Microscopic Theory of Polariton Lasing via Vibronically Assisted Scattering
Leonardo Mazza, Stéphane Kéna-Cohen, Paolo Michetti, Giuseppe C. La Rocca
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Microscopic Theory of Polariton Lasing via Vibronically Assisted Scattering
L. Mazza, ∗ S. K´ena-Cohen, P. Michetti, and G. C. La Rocca Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126, Pisa, Italy The Blackett Laboratory, Department of Physics,Imperial College London, London SW7 2AZ, United Kingdom Institute for Theoretical Physics and Astrophysics,University of W¨urzburg, D-97074 W¨urzburg, Germany Scuola Normale Superiore and CNISM, Piazza dei Cavalieri 7, I-56126, Pisa, Italy
Polariton lasing has recently been observed in strongly coupled crystalline anthracene microcavi-ties. A simple model is developed describing the onset of the non-linear threshold based on a masterequation including the relevant relaxation processes and employing realistic material parameters.The mechanism governing the build-up of the polariton population - namely bosonic stimulatedscattering from the exciton reservoir via a vibronically assisted process - is characterized and itsefficiency calculated on the basis of a microscopic theory. The role of polariton-polariton bimolecularquenching is identified and temperature dependent effects are discussed.
I. INTRODUCTION
In strongly coupled semiconductor microcavities thecavity mode and the excitonic resonance mix and formnew bosonic quasiparticles, the polaritons. Their prop-erties differ significantly from those of the bare uncou-pled excitations, though they originate from them. Thelower polariton (LP) has a peculiar dispersion law witha deep minimum at small wavevectors, characterized bya tiny mass. At high densities, the build-up of a largepopulation at the bottom of this branch is favoured bybosonic final-state stimulation as soon as the occupationper mode of the lower polariton states exceeds unity.Coherent light-emission, called polariton lasing, resultsfrom this macroscopic population due to the finite life-time of the polaritons which leak out of the cavity viatheir photonic component. This is only one of the manyoutstanding phenomena that have attracted more andmore attention to the field of polaritonics in inorganicsemiconductor microcavities since the pioneering obser-vation of the strong coupling regime. The weak binding energy and oscillator strength ofWannier-Mott excitons characteristic of inorganic semi-conductors are limitations that can be overcome em-ploying organic semiconductors having strongly boundFrenkel excitons with a large oscillator strength. Thestrong coupling regime in an organic based microcav-ity was first observed at room temperature using a por-phirine molecule (4TBPPZn) dispersed in a polysterenefilm as optically resonant material at room tempera-ture, and later in a variety of organic materials in-cluding polyacene molecular crystals. The latter arealso characterized by the presence of well developed vi-bronic replicas that participate in polariton formationand affect their luminescence. In contrast to the case ofinorganic microcavities, manifestations of bosonic stim-ulation using organic cavity polaritons have been quiteelusive. Recently, however, several non-linear phenom-ena were reported: room temperature polariton lasingin an anthracene single crystal microcavity, indirect pumping of J-aggregate lasing microcavities, and non-linear emission in polymer-based microcavities. In an-thracene, the observation of a threshold for nonlinearemission was accompanied by a significant line narrow-ing and by a collapse of the emission lifetime. In thatcase, a comparison with the best-case estimate of thethreshold for conventional lasing inferred from amplifiedstimulated emission measurements shows that the lasingthreshold observed in the strongly coupled microcavityis slightly lower than that anticipated for a conventionallaser. The temperature dependence of the polariton las-ing threshold has also been investigated and shows anorder of magnitude decrease from room temperature tolow temperatures. These experiments demonstrate thehigh excitation density regime of polariton bosonic stim-ulation, which could pave the way to the observation inorganic based microcavities of other phenomena relatedto polariton fluidics where weak polariton-polariton in-teractions may also manifest. In the present work, we develop a semiclassical kineticmodel to describe the onset of the non-linear thresholdfor polariton lasing in anthracene-based microcavities.We show, in particular, that the mechanism providingthe bosonic final-state stimulated formation of the en-semble of lower cavity polaritons is the vibrationally as-sisted radiative decay of incoherent excitons, previouslypopulated by non-resonant pumping. In Section II, weset up a minimal master equation to describe the polari-ton population dynamics, we make a realistic choice ofmaterial parameters and we fit the experimental data onthe pump dependence of the polariton emission, pointingout the relevance of bimolecular quenching processes. InSection III, we calculate microscopically the efficiency ofthe relevant scattering process justifying the value ob-tained from the fit. In Section IV, we consider withinour model the dependence of the polariton lasing thresh-old on temperature. Finally, in Section V, we present ourconclusions.
II. TWO-LEVEL MODEL
We model the dynamics of the lasing process usinga minimal rate-equation approach. In this section, weestimate the typical time-scale of the mechanism whichselectively transfers excitations from the reservoir to thebottom of the polariton branch, without any assumptionsregarding its microscopic nature.
A. The Master Equation
The anthracene crystal has two molecules per unitcell and strongly anisotropic optical properties.
