Introduction to Organic Semiconductors Using Accessible

Article Cite This: J. Chem. Educ. XXXX, XXX, XXX−XXX

pubs.acs.org/jchemeduc

Introduction to Organic Semiconductors Using Accessible Undergraduate Chemistry Concepts Kevin L. Kohlstedt,*,† Nicholas E. Jackson,‡ Brett A. Savoie,§ and Mark A. Ratner*,† †

Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, United States Institute of Molecular Engineering, University of Chicago, 5640 Ellis Avenue, Chicago, Illinois 60637, United States § Davidson School of Chemical Engineering, Purdue University, 480 West Stadium Drive, West Lafayette, Indiana 47907, United States Downloaded via UNIV OF CALIFORNIA SANTA BARBARA on July 6, 2018 at 02:29:55 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

‡

S Supporting Information *

ABSTRACT: Organic semiconductors (OSCs) are making great progress as active components in alternative energy and flexible electronics technologies that are of interest to many undergraduates. However, in materials governed by the confluence of multiple length scales (molecular (Angstrom), intermolecular (nm), domain (100s nm)), providing a pedagogically accessible pathway to incorporating OSCs into undergraduate education can be difficult. Here, we provide a multiscale description of OSCs that relies only on concepts covered in typical undergraduate chemistry and chemical engineering curricula: a tight-binding description of molecular orbitals using Hückel theory, the miscibility of intermolecular domains using thermodynamic principles based on the Ising spin model, the incorporation of thermal disorder of both the electronic and molecular states using the Boltzmann distribution, and a simple description of charge percolation using low-level graph theory. We illustrate these topics on a small organic molecule, 1,3,5-hexatriene, simple enough to be utilized in an undergraduate course without resorting to opaque metrics. KEYWORDS: Upper-Division Undergraduate, Interdisciplinary, Polymer Chemistry, Physical Chemistry, Analogies/Transfer, Conformational Analysis, Molecular Modeling, Transport Properties

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INTRODUCTION

over 75 years of pedagogical developments in chemistry and chemical engineering applied to hydrocarbon derivatives.9 Organic semiconductor function is inherently multiscale, with both molecular and mesoscopic scale phenomena affecting how charge moves through these materials. The chemical processes occurring on multiple length scales in OSCs,10 while providing a rich area for research,11,12 can be challenging to teach in the laboratory or classroom without oversimplification. Here we show that despite their complexity, much of the physics and chemistry involved with OSCs can be described in an accessible formalism using only the tools from undergraduate general, organic, and physical chemistry courses.13 In order of presentation, we show how the electronic structure of organic molecules, and their aggregates, can be approximately described by Hückel theory as learned in an organic chemistry course;14 the formation of large-scale films of OSCs can be approximately described by a simple Ising model as learned in a physical chemistry course; and the Boltzmann distribution can be integrated into the two latter ideas to aid in understanding how electronic and morphological aspects of OSCs are strongly influenced by disorder via temperature and processing.15 OSCs represent a prime opportunity to demonstrate not only the application of the individual topics to an applied problem but more importantly the integration of ideas from

Polymer science has been presented in undergraduate chemical engineering courses for many decades, with topics as diverse as synthesis,1 solubility,2 mechanical properties,3 and processing,4 all incorporated in a unified fashion.5 Despite the wide variety and apparent complexity of these many polymer subfields, effective pedagogy has been built from simple, intuitive models,6 as well as an approach that embraces the inherent disorder of polymeric materials.7 In contrast, the manner in which organic electronic devices have been incorporated into chemistry textbooks has not benefitted from the intuitive models that are presented in both organic and physical chemistry undergraduate textbooks. Instead, many of the organic electronic principles have been borrowed from the solid-state semiconductor canon.8 We believe, akin to the way polymer science is taught in engineering coursework, that an effective pedagogical presentation of organic electronics can utilize a similar synthesis of simple models with an emphasis on the inherent disorder of these systems. Here, we extend the pedagogical approach traditionally used in polymer science to illustrate the transport properties of organic semiconductors (OSCs). Specifically, our goal is to show that simple models like Hückel theory, the Boltzmann distribution, and the Ising model, that are covered in a typical chemistry undergraduate curriculum, can be applied to OSCs to develop a semiquantitative understanding of charge transport. More broadly, the integration of renewable energy technologies in chemistry curricula, which OSCs are central to, can benefit greatly from © XXXX American Chemical Society and Division of Chemical Education, Inc.

Received: January 30, 2018 Revised: June 4, 2018

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DOI: 10.1021/acs.jchemed.8b00064 J. Chem. Educ. XXXX, XXX, XXX−XXX

Journal of Chemical Education

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seemingly disparate undergraduate chemistry coursework to tackle a technologically relevant problem of great interest to many undergraduates. The remainder of this manuscript details the undergraduate-level concepts that can be leveraged to describe charge transport in OSCs while providing students with an example of clean-energy technology (as an example, see University of Washington’s Clean Energy Institute’s educational solar devices)16 with physical processes that can be tied back to classroom concepts. The examples provided here can be used in part or whole to introduce students to some of the molecular processes relevant to renewable energy technology while providing a blueprint for engaging students in chemistry-related courses on renewable energy processes and materials.

