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Comparison of different rate constant expressions for the prediction of charge and energy transport in oligoacenes V. Stehr,1* R. F. Fink,2 M. Tafipolski,1 C. Deibel3 and B. Engels1 Charge and exciton transport in organic semiconductors is crucial for a variety of optoelectronic applications. The prediction of the material-dependent exciton diffusion length and the charge carrier mobility is a prerequisite for tailoring new materials for organic electronics. At room temperature often a hopping process can be assumed. In this work, three hopping models based on Fermi’s Golden rule but with different levels of approximation—the spectral overlap approach, the Marcus theory, and the Levich–Jortner theory—are compared for the calculation of charge carrier mobilities and exciton diffusion lengths for oligoacenes, using the master equation approach and Monte Carlo simulations. © 2016 John Wiley & Sons, Ltd How to cite this article:

WIREs Comput Mol Sci 2016. doi: 10.1002/wcms.1273

INTRODUCTION

O

ver the years, organic semiconductor devices have gained more and more attention due to their low production costs and easy processability compared with conventional inorganic semiconductors. Important fields for application are organic light emitting diodes (OLEDs),1–4 organic field effect transistors (OFETs),5–9 radio frequency identification tags (RFIDs),10–12 and organic photovoltaics (OPV),13–22 to mention just a few. The field has grown rapidly since it was shown in 1977 that organic polymers can be made highly conductive by chemical doping.23 For inorganic covalently bonded materials, the band theory is well established, however, it is not *Correspondence to: [email protected] 1

Institut für Physikalische und Theoretische Chemie, Universität Würzburg, Würzburg, Germany

2

Institut für Physikalische und Theoretische Chemie, Universität Tübingen, Tübingen, Germany

3

Institut für Physik, Technische Universität Chemnitz, Chemnitz, Germany Conflict of interest: The authors have declared no conflicts of interest for this article.

particularly appropriate for organic conductors, because organic molecular crystals are only weakly bound by van der Waals interactions. Owing to the complex nodal structure of the molecular orbitals, the transfer integrals between the monomers are very sensitive to even small molecular displacements, and therefore, lattice vibrations play a more important role in organic than in inorganic materials, because they destroy the long-range order and cause the charge carrier to localize.24 In order to take these vibrations into account, a variety of models has been proposed that incorporate the local (Holstein)25,26 and the nonlocal (Peierls)27 electron–phonon coupling. The latter leads to a polaron model where the charge carrier is partially localized and dressed by phonons.28–31 As the fluctuations of the coupling between the molecules are of the same order of magnitude as the average coupling,32 this leads to a rather strong localization. Further models have been developed, where the charges are assumed to be localized and the inter- and intramolecular vibrations are treated classically.33–35 In this work a hopping mechanism for the movement of the excitons and charge carriers is assumed. The band model and the hopping model lead to different temperature dependencies of the transport parameters. While the band model

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leads to decreasing charge mobilities and exciton diffusion coefficients with increasing temperature because of the scattering off the phonons, the hopping model leads to an increase of the transport parameters as the hopping process is thermally activated. The description of the crossover region between these two models is complicated and not yet fully understood. The models used here are only appropriate in the hopping regime and should not be used for transport calculations at low temperatures, where band transport dominates. Three different thermally activated hopping models based on time-dependent perturbation theory—the spectral overlap approach, the Marcus and the Levich–Jortner theory—are compared. In the second section, their derivations are presented coherently in order to emphasize the different levels of approximations and their respective prerequisites. Third section explains the calculations of the electronic couplings and the reorganization energies. Next section presents the master equation and the kinetic Monte Carlo method as two approaches that are frequently used for transport simulations. Finally in the sixth section, the results for exciton diffusion lengths and charge carrier mobilities are presented.

THE HOPPING MODELS

D E ^ ♯ ΨAm ΨDmD Ψ♯AnA jVjΨ D D Dn A ! E ! = VAD χ DmD RD jχ ♯DnD RD D ! ! E χ ♯AnA RA jχ AmA RA

ð2Þ

with the Coulomb matrix element ! VAD = ψ DR rD ψ ♯ !

!

ARA

D

! ^ ♯ rA jVjψ

!

DRD

! ! rD ψ AR rA

!

A

ð3Þ where χ is the nuclear and ψ the electronic wave !

function with the nuclear coordinates R and the elec! tronic coordinates r. The overlap and exchange interaction is neglected here which is usually justified for the dimer distances that appear in molecular crystals. The delta function in Eq. (1) ensures the energy conservation of the system during the charge/exciton transfer. By introducing an energy integral, it can be divided into a donor and an acceptor part: h i h i EDmD −E♯DnD + E♯AnA −EAmA ∞ ð ð4Þ dEδ EDmD −E♯DnD + E δ E♯AnA −EAmA −E =

δ

The Spectral Overlap Approach A system containing two molecules that do not need to be identical is regarded. One molecule is initially charged/excited (the donor, D) and the other one is in a neutral/ground state (the acceptor, A). If the coupling between the molecules is sufficiently small, time-dependent perturbation theory can be applied, where the coupling appears as the perturbation. This leads to Fermi’s Golden rule,36–38 which reads νAD =

Applying the Born–Oppenheimer39 and the Condon approximation,40–42 the coupling between the donor and the acceptor is

2π X X ♯ fDnD fAmA ℏ D mD nD mA nA E ♯ ^ ♯ ΨAm 2 ΨDmD ΨAnA jVjΨ A DnD ♯ ♯ δ EDmD − EDnD + EAnA − EAmA

ð1Þ

where the tuples m and n designate all vibrational quantum numbers of the neutral/ground state and the charged/excited state, respectively, where the latter is indicated by #. The thermalized state manifold of the initial state with respect to the vibrational states is ♯ taken into account by the distribution functions fDn D and fAmA , see Eq. (13), for the vibrational equilibrium of the donor in the charged/excited state and the acceptor in the neutral/ground state, respectively.

−∞

By defining the auxiliary function ! ! E X ♯ D fDnD j χ DmD RD jχ ♯DnD RD j2 DD ð E Þ : = mD nD δ EDmD −E♯DnD + E

ð5Þ

for the donor deexcitation/neutralization and the analogue equation for the acceptor excitation/ionization respectively, the hopping rate, Eq. (1), can be written as 2π 2 νAD = VAD ℏ

∞ ð

dE DD ðEÞDA ðEÞ = −∞

2π 2 V JAD ℏ AD

ð6Þ

with the Franck–Condon weighted density of states (FCWD)40,41,43


∞ ð

dE DD ðEÞDA ðEÞ

JAD = −∞

ð7Þ

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Prediction of charge and energy transport in oligoacenes using rate constant expressions

which accounts for the vibrations of the molecules and contains the spectral overlap of the densities of states DD of the donor and DA of the acceptor. If the donor and acceptor are not at the same energy level, caused by environmental effects, or caused by an external electric field in the case of charge carriers, this can be taken into account by an energetic shift ΔEAD of DD(E) and DA(E) (i.e., the intensities, see below in the section Calculation of the Densities of States) with respect to each other:44,45 ð

eJ = dE DD ðE − EAD ÞDA ðEÞ ð = dE DD ðEÞDA ðE + EAD Þ

ð8Þ

J0 ≈J 1 + EAD J

ð9Þ

Here, a linear approximation for eJðEAD Þ is used, where J0 : = deJðEAD Þ=dEAD jEAD = 0 and J : = eJð0Þ, and therefore e νAD = νAD 1 +

0

J EAD J

ð10Þ

Equation (6) is frequently used in the literature to calculate exciton transfer rates.46–49 However, it becomes clear that this approach works just as well for charge transport. However, it is important to note that in the case of charge transport not only energy is exchanged, even though the picture of energy emission and absorption is used here to ensure energy conservation. But, despite this fact, the mathematical approach by defining the auxiliary functions DD(E) and DA(E) and calculating their overlap is perfectly approvable.

The density of states D(E) can be obtained from the multidimensional vibrational spectrum which was calculated in this work with the parallel mode approximation (see below). Here, the relative intensity is I ðE Þ =

m1 , …,, np

δ

" eI m1 ,n1 ,m2 ,n2 ,…,mp ,np

p X j=1

p Y

eI m1 , n1 , m2 ,n2 ,…, mp ,np = j χ n jχ m j2 f ðmi Þ i i i=1

ð12Þ p is the number of the vibrational modes, ω is their frequency, m are their initial, and n their final vibrational quantum numbers.

ℏωi ℏmi ωi exp − f ðmi Þ = 1− exp − kB T kB T

is the (Boltzmann type) distribution function mentioned in

Eq. (1) and the Franck–Condon factor j χ ni jχ mi j2 is the overlap integral of the wave functions of two harmonic oscillators. By assuming that the frequency of both oscillators is the same, it results in51 i2

mi ! ni − mi −Si h ni − mi Si j χ ni jχ mi j2 = e Lmi ðSi Þ ni !

