Bose glass and superfuid phases of cavity polaritons G. Malpuech,1 D. D. Solnyshkov,1 H. Ouerdane,1 M. M. Glazov,2 and I. Shelykh3 1LASMEA, CNRS-Universit´e Blaise Pascal, 24 Avenue des Landais, 63177 Aubi`ere Cedex France 2A. F. Ioffe Physico-Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia 3ICCMP, Universidade de Bras`ılia, 70919-970 Bras`ılia-DF, Brazil 7 (Dated: February 6, 2008) 0 We report the calculation of cavity exciton-polariton phase diagram which takes into account 0 the presence of realistic structural disorder. Polaritons are modelled as weakly interacting two- 2 dimensional bosons. We show that with increasing density polaritons first undergo a quasi-phase n transition toward a Bose glass: the condensate is localized in at least one minimum of the disor- a der potential, depending on the value of the chemical potential of the polariton system. A fur- J ther increase of the density leads to a percolation process of the polariton fluid which gives rise 9 to a Kosterlitz-Thouless phase transition towards superfluidity. The spatial representation of the 2 condensate wavefunction as well as the spectrum of elementary excitations are obtained from the solution of theGross-Pitaevskii equation for all thephases. ] i c PACSnumbers: 71.36.+c,71.35.Lk,03.75.Mn s - l r Exciton-polaritons (polaritons) in microcavities are a = ~2/mV [12]. In two dimensions E is of the or- t 0 0 c m composite2-dimensionalweaklyinteractingbosons[1,2]. der ofpmean potential energy (i.e. 0 in our case), and . Despite their short radiative life time (∼ 10−12 s) stim- s ≈ 0.75 [11]. The quasi-classical density of states is t a ulated scattering towards their ground state has been D0(E) ∼= M/4π~2[1 + erf(E/V0)] [13]. However, an m demonstrated [3, 4], and quasi-thermal equilibrium was exciton-polariton is a composite boson also containing - recently reported [5]. In this quasi-equilibrium regime a photonic component which makes its trapping in the d cavitypolaritonsareexpectedtogiverisetoaKosterlitz- state with the localization radius a = a . λ (where n c o Thouless (KT) phase transition towardsuperfluidity [6]. λ is the wavelength of the incident light) not possible. c Thecorrespondingphasediagramhasbeenestablisheda Thus the resulting effective density of states shows an [ few years ago [7], and recently refined to fully take into abruptcut-offatE =E whichisself-consistentlydeter- 0 1 account the non-parabolicshape of the polariton disper- mined from a(E0)=ac: D(E)=D0(E) for E >E0 and v sion [8]. Because of their light effective mass (typically D(E)=0 otherwise. 3 10−4 times the free electron mass) polaritons show ex- Clearly, non-interacting bosons cannot undergo Bose 1 tremely small critical density and high critical tempera- Einstein Condensation (BEC) as the number of parti- 7 tures that can be larger than room temperature in some cles which can be fitted to all the excited states of the 1 0 cases. However,semiconductorswereassumedtobeideal system (E >E0) is divergent. The situation is thus dif- 7 intheapproachusedinRefs.[7]and[8]whileexperimen- ferent from those for cold atoms trapped by power-low 0 taldataclearlyshowstronglocalizationofthecondensate potential in 2D, where the renormalization of the den- / t because of structural imperfections. The phase observed sityofstatesmakes“true”BECpossible[14]. Therefore, a isinfactcharacteristicofaBoseglass[9]andnosignature even in the presence of disorder, BEC cannot take place m of superfluidity has been reported thus far. In this Let- strictly speaking for cavity polaritons. However, it is - ter we propose the derivation of a new polariton phase possible to define a quasi-phase transition which takes d n diagram taking into account structural disorder whose place in finite systems [7]. Indeed, for the finite size R o