Theoretical study of the charge-transfer state separation within Marcus ...

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Theoretical Study of the Charge-Transfer State Separation within Marcus Theory: The C60-Anthracene Case Study Riccardo Volpi,†,§ Racine Nassau,† Morten Steen Nørby,†,‡ and Mathieu Linares*,†,§ †

Department of Physics, Chemistry and Biology (IFM), Linköping University, SE-581 83 Linköping, Sweden Department of Physics, Chemistry and Pharmacy, University of Southern Denmark, DK-5230 Odense M, Denmark § Swedish e-Science Research Centre (SeRC), Linköping University, SE-581 83 Linköping, Sweden ‡

S Supporting Information *

ABSTRACT: We study, within Marcus theory, the possibility of the charge-transfer (CT) state splitting at organic interfaces and a subsequent transport of the free charge carriers to the electrodes. As a case study we analyze model anthracene-C60 interfaces. Kinetic Monte Carlo (KMC) simulations on the cold CT state were performed at a range of applied electric fields, and with the fields applied at a range of angles to the interface to simulate the action of the electric field in a bulk heterojunction (BHJ) interface. The results show that the inclusion of polarization in our model increases CT state dissociation and charge collection. The effect of the electric field on CT state splitting and free charge carrier conduction is analyzed in detail with and without polarization. Also, depending on the relative orientation of the anthracene and C60 molecules at the interface, CT state splitting shows different behavior with respect to both applied field strength and applied field angle. The importance of the hot CT in helping the charge carrier dissociation is also analyzed in our scheme. KEYWORDS: organic solar cell, charge transfer state, splitting, separation, interface, Marcus theory, kinetic Monte Carlo

I. INTRODUCTION The field of organic electronics has gained much attention in recent years. Organic electronics can provide new functionalities1−4 that traditional silicon-based electronics cannot, such as flexibility and transparency. Organic electronics also have the potential to be cheaper and more environmentally friendly than their inorganic counterparts. Organic conjugated molecules have been used to create various devices, such as organic solar cells, organic light-emitting diodes (OLEDs), and organic fieldeffect transistors (OFETs). Organic solar cells are currently of great interest as climate change and energy security have become pressing issues. While organic solar cells have been improving rapidly, they are still not efficient or stable enough to be competitive with inorganic solar cells.5,6 Currently, there is a consensus on the functioning of organic solar cells, as has been described in several articles.7−11 When a photon is absorbed in an organic solar cell, an exciton is formed. If that exciton reaches an interface between an electron-donor material and an electron-acceptor material before decaying, the exciton can split, with the electron and hole on adjacent acceptor and donor molecules, respectively.12 This is called a charge-transfer (CT) state.11 The charge carriers can then either recombine or separate. If the charge carriers separate into free charges, they can move toward the electrodes and be collected or, if they travel toward an interface, © 2016 American Chemical Society

they can recombine with other charge carriers at the interface. The processes that occur at the interface greatly impact the efficiency of the organic solar cell, so the morphology of the interface is important in improving the efficiency. Previous theoretical studies have demonstrated the impact of interface morphology on organic solar cell efficiency.13−16 The simplest interface structure is that of a bilayer cell, consisting of a layer of donor material and a layer of acceptor material joined together.17 However, since excitons can generally only diffuse up to 10 nm before decaying, if a photon is absorbed at a distance greater than 10 nm from the interface of these two layers, the exciton will likely decay. On the other hand, increasing the thickness of the layers increases the photon absorption, a trade-off between light absorption and exciton decay must be made. A compromise can be achieved by the bulk heterojunction (BHJ) solar cell, in which the two layers are intermixed.18 In this structure, the solar cell can be thicker without sacrificing distance to the interface. However, a tradeoff must also be made in BHJ solar cells, as free charge carriers coming from a given exciton can still recombine with other charge carriers at the interface. While greater intermixing of the Received: June 7, 2016 Accepted: August 26, 2016 Published: August 26, 2016 24722

DOI: 10.1021/acsami.6b06645 ACS Appl. Mater. Interfaces 2016, 8, 24722−24736

Research Article

ACS Applied Materials & Interfaces

Figure 1. MD simulated (01-1) interface (a) and (001) interface (b) of C60 and anthracene. The orientation of anthracene at the interface is different for the two boxes.

state (or cold CT state), that is, the CT state composed of an electron on the lowest unoccupied molecular orbital (LUMO) of the acceptor molecule and a hole on the highest occupied molecular orbital (HOMO). The role of the hot CT state (i.e., electron and/or hole on a higher energetic level) is highly debated in literature and the precise mechanism favoring efficient exciton dissociation is still unknown, in particular the role of the excess energy resulting from absorption of a photon with energy higher than the HOMO−LUMO gap. Several studies suggest how this excess energy is useful for the exciton dissociation, opening up new paths (on higher energy levels) for charge dissociation.47−53 Other studies, focused on polymer−fullerene solar cells, instead point out how the exciton dissociation in these systems is independent of the excess energy.54−56 In this Research Article, we will show how, thanks to the CT state splitting diagram proposed here, it is possible to rationalize the role of polarization on the cold CT state dissociation as well as the importance of the contribution of the hot CT states to the charge dissociation mechanism. To support our analytical considerations, results of kinetic Monte Carlo (KMC) simulations of charge carriers splitting from the cold CT state at the interface between anthracene and C60 were studied. Molecular Dynamics (MD) was used to simulate the geometry of this system, and then KMC simulations were used to model the cold CT state splitting at the interface and the subsequent transport of the dissociated charge carriers.

layers results in shorter distances to the interface and therefore a greater percentage of the excitons able to reach the interface before decaying, the separated charge carriers are more likely to travel near an interface and recombine with another charge carrier before reaching the electrode. Charges hop between molecules or polymer segments, and this hopping occurs with a certain probability, making probabilistic methods such as KMC appropriate for modeling charge transport. The simulation scheme used in this article was inspired by previous Monte Carlo studies on charge transport in organic materials. In early work of Bässler19 the molecules of an organic material were modeled only as a lattice of molecular sites, and the structure of the molecules was neglected. More recent works have improved upon these models by including an atomistic description of the molecules. For example, we can cite the work of Olivier,20,21 Castet,22 and Andrienko.23−27 In previous articles,28−30 we have used our kinetic Monte Carlo code to study transport of a single charge carrier within a single organic material, proposing a new way to calculate the external reorganization energy and model the polarization in organic materials.30 The CT state dissociation at the interface has attracted considerable attention31−33 since the CT state is the main intermediary between exciton and free charge carriers (polarons). A study on the possibility of CT state dissociation at bilayer interfaces is presented in this article. Studies of BHJ interfaces are also present in literature, studying the efficiency of transport and CT state separation as a function of the interface morphology. These simulations employ a site-based KMC with morphology generation algorithms based on the Ising model34,35 or Cahn−Hilliard equation.36 The study of a whole BHJ interface with an atomistic description is beyond the scope of the present article, but an analysis of the effect of the electric field angle on the CT state separation will be presented (to simulate the action of the electric field in these kinds of interfaces). We introduce a theoretical model to study the possibility of CT state splitting of two charge carriers (an electron and a hole) at the interface between two organic materials. As a case study we perform KMC simulations of CT splitting at C60anthracene interfaces. These materials were chosen because they are simple and well studied acceptor and donor materials. C60 has already been studied in our previous article.30 Theoretical studies on the interface of C60 with pentacene and other oligoacenes are present in literature,37−42 together with studies on charge transport and polarization in anthracene and other oligoacene crystals.43−46 These previous works (as well as the present one) mainly focus on the lowest energy CT