ð14Þ

with the Huang–Rhys factor S51,52 (see below) and the Laguerre polynomial Lnα ðxÞ.53,54 For the emission spectrum m and n must be interchanged in Eq. (12). The spectrum is convoluted with a Lorentzian function l ðE Þ =

1 σ π σ 2 + E2

ð15Þ

(σ is the half width at half maximum) to account for natural and collision broadening and to finally obtain the density of states: ∞ ð

DðEÞ =

IðE0 Þl ðE− E0 Þ dE0

ð16Þ

−∞

In order to determine the Huang–Rhys factors Si which are needed for the Franck–Condon factors, !

Eq. (14), the difference vector Δd of the equilibrium structures of the charged/excited state and the neutral/ground state is determined. This vector is trans! formed from Cartesian coordinates into a vector Δq in the normal mode basis via !

nj − mj ℏωj − E

ð13Þ

Calculation of the Huang–Rhys Factors

Calculation of the Densities of States

∞ X

with50

!

AΔq = Δd

!# ð11Þ

ð17Þ

where the matrix A contains the normal mode vectors. The molecular vibrations are treated as harmonic oscillators.50,55 Therefore, the relaxation


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energy after ionization/excitation or neutralization/ deexcitation into the new equilibrium structure is (cf. Figure 1) 1 1 !2 !2 λi = ki Δqi = Mi ω2i Δqi 2 2

ð18Þ

ki is the force constant, which is the eigenvalue of the force matrix with respect to the normal mode i. Mi is the reduced mass which is calculated as X n2i, j mj j

Mi = X

ð19Þ

n2i, j

j

ni,j is the jth coordinate in the ith normal mode vector and mj is the mass of the respective atom. The Xp sum λ over all vibrational modes of the neui=1 i tral/ground state and the charged/excited state results in the reorganization energy used in the Marcus theory, see The Marcus Hopping Rate section. The Huang–Rhys factor of the vibrational mode i is the ratio between the contribution of this mode to the relaxation energy and its vibrational energy ℏωi: Si =

rffiffiffiffiffiffiffiffiffiffiffi ! 2 λi 1 Δqi ℏ = with βi : = ℏωi 2 βi Mi ωi

ð20Þ

βi is a parameter with the dimension of a length so that Si is a dimensionless quantity. E

Charged/ excited state

In this approach, it is assumed that the normal modes in the neutral/ground state and the charged/ excited state are aligned in a parallel manner so that the vibrations in both states can be described by the same normal mode basis. However, in the extreme case, the molecular conformation could change completely due to the ionization/excitation. But, usually the normal modes from one state can be written as a linear combination of the normal modes of the other state. This can be expressed by a Duschinsky rotation.56,57 In many cases, however, the Duschinsky rotation can be neglected,58 which is known as the parallel mode approximation. For oligoacenes, some Duschinsky mixing was found for vibrations in the region of 1200–1600 cm−1. However, its neglect led to good general agreement between the experimental and theoretical spectra.50 If initially only the vibrational ground state is occupied (m = 0), the Franck–Condon factor, Eq. (14), turns into a Poisson distribution, jhχ n jχ 0 ij2 =

S n −S e n!

ð21Þ

with the expectation value and the variance S. If the neutral/ground state and the charged/excited state are ! not shifted against each other, Δq (see Figure 1) and therefore S become zero and the 0 ! 0 transition has the probability 1. In this case, the occupation of the vibrational modes does not change during ionization/ ! excitation. However, if Δq 6¼ 0 and therefore S 6¼ 0, the electronic and the vibronic excitations couple. Therefore, the Huang–Rhys factor S can be regarded as a measure for the electron-vibration coupling.

The Marcus Hopping Rate Neutral/ ground state

*i

i q Δ qi

F I G U R E 1 | The potential curves for one vibrational mode in the neutral/ground state and the charged/excited state depending on the normal mode coordinate q, based on the harmonic approximation. Δq is the difference vector between the two equilibrium structures, λ indicates the relaxation energies.

If the thermal energy exceeds the vibrational energy, kBT ℏω (high-temperature limit), the vibrations can be treated classically. In this case, the system reduces to a simple two-level system (the neutral/ ground and the charged/excited state), whose energetic positions are modulated time dependently by the molecular vibrations. These vibrations are described by classical harmonic oscillators and the collective coordinates qD and qA for the donor and the acceptor geometry, respectively. In the quantum mechanical the D ! picture, ! E ♯ Franck–Condon factor χ AnA RA jχ AmA RA 2 equals the probability that the acceptor with the vibrational quantum number mA in the neutral/ ground state is in the vibrational state nA after the ionization/excitation. As the vibrations are treated




classically here, there are no different vibrational states in the neutral/ground state and the charged/ excited state respectively. Therefore, the Franck– Condon factor vanishes. The same holds for the Franck–Condon factor of the donor. Furthermore, the summation over the vibrational quantum numbers changes to an integration over the vibrational coordinates qD and qA, so that Eq. (5) turns to ð DD ðEÞ = dqD fD♯ ðqD Þδ ED ðqD Þ − E♯D ðqD Þ + E ð22Þ The equation for DA(E) changes analogously. In the following calculations for the donor, the index D is omitted. Regarding an ensemble of donor–acceptor pairs, the vibrational equilibrium is described by a Boltzmann distribution: 0 2 1 sffiffiffiffiffiffiffiffiffiffiffi ♯ K q− q 0 K B C exp@ − f ♯ ðq Þ = A πkB T kB T

2 ΔE0 + E + K q♯0 −q0 e q= − + q♯0 ♯ 2K q0 −q0

between the parabolas for the neutral/ground state and the charged/excited state, see Figure 2. The activation energy, which appears in the enumerator of the exponential function in Eq. (25), is defined as the energy difference between the intersection of E#(q) with E(q) at q = e q and the minimum E#(q0): 2

Eact = E♯ ðe qÞ −E♯ ðq0 Þ = Kðe q − q♯0 Þ 2 2 ♯ ΔE0 + E + K q0 − q0 = 2 4K q♯0 −q0

0

2 1 ♯ e K q − q 0 1 1 B C ffi exp@− DðEÞ= rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ð25Þ 2 k T 2 B ♯ πkB TK q0 − q0

ð27Þ

The reorganization energy, which is depicted in Figure (2), is 2 λ : = K q♯0 −q0

ð24Þ

It is assumed that the curvature 2K is the same for both the neutral/ground state and the charged/ excited state. (However, K can be different for the donor and the acceptor.) This leads to

ð26Þ

ΔE0 = E0 − E♯0 = Eðq0 Þ − E♯ q♯0 is the energetic shift

ð23Þ

Here, a harmonic potential is assumed (see Figure 2), which is described by 2 E♯ ðqÞ = E♯ q♯0 + K q− q♯0

with

ð28Þ

Substituting Eact in Eq. (25), using the definition of λ and inserting the index D again results in 1 ðΔE0D + E + λD Þ2 DD ðEÞ = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp − 4kB TλD 2 πkB TλD

! ð29Þ

The same calculation for the acceptor molecule yields

E

E #

#

ED

EA EA

EA

λA λD

#

E0D E0D q0D

~ qD

#

# E0A qD

E0A

q0D

q0A

~ qA

#

qA

q0A

FI GU RE 2 | The potential curves for the neutral/ground state and the charged/excited state (indicated by #) of the donor (index D, left) and the acceptor (index A, right) molecule, depending on the geometry coordinate q. λ indicates the reorganization energy. © 2016 John Wiley & Sons, Ltd

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1 ðΔE0A − E + λA Þ2 DA ðEÞ = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp − 4kB TλA 2 πkB TλA

! ð30Þ

Inserting these expressions into Eq. (6), integration over E and using the definitions ð31Þ λ : = λD + λA ΔE : = ΔE0D + ΔE0A = E0D + E♯0A − E♯0D + E0A ð32Þ results in the Marcus equation: V2 e νAD = AD ℏ

! rffiffiffiffiffiffiffiffiffiffiffi π ðΔE + λÞ2 exp − 4kB Tλ kB Tλ

ð33Þ

The FCWD is within this theory 1 λ2 JMarcus = pffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp − 2ð2λkB T Þ 2π 2λkB T

ð34Þ

The Marcus theory was originally derived for outer sphere electron transfer in solvents.59,60 Customarily the whole donor–acceptor complex is described by only one collective coordinate. Here, the donor and the acceptor are described separately by one collective coordinate each, which allows for the formal analogy between Marcus and Förster theory and the separate calculation of the reorganization energy. Treating the coupling as a perturbation requires that VAD is small compared to λ/4, which corresponds to the total activation energy for the charge/ exciton to move from one molecule to another (for ΔE = 0). Because of the smaller couplings and the larger reorganization energies, this condition is satisfied even better for excitons than for charges. Furthermore, the relaxation (the geometric reorganization) has to be fast in comparison with the transfer so that the system can be assumed to be in thermal equilibrium during the transfer. In addition, the theory is restricted to the high-temperature case since tunneling is neglected completely and the molecular vibrations are treated classically, which requires kBT ℏω.

