We examine the possibility of observing Bose condensation of a confined two-dimensional polariton gas in an organic quantum well. We deduce a suitable parameterization of a model polynomial Hamiltonian based upon the cavity geometry, the biexciton binding energy, and similar spectroscopic and structural data. By converting the sum-over-states to a semiclassical integration over D-dimensional phase space, we arrive at a principle of correspondence between ideal and non-ideal Bose gases that share a common critical exponent. Using our results, we can calculate the properties for a model cavity containing an anthracene thin film.
I. INTRODUCTION
Over the past few years, considerable progress has been achieved in creating Bose Einstein condensates (BEC) of weakly interacting particles. The most recent breakthrough is where BEC of photons was realized by pumping an optical cavity filled with a dye and bounded by two reflective mirrors.1 Within the optical cavity, the photons acquire an effective mass as determined by the cut-off frequency of the cavity that can be 6–7 orders of magnitude less than mass of an electron. Depending upon the density, this allows for a BEC transition temperature that can approach room temperature. Polaritons are also ultra-light quasiparticles that are known to condense in systems composed of a semiconducting quantum well sandwiched between two reflective mirrors.2–6 In this case, however, the polaritons act as hard-core Bosons and scattering at high density allows for a rapid thermalization of the gas.
In the experiments by Snoke and co-workers,6,7 translational symmetry is broken by putting a local strain on the cavity. It has also been argued that the finite size of the pumping laser is sufficient to break the 2D translational symmetry. Likewise, in the photon condensation reported by Klaers et al., condensation occurred within a finite 2D cavity with parabolic mirrors.1 For the case of a molecular crystal, translational symmetry can be broken by the presence of microcrystalline domains within the sample itself. Within a given domain, the electronic transition moments are aligned along the crystallographic axes. The grain boundaries impose a hard-wall trapping potential. This should allow for a much higher density of polaritons since excitons created within a given domain will be confined to that domain rather than being free to diffuse throughout the sample. However, as we show next, grain boundaries alone are insufficient to break 2D symmetry and one needs to consider the effect of the interactions.
In this paper, we consider the formation of a polariton BEC in an organic quantum well consisting of a layer of a semiconducting polyacene molecular crystal sandwiched between two reflective mirrors. In contrast to inorganic quantum wells, organic quantum wells can be fabricated on the bench-top without the need for ultra-clean environments. Recent experiments by Forrest and co-workers indicate that exciton polaritons in organic quantum wells can exhibit a lasing transition at room temperature,8,9 and our recent work indicated that parametric amplification and the transition to superfluid states should be possible in these systems even in the presence of some disorder within the crystal lattice itself.10,11 Furthermore, as we will discuss, the fact that molecular crystals may be glassy or have finite-sized microcrystalline domains breaks translation symmetry and may enhance the local density of polaritons thereby resulting in a higher BEC transition temperature. We consider the nature of the polariton/polariton scattering interaction by relating the biexciton binding energy to the S-wave scattering length for excitons in the cavity. Finally, taking molecular-level data as input to our theory, we estimate the transition temperature and critical density needed to achieve polariton BEC in a quasi-two-dimensional molecular crystal.
II. POLARITONS IN MICROCAVITIES
Polaritons are quasiparticles formed by strong coupling between a photon field and excitations within a material. We briefly recapitulate the quantum mechanical model for the formation of polaritons in a microcavity. Within the Frenkel exciton model, we can write a model Hamiltonian for this assuming that excitations are local to molecular sites, that excitons can hop to other sites via dipole-dipole coupling of their transition moments, and that the exciton/photon interaction is mediated via single quanta exchanges. Since polaritons are very lightweight quasiparticles, it is best to work in the long-wave, continuum limit rather than within a molecular-site representation more suitable for Frenkel excitons. With this in mind, we write the exciton dispersion as
where
Within the cavity, the photon dispersion is given by
where L is the spacing between the cavity mirrors and
where Δ = 2πℏc/Lη is the cut-off energy for the cavity and meff = 2πη/cL is the effective photon mass. Taking L = 230 nm, η = 2.99 (Ref. 15) one obtains Δ = 3.11 eV and a polariton effective mass of meff/me = 3.60 × 10−5. These conditions put the cavity cutoff in resonance with the exciton transition energy, thereby producing the strongest coupling between the photon field and the excitons. Figure 2 shows the dispersion curves for a L = 230 nm anthracene cavity based upon our model. We also created an on-line “calculator” for computing the spectroscopic properties of a polariton cavity based upon user-input.16
Polariton dispersion for a model anthracene slab in a resonant cavity (L = 230 nm). Solid curves: Upper and lower polariton dispersions. Dashed curves: Bare cavity and exciton dispersions.
