How do Triplets and Charges Move in Disordered ... - ACS Publications

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How do Triplets and Charges Move in Disordered Organic Semiconductors? A Monte Carlo Study Comprising the Equilibrium and Nonequilibrium Regime Sebastian T. Hoffmann,†,‡ Stavros Athanasopoulos,†,‡ David Beljonne,‡ Heinz Bas̈ sler,† and Anna Köhler*,† †

Experimental Physics II and Bayreuth Institute of Macromolecular Science (BIMF), Department of Physics, University of Bayreuth, Bayreuth 95440, Germany ‡ Laboratory of Chemistry of Novel Materials, University of Mons, Place du Parc 20, B-7000 Mons, Belgium ABSTRACT: We have investigated how electronic excitations that couple via short-range interaction, i.e., triplet excitations and charge carriers, move in a disordered organic semiconductor. In this systematic study, we paid special emphasis to the transition from quasi-equilibrium to nonequilibrium transport as the temperature is lowered from 300 to 10 K. As a method, we used Monte Carlo simulations employing both Marcus as well as Miller−Abrahams (MA) transition rates. The simulation parameters are the degree of static energetic disorder, the geometric reorganization energy, and the degree of electronic coupling among the hopping sites. In the case of conjugated polymers, the effects of intrachain versus interchain transport are taken into account. In the simulations, we monitor the spectral relaxation of excitations as well as their diffusivity. We find that, below a disorder controlled transition temperature, transport becomes kinetically frustrated and, concomitantly, dispersive. In this temperature regime, transport is controlled by single phonon tunneling, tractable in terms of MA rates, while in the high temperature regime multiphonon hopping, described by Marcus rates, prevails. The results also provide a quantitative assessment of dispersive excitation transport within the intermediate temperature regime in which no analytic theory is available so far. Quantitative agreement between simulation and previous experiments allows one to extract system parameters such as the minimum hopping time and to delineate the parameter range in which Marcus and MA rates should be used in transport studies.

I. INTRODUCTION The operation of organic semiconductor devices such as organic light-emitting diodes, solar cells, and transistors is based on the transport of charges, spin-singlet and spin-triplet excited states. While the motion of singlet states proceeds by a longrange mechanism based on dipole coupling, excitations such as charges and triplet states are transported by short-range processes, for example, by wave function overlap, exchange interaction and related mechanisms.1−6 A charged state and a spin-triplet excited state differ regarding their extent and, naturally, their charge. As a result, parameters such as the associated energetic disorder and geometric reorganization energy take different values.7 However, both excitations are transferred by a short-range mechanism that involves wave function overlap and, for triplets, also exchange interaction.2−4 To optimize the efficiency of organic semiconductor devices, the transport of charges and triplets needs to be well understood and controlled. For example, the performance of field-effect transistors is largely dependent on the charge carrier mobility.8−11 Similarly, charge separation and extraction in solar cells requires suitable carrier mobilities,12−17 and further, the role of a balanced charge transport for efficient light-emitting diodes has long been recognized.18 The diffusion of triplet states can control the efficiency of devices in a similar way. For © 2012 American Chemical Society

example, the roll-off of LED efficiency observed at high current densities is attributed to triplet-charge annihilation (TCA) and triplet−triplet annihilation (TTA), and thus being ultimately governed by triplet diffusion.19−29 Similarly, devices based on TTA-induced delayed fluorescence, sometimes referred to as nonresonant up-conversion,27,30−32 rely on triplet diffusion. A thorough quantitative understanding of the material parameters controlling the diffusion rate of triplets and charges is therefore needed for a systematic development of such devices. The transport of charges and triplets may take place in thermal equilibrium or under nonequilibrium conditions. While the former is reasonably well understood,6,33 a general quantitative understanding of nonequilibrium transport has, to the best of our knowledge, not yet been developed. Experimentally, this nonequilibrium regime is frequently encountered in common measurements. For example, it is well-known that the transport of charges becomes dispersive below room temperature, indicating that it is no longer in thermal equilibrium. 34−36 This can be observed when measuring the mobility of charge carriers that are generated Received: May 24, 2012 Revised: July 5, 2012 Published: July 5, 2012 16371

dx.doi.org/10.1021/jp305062p | J. Phys. Chem. C 2012, 116, 16371−16383

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energy of an excitation after spectral diffusion and concerning the diffusivity of excitations. These results are parametrized in energetic disorder, geometric reorganization energy, anisotropy of electronic coupling, and the ratio between hopping time and excitation lifetime. Our simulations agree well (i) with the analytical model by Movaghar in the available limiting cases of zero temperature and thermal equilibrium44 and (ii) with the experimental data on frustrated spectral diffusion that exist for the case of triplet excitons.39 We discuss the implications arising for the description of charge transport and triplet exciton transport. This paper is structured as follows. Section II briefly reviews the exisiting experimental data against which the Monte Carlo results will be compared. Details of the simulation technique are given in section III. The results pertaining to the spectral relaxation are presented and discussed in section IV, while section V comprises results and discussion on the diffusivity. A more general and summarizing discussion in section VI concludes the paper.