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DISTINCT AT MULTIPLE SCALES In the remainder of the work we will work through OSCspecific phenomena at different scales, beginning with the molecular and finishing with the device scale. While some students may be familiar with traditional inorganic semiconductors such as silicon, the differences between these systems and soft OSCs often significantly outweigh their similarities and fundamentally different physical models are required. For example, in comparison with traditional photovoltaic-semiconductors such as crystalline silicon (c-Si) and CdTe, the structural and chemical impurity of OSCs is extreme.17 Photovoltaics grade c-Si exhibits chemical purities in the range of ppm-ppb (parts per million/parts per billion), whereas photovoltaics grade OSCs have purities in the range of parts per hundred/parts per thousand. The average crystalline grain size of c-Si is typically > mm (visible to the eye), whereas “pure” crystalline domains of OSCs are difficult to form and do not always correlate with best performance. These differences affect all aspects of the design, characterization, and optimization of OSCs. The physics of OSCs is inherently multiscalemeaning that not only do the orientations of individual atoms relative to one another matter, but the orientations of individual molecules relative to other molecules are also vitally important. Here, we delineate three approximate scales: (1) the molecular (Angstrom), (2) the intermolecular (nm), and (3) the device (100s nm). Throughout these examples, we will use the small organic semiconductor 1,3,5-hexatriene (Figure 1) as a model, first starting with a molecular description and gradually working toward a description of its behavior in a semiconducting film. We choose 1,3,5-hexatriene, in part, to pay homage to a building block of one of the first ever successful organic semiconducting materials, polyacetylene,18 which is simply the polymerized version of 1,3,5-hexatriene. While 1,3,5-hexatriene is a simple small molecule, it does bear many of the features (e.g. “band-gap”, conjugation length, and bond length alternation) present in more complex polymers.11

Figure 1. trans-1,3,5-Hexatriene molecule shown with three dihedral angles ψ1, ψ2, and ψ3 that control the conformational degrees of freedom.

arrangements of hexatriene’s atoms in space relative to one another. In Figure 1 we have drawn the all-trans conformation of 1,3,5-hexatriene (shortened hereafter to hexatriene). However, in OSCs, it is common that a single conformer does not accurately represent the ensemble of thermodynamically accessible structures due to the existence of conformational isomers. These conformational isomers are often attributable to “soft” degrees of freedom, which are easily affected by thermal energy, causing the geometry to adopt other thermally activated conformations. In OSCs these soft degrees of freedom are often the dihedral angles controlling the orientations of neighboring carbon p-orbitals (C 2p); in the case of hexatriene, there are three relevant dihedrals {ψ1:1,2,3,4; ψ2:2,3,4,5; ψ3:3,4,5,6}, denoted in Figure 1. A 180° rotation around any of these bonds changes the conformation of the molecule. ψ1 and ψ3 are considered “soft”, whereas ψ2 involves a higher energy barrier rotation about a double bond that may be affected by synthetic conditions, and to a lesser degree, thermal energy. These dihedral degrees of freedom are analogous to those found in the highest performing, more synthetically complex OSCs used in technological applications.19 How many unique conformations can we enumerate for hexatriene by modifying the three dihedral angles? Simple combinatorics tells us there are 23 dihedral angles (3 dihedrals each with a cis or trans conformation), but two of them are nonunique due to symmetry, resulting in 6 unique conformations. At room temperature, any of these conformations can exist which leads to conformational dispersion. With a few assumptions, we can also understand which conformations will be the most probable to exist within the film. The distribution of these conformations at thermodynamic equilibrium is governed by the Boltzmann distribution learned in undergraduate physical chemistry:

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MOLECULAR SCALE While films of OSCs are composed of billions of copies of a few small molecules or polymers, it is necessary to first understand the behavior of a single molecule. Since OSCs utilize the mobility of valence electrons, an analysis of bulk 1,3,5-hexatriene begins by characterizing the electronic structure of the π-electron system of a single 1,3,5hexatrieneto do this we must first characterize the likely

p(configuration) =

e−U (configuration)/ kBT ∑ e−U (configuration)/ kBT

(1)

where p(configuration) is the normalized probability of a particular configuration of hexatriene’s atoms, U is the energy of the configuration, and kBT is the thermal energy of the configuration. ∑... is a sum over all possible configurations (in this case there are 6) or partition function, which normalizes B



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Figure 2. (a) Six lowest energy 1,3,5-hexatriene conformations. Three dihedral angles (ψ1, ψ2, ψ3; see Figure 1) control the hexatriene conformational degrees of freedom, and relative energies ΔE using extended-Hückel are given for each conformer. HOMO energy values are provided from the Hückel orbital calculations (by color). (b) Torsion parameter values (aij) for each CC or CC bond are listed for each of the six conformations (by color). The trans-1,3,5-hexatriene is shown as the first entry (black font).

the probabilities. Assuming that the molecules in the film are weakly interacting, U can be approximated by the relative energy in the gas phase of the various cis and trans hexatriene conformers. In Figure 2 we have calculated the relative energies (with the trans configuration being the ground state) in terms of kBT at room temperature (T = 300 K) for these different configurations. We executed the calculations utilizing the simple Hückel molecular orbital model (described below) of hexatriene that could be performed by an undergraduate using any simple linear algebra solver.20 Since the Hückel calculation provides the energy, U, of each configuration (the necessary input for eq 1), one can now compute the relative probabilities for all conformations within the film. In contrast to polymers and more complex OSCs, the distribution governed by eq 1 of the hexatriene conformers would be narrow according to the energies of Figure 2, with only the trans and cis conformations being readily accessible at room temperature.21,22 While all other conformations would reside in the low probability tails of the equilibrium distribution, OSC films are often kinetically trapped into disordered morphologies and high energy molecular conformations can be prevalent. For this reason, charge transport is usually highly sensitive to the processing protocols (spin-coating, blade-coating, vapor deposition, etc.) used to make the semiconducting film. The ability of these soft organic molecules to adopt many different configurations is ultimately one of the prime difficulties in controlling their function in devices. Depending on which conformations hexatriene adopts (Figure 1), we note that the electronic energy varies dramatically. Moving forward, we will see that the way electrons move through OSC films will also be strongly affected by changes in the conformations of individual molecules and aggregates.23,24

