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Polariton dynamics and Bose-Einstein condensation in semiconductor microcavities

D. Porras1, C. Ciuti2, J.J. Baumberg3, and C. Tejedor11 Departamento de Fısica Teorica de la Materia Condensada. Universidad Autonoma de Madrid. 28049 Cantoblanco,

Madrid, Spain.2 Physics Department, University of California San Diego. 3500 Gilman Drive, La Jolla, CA 92093. USA.

3 Department of Physics & Astronomy, University of Southampton, Southhampton, SO17 IBJ, United Kingdom.

(February 1, 2008)

We present a theoretical model that allows us to describe the polariton dynamics in a semi-conductor microcavity at large densities, for the case of non-resonant excitation. Exciton-polaritonscattering from a thermalized exciton reservoir is identified as the main mechanism for relaxationinto the lower polariton states. A maximum in the polariton distribution that shifts towards lowerenergies with increasing pump-power or temperature is shown, in agreement with recent experi-ments. Above a critical pump-power, macroscopic occupancies (5 × 104) can be achieved in thelowest energy polariton state. Our model predicts the possibility of Bose-Einstein Condensation ofpolaritons, driven by exciton-polariton interaction, at densities well below the saturation density forCdTe microcavities.

PACS numbers: 71.35.+z

I. INTRODUCTION

Polaritons are quasiparticles created by the strongcoupling between excitons and photons, and behave ascomposite bosons at small enough densities. In thelast decade a huge experimental and theoretical efforthas been dedicated to the search for quantum degener-acy effects in microcavity polaritons. They were soonproposed as candidates for the formation of a Bose-Einstein condensate, due to their small density of states(DOS) [1]. However, a bottleneck effect due to the slowpolariton-acoustical phonon scattering was predicted [2]and observed in angle-resolved experiments [3]. Stim-ulated scattering due to the bosonic nature of polari-tons has been demonstrated [4] in pump-probe experi-ments. Parametric amplification and parametric oscilla-tion, under resonant excitation, has been observed [5,6]and was well explained by coherent polariton-polaritonscattering [7–9]. In GaAs microcavities, experimentshave shown the role of the polariton-polariton scatteringin the nonlinear emission after non-resonant excitation,either in continous wave [10–12], or time-resolved exper-iments [13]. The growth of II-VI microcavities leads tonew exciting possibilities due to the larger stability ofthe exciton. The study of these samples has allowedfor the observation of stimulated scattering in the strongcoupling regime under non-resonant excitation conditions[14]. The robustness of the polariton in these samples hasbeen recently shown by the demonstration of parametricamplification at high temperatures [15].

The theoretical study of semiconductor microcavitieshas followed two different lines. A fermionic formalism,including carrier-carrier correlation, explained the lasingby population inversion at densities above the saturationdensity, where the coupling between the exciton and thephoton disappears (weak coupling regime) [16,17]. On

the other hand, a bosonic picture, including polariton-polariton interaction, has been developed. It must bestressed that recent experiments agree with the predic-tions made by the bosonic formalism (bottleneck effect,stimulated scattering) under excitation conditions suchthat the density of polaritons remains below the satura-tion density. The polariton-polariton interaction, givenby the usual Coulomb exchange interaction between ex-citons, and the saturation term, has also explained quan-titatively many recent observations [4–8].

In this paper we present a theoretical study of the evo-lution of the polariton population at large densities witha bosonic description. We use the semiclassical Boltz-mann equation, developed by Tassone et al. [18], as thestarting point for the description of the polariton dy-namics. Exciton-polariton (X-P) scattering is identifiedas the most important mechanism for relaxation towardsthe lower polariton branch. We deduce a simplified modelin which the high energy excitons are considered as athermalized reservoir. This simplification allows us todescribe all the microscopic lower polariton states, andthe evolution of the polariton population towards Bose-Einstein Condensation (BEC) [19].

X-P scattering is shown to produce a polariton pop-ulation with an occupancy that has a maximum at agiven energy below the exciton energy. The energy atwhich the polariton distribution peaks is shown to de-crease linearly as pump-power or sample temperatureis increased, in such a way that the dip in the polari-ton dispersion behaves as a polariton trap in phase-space[20]. This conclusion is valid for all semiconductor mi-crocavities. In particular, the same behavior has beenobserved in a recent experiment in a GaAs microcavity[10]. However, in GaAs, densities comparable to the sat-uration density are created, as one increases the pump-power, before the threshold for BEC is reached. Thus, for

1

a clean observation of all the phenomenology describedin this paper, better material parameters than those ofGaAs microcavities are desirable. We predict that inCdTe microcavities, where excitons are more stable dueto their smaller Bohr radius, the threshold for BEC canbe reached for densities well below the saturation den-sity. Thus, we choose, for concreteness, CdTe parametersfor the detailed calculations shown below. Our modelshows that the relaxation towards the ground state isonly possible for densities comparable to the saturationdensity in GaAs microcavities. In order to achieve BECwith smaller polariton densities, an additional scatteringchannel that those considered here should be included.Electron-polariton scattering in doped microcavities hasbeen recently proposed [21] as an efficient mechanismthat allows BEC for such small polariton densities.

The paper is organized as follows. In section II wepresent the theoretical framework. A set of differentialequations is obtained for the description of the micro-scopic lower polariton levels, and the exciton density andexciton temperature. In section III, we apply this modelto the study of the dynamics of the polariton popula-tion for different pump-powers and temperatures, afternon-resonant, continuous excitation. The case of Bose-Einstein Condensation into the lowest energy polaritonmode is considered in section IV. Finally, in section V,we list our main conclusions, and discuss the comparisonwith experiments.