Ex-citations in this material are well-described within theFrenkel-exciton framework, which is based on the in-tramolecular promotion of an electron from the highestoccupied molecular orbital to the lowest unoccupied one.Because of molecular dipole-dipole interaction, the exci-tation can propagate, resulting in two orthogonal transi-tion dipole moments, ~µ a,b , directed along the in-plane a and b axes. When a thin anthracene crystal is placedbetween two mirrors, light couples to both a - and b -polarized excitons and creates two orthogonally-polarizedlower polariton branches. Measurements are usually re-ported for light polarized along a and b : in these casesthe p and s in-cavity light polarizations separately cou-ple to the dipole moments ~µ a,b and no mixing effect ispresent.We focus only on b -polarized excitons, i.e. thosewith largest oscillator strength, for which lasing has beenreported and neglect other polaritonic and excitonicstates. The initial relaxation of the pump excitationsis also neglected, and the presence of an effective exci-tonic reservoir at a fixed energy independent on the cav-ity properties is considered. We note that the experi-mental photoluminescence (PL) from anthracene micro-cavities shows always a clear maximum at energy ∼ . and indeed las-ing has been achieved in a cavity where the minimum ofthe LP is exactly at 2 .
94 eV. This is a signature that themicroscopic dynamics resulting in the lasing phenomenonis that of a two-level process rather than that of the well-known polariton bottleneck. We thus develop a two-levelmaster equation for ν e ( t ) and ν p ( t ), the surface densityof reservoir excitons and of lasing polaritons located near k = 0, respectively.We denote with A the subregion of the Brillouin zonelocated around k = 0 which is occupied by the lasing po-laritons. Because states at the bottom of the LP branchdo not have a well-defined wavevector k , we consider allof the localized wavepackets with energy ∼ E LP ( k = 0)as equally contributing to the lasing process. N pol is thenumber of such polaritonic states, while N exc is the num-ber of excitonic states. The polariton and exciton decayrates are Γ p = | c ( e ) p | /τ e + | c ( p ) p | /τ p and Γ e = 1 /τ e , re-spectively, where τ p ( τ e ) is the bare photon (exciton) life- Pump P ′ ( t ) Efficient transfer Excitonquenching γ ′ Bimoleculardecay Γ e Decay topolaritonsmechanism W e → p Lowerpolaritonbranch reservoir, ν e Exciton Z e → Lasingpolaritons, ν p Bimolecularquenching γ ′ Luminescence Γ p FIG. 1. Sketch of the LP branch and of the physical processesand scattering mechanisms included in master equation (1). time and c ( p ) p (cid:16) c ( e ) p (cid:17) is the photonic (excitonic) Hopfieldcoefficient for the lasing polaritons.The parameter Z e → is the decay rate via other chan-nels, such as phonons, lower polaritons outside the A region and leaky modes, whereas bimolecular quenchingprocesses are treated separately, with a rate γ ′ . A stan-dard pump term proportional to P ′ ( t ) is included; inorder to take into account possible saturation effects theterm (1 − ν e / ¯ ν e ) has been considered (¯ ν e = N exc /A is thesurface-density of excitonic states and A is the area ofthe sample).The rate of resonant excitation transfer from the reser-voir to the lasing polaritons is W e → p . We retain thebosonic enhancement term (1 + ν p / ¯ ν p ) responsible forlasing effects, where ¯ ν p = N pol /A is the surface-densityof polaritonic states.The master equation for ν e ( t ) and ν p ( t ), whose physicsis sketched in Fig. 1, reads:˙ ν e = − Γ e ν e − W e → p ν e (cid:18) ν p ¯ ν p (cid:19) − Z e → ν e + − γ ′ (cid:16) ν e + | c ( e ) p | ν p (cid:17) ν e + (cid:18) − ν e ¯ ν e (cid:19) P ′ ( t ) (1a)˙ ν p = − Γ p ν p + W e → p ν e (cid:18) ν p ¯ ν p (cid:19) + − γ ′ (cid:16) ν e + | c ( e ) p | ν p (cid:17) | c ( e ) p | ν p (1b)The full derivation is given in appendix A. Note that theresulting equations are completely analogous to those de-scribing conventional lasing, with the important differ-ence that the lasing state is a polariton and thus retainsan excitonic component. B. Parameters
We relate Eq. (1) to the experimental system inRef. [12] using the following parameters.
Simulation Parameters ρ = 4 . × cm − L z = 120 nm¯ ν e = 5 . × − cm − q = 2 . × cm − τ p = 85 fs ∼ τ e = 2 ns c ( p ) p = 0 . c ( e ) p = 0 . τ p W e → p γ ′
85 fs 4 × s − . × − cm s − . × s − . × − cm s − TABLE I. Parameters for the numerical simulations. (top)List of the most important simulation parameters used in thenumerical simulations. (bottom) Results of the fit reportedin Fig. 3 a. Anthracene Crystal.