In inorganic semiconductors, these considerations are much less important, because the bonds holding all atomic orbitals in the system are so much stronger than the energy available at room temperature (kBT) that temperature fluctuations have a minimal effect on the system’s spatial structure. Conversely, in organic materials, these bonds, especially intermolecular, are much weaker, and thus temperature fluctuations can have a dramatic impact on the electronic properties of the molecules, and consequently the device performance. To characterize the electronic structure of hexatriene, we will make use of the Hückel theory of molecular orbitals studied in undergraduate organic chemistry.25 In this approach, the π-orbital system of the conjugated material is described using a linear combination of the atomic 2p-orbitals associated with the moleculethis is a good approximation to the valence band electronic structure of many conjugated systems. It is vital to note that these p-orbitals are the same ones that govern the absorption of light and the movement of electricity in f ilms of organic semiconducting devicesthus, the same methodology used in introductory organic chemistry can be used to qualitatively understand the optoelectronic f unctionality of organic semiconducting molecules. For OSCs we are typically concerned with the energies of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) as these are usually the most reactive and capable of gaining/losing electrons and holes. To understand the dependence of the HOMO/LUMO of hexatriene on its atomic configurations (Figure 2), we build a Hückel matrix of the same size as one of the canonical organic chemistry examples, benzene, but with different connectivity of the matrix elements. In this matrix, α is the energy of the individual atomic orbital, and β is the electronic coupling between orbitals, i.e. the energy associated with the spatial overlap of the orbitals. The energy of the C



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Figure 3. Three 1,3,5-hexatriene dimer configurations: head-to-tail, perpendicular, and side-to-side (top down) with corresponding center-of-mass distances shown above each of the three dimer geometries (left column). Intermolecular binding energies EAB are given for each pair (middle column). Boltzmann weighted probabilities (blue; right column) are also reported for each dimer configuration.

atomic orbital, α, is dependent on the atomic identity of that atom (carbon, nitrogen, oxygen, etc.), whereas the electronic coupling βij between two neighboring bonded atoms depends exponentially on the distance between the two atoms (remember the wave function solutions for the hydrogen atom from physical chemistry), as well as their atomic identity. The Hückel matrix for hexatriene can be written as follows: ij α β12 0 0 0 0 yz jj zz jj zz jj β zz α β 0 0 0 jj 12 zz 23 jj zz jj 0 β zz α β 0 0 j zz 23 34 j zz H AO = jjjj z jj 0 0 β34 α β45 0 zzz jj zz jj zz jj 0 0 0 β zz α β jj z 45 56 z jj zz jj z j 0 0 0 0 β56 0 zz k {

We can add physical realism to the description by calculating α and β using a method developed by Roald Hoffmann, called “extended-Hückel”, that makes use of the geometry of the molecule, the elements, and orbital types (i.e., carbon 2p orbitals for hexatriene) to calculate these values.26 In the extended-Hückel formalism α = 11.4 eV for C 2p orbitals, and for C 2p-C 2p bonds, the electronic coupling can be written approximately as βij = Be−ζrijFij(ψ )

(3) −1

with the value of ζ being 1.5679 bohr for C 2p bonds and rij is the distance between two p orbitals.27 In eq 3 we distinguish between 0° or 180° (cis- or trans-) rotations using a torsional response with a factor Fij(ψ) that weights β by a simple periodic function Fij(ψ) = aij cos ψ. The coefficient aij weights the trans rotations to be slightly more energetically favorable (by 10%) than cis rotations since we know the minimum energy configuration to be a trans conformer (Figure 2b). The parameter B is a coupling constant (16.98 eV) that depends on

(2) D



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Figure 4. Intermolecular charge transport in a hexatriene cluster. Eight hexatriene molecules are classified into electrically coupled networks (a). Molecules are coupled only if they are coupled to all other molecules in the network. The Hamiltonian in eq 5 is color coordinated to the molecules shown above the matrix. The density of states projected onto the molecular orbitals of the cluster is plotted over the energies that span the HOMO and LUMO of the molecules (b). The energy scale is subtracted by the Fermi energy (EF) to center the band gap at 0 eV. Inset: The HOMO and LUMO orbitals are illustrated next to their peak in the density of states.

the atomic constituents of the molecule and can be rigorously defined by the dihedral angles ψij of hexatriene (Figure 1) and bond lengths lij (see SI Section S1; Figure S1). The spectrum of molecular orbital energies for the π-electron system can be derived by diagonalizing the Hückel matrix for hexatriene (eq 2). After filling these energy levels with the 6 π-electrons according to Hund’s rule, the HOMO is associated with the highest filled orbital, and the LUMO with the lowest unfilled orbital. We will perform this procedure for each conformation, recording the HOMO and LUMO values. Calculations of β can be performed using the torsional coefficients (and interatomic distances; see SI Figure S1) specified in Figure 2b. The relative conformational energies via Extended-Hückel are given in Figure 2a for the six lowest energy conformations as well as the HOMO energy values for each conformer. Alternatively, one could use the output of a density functional calculation or Hatree-Fock method to calculate the ground state energies of the 1,3,5-hexatriene configurations for a more accurate characterization of the molecular orbital energy levels (not shown). With this information, we have developed a complete qualitative model for the electronic properties of hexatriene and how they vary with conformation. Assuming the states are thermodynamically accessible and Boltzmann distributions apply, we can assess the distribution of electronic

energy levels in the system and how it changes with temperature.

