A graduate student summary of the public lecture
Bose-Einstein Condensation of Excitions in Quantum Hall Systems
by Allan MacDonald
Winning Showcase essay by
Bose-Einsten condensation of excitons is no longer just a theoretical
prediction but is already evident in bilayer semiconductors. Such a
condensation has been predicted many years ago but its realization has become
possible only recently thanks to a technological improvement in producing and
probing currents in bilayer systems in the quantum Hall regime.
Excitons are pairs of particles and holes, typically found in
semiconductors where the hole belongs to the nearly full valance band and the
electron belongs to the nearly empty conduction band. Indeed, being composed
of two fermions, the excitons are bosons. Like any other bosonic system, one
can expect that excitons will undergo a Bose- Einstein condensation (BEC),
i.e., at low temperature a macroscopic number of excitons will occupy the
However, unlike molecules of 4He, the number of excitons in a system
is generally not conserved - they may be annihilated and created by various
processes. Processes that create excitons require energy (of the order of
the band gap) and are not likely to occur at the low temperature required for
One can imagine creating excitons by shining light on the system
instead of relying on thermal fluctuations. In semiconductor systems,
however, the excitonic lifetime is rather short due to recombination
processes. The short lifetime is the main obstacle for the formation of an
excitonic BEC in band-gap semiconductors.
The above reasoning has led to the search for systems in which
electrons and holes are strongly interacting but have a large energy barrier
for recombination. Such systems are the bilayer semiconductors in the Hall
regime. In bilayer systems the pairing of electrons and holes occurs between
electrons in one layer and holes in the other. The energy levels from which
the electrons and holes are taken are Landau levels (instead of the usual
semiconductor energy bands) which result from an applied magnetic field.
The advantage of using Landau levels is the flexibility to tune the
filling of these "bands". Ideally, the condensate will contain all of the
particles in the system. This can be achieved only when the number of holes
in one layer is equal to the number of electrons in the other. Since the
filling is determined by the total flux it is the same in both layers. This
means that 1/2-filling is favourable for the BEC formation.
In the bilayer system, the magnetic field can be tuned to produce a
half filled Landau level in both layers. In this situation one can describe
each layer equally well as either a liquid of electrons or a liquid of holes.
Let us think of electrons in one layer and holes in the other.
Another tunning handle in the bilayer Hall systems is the separation
between the layers. When the separation is not too large the interaction
between the levels is such that the electron-hole bound state is favoured
energetically and thus excitons are formed. When the separation between the
layers is not too small, the tunnelling of particles from one layer to the
other is suppressed and therefore the annihilation of excitons is suppressed.
An intermediate distance between the levels that satisfy both requirements is
the key to the formation of the BEC.
Experimental evidence for the existence of the exciton BEC was found
in a measurement of the tunnelling current between layers as a function of an
applied bias voltage. At large interlayer separation the tunnelling between
layers is suppressed at low voltage. This can be understood as a state of
uncorrelated layers. In each layer the fermionic particles avoid each other
in order to obey the Pauli exclusion principle. A particle from the other
layer is unaware of this delicate ‘dance’ and therefore cannot easily join
the layer. On the other hand when the interlayer distance is reduced a peak
in the tunnelling appears around zero bias. This peak in the tunnelling rate
is interpreted as evidence for a highly correlated state where all of the
particles in the system participate in an excitonic bound state. The
particles in the two layers are highly correlated and can easily tunnel since
they are already aware of the ‘dance’ performed by the particles in the other
layer. At low temperature (the experiment is done at around 400mK) the
excitons are all condensed in the ground state.
Another experiment may provide evidence for the BEC. A measurement
of the Hall current in both layers separately should reveal the charge of the
carriers in each layer. The BEC should appear at some non-zero magnetic
field where the Hall current is expected to vanish due to a uniform flow of
correlated holes and electrons in the same direction.
For more details see: J.P. Eisenstein & A.H. MacDonald, Nature 432, 691 (2004)