impact on the spatial shape of the wavefunction and the system there is a finite number N of potential traps trap c dispersion of elementary excitations, is analyzed within for polaritons, thus, there is an energy spacing between : v the framework of the Gross-Pitaevskiitheory [10]. the single particle states. The typical energy distance i between the ground and excited states of the finite-size X To give a qualitative picture of the model, we assume system levels δ under the assumption of long-range po- r that the polaritons are moving in a random potential a V(r) whose mean amplitude and root mean square fluc- tential is approximately given by V0/Ntrap or ~2/2MR02 tuation are given by hV(r)i = 0 and hV2(r)i = V whichever is smaller. In this framework the critical den- 0 sity is givenby the totalnumber of polaritonswhich can respectively. The correlation length opf this potential be accomodated in all the energy levels of the disorder is R0 = hV(r)V(0)idr/V02. As in any disordered potential V(r) except the ground one [7]: q system, therRe are here two types of polaritons states [11]: the free propagating states and the localized states n (T)= f (E ,E ,T), (1) with energy E < E , where E is the critical “delo- c B i 0 c c calization” energy. The localization radius scales like Xi6=0 a(E) ∝ a Vs/(E − E)s, s being a critical index and where f (T,µ,T) is the Bose-Einstein distribution func- 0 0 c B 2 tion, µ is the chemical potential. To evaluate the critical density n (T,R), the discrete c sum is replaced by an integral in Eq. (1), and we find n (T) ≈ D(E )k T ln 1/(1−eδ/kBT) assuming D(E) c 0 B is a smooth function. A(cid:2) bove this dens(cid:3)ity all additional particles are accumulating in the ground state and the concentration of condensed particles n satisfies n ≥ 0 0 (n−n ), where n is the total density of polaritons. It is c notarealphasetransitionsincethesystemhasadiscrete energy spectrum and the value of the chemical potential never becomes strictly equal to E . 0 Interactions between particles start to become dom- inant once the polaritons start to accumulate in the ground state. The situation can be qualitatively de- scribedasfollows: particlesstarttofillthelowestenergy state which is therefore blue shifted because of interac- FIG.1: (Coloronline)Spatialimages(toppanels)andquasi- tions(µ−E >0). Thus,forsomeoccupationnumberof 0 particlespectra(bottompanels)forarealisticdisorderpoten- thecondensatethechemicalpotentialreachestheenergy tial. Thefiguresshowncorrespondtodensities0,6×1010 and ofanotherlocalizedstate,andthis statestartsinturnto 2×1012 cm−2. Theredlinesareonlyguidestotheeye,show- populateandtoblueshift. Thecondensate,likealiquid, ing parabolic, flat and linear-typedispersions. The colormap fills several minima of the potential. It gives rise to the of the panel1.b is thesame as 1.c. spatial and reciprocal space pictures of Refs. [5, 6]. A few localized states, covering about 20 % of the surface of the emitting spot are all emitting light at the same m = 5×10−5m0, where m0 is the free electron mass, energy and are strongly populated. This characterizes a and the interaction constant g = 3Eba2B/Nqw, where Bose glass [9]. This situation occurs up to the achieve- Eb is the exciton binding energy (25 meV in CdTe), ment of the condition µ=Ec. This condition should be aB = 34 A˚ is the exciton Bohr radius and Nqw = 16 accompaniedbyapercolationofthecondensatewhichat is the number of wells embedded in the microcavity. We this stage should cover 50 % or more of the sample (in haveincluded arandomdisorderpotentialwith V0 =0.5 the semiclassical representation). The delocalized con- meV and R0 = 3 µm. Figure 1.a corresponds to the densate becomes at this stage a KT superfluid. More non-condensed situation. The spatial profile is given precisely, the