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AGGREGATE SCALE (10S NM) Now that we have a grasp on how a single hexatriene molecule behaves, we are concerned with how multiple hexatriene molecules behave when aggregated together, and more importantly how electrons are transferred between molecules and ultimately traverse a semiconductor film. First, we consider what some of the most likely configurations of a pair of neighboring molecules will be (just as we needed to understand the likely configurations of a single molecule in the previous section). At equilibrium, the analysis proceeds similarly to the intramolecular case, with the exception that there are many possible arrangements of the molecules. Similar to how the individual conformations of hexatriene had different energies, now the different arrangements of hexatriene molecules in space will have different energies, which at thermodynamic equilibrium will also follow the Boltzmann distribution. As an illustration, we’ve computed the energies of three different pairwise configurations in Figure 3. The corresponding Boltzmann distribution of their configurational probabilities is also provided. Based on this analysis, we can see that it is energetically favorable for the π systems of the two E



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molecular site (HOMO) energy of molecule A is stored along the diagonal as αii, while the intermolecular coupling is stored on the off-diagonal as βij. The intermolecular couplings tell us the pairs of molecules that are capable of transporting charge and is related to the spatial overlap of the two molecular orbitalscharges can jump short distances between molecules more effectively than long distances. One can then inspect the couplings and classify the molecules into networks of electrically coupled molecules. In Figure 4a, we classify the cluster of 8 hexatriene molecules into 4 coupled networks and we distinguished the networks by color. For illustrative purposes, we chose molecules to be in a network only if they are connected to every other molecule in the network. One could define networks based on other factors such as the strength of the coupling or amount of static/dynamic disorder in the site energy or coupling.29 A common characterization of intermolecular transport is through the density of states (DOS) the electron (hole) has access to when traversing the network. In this context, the DOS is tabulated from all of the atomic orbitals (occupied or not) in the molecular cluster and then projected onto molecular orbitals. During incoherent (Marcus hopping) transport25,30 through disordered molecular systems it is necessary to have a coupled network for the electron to traverse (large values of the intermolecular coupling between all molecules in the network), but it is not sufficient. Having a series of tightly energetically spaced states is crucial for the electron to find states it can hop into, as energy must be conserved in each hopping event; to hop effectively, two orbitals must be close in energy and have a large electronic coupling.31 To this end we must have an understanding of how the energies of the molecular orbitals vary from molecule to molecule. In the simplest picture where all intramolecular geometries are static, the HOMO and LUMO energies of all molecules only depend on their conformation.32 To aid in characterizing this landscape of energy levels, one can compute the density of states of all of the molecular orbital energy levels in the systemthis calculation will describe whether many molecular orbitals in the system are energetically near to each other, thus implying efficient transport, or not. In Figure 4b we show the density of states projected onto the molecular orbitals of the eight hexatriene molecules in the cluster. Visual inspection of the low-lying LUMO, LUMO + 1, ... orbitals shows there are delocalized states around 1.5 eV for an electron to occupy (see Figure 4b inset). The breadth of the LUMO (and HOMO) peaks shows that there is disorder in the aggregate around that electronic state and, actually, there are a set of energy levels centered around the individual molecule’s LUMO energy. This is the hallmark of weakly coupled electronic states, where thermal fluctuations within the aggregate lead to broadened electronic energies. The DOS in Figure 4b was calculated using a periodic DFT solver, but for disordered systems DOS calculations via DFT become intractable for more complex molecules or for large aggregates, and statistical means must be used to find the DOS. Based on the results of Figure 4b, at 0 K, the electrons in the system will fill the available orbitals up to the Fermi level. The Fermi level has its roots in the free electron gas model for metals proposed by Sommerfeld,28 but one can think of the Fermi energy level in a molecular system as the energy of the highest occupied molecular orbital for all eight molecules, when filled according to Hund’s rule. However, at room temperature, electrons (holes) will have access to higher

molecules to be aligned and approximately eight times more likely to be observed than the next most probable structure. Extending this approach to more molecules is challenging because we have been relying on a direct evaluation of the partition function in these examples. In practice, methods like Monte Carlo and molecular dynamics are used to sample configurations in more complicated systems for which the partition function cannot be directly calculated. Likewise, developing methods to characterize and predict kinetically trapped structures in soft materials such as OSCs are ongoing research activities. For discussing the question of how electrons hop between molecules in these aggregates, we will use the simple example of an aggregate of eight hexatriene molecules shown in Figure 4a. Whereas for calculating molecular orbitals it was useful to use a linear combination of atomic orbitals, for calculating the orbital structure of collections of molecules, it is useful to use a linear combination of molecular orbitals. For organic molecules that interact primarily through van der Waals forces and weak electrostatic interactions, it is an excellent assumption that only the first few molecular orbitals of each molecule interact with the neighbors. In this context for organic molecules, this is commonly referred to as a tight-binding model,28 which in reality is identical to a Hückel model in form but can be applied to any basis of orbitals, not just atomic orbitals, as in Hückel theoryin this way, we are following the exact same mechanics of Hückel theory used in the previous section, but now entire molecular orbitals (HOMO/LUMO) form the basis, instead of single atomic orbitals (C 2p). From now on, we will pick the HOMO of each hexatriene molecule as the relevant molecular orbital (whereas previously we picked the atomic 2p orbitals of carbon) and form a new Hamiltonian from these. In our new molecular orbital model, αii will be the energy of the HOMO of molecule i. Similar to how eq 2 expressed the electronic coupling between two π-orbitals as a function of distance, we can express the intermolecular electronic coupling between two molecules A and B as a function of distance as well:

βAB = H0e−ζrAB

(4)

Here a typical value of ζ in organic semiconducting systems is 2.0 A−1, and typical values of H0 are between 1 and 25 eV. Once we form this Hückel matrix, we can diagonalize it to derive the energies of the system. ij α11 β12 0 0 jj jj jjj β12 α22 0 0 jj jj jj 0 0 α33 β 34 jj jj jj 0 0 β α jj 44 34 HAB = jjj jj 0 0 0 β jj 45 jj jj jj 0 0 0 0 jj jj jjj 0 β27 0 0 jj jjj k 0 0 0 0

