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Ultrastrongly Coupled Exciton–Polaritons in Metal-Clad Organic Semiconductor Microcavities

Authors


Abstract

Ultrastrong exciton–photon coupling of Frenkel molecular excitons is demonstrated at room temperature in a metal-clad microcavity containing a thin film of 2,7-bis[9,9-di(4-methylphenyl)-fluoren-2-yl]-9,9-di(4-methylphenyl)fluorene. A giant Rabi splitting of Ω ∼ 1 eV is measured using angle-resolved reflectivity and bright photoluminescence is observed from the lower polariton branch. To obtain the virtual photon and exciton content of the polariton ground state, the results are interpreted in terms of the full Hopfield Hamiltonian, including anti-resonant terms. Also included is an analytical treatment of the often ignored and sometimes misinterpreted TM-polarized metal–insulator–metal plasmon–polariton.

1 Introduction

The large exciton binding energies and oscillator strengths characteristic of organic semiconductors make this class of materials uniquely suited for the study of the strong exciton–photon coupling regime at room temperature.[1] In this regime, where the exciton-photon interaction exceeds the photon and exciton damping, new coherent light-matter excitations are formed called microcavity polaritons. In inorganic semiconductors, polaritons have been the source of a wealth of fascinating phenomena such as parametric amplification, Bose–Einstein condensation and superfluidity.[2] These stem principally from the exciton character of the resulting excitation.[3] In organic semiconductors, macroscopic occupations of the lower polariton branch minimum have been demonstrated,[4] but nonlinear phenomena resulting directly from polariton–polariton interactions remain to be shown.[5] To allow the buildup of a polariton condensate, all of these demonstrations have sandwiched the active material between two dielectric mirrors, each providing almost 100% reflectivity and thus reducing the decay of polaritons via their photon component.

The use of a metal mirror, however, can be beneficial, principally for three reasons: it reduces the mode volume below that achievable with Bragg mirrors which results in a larger Rabi splitting, it provides the stop band width then required to accommodate both the upper and lower polariton branches and it provides a simple means for electrical injection as demonstrated in the pioneering work by Tischler et al.[6] and further investigated in other structures.[7] The enhancement in Rabi splitting was first explicitly demonstrated by Hobson et al., who observed a Rabi splitting of math formula ∼ 320 meV in an all-metal cavity.[8] Since then, Rabi splittings of math formula∼ 430 meV, math formula∼ 450 meV and math formula∼ 650 meV were observed in reflectivity from polysilane containing metal/dielectric cavities,[9] from polycrystalline tetracene[10] and photoisomerized merocyanine[11] all-metal cavities, respectively. Such values are comparable (>20%) to the bare exciton energy Eex. This regime where math formulaEex has been termed the ultrastrong coupling regime and possesses a number of interesting properties.[12] For a correct treatment of the resulting polaritons, anti-resonant terms and the contribution arising from A2, the squared magnetic vector potential, must be included within the usual Jaynes-Cummings-type Hamiltonian. In the context of bulk exciton–polaritons, the full light–matter Hamiltonian was considered in the seminal work of Hopfield and Agranovich.[13, 14] There, it was shown that polaritons are the lowest lying excitations of the coupled light–matter system. One can also show that if the full Hamiltonian is considered, the ground state of the coupled system is modified by the light–matter interaction. This ground state was later explicitly constructed by Quattropani et al.[15] One of its most interesting aspects was recently highlighted by Ciuti et al. in the context of cavity exciton–polaritons: if the coupling can be modified non-adiabatically, the virtual photon content of the ultrastrongly coupled ground state can, in principle, be released.[12, 16] That work also highlighted the advantages of using intersubband polaritons for reaching the ultrastrong coupling regime.[17] In this paper, we demonstrate ultrastrong coupling using a thin film of the organic molecule 2,7-bis[9,9-di(4-methylphenyl)-fluoren-2-yl]-9,9-di(4-methylphenyl)fluorene (TDAF)[18] sandwiched between two Al mirrors. We observe a Rabi splitting of ∼1 eV and bright photoluminescence (PL) from the lower polariton branch. In addition to its high photostability and quantum yield, as compared to J-aggregated cyanine dye molecules, the use of TDAF allows for the possibility of direct electrical injection.[19] In contrast to previous reports, ultrastrong coupling is treated using the full Hopfield Hamiltonian allowing for estimates of the virtual photon and exciton content of the polariton ground state. The results are also shown for the TM polarized metal–insulator–metal (MIM) plasmon–polariton, which possesses a significantly modified dispersion relation due to its plasmonic character.[20] A simple model is used to describe its dispersion relation below the light line in a form analogous to that of the commonly studied TE mode.[8, 10, 11]