II. KMC SIMULATION METHOD Two types of C60-anthracene interfaces have been built depending on the relative orientation of the C60 and anthracene molecules at the interface: the anthracene (01-1)/C60 interface and the anthracene (001)/C60 interface. At the anthracene (011)/C60 interface (Figure 1a), anthracene molecules are πstacked in the direction orthogonal to the interface. Instead, at the anthracene (001)/C60 interface (Figure 1b), the anthracene molecules are π-stacked parallel to the interface. These interfaces were constructed with molecular dynamics (MD) simulation as described in subsection V.A. In this article, we will refer to the MD-simulated anthracene (01-1)/C60 interface as the (01-1) box, and the MD-simulated anthracene (001)/C60 interface as the (001) box. To simulate a CT state, an electron and a hole are placed on adjacent C60 and anthracene molecules, respectively. Let M = (i,m) identify a molecular orbital, composed of the index of a molecule i and the quantum numbers m of the orbital level on that particular molecule. To determine the probabilities at 24723


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ACS Applied Materials & Interfaces which a charge carrier hops from molecular orbital M = (i,m) to molecular orbital N = (j,n) we use the Marcus formula57

where p is the collection of all the atomic dipoles p⃗b and E is the collection of the electric fields E⃗ b (i.e., the electric fields felt at the atomic positions rb⃗ ). The matrix B can be split into Ni rectangular matrices 3 × 3Ni thus obtaining

2π 1 |HNM|2 ℏ 4πkBTλNM

wNM =

⎛ ( −eE ⃗ ·Δ r ⃗ + ΔE + λ )2 ⎞ NM NM NM ⎟ exp⎜ − 4λNM kBT ⎝ ⎠

pb⃗ = Bb E̲

From the total electric field E we can extract the contribution of the hopping charge c, obtaining

(1)

where HNM is the transfer integral, λNM is the reorganization energy, and ΔENM is the site-energy difference. The action of the electric field E⃗ has been explicitly reported, and ΔrN⃗ M is the difference rN⃗ − rM⃗ . KMC simulations will be performed only on the cold CT state, so only LUMO and HOMO levels are considered for the electron on C60 and the hole on anthracene, respectively. M and N thus always represents the LUMO level for a C60 and the HOMO level for an anthracene.

E̲ = Eext + Ec

IV. POSSIBILITY OF SPLITTING THE CT STATE Once the parameters for KMC simulation have been calculated, an iterative process based on trial and error is needed to establish which are the fields at which the CT state splits. In general this process is slow to perform since many simulations are needed at each field value to obtain a statistically significant ensemble to analyze. Here we present a method to analyze the qualitative behavior of a bilayer interface, based on the parameters calculated for Marcus formula, but without having to run KMC simulations. It is therefore possible to determine a priori the interesting fields for the system under consideration and whether the two chosen donor and acceptor materials are suitable to be used together in an OPV device. The Marcus formula (eq 1) is used to estimate the hopping probabilities. An exhaustive description on how the main parameters of this formula are calculated is explained in ref 30. Both the transfer integrals and the internal reorganization energies are calculated from QM calculations without any external field or extra charges. Theoretical studies support the independence of transfer integrals and internal reorganization energy from the external electric field.20,59 The external reorganization energy (eq 2) is also independent of external field and additional charges, as demonstrated in section III. The only parameter influenced by the field is thus the energy difference, eE⃗ · ΔrN⃗ M, at least in a first approximation. Considering the interaction of atomic charges with the external electric field, ΔrN⃗ M can be written as ⎯r − ∑ q (s)→ ⎯r ∑b ∈ j qb(s)→ b a a∈i a Δ rNM ⃗ = Δ rji⃗ = (7) e

λout = Eiind( s2 , s1) + Ejind( s2 , s1) − Eiind( s2 , s2) (2)

that is, as the difference in energy of the molecules i and j before and after the polarization rearrangement. Here, Eind i ( s2 , s1) is the energy of molecule i whose atomic charges are given by the state s2 (after the charge hopping), embedded in the polarization given by the state s1 (before the charge hopping) Eiind( s1 , s2) =

1 4π ϵ0

∑ ∑ a∈i

b∈h h ∈ Ip(i)

qa(s1i)pb⃗ ( s2) ·( ra⃗ − rb⃗ ) | ra⃗ − rb⃗ |3 (3)

leading to the field contribution

Here, h is a molecule in the polarization interaction set of i, while a and b are atoms belonging respectively to molecule i and molecule h, respectively. See ref 30. The induced dipoles in eq 3 are calculated with the Thole model.58 If there are Ni atoms inside the polarization interaction set Ip(i), the dipoles are obtained as the solution of a system of 3Ni linear equations with 3Ni unknowns

p̲ = BE̲

(6)

where Eext contains the external electric field and the field of all the other charges in the material. In a KMC scheme only one hop at a time is allowed, thus the external part Eext remains the same during the hop of charge c. The reorganization energy (eq 2) involves only differences of Eind i and, as a consequence of the linearity of eq 3 in the electric field E, it depends exclusively on the Ec part of the field. The reorganization energy is thus independent of the external electric field and of any other additional charge in the material (aside from the hopping charge c), and is thus dependent solely on the morphology of the system.

