Nanosecond pump and probe observation of bimolecular exciton effects in rubrene single crystals

Photoexcited singlet states in organic molecular crystals can transition into two triplet states with a total spin of zero, in addition to radiatively or nonradiatively decaying back to the ground state. The former can be seen as a first step in a fission process that can lead to the two triplet excitons independently diffusing in the crystal—a process that could play a key role in organic photovoltaics.¹ Rubrene single crystals are characterized by a particularly efficient fission mechanism that leads to a large majority of photoexcited singlet states ultimately transforming into triplet excitons.^2,3

Singlet fission in rubrene single crystals has not been consistently described yet. Tao et al.⁴ have observed a ∼100 ps fast decay of a photoinduced infrared absorption band which they assigned to singlet excitons, but they interpreted the decay as caused by exciton dissociation, not fission. Furube et al.⁵ observed transient absorption kinetics in the near infrared which they interpreted as singlet exciton fission into triplets with a time constant of 10 ps. Ma et al.^6,7 identified photoinduced absorption bands in the visible spectral region, which they assigned to singlet and triplet excitons, and they derived an ∼2 ps time for singlet fission from higher vibrational states and an ∼20 ps time for singlet fission from the lowest vibrational state. With the exception of Ref. 4, transient absorption measurements have been largely interpreted as a singlet-to-triplet fission process that occurs on a time scale on the order of ∼10 ps, sometimes even in nominally amorphous films.^5–8 However, Piland et al.⁹ measured photoluminescence transients in rubrene films that they interpreted as indicative of a fission time of 2 ns. The inability to detect tell-tale quantum beats in the rubrene photoluminescence¹⁰ led them to hypothesize an insufficient fission rate. In general, time resolved photoluminesence measurements have not shown a clear transition from an initial singlet luminescence transient on the picosecond time scale (matching the singlet lifetime) to the typical power-law decay expected for a dense triplet population.^2,11–14 On the contrary, photoluminescence transients have shown a multiexponential decay with time-constants ranging from 80 ps to a few nanoseconds.^6,8,14

It is worth noting that in general a singlet exciton transitions into a quantum superposition of the singlet state with two triplet states having a total spin of zero.¹⁰ This superposition state can then decay by radiative recombination of the singlet component, or else it can decay into two triplet excitons that can diffuse independently. It is the latter possibility that amounts to a singlet exciton fission into two independent triplet excitons.¹⁵ It follows that the ∼10 ps lifetime mentioned above can also be interpreted as the time needed to reach an equilibrium singlet-triplet superposition state, and not necessarily the time to create two independent triplet excitons, which could be much longer, similar to the nanosecond-scale decays observed in photoluminescence transients.⁹ Further investigation of excited state dynamics in rubrene on the time-scale of a few nanoseconds seems to be necessary in order to gain more insights into the fission process that ultimately leads to two independent triplet excitons.

In this work, we use pump and probe transient grating experiments to investigate excited-state dynamics on a time scale between 20 ps and 10 ns in vapor-transport grown rubrene crystals. Such crystals have an orthorhombic unit cell containing four molecules and lattice parameters a = 1.44 nm, b = 0.718 nm, and c = 2.69 nm,¹⁶ where we define the crystallographic axes in the same way as Refs. 17–21. Investigating triplet dynamics in rubrene is particularly interesting because of its large triplet diffusion length of 4 μm,¹⁷ long triplet lifetime of 100 μs,^2,14 large hole mobility,^19–22 large delayed photocurrent^23,24 due to triplet dissociation, and large triplet fusion probabilities.^2,3

Transient grating experiments are based on the photoexcitation of a spatial excited state distribution using the interference pattern between two optical pulses, followed by the detection of the resulting absorption or polarizability changes by diffraction of a time-delayed probe pulse from the resulting grating. Compared to the straight detection of a photoinduced absorption, transient grating experiments are characterized by a backgroundless signal that allows the detection of both absorption and refractive index changes with a higher sensitivity.

We used an excitation and detection wavelength of 532 nm, which allows excitation to the ground vibrational state of the singlet exciton,¹⁸ and falls at the long-wavelength edge of a transient absorption band that has been assigned to triplet absorption.^5,6,25 We used 20 ps laser pulses crossing at angles ranging from 4.8∘ to 90∘. This created transient gratings with spatial periods between 6.3 μm and 0.4 μm, respectively. The transient grating was detected by measuring the diffraction efficiency of a time-delayed pulse incident under the Bragg angle. All data shown in this work were obtained using a pristine rubrene sample, verified by measuring its photoluminescence spectra,¹⁸ with dimensions of ∼1 mm along the a and b directions, and 12 μm along the c direction. This crystal was mounted with the c-axis along the bisectrix between the two excitation beams (see Fig. 1, inset), and the b-axis parallel to the grating wavevector. We chose the excitation light to be a-polarized, which gave the strongest signal because of the relatively long absorption length of ∼15 μm at 532 nm,¹⁸ leading to a thicker grating and less attenuation of the counterpropagating probe pulse. We also used other samples, different orientations, and different polarizations, and—in all cases—the observed dynamics were the same as what we report below.

