On Bose condensation of excitons in quasi-two-dimensional semiconductorheterostructuresV. B. Timofeev Citation: Low Temperature Physics 38, 541 (2012); doi: 10.1063/1.4733681 View online: http://dx.doi.org/10.1063/1.4733681 View Table of Contents: http://scitation.aip.org/content/aip/journal/ltp/38/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Strong coupling at room temperature in ultracompact flexible metallic microcavities Appl. Phys. Lett. 102, 011118 (2013); 10.1063/1.4773881 Suppression of cross-hatched polariton disorder in GaAs/AlAs microcavities by strain compensation Appl. Phys. Lett. 101, 041114 (2012); 10.1063/1.4739245 Bose-Einstein condensation of dipolar excitons in lateral traps Low Temp. Phys. 37, 179 (2011); 10.1063/1.3570931 Phase effects on the exciton polariton amplifier Appl. Phys. Lett. 91, 191112 (2007); 10.1063/1.2807280 Polariton parametric luminescence in a single micropillar Appl. Phys. Lett. 90, 051107 (2007); 10.1063/1.2435515
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 142.150.190.39
On: Sat, 20 Dec 2014 13:32:48
OPTICS AND MAGNETOOPTICS
On Bose condensation of excitons in quasi-two-dimensional semiconductorheterostructures
V. B. Timofeeva)
Institute of Solid State Physics, Russian Academy of Sciences, Chernogolovka 142432, Russia(Submitted March 19, 2012)
Fiz. Nizk. Temp. 38, 693–702 (July 2012)
Two semiconductor systems, quantum wells with spatially indirect dipolar excitons and exciton
polaritons in semiconductor microcavities, exhibiting the Bose condensation of excitons, are dis-
cussed. VC 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4733681]
1. Introduction
Hydrogen-like excitons in semiconductors are lowest
and electrically neutral intrinsic electronic excitations. For
nearly half a century, the excitons are successfully used as a
convenient physical object that allows to simulate the behav-
ior of matter at variation of density and external factors—
temperature, pressure, electric and magnetic fields, etc.
Depending on the concentration of electron-hole excitations
and the temperature, in an experiment there can be realized
situations of a weakly interacting exciton gas, a molecular
exciton gas (or gas of biexcitons), a spin-reoriented exciton
gas, a metallic electron-hole liquid, or drops of the EHL, an
electron-hole plasma, etc.
An exciton is composed of two Fermi particles, an elec-
tron and a hole, coupled by the Coulomb attraction, so the
resulting exciton’s spin is integer, and the hydrogen-like exci-
ton is a composite boson. That was a basis of the hypothesis,
formulated in the early 60s of last century, that the Bose-
Einstein condensation (BEC) is possible in a weakly nonideal
and diluted gas of excitons (nadex � 1, n is the exciton density,
aex is the exciton Bohr radius, d is the dimension of the sys-
tem), cooled to sufficiently low temperatures.1–5 According to
Einstein’s works6 (see also Refs. 7 and 8), in an ideal gas of
identical and non-interacting Bose particles, BEC occurs
when the de Broglie wavelength exceeds the mean interpar-
ticle distance, kdB ¼ ð2p�h2=mexkBTÞ1=2 � n�1=2, n is the gas
density. Under these conditions, the total free energy of a sys-
tem of bosons is minimized. Bose condensation is accompa-
nied by a macroscopic population of the ground state with
zero angular momentum and the emergence of spontaneous
order parameter (coherence) in the condensate, which is
destroyed by thermal fluctuations.8,9 Bose particles condensed
into such a state form a collective state, called the Bose-
Einstein condensate, which is a large-scale coherent matter
wave.9 Due to quantum mechanical effects of the interparticle
exchange interaction the emerged quantum state is stable,
since contributions of exchange interactions are added coher-
ently. Individual properties of Bose particles in the condensate
are lost, and the condensate demonstrates the collective coher-
ent properties at macroscopic scale.
In the limit of large electron-hole density (na3ex � 1)
excitons have been considered in direct analogy to Cooper
pairs, and the condensed excitonic state itself, or the state of
an excitonic insulator, have been described in a mean-field
approximation, similar to the superconducting state of
Bardeen-Cooper-Schrieffer (BCS) with the difference being
that a pairing in the excitonic insulator is determined by the
electron-hole interaction, and excitons themselves play a
role of Cooper pairs.3–5 The transition between the high and
low density, according to theoretical studies performed in
Ref. 5, is smooth.
It is worth recalling here that dissipationless flow of a
matter - superfluidity of 3He, 4He, and superconductivity in
metals - are directly related to the Bose condensation of com-
posite bosons, in the case of superconductors—to fermions
bound in Cooper pairs which are also composite bosons.9 The
possible relation of 4He superfluidity with the BEC phenom-
enon has first been noticed by London in 1938.10
Since Einstein predicted the Bose condensation phenom-
enon, it took about 70 yr before the BEC was found experi-
mentally in diluted and highly cooled gases of atoms of
alkali metals with a resulting integer spin (see the review11).
This remarkable achievement, recognized by the Nobel
Prize, was made possible thanks to an elegant implementa-
tion of a technique of laser and evaporative cooling of gas of
Bose atoms, selectively accumulated in limited volumes—
magneto-optical traps.11 The transition temperatures TC in
the case of atoms of Bose gas have appeared to be extremely
low, at microkelvin scale and below, that is due to large
masses of atoms and relatively low gas densities because of
the inevitable losses at accumulation of atoms in traps and
their evaporative cooling.