II. RATE EQUATION SIMPLIFIED MODEL

A. Simplified model for the exciton-polariton

scattering

At densities such that na2B << 1, where aB is the

2-D exciton Bohr radius, excitons can be described asinteracting bosons. From typical parameters for CdTeQW’s (exciton binding energy EB ≈ 25 meV , and elec-tron and hole effective masses me = 0.096, mh = 0.2),we get a 2-D exciton Bohr radius aB = 47 A, anda−2

B = 4.5 × 1012cm−2. On the other hand, at large ex-citon densities, carrier exchange interactions and phase-space filling effects destroy the coupling between excitonsand photons. The density at which the exciton oscillatorstrength is completely screened is the saturation density,nsat. Following Schmitt-Rink et al. [22], we can estimate,for a CdTe microcavity,

nsat =0.117

πa2eff

= 6.7 · 1011cm−2, (1)

where a2eff = aB/2. In InGaAs and GaAs QW’s, nsat

would result to be 6.6 × 1010cm−2 and 1.3 × 1011cm−2,respectively. Along this paper we consider the case ofCdTe in a range of polariton densities well below nsat,

so that the microcavity remains in the strong couplingregime.

In the strong coupling regime, the Hamiltonian in-cludes the coupling between excitons and photonsthrough the polariton splitting, ΩP [23]:

H0 =∑

k

[

ǫxkb†kbk + ǫc

ka†kak +

ΩP

2

(

a†kbk + b†kak

)

]

. (2)

b†k, ǫxk (a†

k, ǫck) are the creation operator and dispersion re-

lation for the bare exciton (photon) mode. The residualCoulomb interaction between excitons, and the satura-tion of the exciton oscillator strength, can be describedby the effective exciton-exciton, and exciton-photon in-teraction Hamiltonian [24]:

HI =∑

k1,k2q

Mxx

2Sb†k1

b†k2bk1+qbk2−q +

σsat

Sb†k1

b†k2bk1+qak2−q + h.c. (3)

where S represents the quantization area. We considerhere the small momentum limit for HI , an approximationthat is valid for the range of small wave-vectors consid-ered below. The matrix elements in HI are given by:

Mxx ≈ 6 EBa2B

σsat ≈ 1.8 ΩP a2B (4)

H0 can be diagonalized in the polariton basis:

plp,k = X lpk bk + Clp

k ak

pup,k = Xupk bk + Cup

k ak (5)

where Xlp/upk , C

lp/upk are the Hopfield coefficients, that

represent the excitonic and photonic weights in the po-lariton wave-function for the lower/upper branch, respec-tively [25]. By means of the transformation (5) we can ex-press HI in terms of polariton operators. We will neglectthe upper polariton branch, an approximation that willbe justified later. We express the operators bk, ak in (3)in terms of plp,k, pup,k, and retain the terms correspond-ing to interactions within the lower polariton branch only,

H =∑

k

ǫlpk

p†lp,kplp,k

+∑

k1,k2k3,k4

1

2SV lp−lpk1,k2,k3,k4

p†lp,k1p†lp,k2

plp,k3plp,k4

, (6)

where

1

2V lp−lpk1,k2,k3,k4

=(

1

2Mxx X lp

k1X lp

k2X lp

k3X lp

k4+

σsatClpk1

X lpk2

X lpk3

X lpk4

+

σsat X lpk1

X lpk1

Clpk3

X lpk4

)

δk1+k2,k3+k4. (7)

2

ǫlpk is the dispersion relation for the lower polariton

branch. It is plotted, assuming typical parameters fora semiconductor microcavity, in Fig. 1. For energieshigher than the bare exciton energy, it matches the bareexciton dispersion.

The theoretical model developed by Tassone et al.[18] describes the evolution of the polariton popula-tion by means of a semiclassical Boltzmann equationthat includes polariton-acoustical phonon and polariton-polariton scattering. A grid uniform in energy is consid-ered for the description of the polaritonic levels. Here,we are interested in the evolution of the system towardsa Bose-Einstein condensate. It is well known that in 2-D systems, BEC is possible at finite temperatures, onlyfor systems with a finite quantization length, Lc. As aresult, the critical density and temperature depend log-arithmically on this length scale. Due to the peculiarityintroduced by the 2-D character of semiconductor mi-crocavities, the grid uniform in energy considered in [18]is not well suited for the study of the transition towardsBEC. Instead, a uniform grid in k-space, related with theinverse of the quantization length, L−1

c , has to be consid-ered. Unfortunately, the number of levels that one hasto consider for the description of the polariton dynam-ics is dramatically increased, and a numerical calculationthat includes all the possible polariton-polariton scatter-ing processes is not possible. Thus, some simplificationhas to be done, which still allows to describe satisfacto-rily the polariton dynamics, and to predict the possibilityof Bose-Einstein Condensation.

In order to develop a simplified model for the case ofnon-resonant excitation, we make the following assump-tions:

(1) The exciton population above the bare exciton en-ergy can be considered as a reservoir at a given temper-ature, that follows a Maxwell-Boltzmann distribution.

(2) The main scattering mechanism at large densi-ties for relaxation into the lower polariton modes is theone shown in Fig. 1, in which two excitons scatter inthe reservoir, and the final states are a lower polariton,and another reservoir exciton. We will call this processexciton-polariton scattering.

A scattering process in which two excitons are initiallyin the exciton reservoir, and a lower and upper polaritonare the final states is also possible, but has a much smallerprobability. This is due to the fact that in such a colli-sion, an upper polariton, instead of an exciton, plays therole of a final state. Thus, the scattering rate is reducedby the ratio between the upper polariton and bare exci-ton DOS, ρup/ρx ≈ 4×10−5. In fact, for negative or zerodetunings, the upper polariton population remains negli-gible, as shown experimentally [14] and theoretically [18].The scattering of excitons with acoustical phonons goinginto lower polariton states is also much slower than theX-P scattering already at exciton densities of 109cm−2,and the lattice temperatures considered here.

The phase-space is, thus, divided into two pieces.

• The exciton reservoir, which describes the excitonsdirectly created by the non-resonant pump. Wetake as zero energy the bare exciton energy, so thatthe exciton reservoir includes all the levels withǫ > 0. These levels are weakly coupled to the pho-tonic modes, so that they can be approximated asbare excitons. The occupation numbers are givenby,

Nxi = Nxe−ǫx

i /kBTx . (8)

Nx is given by

Nx =2πnx

ρxkBTx, (9)

where nx is the exciton density, ρx is the bare ex-citon density of states,

ρx = (me + mh)/h2, (10)

and Tx is the temperature of the exciton reservoir.The assumption of a rigid thermal exciton distri-bution is well justified, as discussed below, for lev-els above ǫ = 0, due to the fast exciton-excitonscattering that thermalize the exciton populationat large densities [18,26]. On the other hand,a Maxwell-Boltzmann distribution, rather than aBose-Einstein distribution can be used to describethe exciton reservoir, provided that Nx << 1.Nx ≈ 0.2 for the largest densities considered here,so that quantum degeneracy effects are negligible.The main mechanism for radiative losses in the ex-citon reservoir is due to the coupling to the leakymodes, that we include by considering an excitonlifetime, τx = 100ps [27].