different sides of the finite-size system are by the statistical averaging over the all occupied states, linked by the phase coherent path. Therefore we predict n(r) = jfB(Ej,T,µ(T))|Ψj(r)|2. Here the tempera- two quasi-phase transitions driven by temperature and ture is sPet to T = 19 K which corresponds to the effec- particle density: first, with an increase of the polariton tive polariton temperature measured in [5]. In this case density beyond n (T) the system enters the Bose glass the total number of particles is small and thus nonlinear c phase, then with a further increase of the density the terms in Gross-Pitaevskiiequation can be neglected. polaritonsystem becomes superfluid. The critical condi- Oncethequasi-condensateisformed,andformoderate tion µ = E is valid only at low temperature where the temperatures,one can neglectthe thermaloccupationof c thermal depletion of the condensate is negligible. the excited states and the spatial image of the polariton The quantitative analysis can be carried out in the distribution is given by the ground state wavefunction framework of the Gross-Pitaevskii equation for the con- which corresponds to solution of Eq. (2). We show the densate wavefunction Ψ(r,t) which reads resultingdensitybelowandabovethepercolationthresh- old on Figs. 1.b and 1.c respectively. As expected, the condensate is localized in a few minima of the random ∂ ~2 potential as shown on Fig. 1.b. On Fig. 1.c the conden- i~ Ψ(r,t)= − △+V(r)+g|Ψ(r,t)|2 Ψ(r,t), ∂t (cid:18) 2M (cid:19) satewavefunctionstillexhibitssomespatialfluctuations (2) connected to disorder,but the condensate is nonetheless where g is a constant characterizing the weak repul- well delocalized, covering the whole sample area. sive interaction between polaritons. The solution of To calculate the quasiparticle spectra shown in lower the Gross-Pitaevskii equation takes the form Ψ(r,t) = panels 1.d, 1.e and 1.f of Fig. 1 we introduce a single- Ψ (r)exp(−iµt/~). Toppanels(a),(b),and(c)ofFig.1 particle Green’s function which takes the form 0 show the real space distribution of the polaritons ob- tainedfromthesolutionoftheGross-Pitaevskiiequation. Ψ (r)Ψ†(r ) The parameters are those of a realistic CdTe microcav- G (r,r )= j j 0 , (3) ω 0 ~ω−E ity at zero detuning. We have taken the polariton mass Xj j 3 where E and Ψ (r) are energies and eigenfunctions of j j the elementary excitations [15], found numerically from 1.0 Eq.(2). The spectrumof elementaryexcitationsis given by the poles of the Green’s function in the (k, ω) repre- n0.8 o sentation, and shown on the lower panels of Fig. 1. The cti leenfetdpabnyelth1e.ddsihsoorwdsertyppoicteanltpiaalr.abTolhice dmisipdderlesiopnanberloa1d.e- d fra0.6 ui shows parabolic dispersion with a flat part produced by erfl0.4 the localization of the condensate. The linear spectrum p u S0.2 on the right panel 1.f is the distinct feature of the su- perfluid state of the system. Only the upper Bogoliubov 0.0 branch is shown. Figures 1.b and 1.e reproduce quite 1011 1012 1013 well the experimental observations of Ref. [5] which are Density of particles (cm-2) characteristics of the formation of a Bose glass. It is instructive to analyze both the variation of the FIG.2: (Coloronline)Superfluidfractionasafunctionofthe emissionpatternandthequasi-particlespectrumincom- density of particles, obtained from twisted boundary condi- parisonwiththebehaviorofthesuperfluidfractionofthe tions (black curve) and from the perturbative approach (red polariton system. The latter quantity can be calculated curve). using the twisted boundary conditions method [16]. Im- posingsuchboundaryconditionsimpliesthattheconden- sate wavefunction acquires a phase between the bound- T =0 K. Due to the finiteness of the system considered aries, namely the superfluid fraction remains