As a consequence of the discovery of the BEC phenom-
enon in highly cooled and diluted gases of Bose atoms an in-
terest in excitons, as a fundamentally different and new
object of experimental investigations in this rapidly evolving
field, has increased significantly and assumed new impor-
tance. The attractiveness of such an object is caused primar-
ily by the fact that translational effective masses of excitons
in semiconductors are by several orders of magnitude
smaller than atomic masses, so it was expected that the BEC
in an exciton gas can occur at standard cryogenic tempera-
tures, comprising several units or even tens of Kelvin. How-
ever, on the other hand, in contrast to atoms, excitons are
metastable with finite lifetimes and a dissipative and thermo-
dynamically nonequilibrium system of interacting Bose
1063-777X/2012/38(7)/8/$32.00 VC 2012 American Institute of Physics541
LOW TEMPERATURE PHYSICS VOLUME 38, NUMBER 7 JULY 2012
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 142.150.190.39
On: Sat, 20 Dec 2014 13:32:48
quasiparticles. Could Bose condensation of excitons take
place in such conditions? An answer to the question could be
given only by an experiment.
In recent years, an interest in this issue is being focused
on excitons in quasi-two-dimensional systems. The most
suitable for the detection of BEC are objects in which the
rate of radiative exciton annihilation is by several orders of
magnitude lower than the relaxation rate of excitons along
an energy axis. In this area there have been studied most
effectively quasi-two-dimensional excitons in semiconductor
heterostructures with spatial separation of electron and hole
layers,12–28 as well as two-dimensional exciton polaritons in
quantum wells located in microcavities.29,30
It is worth recalling that in an ideal and spatially infinitetwo-dimensional system, where the density of one-particle
states is constant, the BEC at finite temperatures cannot
occur due to fundamental reasons: the divergence of the
number of states when the chemical potential l! 0 (i.e., the
states with momentum K% 0 may accumulate an unlimited
number of bosons). It is also relevant to recall a theorem
according to which on a base of inequalities established by
Bogoliubov it has been proved rigorously that in an ideal
and infinite 2D system there is no non-zero order parameter,
which is destroyed by fluctuations diverging logarithmically
with increasing size of the two-dimensional system.31 This
proof applies to both the superfluid liquid and the supercon-
ductivity in ideal 2D systems, as well as to the 2D Heisen-
berg ferromagnet.32 However, this problem is removed by
the spatial restriction of free movement of two-dimensional
excitons and their accumulation in lateral traps, prepared
artificially or in natural traps associated with large-scale fluc-
tuations of the random potential.
2. Bose condensation of spatially indirect,dipolar excitons
In the case of double quantum wells in an electric field
transverse to heterolayers, photoexcited electron and hole
appear to be spatially separated. These excitons are called
spatially indirect excitons, or dipolar. Fig. 1 shows a diagram
of optical transitions for direct (D) and spatially indirect (I)
excitons in double GaAs quantum well, separated by
AlGaAs barrier, under applied electric bias perpendicular to
heterolayers. This figure also demonstrates the behavior of
luminescence spectra of an indirect exciton when varying
the bias voltage. Because of the broken central symmetry
spatially indirect excitons have a static dipole moment in the
lowest state. Therefore, due to the dipole-dipole repulsion
such excitons are not bound into molecules or other complex
molecular systems and do not condense into a liquid.
Because the overlap between the wave functions of electrons
and holes is limited in the direction of the applied electric
field, radiative lifetimes of dipolar excitons are by several
orders of magnitude higher than the thermalization times
and the lifetimes of direct excitons in these structures. There-
fore, these excitons are easier to accumulate and cool to suf-
ficiently low temperatures close to the temperature of the
crystal lattice. The spatial separation of electrons and holes
in the applied bias occurs in single, rather wide quantum
wells where spatially indirect dipolar excitons can also be
excited by light. Recall that an interest in the study of sys-
tems with spatially separated electron-hole layers has been
stimulated by Refs. 13 and 14.
The BEC of 2D exciton gas of dipolar excitons takes
only place under conditions of spatial constrain, e.g., with
the excitation of excitons in lateral traps prepared by one
a
ID
c
1.55
1.54
1.53
1.520 –0.5 –1
D
T
I
U, V
E, e
V
–1.2
–1.0
–0.8
–0.6–0.4
1.52 1.525 1.530 1.535 1.540 1.545E, eV
b
FIG. 1. A scheme of optical transitions (a);
dependences of spectral positions of lines for
a direct exciton (D), a charged exciton com-
plex, trion (T), and a spatially indirect, dipolar
exciton ME (I) on the electric displacement U
(b); a behavior of luminescence spectra of
dipolar excitons when changing the applied
voltage, the numbers on the left of the spectra
correspond to the electrical voltage in volts
(c), T¼ 2 K.