• The lower polariton branch (LPB), where the en-

ergy levels are below the bare exciton energy (ǫlpk <

0). Exciton and photon levels in the LPB arestrongly coupled, and radiative losses are muchmore important, due to the photonic weight ofthese states. Thus, we expect a nonequilibrium po-lariton distribution that depends strongly on tem-perature and density. On the other hand, theabrupt dispersion relation implies a very small ef-fective mass for the LPB states, mlp ≈ 4×10−5mx,leading to the possibility of Bose-Einstein Conden-sation.

With this separation of the phase-space we are neglect-ing the details of the polariton population near ǫ ≈ 0,that is, at the knee in the polariton dispersion relation,where the LPB and the quadratic dispersion of the bareexcitons merge together (see Fig. 1). The contribution

3

from these states is negligible already at the lowest tem-peratures considered here (4K, kBT ≈ 0.35meV ), be-cause the main part of the exciton distribution is in theweak coupling region.

εlp

εi

x

|εlp|/2

2|εlp|

εf

x

k||

lower polaritonicstates (ε<0)

Energyexciton reservoir(ε>0)

FIG. 1. Exciton-polariton scattering process from a ther-malized exciton reservoir into the lower polariton branch.

We describe the LPB states by a grid in k-space. Theexact quantization area is not well determined in exper-iments, due to inhomogeneities in the spot, and surfacedefects. The relevant quantity is the quantization length,that gives the order of magnitude for the distance be-tween the energy levels. We consider first Lc = 50µm(a typical excitation spot diameter), and plane waves forthe microscopic polariton states, so that the quantiza-tion area is simply S = L2

c . The allowed wave-vectorsare given by

klp = (nxx + nyy)2π

Lc, (11)

where nx, ny are integers. A similar method has beenemployed to describe the microscopic polariton levels ina microcavity etched into a microscopic post structure(Lc = 2µm) [28]. In that case, the LPB is reduced toa single microscopic level. On the contrary, in our case,Lc is larger, and the number of lower polariton levels re-sults to be of the order of 103. The main results of ourwork do not depend strongly on the exact value of Lc,but only on its order of magnitude, as we show in thelast section. The weak dependence of our results on Lc

can be easily understood from the peculiarities of BECin 2-D (see Appendix B).

We write now a set of rate equations for N lpk (occupa-

tion numbers in the lower polariton branch) and nx (thedensity of the thermalized exciton reservoir). We con-sider the injection of polaritons from isotropically dis-tributed excitons, so that the occupation numbers N lp

k

will depend on the absolute value of the in-plane wave-vector only. The rates for scattering from the thermalized

reservoir into and out of the LPB, due to the interactionHamiltonian (3) are calculated in Appendix A. They leadto a simplified version of the semiclassical Boltzmannequation:

dN lpk

dt=

W ink n2

x(1 + N lpk ) − W out

k nxN lpk − Γlp

k N lpk

dnx

dt=

− 1

S

∑

k

dglpk (W in

k n2x(1 + N lp

k ) − W outk nxN lp

k )

− Γxnx + px. (12)

Γlpk , Γx are the radiative losses in the LPB and exciton

reservoir, respectively:

Γlpk = |Clp

k |2 1

τc

Γx =1

τx, (13)

where τc is the lifetime for the bare photon within thecavity. We consider τc = 1ps, a typical value for highquality microcavities. px represents the non-resonantpump that injects excitons directly into the exciton reser-voir. If relaxation into the LPB is slow enough, the life-time for an exciton in the exciton reservoir is given byτx, so that a given pump px = nx/τx = nx/100ps cre-ates an exciton density nx. If scattering into the LPB isfast enough, however, the density created by the pump issmaller, as we will see below. dglp

k is the degeneracy forthe polariton level corresponding to the wave-vector k,and is calculated according to the distribution obtainedfrom Eq. (11).

W ink n2

x, W outk nx are the rates for X-P scattering into

and out of the LPB. They are given by the expressions(see Appendix A):

W ink =

2π

hkBTxM2

keǫlp

k/kBTx

W outk =

1

hM2

kρxe2ǫlp

k/kBTx , (14)

where

Mk = MxxX lpk + σsatC

lpk . (15)

The dependence of the rates on temperature can bequalitatively understood by the phase-space restrictionsfor relaxation into and out of the LPB (Fig. 1). Letus consider two excitons in the exciton reservoir witha wave-vector kx

i and energy ǫxi , that scatter to a LPB

state with energy ǫlpk and to a high energy state with kx

f ,ǫxf . Taking into account that the wave-vector of the low-

energy polariton is negligible, when compared with the

4

wave-vectors in the reservoir, simultaneous energy andmomentum conservation implies that:

2 |kxi | = |kx

f | → 4 ǫxi = ǫx

f

2 ǫxi = −|ǫlp

k | + ǫxf → ǫx

i = |ǫlpk |/2, ǫx

f = 2 |ǫlpk |. (16)

Eq. (16) means that two excitons need an energy |ǫlpk |/2

to be the initial states of a process in which one of themfalls into the LPB. On the other hand, an exciton needsan energy 2 |ǫlp

k | in order to scatter with a lower polaritonand take it out of the LPB.

The polariton distribution in the steady-state regimewill be given by the set of equations (12) with dN lp

k /dt =0, dnx/dt = 0. In the steady-state, the occupation num-bers in the LPB are given by:

N lpk =

W ink n2

x

W outk nx + Γlp

k − W ink n2

x

. (17)

If we neglect the dependences on k of the matrix elementMk and the radiative losses Γlp

k , then the maximum inthe polariton distribution has a peak at the energy:

|ǫlpmax| =

1

2kBTxlog

(

1

hΓlpM2ρxnx

)

. (18)

Eq. (18) implies an approximately linear relation be-tween the exciton reservoir temperature and the energyposition at which the system prefers to relax into theLPB (that is, the maximum energy loss in the relaxationprocess).