The Levich–Jortner Hopping Rate Whereas in the spectral overlap approach (The Spectral Overlap Approach section) the molecular vibrations are treated quantum mechanically, they are treated as classical oscillations in the Marcus theory (The Marcus Hopping Rate section). This is

strictly speaking only acceptable in the hightemperature limit if the thermal energy exceeds the vibrational energy. Most intramolecular vibrational frequencies, however, are high above the thermal energy at room temperature (≈ 26 meV). On the other hand, the reorganization of the environment (the solvent, or, in the case of crystals, the movement of the surrounding molecules) can often be treated classically. This is taken into account in the Levich–Jortner theory.61–63 The quantum mechanical vibrational sublevels are taken into account for those vibrations that are energetically higher than the thermal energy. For that reason, only the vibrational ground state is occupied in the thermal equilibrium. That is why the Franck– Condon factor in Eq. (5) simplifies to a Poisson distribution, Eq. (21), containing an effective Huang– Rhys factor, because all vibrations are merged to only one effective vibrational mode per molecule with an effective frequency ωeff. Since one regards both classical low-energy and quantum mechanical high-energy vibrations, Eq. (5) changes to Xð

Sm DD ð E Þ : = dqD fD♯ ðqD Þ D e − SD m! mD × δ EDmD ðqD Þ−E♯D ðqD Þ + E

ð35Þ

As in the Marcus theory, the classical oscillations are described by the collective vibrational coordinates qD and qA, cf. Eq. (22). Before the charge/exciton transfer the energy of the donor is 2 E♯D ðqD Þ = E♯D q♯0D + K qD −q♯0D

ð36Þ

because initially only the classical vibrations (ℏω kBT) are excited (nD = 0), and after the transfer, the energy is EDmD ðqD Þ = ED ðq0D Þ + KðqD −q0D Þ2 + mD ℏωeff , D ð37Þ The zero point energy is neglected. Analogue equations hold for the acceptor. In the following calculations for the donor, the index D is omitted. As in the thermal equilibrium only the classical vibrations are excited, the distribution function f#(q) in Eq. (35) is the same as in the Marcus case, Eq. (23). However, due to the excitation of the high-energy vibrations upon the transfer, the delta term in Eq. (35) now changes slightly to




δ EðqÞ− E♯ ðqÞ + E 1 = 2K q♯0 − q0 0 1 + mℏωeff + E ΔE0 + K q20 − q♯2 0 A × δ@ q + 2K q♯0 − q0

ΔE + λcl + mD ℏωeff , D + nA ℏωeff , A × exp − 4kB Tλcl ð38Þ

As in the Marcus theory, ΔE0 is the energetic shift between the parabolas that describe the classical part of the vibrations, see Figure 2. With Eqs. (23) and (38) and integration over q Eq. (35) becomes 1 DðEÞ = rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2

πkB TK q♯0 − q0 0 2 1 ♯ K q ~− q X Sm 0 B C expð − SÞexp@ − × A m! k T B m 2

ð39Þ

ð43Þ This equation simplifies further if the effective frequencies are assumed to be the same in the neutral/ ground state and the charged/excited state, so that mD ℏωeff , D + nA ℏωeff , A = ðmD + nA Þℏωeff

This presumption is consistent with the classical part of the vibrations where the curvature 2K ( = 2Mω2cl with the reduced mass M) is also set to be the same in both states. If P(X) is the Poisson-distributed probability for a certain vibrational state and m + n =: p (the indices D and A are omitted again) then

ð40Þ

This is again a Poisson distribution with Seff := SD + SA. Equation (43) results in the Levich–Jortner equation63

Equivalent to Eq. (27), the activation energy is Eact =

ΔE0 + λcl + mℏωeff 4λcl

2 +E

e νAD = ð41Þ

where the definition of the reorganization energy, Eq. (28), is used (see also Figure 2). The index ‘cl’ are added to indicate that only the classically treated low-frequency vibrations enter λ here, as explained above. With the definition of λ and Eact and inserting the omitted index D, Eq. (39) results in

ð42Þ An analogue equation holds for DA(E). Inserting these equations into Eq. (6), performing the integration with respect to E and using the definitions (31) and (32) yields 2 VAD ℏ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffi X mD SnA SD π expð − SD Þ A expð − SA Þ nA ! kB Tλcl mD , nA mD !

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiX Sp π eff exp − Seff kB Tλcl p p! 2 ! ΔE + λcl + pℏωeff × exp − 4kB Tλcl

2 VAD ℏ

ð45Þ

The effective frequency and Huang–Rhys factor are calculated as

ωeff

X SmD 1 D expð − SD Þ DD ðEÞ = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 πkB Tλcl, D mD mD ! 2 ! ΔE0D + λcl, D + mD ℏωeff , D + E × exp − 4kB Tλcl, D

e νAD =

ð44Þ

∞ ∞ X X PðXD = mÞPðXA = nÞf ðm + nÞ = PðX = pÞf ðpÞ m, n p=0

with

2 ΔE0 + K q♯0 − q0 + mℏωeff + E e + q♯0 q= − ♯ 2K q0 − q0

2 !

X ωi Si i = X Si

ð46Þ

i

Seff =

λqm, D λqm, A λqm + = ℏωeff ℏωeff ℏωeff

ð47Þ

The summation with respect to i runs over all quantum mechanically treated vibrations of the donor– acceptor complex. It is important to note that the partition between the classical and the quantum mechanical part, and therefore λ, ωeff and Seff, depends on the temperature. For high temperatures, Eq. (45) turns into the Marcus equation (33).63 Concerning the physical picture, this approach is better than the Marcus approach, The Marcus


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Hopping Rate section, because at room temperature, the vibrations are correctly treated quantum mechanically. The advantage over the spectral overlap approach, The Spectral Overlap Approach section, which also treats the molecular vibrations quantum mechanically, is that this approach allows for the inclusion of surrounding effects of the donor– acceptor complex by including them by means of the external reorganization energy in λcl.

CALCULATION OF THE PARAMETERS The Electronic Couplings Exciton Couplings

If the donor and acceptor orbitals, φD and φA, are assumed to be orthogonal and triplet, double and charge transfer excitons are neglected, the exciton coupling is ðð ! ! 1 ♯* ! ! e2 ! ! dr1 dr2 φ*D r1 φ♯D r1 φ r2 φ A r2 VAD = 2 4πε r12 A 0 2 ðð ! ! 1 ♯* ! ♯ ! e ! ! dr1 dr2 φ*D r1 φA r1 φ r2 φD r2 − 4πε0 r12 A ð48Þ !

!

where r12 = jr1 − r2 j. φ and φ# are the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital), respectively, of the donor (index D) and acceptor (index A) and are used here as an approximation for the adiabatic monomer states of the ground and excited state, respectively. The first summand ðð ! 1 ! e2 ! ! F dr1 dr2 ϱD r1 : =2 ϱ r2 ð49Þ VAD 4πε0 r12 A describes the Förster transfer64,65 with the transition densities ! ! ! ! ! ! ϱD r1 : = φ*D r1 φ♯D r1 and ϱA r2 : = φ♯* A r2 φ A r2 ð50Þ

(a)

(b)

at the donor and acceptor monomer.66 As an example, Figure 3 shows the HOMO and LUMO orbitals of ethene and the transition density resulting as a product of these two orbitals. The Förster transfer can be interpreted as caused by a coulombic interaction between the charge distributions eϱD and eϱA. While the acceptor changes to the excited state, the donor changes to the ground state due to the interaction of the transition densities. The second summand in Eq. (48) describes the Dexter transfer.67 While an electron moves from the LUMO of the donor to the LUMO of the acceptor, another electron moves from the HOMO of the acceptor to the HOMO of the donor. For Dexter transfer, the wave functions of the donor and the acceptor must overlap. However, if the monomer distance becomes large, the differential ! ! ! ♯ ! overlaps between φ*D r1 φA r1 and φ♯* A r2 φD r2 are rapidly approaching zero as it decays exponentially. In contrast to that, Förster transfer is also possible ! ! for larger distances because ϱD r1 and ϱA r2 do not depend on the distance. Therefore, the Dexter transfer can often be neglected for sufficiently large distances. However, it is important for the transfer of triplet excitations and of dipole forbidden excitations F = 0. because there VAD With a Taylor expansion of r12 up to second !

order around the monomer distance R, the Förster coupling, Eq. (49), becomes68 ! ! ! !1 3 pD R pA R 1 @pD pA A VAD = − ! ! 4πε0 jRj3 jRj5 0

!

