Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

DOI: 10.1039/C9SC04367E
(Edge Article)
Chem. Sci., 2020, Advance Article

Pavel Malý^{a},
Julian Lüttig^{a},
Arthur Turkin^{b},
Jakub Dostál^{a},
Christoph Lambert*^{bc} and
Tobias Brixner*^{ac}
^{a}Institut für Physikalische und Theoretische Chemie, Universität Würzburg, Am Hubland, 97074 Würzburg, Germany. E-mail: brixner@phys-chemie.uni-wuerzburg.de
^{b}Institut für Organische Chemie, Universität Würzburg, Am Hubland, 97074 Würzburg, Germany. E-mail: christoph.lambert@uni-wuerzburg.de
^{c}Center for Nanosystems Chemistry (CNC), Universität Würzburg, Theodor-Boveri-Weg, 97074 Würzburg, Germany

Received
29th August 2019
, Accepted 18th November 2019

First published on 18th November 2019

Exciton transport and exciton–exciton interactions in molecular aggregates and polymers are of great importance in natural photosynthesis, organic electronics, and related areas of research. Both the experimental observation and theoretical description of these processes across time and length scales, including the transition from the initial wavelike motion to the following long-range exciton transport, are highly challenging. Therefore, while exciton dynamics at small scales are often treated explicitly, long-range exciton transport is typically described phenomenologically by normal diffusion. In this work, we study the transition from wavelike to diffusive motion of interacting exciton pairs in squaraine copolymers of varying length. To this end we use a combination of the recently introduced exciton–exciton-interaction two-dimensional (EEI2D) electronic spectroscopy and microscopic theoretical modelling. As we show by comparison with the model, the experimentally observed kinetics include three phases, wavelike motion dominated by immediate exciton–exciton annihilation (10–100 fs), sub-diffusive behavior (0.1–10 ps), and excitation relaxation (0.01–1 ns). We demonstrate that the key quantity for the transition from wavelike to diffusive dynamics is the exciton delocalization length relative to the length of the polymer: while in short polymers wavelike motion of rapidly annihilating excitons dominates, in long polymers the excitons become locally trapped and exhibit sub-diffusive behavior. Our findings indicate that exciton transport through conjugated systems emerging from the excitonic structure is generally not governed by normal diffusion. Instead, to characterize the material transport properties, the diffusion presence and character should be determined.

The usual method of choice for measuring the initial dynamics is time-resolved spectroscopy of transient absorption (TA) type. There, a pump pulse triggers one-exciton dynamics, which are subsequently interrogated by the probe pulse.^{7} In the description of perturbation theory, this is a third-order spectroscopy with respect to the number of interactions with the laser electric field.^{8,9} While traditional TA spectroscopy employs one pump and one probe pulse, an extension using a pair of pump pulses is offered by coherent two-dimensional electronic spectroscopy (2DES).^{10–15} The resulting frequency resolution of both the pump and the probe step allows the analysis of lineshapes,^{16} transition couplings,^{17} coherent dynamics,^{18} state-to-state population transfer kinetics,^{19–21} and coupling to dark states such as charge-transfer states.^{22} In the double-quantum coherence variant of third-order 2DES the energetic positions and lineshapes of higher excited/biexciton states can be probed as well, though their kinetic evolution is not accessible for lack of another time variable.^{23}

Despite the power of the technique, third-order spectroscopy is not well-suited for measurement of long-range exciton transport and exciton–exciton interaction. The reason is that, in the interpretation of the dynamics, one assumes an observed exciton to be independent of all other excitons in the sample, i.e., an interaction-free scenario. Because long-range transport of a single exciton through an extended aggregate (polymer, photosynthetic membrane, etc.) does not lead to spectral changes to the aggregate absorption, it is invisible for third-order spectroscopy. Concerning EEI, if it is observed at all, this is typically viewed to be an undesired artifact arising from too high excitation density. An exception is the biexciton decay visible in the two-exciton lineshape, as it contributes to rapid dephasing.^{24} However, to disentangle two-exciton kinetics from the lineshape alone is a formidable task, unfeasible for any interaction on longer timescales such as diffusive motion.