B. Model for the evolution of the exciton reservoir

temperature

The strong dependence of the X-P scattering into theLPB on Tx implies that, in order to complete our model,we have to describe properly the evolution of the exci-ton reservoir temperature. During the relaxation processinto the LPB, shown in Fig. 1, a high energy exciton isinjected into the exciton reservoir. This way, during therelaxation process, the exciton reservoir is heated. Onthe other hand, the scattering with acoustical phonons,tries to keep the exciton reservoir at the lattice temper-ature. In order to describe the evolution of the excitonreservoir temperature, a new variable has to be addedto this set of equations. We introduce ex, the densityof energy for the exciton reservoir, that is given, for aMaxwell-Boltzmann distribution, by the expression:

ex =1

S

∑

k

Nxk ǫx

k = nx(kBTx), (19)

The evolution of the energy allows us to obtain the ex-citon temperature, at each step in the evolution of thepolariton population.

However, we still need to make an assumption aboutthe exciton distribution that the pump directly injectsinto the reservoir. Non-resonant experiments usually cre-ate a population of high energy electron-hole pairs, thatrelax towards low energy states, after the process of ex-citon formation [2,29]. It must be stressed that for thecase of CdTe QW’s this process is particularly fast, dueto the possibility of exciton formation through the emis-sion of LO phonons, whose energy is 21 meV, below theexciton binding energy (25 meV). The time scale for ther-malization due to exciton-exciton scattering is very fast(≈ 1ps), as compared with the time scale for relaxationinto the LPB (≈ 20ps, for the larger exciton densitiesconsidered here). Thus, it is very well justified to de-scribe the non-resonant pump, as a source that injectsthermalized excitons at a given pump-temperature intothe exciton reservoir. We still need to specify whichis the initial pump-temperature. Due to the large un-certainties in non-resonant experiments it is very dif-ficult to make predictive calculations about this initialcondition. We choose instead to make the simplest as-sumption: we consider that the pump injects excitons atthe lattice temperature, TL. For large exciton densities(above 5 × 1010cm−2) the processes of reservoir heatingby X-P scattering and cooling through exciton-phononscattering, govern the final exciton temperature, and ourresults do not depend on our assumption for the pump-temperature. For small temperatures (below 10 K), anddensities, our results depend on this assumption. In areal experiment, the exciton temperature can be largerthan the one considered here, if the cooling of the initialnon-resonantly created excitons is not fast enough.

It must be pointed out that a very efficient experi-mental setup for non-resonant excitation of a cold exci-ton distribution in a semiconductor microcavity has beendemonstrated by Savvidis et al. [10]. In this experiment,non-resonant excitation has been achieved with small ex-cess energies (below the stop band), due to the finitetransmittance of the Bragg reflectors. The experimentalresults presented in [10] are in agreement with a model inwhich excitons relax from a thermalized distribution intothe lowest energy states, and justify all the assumptionsmade above.

We have to quantify the three contributions for theevolution of the exciton temperature listed above:

(i) Heating by the scattering to high energy exciton lev-els.

The total energy is conserved in the X-P scattering to-wards the LPB. Thus, the energy lost by the polaritonsfalling into the lower energy states must be gained by theexciton reservoir. The following expression gives the rateof change of ex by each X-P scattering process:

dex

dt|X−P = − 1

S

∑

k

ǫlpk dglp

k

5

× (W ink n2

x(1 + N lpk ) − W out

k nxN lpk ). (20)

The heating of the exciton gas in the relaxation to-wards the LPB was also shown theoretically in [28], forthe case of a microcavity post.

(ii) Cooling produced by acoustical phonons.

The exciton-phonon scattering tries to keep excitonsat the lattice temperature, TL, against the heating con-sidered above [26]. We divide the exciton Maxwell-Boltzmann distribution into levels equally spaced in en-ergy with a given ∆E << kBTL, kBTx. The probabilityfor absorption or emission of acoustical phonons givesa scattering rate W ph

j→i(TL) for the transition betweenstates with energies ǫx

i , ǫxj . These rates depend on the

phonon distribution through TL, and can be evaluatedfrom the exciton-acoustical phonon interaction Hamilto-nian. The evolution of the occupation numbers, Nx

i , inthe exciton reservoir leads to an expression for the rateof change of the total energy:

dex

dt|ph=

ρx

2π

∑

i

∆E ǫxi

dNxi

dt|ph=

=ρx∆E

2π

∑

i

ǫxi

∑

j

(

W phj→i(TL)Nx

j − W phi→j(TL)Nx

i

)

=∆E nx

kBTx

∑

i

ǫxi ×

∑

j

(

W phj→i(TL)e−ǫx

j /kBTx − W phi→j(TL)e−ǫx

i /kBTx

)

.

(21)

For the calculation of W ph we have used the exciton-acoustical phonon coupling through the deformation po-tential interaction. These rates have been presented inseveral previous works [18,26,29], and we will not re-write them here. We consider deformation potentialsae = 2.1eV , ah = 5.1eV for CdTe [30], and an 80A QW.In Eq. (21), stimulated terms are neglected, due to thesmall occupation numbers in the exciton reservoir. Therate of change for ex results to be linear in nx, and de-pendent on both TL and Tx.

(iii) Cooling to the pump temperature.

The injection of excitons into the reservoir by a pumpthat is considered to be at the lattice temperature triesto keep the exciton reservoir at TL. The terms that cor-respond to pumping and radiative losses in Eqs. (12),

dnx

dt|pu= px − Γxnx, (22)

imply that the rate for the evolution of the occupationnumbers has the contribution:

dNxk

dt|pu=

2πpx

ρxkBTLe−ǫx

k/kBTL − Γx2πnx

ρxkBTxe−ǫx

k/kBTx ,

(23)

where the condition that the pump injects excitons at TL

into a Maxwell-Boltzmann distribution at Tx is included.The rates in Eq. (23) leads to a new contribution to theevolution of ex:

dex

dt|pu=

1

S

∑

k

ǫxk

dNxk

dt|pu= pxkBTL − nx

τxkBTx. (24)

Note that this contribution cools the exciton reservoir ifTx > TL.