542 Low Temp. Phys. 38 (7), July 2012 V. B. Timofeev
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 142.150.190.39
On: Sat, 20 Dec 2014 13:32:48
way or another. In this case, we can avoid “scattering” of
these excitons due to the dipole-dipole repulsion. To prepare
the traps, which can accumulate dipolar excitons, non-
uniform electric and deformation fields are used.22,24,25
Let us consider collective properties of dipolar excitons
and their Bose-Einstein condensation when excitons are
accumulated in electrostatic ring traps.26,27 The work was
carried out with spatially indirect dipolar excitons in a wide
enough (25 nm) GaAs/AlGaAs single quantum well placed
in an electric field transversed to heterolayers. A voltage was
applied between a metal gate on the structure’s surface
(Schottky gate) and a conducting electronic layer inside the
heterostructure (an integrated electrode in the form of the
epitaxially grown silicon-doped layer). The excitation of
dipolar excitons with a laser, and observation of their lumi-
nescence with high spatial resolution of about 1 lm was car-
ried out through the round window 5–10 lm in diameter in a
metal mask. The windows were etched in the mask by using
electron-beam lithography. Photoexcited excitons were
accumulated in the lateral ring trap, which appeared along
the perimeter of the window because of the strongly inhomo-
geneous electric field.26
Upon reaching critical conditions of condensation in
pumping and temperature, a narrow line of dipolar excitons,
corresponding to the excitonic condensate, grows up in a
threshold manner in the spectrum of spontaneous lumines-
cence (Fig. 2(a)).21 In the vicinity of the condensation
threshold, a spontaneous luminescence spectrum exhibits
strongly nonlinear behavior: the line width narrows almost
by half (Fig. 2(c)), and its intensity I increases exponentially
(Fig. 2(b)) in accordance with macroscopic occupation of
the lowest state in the trap. Close to the threshold the line
maximum first shifts towards lower energies by the value of
about kT, which is also in agreement with the macroscopic
population of the lowest state in the trap (Fig. 2(c)). When
observing the spatial distribution of the exciton lumines-
cence in the far field, the angular size of the luminescence
spot, close to the threshold of condensation, significantly
narrows due to accumulation of excitons near the wave vec-
tors K% 0 (the inverse peak width DK is equal to the trap
width� 1 lm (Fig. 2(d))). This behavior is a consequence of
processes of stimulated scattering of dipolar excitons to the
lowest state in the trap with zero momentum, accompanied
by a macroscopic population of this state (the rate of stimu-
lated scattering is proportional to a number of excitons in the
final state, i.e., cS!(nqþ 1), nq is the occupation number of
excitons, which is larger than unity above the condensation
threshold). Measurements of radiated power, taking into
account the experimental geometry and the lifetime of exci-
tons, give the occupation numbers near the threshold nq� 1
to within a factor of 2. With further increase in the photoex-
citation power P the exciton line width starts to increase, and
the spectral position of its maximum is shifted towards
higher energies in accordance with the increasing repulsive
interaction between dipolar excitons upon increasing their
density.
The observed phase transition associated with the BEC of
dipolar excitons occurs at quasi-equilibrium conditions.
“Supracondensed” excitons are distributed in a quasi-
equilibrium way, that is reflected in the exponential behavior
of the luminescence intensity in the region of high-energy
“tails” of the exciton line, where the energies of translational
motion of excitons exceed kT (Fig. 3). The temperatures of
supracondensed excitons, found without adjustable parame-
ters, exceed the lattice temperature by several degrees, and
increase with pumping (Fig. 3). This is due to the finite life-
time of dipolar excitons (about a few nanoseconds in this
case).
Under conditions of the Bose-Einstein condensation of
excitons, a spatially periodic axially symmetric structure of
equidistant excitation spots appears in the luminescence
image, which was observed through the window with the
1 10 100
1.5083
1.5084
1.5085
1.5086
0.50
0.55
0.60
0.65
0.70
10
100
1000
P, μW Angle, deg1.506 1.508 1.510
0
1000
2000
3000
a–2 –1 0 1 2
–15 –10 –5 0 5 10 150
1000
2000
3000
I, nu
mbe
r of p
hoto
ns/s
line
wid
th, m
eV
I, nu
mbe
r of p
hoto
ns/s
I, nu
mbe
r of p
hoto
ns/s
EVe ,
E, eV
K||, 10 cm–4 –1
c
b d
FIG. 2. A behavior of luminescence spectra of dipolar excitons (a), an intensity of the line I (b), its width and spectral position (c) as well as an angular size of
the luminescence spot in the far field (in the space of wave vectors K, cm�1) (d) on the power of Ne-He laser. T¼ 1.7 K.
Low Temp. Phys. 38 (7), July 2012 V. B. Timofeev 543
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 142.150.190.39
On: Sat, 20 Dec 2014 13:32:48
resolution of about 1 lm and spectrally selected by an inter-
ference filter (Fig. 4(a)). An optical Fourier transformation
of spatially periodic structures (transformation from r- to
K-space) was realized experimentally in situ. The resulting
Fourier transforms, reproducing the pattern of distribution of
the luminescence intensity in the far field (see Fig. 4(b)),
demonstrate the result of destructive and constructive inter-
ference as well as the spatial directionality of the lumines-
cence along the normal to heterolayers. At temperatures
above the critical (T� 5 K) the visibility of spatially periodic
structures of luminescence as well as the directionality of the
illumination disappear. These results are a natural conse-
quence of the large-scale coherence of the condensed exciton
state in the lateral ring trap.