0

0

0

0

0

β27

0

0

0

β45

0

0

α55 β56 β57 β56 α66 β67 β57 β76 α77 0

0

0

0 yz zz z 0 zzzz zz z 0 zzzz zz z 0 zzzz zz z 0 zzzz zz z 0 zzz zz z 0 zzzz zz z α88 zz{

(5)

Equation 5 shows the undiagonalized HAB for a representative cluster of eight interacting hexatriene molecules (see Figure 4a) that could conceivably be drawn from a distribution obeying the Boltzmann probabilities shown in Figure 3. The F



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Figure 5. Calculating transport networks over longer length scales. (Left) Schematic illustration of a multidomain film sample of hexatriene. (Middle) Calculation of the electrically connected components (i.e., transport networks) that satisfy kct > kgrain, kct is the largest intermolecular charge transfer rate for a given molecule in the network, and kgrain is the rate of intergrain charge transfer. By coloring the connected components, the existence of grain boundaries and molecular defects (dark gray) becomes obvious. (Right) Calculation of the electrically connected components that satisfy kct > kpara, where kpara is the intragrain hopping rate and kpara > kgrain. Paracrystallinity contributes to reduced electron and hole mobility within a grain.

As the network becomes large, we can begin to use concepts from graph theory15,37 to evaluate if the networks percolate the film and characterize the spatial structure of the transport networks. Graph theory also provides tools to evaluate the least-resistive pathways through the film and identify physical bottlenecks to charge transport. We hope the above analysis provides a template for the reader to bridge the gap between molecular level quantum methods and transport calculations over disordered aggregates of molecules. We do not want to lose sight of the fact that all of the preceding electronic analysis is just an extension of the Hückel theory ideas taught in organic chemistry−forming model Hamiltonians with different atomic (C 2p) or molecule (HOMO) orbitals, then diagonalizing or analyzing these Hamiltonians directly to understand the layout of energy levels in the system. The complications arise as one incorporates structural disorder into the magnitudes of the couplings and energy levelsin rigid molecules (such as benzene), such considerations are irrelevant because all conformations fluctuate around a local minimum. In soft, organic semiconducting molecules, multiple conformations exist around kBT and can thus be accessed, changing the electronic structure of the system as a function of temperature or processing.

(lower) energy orbitals in the DOS, as thermal energy will allow the electrons/holes to occupy a distribution of orbitals. We can predict the electronic states the electron/hole would have access to by using an ensemble formulation of the structures present in the aggregate using the statistical measure, the probability the electron occupies an energy of E. Once again, the Boltzmann distribution provides such a probability distribution, allowing us to choose the energy from an ensemble of energies based on the molecular disorder being in equilibrium with a thermal bath. As with intramolecular disorder, processing can have important influences leading to non-Boltzmann distributions, such as polymers can have degrees of freedom frozen when they are below the glass transition temperature,33 small molecules can organize into liquid crystalline domains with defect boundaries,34 and large intermolecular dispersion interactions promote kinetically trapped clusters;35 all these phenomena may lead to disordered structures.36 The molecular organization governed by equilibrium thermodynamics may lead to either electronically highcoupling or low-coupling arrangements. In the case of hexatriene, from Figure 3 we see that side-to-side dimer geometries are favored. If this geometry also gave the highest electronic coupling, we would want to promote that arrangement in the device. The promotion of geometries can be achieved through polymerization or nonbonded interactions. Now that we have a mechanism to handle the energetic disorder, we can turn our attention to predicting the transport behavior of electrons in aggregates. Expanding the network approach from small aggregates to domain-sized aggregates in films is straightforward (Figure 5). One just needs to develop the appropriate physical criterion to determine connected members of the transport network. Perhaps the most straightforward criterion is to use the rate of charge transfer (kct) to determine if orbitals are electrically connected in any physically meaningful sense. Since the charge transfer rate is a continuous value, the relevant threshold will be application specific, and evaluating connectivity for different threshold values can give insight into different sources of resistance in a semiconducting film. A schematic illustration is presented in Figure 5 showing how grain boundaries and paracrystalline disorder affect the transport networks by evaluating the network distribution at different values of kct.

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DEVICE SCALE (100S NM) Now that we have a description of the aggregate scale, we need to understand how aggregates of molecules interact over length scales relevant to an actual device. In many common applications, especially organic photovoltaic devices, a mixture of two slightly miscible molecules is used, typically with different molecular structures and thus different electronic structures. The performance of a device may vary drastically depending on how these different molecules interact with each other. An important part of the descriptive process is being able to determine what the optimum formation of domains might be for this mixture of molecules. To this end, we can introduce a simple physical model, known as the Ising model and covered in the statistical mechanics part of a physical chemistry course, to predict domain formation in mixtures. The Ising model, originally developed to predict the magnetic behavior of ferromagnets, is a statistical means to generate and G



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Figure 6. Two-component domain decomposition illustrated with the 2D spin Ising model. Blue pixels show down-spins, red pixels show up-spins, and the system is 550 × 550 spins. High-temperature (annealed), with a reduced temperature Tr = 2.7 and an external field M = 0.1, shows coalesced domains (a). Low-temperature (quenched), Tr = 0.25; M = 0.1, shows homogeneous small domains (b). Domain coarsening time series of the time-dependent Ginzburg−Landau model. Three snapshots of spin densities for increasing time t (in # of time steps): 102, 103, 104. The system size is 150 × 150 density points.