The experimental microcav-ity embeds a crystal of anthracene with thickness L z =120 nm; the molecular density is ρ = 4 . × cm − :we ignore the monoclinic structure of the unit cell andinstead estimate its linear size as a = ( ρ / − / =7 . × − cm, including the presence of two moleculesper unit cell. The number of layers is estimated as N = L z /a ≈ E = 3 .
17 eV. The excitonmeasured lifetime is of the order of τ e ∼ − τ e = 2ns. The contribution of Z e → is neglected because it canbe included into τ e without any substantial difference aslong as τ e < /Z e → , which can be safely assumed. b. Microcavity and Polaritons. If we assume homo-geneous broadening, the cavity lifetime can be estimatedfrom the polariton linewidth at k = 0, where it is mostlyphoton-like. Using this approach, we obtain a lowerbound τ p = 85 fs. An exact calculation assuming per-fect interfaces for the mirrors results in an upper bound τ p = 1 ps. We will estimate W e → p corresponding to bothextrema. The Hopfield coefficients of the LP branch are : c ( p ) p = 0 .
92 and c ( e ) p = 0 . | k | , the A region has cylindrical sym-metry, . Its radius, q , can be estimated using E LP ( q ) − E LP ( k = 0) = Γ /
2, where Γ = 15 meVis the linewidth of polaritons at k = 0 below threshold; we obtain q = 2 . × cm − . c. Pump. The pump density is: P ′ ( t ) = P ′ exp (cid:20) − t σ (cid:21) ; σ = 1502 √ ≈
64 fs , with P ′ = P / ( πr ~ ω pump ) where r = 110 µ m is theradius of the pump spot and ~ ω pump = 3 .
45 eV is theenergy of the pump photons. Because E tot = R P ( t ) dt = √ πP σ and because E tot , the total absorbed energy,and σ are experimentally known, P is also known. d. Bimolecular Quenching Rate. To the best of ourknowledge there are no measurements of the bimolecularquenching rate, γ ′ , for two-dimensional anthracene crys-tals. According to the standard theory for bimolecular Absorbed Pump Fluence (nJ) I n t eg r a t ed I n t en s i t y ( a r b . un i t s ) Experimental= 1 ps τ p τ p = 85 fs FIG. 2. Time-integrated surface density of polaritons R ν p ( τ ) dτ calculated from solution of Eq. (1) (lines) and fromexperimental data (squares). The bimolecular quenching rateis taken from the measured 3D bulk value: γ ′ = γ /L z .The calculation parameters are: (dashed line) τ p = 85 fs, W e → p = 7 × s − ; (solid line) τ p = 1 ps, W e → p = 7 × s − . Because the experimental data is in arbitrary units, hereand in the following fits the experimental data is normalizedso that the first experimental and theoretical points coincide. quenching , γ = 8 πRD , where R is the F¨orster radiusof the exciton and sets the volume around the exciton inwhich annihilation happens, while D is the diffusion co-efficient of excitons. Measurements for three-dimensionalanthracene crystals have yielded values of γ = 10 − cm s − and D ∼ − × − cm s − . The cor-responding diffusion length ℓ = ( τ e D ) / ∼ − × − cm is smaller than L z = 1 . × − cm and suggeststhat excitons can be treated as diffusing in a three-dimensional environment. As a result, we initially fix γ ′ = γ /L z = 7 × − cm s − .Parameters used in the numerical simulations arebriefly summarized in Table I. C. Results
Since all other parameters are known, we leave only W e → p as a fit parameter. We numerically integrateequations (1a) and (1b) and once the complete time-dependent functions ν e,p ( t ) are known we compute theintegral R ν p ( τ ) d τ and compare it with the experimentalvalues.In Fig. 2 the fits obtained for the extreme values of τ p = 85 fs and 1 ps are shown. The value of W e → p hasbeen fit to the experimentally observed threshold value.In both cases, W e → p is of the order 10 s − . The agree-ment with the experiment is poor and it is apparent thatthe chosen value of γ ′ does not properly describe the tran-sition between linear and sublinear region below thresh-old. Note that the exciton lifetime τ e ∼ τ e, bulk ∼
10 ns; surface in-teractions or defects within the layers could explain thisdiscrepancy. In this situation, the excitonic diffusion co-efficient can be smaller, resulting in a reduced possibility I n t eg r a t ed I n t en s i t y ( a r b . un i t s ) Absorbed Pump Fluence (nJ)
Experimental= 1 ps τ p τ p = 85 fs FIG. 3. Time-integrated surface density of polaritons R ν p ( τ ) dτ calculated from solution of Eq. (1) (lines) and fromexperimental data (squares). The bimolecular quenching pa-rameter γ ′ is used to fit the below-threshold behavior of theexperimental data. The calculation parameters are: (dashedline) τ p = 85 fs, γ ′ = 1 . × − cm s − , W e → p = 4 × s − ; (solid line) τ p = 1 ps, γ ′ = 1 . × − cm s − , W e → p = 3 . × s − . for excitons to pairwise annihilate.Because a fit of γ ′ which determines the onset of bi-molecular quenching can be readily decoupled from thatof W e → p , both parameters are allowed to vary and the re-sulting fits are shown in Fig. 3. We obtain γ ′ ≈ . × − cm s − independently of τ p , as expected. Note that thisvalue is two orders of magnitude smaller than γ /L z .The resulting values for W e → p are 4 × s − and3 . × s − for τ p = 85 fs and 1 ps, respectively. Even ifthe scattering process acts on a sensibly longer timescalecompared to the exciton and polariton lifetimes, it canlead to observable effects in presence of