Limit of Exciton Diffusion in Highly Ordered π-Conjugated Systems

Article pubs.acs.org/JPCC

Limit of Exciton Diffusion in Highly Ordered π‑Conjugated Systems Pedro Henrique de Oliveira Neto,† Demétrio A. da Silva Filho,† Wiliam F. da Cunha,† Paulo H. Acioli,*,†,‡ and Geraldo Magela e Silva† †

Institute of Physics, University of Brasilia, Brasilia, 70.919-970, Brazil Department of Physics, Northeastern Illinois University, Chicago, Illinois 60625, United States

‡

ABSTRACT: There are many limiting factors in the efficiency of organic photovoltaic devices. One of these factors is their diffusion length or alternatively the diffusivity of excitons, the quasi-particles responsible for the energy conversion. In this work, we investigated the limit of diffusion of excitons in πconjugated systems. Using a model Hamiltonian that included electronic and strong electron−phonon coupling, and including quantum corrected thermal effects through Langevin dynamics, we determined how excitons diffuse within a highly ordered polymer composed of many identical monomers. We established that the exciton follows a typical 1-dimensional random walk diffusive behavior. The diffusivity followed a Marcus behavior with a very low activation energy of 15 meV and a room temperature diffusivity constant of 6.12 × 10−2 cm2/s. We obtained the maximum diffusion constant of 1.1 × 10−1 cm2/s. This is the upper limit of exciton diffusivity in π-conjugated systems as the systems modeled are highly ordered and contained no impurities and the excitons are very delocalized, justifying the low activation barrier and the high diffusion constant.

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INTRODUCTION In the past decade, the performance of organic photovoltaic devices based on π-conjugated systems has increased steadily,1−8 making these devices promising candidates in the search for alternative energy sources. As an alternative to inorganic photovoltaics, the organic analogue has the advantage of low production cost, low weight, flexibility, etc. The major challenge in the field of organic photovoltaics is related to the efficiency. While in inorganic based devices, the absorption of photons with energy greater than the band gap leads to a direct creation of free charge carriers, in organics, the nature of an optically excited state is quite different. Organic materials have a dielectric constant much smaller than their inorganic counterparts, therefore, electrons and holes created by photon absorption are subjected to a strong Coulombic attraction. This leads to a strongly bound state (∼300 meV). In inorganic materials, an exciton created by the absorption of a photon has binding energy similar to the thermal energy inherent to the material (∼30 meV), leading to sudden charge carrier formation. In organic materials, excitons are localized (in the tens of angstroms) and are much harder to dissociated into two free carriers. Accordingly, OPVs rely on two-component systems containing an electron-donor and an electron-acceptor that is similar to that of a p−n junction. An exciton, usually formed in the donor molecule,9 should migrate to the donor−acceptor (D−A) interface. When the energy difference between the donor and the acceptor LUMO (lowest unoccupied molecular orbital) is greater than the binding energy of the exciton, the dissociation is favored. Consequently, the efficiency of OPVs depends, among other factors, on the exciton’s ability to © 2015 American Chemical Society

migrate to the interface region. In other words, an exciton has to travel a distance before suffering a recombination process. Hence, the diffusion length, Ld), is paramount to the efficiency determination. Great attention has been devoted to reducing the distance an exciton must migrate to reach the dissociating interface. A gain in efficiency has been observed by optimization of the active-layer film morphology in D−A blends.10 Understanding the mechanisms that determine the magnitude of the Ld were less explored, but is a promising alternative in the attempt to optimize OPVs efficiency. In general, the exciton diffusion length is defined as Ld = τD where τ is its lifetime, and D is the coefficient of diffusion or diffusivity. Essentially, τ is related to photocreation, dissociation and recombination processes, while D characterizes the intrinsic ability of the quasi-particle to move in the material. This quantity is related to morphological and thermodynamical factors. In this sense, the physics of an exciton diffusion is a key issue for the understanding of the underlying science on organic solar cells. Specifically, the characterization of D and Ld and the dynamics of these quasi-particles are critical for engineering highly performing OPVs based on π-conjugated compounds. Recently, a report on the transport of excitons in tetracene11 (both crystal and thin film) has shown that the exciton dynamics is similar to a random walk on a time scale of τ < 2 μs. Assuming that diffusion in each direction of the crystal is Received: June 9, 2015 Revised: August 11, 2015 Published: August 11, 2015 19654

