A Simple Index for Characterizing Charge Transport in Molecular

Letter pubs.acs.org/JPCL

A Simple Index for Characterizing Charge Transport in Molecular Materials Nicholas E. Jackson,† Brett M. Savoie,†,§ Lin X. Chen,†,‡ and Mark A. Ratner*,† †

Department of Chemistry, Northwestern University, Evanston, Illinois 60208, United States Chemical Sciences and Engineering Division, Argonne National Laboratory, Lemont, Illinois 60439, United States

‡

S Supporting Information *

ABSTRACT: While advances in quantum chemistry have rendered the accurate prediction of band alignment relatively straightforward, the ability to forecast a noncrystalline, multimolecule system’s conductivity possesses no simple computational form. Adapting the theory of classical resistor networks, we develop an index for quantifying charge transport in bulk molecular materials, without the requirement of crystallinity. The basic behavior of this index is illustrated through its application to simple lattices and clusters of common organic photovoltaic molecules, where it is shown to reproduce experimentally known performances for these materials. This development provides a quantitative computational means for determining a priori the bulk charge transport properties of molecular materials.

rganic semiconductors is a field overflowing with diverse molecular structures.1 For most practical applications, a molecule’s optical/transport gap and its ability to effectively conduct charge in the bulk are its two most prized molecular attributes. While significant progress has been made in predicting the former via advances in quantum chemical computation,2,3 predicting the latter has proven far more challenging:4 there exists no simple and rapid means for the a priori differentiation between structures with “good” or “bad” charge transport characteristics. Given the many extant rational design schemes,5,6 it is desirable to screen not only for molecular energetics, but also for the mesoscale, multimolecule charge transport ability of a molecular species. While the disordered, multimolecule system is daunting for conventional theoretical methods, mathematical graph theory is well-equipped to study the properties of networks composed of numerous, varied subcomponents. The utility of graph theoretical approaches in chemistry is well-known, with notable contributions by Kekule, Wiener,7 Randic,8 and numerous others. Current chemical applications of graph theory often involve topological descriptors that, when parametrized with a large data set, allow for the accurate correlation between molecular descriptors and physical observables.9−11 In this report, we formulate the problem of charge transport in disordered molecular materials in terms of a variant of the Kirchoff index,12 a common graph theoretical descriptor. Conventionally, the Kirchoff index characterizes the resistivity of a network of classical resistors, utilizing the admittance between pairs of nearest neighbor resistors to compute the effective resistance (resistance distance) between any two arbitrary sites on the resistor network. Here, this concept is adapted to the molecular scale: we define an effective resistance

O

© XXXX American Chemical Society

distance between two localized charge transport states in terms of the absolute value of the intermolecular transfer integral between nearest neighbor states. In doing this, we treat the molecular bulk as a network of classical resistors, where the admittance between two sites is replaced by the electronic coupling between two states. We define the Kirchoff Transport index (KT) as the normalized sum of the inverse resistance distances between all states of the charge transport network. KT is shown to effectively characterize the charge transport networks of a simple lattice model, as well as a set of commonly used organic photovoltaic (OPV) acceptor materials, providing a simple means of discriminating a molecule’s ability to conduct charge through the noncrystalline molecular bulk. Adapting Classical Resistor Theory. We define a weighted, connected graph G, with vertex set φ and edge set H corresponding to the charge transport network of a collection of molecules. Here, φ corresponds to the set of localized charge transport states in the molecular ensemble, and H corresponds to the electronic couplings between those states. The weighted adjacency matrix A of G is defined according to eq 1, ⎧ if i ≠ j ⎪|Hij| A ij = ⎨ ⎪ ⎩ 0 if i = j

(1)