!

ð51Þ

where pffiffiffi D ! E pffiffiffi D ! E ! ! pD = 2e φ♯D jrjφD and pA = 2e φ♯A jrjφA ð52Þ are the transition dipole moments. Equation (51) corresponds to the energy of two interacting dipoles. As mentioned above, the Dexter transfer is neglected

(c)

F I G U R E 3 | (a) HOMO orbital φ of ethene, (b) LUMO orbital φ#, (c) transition density ϱ, cf. Eq. (50). HOMO, highest occupied molecular orbital; LUMO, lowest unoccupied molecular orbital.




here which is not justified for short distances. In addition the actual distribution of the charge in the molecule is not taken into account in detail but only approximated as a dipole since the Taylor expansion is only up to second order. Therefore the dipole approximation should only be used for distances which are large compared to the molecule size. The wave functions and molecular properties such as the transition density and the transition dipole moment can be obtained without the presence of the other molecule. This significantly reduces the computational effort of the coupling calculation, though it necessitates additional approximations. The donor and acceptor orbitals are not orthogonal, however, their overlap is simply neglected. This can be avoided by an adiabatic computation of the complete donor– acceptor unit where the orbitals are delocalized over the whole dimer, which is therefore referred to as supermolecular approach. However, it must be born in mind that despite the adiabatic computational approach, the physical picture behind is still diabatic. The interaction of two degenerated monomer excitations results in two dimer excitations which are split by a value of 2VAD: E2 − E1 VAD = 2

ð53Þ

E1 and E2 are the energies of the first and second dimer excitation, respectively. This is the so-called Davydov splitting,69 Figure 4, and corresponds to the energetic distance of the adiabatic potential energy surfaces at the avoided crossing point. At this point, the excitation is equally delocalized over both monomers if the dimer is symmetric. Strictly speaking, the dimer geometry at this transition point has to be taken for the calculation of the coupling. As a simplification, normally, the geometry of the ground state dimer is used instead.

Charge Carrier Couplings

The energy splitting in dimer method70,71 for calculating the couplings for charge transport is very E ED2 Ei = E(i*,j)

2Vji

Ej = E(i,j*)

ED1

FI GU RE 4 | The Davydov splitting between the first two excited states of the dimer with the energies ED1 and ED2. The splitting is twice the exciton coupling Vji. Ei and Ej are the monomer excitation energies.

similar to the supermolecular approach for exciton coupling. The charge is supposed to be localized at one monomer in the dimer. Instead of calculating the ionization energies, the Koopmans’ theorem72,73 can be applied which states that the ionization energy of the HOMO equals the negative orbital energy. It is also approvable to approximate the electron affinity by the negative LUMO energy.70 The splittings of the adiabatic energies for the corresponding ionized systems can therefore be approximated by the splittings of the HOMO and LUMO, respectively. Generally, the calculations are based on the geometry of the neutral system instead of the geometry at the transition structure. However, the coupling does not depend strongly on the geometry.70 The interaction between the HOMOs of the two monomers with each other and likewise the LUMOs leads to an energy level splitting in the dimer. The linear combination of the monomer HOMOs leads to the HOMO and HOMO-1 orbitals in the dimer, the linear combination of the monomer LUMOs results in the dimer LUMO and LUMO+1. The coupling is then calculated with 1 VAD = ðEHOMO −EHOMO −1 Þ 2

ð54Þ

for holes and 1 VAD = ðELUMO + 1 −ELUMO Þ 2

ð55Þ

for electrons. As mentioned above, these equations only hold if the monomer HOMO and LUMO energies respectively are the same. This is only the case if the dimer is symmetric, i.e., if one monomer is shifted in a parallel manner relative to the other (and if it is rotated only around the axis perpendicular to the molecular plane if the monomer is planar). However, this is not the case for many of the monomer pairs in crystals, which are regarded in this work. It was shown before74 that for monomers which are tilted with respect to each other, the couplings deviate significantly from the ones calculated with Eqs. (54) and (55). Therefore, these equations cannot even be used as an approximation if the charge transport in crystals is studied. Taking into account, the different monomer site energies in unsymmetric dimers and the nonzero overlap when using localized monomer orbitals lead to71,75


VAD =

HAD − 12 ðHDD + HAA ÞSAD 1− S2AD

ð56Þ

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External Reorganization Energy

with D ^ Di HAD = φA jHjφ

and

SAD = hφA jφD i

ð57Þ

φD and φA are the HOMO or LUMO orbitals (for hole and electron transport respectively) of the iso^ is the one-particle Hamilton lated monomers and H operator of the neutral dimer system, which is conveniently approximated with the (Fock type) Kohn–Sham operator in Density Functional Theory (DFT). HDD and HAA are the site energies of the two monomers, SAD is the spatial overlap, and HAD is the charge transfer integral in the nonorthogonalized basis. The energy difference between the monomer HOMO and HOMO-1 and between LUMO and LUMO+1 has to be sufficiently large so that only the HOMO and LUMO orbitals of the monomers contribute significantly to the HOMO and LUMO orbitals of the dimer and the other orbitals can be neglected.75

The Reorganization Energy Internal Reorganization Energy

The reorganization energy λ needed for the Marcus theory is defined in Eqs. (28) and (31). In order to calculate λ for charge transport, the geometry of the isolated monomer is optimized for the charged and the neutral state. The energies E0 and Ec of the neutral and the charged monomers in their lowest energy geometries and the energies E*0 and E*c of the neutral monomer with the ion geometry and the charged monomer with the geometry of the neutral state are calculated to get the intramolecular reorganization energy50,58,76 ð58Þ λ = λc + λ0 = E*c − Ec + E*0 − E0 It was shown that this approach using the diabatic potential surfaces of the neutral and the charged state leads to the same λ as summing up the λi, Eq. (18), for all vibrations which are obtained by a normal mode analysis.77 The reorganization energy for exciton transport is calculated analogously to the λ for charge transport: The monomer geometry is optimized for the ground and the excited state. The energies Eg and Eex of the ground and excited state in their respective minimum geometries and the energies E*g and E*ex of the ground state in the excited state geometry and vice versa are calculated to obtain the intramolecular reorganization energy ð59Þ λ = λex + λg = E*ex − Eex + E*g − Eg

The charge carrier does not only influence the geometry of the molecule it is located at, but also the surrounding due to polarization effects. This intermolecular effect can be taken into account within the Marcus theory by calculating the external reorganization energy and adding this to the internal (intramolecular) part. The external λ and therefore its impact on the transport are expected to be small.78,79 However, the external reorganization energy is very important in the framework of the Levich–Jortner theory, since there the low-frequency vibrations play the major role for the charge transport, where the external reorganization has a significant percentage. The calculations of the external reorganization energies are conducted with polarizable force fields using the AMOEBA polarizable force field80 within the Tinker program package.81 Distributed atomic multipole moments up to and including quadrupoles are employed to treat permanent electrostatic interactions. They are obtained by means of Gaussian Distributed Multipole Analysis (GDMA)82 from wavefunctions calculated by density functional theory (PBE0/aug-cc-pVTZ) of the isolated monomer (neutral or cationic). Molecular polarization is achieved via an interactive induction model with distributed atomic isotropic polarizabilities based on Tholes damped interaction method in order to avoid a polarization catastrophe at very short range.83 Iterating the dipole-induced dipole interaction to selfconsistency captures the many-body effects. A crystalline cluster consisting of 6 × 6 × 3 unit cells in a, b and c direction and with periodic boundary conditions is geometrically optimized with a charge carrier positioned in the middle of the cluster, leading to the energy E. The geometry of the central molecule is kept frozen during the optimization in order to isolate the external part of the reorganization. In a next step, the energy E* of the same cluster geometry, but with the charge at a neighbor molecule, is calculated. The reorganization energy is then λext = E* −E

ð60Þ

However, depending on the crystal symmetry λext depends on the regarded donor–acceptor pair, because the different monomers have a different crystal surrounding. The resulting λext is taken as the average of these values. Another approach which is also found in litera79 ture is to optimize the cluster in its neutral state and with a charge carrier positioned in the middle of the cluster. The energies E0 of the neutral cluster and Ec of the cluster containing the excess charge are determined




and additionally the energy of the cluster with neutrally optimized geometry but with additional charge, E*c , and the energy of the neutral cluster with the geometry optimized for the charged state, E*0 , are determined. The external reorganization energy is then calculated analogously to the internal λ, λ0ext = E*c − Ec + E*0 − E0

νji : =

For sufficient small Δti it follows the master equation dpi ðtÞ X = νij pj ðt Þ− νji pi ðtÞ dt j