domain size is a very sensitive balance of chemical identity and “processing”, which we can loosely think of as the kinetic path toward equilibrium that the simulation takes. A simple, commented program written in Python is available in the SI for a straightforward implementation and visualization of the Ising model results. The kinetic path the domains take can be modeled using a mean-field approach taken from the Ising model. The methodology described below can be easily replicated using a simple Python code provided in the SI that allows for exploration of the parameters in the model and creates the plots we show in Figure 6. The Ginzburg−Landau equation uses a two-component density field to model the phase behavior during the equilibration process. We write the timedependent Ginzburg−Landau equation as38

predict the phase behavior of materials based on discrete interactions, be it atomic magnetic spins or enthalpic interactions of molecules. Although the model is deceptively simple, just pairwise spins interacting on a lattice (see Figure S2), it readily captures the domain formation of two miscible molecules in a film quite well. We have visualized the Ising interaction on a lattice in the SI in section S2. Randomized simulations, like Monte Carlo, are usually used in conjunction with the Ising model to generate the thermodynamically favored domains for a given interaction between two molecular species. By picking different values of the molecular interactions, we can run the simulation to generate different morphologies. We show in Figure 6a,b the Ising model predictions for two temperatures. The more chemically dissimilar the molecules, the higher the interaction energy between dissimilar molecules, and the larger the domain formationloosely speaking, one can think of this in the same framework as “like-dissolves-like” concepts of solubility, as one can think of the solubility of one molecule, within the bulk of the other. The longer the simulation is left to run (the closer to equilibrium the simulation gets), the coarser (larger) the domains become. Consequently, the formation of a desired

φ̇ =

1 2 ∇ φ + A(Tr )φ − Bφ3 τ

(8)

where φ is the density field, A(Tr) is a temperature dependent parameter that is positive when the system temperature T is greater than the critical temperature Tc, Tr is the unitless reduced temperature (T/Tc), B is a constant, and τ is a time H



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have a distribution of electronic states and statistical models provide a way to sample those states without knowledge of the structural details of every conformation.23 Yet, because there is a direct relationship between the molecular structure and electronic energy one can use statistical models to give guidance on how to tune the interatomic interactions to limit the energetic dispersion,23,24 such as how weak, noncovalent bonds can be used to lock-in conformations to limit the energetic dispersion seen in a variety of copolymers.42 At the aggregate level, enumerating the electronic states becomes even more impractical so statistical techniques to account for the energetic dispersion due to intermolecular disorder must be employed. These include a statistical treatment to describe the density of states the hopping charges have access to as well as the intermolecular organization the molecules arrange themselves into. As with the molecular level disorder, the Boltzmann distribution can be utilized to describe the intermolecular degrees of freedom within an aggregate. Using our test molecule 1,3,5-hexatriene, we enumerated the intermolecular binding energies and detailed the energetic broadening on the electronic density of states. One possible conclusion from the results is that if we forced the hexatriene to aggregate into parallel stacks the charge could more efficiently hop between the molecules without getting stuck in a trapped state. The formation of domains in actual devices requires an understanding of the domain formation process. Without that understanding it is impossible to predict the domain morphologies of the device to assess any given material’s performance. A straightforward way to connect the miscibility of two organic materials and the domain morphologies is through the Ising model. For condensed materials such as organic electronics, the Ising model provides a construct to choose domain morphologies. Statistical simulations, such as Monte Carlo, used in conjunction with an Ising model can efficiently generate domains at the device scale (10−100 μm in size). In addition to predicting morphologies of devices, these models can give insight into the processability of these materials by elucidating the kinetic pathways of the domain coarsening. The majority of organic electronic films are spin coated and thermally annealed,43,44 such that during the annealing time, which is almost always system dependent, the domains coarsen and grow. There is limited understanding at the molecular level how the coarsening affects the performance, but connecting a domain size and shape molecular level parameters would go a long way to rationalizing the design and processability of the materials in the devices. In an advanced undergraduate studies class that incorporates transport phenomena, the goal should not be to just describe the transport mechanisms but to bridge the descriptions of the charge transport for each length scale (molecular, aggregate, and device). The utility of using detailed molecular descriptions to predict the charge transport properties of OSC devices is the direct physical connection between molecular and electronic structures and transport observables. While simple metrics make device performance across any material deceptively easy, it can reinforce the thought that the charge transport phenomenon is the same. It is not! Transport in OSCs is vastly different from its solid-state counterpart. We present the physical process of electrical current generation (or recombination in organic LEDs) in OPV devices based on a framework of molecular interactions governed by the structural organization of the system. Using this framework, we describe

constant. From the solution of eq 8, we show in Figure 6c the coarsening process of a miscible mixture near the critical temperature of the phase transition. The boundaries between the materials are diffuse and are shown in white where the materials are mixed together. As long as the temperature is near the critical temperature this mean-field approach predicts the phase behavior of mixtures. The utility of these models in films of organic electronics is in generating device scale morphologies for charge transport calculations and has recently been used to generate bulk heterojunction morphologies.39 Essentially these models are predicting morphologies that incorporate the opposing forces between the enthalpic desire of two molecules to self-aggregate and their entropic desire to mix. As we show in Figure 6, how one makes these films can have a dramatic impact on how these domains look. Using this larger scale means of domain formation in conjunction with the other length scale characterizations outlined in the preceding two sections, one can semiquantitatively characterize the electronic transport characteristics of an entire, noncrystalline film.