randomly in the density of states, as is typically done in time-offlight (TOF) experiments.6,37,38 Correct derivations of the activation energy based on such time-of-flight measurements are therefore inherently difficult. Similarly, when measuring the 0−0 position of phosphorescence as a function of temperature, one observes first a red shift upon lowering the temperature.39,40 This spectral relaxation is in full agreement with the theory of random walks within a static Gaussian density of states (DOS) distribution under conditions of quasi-equilibrium.41 However, below a certain temperature, the course of spectral relaxation is reversed. After passing through a minimum, the normalized spectral relaxation energy Δε/σ increases upon lowering the temperature further.39,42 This blue shift with reducing temperature has been taken as evidence that the spectral relaxation is progressively frozen out, i.e., it becomes frustrated, implying that triplet diffusion proceeds out of thermal equilibrium.39,43,44 Both phosphorescence and time-of-flight measurements on disordered organic semiconductors are well-established techniques. Thus, experimentally, phenomena associated with a nonequilibrium transport of excitation can be observed with ease. The critical temperature Tc below which transport proceeds out of thermal equilibrium depends on the ratio between the lifetime of the excitation (or transit time for charges) and its minimum hopping time. Experimentally, one observes the Tc value to occur at about 100 K for triplets,39,45 while, for charges in a TOF experiment, this is at about 250 K.46−48 Since the disorder parameters are roughly σ ∼ 30−50 meV for triplets28 and around σ ∼ 100 meV for charges, this implies kTc ∼ σ/3 for triplets, while this is at approximately kTc ∼ σ/4 for charges. While organic semiconductor devices usually operate above these temperatures, experiments around and below these values are not unusual for scientific investigations. For the interpretation of measurements associated with charge or triplet transfer near or below the critical temperature, a quantitative understanding of the transport of an excitation outside thermal equilibrium is needed. This pertains in particular with respect to material dependent parameters such as the energetic disorder, geometric relaxation energy, size and anisotropy of the electronic coupling, and lifetime of the excitation. From a theoretical perspective, the quantitative description of nonequilibrium transport at arbitrary temperature is not trivial. Analytical equations that describe the transport of excitations in the nonequilibrium regime have, in principle, been developed for the case of a Förster-type dipole coupling49,50 as well as for exchange-based coupling.44 However, the nonequilibrium descriptions are restricted to the singular case of zero temperature when any thermally activated jump of an excitation is frozen out completely. To the best of our knowledge, to date, there is no general theory available suited to portray the transport of excitations in amorphous organic semiconductors at arbitrary temperature outside thermal equilibrium. We have therefore adopted the approach of using a “computer experiment” by employing Monte Carlo simulations. In this paper, we develop a clear and unified phenomenological description for the diffusion of excitations that couple by a short-range mechanism. Using Monte Carlo simulations, we are able to cover the entire temperature range between 300 and 10 K. In particular, we use the simulations to elucidate the transport mechanism at low temperatures, where the excitations are not in thermal equilibrium so that kinetic frustration occurs. We obtain values for the nonequilibrium regime concerning the

II. EXPERIMENTAL BACKGROUND In order to evaluate the Monte Carlo simulations, we compare them against experimental data on the diffusion of triplet excitons. Triplets are neutral excitations that are amenable to optical spectroscopy. Quantities such as the energetic disorder of the DOS or the geometric relaxation energy associated with the excitation can easily be derived from the optical spectra, and they can be related to the temperature dependent dynamics of triplet excitons.28,33,42 This is in contrast to charges, where such parameters can only be derived indirectly, for example, by transport measurements. This arises from the fact that there is no allowed optical transition from the neutral state to the ionized state of a charge transporting molecular entity. A study of triplet diffusion allows us to draw on models originally developed for charge transport, and conversely, the insight gained by the optical studies available from triplets can again be applied to charges, subject to considering a suitable parameter range. As the results of the simulation will be compared against available data on triplet diffusion, it is appropriate to briefly summarize the pertinent features. Figure 1 shows the spectral diffusion of singlet and triplet excitons as a function of temperature for a few representative compounds, that is, two polymers with different degrees of energetic disorder and a related oligomer. The shift in the 0−0 peak, Δε, is indicated for fluorescence and phosphorescence. For ease of comparison, it is normalized to the energetic disorder estimated from the Gaussian width σ of the highenergy tail of the emission spectra. Upon lowering the temperature, the fluorescence (filled symbols) follows the Δε/σ = −σ/kT dependence predicted for thermal equilibrium until it saturates due to the limited lifetime of the excitation.39 In contrast, the phosphorescence (open symbols) exhibits a minimum at the temperature where the fluorescence saturates and shifts to the blue for lower temperatures. This has been interpreted as a signature that transport is no longer in thermal equilibrium.39 The critical temperature where Δε/σ deviates from −σ/kT for fluorescence and phosphorescence, and thus the depth of the minimum reached varies between the compounds, even though the energy shift and the temperature are normalized to disorder. A second and complementary data set is provided by the temperature dependence of the triplet diffusion coefficient (diffusivity). Experimentally, one can derive the triplet 16372



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III. SIMULATION METHODS The dynamics of triplet exciton diffusion has been studied with a stochastic random walk model. In detail, to tackle the problem of spectral diffusion, a kinetic Monte Carlo method was employed that allows one to monitor the motion of the quasiparticles as hopping events. The materials under study are disordered in nature; therefore, triplet exciton transport is expected to take place in the hopping regime. This is also supported by recent experiments.7 The excitations move incoherently between lattice sites that represent conjugated molecular segments of a few repeat units. A rectangular lattice of extended polymer chains with an interchain separation of 1.5 nm has been used with each lattice point corresponding to a conjugated segment and an intrachain lattice constant a = 1.68 nm. The intrachain lattice constant has been chosen to accommodate a fluorene dimer at each lattice point along the chain axis. Independent trajectories start by randomly placing an excitation on a segment. At each Monte Carlo step, the triplet exciton state at site i can either hop to a neighboring site j or decay to the ground state. To allow for infrequent but nevertheless possible hopping events, we expand the hopping space to the 18 nearest neighbor sites. Each hopping site is assigned an energy drawn from a Gaussian distribution with a variance σ2. In our simulation, disorder is therefore solely energetic. For the jumps between sites, a dwell time is calculated from an exponential distribution according to τij = −

1 ln X kij

(1)

with X being a random number from a box distribution and kij being the transfer rate. Similarly to eq 1, a stochastic lifetime is calculated as τ = −ln X/τex, where τex is the triplet exciton lifetime. The event requiring the smallest time is selected and executed. This may be a jump to a particular neighbor or a decay process. The exciton migration terminates when a decay process occurs. Monitoring the spatial positions and energies of the initial and final site the exciton has visited, we can deduct values for the spectral shift Δε and the effective diffusion coefficient D = Δx2/t. These values are meaningful only when averaging over a few thousand trajectories. To describe the transport of excitations that couple by exchange interaction, a Marcus rate as well as a Miller− Abrahams rate has been used.5−7,40,54−58 In the current work, we use both approaches in order to assess which transport mechanism is most appropriate for a given experimental situation that may be characterized by parameters including temperature, energetic disorder, and geometric reorganization energy. (i) A single phonon assisted tunneling process between sites of energy Ei and Ej is considered when adopting a Miller− Abrahams type of rate55

