Anisotropic light-matter coupling and below-threshold excitation dynamics in an organic crystal microcavity.

Organic semiconductors are promising candidates as platforms for room temperature polaritonic devices. An issue for practical implementation of organic polariton devices is the lowering of condensation threshold. Here we investigate anisotropic light-matter coupling characteristics in an organic crystal microcavity showing strong molecular orientation. Furthermore, the below-threshold excitation dynamics are investigated to clarify the spontaneous transition pathways from reservoir to polariton states. Time-resolved photoluminescence measurements reveal that photonic/excitonic hybrid transition processes coexist in the microcavity system. This finding provides valuable insights into a detailed understanding of polariton dynamics and help in the design of polaritonic devices showing a low-threshold condensed phase.

Anisotropic light-matter coupling and belowthreshold excitation dynamics in an organic crystal microcavity: supplemental document

Raman scattering
To characterize the vibrational modes in BP1T-CN molecule, we performed Raman scattering measurement as shown by a solid curve in Fig. S1. The excitation light source was Kr + laser (646.4 nm). Strong peaks observed at ~180 meV (1450 cm -1 ) and ~200 meV (1600 cm -1 ) show the molecular vibronic modes A 1 and B 1 , respectively. We also show the PL spectrum of BP1T-CN crystal by a dashed curve and compare it with the Raman scattering result. When the exciton energy of 2.68 eV is defined as the origin of the Raman spectrum, the PL peak position exhibits a good agreement with the Raman shifts of A 1 and B 1 mode. This result shows that the main PL emission peak of BP1T-CN is attributed to the 0-1 vibronic transition.

Anisotropic dispersion
Anisotropic features of microcavities fabricated using BP1T-CN crystal can be investigated by polarization-dependent angle-resolved PL measurements. Figure 3 of the main manuscript shows the results for Microcavity-A having a crystal thickness of ~400 nm.. To study this anisotropic feature in more detail, we show polarization dependent angle-resolved PL results for Microcavities-B and -C (crystal thickness ~90 and ~440 nm, respectively).
As exhibited in Fig. S2(a), Microcavity-B shows one LPB (~2.36 eV at θ = 0 º) under the ∥ configuration. The small thickness of the BP1T-CN crystal causes the appearance of single LPB mode. On the other hand, under the ⊥ configuration [ Fig. S2(b)], the emission intensities from the LBP modes are very weak as compared to the ∥ configuration. This is because the intrinsic emission intensity of the BP1T-CN crystal is small in this polarization direction. In addition, another dispersion mode having a rapid angular dependence appears. The small refractive index at this polarization direction is the origin of this rapid angular dependence. Furthermore, the p-polarized mode becomes clearer at the crossing angle with LPB mode owing to the polarization mixing effect. The strong coupling is expected to occur only for the s-polarized mode as the sufficiently large | | is obtained.  Figure S3 shows the s-polarized PL results for Microcavity-C at the ∥ configuration. The p-polarized data is not shown here. In this sample, the polarization mixing effect is observed more visibly; the anti-crossing feature between the polariton and cavity photon modes are clearly observed. As the BP1T-CN crystal shows the low symmetric triclinic crystal structure, the dielectric tensor cannot be diagonalized, and thus the two orthogonal polarization modes are not independent of each other. It is expected that the polarization mixing effect does not strongly affect the reservoir dynamics, as the cavity photon and/or polariton lifetime is still much shorter than the reservoir dynamics. Therefore, we can find no clear causal relationship between the birefringence and the PL lifetime.

Complex refractive index of BP1T-CN crystal
As shown above, the BP1T-CN crystal shows the strong anisotropy and birefringence. The light-matter strong coupling is possible only for cavity photon modes having the same polarization direction with the transition dipole moment. To characterize the coupling parameter, it is required to know the complex refractive index of BP1T-CN crystal, ( ) + ( ). We employed the Kramers-Kronig (KK) analysis for this purpose. The imaginary part of refractive index, ( ), is calculated from the absorption coefficient spectrum ( ) at the measurement configuration of ∥ using relation ship of The real part of refractive index, ( ), can be obtained from the KK relationship of The calculated result is shown in Fig. S4(a). The background refractive index is estimated to be ~3.8. This result is used for transfer matrix simulation of active microcavities.
When we simulate passive cavities, it is necessary to subtract the contribution of an oscillator involved in the strong coupling from the complex refractive index spectra shown in Fig. S4(a). This is possible by extracting the absorption band at ~2.68 eV that shows the strong polarization dependence (see also Fig. 1 in the main manuscript). As exhibited in Fig. S4(b), the complex refractive index used for simulating the passive cavity is obtained as a result of this operation, whereas the subtracted index terms associated with the oscillator contributing to the strong coupling is shown in Fig. S4(c). Note that the and spectra in Fig. S4(a) are sums of the spectra in Figs. S4(b) and S4(c).