The set of first-order differential equations (12), to-gether with (20, 21, 24), leads to a closed descriptionfor the dynamics of the exciton reservoir and lower po-laritons. We have integrated numerically these equa-tions until the stationary solution (dN lp/dt = dnx/dt =dex/dt = 0) is reached. The obtained solution describesthe steady-state for the case of a continuous non-resonantpump.

C. Comparison between our simplified model and

the complete semiclassical Boltzmann equation.

We have checked whether the polariton distributionobtained with this simplified model agree with the oneobtained in a detailed calculation performed with aBoltzmann equation in which the simplifying assump-tion of a Maxwell-Boltzmann distribution for the lev-els in the exciton reservoir is not made. In Fig. 2 wepresent the comparison between a complete calculationthat follows reference [18], with a grid uniform in energy(∆E = 0.1meV ), in which all the polariton-polariton andpolariton-phonon scattering processes are included, andour simplified model. We expect that the results fromboth calculations coincide under conditions for whichBose-Einstein Condensation (that is, the macroscopic oc-cupancy of a single microscopic mode) is not possible.On the contrary, in the range of densities near or aboveBEC, comparison between both models would not be al-lowed because the system would not be free to relax intoa single microscopic mode. Thus, a continuum density ofstates would not be able to describe properly the LPB,and a grid uniform in energy could not be applied. InFig. 2 a few pump-powers have been chosen, for whichthe comparison is possible. It can be clearly seen thatboth calculations yield identical results, so that, we canconclude that our simplified model captures the essen-tial physics in the relaxation from the exciton reservoirtowards the lower energy polariton states.

6

−6 −4 −2 0 2 4Energy (meV)

10−4

10−3

10−2

10−1

100

101

102

Occu

pa

tio

n n

um

be

r

px = 1

px = 5

FIG. 2. Polariton distribution calculated by means of thecomplete semiclassical Boltzmann equation, as in Ref. [18](points), and polariton distribution calculated by means ofour simplified model (continuous line). We have considereda CdTe microcavity with zero detuning and TL = 10K. Thecurves, from bottom to top, refer to pump-powers from 1 to5, in units of 1010cm−2/100ps.

III. EVOLUTION OF THE POLARITON

DISTRIBUTION WITH PUMP-POWER AND

TEMPERATURE

A. Evolution with temperature

In the previous section, we obtained an estimation forthe evolution of the polariton population as the excitontemperature Tx is increased. A rough linear relation be-tween the position of the maximum in the distributionand Tx was predicted. Now we present a complete calcu-lation with our model that confirms this estimate. Thiseffect can be observed for a different range of polaritonsplittings and detunings. However, the main parameterthat governs the relaxation from the exciton reservoir isthe energy difference between the bare exciton and thebottom of the LPB. This energy distance can be inter-preted as the depth of the polariton trap formed by thedip in the polariton dispersion. Within this paper we willconsider, for concreteness, the case of zero detuning (thatis, Ex (bare exciton energy) = Ec (bare photon energy)),so that the lowest polariton mode has an energy ΩP /2below EX .

In Fig. 3 we show the evolution of the polari-ton distribution for ΩP = 10meV . We choose px =1010cm−2/100ps. Under such excitation conditions, X-Pscattering is slower than radiative recombination in theexciton reservoir, and nx ≈ τxpx. The heating of the ex-citon gas is also negligible, so that Tx ≈ TL. The peak inthe polariton distribution moves towards lower energiesas temperature is increased. Due to the fact that the rategiven in Eq. (14) for scattering into the LPB decreases

with Tx, smaller occupation numbers are predicted forhigher temperatures. At high Tx, the bottleneck is sup-pressed and the polariton distribution becomes almostflat. The difference between high and low Tx was alreadyobserved in [11], where a flat distribution was evidencedfor pump-powers where the X-P scattering is the domi-nant relaxation mechanism.

−5 −4 −3 −2 −1Energy (meV)

10−3

10−2

10−1

100

101

Occu

pa

tio

n n

um

be

r

TL = 4 K

TL = 30 K

FIG. 3. Evolution of the polariton distribution forTL = 4, 10, 15 . . . 30K, ΩP = 10 meV, zero detuning, andpx = 1010cm−2/100ps.

The evolution of the peak in the polariton distributionis shown in Fig. 4 for a range of polariton splittings. Thelinear dependence predicted by the estimation in Eq.(18)is clearly observed. For high enough temperatures andsmall energy distance between the lowest polariton en-ergy and the exciton reservoir, the maximum in the po-lariton distribution can reach the bottom of the LPB.However, the occupation numbers are still smaller thanone, and the situation is not able to produce BEC.

0 10 20 30 40 50 60TL (K)

0

2

4

6

8

10

12

|εlp

ma

x| (m

eV

)

ΩP=6 meV

ΩP=10 meV

ΩP=26 meV

FIG. 4. Dependence of the energy of the peak in the po-lariton distribution on TL for px = 1010cm−2/100ps, and dif-ferent polariton splittings.

7

Due to the isotropy of the lower polariton distribution,the emission at a given energy ǫlp

max, corresponds to emis-sion at a given angle with the growth axis. The movementof ǫlp

max towards lower energies implies that the sampleemits light preferentially in a ring, at a given angle thatdecreases with temperature. The recent experiment re-ported by Savvidis et al. [10] has observed the ring emis-sion with the linear dependence on lattice temperaturedescribed above, and shows that the X-P scattering froma thermalized exciton reservoir considered here, is themain scattering mechanism for relaxation into the LPB.

B. Evolution with Pump-Power

As shown in Eq. (18), ǫlpmax depends logarithmically

on the exciton density, and linearly on Tx. At large den-sities, however, the heating of the exciton reservoir mustbe properly taken into account. When occupation num-bers N lp

k > 1 are achieved, the X-P scattering mechanismshown in Fig. 1 is stimulated, and the injection of highenergy excitons becomes more and more important. Inthis situation, the heating of the exciton reservoir gov-erns the evolution of ǫlp

max as px is increased, becausefor higher Tx, the polariton distribution has a maximumat lower energies. This way, the heating of the excitonreservoir allows the excitons to fall into the polariton trapformed by the dip in the polariton dispersion.