high excitonicdensities. We can roughly estimate the surface densityof excitons at threshold via Γ p ν p = W e → p ν e (1 + ν p / ¯ ν p ).Assuming that at threshold ν p = ¯ ν p , we obtain: ν e,th ¯ ν e = ¯ ν p ¯ ν e Γ p W e → p ∼ . ν e,th / ¯ ν e does notdepend on the value of q , because both W e → p and ¯ ν p depend linearly on the size of the A region.Although the fit below threshold is excellent, the re-gion above threshold is poorly described. It can be seenin Fig. 4 and Fig. 5, which shows the time dependenceand peak of the normalized surface exciton and polaritondensities, that at threshold the exciton density reaches afew percent of the total molecular density. Such highexcitation densities may require a more refined descrip-tion of the annihilation process. Indeed, our calculationabove threshold seems to be in better agreement with re-cent low-temperature data, where the threshold occursat lower excitation density. Moreover, above thresh-old, when the polariton density becomes important, the - - H n s L Ν e H t L Ν e , Ν p H t L Ν p FIG. 4. Time-dependence of the normalized surface density ofexcitons ν e ( t ) / ¯ ν e and of polaritons ν p ( t ) / ¯ ν p below threshold( E tot = 17 nJ, dotted line and solid line respectively) and atthreshold ( E tot = 150 nJ, dashed line and dashed-dotted line)plotted for τ p = 85 fs. Note that this time dependence is ingood agreement with that reported in Ref. [12]. Absorbed Pum p Fluen ce H n J L S u rf ace D e n s it y H c m - L FIG. 5. Maximal population density of excitons max t ν e ( t )(squares) and of lasing photons max t ν p ( t ) (circles). See Fig. 3for the parameters; τ p = 85 fs. details of the theoretical model used for the polariton-polariton bimolecular quenching become important. Note that the mean-field polariton-polariton interac-tion has not been included as no blue shift has beenresolved in the experiments, which feature a relativelybroad linewidth. In conclusion, using our simple two-level model we haveextracted an estimate for the scattering process W e → p relevant to polariton lasing in anthracene. Furthermore,we believe that the strongly reduced rate of bimolecu-lar annihilation observed should motivate further exper-imental and theoretical studies of this process. III. THE SCATTERING MECHANISM
In this section we focus on the microscopic origin ofthe excitation transfer of Sec. II. In particular, we pro-pose as the relevant mechanism the radiative recombina-tion of a molecular exciton assisted by the emission of avibrational quantum of the electronic ground state. Weshow that the resulting scattering rate is in good agree-ment with that obtained in the previous Section. Finally,we also consider an alternative and possibly coexisting
Final stateProcessScattering emissionradiative
ElectronicexcitedstatesElectronicgroundstates
Initial state
ElectronicexcitedstatesstatesgroundElectronic
Molecularstates Molecularstates (photon)PolaritonassistedVibronically
FIG. 6. Sketch of the radiative microscopic mechanism re-sponsible for the efficient excitation transfer in the Franck-Condon approximation. model based on the non-radiative emission of an opticalphonon. A. Radiative Transition
The absorption and PL spectra of anthracene show sev-eral vibronic resonances. The resonances observed inabsorption correspond to the molecular vibrations of thefirst electronically excited state, and those in PL to thevibrations of the electronic ground state. Strong light-matter coupling has only been demonstrated for the for-mer, since the fraction of vibrationally excited ground-state molecules is negligible at room temperature. How-ever, as shown schematically in Fig. 6, the transitionsresponsible for the vibronic structure in PL result in thescattering of excitons to lower energy polaritons, wherethe missing energy exactly corresponds to that of the vi-brational quantum, E ∼
173 meV. In appendix B, we discuss the approximations neededto apply the known microscopic theory to the presentsystem . For instance, the microscopic model consid-ers a thin organic material comprising N ∼ O (1) lay-ers and placed in the middle of the microcavity whereas the experimental sample embeds an organic ma-terial which fills the whole region between the two mir-rors ( N ∼ O (10 )) and has no planar translational in-variance. Moreover, the theory assumes the presence ofperfect mirrors, whereas in experiment τ p is always finite.In equation (B6) the scattering rate W j → k from onemolecular exciton (labelled by j ) to a lasing polariton (la-belled by k ) is related to the parameters of a anthracenemicrocavity. As discussed in appendix A, the scatter-ing rate appearing in the master equation is W e → p = P k ∈A W j → k . Working in the energy space and definingthe spectral region of lasing polaritons E ∈ [ E A inf , E A sup ]and the polariton density of states D ( E ), we get: W e → p = Z E A E A V ± ~ π S | c ( p ) p | M N f ( E − E − E ) D ( E )d E (2) M the number of unit cells in the two-dimensional quan-tization surface and V is the fit light-matter coupling. S is the Huang-Rhys parameter, which is approximately ∼ See appendix B for more details. The 2D densityof state is D ( E ) = mM a π ~ θ [ E − E LP ( k = 0) ] , where the effective mass m can be obtained from thefits of the dispersion relations: m ∼ . × − m e ;moreover ¯ ν e = 2 N/a . The normalized linewidth of(0-1) photoluminescence f ( E ) is a Lorentzian centeredin zero with FWHM Γ = 100 meV; we also assume E A inf = E − E = E LP ( k = 0) whereas E A sup = E A inf +7 . W e → p = πS | c ( p ) p | mV ¯ ν e ~ Z E A E A f ( E A inf − E )d E. (3)The rate before the integral is equal to ≈ . × s − ,while the contribution from the integral, which comesfrom the lineshape, is ≈ . W e → p ≈ × s − .Because the theoretical model neglects effects whichcan possibly lower the efficiency of the resonant scatter-ing, we consider our estimate to be in good agreementwith the values estimated from data in Sec. II. B. Non-radiative transition