2 Experimental Results

Three microcavities consisting of 67, 73, and 78 nm thick layers of TDAF sandwiched between 70 nm Al bottom mirrors and 30 nm Al top mirrors were fabricated. All of the layers were thermally evaporated at a base pressure of ∼10−7 mbar on cleaned fused silica substrates and a 1 nm-thick layer of LiF was inserted prior to top mirror deposition o reduce interdiffusion of Al atoms into the TDAF.[21] The schematic microcavity and TDAF molecular structures are shown in Figure 1(a).

Figure 1.

(a) Schematic representation of the microcavity structure showing the molecular structure of TDAF and the definition of θ, the angle of incidence. (b) Absorbance (right) and PL (left) of a 60 nm thick TDAF film. The absorbance shows a strong, inhomogeneously broadened, exciton resonance around 3.5 eV and the PL exhibits three well defined vibronic peaks.

The absorbance and PL of a bare 50 nm-thick TDAF film are shown in Figure 1(b). The absorbance exhibits a strong, inhomogeneously broadened exciton resonance around 3.5 eV, while the Stokes-shifted PL shows resolved vibronic structure and has a measured quantum yield of 0.43. To represent the experimentally measured dispersion, contour plots of the angle-resolved reflectivity are shown in Figure 2 for the 67 nm and 78 nm-thick microcavities. The bare exciton and photon resonances are shown as white dashed lines; the horizontal lines in these pairs are the exciton resonances. The lower (LP) and upper (UP) polariton branches exhibit clear anti-crossing behavior around the bare exciton resonance located at 3.5 eV. Note that slight Fano-like asymmetry is observed in bare Al resonators due to the peculiarities of its dielectric constant and that this asymmetry is also apparent in the shape of the LP resonance for the thicker cavities. We find that for both samples, the UP and LP dispersions are flat, characteristic of the ultrastrong coupling regime[11] and of potential benefit for display applications where angle-dependent color shifts are undesirable.[9] As can be seen from the dashed white lines, however, the flatness of the TM-polarized polariton is also a result of the high group index of the uncoupled metal–insulator–metal plasmonic cavity mode. The cavity quality factor extracted from the low energy linewidth of the 78 nm-thick microcavity is Q∼ 30. In these microcavities, the exciton damping exceeds the photon loss. When resonance is approached, however, a slight reduction in the LP linewidth is observed, attributed to “motional narrowing” due to the inhomogeneously broadened linewidth, despite the increasing exciton character of the LP branch.[22]

Figure 2.

False color angle-resolved reflectivity maps of the 67 nm (a,b) and 78 nm (c,d) thickness microcavities. The spectra are shown for TE (a,c) and TM (b,d) polarization. The dashed white lines show the uncoupled photon and exciton (Eex = 3.534 eV) resonances. The dashed black lines trace the positions of the reflectivity minima that correspond to the lower and upper polariton branches. Note that the bare photon dispersion of the TM modes is considerably flatter than that of the TE modes and that the polariton dispersions exhibit only a modest degree of curvature.

To facilitate a comparison with theory, the angle-resolved reflectivity spectra are individually shown for the 67 nm-thick cavity in Figure 3(a) and (b). As expected, the spectra can be fully reproduced using linear dispersion theory[23] and the calculated spectra using a transfer matrix scheme which includes the measured refractive index of TDAF are shown in Figure 3(c) and (d). Using spectroscopic ellipsometry, we have identified weak uniaxial anisotropy with the optical axis along the growth direction in agreement with previous reports suggesting that the molecules lie principally within the plane of the film (see Supporting Information).[24] Although the TE-polarized reflectivity depends only on the ordinary refractive index, both the ordinary and extraordinary refractive indices were included for the calculation of the TM-polarized reflectivity.

