fields this can be zero, at hich point the Hall resistance itself becomes zero!aaPAGE PAGE 4I0JmHnHu0Jaj0JUajEUaj!UajUpqslm2344!45DEJKpqIa...
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