We now consider an alternative and possibly coexistingrelaxation channel, which is non-radiative. An excitonis scattered from the reservoir to one polariton state bythe emission of a molecular vibration of the electronicexcited state. This is due to the intramolecular exciton-phonon coupling which has been demonstrated to playa key role in the modeling of the PL of J-aggregates mi-crocavities.
Note that in this case the considered phonon belongs tothe electronic excited state, whereas in the radiative caseit was related to the electronic ground state. Moreover,the resulting scattering element W e → k includes the exci-tonic content of the outcoming polariton, whereas Eq. (3)is weighted by the photonic Hopfield coefficient.The scattering rate from one molecular exciton (la-belled by j ) to one lasing polariton (labelled by k ) isgiven by: W j → k = 2 π ~ g E | c ( e ) p | N M δ ( E − E LP ( k ) − E ) (4)where g = √ S ∼ E is the energy quantum of a vibrationof the excited state. Even if the Franck-Condon modelwhich we are using prescribes E = E , this is not nec-essarily true in general. The factor | c ( e ) p | / (2 N M ) is theHopfield coefficient for the exciton of the molecule j rel-ative to the polariton k . Because c ( e ) p = c ( e ) k = P j c ( j ) k ∀ k , we are assuming that the exciton is equally dis-tributed among all the molecules. This is consistent withthe assumptions used in the derivation of the masterequation (see appendix A).The comparison of Eq. (4) with Eq. (B5) for the ra-diative case shows that the two processes have a similarefficiency. Indeed, using Eq. (B3): W j → k RAD W j → k NON − RAD = πV E · | c ( p ) p | | c ( e ) p | ; (5)because both V and E are of the same order of mag-nitude, 100 meV, the efficiency ratio mainly depends onthe Hopfield coefficients of the bottom polaritons. Thus,as in our case | c ( p ) p | / | c ( e ) p | ≈
5, we expect the radia-tive mechanism to be the main origin of the excitationtransfer which results in lasing, even if to understand theimportance of the non-radiative transfer a more detailedanalysis is necessary.In conclusion, we have studied two physical mecha-nisms which can possibly induce the excitation transferstudied in Sec. II. Using simple models, we have obtainedestimates which are in good agreement with those fromthe data. The photonic and excitonic components of thebottom polaritons are crucial for determining the impor-tance of the two mechanisms. We thus expect that inmaterials requiring different cavity detunings to matchthe condition E − E = E LP ( k = 0), the relevance ofthe two processes could be reversed. An experimentalanalysis exploring several organic crystals would thus beof the greatest interest. IV. TEMPERATURE
Reported data for anthracene microcavities show a re-duction of the lasing threshold of slightly less than an or-der of magnitude once temperature is lowered from 300 Kto 12 K. In this section we discuss temperature effectswithin the framework of the developed model, and therelated consequences on the lasing properties.Experimental studies on the PL from bulk anthracenecrystals have shown a strong temperature dependencecharacterized by considerable spectral narrowing. Thetemperature dependence obtained using thin crystalsgrown from solution is shown in Fig. 7. Here, the crystalswere grown on silicon substrates to ensure good thermalcontact to the cryostat cold finger and were excited us-ing 1 ns-long pulses at λ =337 nm. A composite vibronicstructure emerges, which can be understood in terms ofa high-energy phonon (considered in this work) and of alow-energy phonon, which is not resolved at room tem-perature because of thermal broadening. Such a systemrequires the use of two-phonon states in order to exactlyreproduce the spectra; however, we ignore this compli-cation because we are only interested in the phenomeno-
150 K100 K50 K
Energy (eV) P L I n t en s i t y ( a r b . un i t s )
13 K
FIG. 7. Photoluminescence spectra of anthracene layers fortemperatures between 13 K and 150 K. logical properties of the line which is responsible for las-ing.The scattering rate W e → p in Eq. (3) depends on tem-perature via f ( E ). On the one hand, at low tempera-ture the Lorentzian is narrower, and thus a smaller frac-tion of the oscillator strength is dispersed into non-lasingmodes. On the other hand, only a fraction of the oscilla-tor strength of the (0-1) transition contributes to lasing,because the other lines are far detuned. Additionally,both the quantum yield, estimated at room temperatureto be 0 .