DOI: 10.1021/acs.jpcc.5b05508 J. Phys. Chem. C 2015, 119, 19654−19659

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where n labels the chain sites. Here, yn ≡ un+1 − un in which un is the lattice displacement of a particular site n. The conjugated momentum to un is pn, K is the harmonic constant for a σ bond and M is the mass of a monomer. The operator C†n,s (Cn,s) creates (annihilates) a π-electron with spin s at the nth site. The hopping integral assumes tn,n+1 = [(1 + (−1)nδ0)t0 − αyn], t0 being the hopping integral of a π-electron between neighboring sites in the fully undimerized chain and α the electron−phonon coupling. In order to simulate armchair π-conjugated polymers we introduce the Brazovskii-Kirova symmetry-breaking term δ0. The parameters used here are recognized as standard and have the following values: t0 = 2.5 eV, M = 1349.14 eV × fs2/Å2, K = 21 eV Å−2, δ0 = 0.05, α = 4.1 eV Å−1, and a = 1.22 Å. These values have been used successfully in previous simulations.26−32 In the scope of a non adiabatic time evolution method, we prepare an initial self-consistent stationary state in all the degrees of freedom of electrons and phonons. The lattice backbone dynamics is described classically by a Newtonian equation Mün = Fn(t). Where,

uncorrelated they measured a normal diffusion process with time independent diffusion constants that range 1 order of magnitude from 1 × 10−4 to 1 × 10−3 cm2/s depending whether the diffusion was in the directions defining the herringbone plane or the third crystalline direction. This work demonstrated that the decrease in nanoscale disorder generates an exciton diffusion behavior that can achieve an upper limit, along the a axis, of Ld on the order of 4 μm in tetracene. This results are also in line with previous work12,13 that showed that the diffusion length can be increased by increasing the crystalline order. The role of intrachain versus interchain effects is studied in depth in ref 13. They showed that increasing the size of the monomer increases its diffusion length, highlighting the importance of intrachain effects in exciton diffusion. In fact, in another study, it was observed, by transient absorption spectroscopy the exciton dynamics in crystalline domains of regioregular poly(3-hexylthiophene) (P3HT),14 a higher diffusion coefficient of the order of 1 × 10−3 cm2/s. The one-dimensional exciton diffusion was, also, strongly supported by anisotropy decay measurements. Both studies also observe one-dimensional random walk behavior for the diffusion of excitons. Many models of exciton diffusion15 are concerned about increasing the efficiency by optimizing the interchain diffusion processes, i.e., the diffusion between molecules in the OPVs. However, in ref 13, the authors show that when the order in the crystal increases, the intrachain processes become more important. In this context, there is the need to predict how far changes in morphology can modify the diffusion coefficient and hence the exciton diffusion length. A few other studies are devoted to studying the diffusion of charge carriers in organic semiconductors.16−23 They use a Hamiltonian similar to the one used in this work; however, they are all focused in the interchain diffusion on systems with weak electron−phonon coupling. As described below, the focus of this work is on the exciton diffusion in long polymeric chains as there is strong evidence that this is one of the routes for increasing the diffusivity. In this work, we report molecular dynamics on a highly ordered π-conjugated chain containing an intrachain triplet exciton in the presence of thermal effects. By means of a model Hamiltonian combined with a nonadiabatic time evolution method, we determine the behavior of the exciton diffusion coefficient as a function of the temperature of the system. Our model predicts a diffusion coefficient of D = 6.12 × 10−2 cm2/ sat room temperature. More importantly, our model predicts an upper level for the exciton diffusion coefficient of D0 = 1.1 × 10−1cm2/s for highly ordered organic systems. Our results corroborate the experimentally observed one-dimensional diffusive random walk behavior, and supports the potential of materials engineering to increase LD in order to achieve maximum efficiency in organic photovoltaic devices by creating systems with longer chains and lower disorder.

