Quantitative Investigation of the Rate of Intersystem Crossing in the Strong Exciton–Photon Coupling Regime

Strong interactions between excitons and photons lead to the formation of exciton-polaritons, which possess completely different properties compared to their constituents. The polaritons are created by incorporating a material in an optical cavity where the electromagnetic field is tightly confined. Over the last few years, the relaxation of polaritonic states has been shown to enable a new kind of energy transfer event, which is efficient at length scales substantially larger than the typical Förster radius. However, the importance of such energy transfer depends on the ability of the short-lived polaritonic states to efficiently decay to molecular localized states that can perform a photochemical process, such as charge transfer or triplet states. Here, we investigate quantitatively the interaction between polaritons and triplet states of erythrosine B in the strong coupling regime. We analyze the experimental data, collected mainly employing angle-resolved reflectivity and excitation measurements, using a rate equation model. We show that the rate of intersystem crossing from the polariton to the triplet states depends on the energy alignment of the excited polaritonic states. Furthermore, it is demonstrated that the rate of intersystem crossing can be substantially enhanced in the strong coupling regime to the point where it approaches the rate of the radiative decay of the polariton. In light of the opportunities that transitions from polaritonic to molecular localized states offer within molecular photophysics/chemistry and organic electronics, we hope that the quantitative understanding of such interactions gained from this study will aid in the development of polariton-empowered devices.


Film preparation.
Glass substrates (2.5 cm×2.5 cm) were immersed in alkaline solution (0.5 % of Hellmanex in distilled water) and sonicated there for 15 min. Afterwards the substrates were sonicated in water and ethanol, for 1 h, respectively. The cleaned glass substrates were dried in an oven overnight before cavity preparation. The ErB films were deposited on the cleaned glass substrates by spin-coating (Laurell Technologies WS-650).
15 wt% ErB/PVA film: 4.6 mg of ErB molecules were dissolved in 1 ml of water containing PVA (88% hydrolyzed, Acros Organics, 22 mg mL −1 ) 1-3 . The solution was filtered through a PHENEX syringe filter (pore size, 0.45 m) and 250 L of the solution was deposited by spin coating on a clean glass plate at various rpm (Table S1) for cavity preparation.

Cavity preparation.
The Fabry-Pérot cavities were built on glass substrates with a size of 2.5 × 2.5 cm 2 . The Ag mirrors were fabricated by vacuum sputtering deposition (HEX, Korvus Technologies). A 100 nm thick Ag film was sputtered on top of the cleaned glass substrate and the molecules were then deposited by spin-coating (Laurell Technologies WS-650) on top of the Ag layer. The spin-coating speed was varied from 1200 to 3000 rpm to achieve different film thicknesses. A semitransparent 30 nm thick Ag film was then sputtered on top of the molecular film to complete the cavities. The typical cavity configuration and detailed parameters for different cavities are listed in Table 1.
Scheme S1. A typical Fabry-Pérot cavity used to study strong light-matter interactions in this study (glass support/100 nm Ag/ErB-PVA film/30 nm Ag).

Steady state absorption, reflection, and emission spectroscopy.
Steady state absorption spectra of bare films and cavities were recorded on a Perkin Elmer LAMBDA 950 spectrometer. Steady-state angle-dependent reflectance spectra were measured in 2 degree intervals using a spectrophotometer equipped with a small-angle specular reflectance accessory (Lambda 950, Perkin Elmer). The spectra were measured relative to a standard reflectance mirror using a horizontal Glan-Taylor polarizer. The reflectance spectra were collected in TE mode using the polarizer. The absorbance of these cavities were determined by using Abs = log(1/R), assuming that transmission and scattering by the cavities are zero. The quality factor (Q = r/Δ ) of an empty cavity (glass support/100 nm Ag/PVA film/30 nm Ag) was around 20 ( Figure S1). Figure S1. Absorbance of bare cavity (glass support/100 nm Ag/PVA film/30 nm Ag). The quality factor is 20.
Steady state emission spectra of the films and the cavities were measured on an Edinburgh Instruments FLS 1000 spectrofluorometer with a Xenon lamp as excitation source. Angular resolved emission spectra of the cavities were measured on the same instrument using a fiber induced angle-resolved platform.

