Bose-Einstein condensation of trapped polaritons in two-dimensional electron-hole systems
in a high magnetic field
Oleg L. Berman,
1
Roman Ya. Kezerashvili,
1,2
and Yurii E. Lozovik
3
1
Physics Department, New York City College of Technology, The City University of New York, Brooklyn, New York 11201, USA
2
The Graduate School and University Center, The City University of New York, New York, New York 10016, USA
3
Institute of Spectroscopy, Russian Academy of Sciences, 142190 Troitsk, Moscow Region, Russia
Received 25 May 2009; published 1 September 2009
The Bose-Einstein condensation BEC of magnetoexcitonic polaritons magnetopolaritons in two-
dimensional 2D electron-hole system embedded in a semiconductor microcavity in a high magnetic field B is
predicted. There are two physical realizations of 2D electron-hole system under consideration: a graphene layer
and quantum well QW. A 2D gas of magnetopolaritons is considered in a planar harmonic potential trap. Two
possible physical realizations of this trapping potential are assumed: inhomogeneous local stress or harmonic
electric field potential applied to excitons and a parabolic shape of the semiconductor cavity causing the
trapping of microcavity photons. The effective Hamiltonian of the ideal gas of cavity polaritons in a QW and
graphene in a high magnetic field and the BEC temperature as functions of magnetic field are obtained. It is
shown that the effective polariton mass M
eff
increases with magnetic field as B
1/2
. The BEC critical tempera-
ture T
c
0
decreases as B
-1/4
and increases with the spring constant of the parabolic trap. The Rabi splitting
related to the creation of a magnetoexciton in a high magnetic field in graphene and QW is obtained. It is
shown that Rabi splitting in graphene can be controlled by the external magnetic field since it is proportional
to B
-1/4
while in a QW the Rabi splitting does not depend on the magnetic field when it is strong.
DOI: 10.1103/PhysRevB.80.115302 PACS numbers: 71.36.c, 03.75.Hh, 73.20.Mf
I. INTRODUCTION
In the past decade, Bose coherent effects of two-
dimensional 2D excitonic polaritons in a quantum well em-
bedded in a semiconductor microcavity have been the sub-
ject of theoretical and experimental studies.
1,2
To obtain
polaritons, two mirrors placed opposite each other form a
microcavity and quantum wells are embedded within the
cavity at the antinodes of the confined optical mode. The
resonant exciton-photon interaction results in the Rabi split-
ting of the excitation spectrum. Two polariton branches ap-
pear in the spectrum due to the resonant exciton-photon cou-
pling. The lower polariton branch of the spectrum has a
minimum at zero momentum. The effective mass of the
lower polariton is extremely small and lies in the range
10
-5
–10
-4
of the free-electron mass. These lower polaritons
form a 2D weakly interacting Bose gas. The extremely light
mass of these bosonic quasiparticles, which corresponds to
experimentally achievable excitonic densities, result in a
relatively high critical temperature for superfluidity, of 100 K
or even higher. The reason for such a high critical tempera-
ture is that the 2D thermal de Broglie wavelength is in-
versely proportional to the mass of the quasiparticle.
While at finite temperatures there is no true Bose-Einstein
condensation BEC in any infinite untrapped 2D system, a
true 2D BEC quantum-phase transition can be obtained in
the presence of a confining potential.
3,4
Recently, the polari-
tons in a harmonic potential trap have been studied experi-
mentally in a GaAs/AlAs quantum well embedded in a
GaAs/AlGaAs microcavity.
5
In this trap, the exciton energy
is shifted using a stress-induced band gap. In this system,
evidence for the BEC of polaritons in a quantum well has
been observed.
6
The theory of the BEC and superfluidity of
excitonic polaritons in a quantum well without magnetic
field in a parabolic trap has been developed in Ref. 7. The
Bose condensation of polaritons is caused by their bosonic
character.
6–8
While the 2D electron system was studied in quantum
wells
9
in the past decade, a novel type of 2D electron system
was experimentally obtained in graphene, which is a 2D hon-
eycomb lattice of the carbon atoms that form the basic planar
structure in graphite.
10,11
Due to unusual properties of the
band structure, electronic properties of graphene became the
object of many recent experimental and theoretical
studies.
10–16
Graphene is a gapless semiconductor with mass-
less electrons and holes which have been described as Dirac
fermions.
17
The unique electronic properties in graphene in a
magnetic field have been studied recently.
18–21
The electron-
photon interaction in graphene was discussed, for example,
in Ref. 22. The energy spectrum and the wave functions of
magnetoexcitons, or electron-hole pairs in a magnetic field,
in graphene have been calculated in interesting works.
23,24
The spatially indirect excitons in coupled quantum wells
CQWs with and without a magnetic field B have been stud-
ied recently experimentally in Refs. 25–28. The experimental
and theoretical interest in these systems is particularly due to
the possibility of the BEC and superfluidity of indirect exci-
tons, which can manifest in the CQW as persistent electrical
currents in each well and also through coherent optical prop-
erties and Josephson phenomena.
29–32
Since the exciton bind-
ing energies increase with magnetic field, 2D magnetoexci-
tons survive in a substantially wider temperature range in
high magnetic fields.
33–39
The BEC and superfluidity of spa-
tially indirect magnetoexcitons with spatially separated elec-
trons and holes have been studied in graphene bilayer
40
and
graphene superlattice.
41,42
The electron-hole pair condensa-
tion in the graphene-based bilayers have been studied in.
43–46
However, the polaritons in graphene in high magnetic field
PHYSICAL REVIEW B 80, 115302 2009
1098-0121/2009/8011/11530211 ©2009 The American Physical Society 115302-1