The direct measurements of a two-beam interference of
pairs of luminescent spots along the diameter in the ring by a
Michelson interferometer allowed to determine the coher-
ence length in the plane of the structure, which turned out to
be equal to 4 lm and close to the perimeter of the ring trap,
which is more than an order of magnitude greater than the
thermal length of the de Broglie wave in these same condi-
tions (kdB% 0.17 lm). Such a large-scale spatial coherence
of the Bose condensate of dipolar excitons, found for the first
time, indicates that the experimentally observed spatially
periodic structure of luminescence under conditions of the
Bose-Einstein condensation of dipolar excitons in lateral
ring traps are described by a single wave function. Finding
the spatially periodic structures in the near and far field is a
direct indication that the macroscopic coherent phase of the
Bose condensate appears spontaneously in the reservoir of
incoherent excitons. An analysis of the interference pattern
visibility and direct measurements of the first-order correla-
tor (experimentally measured value g(1)(r,r0)% 0.2) give evi-
dence that the Bose condensate of dipolar excitons is
depleted in the case under consideration.
There is another interesting phenomenon related to the lu-
minescence. It turned out that the luminescence of spots in spa-
tially periodic structures is linearly polarized28 (see Fig. 5).
Due to quantum mechanical effects of interparticle exchange
interaction, the appearing ground state of the Bose condensate
with the spin degrees of freedom turns out to be the most stable
for the same number of bosons, which are distinguished by
their spin projections (DS¼61), since contributions of
exchange interactions are added coherently. As a result, the
condensed exciton phase is linearly polarized. It was also
found that in the overwhelming majority of the experiments
performed, the plane of linear polarization was “bound” (pin-
ning effect) to the crystallography of the structure (as a rule, to
the direction [011] within the plane (001) of the heterostruc-
ture), which is caused by a strong anisotropy of the random
potential associated with features of the structure and structural
imperfections. The observed linear polarization of the lumines-
cence can be a direct consequence of spontaneous symmetry
breaking in conditions of the Bose-Einstein condensation.
3. Bose condensation of exciton polaritonsin a microcavity
A phenomenon of Bose-Einstein condensation of exci-
ton polaritons, which are composite bosons too, has recently
10000
1000
100
E, eV1.509 1.510
(8.7 1.5) K
8.2 K
6.7 K
6.5 K
(5.5 1) K
I, nu
mbe
r of p
hoto
ns/s
FIG. 3. Exponential “tails” of the luminescence intensity I, reflecting the
distribution of the occupation numbers of the supracondensed excitons. A
slope of “tails” in the scale lnI–E was used to find the temperature of an
exciton gas. The temperatures defined in this way for different pumping are
shown to the right of the curves. The dotted line shows linear approxima-
tions of the logarithmic “tails.” The arrow indicates the position of the dipo-
lar exciton. T¼ 1.7 K.
FIG. 4. The spatial (a) and the angular (b) (result of optical Fourier transfor-
mation) distributions of the luminescence intensity of dipolar excitons in a
ring trap of the diameter of 5 lm for the integral power of photoexcitation
P633 nm¼ 10 lW. An angular size in the far field for the image (b) is of 32�
or 4� 104 cm�1 for the planar component of the wave vector K||. Lumines-
cence of dipolar excitons was observed with use of an interference filter cut-
ting out a narrow spectral range corresponding to the line of exciton Bose-
condensate. T¼ 1.7 K.
7300
5500
3700
E
E
I, nu
mbe
r of p
hoto
ns/s
FIG. 5. A spatial intensity distribution I of luminescence of dipolar excitons
in a single quantum well GaAs/AlGaAs (25 nm) observed in a window of
the diameter of 5 lm in a metal electrode on the surface of the sample at two
orthogonal directions of linear polarization indicated by arrows. The obser-
vation conditions are similar to Fig. 4. T¼ 1.7 K.
544 Low Temp. Phys. 38 (7), July 2012 V. B. Timofeev
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 142.150.190.39
On: Sat, 20 Dec 2014 13:32:48
been found in a quasi-two-dimensional semiconductor heter-
ostructures placed in a microcavity.29 Recall that the polari-
ton is a quasiparticle, which is a quantum superposition of
the electromagnetic (photonic) and polarization (excitonic)
excitations in solids.33,34 It has been proposed by Hopfield to
represent the wave function of such a superposition as
follows:34
jw6i ¼ gcjwci6gxjwxi: (1)
Here, gc,x are the coefficients determining partial fractions of
light (photon) and matter (exciton) contributions to polariton
substance, and wc, wx are the wave functions of photon and
exciton respectively. For equal partial contributions the coef-
ficients are gc;x ¼ 1=ffiffiffi
2p
, and the polariton at this partial ratio
is a half-light and half-matter particle. By varying the coeffi-
cients gc,x it is possible in principle to interpolate smoothly
between light and matter limits that is, by itself, a unique op-
portunity for Bose systems.
If a quantum well (or a few of quantum wells) is located
in the antinode of a standing electromagnetic wave in a fairly
high-Q resonator and the exciton energy of the two-
dimensional exciton is equal to the energy of the photon
mode in the microcavity at K¼ 0, then at these conditions
states of transverse exciton and photon interact strongly with
each other in the region of light wave vectors, i.e., in the
electrodynamic region. As a result of this interaction, as
shown in Fig. 6, there occurs a quantum-mechanical mixing
of the photon and exciton states (Eq. (1)), and degeneracy in
the vicinity of zero planar wave vectors is removed. Finally,
there are two new intrinsic one-particle states appear, each
of which is a linear combination of the transverse photonic
and polarization excitonic modes. These intrinsic states have
come to be known as exciton polaritons. Fig. 6 shows the
upper and lower branches, split due to the exciton-photon
interaction, of exciton polaritons in a microcavity.