When applied carefully, studying the excitation power dependence of transient absorption can reveal valuable information about EEI.^{25–30} The EEI signal is, however, present merely as a power-dependent perturbation of the single-exciton time-dependent spectra. It is here that one of the key advantages of higher-order nonlinear spectroscopy comes into play. An extended nonlinear spectroscopy such as 2DES can in higher orders directly observe the biexciton dynamics in time.^{24,31–36}

Next to power-dependent annihilation, spectroscopy methods to observe the long-range exciton transport include spatially resolved transient absorption or emission^{37,38} and surface or bulk quenching.^{39,40} Despite being powerful and widespread, all these techniques have their limitations. In transient absorption or photoluminescence microscopy, the temporal and spatial resolution are intertwined, limited by observable changes within the diffraction limit.^{37} Surface quenching relies on well-defined, homogeneous sample morphology, and, together with the bulk quenching, measures the average transport behavior.^{41}

The mentioned techniques are commonly used to determine exciton transport properties such as the diffusion coefficient and diffusion length in a wide spectrum of materials.^{3,42} The standard procedure for evaluating the data from the various measurements is to assume normal diffusion of the excitons,^{43,44} often accompanied by a calculation of a (generalized) Förster radius.^{42} Although there is both experimental^{45} and theoretical^{46,47} evidence for occurrence of anomalous diffusion in the presence of exciton delocalization and/or energetic disorder, the normal diffusion assumption is rarely questioned in practice. In this work we challenge this assumption, observing anomalous, trapped diffusion of excitons. In the present text we employ the term “diffusion” in a general way to signify the process by which excitons move through the system as true quasiparticles, but we will then go on to show that the kinetic behavior does not, in general, follow the normal diffusion equation.

In our approach we probe the interaction of exciton pairs, using our recently developed fifth-order exciton–exciton-interaction two-dimensional (EEI2D) spectroscopy. This technique facilitates a direct observation of exciton–exciton interactions.^{48–50} In EEI2D, exciton–exciton annihilation is observable in the evolution of the signal amplitude as a function of population time. This technique can therefore be used not only to study EEI, but one can utilize the annihilation as a tool to probe single-exciton propagation dynamics even in the absence of spectral changes during transport.

We employ EEI2D spectroscopy to study biexciton dynamics and interactions in squaraine-based copolymers.^{51} Squaraine copolymers provide an excellent combination of polymer and J-aggregate properties. Previous studies have shown that these copolymers support large exciton diffusion lengths^{51} and can serve as efficient electron donors.^{52} This makes them interesting for various applications such as in heterojunction solar cells,^{52} thin-film transistors,^{53} or OLEDs.^{54} The optical properties of such conjugated polymers are determined by the excitonic structure.^{55} Locally, the exciton delocalization leads to wavelike dynamics, connected to phenomena such as supertransfer.^{56} On the long range, however, the exciton transport is typically described as diffusive. The current paradigm is that after photoexcitation the excitonic states are established, possibly with accompanying polaron formation.^{57} The excitons then diffuse by a classical random walk through the polymer.^{5} While separately the diffusive transport and wavelike motion have been described, how the former emerges from the latter is a subject under debate,^{56} typically ignored when discussing the exciton migration.^{2,51} Unlike in previous studies,^{58} we are able to systematically vary the length of the synthesized polymers. This additional degree of control enables us to study the transition from mostly wavelike to predominantly diffusive excitation dynamics.

Label | M_{n}/g mol^{−1} |
X_{n} |
Đ |
---|---|---|---|

P19 | 26200 | 19 | 1.62 |

P18 | 24700 | 18 | 1.54 |

P11 | 15200 | 11 | 2.11 |

P5 | 7200 | 5 | 2.88 |

The system was described by a Frenkel exciton model. Each squaraine molecule was represented by a three-level system with one ground state, one first excited state, and one higher excited state. We considered electronic coupling only between transitions of neighboring chromophores, both within and between the dimers. The resulting Hamiltonian of N dimers was diagonalized (see Section S1 in the ESI† for the full procedure and Fig. S1† for an overview of the assumed couplings and the resulting energy structure), yielding 2N one-quantum states (i.e., one-exciton states (|e〉)), two-quantum states (i.e., two-exciton states (|ee〉), mixed with 2N higher-excited molecular states (|f〉)), and three-quantum states (i.e., three-exciton states (|eee〉) mixed with 2N(2N − 1) combined higher- and one-exciton states (|fe〉)). The strength of the transitions between the state manifolds was obtained by transforming the transition dipole-moment operator into the eigenstate basis. The exciton delocalization length was calculated in the standard way from the inverse participation ratio as , where the c_{m}^{i} are coefficients of the transformation from the site (index m) to the excitonic (index i) basis and N is the polymer length. The system environment was described as a weakly interacting vibrational bath. The vibrations consisted of a continuum of low-frequency modes and one underdamped, intramolecular mode at 1280 cm^{−1}. This relatively pronounced vibrational mode is visible in the absorption spectra and contributes to the energy transfer, as its frequency is close to the energy gap between the squaraine A and squaraine B transitions (1200 cm^{−1}). The one- and two-quantum state dynamics were described by Redfield theory, while the relaxation from two-quantum to one-quantum states was described within a Lindblad formalism.^{9} All the equations used for the calculations can be found in the ESI.†