Δti 1 1 τi : = Δti = X = X αji = X αji νji Δti j

The Master Equation The master equation is a phenomenological firstorder differential equation, which describes the time evolution of the probability for a system to be in a certain state out of a discrete set of states. It is often used in physics and chemistry, e.g., to describe diffusion processes (random walks), population dynamics, and chemical kinetics.49,84–92 If a charge carrier or an exciton is at a certain position at time t and hops to other positions with a certain transition probability α, there is a probability pi(t + Δti) that the particle is at the position i at the time t + Δti. It is ð62Þ

j

pj(t) is the probability that the site j is occupied by a particle and αij is its transition probability from j to i. The summation index j runs over all positions from where a hop to i is possible. As it is assumed that the particle definitely jumps after the time Δt (this dwell time will be determined below) it is ð63Þ

νji αji = νji Δti = X νki

ð68Þ

The master equation describes a so-called Markov chain,93 where the next state of the system only depends on the current state but not on the preceding states, i.e., p(t + Δt) only depends on p(t) but not on p(t − Δt), see Eq. (62). The master equation is also known as Pauli master equation since Wolfgang Pauli is said to have been the first who derived this type of kinetic equation.94 He used this approach to study the time evolution of a many-state quantum system95 where the state probabilities correspond to the diagonal elements of the density matrix. The coupled Eq. (66) for all sites can be written as a matrix equation !

dpðtÞ ! = Np dt

ð69Þ

where the matrix N contains the hopping rates νji ! and the vector p contains all occupation probabilities. Its solution is96 !

p ðt Þ =

so that

X! ci e li t

ð70Þ

i

X j

,

j

k

j

pi ðt + Δti Þ −pi ðtÞ =

ð67Þ

Combining Eqs. (65) and (67), the transition probability is

SIMULATION OF THE DYNAMICS

X αji = 1

ð66Þ

Using Eqs. (63) and (65) the average dwell time after which a jump occurs can be expressed via the hopping rates:

j

X αij pj ðt Þ

ð65Þ

ð61Þ

Although, the reorganization is probably overestimated because for the calculation, the crystal structure of a cluster containing a charge in the middle is compared with the structure of a completely neutral cluster.

pi ðt + Δti Þ =

αji Δti

αij pj ðtÞ −

X αji pi ðt Þ j

ð64Þ X αji pi ðt + Δti Þ − pi ðtÞ X αij = pj ðt Þ− pi ðt Þ Δti Δti Δti j j

One now can define the hopping rate:

with the eigenvalues li and the respective eigenvectors ! ci of N. In the steady state, the occupation probabilities pi(t) do not change anymore and Eq. (69) equals zero, which therefore turns into a homogeneous linear system of equations


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!

ð71Þ

Np = 0

It can be solved by standard numerical procedures.97 The charge carrier mobility can then be calculated with88,96 !

μ=

1X ! ℰ pi e νji rji ℰ ℰ ij

ð72Þ

where ℰ is the external electric field and rji is the distance between the sites i and j. The calculation of the diffusion constant is a bit more complicated. Quite often the equation D=

! !2 1X pie νji rji e 2n ij

ð73Þ

is found in the literature. However, even in perfectly ordered crystals where all jump rates are symmetric, i.e., νji = νij, it can happen that the charge or exciton jumps back and forth very often between the same two monomers. This takes place if their coupling is large and the coupling to the other neighboring monomers is smaller. In this case, these back and forth jumps are all summed up in Eq. (73) even though they do not contribute to a macroscopic spreading of the occupation probability of the charge or exciton. As a consequence, the diffusion constant is overestimated. This numerical artifact can be overcome by using the equation44 !

J 1X ! F D= − 0 pie νji rji F 2J F ij

analytically or numerically hard to handle. The kinetic Monte Carlo method, which allows for the simulation of the time evolution of a system, is frequently used to study charge102–124 and exciton125–129 transport in organic solids. Instead of solving the master equation (66) directly in order to obtain the (probability) distribution, one can also solve this equation stochastically. The algorithm works as follows:112 The particle is assumed to be at a certain site i. A cut-off radius around i is defined which contains all possible target sites k. For all targets, the transition probability αki is calculated from the hopping rates via Eq. (68). Then a uniformly distributed random number ξ is generated and the hopping target j is determined by

ð74Þ

j− 1 X k=0

!

−

JMarcus = kB T 0 2JMarcus

ð75Þ

In this case, the relation between Eqs. (74) and (72) results in the Einstein relation.98,99

The Kinetic Monte Carlo Method The Monte Carlo method100,101 is a stochastic approach to tackle mathematical problems which are

ð76Þ

k=0

* !+ hvi 1 d ! ℰ = rðt Þ μ= ℰ ℰ ℰ dt

ð77Þ

Analogously, the diffusion constant is calculated with

which is based on the equation of the mobility. For !

j X αki

The particle moves from i to j and the time is advanced by the dwell time τi, Eq. (67). By repeating this process, a trajectory of the particle is simulated. Averaging over a sufficient number of trajectories leads to the same probability distribution as obtained by solving the master equation (66). The charge carrier mobility can be calculated via

D=

charge carriers, the force is F = qℰ. In the case of excitons, this force is purely fictitious and formally the limiting case F ! 0 has to be regarded. However, if F is sufficiently small, D does not depend on F.44 For the Marcus theory, it can be shown that44

αki < ξ ≤

1 d D! ! 2 E ! !2 rðtÞe − rðt Þe 2 dt

ð78Þ

!

where e is the unit vector in the regarded direction. For N simulated trajectories, the mean square deviation of the arithmetic mean of the occupation probability p(i, t) of site i at time t0 is vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u N N X uX 1 u pn ði,t Þ − N pm ði,tÞ u tn = 1 m=1 σ pði, tÞ = N ðN − 1 Þ σ~pði, tÞ = pffiffiffiffiffi N

ð79Þ

where σ~pði, tÞ is the mean square deviation of a single trajectory, which is independent of N. This means that the statistical error of the Monte Carlo simulapffiffiffiffiffi tion converges only with N and therefore, many




simulation runs are needed in order to achieve an acceptably low statistical error. Furthermore, the steady state is obtained only after a certain transient time that may possibly lead to an untenable longsimulation time. However, especially for large systems the Monte Carlo approach can be advantageous for describing the time-dependent motion of particles. The number of coupled equations (66) which have to be solved for the master equation approach increases with ld, where d is the dimensionality of the system and l is the length of the regarded region in one dimension, while the convergence behavior of Monte Carlo, Eq. (79), does not depend on the size and the dimension. The Monte Carlo approach allows to simulate the motion of many particles at the same time and to include interaction forces between them. For charge carriers, the long-range Coulomb interaction is important in the case of higher charge carrier densities. The charge distribution causes the lattice sites to have different energies, which influences the hopping rates. Because of this interaction, the motions of the charges are not independent from each other, and one has to furthermore identify in each cycle of the simulation which charge is the one to hop next either by determining this from their respective dwell times and the simulation time112 or by choosing it randomly.130

QUANTUM CHEMICAL METHODS The electronic couplings for exciton transport are calculated with the spin-component scaled131,132 algebraic diagrammatic construction through second order133 (SCS-ADC(2)) together with the correlation consistent polarized triple ζ basis sets134 (cc-pVTZ). It was shown that the time-dependent Hartree-Fock (TDHF) slightly overestimates the coupling compared to the more accurate SCS-ADC(2) and the very similar approximate coupled cluster singles and doubles135–138 (SCS-CC2).139 In contrast to that, the reorganization energy is very sensitive to the used quantum chemical method139 (cf. Exciton Transport section). Therefore, λ was calculated with SCS-CC2 for both the geometry optimization and energy calculation and with the cc-pVTZ basis sets. The same holds for the frequency calculations. The influence of the basis sets is negligible.139 In contrast to exciton transport, the reorganization energy and frequency calculations for charge carriers do not depend strongly on the method. Therefore, all calculations (including the coupling) are conducted via density functional theory using the hybrid generalized gradient functional B3-LYP140–145

with the correlation consistent polarized valence double ζ basis set (cc-pVDZ)134 for all atoms. This functional is chosen because it has been shown that it leads to quite good results for describing the ionization-induced geometry modifications of oligoacenes.58,77 All quantum chemical calculations are performed with the Turbomole program package.146–148

RESULTS Exciton Transport Table 1 shows a comparison of the exciton diffusion constants and lengths of naphthalene between the ones obtained with the spectral overlap approach,144 the Marcus theory,139 and experimental values.149,150 While the values calculated with the spectral overlap fit very accurately to the measurements, the Marcus theory yields values which are too small. The FCWD, Eq. (7), for naphthalene is J = 0.71 eV−1, and the corresponding value in the Marcus theory, Eq. (34), is JMarcus = 0.10 eV−1 (with λ = 347 meV, both calculated with SCS-CC2/cc-pVTZ). JMarcus is smaller than J because in the Marcus theory a classical approximation with only one collective vibrational mode is employed. This corresponds to narrow absorption and emission spectra, resulting in a smaller overlap of the spectra. Therefore, the diffusion constants calculated with the more precise spectral overlap are bigger by a factor of 6.9 and the diffusion lengths are larger by a factor of 2.6. Figure 5 TABLE 1 | The Exciton Diffusion Constants D and Diffusion Lengths L for Naphthalene44,139 Diffusion Constant D (in 10− 9 m2/s) Overlap

Marcus

a

14.4

2.0

b

33.6

4.9

c

10.3

1.5

c’

9.0

1.3

Measured149

5

Diffusion Length L (in nm, τ = 78 ns150,165,166) Overlap

Marcus

a

48

18

b

72

28

c

40

15

c0

37

14

Measured150 501

a, b, and c are the directions of the respective lattice vectors, c0 is the direction perpendicular to the ab plane. (Vji was calculated with SCS-ADC(2)/ccpVTZ, λ and J were calculated with SCS-CC2/cc-pVTZ.) 1 Direction unknown.