Fn(t ) = − K[2un(t ) − un + 1(t ) − un − 1(t )] + α[Bn , n + 1(t ) − Bn − 1, n(t ) + Bn + 1, n(t ) − Bn , n − 1(t )].

Here, Bn,n′(t) = ∑k,s ′ ψk,s * (n,t), where ψk,s(n′,t) is the term that couples the electronic and lattice parts of the model Hamiltonian. The primed summation represents a sum over occupied states. In order to simulate an excited state, we change the occupation to appropriate values. The Newtonian equation was numerically integrated using the euler method. The electronic part is governed by the Schrödinger equation and evolves through a linear combination of instantaneous eigenstates of the electronic Hamiltonian that can be put in the form ψk , s(n , t j + 1) =

l

×e

+

∑ n

pn

(3)

with the following properties: ⟨ζ(t)⟩ ≡ 0 and ⟨ζ(t)ζ(t′)⟩ = Bδ(t − t′). Here, Kb is the Boltzmann constant and γ the damping term. The relationship between ζ, γ, and the temperature T of the system is given by the fluctuation−dissipation theorem B = 2KbTγ M. We use γ = 0.06 fs−1. This value was chosen, taking into account the typical frequency of a C−C bond oscillation on π-conjugated systems (∼1500 cm−1). In an earlier work,35 we have shown that this methodology of including thermal

2

2

2M

ϕl , s(n , t j),

Mun̈ = −γuṅ + ζ(t ) + Fn(t )

∑ (tn,n+ 1Cn†+ 1,sCn,s + h.c.) + ∑ K yn2 n

m

where {ϕl(n)} and {εl} are the eigenfunctions and the eigenvalues of the electronic part for the Hamiltonian at a given time tj. At time tj the wave functions {ψk,s(n,tj)} are expressed as an expansion of the eigenfunctions {ϕl,s}: ψk,s(n,tj) s s = ∑Nl=1Cl,k ϕl,s(n), in which Cl,k are the expansion coefficients. In order to take thermal effects into account, we make use of a canonical Langevin dynamics as implemented in previous work33,34 and widely applied in the literature. Is this sense, the temperature influence on the electronic part of the system is considered by means of the coupling terms. The used Langevin equation is characterized by a white stochastic signal ζ(t) as the fluctuation term and a Stokes like dissipation term. Thus, we rewrite the site equation as

METHODOLOGY In order to simulate a fully π-conjugated polymer chain we use the following model Hamiltonian24,25

n,s

∑ [∑ ϕl*,s(m , t j)ψk ,s(m , t j)]

(−iεl Δt / ℏ)

■

H=−

(2)

, (1) 19655

DOI: 10.1021/acs.jpcc.5b05508 J. Phys. Chem. C 2015, 119, 19654−19659

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RESULTS AND DISCUSSION All simulations were performed on a polymeric chain with 80 sites and periodic boundary conditions. Our goal is to analyze the diffusivity behavior as a function of temperature. We use the following molecular dynamics temperatures (TMD): 10, 20, 50, 100, 150, 200, 300, 400, and 500 K. In order to locate the exciton in the chain we analyzed the C−C bond length pattern time evolution, as excitons are neutral excitations and therefore do not have a characteristic charge profile. However, in πconjugated systems, the excitons generate a change in the binding pattern that extend for about 20 sites. To study the diffusivity of the exciton we simulated 200 realizations for each temperature. A realization is defined by the initial seed of the random number generator used in the Langevin dynamics. In Figure 2, we show one of the C−C bond length time evolutions