Experimental set-up for angle resolved excitation measurements.
Steady-state excitation spectra were measured with a spectrofluorometer (FLS1000, Edinburgh Instruments). The optical properties of the cavities were studied in front-face geometry.
Excitation and the emission paths were fiber-coupled to the rotating arm of a goniometer. The experimental setup is shown in Scheme S2. During the measurement, the excitation angle was changed in 5 o interval, and the emission angle was fixed. The excitation was polarized in the TE plane. Due to the experimental geometry, the smallest angle at which the emission was collected is around 20 o . The emission was monitored about 20 nm towards a longer wavelength as compared to the Pemission. To probe prompt and delayed spectra, a μs-pulsed lamp was used (Edinburgh Instruments). The "prompt" spectra were recorded at times when the lamp was on, and the "delayed" spectra were recorded 210 μs after the lamp flash was over (gate time was 1000 μs).
Scheme S2. Angle resolved excitation spectra setup (S is the sample, P is the polarizer). The flash lamp (pulse period is in microsecond) operates as the excitation source and the emission is collected by a PMT.

Modelling the angle-resolved reflectivity spectra.
In order to fit and extract parameters from angle-resolved reflectivity spectra, we used the coupled harmonic oscillators (CHO) model. The coupling is described by a 2 × 2 Hamiltonian matrix describing the interaction between a photon and an exciton inside the cavity, as shown below: Where is the cavity energy, is the exciton energy, and is the coupling strength (corresponding to a Rabi splitting of ℏΩ = 2 at resonance). and are the Hopfield coefficients. Diagonalization of this Hamiltonian gives rise to the eigenvalues of the Hamiltonian that represent the polariton energies: here ( ) is the energy dispersion of the cavity, described by Where (0) is the cavity energy at angle zero, is the angle of incidence, and is the refractive index of the medium.

2.5
Angle-resolved excitation of the bare film. Figure S4. Excitation spectra of a film as a function of excitation angle in the prompt and delayed regimes (collected in TE mode). The excitation energy was varied from 2.8 eV to 2.16 eV first and then from 2.09 eV to 1.8 eV, to avoid the scattering from the laser at the emission wavelength (2.12 eV). The two graphs are very similar, suggesting that the differences in the prompt and delayed excitation spectra in Figure 3 are not due to the molecule used.  Figure S6. The extracted and normalized excitation peak counts (here the peaks are normalized at the maximum intensity). For the blue detuned cavities (1 and 2) the P + and Pintensity remains the same in the prompt and delayed regimes. For the red detuned cavities (3,4 and 5), the P + intensity remains similar, but the Pintensity differs in the prompt and delayed regimes. Figure S7. The absorbance peak counts of the cavities, interpolated in 5 degree interval from the absorbance taken in 2 degree interval.

Calculating the rate constants
The radiative decay kr of the polariton is proportional to the photonic fraction Hopph of P -, and inversely proportional to the quality factor of the cavity. For a metal clad cavity, we consider it to be: Where Ec is the energy of a bare cavity and QF equals to the cavity quality factor (=20).
Using these values, the rate is 1.6•10 14 s -1 (without incorporating ℎ ( )). The polariton to exciton reservoir transfer rate, kP -→ER, is estimated as proportional to the energetic penalty for endothermically repopulating the exciton reservoir. 4 Here we consider the Pstate as a delta function in energy scale by assuming that the polariton energy is well-defined. The ER bandwidth is taken into account by an integral that is scaled with an exponential factor, which includes the polariton-exciton energy separation. The exponential function is set to 1 for the energies where the transfer is exothermic. The rate constant is further scaled with the excitonic fraction Hopmol of the Pbranch as kP -→ER signifies the transfer to the exciton reservoir: where E is the energy scale over which the absorbance is measured, EP-is the energy maximum of Pat each angle ( ), and kBT is the energy corresponding to room temperature.
We consider the transfer between the exciton reservoir and P´ states by a radiative pumping mechanism. 5 For small organic molecules this is often the case due to their relatively large Stokes shift. 6 The rate is therefore proportional to the overlap between the emission intensity of the bare film, Emfilm(E) and the Pabsorption, AbsP -(E, θ). It also depends on Hopph as the transfer is photonically mediated.
Transfer from ER to the triplet state, kER→T1, was set to the literature value of ISC in a concentrated film of ErB (kER→T1 = 5.7•10 8 s -1 ), 7 and knr corresponds to the non-radiative decay of the exciton reservoir. This value can be approximated with the non-radiative decay of ErB outside the cavity. At the used concentrations, the emission quantum yield of ErB is low, and the non-radiative decay can therefore be approximated by the inverse excited state lifetime. To probe this value, the emission decay from a neat film was taken, but the lifetime was below the time resolution of our TCSPC instrument. knr was therefore set to the time resolution of the instrument, which is about 100 ps. The set value of knr can therefore be regarded as a lower boundary. We assume that knr is constant with excitation wavelength.
Previous work has shown that the polariton emission lifetime, and thus presumably the nonradiative rate, can be dependent on excitation wavelength. 8 We therefore explored how sensitive our conclusions are on the set value of knr. The direct pathway from Pto T1 becomes enhanced as knr increases and vice versa. Relatively good fits can be performed with knr values in the 1•10 11 -8•10 9 s -1 range. However, most importantly, for all cases where physically sound fits can be achieved (i.e., positive fitting coefficients), does the direct pathway from Pto T1 significantly contribute to the yield of ISC.
We assume that transfer from Pto the triplet state (T1) depends on the energy overlap between these two states. Then, kP -→T1 becomes proportional to the energy overlap between the absorption of the triplet state (AbsT) and P -. The triplet state energy cannot readily be experimentally obtained but an estimate of it was made by first assuming that the spectral envelope of the triplet and singlet states are approximately the same. It was further assumed that the reorganization energy on the singlet and triplet surfaces are about the same. In practice was the T1 energy constructed by translating the S1 absorbance to an energy, one Stokes shift higher as compared to the phosphorescence maximum ( Figure S8).
It should be noted here, that we cannot exclude the possibility that the transfer from Pto T1, is mediated by an optically dark short-lived state, such as an excimeric state, as the observed kinetics would be the same.
When fitting the experimental data to the rate equation model (equations 5 and 6), C1, C2, C3 was used as global fitting parameters. Absorbance/emission Energy (eV) Figure S8. Absorbance (blue line) spectrum of a 15 wt% ErB film in a PVA polymer matrix (glass support/ErB-PVA film). Also shown is the approximated energy of the T1 state (black line), and the emission from a half cavity (red line; glass support/100 nm Ag/ErB-PVA film). Bare Cavity 30nm Ag 100nm Ag ErB half cavity Figure S9. Absorption of bare cavity (green), half cavity (red), 100 nm Ag mirror (blue) and 30 nm (burgundy) Ag mirror. The absorption was calculated as 1-T-R.