The upper and lower polariton branches at K¼ 0 are
split by a quantity, received the name Rabi splitting, which
is a measure of the exciton-photon interaction.35 The Rabi
frequency can be determined as follows:
XRabi ¼ e < jrj > E=�h: (2)
Here, e is the electron charge, <jrj> is the matrix element of
dipole moment of direct allowed transition to the exciton
state, E is the amplitude of electromagnetic wave in a micro-
cavity, and �h is Planck’s constant. Qualitatively the Rabi
splitting determines the frequency with which during the
lifetime of the polariton in a microcavity a photon is trans-
formed into an exciton, and vice versa—an exciton into a
photon. Polariton effects are important when the Rabi fre-
quency is much higher than the characteristic frequencies of
damping in a microcavity, related to the finite lifetime of
polaritons, including tunneling “percolation” of a polariton
through the Bragg mirrors into a vacuum with the transfor-
mation of the polariton to a photon, as well as the processes
of inelastic scattering of polaritons by phonons and structural
imperfections inside the cavity. These conditions are realized
in microcavities with high Q-factor and structural perfection.
In the region of strong exciton-photon interaction
(the lower polariton branch in Fig. 6) an effective mass of
polaritons is extremely small (�10�5 m0) up to wave vectors
� 3� 106 m�1. The mass of an exciton polariton can be
found by using the Einstein relation: E¼ h�¼mc*2, where
c* is the speed of light in medium of a microcavity. So small
mass of an exciton polariton is only determined by photon
admixture to a resulting light-exciton superposition. How-
ever, with the further increase of wave vectors, i.e., when
moving off the electrodynamic region, the dispersion of the
lower polariton branch becomes increasingly closer to that
of mechanical two-dimensional excitons the mass of which
is large and, hence, the density of states is high. For instance,
in the case of GaAs the exciton effective mass mex� 0.3 m0,
i.e., it is by four orders of magnitude larger than the polariton
mass. At the same time, the upper polariton branch with
increasing wave vectors is more “photon-like.”
The Rabi frequency allows us to estimate the maximum
values of the temperature at which the Bose condensation of
exciton polaritons can take place. These critical temperatures
are caused by properties of the material environment under
use, which determines the dipole moment of optical transi-
tion, and the quality of a resonator. In microcavities with the
Q-factor of 105 and the quantum wells based on GaAs the
critical temperature may be about 100 K, in the case of quan-
tum wells based on CdTe this temperature is half as high.
When molecular crystals with Frenkel-type excitons are
used as a material medium the critical temperatures can, in
principle, be even higher than room temperature.
Polaritons in microcavities have a remarkable property
which is needed for experimental studies of their spatial and
1440
1430
1420
Еm
eV,
Photon
Exciton
14101 10 102
Wave number, 106/m
FIG. 6. A schematic representation of the spectrum of exciton polaritons in
a microcavity. The dotted line shows dispersions of two-dimensional pho-
tons and excitons in the absence of interaction. To simplify the consideration
there is shown a situation when the energies of size quantization of photons
in a microcavity and the exciton resonance in a quantum well are the same
in the region K¼ 0. In the conditions of light-exciton interaction, there
appears the upper (photon-like) and the lower (exciton-like) polariton
modes, which at K¼ 0 “repel each other” by the value of the Rabi splitting
(these modes are shown by solid lines). The Rabi splitting XR is defined by
the light-exciton interaction (discussed in the text).
Low Temp. Phys. 38 (7), July 2012 V. B. Timofeev 545
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 142.150.190.39
On: Sat, 20 Dec 2014 13:32:48
temporal evolution, dynamics, scattering (including the proc-
esses of interband and intraband parametric scattering), as
well as damping and relaxation under both resonant and non-
resonant photoexcitation. This property is due to the fact that
the planar momentum of polaritons is conserved, when they
are tunneling through mirrors into a vacuum with the trans-
formation of a polariton to a photon. It follows herefrom that
there is one-to-one correspondence and a relationship
between the quantum state of the polariton in a microcavity
and the emitted photon. Therefore, the distribution of exciton
polaritons and their dynamics can be studied experimentally
as a function of the planar wave vector in reflection, scatter-
ing, transmission and luminescence, performing spectral
measurements of angular distributions of the light intensity,
emitted by a microcavity (in another language, making opti-
cal measurements in the far field).36,37 From a purely experi-
mental point of view, such possibilities are quite unique.
A group of exciton polaritons in a microcavity is an open
and dissipative system of interacting bosons, which is thermo-
dynamically highly nonequilibrium due to the extremely short
lifetimes of polaritons in a cavity (picosecond scale). How-
ever, in the entire set of observed properties, the Bose conden-
sation of exciton polaritons in a microcavity is different from
the effect of laser generation in semiconductor heterostruc-
tures and, despite the strong non-equilibrium of the system,
very similar to the phenomenon of BEC in a group of strongly
cooled atomic Bose gases.11 If we ignore the terminology and
evaluate the entire panorama of the observed exciton-
polariton group effects altogether, exciton polaritons in micro-
cavities are radically new and certainly interesting object for
fundamental studies of collective properties of Bose systems,
where quantum effects manifest themselves at the macro-
scopic scale, while the non-equilibrium of the system of exci-
ton polaritons in microcavities opens up new possibilities and
reveals the properties that may be in demand for technical
applications.