Advanced Review


Overlap, master eq. Marcus, master eq. Marcus, Monte Carlo

b

c c

a

0 20 40 Diffusion length L (nm)

60

80

b


60

a

80


60

80

F I G U R E 5 | The exciton diffusion length in the naphthalene crystal in the (ab), (bc), and (ac) plane. The dashed lines are calculated with the spectral overlap approach and the solid lines are calculated with the Marcus equation, both using the master equation approach. The dots are calculated with the Marcus equation using the Monte Carlo method. (Vji was calculated with SCS-ADC(2)/cc-pVTZ, λ and J were calculated with SCS-CC2/cc-pVTZ, τ = 78 ns.150,165,166) TABLE 2 | The Reorganization Energy of Naphthalene Calculated with Different Methods and Basis Sets (in meV).139 cc-pVTZ

cc-pVQZ

183

187

186

B3-LYP

213

218

216

TDHF

515

527

525

SCS-CC2

351

347

345

compares both approaches using the master equation approach. Additionally, the Monte Carlo results for the Marcus theory are plotted, which fit exactly to the master equation results. One computationally important point that must be taken into account is that the reorganization energy is very sensitive to the quantum chemical method that is used for the calculation, see Table 2.139 Using SCS-CC2 as the presumably most accurate method results in a reorganization energy of 350 meV, while TDHF predicts a much larger value of 520 meV. The tested DFT approaches underestimate the reorganization energy in comparison to SCS-CC2. Because the reorganization energy enters the hopping rate exponentially, variations of λ change the diffusion constant by almost two orders of magnitude (cf. Figure 6). This indicates that at least the reorganization energy has to be computed with a high-level ab initio method to achieve reliable estimates for D. In contrast to that, the influence of the quantum chemical method used for the electronic coupling has a much smaller influence on the result139 (cf. Table 3 and Figure 7).

Diffusion constant D (m2/s)

cc-pVDZ RI-BLYP

10–7

Direction a b c

10–8

10–9

RI−BLYP B3−LYP

SCS−CC2

TDHF

10–10 200

300

400

500

Reorganisation energy (meV)

F I G U R E 6 | The diffusion constant D of naphthalene along the unit cell vectors depending on the reorganization energy λ. The λ values from Table 2 are tagged by arrows. (Vji was calculated with SCS-ADC(2)/cc-pVTZ.) Table 4 shows a comparison of the calculated diffusion lengths for anthracene,44,139 using an exciton lifetime of τ = 10 ns,150–152 with experimental values.153,154 The FCWD, Eq. (6), is J = 0.35 eV−1, while the corresponding expression in the Marcus theory has a value of 0.01 eV−1 (λ = 530 meV, both calculated with SCS-CC2/cc-pVTZ). The diffusion constants and lengths obtained with the spectral overlap approach are therefore larger than the Marcus values by a factor of 24.7 and 5.0, respectively. For the calculation of L, an exciton life time of 10 ns was assumed.150–152 The results of the spectral overlap approach compare qualitatively excellent with the experimental values.




TABLE 3 | The Largest Coupling Vji (in meV) in the Naphthalene Crystal which Is the Coupling between the Two Monomers in the Unit Cell139

TABLE 5 | Internal and External Reorganization Energies for Hole Transport for Different Acenes λext [meV]

λ0 ext [meV]

cc-pVDZ

cc-pVTZ

cc-pVQZ

Reference

λint [meV]

9.3

9.5

9.5

Naphthalene

168

183

35

70

TDHF SCS-CC2

7.5

7.4

7.5

Anthracene

169

136

52

72

SCS-ADC(2)

6.8

7.1

7.1

Tetracene

170

110

19

56

Pentacene

171

92

29

52

Hexacene

172

81

33

44

b

SCS−ADC(2) SCS−CC2 TDHF

200

int

180

int, fit

160

ext

140

100 Change of crystal Symmetry

a

(meV)

120

80 60 40 20 0

2

3

4

5

6

Number of benzene rings

0 10 Diffusion length L (nm)

20

30

40

FI GU RE 7 | The exciton diffusion length in the naphthalene crystal in the (ab) plane (τ = 78 ns150,165,166). The couplings were calculated with SCS-ADC(2), SCS-CC2, and TDHF, respectively. In all cases, the cc-pVTZ basis sets were used and the reorganization energy was calculated with SCS-CC2/cc-pVTZ.139 TABLE 4 | Comparison of Diffusion Lengths for Anthracene (in nm, τ = 10 ns150–152) Calculated with the Spectral Overlap Approach,44 the Marcus theory,139 and Experimental Values

Overlap

Marcus

Measured

a

58

12

60 101

b

132

27

Approximately 1001

c

58

12

c’

57

11

60 52, 49 13, 47 13, 36 201

Vji was calculated with SCS-ADC(2)/cc-pVTZ, J and λ were calculated with SCS-CC2/cc-pVTZ. 1 Ref 153. 2 Ref 154. 3 Ref 167.

Charge Transport Marcus Theory Table 5 lists some acenes and their reorganization energies. The internal reorganization energies λint,

F I G U R E 8 | The internal and external reorganization energies for the acenes depending on the number of benzene rings. The line is a fit, see Eq. (80). Note that the crystal symmetry changes when going from three to four benzene rings: naphthalene and anthracene belong to the space group P21/a, tetracene, pentacene, and hexacene belong to the space group P1.

Eq. (58), decrease with increasing number n of the benzene rings which can be described by the fitted power law λint ≈0:31eVn − 0:74

ð80Þ

see Figure 8. However, because of the larger complexity, there is no such simple correlation for the external reorganization energy λext, Eq. (60). Naphthalene and anthracene belong to the space group P21/a, whereas tetracene, pentacene, and hexacene belong to the space group P1. This impedes to find a similar relationship as for λint. However, the respective crystals with the same symmetry show an increasing λext with increasing number of benzene rings. The reorganization energy for charge carriers is not that sensitive to the quantum chemical method used, as this is the case for excitons (cf. Exciton Transport section). Using density functional theory,


Advanced Review


the long-range corrected functional ωB97X-D155 was found to perform best, especially for push–pull systems. However, for smaller molecules as, e.g., acenes, the differences are negligible.156 Table 5 furthermore lists the alternative reorganization energies λ0 ext calculated with Eq. (61). As expected, these energies are larger than λext obtained with Eq. (60), up to a factor of 3 for tetracene. In contrast to λext, λ0 ext decreases with increasing molecule size for the molecules with P1 symmetry (tetracene to hexacene). Furthermore λ0 ext is almost the same for naphthalene and anthracene, whereas their λext differs by a factor of 1.5. This shows the necessity of further effort to investigate the correct calculational approach for the external reorganization energy, since even the qualitative trend differs for

different approaches. Since the approach of calculating λext appears to be physically more reasonable than λ0 ext, these external reorganization energies are used for the mobility calculations. The calculations are conducted with a relative dielectric constant of εr = 1. However, because of the polarization of the crystal environment, a value of εr = 3 appears to be more appropriate.157–161 Taking this into account, the λ0 ext value of naphthalene changes from 70 meV (εr = 1) to 24 meV (εr = 3) which is about a divisor of three smaller. In the case of λext, this effect is expected to be smaller. However, one has to bear in mind that the external reorganization energy is probably overestimated in trend. Figures 9 and 10 depict the hole mobilities for tetracene and pentacene for three different crystal

Without ext With ext

b

c

c

Measurement

a

0

5

a

10

0

Mobility (cm2/(Vs))

5

b

10

0

Mobility (cm2/(Vs))

5

10

Mobility (cm2/(Vs))

F I G U R E 9 | Hole mobility in the tetracene crystal with the Marcus theory, calculated with and without external reorganization energy. The experimental values are from Ref 162. c

Without ext With ext Measurement

b

c

a

0

5

10

15

Mobility (cm2/(Vs))

20

a

0

5

10

15

Mobility (cm2/(Vs))

20

b

0

5

10

15

20

Mobility (cm2/(Vs))

F I G U R E 10 | Hole mobility in the pentacene crystal with the Marcus theory, calculated with and without139 external reorganization energy. The experimental values are from Ref 163.




planes, calculated with and without the external reorganization energy. For comparison, some experimental values are also drawn. The calculated mobilities are much larger than in the experiment162,163 so that even including λext hardly leads to an improvement. In the literature, different approaches for the calculation of the external reorganization energies are proposed.79,164 However, the large discrepancy between calculation and experiment suggests that improving the calculation of λext is not constructive in order to achieve better agreement with the experiment. For a matching, the external reorganization energy for pentacene, for example, had to be about 180 meV, which is twice the internal reorganization. This is not realistic and it makes clear that the mismatch between the measurement and the Marcus hopping model cannot be explained by surrounding effects.