Figure 3.

Angle-resolved reflectivity spectra for the 67 nm thick cavity measured using TM (a) and TE (b) polarized light shown in increments of 10°. For each increment, the absolute spectra have been shifted down (up) by 2% for TE (TM) polarized light. The calculated reflectivity using the measured anisotropic refractive index of TDAF is shown in (c) and (d) for TM and TE polarized light, respectively. As expected, linear dispersion theory correctly predicts both the position and depth of the polaritonic resonances.

To measure the PL, the microcavities were pumped non-resonantly, at θ = 50°, using a 2 ns long pulses from a λ = 355 nm Nd:YAG laser. The angle-resolved PL measured from the 67 and 78 nm thick cavities is shown in Figure 4 for both detection polarizations. Here, the pump power was kept low so as to lie within the linear PL vs pump intensity regime, thereby avoiding any complications associated with bimolecular annihilation. At this wavelength, corresponding to 3.5 eV photon energy, the exciton reservoir is directly populated. The observed PL corresponds to excitons scattered into the LP branch and their subsequent relaxation via the LP photon component. Despite the structured emission of the bare TDAF PL, the polariton PL always exhibits a clear maximum at the bottom of the LP branch suggesting some form of non-radiative relaxation.[25] The counts have been normalized by the absorbed pump fluence and it can be seen that the maximum intensity is comparable for both cavities despite the significantly different detunings between the bare exciton and photon components of the cavities. Because the pump energy is matched to that of the reservoir, no PL is observed from the upper polariton branch. Indeed, scattering into this branch would require the absorption of phonons with energy much greater than kbT, which are negligibly populated at room temperature.

Figure 4.

Contour plots of the angle-resolved PL for the 67 nm (a,b) and 78 nm (c,d) thick microcavities. In both cases, the TE (a,c) and TM (b,d) polarized emission was measured. Note that the ordinate axis has been rescaled, as compared to Figure 2, to emphasize the dispersion. The observed peak positions are in excellent agreement with those obtained from reflectivity. The counts have been normalized to the absorbed pump power and are shown, for both detunings, to be comparable. In all cases, the strongest signal is observed from the bottom of the LP branch, despite the structured nature of the bare TDAF PL.

The dispersion relations extracted from the minima in reflectivity for angles θ ≥ 15°, and from the maxima in PL for angles θ < 15°, are plotted together as squares in Figure 5 for both polariton polarizations. The solid lines correspond to a least-squares fit to the full Hopfield Hamiltonian.[14] A Rabi splitting, math formula∼ 1 eV, is evident, which to our knowledge, is the largest in absolute value observed to date.

Figure 5.

Measured dispersion relation (squares) for (a) TE and (b) TM polarization. The squares correspond to the reflectivity minima, except for θ < 15°, where the peaks are taken from PL maxima. The lines show a least-squares fit to the full Hopfield Hamiltonian with Ω0, neff and Ecav(0) as fit parameters. The resulting parameters are summarized in Table 1.

3 Theory

3.1 Ultrastrong Coupling

In quantum theory, the strong coupling Hamiltonian can be written in terms of the photon and exciton creation operators with in-plane wavevector q, where aq creates a cavity photon at frequency ωcav(q) and Bq creates an exciton at frequency ωex(q):

display math(1)

where the constant terms have been lumped into the ground state energy. This Hamiltonian is easily diagonalized by introducing polariton operators which are a linear combination of the exciton and photon operators. It has the Jaynes-Cummings form with the important exception that it contains no anharmonicity due to the presence of N bosonic exciton states, where N is the total number of molecules. In this case, the ground state remains the exciton and photon vacuum, but the resulting excitations are polaritons. The rotating wave approximation, valid if Ω < ωex,ωcav, is implicit in this form of the Hamiltonian. The full Hamiltonian, however, includes anti-resonant interaction terms as well as a contribution from A2 containing both types of terms:[12-15]

display math(2)
display math(3)

where math formula quantifies the contribution arising from the magnetic vector potential. The full Hamiltonian including these terms was diagonalized by Hopfield and Agranovich by introducing polariton annihilation operators math formula. The eigenvalue problem can be written in matrix form as:

display math(4)