Figure 1. (a) The spectral diffusion for fluorescence (filled symbols) and phosphorescence (open symbols) for the polymers DOOPPP (blue squares), PIF (black circles), and for the PF2/6 trimer (green triangles). The ordinate shows the relative energy shift Δε between the 0−0 emission peaks and the center of the DOS (as derived from the 0−0 transition in absorption), normalized to the energetic disorder σ and plotted against the disorder normalized temperature scale kT/σ. For reference, the solid red line indicates the (Δε/σ) = −(σ/kT) curve expected for thermal equilibrium. The data is replotted from ref 39. (b) The triplet diffusion coefficient D as a function of inverse temperature for DOOPPP, PIF, and the PF2/6 trimer determined by phosphorescence lifetime measurements. Data replotted from ref 45. DOOPPP is a poly(p-phenylene) derivative, PIF is poly(indenofluorene), and PF2/6 is a poly(fluorene). The dotted line is a guide to the eye.

quenching rate kq from the reduction of phosphorescence lifetime τ upon warming up the sample from 10 K to ambient temperature according to kq(T) = τ(T)−1−τ(0 K)−1 .45,51 Since this reduction is caused by the diffusion of triplets toward quenching sites, it is a direct measure of how the triplet diffusion rate kt changes with temperature. Quantitatively, kq = ckt, where c is the relative concentration of quenching sites. Consequently, this lifetime reduction is also proportional to the triplet diffusion coefficient (diffusivity) D, since D ∝ kt. For example, in an isotropic three-dimensional crystal, D = a2kt/6, where a is the lattice constant.52 For reference, the temperature dependent diffusivity is also displayed in Figure 1. These experiments demonstrate that, while D(T) is temperature activated at moderate to high temperatures, below a certain critical temperature Tc, D(T) is only very weakly temperature dependent.45,53

⎧ v e−2γR ij Ei ≥ Ej ⎪0 kij = ⎨ ⎪ v0e−2γR ije−(Ej − Ei)/ kT Ei < Ej ⎩

(2)

where ν0 is the attempt to hop frequency, γ is the wave function localization constant, Rij the hopping distance, and kT the thermal energy.7 (ii) A multiphonon hopping process is represented by a Marcus type of rate 16373


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Jij 2 ℏ

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⎧ (ΔG + 4Ea) ⎫ π ⎬ exp⎨− 4EakT 16EakT ⎭ ⎩

(3)

where Jij is the electronic coupling that decays exponentially with the distance Jij = J0e−γRij, Ea the activation energy, and ΔG the free energy difference between sites i and j.54 To a first approximation, we have estimated the intrachain coupling Jij between two fluorene dimers at the TD-DFT level as half the energy splitting of the two lowest triplet exciton states of the tetramer at the optimized ground state geometry, with a torsional angle between the dimers of ∼39°, using the B3LYP exchange-correlation energy functional with a 6-311G* basis set. This provides a Jij value of ∼95 meV, while the activation energy Ea is taken from experiment.45 To allow for directional diffusion, we distinguish between intra- and interchain motion by assigning different localization constants γ∥ and γ⊥ corresponding to hops along the chain and in between chains, respectively. The ratio γ∥/γ⊥ controls the directionality of the interactions. It can take a maximum value of unity when considering isotropic diffusion, for example, in disordered oligomeric films. The absolute value of γ is an inverse measure of the delocalization of the excitation wave function, i.e., a small γ implies a large delocalization. For triplets, a rough estimate for the attempt to hop frequency ν0 can be obtained by equating the Miller−Abrahams rate equation (2) with the Marcus rate equation (3) between two isoenergetic sites at a critical temperature of 100 K. As we shall detail in sections V and VI of this paper, the two rates can be obtained when developping a Holstein Hamiltonian in a high and a low temperature limit. Equations 2 and 3 should thus give roughly similar rates at an intermediate critical temperature where the transition from one regime to the other takes place. While the critical temperature for this transition depends on a number of parameters, as discussed in sections V and VI, 100 K turns out to be a typical value experimentally found for triplets. Using γ⊥ = γ∥ = 2 nm−1 in eqs 2 and 3, it follows that ν0 = (π/4EakT)1/2(J02/ℏ)e−(Ea/kT). This results to a ν0 ∼ 400 ps−1 and a minimum hopping time between intrachain nearest neighbors t0 = 1/(ν0 exp(−2γa)) ∼ 2 ps. In the following results section, we will systematically explore the influence of each of the above physical parameters to the spectral diffusion.

IV. SPECTRAL RELAXATION (i). Results. In Figure 2a, we show the simulated spectral relaxation energy Δε of triplet excitations normalized to σ as a function of the normalized temperature kT/σ within a simulation time regime of 0.5 × 104 t0 and variable sigma using Marcus rates (eq 3). t0 is taken to be 2 ps. The spectral relaxation energy Δε is the energy difference between the center of the DOS at ε = 0 and the energy ε obtained at the end of the exciton’s lifetime (in experiment) or simulation time (in simulation). The parameters were the activation energy Ea = λ/ 4 = 60 meV, where λ is the reorganization energy and the disorder parameter σ varies from 10 to 100 meV. The concept of random walks within a Gaussian DOS distribution g(ε) = (2πσ2)−1/2 exp(−ε2/2σ2) under the condition that quasiequilibrium can be established predicts40,41 ∞

⟨lim t →∞ ε⟩ =

∫−∞ εg (ε) exp(−ε/kT ) dε ∞

∫−∞ g (ε) exp(−ε/kT ) dε

=−

σ2 kT

Figure 2. Simulated spectral relaxation Δε normalized to the disorder σ as function of the normalized temperature kT/σ for coupling parameters γ∥ = 2 nm−1 and γ⊥ = 4 nm−1, at a simulation time of t = 104t0. (a) Δε/σ obtained using a Marcus rate with variable σ with values as indicated in the figure and with a fixed activation energy Ea = 60 meV; (b) the same but for fixed σ = 35 meV and variable Ea; (c) Δε/σ obtained using a Miller−Abrahams rate with variable σ values. The blue solid line indicates the spectral relaxation expected for thermal equilibrium according to (Δε/σ) = −(σ/kT). The black dotted lines guide the eye.