effects was very successful in determining the characteristic time of creation of a polaron. A similar approach was already used to describe crystalline organic semiconductors.16 However, in this kind of system, the charge transfer mechanism is related with interchain process. Until now, this dynamical disorder model was not used to describe intrachain processes with thermal effects. The main problem is a limitation on the treatment of phonons at low temperature regimes. An intermolecular transport is governed by phonons with lower frequencies (100−200 cm−1). In this thermal regime, the classical approach is a good approximation and provide some physical insights on intermolecular charge transport. At low temperatures, below the Debye temperature, the quantum behavior will be important and the same classical approach seems not to be appropriated. Although the C−C bond oscillation in the studied system is of the order of 1500 cm−1, at low temperatures only the normal modes with low frequencies will be excited. This means, in principle, that we cannot treat the nuclear degrees of freedom classically. However, at low temperatures, only the normal modes with low frequencies will be excited. When we are close to the classical limit, the contributions from the very high optical modes will be important. It is therefore reasonable that we can study the behavior of these systems using classical molecular dynamics and correct the final results for quantum effects. In order to overcome this drawback, we included quantum corrections using the standard method that is described in ref 36 and used in many classical molecular dynamics simulations in the low temperature regime.37−40 The main idea is to equate the molecular dynamics kinetic energy, using the equipartition theorem, with the energy at a temperature T given by NKBTMD =

⎡

∫ ℏωD(ω)⎢⎣ 12

+

1 e

ℏω / KBT

⎤ ⎥ dω − 1⎦

Article

Figure 2. Evolution of the C−C bond lengths in one of the realizations at TMD = 100 K.

in one of the realizations. The hottest colors represent a larger distortion. It is clear that the exciton is performing a Brownian motion. Note that the distortion in the chain is along approximately 20 sites, which is characteristic of an exciton. Also, it is possible to observe some fluctuations along the chain showing the temperature effect. At about 3500 fs, the exciton reaches the end of the chain at the 0-site and appears at the 80site due to the periodic boundary condition. As mentioned above, the exciton trajectory was determined by the center of this fluctuation in each realization. The exciton position evolution for the 200 realizations at TMD = 300 K is shown in Figure 3a. Each line on the graph represents the exciton position as a function of time, relative to its initial position, in a given realization. In Figure 3a, it is possible to visualize the dispersion of the exciton path, suggesting a diffusive behavior. Note that the model used for the inclusion of the temperature does not subject the quasi-particle directly to a thermal bath. The thermal effects are included in the

(4)

where D(ω) is the phonon density of states. Given the desired temperature T we obtain the corresponding simulation temperature TMD by use of the above expression. Since we are mostly interested in phonon excitations we did not include the zero point energy correction (first term in the equation above). In Figure 1 one can see the correspondence between the molecular dynamics (TMD) and the quantum temperature. The main difference occurs in the low temperature regime where the real temperature is much higher than the one used in the molecular dynamics simulation. However, as the temperature increases the two values approach each other.

Figure 3. (a) Position (Å) of the exciton as a function of time (ps) at TMD = 300 K. (b) Distribution of exciton position at 1.2 ps relative to its initial position at TMD = 300 K.

Figure 1. Quantum corrected temperature as a function of the molecular dynamics temperature (TMD). 19656

DOI: 10.1021/acs.jpcc.5b05508 J. Phys. Chem. C 2015, 119, 19654−19659

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behavior for t > 3 ps. These fluctuations increase for higher temperatures. The relationship between ⟨x(t)2⟩ and t is known as Fick’s Law and occurs in normal diffusion processes. In physical systems where other diffusion processes are present, Fick’s law is modified as ⟨x(t)2⟩ = Atα, where α characterizes the diffusion type and A is an empirically observed scaling factor. When α > 1, the process is said to be superdiffusive. For α < 1, the process is subdiffusive. For α = 1, the normal distribution is recovered. We performed a regression as ⟨x(t)2⟩ = Atα. At 300 K, the regression yields α = 1.026. The same behavior where α ∼ 1 was observed at all temperatures featuring, in fact, a process with normal diffusion. From Figure 4 we calculated the diffusion coefficient as a function of the temperature. The results are plotted in Figure 5.

monomers that form the polymer and the energy from the lattice is passed to the excitons through the strong electron− phonon interaction. Thus, the exciton presents a characteristic behavior of a Wiener process. As dc is the displacement of an exciton relative to the origin, we can calculate the probability of finding an exciton at a displacement, dc, from its initial position as a function of time. The graph in Figure 3b shows the distribution of excitons as a function of the distance dc from the origin at 1.2 ps. The Gaussian regression of this distribution provides important quantities to describe the problem of diffusion in this of system. From the symmetry of the problem one expects that the average position of this distribution to be zero. The number of realizations chosen in this work kept the average position of the quasi-particle at zero within an error of at most 6.5%. Figure 4 shows the variance of each distribution as a function of time for several temperatures. The linearity features a typical

Figure 5. Exciton diffusivity D (cm2/s) as a function of temperature T (K). Blue represents the data with the molecular dynamics temperature (TMD) and the red represent the quantum corrected temperature (TQ).