5, and the exciton lifetime τ e are expected toincrease at low temperature.In Fig. 8 we compute the dependence of the integralappearing in (3): I + Z E A E A f ( E A inf − E ) d E (6)on the width of the Lorentzian function f ( E ) which rep-resents the normalized spectrum of the (0-1) PL emission.Whereas at room temperature the FWHM is ≈ . ∼ . − .
02 eV, and thus W e → p increasesof at least a factor of 5.Roughly speaking, the observed thermal reduction ofthe threshold is of less than one order of magnitude, and thus similar to the numbers of our estimates. Thispoints out a possible connection between the tempera-ture dependence of the laser threshold and of the PL ofanthracene crystals. A more systematic analysis, boththeoretical and experimental, goes beyond the scope ofthis work, and will be the focus of future investigations.As long as the thermal linewidth narrowing is considered,we observe that when the radiative transition is not per-fectly resonant with the lasing polaritons it could evenresult in the opposite effect.For the sake of completeness, in appendix C we includethe thermal population of the vibrations of the molecu- H eV L I FIG. 8. Dependence of the integral (6) appearing in (3) onthe width of f ( E ), which is a Lorentzian function. lar ground state in the master equation (1a) and demon-strate that it can be safely neglected. V. CONCLUSIONS
In this work, a minimal model to describe the polari-ton lasing observed in crystalline anthracene microcav-ities has been developed. Only the essential featuresof the physical processes involved have been included:the incoherently pumped exciton reservoir, the vibroni-cally assisted radiative scattering from the reservoir tothe bottom of the lower polariton branch, the onset ofbosonic stimulation and the build-up of the polaritonpopulation with increasing pump intensity, the polari-ton losses through the mirrors and bimolecular quench-ing processes. All the relevant material parameters, ex-cept from the bimolecular quenching rate, have been de-termined independently from the experimental data onthe pump dependence of the polariton emission . Inparticular, the efficiency of the scattering mechanismhere considered - which takes into account the promi-nent role of vibronic replicas in the photophysics of an-thracene microcavities - has been calculated microscop-ically. The numerical simulations obtained are in goodagreement with the data and describe well the onset ofthe non-linear threshold for polariton lasing. A possi-ble reason for the observed temperature dependence ofthe threshold has also been discussed. The presentmodel could be extended to include further ingredients,in particular polariton-polariton scattering, and be ap-plied to other microcavity systems exhibiting pronouncedvibronic replicas. ACKNOWLEDGMENTS
We thank V. M. Agranovich, D. Basko, S. Forrest,L. Silvestri, M. Slootsky for fruitful discussions and R.Fazio for encouragement. Financial support from theEuropean FP7 ICARUS program (grant agreement N.237900) is gratefully acknowledged. LM is funded by Regione Toscana POR FSE 2007-2013. SKC and PM re-spectively acknowledge the Imperial College JRF schemeand the Deutsche Forschungsgemeinschaft for financialsupport.
Appendix A: Derivation of the Master Equations
We present the derivation of the master-equation (1).We focus on the exciton reservoir (excitons are labeled by j ) and on the polaritons in the A region, which are res-onantly populated by the reservoir (labeled by k ). Thedynamics of the system is described by N exc + N pol cou-pled differential equations:˙ n j = − Γ j n j − X k ∈A W j → k n j (1 + n k ) − X k / ∈A W j → k n j + − γ X j ′ n j ′ + X k ′ | c ( e ) k ′ | n k ′ n j + (1 − n j ) P ( t );(A1a)˙ n k = − Γ k n k + X j W j → k n j (1 + n k ) − γ X j ′ n j ′ + X k ′ | c ( e ) k ′ | n k ′ | c ( e ) k | n k . (A1b)The term P k / ∈A W j → k n j describes excitons scattered toother polariton states via other decay mechanism, as forexample, lattice phonons and luminescence. We don’t in-clude a similar term P k ′ / ∈A W k ′ → k n k ′ (1 + n k ) in equa-tion (A1b) because negligible compared to the efficientdirect scattering from the reservoir. The probability ofannihilating an exciton (or polariton) because of bimolec-ular quenching is proportional to the total number of ex-citons P j ′ n j ′ + P k ′ ∈A | c ( e ) k ′ | n k ′ (we neglect the minorcontribution of polaritons k ′ / ∈ A ).In order to derive the master equation for the sur-face density