independent of the mode of coupling, i.e., Δε/σ = −σ/kT. Figure 2a demonstrates that, at higher temperatures, that is, kT/σ ≳ 0.4, the simulated spectral diffusion follows the predicted Δε/σ = −σ/kT curve, independent of energetic disorder, indicating that scaling is observed. However, at lower temperatures, Δε/σ passes through a minimum that depends on the degree of disorder, indicating that in this case there is no scaling with regard to σ. Moreover, for low temperatures, the simulation predicts a drastic reduction of spectral diffusion to

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the extent of no relaxation near 0 K. The weaker the disorder is, the higher is the temperature at which Δε/σ approaches 0. For 10 K, a finite nonzero value of Δε/σ occurs only if σ is increased to about >70 meV. Increasing Ea also impedes spectral relaxation, as evidenced by Figure 2b. Note that vanishing spectral relaxation would imply that all excitations generated at very low temperatures would emit resonant photoluminescence. This is in contradiction to experiment.39,59,60 However, the results change fundamentally if the Marcus rate is replaced by the Miller−Abrahams (MA) rate. For kT/σ > 0.6, Δε/σ follows the same Δε/σ = −σ/kT law, indicating that the thermal equilibrium value eventually reached is independent of the hopping rate needed to obtain it (Figure 2c). For kT/σ ≈ 0.3, a minimum of Δε/σ develops whose depth is invariant with σ and a finite ordinate intercept limT→0 Δε/σ is approached in the T → 0 limit. In this case, scaling regarding the degree of disorder is indeed observed. To confirm that this is not an accidental result due to the short simulation time used, and in order to check how the spectral diffusion evolves with time, we simulated Δε/σ as a function of kT/σ within the time regime 102 t0 to 107 t0 employing both Marcus and MA hopping rates for fixed parameters σ = 35 meV and Ea = 60 meV (Figure 3). In both cases, the minimum of Δε/σ deepens with simulation time, and it shifts to lower temperatures. However, in the Marcus case, limT→0 (Δε/σ) = 0 is preserved, whereas most importantly, in the case of MA rates, limT→0 (Δε/σ) increases with time. Spectral relaxation depends on the anisotropy of the electronic coupling as expressed through the ratio of the inverse wave function localization length along the chain and orthogonal to the chain, γ∥/γ⊥, as demonstrated by Figure 4. The general feature observed applies to Marcus rates and MA rates alike. For example, when MA rates are used, the spectral relaxation minimum near kT/σ = 0.2−0.3 deepens when going from a virtually one-dimensional (1D) system, characterized by a ratio of the strength of intrachain versus interchain coupling γ∥:γ⊥ = 1:10, to a isotropic three-dimensional (3D) system. This is straightforward to understand. For γ∥:γ⊥ = 1:10, hops of an excitation can only take place along a polymer chain. The presence of two sites with a certain energy above the occupied site, i.e., one somewhere before and one somewhere behind the occupied site, can already impede further diffusion. In the isotropic case, there are more nearest neighbors available, thus making a complete blockade through an enclosure by higher energy neighbors less probable.61 As a result, spectral relaxation can proceed further. In passing, we note that the absolute values of γ and of hopping time are complementary. With a lower γ, the same spectral relaxation is reached in a shorter hopping time, since the absolute value of the hopping rate is higher. This allows for scaling of jump time and localization length. The effect of simulation time, or exciton lifetime, on the spectral diffusion is investigated in Figure 5, which illustrates how Δε/σ evolves with normalized time, ln(t/t0). We study an isotropic hopping system (γ∥ = γ⊥) with different values of the disorder-normalized temperature. The Δε/σ values are taken from the temperature on where the relaxation minimum is obtained (and where relaxation proceeds close to thermal equilibrium, approximately at kT/σ = 0.25) down to very low temperatures in steps of kT/σ = 0.05. In the figure, the limiting cases of the time dependence for the relaxation minimum (kT/ σ)Min and for 0 K (kT/σ) = 0 are indicated by full circles and full squares, respectively. We see that at very short times, roughly for ln(t/t0) < 3, the relaxation is independent of

Figure 3. Simulations of Δε/σ performed at variable simulation time t/t0 between 102 and 107 with a constant coupling between and along the chain (γ∥ = 2 nm−1 and γ⊥ = 4 nm−1), an activation energy Ea = 60 meV, and energetic disorder σ = 35 meV (a) for simulations with a Marcus rate and (b) for simulations with a Miller−Abrahams rate. The blue solid line indicates the spectral relaxation expected for thermal equilibrium according to (Δε/σ) = −(σ/kT). The black dotted lines guide the eye.

temperature. This is a plausible result because an excitation initially generated within the DOS distribution at arbitrary energy will, on average, find a neighboring site to which it can jump to without thermal activation regardless of temperature and follow an exponential decay pattern. As time progresses, the relaxation slows down and deviates from the initial (exponential) decay law. When moving from an isotropic 3D system to a system that is anisotropic, we find that the pattern for the relaxation energy Δε/σ as a function of ln(t/t0) experiences a parallel shift along the abscissa scale. This is illustrated in Figure 6. To keep the presentation clear, only the data pertaining to the limiting cases of (kT/σ) = 0 and (kT/ σ)Min are indicated. The solid symbols refer to the data obtained for an isotropic system with γ = 2 nm−1. When the anisotropy of the system is raised by choosing γ∥:γ⊥ = 2:4, the data indicated by the open symbols are obtained. Inspection of the two data sets shows that shifting the data pertaining to the isotropic system by a factor of 150 on the logarithmic time scale would result in values very close to the anisotropic case represented by γ∥:γ⊥ = 2:4. In other words, increasing the anistropy of the system does not fundamentally alter the spectral relaxation process, yet it slows it down 16375



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Figure 5. The simulated normalized spectral relaxation in an isotropic hopping system plotted versus the ln of the normalized simulation time t/t0 for a Miller−Abrahams rate. The parameter varied is the normalized temperature kT/σ. The values for Δε/σ are obtained for the limit of kT/σ = 0 (full black squares), kT/σ in the range 0.05−0.20 (open triangles), and kT/σ corresponding to the value where the spectral relaxation takes its minimum (full red circles). The thick blue upper line indicates the values expected according to the theory of ref 44 for T → 0. The other lines are only guides to the eye.

