Article pubs.acs.org/cm
Cite This: Chem. Mater. XXXX, XXX, XXX−XXX
Role of Exciton Density in Organic Materials: Diffusion Length, Lifetime, and Quantum Efficiency Leonardo Evaristo de Sousa,† Fernando Teixeira Bueno,‡ Luciano Ribeiro,† Luiz Antônio Ribeiro Junior,‡,§ Demet́ rio Antônio da Silva Filho,*,‡,§ and Pedro Henrique de Oliveira Neto‡ †
Theoretical and Structural Chemistry Group, State University of Goias, 75133-050 Anapolis, Brazil Institute of Physics, University of Brasilia, Brasilia, 70919-970 Brasilia, Brazil § International Center for Condensed Matter Physics, University of Brasilia, CP 04455, 70919-970 Brasilia, Brazil
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‡
ABSTRACT: Exciton dynamics are of pivotal importance in the working of organic electronic devices. Under conditions under which efficient exciton diffusion is combined with high excitation densities, exciton−exciton interactions become relevant, affecting significantly the diffusion length and average lifetimes of these excitations. Employing a kinetic Monte Carlo model, we investigate singlet exciton diffusion under conditions that span a wide range of organic materials. The contributions from the different deexcitation pathways are identified by analyzing simulated time-resolved photoluminescence spectra, allowing us to calculate the exciton densities at which exciton−exciton annihilation effects become dominant. The connection with actual materials of interest in applications is made, and the effects on exciton diffusion length and average lifetime are discussed.
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state.19 In practice, this process amounts to the nonradiative recombination of one of the excitons involved. As such, two undesired effects in terms of device efficiency result from exciton−exciton annihilation. First, the exciton diffusion length and lifetime are decreased with the appearance of a second deexcitation pathway, and second, the quantum efficiency of light emission is decreased. Experimentally, exciton dynamics are studied by means of time-resolved photoluminescence (TRPL) spectroscopy.20−23 In this kind of experiment, lasers are used to excite a material and the intensity of emitted light is registered as a function of time. From these experiments, characteristic decay times are obtained. With different pump fluences, exciton densities can reach sufficiently high values for exciton−exciton interaction effects to be observed, which produces changes in the observed spectra. However, the determination of what constitutes high exciton densities depends on the exciton lifetime and the efficiency of diffusion, which vary considerably among organic materials.24 Thus, it stands to reason that exciton densities for which bimolecular phenomena become relevant in terms of experimental results should also span a large range. A better understanding regarding said critical concentration is therefore of the utmost importance for device efficiency and for an adequate interpretation of experimental data.
INTRODUCTION When it comes to organic electronics, devices such as organic photovoltaics (OPVs) and organic light-emitting diodes (OLEDs) have been the subject of extensive research.1−3 In both cases, exciton dynamics play an essential role, being inextricably linked to device efficiency. In the case of OPVs, light absorption generates excitons that must diffuse through the material until they are able to dissociate into free charges after reaching a donor−acceptor interface.4,5 Conversely, in the working of OLEDs, the emission of light depends on the recombination of excitons that originated from injected holes and electrons.6 More broadly, several aspects of exciton dynamics have been the subjects of very recent research,7−10 speaking to its relevance in different applications. For singlet excitons, the diffusion process happens by means of a process known as Förster resonance energy transfer (FRET).11,12 Such a process is a nonradiative energy transfer mechanism, in which the excitation energy from a donor molecule migrates to an acceptor molecule. For this transfer to occur, there must be overlap between the absorption and emission spectra of the molecules in question. In addition, the efficiency of this mechanism is also strongly dependent on exciton lifetime and intermolecular distances.13 In cases in which efficient exciton diffusion is combined with significant exciton density, exciton−exciton interactions become relevant phenomena.14 These interactions may include biexciton generation,15 stimulated emission,16 or exciton− exciton annihilation.17,18 In the particular case of exciton annihilation, two singlet excitons first combine into a higher excited state and then quickly relax back to the first excited © XXXX American Chemical Society
Special Issue: Jean-Luc Bredas Festschrift Received: March 31, 2019 Revised: July 6, 2019 Published: July 8, 2019 A
DOI: 10.1021/acs.chemmater.9b01281 Chem. Mater. XXXX, XXX, XXX−XXX
Article
Chemistry of Materials
