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# Charge transport network dynamics in molecular aggregates

Edited by Jean-Luc Bredas, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia, and accepted by Editorial Board Member Thomas E. Mallouk June 15, 2016 (received for review February 7, 2016)

## Significance

Time-dependent network analysis is used to describe the structural dynamics underpinning electron transport in disordered aggregates of molecular materials, advancing understanding of how charges move through noncrystalline aggregates. Specifically, our methodology allows for the characterization of how collective, dynamic fluctuations in the 3N nuclear degrees of freedom of the disordered multimolecule aggregate impact the statistically averaged charge motion through that aggregate. Our results describe the characteristic timescales over which electron hopping competes with nuclear reorganization, providing insight into the fundamental timescales governing charge transport in disordered systems.

## Abstract

Due to the nonperiodic nature of charge transport in disordered systems, generating insight into static charge transport networks, as well as analyzing the network dynamics, can be challenging. Here, we apply time-dependent network analysis to scrutinize the charge transport networks of two representative molecular semiconductors: a rigid n-type molecule, perylenediimide, and a flexible p-type molecule,

Computational tools using periodic boundary conditions are integral to understanding charge and excitation transport in crystalline semiconducting materials (1, 2). In noncrystalline systems, where Bloch’s theorem is inapplicable and inclusion of structural disorder is required for an accurate description of charge transport, comparable computational strategies are rare. Typically, one uses an approximate solution of the master equation with semiclassical rates derived from a combination of atomistic molecular dynamics (MD) and quantum chemistry. While this strategy has proven effective (3, 4), it possesses obvious deficiencies: it lacks both an explicit inclusion of structural dynamics and an understanding of the topology of charge transport networks.

Work in this group and others has proposed a view of charge transport based in network analysis (5⇓⇓⇓⇓–10). Network analysis represents a powerful means of analyzing structurally disordered charge transport networks, with the unique ability to place different mechanisms of charge transport on the same footing via the selection of a suitable graph metric. Recently, network views of charge transport in structurally disordered systems have provided useful insights: notably, the relationship between a molecule’s topology and the percolation threshold of its charge transport networks (5, 6), and that charge mobility can be independent of the global morphology, being primarily determined by local molecular packing (7).

Although a useful framework, previous applications of network analysis to structurally disordered charge transport networks have neglected the role of dynamic structural disorder: static snapshots of the charge transport network were used for the network analysis. Transport was assumed to occur on a charge transport network defined by time-independent site energies and electronic couplings. Whereas the static picture provides many insights into the nature of charge transport in soft materials, recent work has emerged indicating the importance of time-dependent fluctuations. Throughout the rest of this article we refer to these fluctuations as “dynamic disorder” (11⇓⇓⇓⇓–16).

In this work we use network analysis to examine the impact of dynamic disorder on charge transport networks of aggregates of two representative organic semiconducting molecules, perylenediimide (PDI, Fig. 1) (17) and

We contend that qualitatively useful insights are derived by examining the dynamics of the charge transport network topology. Specifically, this contribution argues the following points: (*i*) The tools of network analysis provide a useful complement to conventional approaches for analyzing charge transport networks in structurally disordered environments over many timescales. (*ii*) The characteristic timescales of local intermolecular coupling correlations are *iii*) Charge transport network topologies in disordered molecular semiconductors decorrelate on a timescale competitive with charge carrier lifetimes. (*iv*) Ordered morphologies maintain intermolecular coupling correlations longer than disordered morphologies. (*v*) Using the time-averaged network in charge transport overestimates the quality of charge transport networks by

## Model and Methods

### Network Analysis.

Because the primary focus of this contribution is on charge transport networks that are nonperiodic, we apply network analysis to characterize disordered multimolecule aggregates. We proceed by defining the time-dependent adjacency matrix of a graph representing the charge transport network. This network has edges (connections), **1**).

For an accurate treatment of charge transport in disordered systems, site-energy disorder must be taken into account (20). However, given that we are primarily concerned with intermolecular coupling-limited transport, we use a simple definition of the adjacency matrix using the intermolecular electronic coupling between molecular transport orbitals. For a complete treatment and experimental comparison, energetic disorder and reorganization energies must be taken into account (4, 21). It is our belief that the conclusions of this work are rendered most straightforwardly by using our simple definition of the adjacency matrix. Most formulations using energetic disorder will render the adjacency matrix nonsymmetric (hence the directed graph as opposed to an undirected graph), which significantly complicates the network analysis.