non-zero for any finite density, but a very clear threshold behavior for densities corresponding to the percolation threshold as observed Ψθ(r+Li)=eiθΨθ(r), (4) on Fig. 1, is also shown. For high values of the chemical potential,whereV2/µg ≪1,perturbationtheoryapplies whereL (i=x,y)arethevectorswhichformtherectan- 0 i and we obtain gle confining the polaritons and θ is the twisting param- eter. The superfluid fraction of the condensate is given by [16] nd = 1V02, (7) n 4 µg ns 2ML2(µθ −µ0) forthenormaldensity[17]. Theresultingcurveisshown f = = lim , (5) s n θ→0 ~2nθ2 in red on Fig. 2. The twisted boundary conditions ap- proach and Eq. (7) give coinciding results for high den- where µ is the chemical potential corresponding to the θ sity of the polaritons. boundary conditions Eq. (4) and µ is the chemical 0 In the rest of the paper, we concentrate on the estab- potential corresponding to the periodic boundary con- lishmentofthecavitypolaritonphasediagram. Similarly ditions (θ =0). In the case of a clean system, V(r)=0, to previous works [7], we roughly define a temperature the plane wave is the solution of Eq. (2) and µ −µ = θ 0 and density domain where the strong coupling is sup- n~2θ2/2ML2: the superfluid fraction is f = 1. On the s posed to hold. The limits are shown on Fig. 3 as thick contrary,for the strongly localized condensate the wave- dotted lines [20]. The transition from normal to Bose function is exponentially small at the system boundaries glassphasecanbecalculatedfromEq.(1)andarealistic and the change of the boundary condition (i.e. varia- realization of disorder. The lower solid line on Fig. 3 tionofθ)does notchangetheenergyofthe system,thus shows n (T) for the same realization of disorder as for f ∼exp[−L/a(µ )]andgoesto0fortheinfinitesystem. c s 0 Fig. 1. The free polariton dispersion is calculated using Due to the exponential tails of the localized wavefunc- the geometry of Ref. [5]. At T = 19 K we find n = tions a smalldegree ofsuperfluidity remains in the finite c 2×108cm−2. sizesystem. Equations(2)and(4)allowtostudythede- We now calculate the density for the transition be- pletion of the superfluid fraction for arbitrary disorder. tween the Bose glass and the superfluid phase. In the Thecontributionofthedisordertothenormaldensityof low temperature domain, this density is approximately polaritons can be represented as given by the percolation threshold µ= E and does not c dependsignificantlyontemperature. Thisconditioncor- nd =(1−f )n. (6) respondswithgoodaccuracytotheabruptchangeofthe n s superfluidfractionf showninFig.2. However,athigher s Figure 2 shows the superfluid fraction calculated as temperaturethe thermaldepletion ofthe condensatebe- a function of the polariton density in the system for comes the dominant effect. In that case the chemical 4 potential of the condensate is much higher than the per- VCSEL colationenergyE andthedepletioninducedbydisorder c 1012 Superfluid can be neglected compared to the thermal depletion of the superfluid. The normal density then reads 2) − m c n0(T)=− 2 E(k) ∂fB[ǫ(k),µ=0,T] dk, (8) sity (1010 Bose Glass LED n (2π)2 Z ∂ǫ en d n and the superfluid density in the system is given by o olarit108 Polariton diode P n (T)=n−n0(T), (9) s n which can be substituted into the Kosterlitz-Nelson for- 106 mula[18]toobtainaself-consistentequationforthetran- 100 101 102 103 sition temperature: Temperature (K) FIG. 3: (Color online) Polariton phase diagram for a CdTe ~2πn (T ) T = s KT . (10) microcavity containing 16 QWs. The horizontal and verti- KT 2M caldashedlinesshowthelimitingtemperaturesanddensities The joint solution of Eqs. (10) and Eq. (9) allows to wherethestrongcouplingholds. Thelowersolidlineshowthe critical density for the transition from normal to Bose glass determine the superfluid phase