Figure 4. Effect of the anisotropy of coupling on the temperature dependence of spectral relaxation simulated by varying the intermolecular coupling between γ⊥ = 2 nm−1 and γ⊥ = 20 nm−1 at a constant intramolecular coupling of γ∥ = 2 nm−1 (a) using the Marcus rate and (b) using the Miller−Abrahams rate. The activation energy is Ea = 60 meV, the energetic disorder σ = 35 meV, and the simulation time t = 104t0. The blue solid line indicates the spectral relaxation expected for thermal equilibrium according to (Δε/σ) = −(σ/kT). The black dotted lines guide the eye.

significantly. In a way, reducing the dimensionality of the coupling reduces the effective 3D coupling. (ii). Discussion. The key observation that can be made from Figures 2−4 is that in the case of thermal equilibrium, i.e., above a critical temperature Tc, energetic relaxation follows the predicted Δε/σ = −σ/kT dependence as observed experimentally. This equilibrium value is reproduced by both Marcus as well as MA jump rates. This is consistent with the notion that the rate by which a final equilibrium value is reached should have no influence on the equilibrium value itself. The situation is different for the temperature regime below Tc, where transport proceeds out of thermal equilibrium. The experimentally observed finite energetic relaxation in the T → 0 limit cannot be reproduced by a Marcus rate for parameters that are consistent with experiments. Note that for the polymers the disorder parameter σ is typically ≲50 meV in the case of triplet excitons. For such values of σ, Marcus jump rates return values close to zero for limT→0 Δε/σ. This is in clear disagreement with experiment, as illustrated exemplary in Figure 1 and discussed in detail in ref 44. Figures 3 and 4 demonstrate further that the failure of Marcus jump rates is not related to the value adopted to γ or for the hopping time, nor to the anisotropy of transport. Rather, the simulation results imply

Figure 6. Illustration of the temporal shift of the pattern of spectral relaxation when going from an isotropic hopping system (solid symbols) with coupling parameter γ⊥ = γ∥ = 2 nm−1 to an anisotropic system (open symbols) characterized by γ∥ = 2 nm−1 and γ⊥ = 4 nm−1. Only the two limiting cases pertaining to the temperature of the relaxation minimum (circles) and to T → 0 (squares) are shown. The dotted curves are guides to the eye, with the curves pertaining to the anisotropic system obtained by horizontally shifting the curves of the isotropic system.

that the transport mode represented by Marcus jump rates, that is, thermally activated multiphonon hopping, is not suited to describe the experimental situation of short-range excitation transport in the nonequilibrium regime below Tc. In contrast, simulations based on a MA rate yield a finite nonzero energetic relaxation for T → 0. The numerical values depend on the parameters chosen such as simulation time t and inverse wave function localization parameter γ, yet they are consistent with experimental observation.39 In contrast to the classical Marcus rate, the MA rate considers a tunneling process between sites. It seems that a correct description for the nonequilibrium range must consider tunneling events. 16376



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To demonstrate the suitability of the MA rate, Figure 7 compares experimental data in red open symbols with data obtained by a MA based simulation in black solid symbols for three polymers and an oligomer. For the polymers, electronic coupling has been taken to be anisotropic, while isotropic coupling is presumed for the dimer. By adjusting γ and simulation time, the experimental data can be reproduced for polymers with low to high disorder. This is not the case when a Marcus rate is employed. From the consideration of the low temperature spectral relaxation, we conclude that triplet transport below Tc must be treated within the framework of the MA model. Spectral diffusion from initial excitation energy to the thermal equilibrium value is a time-dependent process.62 At high temperatures, hopping rates are fast and spectral diffusion is completed within the exciton lifetime, so this time dependence is not manifested. In contrast, at low temperatures, hopping rates become comparable to the exciton lifetime, so that the ultimate value reached before decay depends on the ratio τex/t0 between exciton lifetime and hopping time. Upon lowering the temperature, the increase in hopping time needed implies the energy shift first deviates from the equilibrium curve given by Δε/σ = −σ/kT and then exhibits a minimum and subsequently increases. We note that the temperature at which the transport falls out of thermal equilibrium is not given by a sharp point, but this process is gradually frozen out in the temperature range between the deviation from the equilibrium line and the minimum. For convenience, we use the minimum in the temperature dependence of spectral diffusion as a measure for the critical temperature Tc. We now consider the time dependence manifested in Figure 5 in more detail. In the T → 0 limit, any subsequent jumps that would require thermal activation are frozen out. The subsequent relaxation therefore has to proceed via lower energy sites. Such sites are progressively further away and consequently require more time for the jump to occur. This process has already been described by Movaghar et al.’s theory for frustrated relaxation in the T = 0 limit.44 It predicts that in the long time limit Δε/σ = [3 ln tυ0]1/2, where υ0 is the attempt-to-jump frequency. We find that the present simulations for T → 0 shown in Figure 5 are in quantitative agreement with Movaghar’s theory.44 This is borne out by the sublinear Δε/σ dependence if plotted on a ln t/t0 scale. As the temperature increases, spectral relaxation is progressively assisted by weakly activated jumps that allow an excitation to overcome a shallow energy barrier that would otherwise prevent further relaxation. A signature of this effect is the increasing negative slope of Δε/σ as a function of ln t/t0 when the temperature is increased. This intermediate transport regime is so far not amenable to analytic theory. As the nearequilibrium value of (kT/σ) = 0.25 is reached where the spectral relaxation shows the minimum, Δε/σ asymptotically approaches the ln t/t0 law that is predicted by the theory of random hopping in the quasi-equilibrium limit.41 The slope is in quantitative agreement with this theory. It predicts that the mean energy relaxes by 1 σ per 3 decades of time. It is satisfying to see that, for the two limiting cases, the T → 0 limit and the quasi-equilibrium limit, the time dependence of the spectral relaxation simulated by MA rates is in quantitative agreement with results obtained by an analytical theory. We note that, for the simulations and the analytical theory, the nature of the excitation (triplet exciton or charge carrier) is not relevant as long as the short range of the interaction is taken into account.