The behavior of the diffusion as a function of the temperature is typical of a mass diffusion process in solids and follows an Arrhenius type law: D(T) = D0 exp(−EA/KbT), where D0 is the maximum diffusion coefficient (T → ∞) and EA is the activation energy for diffusion.41,42 The blue data represents the results using the molecular dynamics temperature (TMD) and the red data represents the quantum corrected results. A regression to the data presented in Figure 5 yields D0 = 8.7× 10−2 cm2/s and EA = 8.2 meV for the MD data and D0 = 1.1× 10−1 cm2/s and EA = 14.7 meV for the quantum corrected temperatures. It is clear from these results that the diffusivity is about an order of magnitude higher than the values reported in the literarure.11,14 This can be explained by the low activation energy that in Marcus’ theory for identical donor and acceptor molecules is related to the reorganization energy required to bring the acceptor to the equilibrium configuration. This very low activation energy can also be explained by the fact that in our system the exciton is delocalized in about 20 monomers, and therefore, the reorganization energy is very small. Given the above arguments, we believe that 1.1× 10−1 cm2/s is the limit of diffusivity in π-conjugated systems as the simulated model contains no disorder or impurities and we accounted for quantum effects through the correction scheme described above.

Figure 4. Variance of the exciton distribution (μm2) as a function of time (ps) for TMD = 10, 50, 100, 200, 300, and 500 K.

diffusion process of a one-dimensional random-walk. The same behavior was observed experimentally in P3HT and pentacene samples. We observe that the slope is strongly dependent on the temperature. The higher the temperature, the more energy is given to each monomer, which in turn transmits more energy to the excitons. Therefore, the quasi-particle can acquire a higher speed and be able to diffuse more within the system. This linear dependence of the variance with time is observed for all temperatures in this study. This shows that in this range of temperatures the excitons exhibit a diffusive behavior. In order to quantify the diffusivity we consider the onedimensional diffusion equation: dn(x , t ) d2n(x , t ) = D(t ) dt dx 2

where n(x,t) is the spatially and time-dependent density of excitons and D(t) is the diffusion coefficient. In our case, considering diffusivity as a constant, the solution is n(x , t ) =

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CONCLUSIONS We presented results of simulations of diffusion of low energy excitons in a highly ordered π-conjugated system. The model Hamiltonian that describes the system is well studied and the parameters are widely used in the literature to study transport in organic systems. To study the diffusion, we included thermal effects using Langevin dynamics. This methodology has been

⎛ x2 ⎞ 1 exp⎜ − ⎟ 4πDt ⎝ 4Dt ⎠

The variance of the mean square displacement ⟨x(t)2⟩ of the excitons, as shown in Figure 4, evolves as σ2(t) = 2Dt. The figure shows fluctuations in the variance about the linear 19657

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validated by our study of recombination of polarons and bipolarons, exciton dissociation and photogeneration of free carriers in organic conductors.33−35 We have shown that the exciton follows a normal diffusion behavior in the chain and that the temperature dependence of diffusivity does follow a Marcus behavior. We obtained a diffusivity that is 1 order of magnitude higher than the maximum diffusivity in tetracene and PH3T. This can be explained by the highly ordered character of our model system and the fact that the exciton is very delocalized in this system. This implies a very low reorganization energy and therefore a very low activation energy. We predict that the maximum diffusivity of low energy excitons of 1.1× 10−1 cm2/s is the upper limit for OPVs.

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AUTHOR INFORMATION

Corresponding Author

*(P.H.A.) Telephone: +1 (773) 442 4733. E-mail: p-acioli@ neiu.edu. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support from the Brazilian Research Councils CNPq, CAPES, and FINATEC. D.A.d.S.F. gratefully acknowledges the financial support from the Brazilian Research Council CNPq, grant 306968/2013-4.

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