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DISCUSSION The chemical research community has made substantial progress toward the understanding of the physical processes of charge transport in OSCs. Yet, we can do better to provide the pedagogically accessible description of the physical processes going on in an OSC device, specifically in relating the processes to fundamental constructs such as thermodynamics, reaction kinetics, and statistical mechanics. The inclusion of OSCs in classroom discussions must go beyond thinking of devices only in terms of a single parameter that captures the efficacy of the material without much connection to physical processes. While an efficiency metric (or goal posts) to compare devices across a variety of materials and photophysical processes can be motivational, such a metric is generally an opaque physical descriptor that covers-up a multitude of physical processes. For a new student entering the field of renewable energy research, such an emphasis on global efficiency metrics can obscure the fundamental multiscale processes occurring within the device that must be rationally controlled to create more efficient future devices. As such, while an efficiency metric can serve as useful practical motivation for new students entering a field, we hope that the simple models outlined herein will serve as a practical introduction to the fundamental processes underpinning these devices such as intermolecular couplings and site energies, solution processing conditions, charge mobilities, and midgap (Coulomb) trap states.40 We have described in this article a simple multiscale framework for thinking about how organic electron materials transport charge in the presence of thermal disorder that is suitable for use at the advanced undergraduate level. At the smallest length scale, we described the effect disorder has on the electronic orbitals the charge carriers occupy, e.g. HOMO and LUMO. The dispersion of the orbital energy levels is especially evident in polymeric materials where often conjugation of the pi−pi bonds between monomers follows a thermal distribution. It is infeasible to enumerate all of the electronic states for large molecules like polymers, so statistical models are employed to predict the thermal distribution. In previous studies, we have shown that HOMO energy levels of short, semiconducting fragments can differ by over 0.25 eV due to multiple conformations!41 Clearly, larger, more disordered, molecules such as polymers I



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macroscopic domains of molecules may be more difficult to connect with the statistical thermodynamic background of the student. As a supplement, we have two python programs to give the students a hands-on experience with making domains based on interacting spinsone based on Ising interactions and the other solving the time-dependent Ginzburg−Landau equation. Both programs are fully documented and automatically export images that contain snapshots of the domains that help visualize the macroscopic domains. Additionally, we include a Discussion section in the Supporting Information (SI-3) on the thermodynamics of mixing two miscible molecules using the Flory parameter and a mean-field approach. As we mentioned above, we have presented the topics in a modular fashion such that they could be added in a piecewise manner to either Organic Chemistry I and Physical Chemistry I as extended topics building on the principles detailed above. The sacrifice would be the students would no longer have a holistic view of organic electronics in chemistry, but, instead, OSCs would serve primarily as the motivating example to underline the principles in the class while no longer standing alone as a topic. A possible outline of this implemented in a special topics course would be 6 weeks of lecture divided into two week sections, each focusing on the three length scales: molecular, aggregate, and device. We imagine each week having at least 2 h of lecture time available as well as at least the same amount of time for outside group work. The group work would consist of using the Hückel matrix to predict molecular configurations and to enumerate their relative probabilities. Each week the project work would build on itself e.g. using the intermolecular Hückel model to predict the electronic couplings while using the same molecules throughout each length scale. If implementing specific topics piecemeal is the only option, both the supplemental Flory− Huggins and Ising model/Ginzburg−Landau python codes can be assigned as project work, where the students can visually see how the strength of model “molecular” interactions changes the domain size and shape. At the smaller length scale, assigning the above Hückel examples as outside work allows the students to work through populating a Hückel matrix for a specific OSC molecule. The Hückel model then guides the student to critically think about how the electronic properties of a OSC molecule affects the structural diversity and hence intermolecular electronic transport. As laid out above, we hopefully have shown the reader that the physical understanding of electronic processes in OSCs is built upon constructs that are intuitive and connect with each other in a multiscale way. The principles we use are based on the Conceptual and Practical Topics that the ACS Curriculum guide advises accredited departments cover in the organic and physical chemistry sequence. While the three length scales are modular and could be plugged into the courses separately, we recommend this topic be handled as a cohesive unit in a Senior-level topics course.

charge transport using foundational undergraduate theoretical constructs. Specifically, molecular disorder can be described using statistical mechanics via the Boltzmann distribution. The rate of incoherent electron transfer between neighboring molecules can be described with Fermi’s Golden Rule, electronic structure via tight-binding models, and electron or hole mobilities via molecular percolation networks. All of these constructs are grounded in well-known physics and chemistry topics and are accessible to upper-level undergraduate chemistry and chemical engineering students.

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COURSE IMPLEMENTATION The topics presented here would benefit from an extended and immersive course discussion; not because of the undergraduate student’s unfamiliarity with the ideas but because the topics build off one another and would be more cohesive as a single course unit. Ideally, this topic would be part of a “physical processes in organic electronics” unit in a senior-level special topics course. An undergraduate senior would have the additional benefit of having taken the full organic and physical chemistry course sequence; courses which underpin the multiscale methodology described above. On the other hand, we have broken up the sections in a modular fashion and they could be added separately into core courses as an organic electronics application, e.g. the use of Hückel theory to calculate the electronic couplings of OSC small molecules in an organic chemistry course. There are certainly downsides to breaking up the topics into separate courses, but below we outline what assigned work outside of class can help ameliorate the disadvantages. Here, we aim to show how this construct can be presented to upper-level undergraduates in an ACS curriculum, either as a unit in an in-depth (Senior) Special Topics course or as modular units in core foundational courses such as organic45 and physical chemistry46 courses. As part of an in-depth Special Topics course, we have designed the elements of this OSC construct to integrate with the foundational aspects of Organic Chemistry I and Physical Chemistry I. Linking with organic chemistry principles would require reminding the students about pi-stacking of conjugated molecules. We use 1,3,5-hexatriene as a motivating example above, but mixed aromatic molecules could be just as easily used. Pi-stacking would require the students to think about the single molecule pi-orbitals, which build into the core organic concept of the interplay of electronic and orbital interactions and how they can describe the behavior of molecules. We have used the Hückel matrix to organize the atomic orbital energies and electronic bonding between the adjacent atoms. The key here is for the students to think about how the atomic orbitals interact via the Hückel matrix to give insight into what the stable configurations of the molecule will be. If this concept is well-understood, then generalizing to aggregates of molecules will be more natural, again using the Hückel formulation for intermolecular electronic coupling. Connections with Physical Chemistry I topics mostly correspond to a statistical treatment of thermodynamics; using Boltzmann ensembles to describe the probability of molecular configurations and then generalizing to configurations of clusters of molecules. We believe that as long as the student is comfortable with a statistical model, the connection to using the Boltzmann distribution to compute the configuration probabilities will be intuitive. At the device (macroscopic) scale the use of statistical models such as the Ising (or Flory−Huggins) and Ginzburg−Landau to describe

