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

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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

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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

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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

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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

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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

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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

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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: timofeev@issp.ac.ru

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Translated by A. Sidorenko

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