which vary considerably in different experimental setups. As such, these effects can be indirectly and generically taken into account by considering the Förster radii we employ in the simulations as the corresponding temporal averages of the actual time-dependent radius. At the beginning of the simulations, excitons are randomly generated in the grid in accordance with the desired concentration. At each time step, the rates of Förster transfer for an exciton are calculated with respect to each one of its eight neighboring sites using eq 1. The probability of a particular site being chosen is obtained by dividing the rate for this particular site by the sum of all computed rates (pi = ki/∑jkj). It is worth noting that because of the inverse sixth-power dependence of the Förster rate, hops toward second nearest neighbors become considerably less likely. After the hopping site is chosen, a random number is then drawn, selecting the hopping site. Once this procedure is done, a second random number is drawn to decide whether the transfer is successful or the exciton undergoes radiative recombination. This number is compared to the efficiency of Förster transfers, which is given by the ratio between the Förster rate and the sum of the Förster and recombination rates.13
Following these observations, in this work we perform simulations of exciton diffusion to determine the effects of excitation density on key aspects of exciton dynamics, namely, diffusion length, average lifetime, and radiative recombination. The simulations employ a kinetic Monte Carlo (KMC) model that includes the possibility of annihilation of excitons that reach the same site simultaneously. This kind of simulation has been thoroughly employed in theoretical studies of exciton diffusion and charge carrier transport.25−30 With the KMC model, we are able to simulate TRPL spectra covering a wide range of organic materials. The influence of the different deexcitation pathways is analyzed by considering fits to the simulated spectra. From these results, we calculate the exciton densities for which bimolecular contributions begin to dominate and relate these results to materials of interest in optoelectronic applications. Finally, we investigate the effects of annihilation on diffusion length and lifetime.
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METHODS
kF =
1 ij RF yz jj zz tM k r {
1+
9c 4κ 2τemi 8π
(1)
∫0
∞
dω ID(ω)αA(ω) ω4
6
( ) r RF
(3)
In our model, exciton−exciton interactions in the form of exciton− exciton annihilation are considered. As such, when two excitons occupy the same site, one of them recombines nonradiatively, leaving a single excited state at the site. This remaining exciton is free to resume diffusion afterward. In this sense, we do not consider in the model bimolecular processes that may result in radiative recombination of excitons, which may be relevant for some particular system but are overall less common.32 To understand the effects of exciton concentration on diffusion length, we performed simulations considering Förster radii ranging from 20 to 40 Å and exciton lifetimes ranging from 500 to 5000 ps. These values are enough to cover the behavior of a large range of molecules typically employed in optoelectronic applications. Exciton concentrations, measured in excitons per lattice site, varying from 0.1% to 4.9% were examined. Each simulation round is run until no excitons are left, and the process is repeated 1000 times with final results corresponding to averages over all rounds. In each simulation round, when exciton recombination or annihilation takes place, the time of the event is registered along with the exciton displacement up until that point. From these data, we are able to calculate exciton diffusion lengths from the root-meansquare displacement of the excitons as well as average exciton lifetimes. In addition, by plotting a histogram of the number of radiative recombination events as a function of time, we are able to simulate TRPL spectra. Importantly, because the annihilation process is nonradiative in nature, annihilated excitons do not contribute to the TRPL spectrum. However, the presence of such a bimolecular process may be inferred from the way that the TRPL spectrum deviates from the monoexponential behavior that is expected when monomolecular recombination is the only relevant mechanism at play.32 With this idea in mind, we perform fits of the simulated TRPL spectra with a biexponential function
6
where r is the intermolecular distance, tM is the radiative exciton lifetime, and RF is the so-called Förster radius. Such a radius is defined as the distance at which the probability of spontaneous recombination equals the probability of transfer. The Förster radius is calculated as13
RF6 =
1
P(r ) =
As mentioned previously, singlet exciton transfer is due to the Förster mechanism. The rate with which this kind of transfer occurs is given by31
(2)