III. INDEPENDENCE OF THE EXTERNAL REORGANIZATION ENERGY FROM EXTERNAL FIELD AND EXTRA CHARGES Following the notation of our previous work,30 s denotes the collection of the quantum states of all the molecules in the system. When a charge carrier hops from molecular orbital M of molecule i to molecular orbital N of molecule j, the states of molecules i and j change. The state of the system s = (s1, s2, ..., si, sj, ...), composed of the quantum states of all the molecules, will thus change according to the charge carrier hop. For example, when an electron jumps from the LUMO of molecule i to the LUMO of molecule j, the system passes from s1 = (0, 0, ..., LUMO, 0, ...) before the jump, to s2 = (0, 0, ..., 0, LUMO, ...) after the jump. The external reorganization energy of this process can be estimated as30

− Ejind( s2 , s2)

(5)

⎛ → ⎯ −eE ⃗ ·Δ rNM ⃗ = −E ⃗ ·⎜⎜∑ qb(s) rb − ⎝ b∈j

⎞

∑ qa(s)→⎯ra⎟⎟ a∈i

⎠

(8)

This formula is valid for both electrons and holes, so the state s represents an extra electron or an extra hole on the molecule. qa(s) is the atomic charge on the atom a when the state of the molecule is s and ra⃗ is the atom position. The atomic charges

(4) 24724


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Figure 2. Construction of the CT state splitting diagram. The average distance between two layers in the direction of the field is represented in Figure (a) for donor (ΔzD) and acceptor (ΔzA) materials, respectively. ΔzD and ΔzA determine the angular coefficient of the straight lines representing the energy gain in hopping to the next layer in the field direction (panels b, c, and d on the bottom). Panels b and c represent the efficiency diagrams (for conduction close and away from the interface) for the electron and for the hole, respectively. The efficiency diagrams for the electron and the hole forming the CT state can be combined to form the CT state splitting diagram (d).

will then be negative or positive depending on s and they will define the preferential direction of the charge (following the field or going against it, for example). The energy of a charge carrier on the molecular orbital M = (i, m), i.e. the orbital m on molecule i, is expressed (at 0 field) as uM = ±ϵM + Eiperm( s ̲ ) + Eiind( s ̲ )

wNM =

2π |HNM|2 ℏ ⎛ (qEΔz + ΔE + λ )2 ⎞ 1 NM NM NM ⎟ exp⎜ − 4πkBTλNM 4 k T λ ⎠ ⎝ NM B (10)

where now ΔzMN is the z component of the M → N hopping movement and q is the charge of the charge carrier performing this hop. The standard deviation of the hopping rate in eq 10 is

(9)

where ϵM is the energy of the orbital M and the ± sign depends on the type of charge carrier (electron or hole). If an electron is added to the molecule, the orbital energy ϵM is added. If a hole is added (removing an electron), ϵM is instead subtracted. In our scheme, ϵ is dependent on M = (i,m) and not only on m, since we perform one QM calculation for every molecule in the box. The energy of the orbital m therefore depends on the particular molecule i considered and its deformation. In eq 9, we also have the energy of the interaction between the atomic charges on i and the nearby molecules: the permanent atomic ) and the induced charges on the nearby molecules (Eperm i ind dipoles on the nearby molecules (Ei ). To find the electric field values at which the charge carriers are expected to separate, let us consider a flat interface in the xy plane with the electric field applied in the −z direction (Figure 2a). The hopping probability (eq 1) has the form of a Gaussian

σNM =

2λNM kBT eΔzNM

(11)

on the electric field axis. The maximum rate for the movement of the charge carrier M → N is obtained when qEΔzNM + ΔENM + λNM = 0. Nonzero rates are obtained when the electric field E is oscillating by ±3 σNM. At the extremes of this interval the electron hopping rate in the z direction is reduced by roughly 99.3% of its initial value, thereby drastically reducing conduction. If ΔENM belongs to the interval (−λNM − 3 σNM, − λNM + 3 σNM) we will express it as qEΔzNM ≏ − ΔENM − λNM

(12)

or equivalently qEΔzNM + ΔENM + λNM ≏0

(13)

The symbol ≏ assumes a precise meaning in this context, specifying that the first member of the equation belongs to the interval identified by the second member ±3 σNM (on the field 24725


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ACS Applied Materials & Interfaces −eEΔzA≏ − (ΔE Fe0−S0int + ΔE Fe0−S0h) − λ Fe0S0

axis). In this way we will obtain shorter formulas, but the reader should keep in mind that the standard deviation σNM is implicit in the notation used. The fields inside the interval specified by eq 13 favor conduction in Marcus Theory. Considering first the movement of the electron, two interesting situations can be analyzed: when the electron is in its initial position at the interface and when the electron is far from the interface. In both these cases, the hole is considered in its initial position at the interface. When the electron is at the interface, the difference in energy for the electron hop from the first (F) to the second (S) layer of molecules in the opposite direction of the field is ΔE Fe0S0 = ΔE Fe0−Sint + ΔE Fe0−Sh0 0

−eEΔzA≏ −

(14)

2π e 2 |H F0S0| ℏ

(15)

with standard deviation (on the electric field axis) 2kBTλ Fe0S0 eΔzA

(16)

Equation 15 represents the efficiency of the process considered (the electron splitting the CT state by passing from the first to second layer) as a function of the external electric field. With the symbol defined in eqs 12 and 13, this can be written concisely as −eEΔzA≏ − (ΔE Fe0−S0int + ΔE Fe0−S0h) − λ Fe0S0

(17)

When the electron is instead far from the interface, the energy difference of a hop between two layers in the z direction can be simplified as e ≈ −eEΔzA ΔE Be B ′0 = ΔE bulk 0

(20)

−eEΔz D≏ − (ΔEOh0−Tint + ΔEOh0−Te0) − λOh0T0 0

(21)

h −eEΔzD≏ − λbulk

(22)

where O and T identify the layers minus one and minus two, respectively. Since we are considering a hole (hence the superscript h), the 0 at the subscript of the layers now specifies the ground state for the hole (HOMO). The conditions imposed by these two equations can be illustrated in the efficiency diagram for the hole, Figure 2c. To have both efficient CT state splitting and free charge carrier collection at the electrodes the field should respect eqs 19−22. This results in a combination of Figure 2b and c in what will be called a CT state splitting diagram, Figure 2d. The CT state splitting diagram plots the electric field strength, the energy of the field gained in jumping to the next layer and the efficiencies of the 4 processes of interest in eqs 19−22. An analysis of the CT state splitting based on these considerations will be presented in section VI.F for the C60-anthracene case study. Introducing excited levels for the electron and the hole leads to the appearance of new gaussians in the CT state diagram. We argue that in the bulk the main transport is given by the ground levels, thus the free conduction gaussians will not be affected significantly and the main effect of the excited molecular levels is on the charge dissociation at the interface (hot CT states). Let us examine the case of the electron, as the reasoning for the hole is analogous. The transfer integrals HeFmSn are typically expected to increase with increasing m and n due to the increasing delocalization of the molecular orbitals. This will result in gaussians with higher amplitudes (see eq 10). Neglecting the change of the reorganization contribution (λeFmSn), we focus on the energy difference (ΔEeFmSn), which is the main factor determining the overlaps. For different values of ΔEeFmSn, we will have hot CT gaussians centered at different positions on the field axis, determining the possibility of the overlap. It will thus be possible to find an energy difference ΔEeFmSn for which the CT splitting Gaussian from Fm to Sn is overlapping with the free conduction Gaussian in the bulk, or with another CT splitting Gaussian that in turn overlap with the free conduction Gaussian in the bulk. In this way it will be possible to establish, at a particular field, which is the most

⎛ ( −eEΔz + ΔE e + λ e )2 ⎞ 1 A F0S0 F0S0 ⎜− ⎟ exp e ⎜ ⎟ 4 λ 4πkBTλ Fe0S0 k T F0S0 B ⎝ ⎠