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.8b00064. Ising model and time-dependent Ginzburg−Landau solver attached as a Python program. (TXT, TXT) J



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(15) Savoie, B. M.; Kohlstedt, K. L.; Jackson, N. E.; Chen, L. X. a. Mesoscale molecular network formation in amorphous organic materials. Proc. Natl. Acad. Sci. U. S. A. 2014, 111 (28), 10055−10060. (16) Clean Energy Institute: Lesson Plans. http://www.cei. washington.edu/education/lessons/. (17) Forrest, S. R. The path to ubiquitous and low-cost organic electronic appliances on plastic. Nature 2004, 428, 911. (18) Longuet-Higgins, H.; Salem, L. The alternation of bond lengths in long conjugated chain molecules. Proc. R. Soc. London, Ser. A 1959, 251, 172−185. (19) Usta, H.; Facchetti, A.; Marks, T. J. n-Channel Semiconductor Materials Design for Organic Complementary Circuits. Acc. Chem. Res. 2011, 44 (7), 501−510. (20) Gilbert, A. T. B. Calculating Excited States. In Q-Chem. Computational Labs; Krylov, A. I., Ed.; 2017; Vol. 1. (21) Hayes, S. C.; Silva, C. Analysis of the excited-state absorption spectral bandshape of oligofluorenes. J. Chem. Phys. 2010, 132 (21), 214510. (22) Clark, J.; Nelson, T.; Tretiak, S.; Cirmi, G.; Lanzani, G. Femtosecond torsional relaxation. Nat. Phys. 2012, 8, 225. (23) Noriega, R.; Rivnay, J.; Vandewal, K.; Koch, F. P. V.; Stingelin, N.; Smith, P.; Toney, M. F.; Salleo, A. A general relationship between disorder, aggregation and charge transport in conjugated polymers. Nat. Mater. 2013, 12 (11), 1038−1044. (24) Nelson, J.; Kwiatkowski, J. J.; Kirkpatrick, J.; Frost, J. M. Modeling charge transport in organic photovoltaic materials. Acc. Chem. Res. 2009, 42 (11), 1768−1778. (25) Atkins, P. W.; De Paula, J. Physical chemistry, 9th ed.; W.H. Freeman and Company: New York, 2010; p 564−619. (26) Hoffmann, R. An Extended Hückel Theory. I. Hydrocarbons. J. Chem. Phys. 1963, 39 (6), 1397−1412. (27) Treboux, G.; Maynau, D.; Malrieu, J. P. Combining Molecular Mechanics with Quantum Treatments for Large Conjugated Hydrocarbons. 1. A Geometry-Dependent Hueckel Hamiltonian. J. Phys. Chem. 1994, 98 (40), 10054−10062. (28) Ashcroft, N. W.; Mermin, N. D. Solid state physics; Holt: New York, 1976; p xxi, 826 p. (29) Troisi, A., Dynamic disorder in molecular semiconductors: Charge transport in two dimensions. J. Chem. Phys. 2011, 134 (3), 034702 (30) Marcus, R. A. On the Theory of Oxidation-Reduction Reactions Involving Electron Transfer 0.1. J. Chem. Phys. 1956, 24 (5), 966−978. (31) It is very much an open question on what necessary molecular conditions must be present in disordered organic films in order to have strong electronic coupling and narrowly dispersive orbital energies. In a majority of proposed OSCs, the rationale behind the transport efficiency of one molecule over another is inconclusive. It is clear though that given the weakness of intermolecular interactions (∼kBT) where van der Waals is the dominate interaction, some energetic dispersion along with transport pathway (via electronic coupling) redundancy must be embraced during the design of OSC devices. (32) Jackson, N. E.; Kohlstedt, K. L.; Chen, L. X.; Ratner, M. A. A nvector model for charge transport in molecular semiconductors. J. Chem. Phys. 2016, 145 (20), 204102. (33) Kim, C.; Facchetti, A.; Marks, T. J. Probing the Surface Glass Transition Temperature of Polymer Films via Organic Semiconductor Growth Mode, Microstructure, and Thin-Film Transistor Response. J. Am. Chem. Soc. 2009, 131 (25), 9122−9132. (34) Chen, J.; Tee, C. K.; Shtein, M.; Anthony, J.; Martin, D. C., Grain-boundary-limited charge transport in solution-processed 6,13 bis(tri-isopropylsilylethynyl) pentacene thin film transistors. J. Appl. Phys. 2008, 103 (11), 114513 (35) Oh, J. H.; Sun, Y.-S.; Schmidt, R.; Toney, M. F.; Nordlund, D.; Könemann, M.; Würthner, F.; Bao, Z. Interplay between Energetic and Kinetic Factors on the Ambient Stability of n-Channel Organic Transistors Based on Perylene Diimide Derivatives. Chem. Mater. 2009, 21 (22), 5508−5518.

We include a discussion section on the extended Huckel coupling parameter. We also include a schematic on the Ising model and a Flory−Huggins thermodynamics discussion section. (PDF)

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Kevin L. Kohlstedt: 0000-0001-8045-0930 Nicholas E. Jackson: 0000-0002-1470-1903 Brett A. Savoie: 0000-0002-7039-4039 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS K.L.K. thanks Martin Mosquera for thoughtful conversations and discussions on DFT topics. We thank the U.S. DOE-BES Argonne-Northwestern Solar Energy Research Center (ANSER), an Energy Frontier Research Center (Award DESC0001059), for funding this project.

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