The solutions correspond to the lower and upper polariton frequencies math formula. It was shown that the full treatment leads to an asymmetric anticrossing as a function of Ω and that the correct ground state is a squeezed vacuum corresponding to the vacuum of polaritons such that math formula.[12] In this case, the ground state has finite photon and exciton content such thatmath formula.[12]

Although the unique ground state properties of the ultrastrong coupling regime can only be elucidated using a quantum treatment, the necessity for the use of the full Hopfield matrix is also clear using a semiclassical approach. Indeed, the calculated reflectivity plots in Figure 3 show that linear dispersion theory provides an accurate description of the linear optical properties even in the ultrastrong coupling regime. To highlight the equivalence with the quantum theory, we ignore damping although it can be readily included and describe the exciton response using the expression for an oscillating polarization density with a restoring force:[14, 26]

display math(5)

where εb is the background dielectric constant and A is proportional to the oscillator strength. For perfect mirrors, we have from HelmholtzO's equation a bare cavity photon dispersion math formula with kz quantized such that kz = mπ/L, where L is the cavity length and m is a positive integer. For the filled cavity, HelmholtzO's equation gives:

display math(6)
display math(7)

which results in the dispersion equation:

display math(8)

This is precisely the dispersion obtained from the full Hopfield Hamiltonian if we make the correspondence math formula. Note that when math formula, the Rabi frequency is math formula and the polariton frequencies are given by:

display math(9)

which is equivalent to the result obtained in the quantum theory.[12]

3.2 TM Microcavity Photon

As can be inferred from Figure 2, the dispersion of the bare microcavity photon in metal-clad microcavities differs strongly for TM and TE polarized light. Indeed, because the group index of the TE mode closely resembles that of the enclosed material, most reports have only considered the TE mode. Lodden et al. has highlighted the “polarization splitting” between the TE and TM modes[27] and a recent paper has used numerical calculations to describe this splitting as being due to the metal-insulator-metal (MIM) plasmon supported by the structure.[28] This latter explanation is correct and the peculiar characteristics of this mode were described more than 40 years ago by Economou.[20] In particular, it should be noted that the MIM mode does not possess a cut-off as a function of thickness, with a dispersion relation continuing well beyond the light-line supporting waves with very high momenta. As the thickness is reduced, the coupling between the two single-interface surface plasmons increases and the dispersion relation of the antisymmetric (for the electric field component along the cavity length) mode is given by:

display math(10)

where εm is the dielectric constant of the metal and math formula where math formula. For a given thickness and plasma frequency, this mode can exhibit either positive or negative dispersion at small q. A minimum can even occur at finite q and Litinskaya et al. have recently proposed its use as a polariton trap.[29] The dispersion of the symmetric mode is located fully beyond the TDAF light line and not considered further. All of the previously demonstrated all-metal organic microcavities fall in the region of positive dispersion. We now derive a simpler expression for the effective index of the MIM mode below the light-line appropriate for the description of these microcavity polaritons.

Instead of expanding Equation (10), we consider the phase change upon reflection from a metal to draw a direct comparison with the TE mode. The metal dielectric constant is described using a Drude model and we assume that we are operating well below the plasma frequency ωp such that math formula.[30] The reflection coefficient for TM polarized light incident from the cavity material is:

display math(11)

After expanding, the real part is given by:

display math(12)

where math formula The imaginary part is given by:

display math(13)
display math(14)

If we are interested in angles smaller than those defined by the cavity material light line, i.e., math formula where math formula, as is the case for cavity polaritons, then Equation (14) can be further simplified:

display math(15)

We can thus write:

display math(16)

A similar expression is obtained for the TE mode:

display math(17)

Using these expressions, the cavity resonance frequencies at q = 0 can easily be found as solutions to math formula and are given by:

display math(18)

The interpretation is clear: the phase change upon refection corresponds to an effective “plasmonic” penetration depth math formula The cavity mode dispersion is:

display math(19)

where the positive sign is for TE polarization. After some simple algebra and using the same assumptions, this equation can be recast to describe the cavity photon energy, math formula, in the form commonly used in the description of cavity polaritons:[31]

display math(20)

where math formula and neff is the effective refractive index. For TE polarized light, neff = nb, while for TM polarization:

display math(21)