where c is the speed of light, κ is an orientation factor, ω is an angular frequency, ID is the donor’s differential emission rate, αA is the acceptor’s absorption cross section, and the integral quantifies the spectral overlap. In this sense, the two main parameters that determine exciton diffusion are the Förster radius and the exciton lifetime. KMC simulations employ eq 1 to simulate exciton diffusion in a lattice. The simulations presented here were performed in a twodimensional 100 × 100 square lattice with 1 nm spacing between adjacent sites, in agreement with previous studies in both exciton and charge transport.26,32−34Importantly, for such intersite distances, we may neglect contributions to exciton diffusion from the Dexter mechanism,35 which relies on the overlap of donor and acceptor wave functions and decays exponentially with distance. In fact, it has been estimated that the Förster mechanism becomes dominant for distances of 5 Å.36 Furthermore, it mitigates the observed overestimation of transfer rates calculated with eq 1 at short distances.37 The two-dimensional treatment employed here is justified by the fact that exciton diffusion can be highly anisotropic14,38,39 because of the orientation factor present in eq 2 and the particular morphology of each system. This anisotropy may effectively result in one- or twodimensional exciton diffusion, as already observed experimentally.40−42 Furthermore, the lattice size is chosen as to be large enough to prevent border effects. Three parameters serve as input to the simulations, the Förster radius, exciton lifetime, and initial exciton concentration. In the simulations, the Förster radius is kept constant. It is known that disorder plays an important role in exciton diffusion in organic materials.43 Its effects manifest themselves in diffusion constants that decrease over time as excitons hop toward lower-energy sites, eventually becoming trapped in local minima of the energetic landscape. This is consistent with a scenario in which the Förster radius decreases over time. However, the extent of the disorder depends on several factors such as morphology and temperature, 2
I(t ) = AM exp(− t /tM) + ABexp(− t /τ )
(4)
where AM and AB are the amplitudes corresponding to monomolecular (natural decay) and bimolecular (annihilation) mechanisms, respectively. As mentioned above, the parameter tM represents the exciton radiative lifetime, which serves as one of the simulation’s inputs. The characteristic time of the bimolecular (annihilation) phenomenon is represented by τ and is obtained from the fits to the data. This procedure allows us to quantify the relative importance of both phenomena. B
DOI: 10.1021/acs.chemmater.9b01281 Chem. Mater. XXXX, XXX, XXX−XXX
Article
Chemistry of Materials
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RESULTS AND DISCUSSION The general feature of TRPL spectra obtained in concentration regimes for which exciton−exciton annihilation is relevant may be seen in Figure 1. The histogram corresponds to the
contributions exceed bimolecular ones. However, because the bimolecular term increases superlinearly, it eventually matches the monomolecular term. In this sense, we consider that from this critical concentration onward exciton−exciton annihilation becomes the dominant process. At this point, it is worth noting how the Förster radius and exciton lifetime affect the position of this critical concentration. As one can see from eq 2, these two quantities are not completely independent. However, for the purposes of our discussion, the Förster radius is the more relevant parameter. This conclusion is drawn by observing that the critical concentration remains the same for simulations performed with equal Förster radii but different radiative lifetimes. This is shown in the inset of Figure 2, where the relative amplitudes from both mono- and bimolecular contributions are plotted for simulations with a 25 Å radius and lifetimes ranging from 500 to 5000 ps. The reason for this behavior is that ultimately, the amount of exciton annihilation at a given initial concentration depends on only the exciton diffusion length, which in its turn depends on the exciton lifetime only through its relationship with the Förster radius.44 The influence of the Förster radius in the location of the critical concentration is demonstrated in Figure 3. It shows the evolution of the relative amplitudes as a function of exciton density. These amplitudes are calculated by averaging the results from simulations with different radiative lifetimes. The error bars represent the standard deviation of the distribution of results. It can be noted that these error bars are larger at lower densities, which stems from the fact that under these conditions the bimolecular contribution is very small, making fits to biexponential functions less reliable. More importantly, Figure 3 shows that when the Förster radius increases from 24 to 30 Å, the critical concentration decreases from roughly 4% to 1%. This goes to show that the concept of low exciton density strongly depends on the material’s Förster radius. The sixth-power dependence on the exciton hopping rates is responsible for the dramatic changes. Gathering results for the different Förster radii, we are able to map this property to the critical concentration. Figure 4 shows how these two properties are related. In the 20−40 Å radius range, critical concentrations drop from 8% to