Let us first consider how looks qualitatively the conden-
sation of exciton polaritons in momentum space in the study
of angular distributions of light intensity, related to the emis-
sion of polaritons at the output of a microcavity, under the
variation of non-resonant optical pumping. The correspond-
ing measurements are performed in the far field, and the
results, as an example, are shown in Fig. 7. Non-resonant op-
tical pumping with photon energy, somewhat smaller than
the width of the band gap in barriers surrounding a quantum
well, generates in the quantum well of a microcavity none-
quilibrium electron-hole pairs (excitations). These excita-
tions are bound in the “hot” excitons, which relax rapidly
with the participation of phonons, optical and acoustic, to
the lowest exciton band, creating an exciton “reservoir”
from which then the condensation of excitons to the region
of minimum K¼ 0 of the lower polariton band occurs. At
low pumping levels, below the condensation threshold, exci-
tons are accumulated close to the bend in the dispersion
curve of the lower polariton branch, where the single-
particle density of exciton states starts to grow strongly. The
accumulation of the excitons is due to the fact that in the vi-
cinity of this feature for one-phonon relaxation processes of
polaritons in K¼ 0, according to the laws of conservation,
there appears a narrow place (the so-called “bottle neck”).
This phenomenon at small pumping manifests itself in the
observation in the far-field emission of the luminescence
ring (the so-called “Rayleigh scattering ring,” Fig. 7(a)).
Fast processes of transverse, nearly elastic, relaxation pro-
vide a distribution of polaritons over the perimeter of the
ring and thus its emission intensity, close to homogeneous.
In these conditions, the density of polaritons in the vicinity
of K¼ 0 is still extremely low, and the corresponding occu-
pation number of polaritons nk� 1. However, for the pump-
ing level above the threshold, when in the vicinity of the
bottom of the lower polariton band the occupation numbers
are accumulated and begin to exceed nk� 1, the processes of
stimulated scattering to the bottom of the band are involved.
In the conditions of stimulated scattering, which are a direct
manifestation of Bose-Einstein statistics, the density of exci-
ton polaritons begins to grow superlinearly with increasing
the pumping in the vicinity of the wave vectors K¼ 0. In this
regime, the emission intensity of the “Rayleigh scattering
ring” is insignificant compared to the huge intensity of the
luminescence from the condensate of exciton polaritons in
the region of zero momentum (see Fig. 7(d)).
The phenomenon of condensation of exciton polaritons in
microcavities demonstrates practically all significant features
and properties of Bose-Einstein condensates, which were
found in diluted and cooled atomic systems. These properties
include the experimental observation of narrowing of the
polariton momentum distributions in the vicinity of K¼ 0 (the
transformation of Boltzmann, the classical type, distribution
to the Bose, quantum, one) and the macroscopic polariton
accumulation in this area (substantial increase of polariton
occupation numbers nk � 1), due to the processes of stimu-
lated scattering under the pumping above the threshold of
10
0
–10
–20
2
1.5
1
0.5
0
Em
eV,
Em
eV,
1457.51457
1456.51456
1455.51455
1454.51454
1453.50
80007000600050004000300020001000
20
10
0
–10
–20
2
1.5
1
0.5
0
1457.51457
1456.51456
1455.51455
1454.51454
1453.50
80007000600050004000300020001000
2 0 –2 2 0 –2
b
a P = 10 kW/cm2
P = 75 kW/cm2 d
c20
K, μ
m–1
K, μ
m–1
K, μm–1 K, μm–1
I(σ+, σ+)
(σ+, σ+)
FIG. 7. The luminescence of exciton polaritons in a microcavity, which is
observed in the far field, under the non-resonant optical pumping below
(P¼ 10 kW/cm2) and above the condensation threshold (P¼ 75 kW/cm2).
An angular distribution of luminescence of exciton polaritons below (a) and
above the BEC threshold (b). The planar wave vectors are measured on the
axes. A bimodal image of the distribution of luminescence intensity of exci-
ton polaritons below (c) and above the BEC threshold (d). Distributions of
the luminescence intensity of polaritons over the energy (the vertical axis)
are shown as a dependence of the planar momentum (the horizontal axis). A
bimodal image of the distribution of the luminescence intensity directly
reproduces a form of dispersion of exciton polaritons of the lower branch.
The observation conditions see in Fig. 4. The image courtesy of V. D. Kula-
kovskii (unpublished).
546 Low Temp. Phys. 38 (7), July 2012 V. B. Timofeev
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 142.150.190.39
On: Sat, 20 Dec 2014 13:32:48
condensation;29 the observation of spatial compressions of
exciton-polariton condensates accumulated in the natural and
artificial traps;38,39 the detection of large-scale spatial coher-
ence as well as the linear polarization of the luminescence of
the polariton condensate and the relationship of this phenom-
enon with spontaneous symmetry breaking;29 the detection of
the effects of spontaneous and stimulated by light excitation
of quantum singularities—vortices40 and the half-vortices;41
finding an analogue of the Josephson effect42 as well as the
phenomenon of dissipationless, superfluid flow of exciton
polariton condensate.43,44 All these properties have no direct
relationship to the regular laser effect. It is surprisingly that
the Bose condensation of exciton polaritons takes place in
such nonequilibrium conditions. It is just this fact remains a
mystery and a challenge to a theory, which should give a com-
prehensive answer as to why the whole panorama of found
properties, observed in BECs of cooled and diluted Bose
gases in quasi-equilibrium conditions, is also seen fully for
exciton polaritons in microcavities, which are a strongly none-
quilibrium system.