σFe0S0 =

(19)

depending on whether the electron is near or far from the interface. Equations 19 and 20 specify the conditions for the field to obtain maximum conduction in the z direction, when the electron is near and far from the interface, respectively. Equation 19 is related to the efficiency of the CT state splitting initiated by the electron, while eq 20 is related to the transport of the free electron to the electrode. If these two equations can be satisfied simultaneously, we will obtain both CT state dissociation (initiated by the electron) and free electron transport in the acceptor phase. Figure 2b illustrates these facts in what we will call the efficiency diagram for the electron. The 3σ tolerances are illustrated both on the energy and on the field axis. Analogous considerations can be made for the hole, leading to two other equations regarding the efficient CT state splitting initiated by the hole and the transport of the free hole to the electrode

where the 0 at the subscript specifies that we are considering the ground states (LUMO). ΔEe−int F0S0 is the contribution of the electron-interface interactions in passing from the first to the second layer and ΔEe−h F0S0 is the change in energy due to the Coulombic interaction of the electron with the hole. ΔEe−int F0S0 is composed of the shift of the energy levels of the molecules because of their geometrical deformation at the interface (Δϵ), and by the electrostatic interaction of the interface with the charge carrier due to permanent atomic charges (ΔEperm) and induced dipoles (ΔEind), see terms in eq 9. The contribution of the field in passing from F0 to S0 is − eEΔzA, with ΔzA the distance between two layers of the acceptor molecule in the z direction (Figure 2a). From eq 10, the hopping rate for the electron F0 → S0 as a function of the external electric field is wFe0S0 =

e λbulk

(18)

where B and B′ are two consecutive layers in the bulk. In eq 18 we assumed for simplicity that the distance ΔzA between the layers in the z direction stays the same. Furthermore, the hole is too far to interact with the electron ΔEe−h bulk = 0, and the energy shifts due to molecular distortions (Δϵ), permanent charges (ΔEperm) and induced dipoles (ΔEind) will be neglected since they average to 0 in the bulk. Finally we obtain for the electron movement the pair of equations 24726


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dimensions of the boxes are 8.44 × 8.31 × 17.44 nm3 for the (001) interface and 8.40 × 8.41 × 17.58 nm3 for the (01-1) interface. V.B. KMC simulations. To study CT state splitting from different starting points, six pairs of adjacent anthracene and C60 molecules were chosen. For each of these interfacial pairs, 100 MC simulations were run for every value of the electric field considered. This was performed for both interfaces, using 2D periodic boundary conditions in the xand y-directions. Periodic boundary conditions were not used in the zdirection since this is the direction perpendicular to the interface. Every MC simulation was performed at 300 K. Simulations are performed until at least one of the charges reaches a distance of more than 30 Å from the interface, and maintains that distance for at least 70 hops. In this case the CT state is split and at least one charge carrier reaches its electrode. If this condition is not met, the simulation is ended after 1000 hops. An electron (or a hole) is assumed to be both split from its CT state and able to reach the electrode if it reaches a distance of at least 30 Å in the direction of (or opposite to) the field. This condition will be referred to as the “charge collection condition” and we will say that the respective charge carrier has been collected. We observed that if the electron and hole moved approximately 33 Å apart, they generally do not come back close to each other. At this distance the two charges still feel each other since the Coulombic cutoff used is 40 Å. The CT state is therefore considered “split” if the electron and hole reach a distance of 33 Å from each other. Transfer integrals and hopping rates are computed for couples of molecules with edge-to-edge distance lower than 9 Å. The edge-toedge distance between two molecules is the minimum distance between any pair of atoms belonging to two different molecules. The projection method72 was used to calculate the transfer integrals from the ZINDO-calculated73 wave functions. As described in section IV and fully detailed in our previous work,30 the single site energy uM consists of the following contributions: the orbital energy O, the energetic contributions of the permanent charges C, and the polarization P

probable chain of events leading to the free charge carrier conduction, and if it is possible to find one. The definition of ΔEeFmSn is analogous to the ground state case (eq 14) ΔE FemSn = ΔE Fem−Sint + ΔE Fem−Shn n

(23)

that is, the average energy difference in passing from the m orbital level in the first layer F to the n orbital level in the second layer S. Wave functions corresponding to excited states are more delocalized and can also produce a different momentum in the charge distribution on the molecule (dipoles, quadrupoles, etc.). These represent higher order corrections to the electrostatic energies and we can argue that in a first approximation the electron−hole interactions do not vary significantly from the cold CT state case, that is, ΔEe−h FmSn ≈ . Similar reasoning can be done for the interaction of the ΔEe−h F0 S0 charge carrier with the surrounding in terms of permanent charges (ΔEperm) and induced dipoles (ΔEind). Consequently, the contribution of the interface will mainly differ due to the orbital energy difference Δϵ. As a first approximation the differences in energy without the field can be written as ΔE FemSn ≈ ΔE Fe0S0 − ΔϵeF0S0 + ΔϵeFmSn

(24)

ΔEOhm−TnoF ≈ ΔEOh0T0 − ΔϵOh 0T0 + ΔϵOh mTn n

(25)

Selecting appropriate initial and final levels of the hop m and n, it is possible to obtain ΔϵeFmSn and/or a ΔϵhOmTn leading to a hot CT Gaussian overlapping with the free conduction one, thus allowing for dissociation. This analysis has a purely illustrative purpose, if willing to determine the most probable paths for the evolution of the system, the pure decays of a single charge carrier from higher to lower level on the same layer also have to be considered.

uM = O + C + P

(26)

The site-energy difference ΔUNM is the difference between the single site energies of two different electronic levels, M and N

ΔUNM = uN − uM

V. COMPUTATIONAL DETAILS

(27)

The Coulombic interaction radius used for the permanent charges is 40 Å. To calculate the external reorganization energy and the polarization component of the site-energy difference, the linear Thole model58 for polarization was used with a screening length parameter a = 1.7278 a0, as in ref 74. Intramolecular polarization was neglected. In the calculations accounting for polarization effects, polarization radii of 15, 20, and 25 Å were used. As discussed later in the text, a polarization radius of 15 Å was chosen for the KMC simulations. The reorganization energy is composed of an internal and an external part, λ = λint + λout. The internal reorganization energy λint was calculated as in refs 27 and 37, obtaining values of 0.139 and 0.132 eV for anthracene and C60, respectively. These values were calculated with the software Gaussian75 using the B3LYP76−78 functional in combination with the 6-31+g(d) basis set.79−81 The external reorganization λout is calculated with eq 2, as commented in section III and ref 30. The movement of the nearby molecules is a slower process compared to the time scales of an electron movement and thus MD is not coupled to the KMC simulation. The nearby molecular movement, as remarked in previous section V.A, is considered as an average in our simulations since for the KMC we use an energy minimized geometry (thus the most probable conformation). The internal geometry reorganization is instead a fast process, since it represents the reorganization of covalent bonds. This is the only geometry adjustment contribution considered in the reorganization energy. For more details on the approximations used, we refer the reader to ref 30. The atomic charges and polarizabilities were calculated for a single optimized geometry of each of the following: neutral anthracene, anthracene cation, neutral C60, and C60 anion. The atom-centered moments were calculated with MOLCAS82 and utilized the LoProp