of excitations ν e ( t ) = P j n j ( t ) /A and ν p ( t ) = P k n k ( t ) /A we have to make the following as-sumptions. We take W j → k to be independent from j and k ∈ A , renamed W ; the same holds for Γ j , substi-tuted by Γ e , for Γ k , renamed Γ p , and for c ( e ) k ′ , renamed c ( e ) p . We introduce the quantities W e → p = P k ∈A W and Z e → = P k / ∈A W j → k . Finally, n j and n k are notexpected to have a significant dependence on j and k .Clearly, this approach is more justified the more the A region is small. We sum the equations (A1a) and (A1b): X j ˙ n j = − X j Γ j n j − X j X k ∈A W j → k n j (1 + n k )+ − X j X k / ∈A W j → k n j + − γ X j ′ n j ′ + X k ′ | c ( e ) k ′ | n k ′ X j n j ++ X j (1 − n j ) P ( t ) (A2a) X k ˙ n k = − X k Γ k n k + X k X j W j → k n j (1 + n k )+ − γ X j ′ n j ′ + X k ′ | c ( e ) k ′ | n k ′ X k | c ( e ) k ′ | n k (A2b)Using the listed assumptions, we obtain:˙ ν e = − Γ e ν e − W e → p ν e (cid:18) ν p ¯ ν p (cid:19) − Z e → ν e + − γ ′ (cid:16) ν e + | c ( e ) p | ν p (cid:17) ν e + (cid:18) − ν e ¯ ν e (cid:19) P ′ ( t ) (A3a)˙ ν p = − Γ p ν p + W e → p ν e (cid:18) ν p ¯ ν p (cid:19) + − γ ′ (cid:16) ν e + | c ( e ) p | ν p (cid:17) | c ( e ) p | ν p (A3b)which is written in terms of the surface density of exci-tonic states ¯ ν e = N exc /A = L z ρ and of polaritonic states¯ ν p = N pol /A , of the pump rate density P ′ ( t ) = ¯ ν e P ( t )and of the quenching parameter γ ′ = γA . Appendix B: Scattering Rate Due to RadiativeTransition
We compute the scattering rate of a molecular excitonto a lasing polariton state via radiative emission assistedby the emission of a vibration (see Sec. III).Linear optical properties of strongly-coupled microcav-ities can be quantitatively described with a simple modelfor the light-matter interaction which conserves the in-plane momentum: H k = ~ ω k V V V V ∗ E V ∗ E V ∗ E . (B1)The energy of the cavity photon, ~ ω k =( c/n eff ) p | k | + π /L , and the energy of the ex-citon accompained by i vibronic replicas, E i , aremeasurable quantities. The couplings V i can be fit fromthe measured polaritonic dispersion relations, whichare the eigenvalues of (B1). We focus on the b exciton and on light polarized along b ; the microscopic expression of their coupling is: V m1 ( k ) = µe − S/ n eff r π ~ ω k La r N + 1) π s − | k | π L + | k | (B2)where S is the Huang-Rhys parameter, µ is the dipolemoments of the b Davidov branch, n eff is the effective re-fractive index, ~ ω k is the photon energy, L is the effectivelength of the cavity, a is the spacing between molecules, N is the number of monolayer comprising the organicmaterial. Neglecting the dependence on k , we identifythe fit parameter V of equation (B1) with the followingmicroscopic expression: V = V m1 ( k = 0) = 4 µe − S/ ( πc ~ ) / L a n / ( N + 1) / . (B3)As a simple consistency check of (B3), we take n eff =1 .
74 and L = 120 nm and µ ∼ V ∼ Thus, even if the theoretical estimateis based on the assumption of a perfect cavity withoutlosses, whereas the fit value refers to a realistic imperfectsystem, the error is under control.Let’s focus on the light-matter interaction responsibleof the exciton radiative recombination assisted by theemission of one molecular vibration:ˆ V n = − µ (cid:16) −√ Se − S (cid:17) ˆ v † n ˆ B n ·· X k s π ~ ω k La M n ω k = ω k e − i k · n || ˆ a † k p ! + H.c. (B4)where M is the number of unit cells included in the two-dimensional quantization area, v † n is the operator creat-ing a vibronic replica at the ground state of the moleculeplaced at n , B n is the operator destroying an electronicexcitation, ˆ a † k p is the photon field operator with p polar-ization.We are interested in the scattering of the molecular ex-citon at n into the lasing polariton region A . Using theFermi Golden Rule, the scattering rate from one molecu-lar exciton (labelled by j ) and a lasing polariton (labelledby k ) is (see also appendix A): W j → k = 2 π ~ πS µ e − S πc ~ L a n M | c ( p ) p | δ ( E − E LP ( k ) − E )(B5)We are assuming that the scattering process only dependson the energy of the final state accordingly with the pic-ture of bottom polaritons as states with a non-definedwavevector and with similar optical properties. We can make the previous equation more realistic bysubstituting the delta function δ ( E ) with the normalizedlineshape of the (0-1) photoluminescence, dubbed here f ( E ). In this work, we consider a Lorentzian linewidth Absorbed Pum p Fluence H nJ L I n t e g r a t e d I n t e n s it y H a r b . un it s L FIG. 9. Time-integrated surface density of polaritons R ν p ( τ ) dτ calculated from solution of Eq. (C5) (lines) andfrom experimental data (squares). The calculation parame-ters are: (dashed line) τ p = 85 fs; (solid line) 1 ps. The fitparameters are as in Fig. 3. f ( E ) = Γ / (2 π ( E − (Γ / )). Comparing this last ex-pression to (B3) we get ( N + 1 ≈ N ): W j → k ≈ V ~ π S | c ( p ) p | M N f ( E − E LP ( k ) − E ) (B6)This expression links the scattering rate assisted by theemission of one molecular vibration to known parameters. Appendix C: Thermal Population of VibronicReplicas