Figure 7. Comparison between experimental (red open symbols) and simulated (black solid symbols) spectral relaxation for a series of compounds using the Miller−Abrahams rate for a simulation time of t = 104t0. (a) Pt-polymer (γ∥ = 2 nm−1 and γ⊥ = 4 nm−1), (b) PF2/6 (γ∥ 16377



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Table 1. Summary of the Hopping Times th Deduced for Several Organic Compounds When Considering the Exciton Lifetime tex

Figure 7. continued

= 1 nm−1 and γ⊥ = 2 nm−1), (c) DOOPPP (γ∥ = 0.5 nm−1 and γ⊥ = 0.5 nm−1), (d) the dimer of PF2/6 (γ∥ = 4 nm−1 and γ⊥ = 4 nm−1), with the chemical structures shown in the figure. The blue solid line indicates the spectral relaxation expected for thermal equilibrium according to (Δε/σ) = −(σ/kT). By comparing simulation with experiment, we can extract absolute numbers on the hopping time and the strength of electronic intersite coupling. To do this, we associate the simulation time with the lifetime τex of the triplet excitations and we link the reference jump time t0, that has been calculated quantum mechanically, with the minimum jump time th, i.e., t/ t0 = τex/th. The th is the jump time that would be obtained for a disorder-free system. The experimental results reported in ref 45 yield values of limT→0 Δε/σ = 2.0 and (Δε/σ)Min = 2.9 for polyfluorene (PF2/6). The experimentally found disorder for triplets is 39 ± 4 meV,45 and the measured triplet lifetime at low temperatures for PF2/6 is about 1 s.63,64 In order to relate this to simulations, we have to make a choice for the coupling parameters. Since in conjugated polymer on-chain and off-chain coupling must differ, we consider the parameter set γ∥ = 2 nm−1 and γ⊥ = 4 nm−1. On the basis of previous experience with charge carriers, this choice seems realistic.65 The time dependence of a MA based spectral diffusion with σ = 35 meV and the electronic coupling γ∥ = 2nm−1 and γ⊥ = 4nm−1 is reported in Figure 3b. We see that values of limT→0 Δε/σ = 2.0 and (Δε/σ)Min = 2.9 are obtained for a simulation time t = 107t0. This corresponds to the experimentally found values for PF2/6 where the lifetime is 1 s. Using t/t0 = τ/th, we arrive at a minimum jump time of about th = 100 ns. The same result can be obtained from Figure 5 by reading the time needed to get the experimental minimum value (Δε/σ)Min from the ordinate when using the red curve with the open symbols for the anisotropic case as a reference. The value obtained is in favorable agreement with data inferred from triplet−triplet annihilation in PF2/6 that yield th = 70 ± 20 ns42 and supports the choice of the coupling parameters γ∥ and γ⊥. For DOOPPP, limT→0 Δε/σ = 2.5 and (Δε/σ)Min = 4.0 translate into a minimum hopping time of about 50 ps using the same coupling parameters, i.e., γ∥ = 2 nm−1 and γ⊥ = 4 nm−1 and extrapolating Figure 3 or 5. For PIF, t ≃ 30 ns is obtained. In the case of the Pt-polymer, there is, within the limit of experimental uncertainty, no minimum of Δε/σ. However, based upon the experimental value of limT→0 Δε/σ = −1.8 and adopting the above values for the above coupling parameters, one would expect an only very shallow relaxation minimum, limT→0 Δε/σ − (Δε/σ)Min of only 0.2 that may be obscured by experimental error. The measured mean value of Δε/σ translates into th = 4 ns, adopting the measured low temperature triplet lifetime of 60 μs. Since in the dimer of fluorene there is no on-chain motion, we assume that the coupling parameter is that of interchain jumps in the polymers, i.e., γ∥ = γ⊥ = 4 nm−1. Comparison with Figure 7d yields a minimum jump time of 400 μs. For convenience of reference, the hopping times are summarized in Table 1. Between different compounds, the minimum hopping times vary significantly and span over 6 orders of magnitude. We expect the minimum hopping times to depend on factors such as the polaronic binding energy, disorder, and concomitantly excitation delocalization, as well as on the strength of the electronic coupling. This issue deserves

material

th

tex

DOOPPP Pt-polymer PIF PF2/6 PF2/6 dimer

50 ps 4 ns 30 ns 100 ns 400 μs

1s 60 μs 1s 1s 1s

due consideration possibly from a quantum chemical perspective.

V. DIFFUSIVITY (i). Results. To get insight into the mode of diffusion in the temperature regime above Tc, we need to consider a ratedependent process such as the value of the diffusion coefficient (diffusivity). Complementary to the spectral relaxation, we thus simulated the diffusion of the triplet excitations. The diffusivity has been inferred from the total squared three-dimensional displacement of an excitation divided by the simulation time, implying that the computationally determined diffusion coefficient is D = (Δx)2/t. Note that this procedure yields the time-averaged diffusion coefficient and not the instantaneous value that would result from dividing the incremental displacements by incremental time steps.66,67 Since the simulation time t is scaled by an arbitrary minimum jump time t0, the calculated value of D is in arbitrary units. The quantity of interest is therefore the slope of D(T). Figure 8 shows the temperature dependence of D, simulated for an anisotropic lattice with coupling parameters γ∥ = 2 nm−1 and γ⊥ = 4 nm−1 using MA hopping rates (Figure 8a) or using Marcus rates (Figure 8b). For the simulation using the Marcus rate, we used Ea = 60 meV as a parameter value, and in both cases, the disorder potential varies between σ = 10 meV and σ = 100 meV. The simulation time was 104 t0. The data are plotted on a Arrhenius scale with a disorder-normalized temperature. As expected, the diffusivities derived from using a MA rate scale with disorder,49 thus confirming our computational approach. While the diffusivity shows a dependence on temperature below σ/kT ≅ 10, it approaches a nearly temperature independent value at lower temperatures. In contrast, there is no scaling with disorder when using Marcus rates. For low values of disorder and temperature, the strongly temperature activated diffusivity drops to very low values. Increasing disorder improves the diffusivity in the low temperature regime. A closer look at the diffusivities in the higher temperature range shows that they differ regarding their slope. Marcus rates result in a significantly steeper temperature dependence. The temperature dependent diffusivity is subject to the anisotropy of the electronic coupling in the hopping rates, as was also manifested in the spectral diffusion data. Figure 9 shows that going from an isotropic coupling to a nearly onedimensional coupling leads to a reduction in diffusivity by 1 order of magnitude at low temperatures and by somewhat less at elevated temperatures. In the simulation, the variation in anisotropy is implemented by changing the coupling parameters along and between chains γ∥/γ⊥ from 2/2 to 2/ 20. It is interesting to observe that γ⊥ of 4 suffices to imply nearly one-dimensional behavior. 16378



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Figure 9. Effect of the anisotropy of coupling on the temperature dependence of triplet diffusion coefficient for time of t = 104t0 and energetic disorder of σ = 35 meV using (a) the Miller−Abrahams rate and (b) the Marcus rate, plotted versus 1/T2. Dotted lines serve only to guide the eye.