V.A. MD Simulations. Boxes of anthracene molecules and a box of C60 molecules were built by multiplying the unit cell to obtain the desired dimensions. The unit cells used for anthracene and C60 have a herringbone and an FCC structure, respectively. Dimensions for these boxes were chosen considering the lattice dimensions of both materials, so that the dimensions of the anthracene box would be similar to the dimensions of the C60 box at the interface and avoid commensurability issues. Table S1 of the Supporting Information shows the sizes of these boxes and the number of molecules they contain. These boxes were equilibrated using the GROMACS60−64 MD software package. An NPT dynamic at 300 K using a Berendsen thermostat65 was performed on each box for 2 ns with a time-step of 0.001 ps. A Berendsen barostat66 with a coupling constant of 0.2 ps was used to maintain a pressure of 1 atm. Periodic boundary conditions and the OPLS (Optimized Potentials for Liquid Simulations) force field67,68 have been used. The electrostatic and van der Waals interactions were both calculated with a cutoff distance of 1.5 nm. The densities of the anthracene and C60 boxes converged to 1.22 and 1.70 g/cm3, respectively, in good agreement with experimental values of 1.28 g/cm3 for anthracene69 and 1.67 g/cm3 for C60.70 The box of C60 molecules was brought close to the anthracene (01-1) surface to construct the (01-1) box. To construct the (001) box, the C60 box was brought close to the anthracene (001) surface. The NPT dynamic previously described was then again performed on the (01-1) and (001) boxes for 10 ns. An energy minimization using the steepest descent method71 was performed on these resulting (01-1) and (001) boxes. The energy minimization gives us a box in the minimum of the potential energy, so we will consider only the statical disorder, averaging out the dynamical disorder. Final 24727


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ACS Applied Materials & Interfaces localization method.83 These calculations were performed at the B3LYP level of theory with an ANO-type recontraction of the 631+G* basis set. The resulting atomic charges and polarizabilities are reported in Supporting Information. For all our calculations we used the proper atomic charges for neutral molecules, anion (of C60) and cation (of anthracene). Nevertheless the atomic polarizabilities of anion (and cation) roughly differ only 1% from the atomic polarizabilities of the neutral molecules. This fact would in principle allow us to use only neutral polarizabilities. From the definition of the polarization energy P (in ref 30) it is in fact clear that the correction on the P part of the energy is of the same entity (1%). On the external reorganization energy this discrepancy is even smaller due to cancellation of errors, originating from the way in which the reorganization energy is calculated (eq 2). In the following we will use a suffix for every system studied, to specify which contributions are included in our calculations. The DOS of O systems is formed only by the QM orbital energies. For OC the energy is composed of orbital energies corrected by Coulombic interactions, while for OCP all the three contributions are included. When including polarization (P) the external reorganization energy λout will also be included. V.C. CT State Splitting Diagram. The parameters of eqs 19−22 have been calculated for the (001) and (01-1) boxes. The interlayer distances, the transfer integrals, the energy differences and the external reorganization energies are obtained as an average over the six couples considered for the KMC simulation (section V.B). For each couple, first we obtain the parameters for a hopping to the next layer when the electron and the hole are in the initial CT state at the interface. Since we want to consider the dominant paths, in the transfer integrals average we will neglect the |HNM|with values lower than 7 meV. The energy difference and the external reorganization energy are calculated with a weighted average, using the square of the associated transfer integrals as weights (as in the Marcus formula). Once we obtain the parameters for the charges in the initial CT state, we move each charge 50 Å away from the interface and we average the parameters again in the same fashion, obtaining the values for the hop in the bulk. The parameters obtained after averaging over all the six couples in Table I.

the (001) interface, however, the interfacial C60 molecules face hydrogen atoms of the anthracene molecules. Therefore, there is a higher π-orbital overlap between the C60 and anthracene molecules at the (01-1) interface than at the (001) interface. This π-orbital overlap stabilizes the molecules at the (01-1) interface, giving these interactions a lower energy per square nanometer than at the (001) interface. Because of these interactions, the anthracene molecules at the (01-1) interface shape themselves around the C60 molecules during the MD simulations, destroying the crystalline structure of the anthracene molecules near the interface, as shown in Figure 1. The MD simulations therefore introduced greater disorder in the (01-1) box than in the (001) box. VI.B. Density of States. To analyze the charge transport properties of the different boxes, the density of states and transfer integrals were studied. Figure 4a−c shows the density of states (DOS) of an electron on the C60 molecules in the (001) and (01-1) boxes. We will follow the nomenclature used in our previous article30 (see also section V.B) and refer to the orbital energy as O, the orbital energy corrected for permanent charges OC, and the orbital energy corrected for permanent charges and polarization OCP. Since C60 has zero permanent atomic charges in its neutral state, the only C60 molecules that are affected by the correction for permanent charge are those near the interface, as they feel the permanent charges from the nearby anthracene molecules (the green bulk peak in Figure 4b and c). The OC density of states presents opposite corrections (stabilizing or destabilizing) for the two interfaces. At the anthracene (001) interface, the C60 molecules are close to the hydrogen atoms of the interfacial anthracene molecules. Since the hydrogen atoms in the anthracene molecules have a positive partial charge, an electron on a C60 molecule will feel a stabilization effect from the interface. At the anthracene (01-1) interface, as described in section VI.A, the C60 molecules are closer to the negatively charged anthracene carbon atoms. This creates a destabilizing effect on an electron situated on an interfacial C60 molecule. However, different trends can be seen when polarization is considered in addition to the permanent charges (Figure 4a−c and also Figure S3). C60 is a highly polarizable molecule, so a charged C60 molecule surrounded by other C60 molecules can be more stable than a C60 molecule near anthracene molecules. This is especially true for the anthracene (001) interface because the C60 molecules near the interface are facing hydrogen atoms, which are not very polarizable. At the anthracene (01-1) interface, the C60 molecules can have positive or negative dipoles (and thus a stabilizing or destabilizing contribution) depending on its relative position with respect to the permanent quadrupoles of the anthracene interfacial layer. A detailed discussion of this phenomenon for pentacene/C60 interfaces can be found in refs 40 and41. When including effects from both permanent charges and polarization, charged C60 molecules are in general less stable at the interface than in the bulk for both interfaces. However, when polarization is neglected, this occurs only for the (01-1) box. Anthracene molecules in a crystalline arrangement experience a stabilization effect from the quadrupoles of the nearby anthracene molecules. As neutral C60 has no permanent charges, a hole on an interfacial anthracene molecule is less stabilized due to permanent charge than in the bulk. In contrast with C60, for anthracene the permanent charges correction affects all of the molecules. This can be seen in Figure 4d−f, where the entire O curve gets shifted when considering

Table I. Parameters for the CT State Splitting Diagrama at the interface DOS

system

OC

(001) (01-1)

OCP

(001) (01-1)

a

charge carrier

HNM

ΔENM

electron hole electron hole electron hole electron hole

0.014 0.014 0.017 0.039 0.014 0.014 0.017 0.039

0.74 0.75 0.80 0.40 0.50 0.60 0.65 0.10

in the bulk λout

HNM

λout

0.41 1.11 0.41 0.75

0.016 0.012 0.016 0.025 0.016 0.012 0.016 0.025

0.41 1.13 0.40 0.75

All quantities are in eV.