To visualize the three-dimensional charge transport networks of our materials, we consider an equivalent graph of the adjacency matrix defined in Eq. **1**. A circular embedding of the adjacency matrix is used to visualize the time-dependent dynamics of representative MD trajectories characterizing PDI and

The dynamics of the adjacency matrix are analyzed using both local and global approaches. The local characterization is chosen as a normalized time-correlation function (TCF), **2**–**5**). For analysis, a value of

To quantify the global network topology, we examine a metric of the charge transport network known as the “Kirchhoff index” to understand whether fluctuations only cause minor perturbations to the overall network structure, or if they nontrivially modify the topology of the entire charge transport network. In previous work, the Kirchhoff index was used as a robust measure to quantify known differences in charge percolation properties in topologically distinct organic semiconducting molecules (6). It is simplest to think of the Kirchhoff index as a metric quantifying a statistically averaged random walk over the entire network (i.e., an average of mean first passage times). By computing the Kirchhoff index at various MD snapshots, we examine the dynamic nature of the disordered charge transport network. In addition to the time-dependent Kirchhoff index of the instantaneous graph (

### Graph Parameterization with Atomistic Simulation.

To parameterize graphs characterizing the charge transport networks of PDI/*SI Appendix*). The simulation box is then subjected to a series of minimizations, compressions, thermal annealing, and equilibrations (*SI Appendix*), upon which a 20-ps sampling run is performed in the microcanonical (NVE) ensemble. For crystalline

To analyze the properties of the time-dependent networks, we output MD geometries every 10 fs of the 20-ps NVE trajectories, and use semiempirical extended Hückel (5) theory to compute transfer integrals between all nearest-neighbor (

## Results

### Visualizing Dynamics of Aggregate Charge Transport Networks.

To visualize charge transport network dynamics of the molecular aggregates, we plot the time-dependent graph of the adjacency matrix corresponding to a single trajectory using a circular embedding. Fig. 2*A* shows five different time intervals (0, 0.1, 1, 10, and 20 ps) for PDI, as well as the time-averaged graph, whereas Fig. 2*B* shows similar intervals plus the time-averaged crystalline network for

Examination of Fig. 2 provides clear evidence that large fluctuations in transfer integrals are observed over short timescales; by 1 ps, many connections between states have increased significantly, and some have almost completely disappeared. This result is in contrast to the common assumption of a static charge transport network when computing mobilities via a master equation approach (24) in highly ordered crystalline, polymeric systems. Operating under the assumption that charge transport in noncrystalline organic semiconductors occurs in the perturbative limit, the timescales of significant changes in the intermolecular coupling appear to be competitive with the rates derived from a Marcus-like approach. This consideration warrants a reevaluation of the dominant mechanism of charge transport, as suggested by other authors (11, 12, 14, 16).

Examination of longer timescales in Fig. 2 demonstrates that by 10 and 20 ps, the charge transport network has substantially evolved. However, an inspection of the graphs in Fig. 2 provides evidence for connections between molecules which are maintained throughout the duration of the simulation. These dominant pathways are most apparent when compared with the time-averaged “mean” graph in Fig. 2. Whereas certain pathways are maintained throughout the simulation, we note how different the individual snapshots can appear compared with the mean graph. Importantly, the mean graph is more connected than any individual snapshot. It is to be emphasized that the time-dependent graphs presented in Fig. 2 are the results of single trajectories, although the nature of the dynamics should be universal across trajectories.

The circularly embedded graph also provides insight into the overall connectivity of the network structure of a crystalline system, allowing one to straightforwardly compare both crystalline and disordered charge transport networks. In Fig. 2*B* this type of visualization provides the useful insight that despite being highly ordered, the crystalline

### Timescales of Local Intermolecular Coupling Decorrelation.

Whereas the graphical nature of Fig. 2 is inherently qualitative, we now explicitly quantify the timescale of electronic coupling decorrelation between pairs of molecules; what is the timescale behind fluctuations in the edge weights of Fig. 2? We plot the normalized TCFs of Eq. **5** for disordered PDI, disordered

Some caution must be taken in assigning physical meaning to the timescales of electronic coupling decorrelation. However, given the general assumption of independent local (intramolecular) and nonlocal (intermolecular) vibrational couplings in model Hamiltonians for charge transport (25), and the fact that the ubiquitous C=C vinyl stretching mode corresponds to a timescale of 30–50 fs (26), we feel that a qualitative assignment of these modes is valid. Consequently, we assign the *SI Appendix*), with the strongest peaks corresponding to characteristic timescales of