Fig. 5 shows the evolution of the polariton distribu-tion as one increases the pump-power. The distributioncan reach the lowest energy state and large occupationnumbers can now be achieved. For large enough powers,the lowest energy state can have the main part of the po-lariton population, a situation that will be described indetail in section IV. For intermediate pump-powers, oc-cupation numbers larger than one are obtained for modeswith k 6= 0. Thus, our model predicts the stimulation to-wards these modes before BEC is achieved.

−5 −4 −3 −2 −1Energy (meV)

10−2

10−1

100

101

102

103

104

Occu

pa

tio

n n

um

be

r

px = 1

px = 15

FIG. 5. Evolution of the polariton distribution forTL = 10K, ΩP = 10meV , and px = 1, 2, 5, 8,15 × 1010cm−2/100ps (from bottom to top).

In Fig. 6 we show the evolution of ǫlpmax, as one

increases px, for different ΩP ’s. Three stages can beclearly distinguished. From px ≈ 109cm−2/100ps topx ≈ 1010cm−2/100ps, the exciton density created by thepump is small enough for the heating of the exciton reser-voir to be neglected. Under such conditions, the evolu-tion of ǫlp

max is governed by the logarithmic dependence onnx ≈ pxτx (see Eq. 18)). Above px ≈ 1010cm−2/100ps,the heating of the exciton gas is important, and the evo-lution of ǫlp

max is governed by the evolution of Tx, throughthe linear dependence shown in Eq. (18). In this regime,ǫlpmax results to depend linearly on px, until it saturates

to the bottom of the LPB.

0 5 10 15px (10

10cm

−2/100 ps)

0

5

10

15|ε

lpm

ax| (m

eV

)

ΩP=6 meV

ΩP=10 meV

ΩP=26 meV

FIG. 6. Dependence of the energy of the peak in the polari-ton distribution on pump-power, for TL = 10K, and differentpolariton splittings.

The heating of the exciton reservoir is shown in Fig.7. For large pump-powers excitons are no longer at thetemperature TL, at which they are initially injected. Theraise of Tx becomes more important for larger ΩP dueto the highest energies of the excitons injected into thereservoir by the X-P scattering (see Fig. 1).

8

0 10 20 30px (10

10cm

−2/100 ps)

0

20

40

60

80

Tx(K

)

ΩP=6 meV

ΩP=10 meV

ΩP=26 meV

FIG. 7. Dependence of the temperature of the excitonreservoir on pump-power for TL = 10K, and different po-lariton splittings.

At large px, X-P scattering is faster than the radia-tive recombination given by τx, an it becomes the mainmechanism for losses in the exciton reservoir. Thus, theexciton-polariton density, nxp (that is, the density of ex-citons in the reservoir plus the density of lower polari-tons), created by the pump is smaller than pxτx. Fig. 8shows the evolution of the steady-state polariton densityas px is increased. For all the three polariton splittingsconsidered the evolution of the exciton-polariton densityis very similar.

Fig. 5 and Fig. 8 allow to estimate the nxp at whichthe relaxation into the ground state is possible. In Fig.5 pump-powers above 5 × 1010cm−2/100ps correspondto polariton densities larger than 4 × 1010cm−2. Thus,in GaAs microcavities, the weak coupling regime will bereached before ǫlp

max reaches the bottom of the LPB dis-persion, due to the small nsat, and BEC is not possible.For the clean observation of the complete evolution of thepolariton distribution towards the ground state, CdTemicrocavities are promising candidates because nxp re-mains well below nsat, before the system relaxes towardsthe ground state.

0 10 20 30px (10

10cm

−2/100 ps)

0

5

10

15

20

excito

n−

po

larito

n d

en

sity (

10

10cm

−2)

ΩP=6,10 meV

ΩP=26 meV

FIG. 8. Dependence of the total exciton reservoir and lowerpolariton density, nxp, on pump-power, for TL = 10K. Theresults for ΩP = 6, and 10meV are indistinguishable.

The shift of ǫlpmax towards lower energies results also

in the emission in a ring at an angle that decreases aspump-power is increased. In the experiments performedby Savvidis et al. [10], this effect was observed, and a lin-ear relation between ǫlp

max and pump-power was reported.In this experiment, the extrapolation of the linear de-pendence of ǫlp

max on px towards small powers shows anenergy offset, in agreement with the theoretical result(see Fig. 6). Angle-resolved measurements allow, thus,to monitor the evolution of the polariton population to-wards the ground state.

As shown in Fig. 5 for intermediate pump-powers thesystem does not relax down to the ground state, butstimulation (occupancy larger than one) is achieved inthe scattering into intermediate modes with k 6= 0. Thestimulated modes will thus emit light at angles θ 6= 0.Fig. 9 presents the evolution of the occupation numbersfor the modes corresponding to different angles, as pump-power is increased. The evolution of the occupancy ateach angles matches the experimental results in [10].

109

1010

1011

px (cm−2

/100 ps)

10−4

10−2

100

102

104

Occu

pa

tio

n n

um

be

r

θ = 0Ο

θ = 25O

9

FIG. 9. Evolution of the occupancy of the modes corre-sponding to emission angles θ = 0, 5 . . . 25 degrees (fromright to left), as pump-power is increased. TL = 10K,ΩP = 10meV .

A quadratic relation is predicted, followed by an ex-ponential growth that saturates at large powers. Stimu-lation can be observed, at polariton densities below theBEC, at given angles, which correspond to the modeswhere the polariton distribution peaks, as shown in Fig.5. The saturation of the emission at a given angle is re-lated to the evolution of the polariton population that”shifts downwards” in energy, due to the heating of theexciton reservoir. Thus, the emission at a given angle sat-urates when the peak in the polariton distribution turnsto smaller angles, as pump-power is increased.

IV. BOSE-EINSTEIN CONDENSATION OF

POLARITONS

In the previous section it was shown that X-P scatter-ing is efficient enough to create a quasi-thermal popula-tion into the LPB. We will quantify now in detail how theoccupancy of the lowest energy polariton state evolveswhen the polariton density is increased.