The obtained average interlayer distances are ΔzA = 7.2 for C60, ΔzD = 9.6 for anthracene (001), and ΔzD = 4.6 for anthracene (01-1). A more detailed description of how these parameters are calculated can be found in Supporting Information. In Figure 3 are reported the CT state splitting diagrams for the two interfaces with the two different methods used for the simulations (OC and OCP) are reported.

VI. RESULTS VI.A. MD Simulation. The van der Waals and Coulombic interactions of the molecules were studied for both interfaces. In the (01-1) box, the energy per square nanometer of the anthracene−C60 interactions is lower than in the (001) box (see Table II). At the (01-1) interface, the conjugated carbon atoms of the anthracene molecules face the C60 molecules. At 24728


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Figure 3. CT splitting diagram for the systems simulated in the article. The electric field strength is plotted on the x-axis. The y-axis is divided in two: the lower half gives the energy gain given by the field in hopping one layer in the direction of the field, while the upper half gives the Marcus rates as a function of the field.

polarization radius of 15 Å is similar to those with polarization radii of 20 and 25 Å. Therefore, a polarization radius of 15 Å was used in the KMC simulations with polarization. VI.C. Transfer Integrals. Figure 5 shows the relation between the relative arrangement of the anthracene molecules and their transfer integrals. Charge transfer occurs from molecule i to molecule j, where molecule i is shown in black, and the surrounding molecules j are grouped by color according to which ones have the same transfer integrals and edge-to-edge distances. In Figure 5a and c, the transfer integrals and edge-to-edge distances of the pristine anthracene crystal at 0 K are marked with dotted lines corresponding by color to the molecules shown in Figure 5b. The dotted gray lines mark the peaks of the transfer integrals of the second layer of molecules from molecule i (not shown in Figure 5b). While the red molecules have the lowest edge-to-edge distance, they have little π-orbital overlap with molecule i, so their transfer integrals are the lowest of the molecules depicted in Figure 5b. On the other hand, yellow molecules have the highest edge-to-edge distance, but their transfer integrals are the highest because of the large π-orbital overlap with molecule i. In summary, the

Table II. Interactions at the Interface and in the Bulk, Given in kJ mol−1 molecule−1 except for the ant−C60 Interactionsa ant−ant

pure anthracene pure C60 (01-1) box (001) box

VdW

Coul

VdW + Coul

−90.5

−6.3

−96.9

−76.5 −89.6

−17.4 −3.2

−93.9 −92.8

C60−C60

ant−C60

VdW

VdW (kJ mol−1 nm−2

−331.6 −320.3 −323.5

−182.4 −139.5

a

Anthracene is abbreviated as ant. van der Waals and Coulombic interactions are abbreviated as VdW and Coul, respectively. The C60 molecule has no permanent charge, so the Coulombic C60−C60 and ant−C60 interactions are zero.

permanent charges. Unlike for C60, the inclusion of polarization does not change the general shape of the density of state curves for anthracene. The anthracene molecules in the (01-1) box are more disordered at the interface, which is reflected in a broader OC and OCP density of states than for the (001) box. For both anthracene and C60, the density of states calculated with a 24729


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Figure 4. Panels a−c show the density of states for an electron on the LUMO of C60 in pure C60, in the (01-1) box, and in the (001) box, respectively. Panels d−f show the density of states for an hole on the HOMO of anthracene in pure anthracene, in the (01-1) box, and in the (001) box, respectively. Density of states are calculated with O, OC, and OCP method. Polarization radii of 15, 20, and 25 Å are used when polarization is activated. In panel a, the O and OC curves overlap since the neutral C60 has no permanent charges.

relative orientation of the molecules has a significant impact on the transfer integrals, as noted for C60 in ref 30. The effect of thermal vibrations, as given by the MD simulation at 300 K, has a significant impact on the edge distances and transfer integrals (pure systems in Figure 5a and c). The effect of the interface is also visible in the transfer integrals and edge-to-edge distance distributions. As explained in section VI.A, the MD simulation introduces greater deformation of the interfacial anthracene molecules in the (01-1) box and the effect of the interface is stronger for this system. Regarding C60, we can notice a similar broadening due to MD simulation (pure systems in Figure 6) as previously analyzed in ref 30. The effect of the interface is particularly visible in the edge-to-edge distance distribution (Figure 6b), which is broader in the (01-1) and (001) boxes than in the pure C60 box, indicating that the interfaces cause disorder in the arrangement of the C60 molecules. This disorder is again stronger in the (01-1) box. The effect of the interface on the edge-to-edge distances is also reflected in the transfer integral distributions of the (01-1) and (001) boxes. The transfer integral distributions of both interfaces are similar to the transfer integral distribution of the pure C60 box, except for the fact that some of the highest transfer integral values are lowered (Figure 6a). Also, because of strong deformations at the interface, a small group of high transfer integrals (∼150 meV) appears at the (01-1) interface. VI.D. KMC Simulations. On the basis of the CT state splitting diagrams of Figure 3 five fields were chosen to compare the KMC simulations on the different systems with and without polarization: 0.3 × 107, 0.5 × 107, 0.7 × 107, 0.9 × 107, and 1.1 × 107 V/cm. The reader might notice how these fields are pretty high compared to those expected in a real device (about 2 orders of magnitude). This need for high fields

Figure 5. (b) Visualization of an anthracene molecule (black) and the neighbors it has the highest transfer integrals with. (a) Distribution of the transfer integrals for anthracene in a pure anthracene box and in the (01−1) and (001) boxes. Transfer integrals of crystalline anthracene marked with dotted lines. The colors of the dotted lines correspond to the molecules with the same color shown in panel b. The gray dotted lines correspond to molecules further away. (c) Distribution of edge-to-edge distances for anthracene. (d) Close-up of molecules colored orange. While it may not be immediately clear at a first glance, all the couples of molecules i-x with x = 1−4 have the same relative orientation.

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Figure 6. (a) Distribution of the transfer integrals for C60 in a pure C60 box and in the (01-1) and (001) boxes. (b) Distribution of edge-to-edge distances for C60. Transfer integrals and edge-to-edge distances of crystalline C60 marked with dotted lines. The transfer integrals and edge-to-edge distances of the same crystalline neighbors are grouped by color.