Transfer matrix simulation
In Figs. S5(a) -S5(c), we exhibit again contour maps of s-polarized angle-resolved PL spectra for Microcavities-A -C, respectively. To analyze these results, we first simulate the angledependent optical properties using a transfer matrix (TM) method. The microcavity structure   reflectivity spectra. To display the small negative peaks clearly, the reflectivity is described in a form of log(1 -) .

Coupled oscillator model analysis
We analyzed the experimental results using a coupled oscillator model that describes the coupling between a cavity photon mode and an exciton transition dipole moment. In the research community, a 4 × 4 phenomenological Hamiltonian, in which coupling between one exciton mode and multiple cavity photon modes is considered, has been often employed for the analysis. However, the use of sets of 2 × 2 matrices are likely appropriate for this study because this study focuses only on the interaction between the lower-energy exciton mode and cavity mode and the resultant LPB. Furthermore, as the energy separation between the cavity modes are as large as ~200 meV, which is larger than light-matter interaction energy we investigate, the contribution of another photon mode is not significant. Additionally, the 4 × 4 model likely includes some physical problems, e.g. (i) the density of states of the cavity mode is generally sufficiently smaller than that of the excitons, and thus the probability that one exciton will couple with multiple cavity photon modes should be very small, and (ii) as there is orthogonality between cavity modes of different orders, they are less likely to hybridize. For these reasons, we consider that the made of 2 × 2 matrices is more appropriate to explain our experimental results. The discussion on this point has been reported from another group (ref. 31). Furthermore, A very recent study reported that the model with a set of 2 × 2 matrices is preferable to the 4 × 4 model when the coupling parameter is large (ref. 32). A phenomenological 2 × 2 Hamiltonian of the coupled oscillator model is shown as follows.
Here is the coupling parameter between the photon and exciton. The exciton energy is treated as constant because of the large effective mass of the Frenkel exciton. ℎ is the energy of cavity photon mode, which depends on in-plane wavevector || and is shown as is the order of cavity mode. The effective refractive index corresponds to a weighted refractive index calculated for the DBR/BP1T-CN/DBR structure and is determined by the electric field distribution of the cavity photon mode. This equation can be transformed to an expression as a function of observation angle .
is zero at the normal incidence. The energies of polariton modes can be obtained by solving an eigenvalue problem for . As a result, the upper and lower polariton energies ± are written as The Rabi splitting energy ℏ at ℎ = is then represented as (S7) To perform fitting analyses using eq. (S6), we need to know ℎ( ) for each LPB mode. For this purpose, the results of TM simulation shown in Figs. S5(d) -S5(f) are available. As shown in left panels of Figs. S6(a) -S6(c), we perform the TM simulation for passive cavity, in which the BP1T-CN layer is regarded as a passive layer and has the complex refractive index exhibited in Fig. S4(b). The obtained dispersion curves are analyzed by using eq. (S5) to estimate ℎ(0) and and to formulate ℎ( ). Then, eq. (S6) is used for evaluatingin the active cavities, as shown in right panels of Figs. S6(a) -S6(c). A set of three 2 × 2 Hamiltonians showing just a one-to-one coupling between a cavity photon mode and an exciton mode is employed. We find that the experimentally observed curves are well explained by the fitting results as LPB and UPB modes. The fitting parameters obtained in the analyses are summarized in Table 1 in the main manuscript, together with the Hopfield coefficients at = 0 º, | | 2 and | | 2 , which are calculated using relationships of The exciton energy also agrees well with the experimental result. ℏ ( = 2 ) was estimated to be in a range of ~140 -200 meV. This result is comparable to our previous report (ref. 29).

Time-resolved PL
We investigate the below-threshold excitation dynamics in the BP1T-CN microcavity by using time-resolved PL measurements. To ensure that the measured dynamics do not include the nonlinear dynamics caused by the strong pumping fluence, the PL decay profile is evaluated at the different pump fluence as shown in Fig. S7. The decay profiles obtained under the pump fluences of 0.46 and 3.1 pJ/cm 2 are almost identical and exhibit single-exponential profiles, revealing that the excitation dynamics we observe is in a linear regime.

Consistency check of scattering model
To check the consistency of our experimental results, we compare the measured PL intensities with those predicted from other experimental results obtained within this study. Basically, the PL intensity is proportional to the product of reservoir-to-LPB scattering rate, LPB decay rate, and photonic Hopfield coefficient. Using this relationship, we qualitatively calculate the PL intensities of Microcavity-C where three LPB modes can be compared fairly under the same excitation condition. The LPB decay rate is estimated from the PL linewidth of each spectrum. In addition, the calculated values were normalized by absorbance to account for the effects of re-absorption effect of reservoir excitons. Note that this calculation was approximately performed by using the data at θ = 0 º, not angular-integrated data. As shown in Fig. S8, the predicted PL intensity depending on the LPB energy shows the same trend of the experimental result, indicating the validity of the scattering model we propose here.