Exciton–exciton annihilation occurs between spatially overlapping excitons via mixing with and transfer to the higher excited states and subsequent relaxation to the one-exciton manifold. This avoids the commonly used phenomenological description of using an effective annihilation rate for overlapping excitons. All of the calculated quantities were averaged over a disorder in the chromophore energies, where for simplicity we assumed independent Gaussian disorder of all chromophore transition energies. The parameters used for the calculation are presented in the Theory section in the ESI.† There we also describe how we determined the parameters and discuss the robustness of our results against reasonable parameter variation. All the parameters used for the calculations (Table S1 in the ESI†) are well in the range used in previous work on squaraine dimers and copolymers.^{27,51,60}

From the absorption spectral shape, a significant disorder and heterogeneity is apparent, which is largely not present in the fluorescence spectra. From the comparison with theory, it is clear that this additional heterogeneity is not a feature of our excitonic model. A possible explanation is an increased and/or non-Gaussian disorder in the polymer ground state. As we are interested in the excited-state dynamics, this feature is not crucial. The peak splitting and oscillator-strength redistribution are characteristic of excitonic splitting; the small Stokes shift and narrow peak width indicate weak interaction with the environment, justifying the concept of delocalized excitons.

Unlike our previous pump–probe and third-order 2D spectroscopy study on squaraine homopolymers, this work does not investigate in detail the initial ultrafast local exciton relaxation, which occurs within the first 100 fs.^{58} Instead, we follow the transport of the excitons along the polymer across time scales. Let us focus on the changes with varying polymer length. For the shortest polymer, P5, the signal rises very fast, basically starting from the plateau, stays constant from 0.1 to 100 ps, and then decays. In contrast, with increasing polymer length, the exciton diffusion phase becomes increasingly prominent and the signal reaches its maximum at later times. When assessing the magnitude of the observed changes, one has to bear in mind the logarithmic scale of the time axis. The calculated traces agree very well with experimental ones, indicating that our model captures the exciton dynamics and interaction well. The source of the small deviations, mostly apparent at the initial times, is possibly the ensemble distribution of the polymer length (see ESI for details†). Another possibility are the higher-order effects of high excitation intensity, which we include and discuss below. Finally, the comparison of experiment and theory at the earliest times is made harder by the fact that the pump spectrum does not cover fully both absorption bands, which results in a difference in the initial exciton population. We especially emphasize the same trend in both experiment and theory, in that the plateau is reached at later times and the signal plateau is shorter for longer polymer chains (see systematic “nested” behavior of rectangular colored regions).

Based on our Frenkel-exciton model, the delocalization length of the excitons is found to be around 3.8 chromophores, that is, about 1.9 dimeric units (see Fig. 4 and its discussion below for exact values). The time scale of annihilation of two excitons localized on the same dimer is ultrafast (30 fs, see Table S1†). We have confirmed this value independently in another measurement, using SQA–SQB dimers, by evaluating the annihilation seen in fluorescence-detected 2DES for high excitation power (see Fig. S4 in the ESI†). While this represents the maximum annihilation rate found in the system, for any pair of excitons the rate will be effectively weighted by their co-localization. In other words, distant excitons do not interact, but once they meet, they annihilate very efficiently. In Fig. 3c, initially separated excitons are depicted exemplarily in the P5 and P18 polymers. In the short polymers, any two excitons have a larger probability, compared to long polymers, to be partially co-localized, and thus they interact practically immediately. In contrast, in the long polymers any exciton pair has a larger chance of being spatially separated, making exciton transport necessary prior to interaction. The key factor determining the biexciton dynamics is thus the exciton size (i.e., delocalization length) compared to the length of the polymer.