Figure 8. Diffusivity as a function of the inverse normalized temperature (kT/σ)−1 for coupling parameters γ∥ = 2 nm−1 and γ⊥ = 4 nm−1 and for a simulation time of t = 104t0 (a) using the Miller− Abrahams rate and (b) using the Marcus rate with an activation energy of Ea = 60 meV. The disorder σ was varied between 10 and 100 meV as indicated in the figure. Dotted lines serve only to guide the eye.

(ii). Discussion. The different temperature dependencies associated with Marcus and MA rates are well-known and can be understood by considering the physical processes associated with either rate.7 It is instructive to recall these processes and their temperature dependence, in particular with a view to eventually exploring the diffusion outside thermal equilibrium. In the case of a MA rate, tunneling from a lower energy site Ei to a higher energy site Ej may be phonon assisted if thermal energy is available and thus be accelerated at elevated temperatures by a Boltzmann factor of e(Ej−Ei)/kT. In the absence of thermal energy, however, it nevertheless takes place with a slow but finite rate of ν0e−2γRij for isoenergetic or downward jumps (eq 2). This may require jumps over longer distances that are correspondingly slower, but the diffusion still proceeds. In contrast, the classical multiphonon hopping process described by the Marcus rate requires thermal energy to overcome an energy barrier between initial and final site (kij ∝ e(−Ea/kT), eq 3). This additional activation barrier Ea arises because occupied and unoccupied sites have different molecular geometries.54,57,68,69 A jump over this barrier thus requires additional energy to allow for the associated molecular reorganization. Without thermal energy, jumps cannot occur unless an energy difference between the two sites lowers the activation barrier. This explains why disorder paradoxically improves the otherwise low diffusivity for a Marcus rate at low temperatures. In the high temperature regime, the temperature activated contribution by e(−Ea/kT) that is present in the Marcus

It is important to recognize that at low temperatures the diffusion coefficient D is a time dependent quantity.49 This is illustrated in Figure 10 using MA rates. There is a low temperature regime in which D(T) is almost temperature independent (Figure 10a). In this regime, the diffusivity bears out a D(T) ∝ (t/t0)−1 dependence (Figure 10b). This indicates that the diffusion is entirely dispersive.34,62 When raising the temperature, the degree of dispersion diminishes and eventually quasi-equilibrium is attained, the critical parameter being σ/kT. It is remarkable that even for a moderate disorder parameter of σ/kT = 2.7 it take more than 108 times the minimum hopping time t0 to reach quasi-equilibrium. This demonstrates that dispersion effects become noticeable already at temperatures around 200 K. The curvature of the diffusivity on the double logarithmic plot in Figure 10b also shows that describing the time dependence of the diffusivity with an analytic function such as D(t) ∝ (t/t0)−α is only a very rough zero-order approximation, and even at 10 K there is a minor deviation from that law expected in the T → 0 limit. This result is complementary to the deviation of the time dependence of the mean energy of the triplet excitations from the limiting T → 0 limit case (Figure 6). This time dependence is a characteristic signature of the “hopping-down” regime in which excitations relax toward the tail states of the DOS distribution and decay radiatively or nonradiatively before entering the regime of quasi-equilibrium, as discussed in detail by Movaghar et al.41 16379



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Figure 10. (a) Temperature dependence of the triplet diffusion coefficient at variable normalized simulation time between t = 102t0 and t = 108t0 using the Miller−Abrahams rate for σ = 35 meV. (b) Dependence of the diffusion coefficient as a function of t/t0 for various temperatures, corresponding to σ/kT values of 200 K = 2.05, 150 K = 2.70, 100 K = 4.10, 80 K = 5.10, 60 K = 6.70, 40 K = 10.10, and 10 K = 41.00. The red line represents a slope of −1 on the double-logarithmic plot.

rate yet absent in the MA rate dominates the diffusivity. This term leads to a largely increased temperature dependence of the Marcus rate compared to the MA rate. Figure 11 compares the simulated diffusivities with the measured diffusivities45 for the polymers PIF and DOOPPP and for the PF2/6 trimer for the parameters σ = 40, 70, and 35 meV and Ea = 60, 70, and 90 meV, respectively. These parameter values have been chosen by comparison with the optical data.45 As detailed above, the experimentally measured quantity is τ(T)−1 − τ(0 K)−1 which is proportional to D(T). This implies that the experimental data may be shifted vertically when displayed on a logarithmic scale. No other scaling has been done. In all three cases, the experimentally measured diffusivity closely follows the curve simulated with a MA rate at low temperatures up to a transition temperature. Above this value, it adheres closely to the data derived using a Marcus rate. When comparing the diffusivities obtained with either rate to the experimentally derived temperature dependence of the diffusivities, one is forced to conclude that transport at low temperatures proceeds by a tunneling process, consistent with our conclusion from the spectral diffusion data, while at higher temperatures a Marcus-type hopping process takes place. The

Figure 11. Comparison of diffusivity as a function of the inverse normalized temperature (kT/σ)−1 obtained for Miller−Abrahams rates (red full circles) and Marcus rates (black full squares) with the experimental data (open blue triangles) of (a) PF2/6 trimer, (b) PIF, and (c) DOOPPP taken from ref 45. The simulations were done for coupling parameters of γ∥ = 2 nm−1 and γ⊥ = 4 nm−1 and a simulation time of t = 104t0. The disorder parameter and the activation energy are shown in the corresponding graph.