transition temperature phase. The uppersolid line shows thecritical density for the T (n). The resultofthis procedureis shownonFig.3. KT transition from the Bose glass to the superfluid phase. The Below 120 K the critical density is given by the percola- dashed part of the line shows the temperature range where tion threshold and there is no temperature dependence. thevalidity of ourapproximations ceases. Above200Kthesuperfluiddepletionisdeterminedsolely by the thermal effects. In the intermediate regime the crossoverbetweenthethermalanddisordercontributions takes place and our approximations are no longer justi- fied. We also find that the superfluid transition takes place very close to the weak to strong coupling thresh- [1] A. Kavokin and G. Malpuech, Cavity Polaritons, Else- old and for densities 3 orders of magnitude larger than vier, (2003). [2] A.Imamoglu, J.R. Ram,Phys.Lett.A 214, 193 (1996). the one of the Bose glass transition at 19 K. This sug- [3] L. S. Dang et al. Phys.Rev.Lett 81, 3920 (1998). gests that experimental observation of this phenomenon [4] H. Denget al.Science 298, 199 (2002). remains a great challenge. [5] J. Kasprzak et al. Nature443, 409 (2006). In conclusion, we have established the phase diagram [6] J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 of cavity polaritons taking into account the effect of (1973). structuralimperfections. Wepredictthatwithincreasing [7] G.Malpuechetal.Semicond.Sci.&Technol.18,Special density the polariton system first enters the Bose glass issue on microcavities, edited by J.J. Baumberg and L. Vin˜a, S 395 (2003). phase before it becomes superfluid. The Bose glass pic- [8] J. Keeling, Phys.Rev.B 74 155325 (2006). ture is in good agreementwith recent experimental data [9] M.P.A.Fisher,P.B.Weichman,G.GrinsteinandD.S. [5]. The condensate wavefunctions as well as the spec- Fisher, Phys. Rev.B 40, 546 (1989). tra ofelementaryexcitations areobtainedfromthe solu- [10] L. Pitaevskii and S. Stringari, Bose-Einstein Condensa- tionsoftheGross-Pitaevskiiequationincludingdisorder. tion (Oxford UniversityPress, 2003). Ourworkalsoshowsthatthepresenceofdisorderhasno [11] A.L.Efros andB. I.Shklovskii, Electronic Properties of significant impact on the occurence of a bosonic phase Doped Semiconductors (Springer, Heidelberg, 1989). [12] Hereandbelowwedisregardthenon-parabolicityeffects transition for polaritons. This explains why this phe- onthecavitypolaritondispersionwhichareknowntobe nomenonhasbeenobservedinaratherdisorderedsystem small providedthetemperatureandpolariton numberis like CdTe. This also gives good hope for the observation not too high [8]. ofsuchphasetransitioninevenmoredisorderedsystems [13] We disregard the spin of polaritons and omit the spin like GaN [19]. However, since disorder strongly affects degeneracy factor here and below. the occurence of the superfluid phase transition,it could [14] V. Bagnato, D. Kleppner, Phys. Rev. A 44, 7439-7441 bringrenewedinterestincleanersystemslikeGaAsbased (1991) [15] E.M.LifshitzandL.P.Pitaevskii,Statisticalphysics,part structures. 2 (Pergamon Press, New York,1980). We thank K.V. Kavokin for enlightening discussions. [16] A.J. Leggett, Phys.Rev.Lett. 25, 1543 (1970). We acknowledge the support of the STREP ”STIM- [17] O.L. Berman, Y.E. Lozovik, D.W. Snoke, R.D Coalson, SCAT” 517769, and of the Chair of Excellence program Phys. Rev.B 70, 235310 (2004). of ANR. [18] D.R. Nelson, J.M. Kosterlitz, Phys. Rev. Lett. 39, 1201 5 (1977). theexcitonbindingenergy.Themaximumpolaritonden- [19] G. Malpuech, A. Di Carlo, A. Kavokin, J.J. Baumberg, sity is taken 32 times larger than bleaching exciton den- M. Zamfirescu and P. Lugli, Appl. Phys. Lett., 81, 412, sity which is assumed to be1011 cm−2 in CdTe. (2002). [20] The edge temperature is assumed to be 300 K equal to