Fig. 4 The role of relative exciton delocalization length. (a) Calculated absolute (black) and relative (red) delocalization length as a function of polymer length (squares). Vertical lines indicate the studied polymers. Circles on the P11 line indicate the more and less disordered polymer variations of P11 studied in (b). (b) Calculated time dependence of the integrated fifth-order EEI2D signal for varying relative exciton delocalization. The effect is analogous to the varying length observed in Fig. 3. (c) Exemplary cartoon illustrating excitons with relative delocalization length of 9% (more disordered case, top) and 34% (more ordered case, bottom). The more delocalized excitons will meet and interact sooner after photoexcitation than the localized ones. |

We can therefore modify our model to incorporate such additional excitons that are not observed directly but that influence the measured signal. In Fig. 5b the kinetics are calculated with the presence of such excitons, and the rise of the signal due to exciton–exciton annihilation is increasingly pronounced for larger excitation intensity, in agreement with the experimental data. In the model correction the quantity which directly scales with the excitation intensity is the population of the additional excitons, for details see the ESI.† While for weak intensity (green curve) there are practically no such excitons, for larger intensities (3 times larger in yellow and 4.5 times larger in orange), their presence becomes more probable. The cartoon in Fig. 5c illustrates the directly observed excitons in yellow and the additional ones in gray. At high excitation powers the excitons have a large chance to encounter one of these “dark” excitons and suffer annihilation. One could say that higher-order effects (i.e., higher than fifth order) effectively increase the exciton–exciton annihilation rate.

Crucially, as the effects just described emerge from a single microscopic description, they are not independent. Take for example the exciton–exciton annihilation, which depends on the mixing of the two-exciton and higher excited states, and also on the state coupling to the environmental vibrations. There are fixed relations between the transition couplings, transition strengths, and couplings to the bath vibrations for the one-exciton, two-exciton and higher excited states (see Table S1 in the ESI†). This leads to a firm connection between the initial contribution of higher excited states, exciton dynamics, spectral peak shapes and positions, and exciton–exciton annihilation. These inner constraints give us confidence that our theoretical description captures the physics correctly since it corresponds to experimental observations.

In the linear range we can fit the time dependence of the mean-square displacement with a power dependence, σ^{2}(t) = Dt^{α}. We obtain a coefficient of α = 0.4, signifying sub-diffusive dynamics of the disorder-trapped excitons, whereas conventional diffusion would correspond to α = 1. Coming back to the excitation probability distribution, Fig. 6a, sub-diffusive (trapped) diffusion character can be seen from the characteristic cusp shape.^{64} The generalized diffusion coefficient, also obtained from the fit, is D = 140([SQA − SQB])^{2} ps^{−0.4}.This time-dependent diffusion coefficient contrasts with the commonly reported normal diffusion coefficients, used to determine the exciton diffusion length.^{42} From the fitted sub-diffusion equation, we can determine (Fig. 6c) the exciton diffusion length by asking how far the exciton could travel (orange curve) within its typical lifetime of 1 ns (gray line). By this we obtain a diffusion length of about 44 dimeric units, which corresponds to roughly 130 nm. This is in line with the previously reported diffusion length in these polymers,^{27} signifying their excellent transport properties, as this value is much higher than many comparable materials.^{42}

Reporting a value for the diffusion length facilitates comparison with other materials known from the literature. Comparing directly the diffusion constant is not feasible due to the unconventional physical units of length^{2} time^{−0.4} in our case that is different from studies in which normal diffusion is assumed a priori. The main point of the present study, however, is not simply to report a value of diffusion length, but to identify the nature of exciton propagation. As shown above, the transport has a sub-diffusive, trapped character and differs substantially from the commonly assumed normal diffusion. Mathematically, in the absence of disorder-induced traps in the long-time (i.e., long-distance) limit the propagation behavior on an infinite polymer will converge to normal diffusion.^{46,47,65} This transition, however, depends on the system parameters. We have theoretically tested this transition by systematically varying the system parameters, see section “Transition from anomalous to normal diffusion” in the ESI.† Our findings indicate that in realistic, finite-length polymers the trapped excitons might never reach the normal diffusion regime within their lifetime. Contrary to the common assumption, the anomalous diffusion then determines the exciton transport properties.

In a broader sense, our results show how phenomena such as exciton (anomalous) diffusion and annihilation, often treated phenomenologically, emerge from a microscopic excitonic structure. Perhaps surprisingly, in the developed description of the studied polymers the diffusion that emerges is of anomalous character, known from diverse other natural phenomena.^{66} This is in contrast to the commonly used normal diffusion. Future experimental and theoretical work will determine the generality and scope of the trapped, anomalous diffusion regime. This insight into the microscopic nature of exciton transport was made possible by applying the newly developed method of fifth-order EEI2D spectroscopy together with microscopic modeling. This approach is applicable to a wide class of other material systems, including natural and artificial light-harvesting materials. We expect it to become a key tool in future studies of exciton transport.

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## Footnote |

† Electronic supplementary information (ESI) available: Detailed theoretical description, sample characterization, synthesis. See DOI: 10.1039/c9sc04367e |

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