Connection between relaxation efficiency and emission quantum yield.
The relaxation efficiency is proportional to the population of a state before emission. As a result, it is also proportional to the emission quantum yield (QY) of the system and to connect the relaxation efficiency to the QY we measured the emission QY of Cavity 4 to 0.8% using an integrating sphere (Edinburgh Instruments). To mention, the relaxation efficiencies were measured in the prompt and delayed regime. The prompt and delayed parts of the total QY of emission was extracted as follows: 1. The excitation spectra were collected at 20° emission angle using a μs-pulsed lamp, which was incorporated into a commercial spectrofluorometer (FLS1000 Edinburgh Instruments).
The excitation wavelength was selected using the double monochromator of the spectrofluorometer, and all recorded spectra are corrected. This lamp has a temporal resolution of a couple of microseconds, but it afterburns for an additional few 10s of microseconds. The Here, = 1/432 s, which was calculated from the phosphorescence decay of an ErB film. Figure S10. Phosphorescence decay of an ErB film.
After determining , we calculated the total delayed counts from 30 μs to 1210 μs, which is considered as .

Relating intersystem crossing and delayed relaxation efficiency.
Equation 6 expresses the delayed relaxation efficiency ( ) as a function of rate constants.
In section SI 2.5 on the other hand, we measure the delayed emission efficiency. To emit a photon from the triplet state, reverse intersystem crossing followed by polariton emission needs to occur. The triplet state, exciton reservoir, and the polaritonic states are in dynamic equilibrium with each other. However, in our simplified model ( Figure 5) we assume that nonradiative relaxation from the triplet state is much faster than reverse intersystem crossing. Thus, from a population perspective, the small yield of repopulation of Pfrom the triplet state does not affect the population density at constant illumination (although we use the photons emitted from this event in our experiments). Thus, we can take the literature ratio between the ISC and the RISC processes ( ) = 5 in order to scale the measured QYdelayed with that is used in Equation 6. 9 Furthermore, as emission is constantly monitored in the same angle, this value should be constant throughout our dataset, no matter that measurements were conducted in the strong exciton-photon coupling regime.

2.10
The influence of kP -→T1.   Figure 6c), blue denotes 5 angles (every 10 degrees), and green denotes 3 angles (10, 30, and 50 degrees). Note that some points are overlayed and can therefore be difficult to see. The only rate constant that depends on the number of polaritonic states is kER→P -. However, the rate of depopulation from the exciton reservoir is constant. In other words, the sum of kER→Pover all angles is constant, ∑kER→P -=3.7•10 8 s -1 , ∑kER→P -=3.5•10 8 s -1 , and ∑kER→P -=3.3•10 8 s -1 when using 9, 5, and 3 polaritonic states in the fit, respectively. Another way of comparing this rate constant for a different discretization of the polaritonic states would be to involve the step size. This would be a physically sound method, but could give errors in the limit of a very small number of angles (comparing 5 and 3 polaritonic states, the step size relates as 20:10, but the number of states as 3:5).