Whereas the specific example of PDI’s electron transport network represents only a single molecular example, the hole transport networks of

### Dynamic Fluctuations of Charge Pathways in Molecular Aggregates.

To rapidly characterize the dynamics of the entire charge transport network we use the Kirchhoff index as a metric of charge transport network connectivity (6). By computing the Kirchhoff index for 20-ps trajectories of distinct morphologies, we examine the real-time dynamics of the charge transport network topology. This information provides complementary information to the local TCF: if a single connection decorrelates [

In Fig. 4 we display the time-dependent Kirchhoff index of 20-ps trajectories for four disordered PDI morphologies. In addition to the time-dependent Kirchhoff index, we plot the Kirchhoff index of the time-averaged graph and the average value of the time-dependent Kirchhoff index as horizontal dashed lines. The simulations of Fig. 4 show rapid fluctuations in the value of *SI Appendix*). These facts are indicative of not only significant changes in local site correlations, but also in the rapid rearrangement of the dominant pathways for charge transport on an ultrafast timescale. Moreover, the results of Fig. 4 demonstrate that the time-averaged graph overestimates transport by

Whereas the fluctuations of the charge transport network topology on short (

## Discussion

### The Validity of the Static Network Assumption for Molecular Materials.

The previous results suggest that the dynamics of charge transport networks in disordered soft matter are not accurately characterized by a static network topology. Given the rapid timescale of intramolecular vibrations (

Previous work (24, 27) on the structural dynamics of crystalline polymeric materials provides perspective on these conclusions. In these works, transfer integrals between charge transport states were centered around

The synthesis of current results with previous literature indicates that in organic semiconducting molecules, the assumption of a static charge transport network is likely untenable. To accurately incorporate dynamic disorder into future charge transport calculations, semiclassical dynamics or dynamic kinetic Monte Carlo may be useful (19, 28), although the issues with these methods have been described in a recent review (16).

### The Validity of the Average Network Assumption for Molecular Materials.

Although a static network approach is unviable, it is less clear if the utilization of an average network picture is appropriate. The results of Fig. 3 indicate that the value of the long-time intermolecular electronic coupling relative to the maximum is on average

To further explore the influence of the average network assumption, we examine the values of

Given the ability of PDI and

### Optimizing Morphologies for Transport in Noncrystalline Materials.

In brief, we propose a potential application of this combination of techniques to rapidly identify nonequilibrium molecular morphologies with desirable charge transport properties, as well as to dynamically monitor the charge transport network topology. In the case of the 150-ns trajectory of Fig. 5, even over the relatively short timescale of 150 ns, there is a noticeable drift in the average value of

## Conclusion

We have analyzed the dynamic charge transport networks of two molecular semiconductors using a combination of atomistic MD, semiempirical electronic structure theory, and network analysis. Using a circular embedding, we have visualized the charge transport networks, providing insight into the topological proximity between charge transport states in a nonperiodic geometry. We have tracked the time dependence of a useful graph metric, the Kirchhoff index, characterizing the “resistance” of entire molecular aggregates, with the notable conclusion that the utilization of a time-averaged graph, especially in systems with nontrivial fluctuations, overestimates the connectivity of the charge transport network by

## Acknowledgments

The authors thank Brett Savoie and Kevin Kohlstedt for useful discussion. We thank the US Department of Energy-Basic Energy Sciences Argonne-Northwestern Solar Energy Research Center, an Energy Frontier Research Center (Award DE-SC0001059), for funding this project.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: NicholasJackson2016{at}u.northwestern.edu.

Author contributions: N.E.J., L.X.C., and M.A.R. designed research; N.E.J. performed research; N.E.J. analyzed data; and N.E.J., L.X.C., and M.A.R. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. J.-L.B. is a Guest Editor invited by the Editorial Board.

↵*Whereas the Kirchhoff index has been only qualitatively associated with experimental charge mobilities in previous work (6), in

*SI Appendix*we provide explicit kinetic Monte Carlo simulations that demonstrate a strictly monotonically increasing relationship between the zero-field charge carrier mobility and the Kirchhoff index, cementing the use of the Kirchhoff index as a meaningful qualitative metric for charge mobilities in disordered systems.This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1601915113/-/DCSupplemental.

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