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.
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
This is usually writen as the greek letter nu, or v due to the limitations
1/3, 2/5, 3/7, 4/9, 5/11, 6/13, 7/15...
2/3, 3/5, 4/7, 5/9, 6/11, 7/13...
5/3, 8/5, 11/7, 14/9...
4/3, 7/5, 10/7, 13/9...
1/5, 2/9, 3/13...
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
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
it does not explain which daughter state (quasi-electron or -hole) should
be the stronger
after a few layers of the hierarchy there will be more quasi-particles
than there were electrons in the original system
between fractions the system is not well defined
the quasi-particles carry fractional 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
More details can be found in the following articles:
T. Chakraborty and P. Pietilainen, The Quantum Hall Effects-Fractional
and Integer, Springer Series in Solid-State Sciences 85, (Berlin,
D.C. Tsui, H.L. Stormer and A.C. Gossard, Phys. Rev.
Lett. 48, 1559 (1982).
R.B. Laughlin, Phys. Rev. Lett. 50,
F.D.M. Haldane, Phys. Rev. Lett. 51, 605
(1983); B.I. Halperin, Phys. Rev. Lett. 52, 1583 & 2390
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.