As can be seen, the effective index for the latter can be much greater than nb. In Figure 6, we compare the results obtained using Equations (20) for a cavity of length L = 100 nm, with nb = 2, to the exact dispersion given by Equation (10) using a Drude model description of Al (ωp = 1.843 × 1016 rad/s) and to that calculated using the measured real and imaginary parts of the Al dielectric constant. The figure also includes the result for TE polarized light. In this spectral range, the agreement of the model with both calculations is excellent, although the Drude model systematically predicts ∼20 meV higher energies. As expected, the discrepancy between our model and the exact Drude calculation vanishes when the cavity thickness increases such that the resonance moves sufficiently away from ωp.

Figure 6.

The analytical expression for the TM cavity photon (bottom) dispersion obtained using Equations (18) and (20) in the text (solid line) is compared to the exact numerically calculated dispersion relation resulting from a Drude model for Al (dashed line) and from the full refractive index of Al (dotted line) for a L = 100 nm cavity and a background index nb = 2. The analytical result obtained for the TE cavity photon (top), neff = nb, with Equation (18), is also compared to the exact result obtained using a Drude model and the full refractive index of Al.

4 Discussion

Based on the semiclassical results, the dispersion relations from Figure 5 are fit using Equation (4) with a Rabi frequency given by math formula where Ω0 is the Rabi frequency on resonance. This last factor, from Equation (8), is included due to the large energetic range swept by the bare microcavity photon over the range of angles measured. All of the resulting fit parameters are shown in Table 1. We should emphasize that to obtain meaningful fit parameters, the bare exciton energy is fixed to Eex = 3.534 eV, which is obtained from a Lorentzian fit to the measured absorption. In contrast, if the exciton energy is allowed to vary, perfect fits can be obtained in all cases, but with thickness-varying exciton energies and meaningless values of the effective index. In addition to the effects already described for the photon dispersion, the remaining discrepancies arise from the dispersive behavior of the background effective index due to higher-lying resonances, the non-Lorentzian absorption lineshape,[32] the weak anisotropy and the ignored damping. The assumption of a constant effective index, for example, leads to a better agreement of the fit to experiment for the TM polaritons due to the smaller energetic range swept by their photon component. A simple model to account for the asymmetric lineshape due to unresolved vibronic structure was proposed in Ref. [32]. Finally, uniaxial anisotropy, which was ignored in Section 'Theory', modifies the transverse component of the wavevector for TM waves such that math formula,[33] where ε0 and εe are the ordinary and extraordinary dielectric constants. For perfect mirrors, this results in a small change of the TM photon dispersion, but leaves the Rabi splitting unaffected.

Table 1. Hopfield Hamiltonian fit parametersa
ThicknessΩ [meV]neffEcav(0) [eV]
  1. a

    The uncoupled exciton energy is fixed at Eex = 3.534 eV.

67 nm (TE)990 ± 202.10 ± 0.103.45 ± 0.03
73 nm (TE)960 ± 202.10 ± 0.203.31 ± 0.04
78 nm (TE)950 ± 202.00 ± 0.103.22 ± 0.03
67 nm (TM)902 ± 0.0063.30 ± 0.203.46 ± 0.01
73 nm (TM)875 ± 0.0053.00 ± 0.103.289 ± 0.007
78 nm (TM)860 ± 0.0062.67 ± 0.083.153 ± 0.008

It can be seen from Equation (9) that the correction arising from using the full Hamiltonian leads to a blue shift on resonance of math formula. In our experiment, this corresponds to a fractional change in polariton energy of approximately 1%. Indeed, the main advantage of using the full treatment is that it allows the extraction of the anti-resonant coefficients of the polariton wavefunction.

For TE polarization, the effective index decreases slightly as the photon resonance moves to lower energy, consistent with the decreasing math formula, while for TM polarization the decrease principally results from Equation (21). Although our treatment ignores the 1 nm LiF layer and is only strictly accurate for math formula, a simple application of Equation (21) to the case of all three cavities using the value nB ∼ 1.8 obtained from ellipsometry gives neff = 3.1, 2.9, 2.8 for the 67, 73 and 78 nm thick cavities respectively, which is in good qualitative agreement with the values from Table 1. Rigorously, however, one should obtain neff = nB for the TE mode and identical values of Ecav(0) for both polarizations.