Above a critical pump-power, the continuous Bose-Einstein distribution shown in Fig. 5 saturates, andthe additional excitons added by the pump fall into thebottom of the LPB. Thus, the lower polariton distri-bution becomes discontinuous, and an abrupt increaseis achieved in the number of polaritons in the lowestmicroscopic level, N0. Above the threshold for BEC,the macroscopic occupancy of a single microscopic modeopens the possibility for spontaneous symmetry breakingand the appearance of coherence in that mode. The in-teraction between the Bose-Einstein condensate and thelow energy polariton modes would lead then to the re-construction of the polariton spectrum, as predicted inthe well known Bogoliubov approach [31]. These effectscan modify the relaxation dynamics above threshold, sothat our results are not rigorous in this range. Howeverwe can predict the critical density and temperature forBEC, and the most favorable conditions for its observa-tion.

Fig. 10 (left) shows the evolution of N0, as a func-tion of px, for different lattice temperatures and polaritonsplittings. For large px above the critical pump, the rela-tion between N0 and px becomes linear, and occupationnumbers of the order of 5×104 polaritons can be reachedin the ground state. In order to interpret the evolutionof the ground state occupancy, it is interesting to plotN0 (Fig. 10 (right)) as a function of the steady-stateexciton-polariton density, nxp.

5 10 15 20 25 px (10

10cm

−2/100 ps)

2×104

4×104

6×104

N0

2×104

4×104

6×104

N0

2×104

4×104

6×104

N0

0.10 0.15 0.20 0.25 nxp/nsat

ΩP = 6 meV

ΩP = 10 meV

ΩP = 26 meV

ΩP = 6 meV

ΩP = 10 meV

ΩP = 26 meV

4K

10K

20K

4K 10K

20K

4K10K

20K

4K

10K

20K

4K10K

20K

4K

10K

20K

FIG. 10. Left: Evolution of the occupancy of the lowestmicroscopic polariton level, as a function of the pump-power.For each polariton splitting we consider TL = 4, 10, 20K.Right: Same as above, but now the occupancy of the lowestmicroscopic level is plotted as a function of the steady-stateexciton-polariton density.

The abrupt increase in N0 allows us to estimate a crit-ical density nc

xp for BEC, which results to be around0.1nsat. However, the situation is now not so clear asin the case of equilibrium BEC, where one expects thatN0 follows the relation (see Appendix B):

N0

S= nxp − nc

xp. (25)

That is, for densities above ncxp, all the excitons added

to the system should fall into the ground state. Fig. 10(right) shows that, for large densities, the linear relationis satisfied. However, close to nc

xp, N0 shows a superlineardependence on nxp. This fact can be interpreted as fol-lows. The equilibrium BEC relation between N0 and nxp

would be given by the extrapolation of the linear regimeof the curves in Fig. 10 (right), towards N0 ≈ 0. By thismethod, we would obtain the equilibrium nc

xp, which issmaller than the one we get from our complete nonequi-librium calculation. This means that in a real microcav-ity, nc

xp is larger than the one predicted by equilibriumBEC. Obviously, this is due to the radiative losses in theLPB, and also to the fact that the system has to evolve

10

towards the ground state, following the steps describedin the previous section. Thus, the X-P scattering hasto overcome the radiative losses, and the exciton reser-voir has to be heated, so that relaxation into the groundstate becomes efficient enough. The nonequilibrium crit-ical density has to be larger than the equilibrium one, forthese conditions to be fulfilled.

The transition from superlinear to linear dependenceof N0 as a function of px, and nxp, is reflected in theevolution of the exciton reservoir temperature, Tx. Fig.7 shows that the heating of the exciton gas slows downfor the pump-power for which the linear growth of N0

starts (see Fig. 10, left panel, ΩP = 10meV ). The samebehavior in the evolution of Tx has been obtained for allthe ΩP ’s considered.

We can also see in Fig. 10, that for small polaritonsplittings, smaller pump-powers and polariton densitiesare needed for BEC, because the energy distance betweenthe bottom of the LPB and the exciton reservoir is alsosmaller. For large ΩP , the exciton gas reaches highertemperatures in the steady-state regime, and the X-Pscattering rates are slower (thus, larger polariton den-sities are needed in order to get BEC). However, in allcases considered in both figures, the critical density iswell below nsat.

Up to now we have assumed Lc = 50µm, of the orderof typical spot diameters. However, Lc is not a well de-termined quantity in experiments. Equilibrium BEC in2-D systems predicts that both critical density and tem-perature depends only on the order of magnitude of Lc.In order to check what happens in our nonequilibriumsituation, we have calculated the density of polaritons inthe ground state, N0/S, for different quantization lengths(see Fig. 11).

0.05 0.10 0.15 0.20 0.25nxp/nsat

0.0

0.5

1.0

1.5

2.0

N0/S

(1

09

cm

−2)

Lc = 25 µm

Lc = 125 µm

FIG. 11. Evolution of the density of polaritons inthe lowest microscopic state, as the exciton-polaritondensity is increased, for different quantization lengths,Lc = 25, 50 − 125µm, and ΩP = 10meV , TL = 10K.

The occupancy of the lowest polariton mode has a

more abrupt increase for smaller Lc. However ncxp is

of the order of 0.1nsat for all the quantization lengthsconsidered.

V. CONCLUSIONS

Our main conclusions are: (i) At large densities, relax-ation after continuous non-resonant excitation can be de-scribed by a model in which lower polaritons are injectedfrom a thermalized exciton reservoir by X-P scattering.(ii) Our model shows a linear relation between the energyof the maximum in the lower polariton distribution andthe exciton temperature or pump-power. By increasingthe exciton density, the heating of the exciton reservoirallows the excitons to overcome the phase-space restric-tions and fall into the LPB, that becomes a polaritontrap in reciprocal space. (iii) Bose-Einstein Condensa-tion is possible for densities well below nsat for CdTe mi-crocavities. (iv) The most favorable conditions for BECare small polariton splittings, due to the smaller differ-ence between the bare exciton energy and the bottom ofthe LPB. (v) Our results have a weak dependence on thequantization length, which is taken as the excitation spotdiameter.