Received: January 20, 2015 Accepted: March 3, 2015

1018

DOI: 10.1021/acs.jpclett.5b00135 J. Phys. Chem. Lett. 2015, 6, 1018−1021

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The Journal of Physical Chemistry Letters where the weight |Hij| represents the absolute value of the intermolecular transfer integral between states i and j. Subsequently, the diagonal strength (degree) matrix, S (eq 2), can be formed. Sii =

shortest paths between all points on a graph. For our current purposes, the Kirchoff index is preferable to the Wiener index because it takes into account all possible paths between two points on the graph, not just the shortest, forming an effective distance between two points. One can treat the Moore− Penrose pseudoinverse of G as the discrete Green’s function of L, as well as the normalized fundamental matrix of the Markov Chain governing random walks on G;18 it is useful to think of Lij+ as a simple Green’s function, taking into account all possible paths between state i and state j. Characterizing Molecular Transport Networks. To demonstrate the efficacy of KT for characterizing molecular transport networks, we apply this index to a simple lattice model for molecular materials. The model assumes a cubic lattice, where each site represents the center-of-mass of a single molecule whose translational degrees of freedom are frozen. The molecule on each site possesses free rotation along all three Euler angles, and its orientation in space is represented by a coordinate system composed of three orthogonal vectors ⟨a,b,c⟩, centered on each lattice site (Figure S3). As the orientation of the molecule rotates, so equivalently does the coordinate system. Each set of vectors on each lattice site represents the electronic coupling topology of that molecule (Figure 1); ⟨a = 1, b = 0, c = 0⟩ represents a 1D coupling

∑ A ij (2)

j

S contains all relevant information regarding the number and magnitude of edges attached to each vertex of the graph. Using S, we generate the Laplacian matrix L = S − A, which is a central object in network analysis and spectral graph theory.13 After generating L, the generalized Moore-Penrose Inverse, L+, is formed. In Randic’s approach, L+ is used to form the resistance distance matrix Ω of eq 3. Ωij = L+ii + L+jj − L+ij − L+ji

(3)

Similarly, one can form the admittance distance matrix Λ (eq 4). −1 ⎧ if i ≠ j ⎪ Ωij Λij = ⎨ ⎪ ⎩ 0 if i = j

(4)

Λij is then used to compute what we term the Kirchoff Transport index, which is normalized to the total number of charge transport states in the system, squared N2 (eq 5). KT =

1 N2

∑ Λij i,j

(5)

It is apparent that the units of KT are not dimensionless, as might be theoretically desirable for a particular index that characterizes the physical traits of a system. To this point we note that many physically meaningful indices are not dimensionless, and are commonly used to characterize similar systems.14 To compute KT, the classical admittance has been replaced with the electronic coupling between two localized charge transport states. Physically, the transfer integral squared is proportional to the transition rate between states, which we model as analogous to passing a current through a classical resistor. In this formalism, the effects of energetic disorder among the charge transport states have been neglected. Site energy disorder could be incorporated by scaling the coupling in eq 1 by the transmission coefficient for two-state scattering, introducing a competition between energetic disorder and the coupling. One could also formulate L based on explicit rate constants between states. However, the computation of rates involves detailed knowledge of the charge transport mechanism, which typically requires the reorganization energy, λ, and the free energy of charge transfer, ΔG.15 By using only the intermolecular transfer integral, a transport mechanism is not assumed, and KT is generalizable to a variety of electronic transport phenomena. It is important to note that KT has been developed assuming a homogeneous condensed phase molecular system, while many organic semiconductor applications involve multiple molecular components.16 While an adaptation of this approach to multiple component systems is straightforward, here we limit our discussion to homogeneous systems. We briefly note that the traditional Kirchoff index is similar in spirit to the commonly used Wiener index.17 The Wiener index is defined using the distance matrix consisting of the

Figure 1. Description of lattice model and coupling topologies. Shown are example 1D, 2D, and 3D coupling topologies, the spatial directions in which they can couple, and their vector representations.

topology, ⟨a = 1, b = 1, c = 0⟩ represents a 2D coupling topology, and ⟨a = 1, b = 1, c = 1⟩ represents a 3D coupling topology. The coupling between two nearest-neighbor sites on the lattice is determined via a generalized dot product incorporating the lattice translation vector (tL̂ ) between those sites, with a maximum possible coupling between sites of Hmax. The cubic lattice obeys periodic boundary conditions. By adjusting the magnitude of ⟨a,b,c⟩, one can control whether the “molecules” at each site behave like molecules with 1D, 2D, or 3D coupling topologies. From the absolute values of these electronic couplings, the lattice Hamiltonian matrix Q is formed. A is derived from Q by setting all diagonal elements of A equal to zero, and L is determined from A via S as detailed above; Λ is then formed, from which KT is computed. Intuitively, for random orientations of all molecules on the lattice, one expects higher dimensional coupling topologies to 1019