Levich–Jortner Theory

Table 6 lists the effective frequencies νeff = ωeff/(2π), the effective Huang–Rhys factors Seff and the quantum mechanical (λqm) and classical (λcl) part of the reorganization energy for some acenes. Seff slightly decreases with increasing molecular length, which means that the geometric distortion upon the charge transfer becomes smaller with increasing molecular size, cf. Eq. (20), and the distribution of excited vibrational quantum numbers is shifted to lower values. Table 7 lists all vibrational modes with a λi contribution of more than 1 meV for tetracene and pentacene. Only the totally symmetric vibrations are excited upon charge transfer. The number of contributing vibrations increases with increasing molecular size, however, only 5–10 vibrational modes are relevant for the series of acenes listed in Table 6. At room temperature, vibrations up to an angular frequency of about 208 cm−1 are excited. However, all vibrational modes in the acene molecules listed in Table 7 lie energetically higher, i.e., all TABLE 6 | Effective Frequencies νeff, Effective Huang-Rhys Factors Seff, Quantum Mechanical Part of Reorganization Energy, λqm, and Classical Part, λcl, for Some Acenes νeff [cm− 1]

Seff

λqm [meV]

λcl [meV]

intramolecular vibrations have to be treated quantum mechanically. Therefore, the classical part of the reorganization energy is λcl = λext. However, this is not the general case, especially for large molecules, where usually more low-frequency vibrations (molecular bending modes, vibrations of side chains) appear. In the case of the acenes from naphthalene to hexacene, the frequency of the energetically lowest totally symmetric vibration—the stretch vibration in the direction of the long molecule axis—decreases with increasing molecular length from 518 cm−1 for naphthalene to 225 cm−1 for hexacene. Figures 11 and 12 show the hole mobilities for tetracene and pentacene, respectively, calculated with both the Levich–Jortner equation (45) and the Marcus equation (33), including the external reorganization energy calculated in External Reorganization Energy section. For comparison, some experimental values are also plotted. The values calculated with Levich–Jortner are throughout significantly larger than those obtained with the Marcus theory. The

TABLE 7 | Frequencies, Huang-Rhys Factors, and Reorganization Energies of the Vibrations of the Neutral and the Cation Structures of Tetracene and Pentacene Neutral

νi [cm

−1

Cation

Si

λi [meV]

1025

0.005

1

1173

0.0013

1221

0.036

1426

0.014

3

1422

0.030

5

1429

0.071

13

1441

0.092

16

1475

0.016

3

1469

0.001

0

1561

0.024

5

1546

0.045

9

1586

0.119

23

1592

0.080

16

]

νi [cm

−1

Si

λi [meV]

1046

0.002

0

2

1185

0.018

3

6

1234

0.034

5

]

Tetracene

Pentacene 263

0.031

1

262

0.030

1

1021

0.005

1

1040

0.003

0

1170

0.012

2

1182

0.017

2

1197

0.036

5

1207

0.034

5

1336

0.001

0

1331

0.005

1

Naphthalene

1417

1.039

183

35

1425

0.085

15

1426

0.010

2

Anthracene

1484

0.738

136

52

1445

0.008

1

1440

0.098

18

Tetracene

1420

0.626

110

19

1489

0.005

1

1498

0.000

0

Pentacene

1299

0.574

92

29

1563

0.081

16

1552

0.049

9

Hexacene

1159

0.548

81

33

1584

0.018

4

1585

0.039

8

For the molecules regarded here λcl is identical to λext. (Calculated with DFT B3-LYP/cc-pVDZ.)

Only those vibrational modes with λi > 1 meV are listed. (Calculated with DFT B3-LYP/cc-pVDZ.)


Advanced Review


Marcus Levich−Jortner Measurement

b

c

c

a

0

10

20

a

30

0

Mobility (cm /(Vs))

10

20

b

30

0

Mobility (cm /(Vs))

2

10

20

30

Mobility (cm /(Vs))

2

2

F I G U R E 11 | Hole mobility in the tetracene crystal, calculated with the Marcus and the Levich–Jortner hopping rate. The experimental values are from Ref 162. b

c

Marcus Levich−jortner Measurement

c

a

0 20 Mobility μ (cm2/(Vs))

40

a


40

b


40

F I G U R E 12 | Hole mobility in the pentacene crystal, calculated with the Marcus and the Levich–Jortner hopping rate. The experimental values are from Ref 163. crucial difference between the Levich–Jortner equation and the Marcus equation is that in the former, the classical part of the reorganization energy, λcl, takes the place of the total λ in the Marcus equation. Furthermore, the quantum mechanical part p ℏωeff is added in the argument of the exponential function and the exponential part is summed over all final vibrational states, p = mD + nA, weighted by the probability of p, which is a Poisson distribution. This is plotted in Figure 13 for the regarded acenes. In the case of naphthalene, the exponential term in the Marcus hopping rate, exp[−λ/(4kBT)] (for simplicity ΔE = 0 here) has a value of 0.12, using the total reorganization energy of λ = λqm + λcl = 218 meV, Table 6. The same exponential term appears in the first summand of the Levich–Jortner hopping rate

(p = 0), however, since λcl (35 meV, Table 6) replaces the total λ, its value is 0.71. This is weighted with the probability P(p = 0) = exp(−1.039) = 0.35, leading to 0.25, which is still twice the value from the Marcus hopping rate. The summands with p 6¼ 0 in principle further increase the hopping rate with respect to the Marcus rate, which leads to the much higher mobilities obtained with the Levich–Jortner theory. This comparison of numbers shows that the different reorganization energies that are used in the Marcus and the Levich–Jortner hopping rates are the reason for the large differences of their results. In the Levich–Jortner theory, it is taken into account that in essence the vibrations up to an energy of kBT are excited and that higher energetic vibrations do become partially excited upon charge transfer,




0.6

Naphthalene Anthracene Tetracene Pentacene Hexacene

0.5

ext = 29 meV

b

ext = 52 meV ext = 200 meV Measurement

Probability

0.4 0.3 0.2

a 0.1 0

0

1

2

3

4

5

Sum of final vibrational numbers p

FI GU RE 13 | The probability distribution of the sum of the final vibrational numbers p = mD + nA for the regarded acenes. 0

however, play a minor role for the molecular reorganization. In the Marcus theory, all vibrational modes influence the hopping rate, even those that are very high in energy. The vibrational modes highest in energy in the neutral and in the cationic tetracene, respectively, have a frequency of about 1590 cm−1, however, they contribute 39 meV to the total λ of 129 meV in the Marcus theory (see Table 7). The reason for the better results of the Marcus theory than those of the Levich–Jortner theory compared to the experimental values (Figures 11 and 12) is therefore the overestimation of the physically relevant molecular reorganization due to the simplified treatment of the vibrations. The Levich–Jortner theory is explained in The Levich–Jortner Hopping Rate section. The energy p ℏωeff enters the exponential function by integration over the delta function, Eq. (38), in Eq. (39). The appearance of this term in the delta function corresponds to an energetic shift of the auxiliary function DD(E) with respect to DA(E), which both are ‘quasidensities of states,’ leading to a smaller overlap and therefore to a smaller hopping rate. For the acenes listed in Table 6, the exponential functions in the Levich–Jortner hopping rate with p 6¼ 0 are virtually zero and only the summand with p = 0, i.e., the vibrational 0 ! 0 transitions for both donor and acceptor, contribute here to the hopping rate. As explained above, the reason for the higher mobilities of Levich–Jortner theory compared with Marcus is the comparatively small classical part of the reorganization energy, λcl, where mainly (in the case of the acenes even exclusively) the external reorganization energy contributes. The correct way of how to take the nonlocal charge-phonon coupling into account and how to calculate the external

