Fractional Quantum Hall Effect

In high mobility semiconductor heterojunctions the integer quantum Hall effect (IQHE) plateaux are much narrower than for lower mobility samples. Between these narrow IQHE more plateaux are seen at fractional filling factors, especially 1/3 and 2/3. This is the fractional quantum Hall effect (FQHE) whose discovery in 1982 was completely unexpected. In 1998 the Nobel Prize in Physics was awarded to Dan Tsui and Horst Stormer, the experimentalists who first observed the FQHE, jointly with Robert Laughlin who suceeded in explaing the result in terms of new quantum states of matter.

[-Experimental data showing FQHE-]The figure shows the fractional quantum Hall effect in a GaAs-GaAlAs heterojunction, recorded at 30mK. Also included is the diagonal component of resistivity, which shows regions of zero resistance corresponding to each FQHE plateau.

The principle series of fractions that have been seen are listed below. They generally get weaker going from left to right and down the page:

(The fractional quantum Hall effect (FQHE) is concerned centrally with filling factor. This is usually writen as the greek letter nu, or v due to the limitations of HTML.)


Explanation of the Fractional Quantum Hall Effect

Just as in the IQHE, FQHE plateaux are formed when the Fermi energy lies in a gap of the density of states. The difference is the origin of the energy gaps. While in the integer effect gaps are due to magnetic quantisation of the single particle motion, in the fractional effect the gaps arise from collective motion of all the electrons in the system.

For the state at filling factor 1/3 Laughlin found a many body wavefunction with a lower energy than the single particle energy. This can also be adopted at any fraction v=1/(2m+1), but the energy difference is smaller at higher m and hence the fractions become weaker along the series 1/3, 1/5, 1/7....

All tests of Laughlin's wavefunction have shown it to be correct. The difficulty that arises is in accounting for all the other fractions at v=p/q where p>1 and simple wavefunctions can not be written down. It is also necessary to explain why q is always odd.

The original explanation, developed by Haldane and Halperin, used a hierarchical model. Quasi-electrons or quasi-holes excited out of the Laughlin ground state would condense into higher order fractions, known as daughter states e.g. starting from the 1/3 parent state addition of quasi-electrons leads to 2/5 and quasi-holes leads to 2/7. Then quasi-particles are excited out of these daughter states which condense again into still more daughter states..... and so on down the hierarchy.

There are several problems or unsatisfactory features within the hierarchical model:

More recently a model of composite fermions (CFs) has been introduced. A composite fermion consists of an electron (or hole) bound to an even number of magnetic flux quanta. Formation of these CFs accounts for all the many body interactions, so only single particle effects remain. The model exploits the similarities observed in measurements of the IQHE and FQHE to map the latter onto the former. Thus the fractional QHE of electrons in an external magnetic field now becomes the integer QHE of the new composite fermions in an effective magnetic field. The CFs have integer charge, just like electrons, but because they move in an effective magnetic field they appear to have a fractional topological charge.

The composite fermion picture correctly predicts all the observed fractions including their relative intensities and the order they appear in as sample quality increases or temperature decreases. It also shows v=1/2, where the effective field for the CFs is zero, to be a special state with metallic characteristics.

More details can be found in the following articles:

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Last updated 10/02/97 by David R Leadley.

All rights reserved. Text and diagrams from this page may only be used for non-profit making academic excerises and then only when credited to D.R. Leadley, Warwick University 1997.