DOI: 10.1021/acs.jpclett.5b00135 J. Phys. Chem. Lett. 2015, 6, 1018−1021

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The Journal of Physical Chemistry Letters be superior to lower dimensional topologies for bulk electrical conduction; the fact that a molecule can couple in many spatial dimensions enhances its ability to resist structural disorder. Figure 2 demonstrates the behavior of KT for a variety of

Figure 2. Kirchoff transport indices computed using the lattice model.

coupling topologies as a function of the size of the periodic lattice used in the lattice simulation. KT is shown to increase as the dimensionality of the coupling topology increases; the 3D conductor is superior to the 2D, which is superior to the 1D. KT also increases when the maximum value of the intersite coupling, Hmax, is increased. To compare disordered molecular systems to those composed of a rigid, crystalline, covalently bonded lattice, we also simulate a 3D conducting system with an intersite coupling approximately equivalent to that found in rigid inorganic lattices (∼1.5 eV). KT for the rigid lattice is found to be ∼3.3, whereas KT for a bulk of 1D π-electron systems is ∼0.01, 2 orders of magnitude smaller. While at this point there is not a rigorous means to quantitatively connect KT with physical observables, we note this result is in qualitative agreement with the order of magnitude differences in experimentally determined charge carrier mobilities in crystalline silicon19 and the highest performing organic semiconductors.15 Importantly, we note the weak dependence on lattice size (Figure 2), with stronger dependencies resulting from larger intersite couplings, as expected. The rapid convergence of KT for couplings on the order of realistic couplings for organic semiconductors is encouraging for explicit simulations involving a finite number of real molecules. Discriminating Conductivities of Molecular Materials. We choose two families of common OPV acceptors, fullerenes and perylenediimides (PDIs), for which to compute KT (shown in Figure 3a). In previous work, we established a link between the successs of mono- and bis-functionalized fullerenes in OPV and their ability to electronically percolate the disordered bulk material.20 Specifically, we concluded that fullerene derivatives are more electronically percolative than PDI derivatives due to their ability to couple in multiple spatial dimensions. It was also observed that the ability of both families of acceptors to electronically percolate suffered as bulkier solubilizing sidechains were added, disrupting the mesoscale charge transport networks. Here, we analyze this same set of materials using the Kirchoff Transport index, comparing these results to previous work20 in order to assess the performance of KT as an index for characterizing charge transport.

Figure 3. Kirchoff transport indices for studied OPV materials. (A) Molecular structures analyzed in this work. Formal names provided in SI. (B) Kirchoff transport indices for the structures in panel A.

The computational methodology used for simulating the molecular morphologies is identical to our previous work.20 For each of the materials studied, a 64-molecule cluster is simulated using classical molecular dynamics (MD) in the NPT ensemble. Network analysis for all molecules is performed on 50 snapshots from five unique MD trajectories. All intermolecular transfer integrals and site energies of the three lowest unoccupied molecular orbitals of each molecule are determined using the extended Huckel (EH) method to form Q. When computing A from Q, couplings between charge transport states with energy differences greater than 0.3 eV are set to zero. Further details regarding the nature of the simulations may be found in our previous work.20 Figure 3b summarizes the value of KT computed for all of the OPV materials studied in this work. Experimentally and theoretically, PC60BM is a superior OPV acceptor compared to bis-PC60BM, which is superior to tris-PC60BM.21,22 This trend is reproduced by the value of KT shown in Figure 3b. The KT values of fullerene derivatives rapidly fall off as bulkier solubilizing side-chains are attached. A similar effect is observed for the PDI derivatives, where KT decreases as the size of the side-chains increases. Fullerene derivatives exhibit significantly higher values of KT compared to any of the PDI derivatives, suggesting that PDIs are more “resistive” charge transport materials than fullerenes. These results are in close correspondence with those of the lattice model of Figure 2, where 2D and 3D coupling topologies were shown to be vastly superior to 1D coupling topologies. 1020