Polariton dispersion for a model anthracene slab in a resonant cavity (L = 230 nm). Solid curves: Upper and lower polariton dispersions. Dashed curves: Bare cavity and exciton dispersions.
In the limit of low density, the combined exciton/cavity system is described by the energy functional,
where
describes the mixing between the exciton and photon fields. Note, that we have included a trapping potential, V(r), that may include any external effect that breaks translational symmetry. We also have assumed that the Rabi frequency, Ω does not depend upon the photon wave vector.
Polaritons are formed by transforming Ho into a diagonal representation, yielding upper and lower branches with dispersion,
where Ω is the Rabi frequency and
where μ is the chemical potential. In this expression, the ϕc is normalized to unity since we have included the number of excitations in system into the nonlinearity,
Excitations of the condensate are then determined by writing
For g sufficiently large, we can approximate the Bogoliubov spectrum as εk = s|p|, where s is the speed of sound s = (g/mL)1/2 and p = ℏk is the momentum. In this regime where the dispersion is linear in p, the condensate is behaving as a superfluid.
Polariton/polariton kinematic interactions were studied in detail by Zoubi and La Rocca19 using the Agranovich-Toshich transformation20 which converts the Frenkel excitons to Bosons, thereby allowing the exciton/exciton scattering interaction to be derived. In short, they show that in the long-wave limit and in the case where the lower polariton (LP) dispersion is nearly parabolic, the scattering T-matrix between polaritons is identical to that scattering from a square well with radius a. With this in mind, we estimate polariton/polariton coupling by assuming that this is due to exciton/exciton scattering. However, here we will construct an effective square-well potential for exciton/exciton scattering based upon the molecular geometry and available spectroscopic data.
The nonlinear coupling coefficient, gx = 4πℏα/mex is determined by the low-energy S-wave exciton/exciton scattering length, α. To provide a robust estimation of this term, based upon molecular and spectroscopic properties, we assume that stable biexciton states exist as a single bound state in an effective square well potential with binding energy εB and that the range is determined by the nearest neighbor spacing in the molecular lattice, b as sketched in Fig. 1. The excitons in our model are hard-core bosons in that we must restrict their population to at most a single exciton per local anthracene site. With this in mind, we include a hard-core contact radius, a, which we take to be the van der Waals radius of anthracene in the molecular crystal plane. From quantum scattering theory we arrive at a simple expression for exciton/exciton S-wave scattering length,
where
Anthracene lattice layer from molecular dynamics simulation. The repulsive core radius, a, and biexciton binding radius, b, are indicated by the concentric circles. The sketch to the right indicates the effective square-well potential for exciton/exciton scattering.
Anthracene lattice layer from molecular dynamics simulation. The repulsive core radius, a, and biexciton binding radius, b, are indicated by the concentric circles. The sketch to the right indicates the effective square-well potential for exciton/exciton scattering.
III. THERMODYNAMICS OF A NON-IDEAL, D-DIMENSIONAL BOSE GAS
For an ensemble of Bosons, the average number of particles in the gas is given by
where z = exp (μβ) is the fugacity and we have pulled out the term representing the population of the ground state so that 0 ⩽ z < 1. The sum in Nex is over all excitation of the system. An ideal gas of bosons will not form a condensate unless the dimensionality of the system, D > 2. The reason for this is that density of states scales as ω(ε)dε ∝ εD/2 − 1dε and consequently the sum over the excited states diverges and BEC is forbidden by translational symmetry in one and two dimensions for a non-interacting system.
Let us consider an interacting system in the mean-field limit by computing the excited state population and assuming the mean-field Hamiltonian takes polynomial form
where n ⩾ 1 and m ⩾ 1. When n = 2, we have the typical case where the kinetic energy is quadratic in the momentum. When n = 1, we have the linear dispersion of a quantum superfluid close to p = 0. This we will take as the approximate dispersion for an interacting Bose gas. Furthermore, for large values of m, the potential term is close to 0 for r < Ro and increases rapidly beyond this radius. As m → ∞, the potential becomes the square-well potential in D dimensions.