The spatial dependence of the light intensity in the photoexcitation pattern is

where y and z are the coordinates along the b and c axes of the crystal, respectively. α_a is the absorption coefficient for a-polarized light at 532 nm, and k_g is the modulus of the grating wavevector. I₀ = I₁ + I₂ is the sum of the intensity of the two interfering pulses, and m₀ is a modulation index given by $2I1I2/I0$⁠. Since a photoexcited molecule has, in general, a different polarizability than a ground-state molecule, photoexcitation can leave behind both an absorption and a refractive index grating that are initially in-phase with the light-intensity pattern. The diffraction efficiency of a probe pulse incident under the Bragg angle is²⁶ 

where L is the effective interaction length, λ is the light wavelength in vacuum, and A(t) is the amplitude of the first Fourier component of the photoinduced grating. It is, in general, complex-valued, with an imaginary part related to absorption modulation and a real part $Re[A(t)]=Δn$ given by the refractive index modulation.²⁶ Immediately after photoexcitation and in absence of saturation effects, $A(0)∝I0m$ and the grating is purely sinusoidal.

Figure 1 shows the time-dependence of the diffraction from the photoinduced grating. The grating amplitude grows immediately during the length of the excitation pulses, and then relaxes within a few nanoseconds.

The data also clearly show a high frequency oscillatory component riding on top of the slower nanosecond decay. The frequency of the oscillations depends on the grating spacing, as shown in Fig. 2. This means that the oscillations in diffraction efficiency are caused by laser-induced ultrasonic waves.^27,28 The grating amplitude in Eq. (2) can then be written as

where A_exc(t) is the amplitude of the background excited state modulation, A_s is the amplitude of the ultrasound grating, and $ωs=vskg$ is the angular frequency of the ultrasound wave with speed v_s. The solid curves in Fig. 2 are least square fits to the data using this expression with an appropriate choice for the time-dependence of $Aexc(t)$ that we will discuss later below. The acoustic speed extracted from fits at different grating spacings (see Fig. 2, inset) is $vs=2.6×103$ m/s, similar to results obtained in other organic crystals.²⁷ In addition, we find that a good fit of the experimental data is only possible using real-valued $Aexc(t)>0$ and A_s < 0 so that the oscillation caused by the acoustic wave reduces the the diffraction that would be caused by the background grating alone. The data cannot be fitted well if one assumes that $Aexc(t)$ has a large imaginary part as would be caused by an absorption grating, and it can also not be reproduced well when assuming that the ultrasound grating is caused by electrostriction.^27,28

From the fact that our data can only be modeled by Eq. (3) with real-valued amplitudes, and the fact that the ultrasound grating is a refractive index modulation, one must conclude that the grating that produces the nanosecond decay in our experiment is also a refractive index grating. The onset of the ultrasound oscillation corresponds to a dilation of the material in the higher intensity regions of the interference pattern. The amplitude of the longer-lived excitation grating must then be a positive index-change that is periodically reduced by the refractive-index decrease caused by elastic dilation, with the time-dynamics of the background grating obtained by connecting the maxima of the oscillation. We thus reach the important conclusion that the nature of the main signal that we are detecting is not an absorption modulation, but a refractive index modulation, likely caused by a higher polarizability of the excited-state molecules when compared to the undisturbed crystal. From the measured diffraction efficiency and Eq. (2), we estimate the refractive index change induced at an excitation density of 8 × 10¹⁹ photons cm⁻³ to be on the order of 0.008.

Fig. 1 shows that the diffraction efficiency from the photoexcitation grating decays in a few nanoseconds, and that this decay is faster at higher excitation densities. The nanosecond decay is not a single exponential. Curve fitting with multiple exponentials would be possible, but with no clear distinction of decay time-constants. Instead, the fact that the decay becomes faster at higher excitation densities could be an indication that it is caused by the interaction of excited states created close to one another. To test this hypothesis, we first derive the time-dependence of the grating amplitude when its decay is caused by a bimolecular interaction process.

The spatial dependence of the first Fourier component of the photoinduced grating must have the same form as (1)

where $ρ̃(t)$ is the initial average density of (as yet unspecified) excited states in the crystal, which may be a function of depth, and $m(0)=m0$⁠, and we do not explicitly write the slow decrease in the z-direction. Local quadratic recombination as described by $dρ(t,y)/dt=−γρ(t,y)2$ implies

where γ is a bimolecular interaction rate. Developing this into a Fourier series and casting its first two terms in the form (4) gives

where $ρ0=ρ̃(0)$ and

This result is valid for any initial modulation m₀ of the grating, but it is useful to consider the limit of a small modulation index, $m0→0$⁠. In this limit, one finds

The diffraction from the photoinduced grating as described by Eq. (2) is governed by the amplitude $A(t)∝ρ̃(t)m(t)$⁠. The effect of quadratic recombination is seen both on the average density, $ρ̃(t)$⁠, as well as on the modulation index, m(t). As a consequence, A(t) follows a power law with an exponent of 2 instead of the simple decay with an exponent of 1 that appears in Eq. (5). This time-dependence will dominate over the effects of quadratic recombination on a possible inhomogeneous profile of the grating in the direction perpendicular to the crystal's surface.