parameters of σ and Ea are in agreement with values derived from the optical spectra. This result is consistent with earlier experimental work that was based on a comparison between optical and transport data.45 Since the diffusivities are timedependent, the temperature range for the cross-over between the temperature activated diffusion and tunneling depends also on the ratio between minimum hopping time and exciton lifetime. 16380



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The transition temperature between the two regimes, defined by their crossing point, is not only a function of time but also a function of energetic disorder. In this context, it is worth mentioning that the phenomena of dispersive transport and frustrated spectral relaxation at low temperatures are correlated. In the same temperature range in which excitation transport is dispersive, spectral relaxation is frustrated. Likewise, the temperature, at which the spectral relaxation minimum occurs, correlates with the onset of strongly temperature activated transport. This disorder-dependent transition temperature is not limited to triplet excitations. Rather, it is remarkable that, at the corresponding degree of disorder and temperature, charge transport becomes dispersive, too.6 This illustrates the analogy between transport of triplet excitations and charge carriers in a disordered system.70

(ii) Being a computer experiment, the nature of the excitation is not relevant as long as it couples by a short-range mechanism and has a finite lifetime. Thus, our conclusions pertain not only to charges but also to triplet excited states, as evidenced by the comparison with phosphorescent data. One needs to be aware that the parameter range accessed by charges and by triplets is rather different. Triplet excited states are characterized by a large geometric distortion and a low disorder, while the converse applies to charges.7 Consequently, the critical temperature Tc is in the range of 100 K for triplets yet it is near 250 K for charges. For experiments conducted at room temperature, triplet motion proceeds in the Marcus regime, while the rate of charge transport just starts to be affected by polaronic contributions. (iii) Most importantly, our investigation explicitly considers a time dependence. In particular, we were able to quantitatively derive the energetic relaxation out of thermal equilibrium (Figure 5). The limiting values of (T = 0) and (T near the lowest temperature where equilibrium is still obtained) were consistent with analytical theory. We are not aware of an analytical treatment for intermediate temperatures. Our study has shown that motion out of thermal equilibrium takes place by tunneling and needs to be described using a MA rate. In a way, this result is intuitive. If thermal energy is not available for a thermally activated process such as the multiphonon hopping described by the Marcus rate, the excitation can only move eventually by tunneling. This result pertains to both the values of energetic relaxation (see, e.g., Figures 3 and 5) and the values for the diffusivity (see Figures 9 and 10). Further, we have seen that motion out of thermal equilibrium is frustrated motion, and it leads to a time-dependent diffusivity implying dispersion. Thus, dispersive transport needs to be described by a MA rate. Note that the converse is not truetransport that takes place by a MA rate does not necessarily need to be dispersive. (iv) Our studies show that the critical temperature for the cross-over from a MA rate description to a Marcus rate depends on more parameters than just the available phonon energy and thermal energy. Figure 8 shows the diffusivity obtained by a MA rate to be independent of energetic disorder, while for the case of a Marcus rate a strong dependence on disorder is observed. Clearly, this implies that the point where the two diffusivities cross is also a function of disorder. Further, Figure 10 and indirectly also Figure 9 demonstrate the time dependence of the diffusivities, suggesting that the crossing point will also depend on the ratio between hopping time and exciton lifetime. In the case of charge carriers, the lifetime is the time at which they arrive at the exit electrode, the experimental signature being that in a TOF experiment the current pulse loses its inflection point. Thus, the transition from equilibrated nondispersive transport to dispersive transport, and thus from a Marcus to a MA rate, depends on disorder, minimum hopping time, and exciton lifetime. While the minimum hopping time may perhaps be related to phonon energies, this is unlikely for the parameters of disorder or exciton lifetime. On the basis of spectroscopic and computer experiments, we have developed a phenomenological quantitative description

VI. CONCLUDING GENERAL DISCUSSION We first arrive at the conclusion that the simulations correctly reproduce the experimental data for energetic relaxation in the density of states (DOS) and for the diffusivity. This implies that the underlying model is suitable and gives a quantitatively correct description of the physics of excitation transfer for short-range coupling. This applies to both the thermal equilibrium regime and the transport out of thermal equilibrium. We find that, at low temperatures, a MA rate has to be employed, i.e., the underlying transport mode is single phonon tunneling. At higher temperatures, however, a Marcus rate is needed. This insight is not trivial and, by comparison with experimental data, goes beyond a mere mathematical evaluation of limits in a Holstein Hamiltonian.68 The need to consider two different regimes is not always appreciated in different communities. For example, Marcus theory is so firmly established in the field of chemistry that researchers with a chemistry background can be led to neglect the tunneling aspect included in the MA rate. On the other side, excitation transport in amorphous, i.e., disordered, solid semiconductors has traditionally been investigated in a community comprising many physicists. In this context, attention has focused on the effects of energetic disorder as implemented by the MA rate, and the polaronic contribution, manifested in the Marcus rate, has been debated critically.71−75 It is well-known that a Holstein Hamiltonian, set up originally to describe the motion of polarons in a crystalline solid, can be developed into a Marcus-type expression at high temperatures and a MA-type expression at low temperature. The transition temperature is assessed by comparing the available thermal energy kT with the phonon energy ℏω prevailing in the solid. It is therefore worthwhile to point out in which way our results go beyond already existing knowledge. (i) Our conclusions are derived from experiment and not from theory (and they pertain to an amorphous solid, instead of a crystal as in the case of Holstein or a solution as in the case of Marcus).54,69 The experiments are either spectroscopic measurements (Figures 1, 7, and 11 and refs 39 and 45) or computer experiments (Figures 2−11), i.e., Monte Carlo simulations that we have shown to be consistent with physical experiments. By comparison with experiment, we are able to derive quantitative parameters for real materials, for example, regarding the minimum hopping time (Table 1) or the critical temperature when transport becomes dispersive. 16381



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for the transport of charges and triplets in and outside thermal equilibrium.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Support is acknowledged by the Deutsche Forschungsgemeinschaft (GRK1640) and the Bundesministerium für Bildung und Forschung (BMBF − Projekt Trip-Q), the FP7 EU project ONE-P (NMP3-LA-2008-212311), FNRS and FRFC through the Interuniversity Scientific Calculation Facility, ISCF. D.B. is a FNRS Research Director.

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