Up to now the scattering of one polariton to the ex-citon reservoir assisted by the absorption of a replica ofthe ground state has been neglected. However, at roomtemperature, a fraction of the molecules quantified bythe Bose-Einstein distribution is in a vibrationally ex-cited state; taking E ≈
173 meV and room tempera-ture ( k B T ≈ . ν e × e E /k B T − ≈ × cm − × − ≈ × cm − . (C1)Even if 10 − is a small fraction in absolute terms, thedensity of phonon-excited molecules is comparable tothe density of excitons of the previous simulations (seee.g. Fig. 5). Thus, polariton depletion because of back-scattering into the exciton reservoir can affect the gainof the lasing process.In order to study the effect of this process, we includein the right-hand side of equation (A1b) the term: − X j W j → k m j n k , (C2)where the sum is over all the molecules and m j is the pop-ulation of the phonon state of the j − th molecule. We do not consider m j as a dynamical variable but ratherconsider the thermal equilibrium population: m j + ( e E /k B T − − . Consequently, Eq. (A3b) includes theterm: − A X k ∈A X j W j → k m j n k = − W e → p ¯ ν e e E /k B T − ν p ( t )¯ ν p . (C3)The depletion rate is estimated as: − W e → p ¯ ν e e E /k B T − ν p ( t )¯ ν p ≈ − × × − s − × ν p ( t ) . (C4)We compare it to the polariton decay rate, Γ p > s − , and conclude that it is not the dominant polaritondepletion mechanism. This would be the case for micro-cavities with larger Q factors, which thus would bene-fit from lower temperatures freezing the main polaritondecay channel. The contribution of this process on thereservoir population is also negligible, because polaritonicstates N pol are a negligible fraction of the excitonic states N exc .We now take into account the population of phonon-excited molecules ν v ( t ) + P j m j ( t ) /A dynamically. Weconsider the following master equation (the derivation isa generalization of the previous discussion):˙ ν e = − (Γ e + Z e → ) ν e − W e → p ν e (cid:18) ν p ¯ ν p (cid:19) + W e → p ν p ¯ ν p ν v + − γ ′ (cid:16) ν e + | c ( e ) p | ν p (cid:17) ν e + (cid:18) − ν e ¯ ν e (cid:19) P ′ ( t ) (C5a)˙ ν p = − Γ p ν p + W e → p ν e (cid:18) ν p ¯ ν p (cid:19) − W e → p ν p ¯ ν p ν v + − γ ′ (cid:16) ν e + | c ( e ) p | ν p (cid:17) | c ( e ) p | ν p (C5b)˙ ν v = − Γ v (cid:18) ν v − ¯ ν e e E /kT − (cid:19) − W e → p ν p ¯ ν p ν v ++ W e → p ν e (cid:18) ν p ¯ ν p (cid:19) (C5c)Γ v models the relaxation to the vibrational ground state,and the presence of an equilibrium population is takeninto account; we set Γ v = 10 ps. The other two termsof equation (C5c) are due to polariton back-scattering tothe exciton reservoir and to the exciton radiative recom-bination respectively.In Fig. 9 we show the results, which are obtained withthe same parameters used in the main text. No qualita-tive difference with Fig. 3 is observable and this refine-ment can not take fix the above-threshold discrepancy.For τ p = 1 ps (solid line), the situation in which theback-scattering efficiency is most comparable to the po-lariton PL rate, a slight shift of the threshold towardshigher pump fluences is observable.Direct inspection of the time dependence of ν v ( t ) showsthat at threshold the system is driven out of equilibriumon the time-scale of 5 ∼
50 ps (Fig. 10). However, even0 Time H ns L Ν e H t L Ν e , Ν p H t L Ν p , Ν v H t L Ν e Time H ns L Ν e H t L Ν e , Ν p H t L Ν p , Ν v H t L Ν e FIG. 10. Time-dependence of the relative surface densityof exciton ν e ( t ) / ¯ ν e (dash-dotted lines), of lasing polaritons ν p ( t ) / ¯ ν p (dotted lines) and of vibrationally excited molecules ν v ( t ) / ¯ ν e (dashed lines) at threshold ( E tot = 300 nJ, τ p = 85fs). Top: ν v ( t ) is a dynamical quantity; bottom: ν v ( t ) =¯ ν e / ( e βE −
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