Finally, let us consider another very interesting phenom-
enon of Bose condensation of exciton polaritons related to its
spin degrees of freedom. It is known that the ground state of
the exciton-polariton Bose condensation in a microcavity with
a GaAs quantum well is a spinor and corresponds to two pro-
jections of the resulting spin onto the direction normal to het-
erolayers, DS¼61. In the theoretical work45 it is shown that
the properties of spinor, spin-polarized Bose condensates in a
magnetic field perpendicular to the plane of layers, are signifi-
cantly different from those of spinless Bose condensates. In
the spinor condensate, using a Bose condensate of exciton
polaritons in a microcavity as an example, the paramagnetic
(Zeeman) splitting of spin components is suppressed up to
some critical values of the magnetic field, which are deter-
mined by the difference in energies of the interaction between
the Bose particles with equal and opposite orientation of spins
in the condensate. This suppression, or screening of paramag-
netism in the conditions of Bose-Einstein condensation in
spinor systems, is called the spin Meissner effect. The effect
can be understood qualitatively as follows. In magnetic fields
smaller than the critical field, the Zeeman splitting of excitons
in the condensate is exactly compensated by polariton-
polariton interaction in an elliptically polarized condensate.
Simultaneously with the suppression of paramagnetism of the
excitonic Bose condensate, there occurs a destruction of the
linear dispersion relation of excitations in the condensate and
its superfluidity (exciton superfluidity). However, in magnetic
fields exceeding the critical magnetic field, B>BC, the para-
magnetic properties of Bose-condensate are restored.
Recently, the spin Meissner effect and the related suppression
of paramagnetism in a magnetic field have been observed
experimentally in Ref. 46.
4. Conclusion
In experiments with spatially indirect dipolar excitons and
exciton polaritons in microcavities, an optical excitation method
is mainly used. However, even now we can see significant pro-
gress in a field related to electrical injection in terms of exciton-
polariton excitations in semiconductor microcavities.47–49
The electrical injection opens up a whole range of possible
practical applications, such as low-threshold coherent light
sources, optical transistors, in which one light beam modulates
another, as well as emitters with entangled pairs of photons.
These fields of investigations will undoubtedly contribute to
new information technologies.
Until now, semiconductors (III-V, II-VI heterostruc-
tures) are mostly used as the objects of the exciton polariton-
ics in microcavities. However, organic materials contain
large potential of possibilities. In organic crystals, the
dipole-allowed excitons have a small radius (the so-called
Frenkel excitons) and have large oscillator strengths exceed-
ing by many orders of magnitude those of oscillators of
hydrogen-like excitons in semiconductors. Therefore, in or-
ganic systems, in principle, it is easier to provide a strong
exciton-photon coupling, and in such structures a much
wider dynamic range of exciton-polariton densities in com-
parison with semiconductor systems can be realized.
P.S. This brief review has been written in connectionwith the birthday of Viktor Valentinovich Eremenko withwhom the author is tied by warm, friendly relations andfruitful professional collaboration of many years. Viktor Val-entinovitch has always paid a particular attention and inter-est to the exciton topic, and this interest to excitons hasappeared yet at the dawn of birth of this science when exci-ton subject in semiconductors just started to work its wayand upheld its rights. I hope that the hero of the anniversarywill be interested in one of the small topics of physics andoptics of modern semiconductor excitonics set forth in thisshort review. I sincerely want to wish to dear Viktor Valenti-novich good health, life’s joys, successes, inexhaustible curi-osity and insight in the science, and optimism.
a)Email: [email protected]
1S. A. Moskalenko, Sov. Phys. Solid State 4, 199 (1962).2J. M. Blatt, K. W. Boer, and W. Brandt, Phys. Rev. 126, 1691 (1962).3L. V. Keldysh and Yu. V. Kopaev, Sov. Phys. Solid State 6, 2219
(1965).4A. N. Kozlov and L. A. Maksimov, Sov. Phys. JETP 21, 790 (1965).5L. V. Keldysh and A. N. Kozlov, Sov. Phys. JETP 27, 521 (1968).6A. Einstein, Sitzungsber. Preuss. Akad. Wiss. 1, 3 (1925).7Some Strangeness in the Proportion, in A Centennial Symposium on AlbertEinstein, edited by H. Woolf (Addison-Wesley, NY, 1980).
8A. Griffin, Excitations in a Bose-Condensed Liquid (Cambridge University
Press, Cambridge, UK, 1993).9A. J. Legget, Quantum Liquids: Bose-Condensation and Cooper Pairingin Condensed Matter Systems (Oxford University Press, New York, 2006).
10F. London, Nature 3571, 643 (1938).11W. Ketterle, Rev. Mod. Phys. 74, 1131 (2002).12T. Fukuzawa, E. E. Mendez, and J. M. Hong, Phys. Rev. Lett. 64, 3066
(1990).13Yu. E. Lozovik and V. I. Yudson, JETP Lett. 22, 274 (1976); Yu. E. Lozo-
vik and O. L. Berman, JETP 84, 1027 (1997).14S. I. Shevchenko, Fiz. Nizk. Temp. 2, 505 (1976) [Sov. J. Low Temp.
Phys. 2, 251 (1976)].15L. V. Butov, A. Imamogl, A. V. Mintsev, K. L. Campman, and A. C.
Gossard, Phys. Rev. B 59, 1625 (1999).16L. V. Butov, A. V. Mintsev, Yu. E. Lozovik, K. L. Campman, and A. C.