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(9) Balaban, A. T. Applications of Graph Theory in Chemistry. J. Chem. Inf. Comput. Sci. 1985, 25 (3), 334−343. (10) Pogliani, L. From Molecular Connectivity Indices to Semiempirical Connectivity Terms: Recent Trends in Graph Theoretical Descriptors. Chem. Rev. 2000, 100 (10), 3827−3858. (11) García-Domenech, R.; Gálvez, J.; de Julián-Ortiz, J. V.; Pogliani, L. Some New Trends in Chemical Graph Theory. Chem. Rev. 2008, 108 (3), 1127−1169. (12) Klein, D. J.; Randić, M. Resistance Distance. J. Math. Chem. 1993, 12 (1), 81−95. (13) Merris, R. Laplacian Matrices of Graphs: A Survey. Linear Algebra Its Appl. 1994, 197−198, 143−176. (14) Berthier, L.; Biroli, G. Theoretical Perspective on the Glass Transition and Amorphous Materials. Rev. Mod. Phys. 2011, 83 (2), 587−645. (15) Coropceanu, V.; Cornil, J.; da Silva Filho, D. A.; Olivier, Y.; Silbey, R.; Brédas, J.-L. Charge Transport in Organic Semiconductors. Chem. Rev. 2007, 107 (4), 926−952. (16) Cao, W.; Xue, J. Recent Progress in Organic Photovoltaics: Device Architecture and Optical Design. Energy Environ. Sci. 2014, 7 (7), 2123−2144. (17) Babić, D.; Klein, D. J.; Lukovits, I.; Nikolić, S.; Trinajstić, N. Resistance−Distance Matrix: A Computational Algorithm and Its Application. Int. J. Quantum Chem. 2002, 90 (1), 166−176. (18) Li, Y.; Zhang, Z.-L. Digraph Laplacian and the Degree of Asymmetry. Internet Math. 2012, 8 (4), 381−401. (19) Canali, C.; Jacoboni, C.; Nava, F.; Ottaviani, G.; AlberigiQuaranta, A. Electron Drift Velocity in Silicon. Phys. Rev. B 1975, 12 (6), 2265−2284. (20) Savoie, B. M.; Kohlstedt, K. L.; Jackson, N. E.; Chen, L. X.; de la Cruz, M. O.; Schatz, G. C.; Marks, T. J.; Ratner, M. A. Mesoscale molecular network formation in amorphous organic materials. Proc. Natl. Acad. Sci. U.S.A. 2014, 111 (28), 10055−10060. (21) Lenes, M.; Shelton, S. W.; Sieval, A. B.; Kronholm, D. F.; Hummelen, J. C.; Blom, P. W. M. Electron Trapping in Higher Adduct Fullerene-Based Solar Cells. Adv. Funct. Mater. 2009, 19 (18), 3002− 3007. (22) Guilbert, A. A. Y.; Reynolds, L. X.; Bruno, A.; MacLachlan, A.; King, S. P.; Faist, M. A.; Pires, E.; Macdonald, J. E.; Stingelin, N.; Haque, S. A.; Nelson, J. Effect of Multiple Adduct Fullerenes on Microstructure and Phase Behavior of P3HT:Fullerene Blend Films for Organic Solar Cells. ACS Nano 2012, 6 (5), 3868−3875.

In this Letter we have utilized results from the theory of classical resistor networks to develop a single parameter for characterizing the bulk charge transport characteristics of an arbitrary molecule. Using a simple lattice model, we describe the bounds and behavior of the Kirchoff Transport index as a function of the intermolecular coupling magnitude, coupling topology, and lattice size. This index is then used to accurately characterize the known charge transport characteristics of two families of common OPV acceptor materials. The simplicity of this index presents great possibilities for the ability to categorize and screen the bulk charge transport behavior of molecular materials by a single parameter, an area previously unavailable via conventional theoretical chemistry techniques.

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ASSOCIATED CONTENT

S Supporting Information *

Description of lattice model, molecular dynamics details, formal molecule names, data used in Figures 2 and 3, and extended Huckel methods used are all available online in the Supporting Information. The data reported in this paper are tabulated in the Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Present Address

§ (B.M.S.) Department of Chemistry and Chemical Engineering, California Institute of Technology, 1200 East California Blvd., Pasadena, CA 91125.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the U.S. DOE-BES Argonne-Northwestern Solar Energy Research Center (ANSER), and Energy Frontier Research Center (Award DE-SC0001059) for funding this project. N.E.J. thanks the NSF for the award of a Graduate Research Fellowship (NSF DGE-0824162).

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REFERENCES

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DOI: 10.1021/acs.jpclett.5b00135 J. Phys. Chem. Lett. 2015, 6, 1018−1021