To assess if the time-dynamics shown in Fig. 1 are indeed caused by a quadratic recombination process, we measured the grating decay over a wider range of excitation pulse fluences and, following (2), derived the time-dependence of the grating amplitude A(t) by taking the square-root of the diffraction efficiency. The results are shown in Fig. 3. We then performed a simultaneous least squares fit of the resulting data using $A(t)=Bρ̃(t)m(t)$ and Eqs. (6) and (7). Here, the only fitting parameters were the proportionality constant B (same for all data sets) and the product $γρ0$ for each of the different excitation densities. We stress that in this way the time-dependence of each data set is fit with only one fit parameter, essentially the initial average density ρ₀. The results are plotted in Fig. 3, and they display a very good agreement between data and model, clearly showing that the strength of the diffracted signal at time t = 0 correlates with the grating decay rate in exactly the way predicted by a quadratic recombination model.

The inset in Fig. 3 shows how the product $ρ0γ$ obtained for different data sets relates to the actual average excitation density used in the corresponding experiment, which was calculated from

where F_tot is the the sum of the fluences of the two excitation pulses as obtained from pulse energies and the beam waist at the surface of the crystal, n_a is the index of refraction along the a axis, t is the Fresnel amplitude transmission coefficient calculated from the incidence angle, and α₀ is the absorption coefficient at 532 nm.¹⁸ The linear correlation displayed in the inset of Fig. 3 implies that $γ=(5.4±1.0)×10−12$ cm³s⁻¹. This value is determined only by the observed decay dynamics and measured pulse fluences, and it agrees well with previous estimates.^2,3 Put another way, the measurements in Fig. 3 imply that quadratic recombination leads to the destruction of 50% of the original excitation after ∼3 ns at an average excitation density at the surface of the crystal corresponding to $ρ0∼$ 0.5 × 10²⁰. Such an excitation density corresponds to an average distance between photoexcited molecules of only ∼3 nm. It is therefore not astonishing that interaction between excited states is possible at such small distances.

In addition to the data in Fig. 3, we also performed several experiments at other grating spacings, and in all cases, we observed a behavior compatible with the one described above. Only at the shortest grating spacings of the order of 0.4 μm, obtained at $90°$ crossing angles, we noted an acceleration of the decay. While this could imply that transport may be responsible for washing out the grating in addition to quadratic recombination, the experiment at these short grating spacings was difficult, with a low signal-to-noise ratio, and this possible contribution of transport could not be investigated further.

We finally come to a discussion of the nature of the excited states that we are detecting in the present experiment. Given the fact that photoexcited singlet states are expected to transform into two entangled triplet-state molecules within our pulse length, an obvious candidate for the excited states that we detect are triplet states. Ma et al. identified a photoinduced absorption band peaking near 510 nm to which they associated triplet excitons formed from singlet-to-triplet fission within 20 ps from photoexcitation.⁶ The refractive index change detected in our experiment, performed at 532 nm, could be associated both with this triplet exciton absorption, as well as with the absorption band assigned to singlet excitons in the same publication.⁶ In a separate experiment performed with 1 ps laser pulses, we observed that the build-up time of the signal we detect is on the order of ∼5 ps and reaches a quasi-steady-state value in less than 10 ps, which compares well with the observations in Ref. 6, and implies that the index change we observe originates from triplets.

It is also possible to estimate the average distance traveled by free independent triplet excitons with the observed diffusion length¹⁷ and lifetime² of 4 μm and 100 μs, respectively, to obtain a diffusion length during a time of 3 ns of $(4 μm)×(3ns)/(100μs)≈20$ nm. This estimate shows that it is at least conceivable that what we observe on the nanosecond time scale is caused by the same triplet diffusion that is observed in steady-state; however, this estimate does not properly take into account the quasi-one-dimensional diffusion, and there is a relatively large latitude for different interpretations. As an example, the triplet states observed in the present experiment could well still be in a coherent quantum superposition with a singlet state.

In conclusion, we demonstrated an alternative method for observing exciton dynamics in rubrene with a high sensitivity and a high time-resolution. The experiments have shown a power law decay that is indicative of a bimolecular interaction of excited states with a triplet component in rubrene over a time scale of nanoseconds and a distance of ∼3 nm, corresponding to excitation densities on the order of ∼ 0.5 × 10²⁰. In any case, an important observation is that the decay rate is highly sensitive to the photoexcitation density, and, because of high triplet fusion probabilities in rubrene crystal, this could be a possible explanation for the observed variability of photoluminescence decay times that have been reported in rubrene.

We thank V. Podzorov at Rutgers University for providing the rubrene samples.