The resulting sum over states can also be evaluated within the semiclassical approximation. To do so, we expand the integrand in powers of z, perform the integration term by term, then re-sum the logarithmical series. This results in the following equations for the excited state population and the free energy:
where α = D(1/n + 1/m),
Similarly, the free energy can be obtained from
These results hold true in all dimensions and for all polynomial trapping potentials and dispersions. Importantly, it gives a general criteria for whether or not a system can form a Bose condensate. Since ζ(α) diverges as α → 1, a system can form a Bose condensate so long as the critical exponent α > 1. This suggests a principle of correspondence whereby systems with the same critical exponent α will have the same reduced equation of state. A low-dimensional interacting gas in the mean-field limit and at low temperature will have the same equation of state as a non-interacting or ideal Bose gas in some higher dimension or confined by some trapping potential.
Expressions similar to ours are sprinkled throughout the literature for power-law forms of the trapping potential. The approach here is similar to that taken by Salasnich for ideal Bose gases.21 which was a generalization of the D = 3 results obtained by Bagnato and Kleppner.22 Also, the equation of state for an interacting gas in a harmonic trap was derived by Ramero-Rochín for a cold atomic gas within the Hartree-Fock approximation.23 However, so far as we know, none have considered the effect that the form of the dispersion would play on the ability of a given system to form a stable condensate for the interacting case.
To obtain the transition temperature for the 2D interacting gas, let us rewrite the population equation in terms of the density,
where we have defined Λ = hs/kBT. Rearranging, we can write the average population of the ground state as
for temperatures such that Λ2ρ > π3/3. It is then useful to define a transition temperature such that
for T < To, and
Note that Eq. (14) derived for our interacting system is identical to a non-interacting gas in a 1D harmonic trapping potential, the difference being in the parameterization of the critical temperature.
In order to estimate the transition temperature for a polariton gas in an organic microcavity, let us assume that the number of polaritons is high enough that we can ignore the kinetic energy term in the GP equation and that V(r) is a circular well potential of radius Ro corresponding to the typical size of an crystalline microdomain. Within the well, we can write
Scaled polariton BEC transition temperature versus cavity detuning, (
Scaled polariton BEC transition temperature versus cavity detuning, (
It is interesting to note that as the biexciton binding increases, the transition temperature generally decreases. This is because the attractive part of the exciton/exciton interaction decreases the effective exciton scattering length. Moreover, for strongly bound biexcitons, the scattering length can become negative resulting in an transition from a BEC state to a bright soliton state. In this case, To becomes imaginary and the transition temperature cannot be defined. For the model cavity at hand, this occurs when ɛB > 25.8 meV, which is almost twice the biexciton binding energy reported for anthracene films.24
An alternative and perhaps more pedagogic approach is to consider the criteria for coexistence between an ideal Bose gas in 2D (with n = 2, s = 1/2m*, and α = 1) and a 2D superfluid (with n = 1,
Since all the quantities to the right of the parenthesis does not vanish, we require the terms within the parenthesis to vanish. Thus, we can relate the critical temperature to the mean-field interaction strength appearing in the Gross-Pitaevskii equation via
For 0 ⩽ z < 1, Li2(z)/Li3(z) > 1. So at sufficiently low temperatures such that kBT < g, the free energy of the 2D superfluid will be less than or equal to the free energy of the ideal 2D Bose gas. As the fugacity, z = eβμ → 1, Li2(z)/Li3(z) → π2/6ζ(3). This gives a straightforward estimate of
Recall that g depends upon the mixing between the photon and exciton degrees of freedom as well as upon the photon number density. Consequently, the actual transition temperature will be very much dependent upon quality of the cavity and how strongly the cavity modes are coupled to the excitonic transitions of the material within the cavity.
IV. SUMMARY
In conclusion, we predict that polariton BEC should be readily observable in an organic microcavity system using polyacene thin films. The arguments we present are based entirelyon either molecular or spectroscopic parameters. Our analysis is based upon the notion that the nonlinear interaction in the GP equation can be deduced from the biexciton scattering which we treat in terms of a repulsive inner core surrounded an attractive square-well interaction. For strongly bound biexcitons, the exciton/exciton scattering length can become negative and the resulting polariton ground state will be a bright-soliton rather than a Bose Einstein condensate. The results presented here are not limited to organic molecular crystals and apply equally to J-aggregate systems, such as those recently studied by the Bulović group at MIT.25
ACKNOWLEDGMENTS
The work at the University of Houston was funded in part by the National Science Foundation (NSF) (CHE-1011894) and the Robert A. Welch Foundation (E-1334). C.S. acknowledges support from the Canada Research Chair in Organic Semiconductor Materials.