Up to now, non-resonant experiments performed inGaAs microcavities confirm that the weak couplingregime is reached before X-P scattering allows polaritonsto relax towards the ground state [10,12]. Experimentalevidence for the intermediate stage in the evolution ofthe polariton distribution has been achieved in a GaAsmicrocavity [10], and the experimental results agree verywell with the results presented in section III. In CdTe,experimental evidence for stimulation towards the lowerpolariton states has been achieved under pulsed non-resonant excitation. The angle-resolved measurementsafter pulsed excitation in CdTe presented in [32] quali-tatively agree with our results for the evolution of thepolariton distribution as pump-power is increased. How-ever, the physical situation in pulsed experiments is quitedifferent than the one considered here, because the sys-tem is not allowed to reach the steady-state in the re-laxation process. Continuous wave excitation seems tobe a cleaner experimental situation for the study of theevolution of the polaritons towards a quasi-equilibriumdistribution. Our model allows to predict not only thepossibility of BEC, but also quantitatively describes theevolution of the system towards the ground state. Angle-resolved measurements under continuous non-resonantexcitation would allow to show the macroscopic occu-pancy of the k = 0 mode, and also to monitor the evolu-tion towards BEC as pump-power is increased.

We are grateful to Michele Saba, Luis Vina, FrancescoTassone, Ulrich Rossler, Antonio Quattropani, and PaoloSchwendimann, for fruitful discussions. D. Porras ac-knowledges the hospitality of the Ecole Polytechnique

11

Federal de Lausanne. Work supported in part by MEC(Spain) under contract N PB96-0085, and CAM (Spain)under contract CAM 07N/0064/2001.

APPENDIX A: CALCULATION OF THE X-P

SCATTERING RATES

The calculation of the rates for the scattering of ther-malized excitons into the LPB was performed in [18].Here we reproduce, for clarity, the main steps, and cal-culate also the rate for ionization out of the LPB.

We consider first the process that injects polaritons. Itis given by:

W ink n2

x(1 + N lpk ) =

4

hπ3|Mk|2ρ3

x

∫

R(ǫxi , ǫx

j , ǫxf , ǫlp

k )

Nxi Nx

j (1 + N lpk )dǫx

i dǫxj . (A1)

Nxi , Nx

j are the occupation numbers for the initial exci-

tons in the reservoir, and Nxf , N lp

k correspond to the fi-nal exciton and lower polariton, respectively. The Nx inthe thermalized exciton reservoir follow the relation (8).Due to their large density of sates we can neglect thestimulation terms in the exciton reservoir (Nx << 1).Energy conservation is implicit in Eq. (A1), so that

ǫf = ǫi + ǫj − ǫlpk . The function R describes the phase-

space restrictions for the problem of exciton-exciton scat-tering in 2-D, under the approximation of isotropic ex-citon distributions. It can be expressed in terms of thecorresponding in-plane momenta [18]:

R(ǫxi , ǫx

j , ǫxf , ǫlp

k ) =∫

1

2dx

( [

(kxi +klp)2−x

] [

x−(kxi−klp)2

]

×[

(kxj+kf )2−x

] [

x−(kxj−kf)2

] )− 1

2 , (A2)

where integration is limited to the interval where the ar-gument of the square root is positive. A great simplifica-tion is obtained when we take the limit klp << kx:

R(ǫxi , ǫx

j , ǫxf , ǫlp

k ) =π

8ρx

1√

ǫxi ǫx

j −(

ǫlpi /2

)2. (A3)

If we insert this expression into (A1), we obtain

W ink =

2π

h(kBTx)2|Mk|2

∫

1√

ǫxi ǫx

j −(

ǫlpi /2

)2e−(ǫx

i +ǫxj )/kBTxdǫx

i dǫxj , (A4)

which after integration in energy gives the expression re-ported in Eq. (14).

The calculation of the rates for scattering out of theLPB follows the same way:

W outk nxN lp

k =

4

hπ3|Mk|2ρ3

x

∫

R(ǫxf , ǫlp

k , ǫxi , ǫlp

j )Nxf N lp

k dǫxi dǫx

j . (A5)

The factor R is again given by (A3), because phase-spacerestrictions are the same as in the previous case,

W outk =

ρx

hπ(kBTx)2|Mk|2

∫

1√

ǫxi ǫx

j −(

ǫlpi /2

)2e−ǫx

f /kBTxdǫxi dǫx

j . (A6)

ǫxf can be expressed in terms of ǫx

i , ǫxj , and the expression

in Eq. (14) is obtained.

APPENDIX B: BOSE-EINSTEIN

CONDENSATION IN TWO DIMENSIONS

Let us consider an ensemble of free bosons that fol-low a Bose-Einstein distribution with chemical potentialµ < 0, and finite temperature T . The density of particlescan be evaluated as:

n(µ, T ) =1

2π

∫ ∞

0

ρ(ǫ)dǫ

e(ǫ−µ)/kBT − 1. (B1)

In 3-D, ρ(ǫ) ∝ √ǫ, and the continuous Bose-Einstein

distribution with µ = 0 gives a finite critical densitync(T ). For densities n > nc(T ), all the extra particlesfall into the ground state.

In 2-D, ρ = m/h2 (constant), and the integration inEq. (B1) diverges as µ → 0. For an infinite system, BECis thus not possible at finite temperature. For a finitesystem, we can introduce a quantization length, Lc, suchthat the integration in Eq. (B1) has a low energy cut off,ǫc:

n(µ, T ) =m

2πh2

∫ ∞

ǫc

dǫ

e(ǫ−µ)/kBT − 1. (B2)

For the case ǫc/kBT << 1, Eq. (B2) gives a very simpleresult for the maximum density that the 2-D B-E distri-bution can accommodate:

nc(T ) = −mkBT

2πh2 log

(

ǫc

kBT

)

. (B3)

Whenever condition n > nc(T ) is satisfied, the extra par-ticles added to the system fall into the ground state:

N0(T )

S= n − nc(T ). (B4)

12

For free particles, ǫc ∝ L−2c , and Eq. (B2) allows to pre-

dict a weak logarithmic dependence of the critical densityon the quantization length. This result qualitatively ex-plains the weak dependence of the critical density on Lc,obtained in the case of nonequilibrium BEC discussedabove.

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13