Table III. Analysis of the Monte Carlo Simulations Focusing on CT State Splitting and Collection of Electrons (el) and Holes (ho) CT split (%)

el and/or ho collection (%)

both el and ho collection (%)

el collection (%)

ho collection (%)

CT splitting time (s)

DOS

field (107 V/cm)

(01-1)

(001)

(01-1)

(001)

(01-1)

(001)

(01-1)

(001)

(01-1)

(001)

(01-1)

OC OC OC OC OC OCP OCP

0.3 0.5 0.7 0.9 1.1 0.3 0.5

11 95 100 100 100 47 95

0 3 96 98 6 0 76

9 95 100 100 100 44 95

0 3 8 1 0 0 76

7 33 0 0 0 6 62

0 1 0 0 0 0 50

9 33 0 0 0 44 95

0 1 0 0 0 0 76

7 95 100 100 100 6 62

0 2 8 1 0 0 50

in our KMC scheme has already been noted for the conduction of a single electron in C60, where high fields were needed to exit the percolation region.30 This issue will be further discussed in the Conclusions, together with a detailed analysis of our data. The KMC simulations were first run with the field applied in the −z direction (perpendicular to the interface), as in a bilayer solar cell. The percentages of simulations for which the charge collection condition (explained in section V.B) was met are shown in Table III. A greater percentage of charge carriers is collected in the (01-1) box than in the (001) box, so it can be concluded that the (01-1) interface is better for transporting charge carriers from the CT state to the electrodes when the field is perpendicular to the interface. Accounting for polarization aids the CT splitting in both the (01−1) and (001) boxes, as shown in Table III. The increase in CT splitting because of polarization is higher for the (001) box, and is higher for both boxes at low fields at which few simulations result in CT state splitting. In many of the simulations (especially at high electric fields) the CT state splits, but one or even both of the charge carriers are never collected (Table III). The explanation of this phenomenon, particularly accentuated for (001) interface, resides in the model used for the transfer rates, that is, the Marcus Formula. The Coulombic interaction between the electron and hole in the CT state is strong. To help the CT state separation an electric field strong enough to counteract this Coulombic interaction has to be applied. However, if the applied electric field is too strong, the inverted Marcus region is reached for the conduction of the single charge carrier. This makes the charge transfer more difficult in the field direction and as a result the single charge carrier moves on the plane perpendicular to the electric field (Figure 7 and Supporting Information).

2 5 8 2 3 4 1

× × × × × × ×

10−9 10−10 10−11 10−12 10−12 10−9 10−9

(001) − 2 4 8 3

× × × ×

10−10 10−11 10−12 10−11

3 × 10−9

Figure 7. Trajectory of charge carriers in the (01-1) box at (a) 0.5 × 107 and at (b) 1.1 × 107 V/cm.

V.E. Direction of Applied Electric Field. In BHJ solar cells, the donor and acceptor are intermixed so the interface is often not parallel to the electrodes. Therefore, the electric field is applied at different angles along the interface, and is not always applied perpendicularly. KMC simulations were run with the field applied parallel to the interface and at 45° angles to the interface. These simulations were performed at the fields 0.5 × 107 and 0.7 × 107 V/cm for the (01-1) and (001) boxes, respectively. These fields were chosen because they showed the highest rates of CT state splitting and subsequent possible charge collection for their respective boxes when the field was applied perpendicular to the interface (without polarization). The results are shown in Table IV. 24731


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Table IV. Percentage of Monte Carlo Simulations in Which the Specified Charge Carriers Traveled More than 30 Å in the Direction of the Applied Electric Fielda hole (%)

a

electron (%)

both electron and hole (%)

electron or hole (%)

field direction

(01-1)

(001)

(01-1)

(001)

(01-1)

(001)

(01-1)

(001)

−z x −x y −y −zx −z−x −zy −z−y

95 11 53 81 52 43 93 72 59

8 100 99 100 100 100 98 100 100

33 49 57 81 52 53 83 67 56

0 17 14 12 13 44 49 42 38

33 10 49 80 51 42 82 67 56

0 17 14 12 13 44 49 42 38

95 50 61 81 53 54 94 72 59

8 100 99 100 100 100 98 100 100

DOS calculated with OC (without polarization). Results shown for the (01-1) and (001) boxes at fields 0.5 × 107 and 0.7 × 107 V/cm, respectively.

For the (01-1) box, applying the field at nonperpendicular angles to the interface generally hinders CT state splitting. For the (001) box, however, it increases the percentage of simulations in which the charge collection condition is met from 8% to around 100%. This increase can be attributed to the field being applied more in the anthracene π-stacking direction rather than perpendicular to it. The angle of the applied electric field, will provoke changes in position and height of the four gaussians of the CT state splitting diagram, giving rise to an intermediate situation between the (001) and the (01-1) box of Figure 3. VI.F. CT State Splitting Diagram. Let us focus first on the (001) OC CT state diagram in Figure 3. The 4 gaussians represent the efficiency of: electron hopping away from the CT state, hole hopping away from the CT state, electron conduction in the bulk, and hole conduction in the bulk. All these processes are related to the movement in the field direction for the hole and against the field direction for the electron. Picking a value on the electric field axis of the CT state splitting diagram it is possible to see which processes have a high efficiency at that value. The 4 gaussians of the (001) OC CT state splitting diagram are separated into couples on the field axis. Furthermore, the group of the two gaussians representing the efficiencies of the CT state splitting and the group of the two gaussians representing the efficiencies of the free charge carrier conduction do not overlap with each other. The poor results of the (001) OC interface (Table III) can then be seen in a new light. It is not possible for this interface to find a value of the electric field promoting both efficient CT state splitting and free charge carrier collection. When the CT state can split, the free charge carriers in the bulk will be in the deep inverted Marcus region and they will not be able to be collected at the electrodes. In its journey to the electrodes, the charge carriers will pass gradually from the Gaussian of the CT state splitting to the Gaussian of the conduction in the bulk. Therefore, assuming that both the charges will be collected at the electrodes, the system will pass from the two gaussians on the right (CT splitting), to the two gaussians on the left (free charge carrier conduction in the bulk). For these reasons we expect it to be better to have a field at the extreme right of the free charge carrier conduction. This is because as soon as the CT state splits (even if very inefficiently), the system will gradually move toward the bulk conductions gaussians. For example, this is the case of the fields 0.5 × 107 or 0.7 × 107 V/ cm (Table III). At 0.5 × 107 V/cm, the CT states splits only in 3% of the cases, but in all of these cases at least one of the two charge carriers is collected.