In addition to providing a more accurate description of the dispersion relation, the full Hopfield Hamiltonian allows for the extraction of the ground state exciton and photon content. In Figure 7(a) and (b) the photon and exciton content (along q) of the TE polarized LP branch, obtained by diagonalizing Equation (4), is shown. For increasing q, the LP branch becomes increasingly exciton-like, as expected. In contrast, Figure 7(c) and (d) shows the ground state photon and exciton content which both decrease when the mixing is reduced. It can be seen that the virtual photon content is approximately 0.5% per state for the full range of momenta (angles) shown. A rough estimate of the virtual TE photon density is given by:

display math(22)

where the integral is taken to the edge of the Brillouin zone (BZ). In practice, beyond the edge of the material light line math formula, the excitations are pure photons and excitons and the conventional vacuum is recovered. The release and detection of these photons via a non-adiabatic modulation of the coupling remains an open challenge. The estimate is slightly more complicated for TM polarization. In this case, the dispersion continues beyond the TDAF light line, but eventually wraps around because of the significant damping. The counting of polariton states is indeed a subtle issue in this system. Agranovich has emphasized that polariton states with an uncertainty in momentum Δq (due to the energetic linewidth) greater than their absolute momentum cannot be considered “coherent” quasiparticles with a well defined wavevector and must instead exhibit a significant degree of spatial localization.[26] As can be seen from Figure 2, the flatness of the polariton branches in the ultrastrong coupling regime increases the degree of localization by bringing more states within one Δq, despite the large Rabi splitting. The number of such states can readily be counted from the dispersion relation. Note that Equation (10) from Ref. [26], which predicts fewer localized states when the coupling increases is not applicable to this case due to the importance of including photon damping when metal mirrors are used. We believe that a better theoretical understanding of the ultrastrongly coupled ground state properties in the presence of such localization is thus of the greatest relevance.

Figure 7.

Photon (top) and exciton (bottom) content of the LP branch (a,b) and polariton vacuum (c,d), for TE polarization, obtained using the full Hopfield Hamiltonian. Note that the ordinate scale differs by two orders of magnitude. The content is shown here for θ < 85°, where the use of a single effective index provides a good approximation to the cavity photon dispersion relation. The TE polaritonic states, however, extend beyond the air light line to the edge of the TDAF light line. The photon and exciton the content of the ground state is ∼0.5% and depends only weakly on the wavevector. A rough estimate predicts ∼107 virtual photons/cm2 in the cavity ground state.

5 Conclusion

We have demonstrated ultrastrong coupling of TDAF excitons to the cavity photons in a metal-clad optical cavity. A giant Rabi splitting of ∼1 eV was observed in angle-resolved reflectivity and photoluminescence. This splitting falls within the telecommunications window, which raises the possibility of using organic microcavities to address this spectral range. The dispersion relation was, for the first time, analyzed in terms of the full Hopfield Hamiltonian and the ground state photon and exciton content were extracted. A rough estimate predicts ∼107 photons/cm2 for the TE mode.

This material system readily allows for electrical injection and thus could provide insight into the effect of ultrastrong coupling on the exciton generation via electron-hole Coulomb capture processes. Another attractive feature of ultrastrong coupling is that it provides a narrowed luminescence emission spectrum compared to the molecular exciton bandwidth but unlike conventional microcavity narrowing does so without introducing angle dependent colour shifts.[9] Moreover, the peculiar polariton dispersion of the TM mode, due to its plasmon-derived antisymmetric MIM character, was considered and a simple expression was derived to predict the effective index in terms of a Drude “plasmonic” penetration depth.

Acknowledgements

The authors would like to thank G. C. La Rocca for bringing Ref. [15] to their attention and V. M. Agranovich, D. Basko and L. Mazza for fruitful comments on this work. The authors also gratefully acknowledge funding from the UK Engineering and Physical Sciences Research Council via the Active Plasmonics programme grant (EP/H000917/1). SKC thanks Imperial College London for the award of a Junior Research Fellowship and DDCB thanks Mike and Ann Lee for endowing the Lee-Lucas Chair of Experimental Physics.

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