Gossard, Phys. Rev. B 62, 1548 (2000).17A. V. Larionov, V. B. Timofeev, J. M. Hvam, and K. Soerensen, Zh. Eksp.
Teor. Fiz. 117, 1255 (2000) [JETP 90, 1093 (2000)].18L. V. Butov, A. C. Gossard, and D. S. Chemla, Nature (London) 418, 751
(2002).19D. Snoke, S. Denev, Y. Lui, L. Pfeiffer, and K. West, Nature (London)
418, 754 (2002).
Low Temp. Phys. 38 (7), July 2012 V. B. Timofeev 547
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 142.150.190.39
On: Sat, 20 Dec 2014 13:32:48
20A. V. Larionov, V. B. Timofeev, P. A. Ni, S. V. Dubonos, I. Hvam, and K.
Soerensen, JETP Lett. 75, 570 (2002).21A. A. Dremin, V. B. Timofeev, A. V. Larionov, I. Hvam, and K. Soeren-
sen, JETP Lett. 76, 450 (2002).22D. W. Snoke, Y. Lui, Z. Voros, L. Pfeiffer, and K. West, e-print arXiv:-
cond-mat/0410298.23O. L. Berman, Yu. E. Lozovik, D. Snoke, and R. Coalson, Phys. Rev. B
70, 235310 (2004).24R. Rapaport, G. Chen, D. Snoke, S. H. Simon, L. Pfieffer, K. West,
Y. Liu, and S. Denev, Phys. Rev. Lett. 92, 117405 (2004).25A. V. Gorbunov and V. B. Timofeev, JETP Lett. 80, 185 (2004).26A. V. Gorbunov and V. B. Timofeev, JETP Lett. 83, 146 (2006).27A. V. Gorbunov and V. B. Timofeev, JETP Lett. 84, 390 (2006).28A. V. Gorbunov and V. B. Timofeev, Pis’ma Zh. Eksp. Teor. Fiz. 87, 797
(2008).29J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. M. J.
Keeling, F. M. Marchetti, M. H. Szymanska, R. Andre, J. L. Staehli, V.
Savona, P. B. Littlewood, B. Deveaud, and L. S. Dang, Nature 443, 409
(2006).30A. Kavokin, J. J. Baumberg, G. Malpuech, and F. P. Lassy, Microcavities
(Oxford University Press, Oxford, 2011).31P. C. Hoenberg, Phys. Rev. 158, 383 (1967).32N. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966).33S. I. Pekar, Zh. Eksp. Teor. Fiz. 33, 1022 (1957); Sov. Phys. JETP 6, 785
(1958).34J. J. Hopfield, Phys. Rev. 112, 1555 (1958).35I. I. Rabi, Phys. Rep. 51, 652 (1937).36C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, Phys. Rev. Lett.
69, 3314 (1992).
37M. S. Skolnick, S. Liew, M. Migliorato, M. Hopkinson, and P. Vogl, in
Proceedings of 24 International Conference on Physics of Semiconduc-tors, edited by D. Gershoni (Jerusalem, 1998), p. 25.
38R. Balili, B. Nelsen, D. W. Snoke, L. Pfeiffer, and K. West, Phys. Rev. B
79, 075319 (2009).39D. Snoke and P. Littlwood, Phys. Today 63, 42 (2010).40K. G. Lagoudakis, M. Wouters, M. Richard, A. Baas, I. Carusotto, R.
Andre, L. S. Dang, and B. Deveaud-Pledran, Nat. Phys. 4, 706 (2008).41K. G. Lagoudakis, T. Ostatnicky, A. V. Kavokin, Y. G. Rubo, R. Andre,
and B. Deveaud-Pledran, Science 326, 974 (2009).42K. G. Lagoudakis, B. Pietka, M. Wouters, R. Andre, and B. Deveaud-
Pledran, Phys. Rev. Lett. 105, 120403 (2010).43A. Amo, D. Sanvitto, F. P. Laussy, D. Ballarini, E. del Valle, M. D.
Martin, A. Lemaıtre, J. Bloch, D. N. Krizhanovskii, M. S. Skolnick, C.
Tejedor, and L. Vina, Nature 457, 291 (2009).44A. Amo, J. Lefrere, S. Pigeon, C. Adrados, C. Ciuti, I. Carusotto, R.
Houdre, E. Giacobino, and A. Bramati, Nat. Phys. 5, 805 (2009).45Y. G. Rubo, A. V. Kavokin, and I. A. Shelykh, Phys. Lett. A 358, 227
(2006).46A. V. Larionov, V. D. Kulakovskii, S. Hofling, C. Schneider, L.
Worschech, and A. Forchel, Phys. Rev. Lett. 105, 256401 (2010).47D. Bajoni, E. Semenova, A. Lemaıtre, S. Bouchoule, E. Wertz, P. Senellart,
and J. Bloch, Phys. Rev. B 77, 113303 (2008).48A. A. Khalifa, A. P. D. Love, D. N. Krizhanovskii, M. S. Skolnick, and
J. S. Roberts, Appl. Phys. Lett. 92, 061107 (2008).49S. I. Tsintzos, N. T. Pelekanos, G. Konstantinidis, Z. Hatzopoulos, and
P. G. Savvidis, Nature 453, 372 (2008).
Translated by A. Sidorenko
548 Low Temp. Phys. 38 (7), July 2012 V. B. Timofeev
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 142.150.190.39
On: Sat, 20 Dec 2014 13:32:48