The (01-1) OC interface presents a different behavior. In this system, the holes are more mobile than the electrons and have more efficient processes. This is visible from the taller red gaussians in Figure 3 associated with the holes of this system (and due to the higher transfer integrals in the field direction). Furthermore, the hole CT state splitting Gaussian is slightly overlapping with the free hole conduction Gaussian, suggesting that the hole will split the CT state and conduct first, with the electron eventually following if the field is not too high. This trend is visible in the percentages of CT state splitting and hole and electron collection in Table III. Polarization, as shown by the OCP CT state diagrams of Figure 3, generally helps the CT state splitting. Polarization has three main effects on the Marcus rates (eq 10). The first effect is on the transport barrier ΔENM + λNM. The DOS carries the destabilization contribution of the interface due to polarization (as in eq 14 and as analyzed in section VI.B), opposing the electrostatic interaction of the CT state. The external reorganization energy introduces an additional obstacle to the carrier movement through λout. Since this latter contribution is usually very large (0.5−1 eV), the resulting effect is an increase in ΔENM + λNM that translates into demand for a stronger electric field for the CT splitting. In the bulk, only the external reorganization energy contribution is present and thus, in both cases, more energy from the field is required to reach the maximum efficiency points of eqs 19−22 when polarization is activated. This translates into a shift of the efficiency gaussians in Figure 3 to higher field strengths. Furthermore, the external reorganization energy affects the Marcus formula (eq 1) in two other ways. It lowers the efficiency of the process under consideration, affecting the prefactor of the Gaussian in the Marcus formula 2π 1 |HNM|2 ℏ 4πkBTλNM

This lowers the height of the efficiency gaussians and leads to an increase in the CT state splitting times (Table III). The other effect of the external reorganization is to increase the standard deviation σNM of the hopping rates as a function of the field (eq 11), thereby broadening the Marcus gaussians. In particular, this last effect is the main cause of the increase in the overlap of the gaussians of the different processes, making it possible for fields to be adequate for both CT splitting and free charge carrier conduction (Figure 3, OCP CT state splitting diagrams). A similar effect has been observed on the mobility of pure C60, where the external reorganization energy helps the 24732


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inclusion of polarization, very high fields are needed for the cold CT state dissociation according to Marcus theory. Anthracene is a small molecule, so this might mean that charge delocalization in the polymer plays an important role. Twisting and kinking of polymers has been demonstrated to reduce delocalization significantly in the stacking direction84 and thus we can assume that the main delocalization helping the CT state dissociation is along the polymer chain. Delocalization on the LUMO of C60 is instead greatly hindered by the MD at finite temperature.85 Another cause for the high electric fields could be because, in our model, the transfer integrals are considered independent of the electric field value. This fact (as already discussed in section IV) is supported in literature.20 Nevertheless, this study uses calculations at the ZINDO level without extended basis functions, thus neglecting relevant polarization effects. Moreover, Marcus theory might not be enough to describe charge transport in organic materials and Marcus−Levich−Jortner theory might be more appropriate, allowing tunneling effects across the potential barrier and lowering the strength of the electric fields needed. We plan to further investigate these matters and also to incorporate Marcus−Levich−Jortner type of transport into our code in the near future.

system to not fall into the inverse Marcus region and allows our model to reproduce the experimental results with good qualitative agreement.30

VII. CONCLUSION We have presented a study of the CT state splitting at organic interfaces. The results on C60−anthracene interfaces have been analyzed and explained using the CT state splitting diagrams (section VI.F). This method provides a theoretical tool to analyze organic interfaces prior to KMC simulation, helping provide a better understanding of the charge transport properties at these interfaces. With a preliminary analysis based on the CT state splitting diagram, it is possible to find out if two materials are suitable to be used together in an interface for organic photovoltaics. If the gaussians of the efficiencies of the 4 main processes considered (Figure 2) do not overlap, it is not possible to find an electric field that promotes both CT state splitting and free charge carrier conduction to the electrodes. Polarization has been shown to be a key factor in the modeling of the CT state splitting. Polarization affects DOS and reorganization energies. The efficiency gaussians of eqs 19−22 are thus shifted due to the DOS contribution, shortened due to the reorganization energy contribution to the prefactor of the Marcus formula and broadened due to the reorganization energy contribution to the standard deviation of the Marcus formula. (section VI.F). In particular, the effect of the external reorganization energy on the standard deviation of the Marcus rates favor the overlapping of the Marcus efficiencies of the 4 processes considered, helping CT state dissociation and free charge carrier collection. How much of the external reorganization has to be taken into account in the Marcus formula is still a matter of debate, since it is not clear how much of the environment will repolarize in between two charge hoppings.30 An intermediate picture between the OC and the OCP state splitting diagrams could be obtained with the introduction of a scaling factor on the polarization. The finetuning of such a parameter is outside the scope of the present work and does not change the underlying message. The (01-1) box has been shown to perform better for CT state splitting and charge carrier collection when the electric field is applied perpendicular to the interface. However, when the electric field is applied parallel or at 45 deg angles to the interface, the (001) box has higher rates of CT state splitting. This is because changing the direction of the field gradually modifies the parameters of the CT state splitting diagrams from the parameters of the (001) box to those of the (01-1) box. All sorts of intermediate stages between the conduction in these two boxes can be found in a BHJ solar cell. Furthermore, we also showed how the role of the hot CT state can be rationalized in the CT state splitting diagram setting and how the hot CT can help charge dissociation in Marcus Theory (section IV). Experimental results at polymer−fullerene interfaces54−56 show how excess excitation energy does not affect charge dissociation efficiency. This means that the cold CT state is the main precursor to charge dissociation and hot CT states decay to cold CT states prior to dissociation. A hypothesis might be that this particular class of interfaces possesses a favorable interfacial gradient causing the excess energy to have a negligible impact on the CT state dissociation.48 Our results show that the cold CT state in a C60−anthracene interface does not split at experimental fields. It seems that even with the

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsami.6b06645. Size and number of molecules for MD simulated anthracene and C60 boxes; illustration of anthracene and C60 molecules with the associated atomic charges and polarizabilities for both the neutral and the charged molecular state; analysis of the effect of the interface on the energetic landscape; additional clarifications on the calculation of the parameters needed for the CT state splitting diagrams; illustration of preferential direction for the charge transport in different conditions and at different fields (help in visualizing the effect of the inverse marcus region at strong fields); and example illustrating the independence of the external reorganization energy from additional charge carriers in the material. (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS M.L. thanks SERC (Swedish e-Science Research Center) for funding and SNIC (Swedish National Infrastructure for Computing) for providing computer resources.

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REFERENCES

(1) Forrest, S. R. The Path to Ubiquitous and Low-Cost Organic Electronic Appliances on Plastic. Nature 2004, 428, 911−918. (2) Kim, J.-H.; Han, M. J.; Seo, S. Flexible, Stretchable, and Patchable Organic Devices Integrated on Freestanding Polymeric Substrates. J. Polym. Sci., Part B: Polym. Phys. 2015, 53, 453−460. (3) Facchetti, A. π-Conjugated Polymers for Organic Electronics and Photovoltaic Cell Applications. Chem. Mater. 2011, 23, 733−758.

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