10

20

30

Mobility (cm2/(Vs))

F I G U R E 14 | The hole mobility for pentacene calculated with

λext = 29 meV (dashed), λ0 ext = 52 meV (dotted, for both values see Table 5) and λext = 200 meV (solid), which is chosen so that the mobility fits to the experimental values (green).

reorganization energy is currently a frequently discussed topic.78,79,164 In External Reorganization Energy section, two different approaches to calculate the external λ were developed, leading to λext = 29 and λ0 ext = 52 meV, respectively for pentacene (Table 5). Figure 14 shows the hole mobility for this crystal in the ab plane calculated with both values for the external reorganization energy. The mobility decreases by a divisor of 1.7 when using the higher λ0 ext instead of λext, however, it is still a factor of about eight bigger than the experimental values. In order to obtain mobilities in the same order of magnitude as the measured values, an external reorganization energy of about 200 meV is necessary. In the case of tetracene (λext = 19 and λ0 ext = 56 meV), Figure 15, even an external λ of 500 meV is necessary. However, these high values are not realistic. Therefore, the reason for the much too large mobilities obtained with the Levich–Jortner theory cannot be found in an insufficient consideration of surrounding effects, as was already shown for the Marcus theory.

Spectral Overlap Approach Figure 16 shows the FCWD depending on the energetic shift caused by the electric field for hole transport in tetracene. The red line is calculated with Eq. (8). For comparison the FCWD calculated with the Marcus theory, Eq. (34), is also plotted (blue line).


Advanced Review


6

b

ext = 500 meV

Overlap approach Linear approxim Marcus theory

5 FCWD J (eV−1)

Measurement

a

4 3 2 1 0 −0.6

−0.4

−0.2

0

0.2

0.4

Energetic shift Eji (eV) 2.6 Overlap approach Linear approximation

Mobility μ

0.1

0.2

2.4

(cm2/(Vs))

F I G U R E 15 | The hole mobility for tetracene calculated with

λext = 500 meV, which is chosen so that the mobility approximately fits to the experimental values.

FCWD J (eV–1)

0

2.2

2

The green line is the linear approximation of J, Eq. (9). One clearly sees that this is a quite bad approximation for J, even in the energy range which is relevant for charge transport, which is plotted enlarged in the lower graph. However, it can be shown that the mobility does not depend on J directly but on the derivative J0 = dJ=dEji jEji = 0 defined in Eq. (9), when using the linear approximation:45 μ= −

2π 0 D 2 2 E qJ Vji rji ℏ

ð81Þ

Without the linear approximation, one gets instead:45 μ= −

E 2π D 0 q J sec, ji Vji2 r2ji ℏ

ð82Þ

The only difference is that in the latter equation, an average of the slopes of the secants through the points (−Eji|J(−Eji)) and (Eji|J(Eji)) appears instead of J0 . The slope of the tangent at J(Eji = 0) appears to fit quite well to the slope of the secants through the points (Eji, J(Eji)). Two secants for the energies Eji = 20 and 40 meV are plotted as green dashed lines. The secants are almost parallel to each other and they are almost parallel to the linear approximation. Figure 17 shows the hole mobilities for pentacene, calculated with both the spectral overlap approach (solid) and the Marcus theory (dashed).45

1.8

−0.04

−0.02

0

0.02

0.04

Energetic shift Eji (eV)

F I G U R E 1 6 | The Franck–Condon-weighted density of states e J Eji depending of the energetic shift between the molecules i and j caused by the external electric field for hole transport in tetracene. The red line is calculated with the spectral overlap approach, Eq. (8), the green solid line is the linear approximation used in Eq. (9), the green dashed lines are secants through the points E1 jeJ ðE1 ÞÞ, and the blue line is calculated with the Marcus theory. The lower graph is a magnification of the upper one, which shows the energy interval which is relevant for charge transport. (Calculated with DFT B3-LYP/ cc-pVDZ.)

For comparison some experimental values163 are also drawn. While the values calculated with the Marcus theory are much larger than the experimental values (as already shown in Marcus Theory section), the values calculated with the spectral overlap approach nicely fit to the measured values. Figure 18 depicts the hole mobilities for tetracene.45 Again the Marcus theory overestimates the mobilities, whereas the overlap approach fits much better to the measured values.162

SUMMARY This work investigated the exciton and charge transport properties of oligoacene crystals. Different



b


b

Spectral overlap Marcus

Spectral overlap Marcus

a

a

0

5

10

15


20

10

Mobility μ (cm2/(Vs)) b

b

Spectral overlap Measurement

Spectral overlap Measurement

a a


2

0 0.2 0.4 Mobility μ (cm2/(Vs))

3

FI GU RE 17 | The hole mobilities for pentacene in the ab plane, calculated with the spectral overlap approach (solid lines) and the Marcus theory (dashed line).45 The experimental values (dots) are taken from Ref 163. (Calculated with DFT B3-LYP/cc-pVDZ.)

thermally activated hopping models based on timedependent perturbation theory were studied for the charge and exciton transport, i.e., the spectral overlap approach, the Marcus theory, and the Levich– Jortner theory. Their derivations were presented coherently in order to emphasize the different levels of approximations and their respective prerequisites. The Marcus theory, originally derived for outer sphere electron transfer in solvents, had already been well established for charge transport

0.6

0.8

F I G U R E 18 | The hole mobilities for tetracene in the ab plane, calculated with the spectral overlap approach (solid lines) and the Marcus theory (dashed line).45 The experimental values (dots) are taken from Ref 162. (Calculated with DFT B3-LYP/cc-pVDZ.)

in organic solids. It was shown that this theory fits even better for excitons than for charges compared with the experiment, because for excitons, the reorganization energy is larger and the coupling is smaller, making perturbation theory more warrantable. However, in comparison to the spectral overlap approach and experimental values, the calculated diffusion constants are underestimated. Nevertheless, the Marcus theory appears to be sufficiently accurate to study trends even without the demanding calculation of the molecular vibrational


Advanced Review


spectrum, as this is necessary for the spectral overlap approach. The Levich–Jortner theory strongly overestimates the charge carrier mobilities and the results deviate even stronger from the experiment than those obtained with the Marcus theory. The latter contains larger approximations by treating all vibrational modes classically. It was shown that this approximation leads to a strong overestimation of the significance of the molecular high-frequency vibrations to the molecular reorganization upon charge transfer, which lowers the hopping rates. As a result, the Marcus theory fits better to the experimental data of oligoacenes than the Levich–Jortner theory due to error cancelation. The spectral overlap approach leads to even quantitatively very good results for exciton diffusion lengths compared to experiment. In the case of charge transport, the mobility does not depend on the spectral overlap itself, but only on its derivative with respect to the external electric field. It was shown that this approach leads to better results than the Marcus theory compared with measured values, however, because of the linear approximation of the spectral overlap introduced here, the reliability of this method is more critical than in the case of exciton transport and may depend on the studied molecules.

It was shown that surrounding effects (taken into account by the external reorganization energy) are too small to explain the mismatch between the Marcus and Levich–Jortner theory and the experimental values in general. As mentioned in the introduction, charge and energy transport in organic crystalline materials are not yet fully understood. However, we found a very satisfactory agreement between experimental data and most of the theoretical mobilities obtained with the hopping model if hopping rates were obtained with the spectral overlap approach whose input data were generated with accurate quantum chemical methods. In order to come to a complete understanding of charge and energy transport in organic materials, it is clearly necessary to address coherence effects, thermal reorganization of the molecules within the crystal, and the external reorganization energy in more detail. Some work in this respect is presently conducted in our and other laboratories. However, to the best of our knowledge, the present work shows which results are obtained if the hopping model is consistently employed with state of the art quantum chemical methods. We think that this is an importing starting point to obtain further insight in this field.

REFERENCES 1. Armstrong NR, Wang W, Alloway DM, Placencia D, Ratcliff E, Brumbach M. Organic/Organic’ Heterojunctions: Organic Light Emitting Diodes and Organic Photovoltaic Devices. Macromol Rapid Commun 2009, 30:717–731. 2. Meerheim R, Luessem B, Leo K. Efficiency and Stability of p-i-n Type Organic Light Emitting Diodes for Display and Lighting Applications. Proc IEEE 2009, 97:1606–1626. 3. Kulkarni AP, Tonzola CJ, Babel A, Jenekhe SA. Electron Transport Materials for Organic LightEmitting Diodes. Chem Mater 2004, 16:4556–4573. 4. Yang X, Zhou G, Wong W-Y. Recent design tactics for high performance white polymer light-emitting diodes. J Mater Chem C 2014, 2:1760–1778.

7. Facchetti A. Semiconductors for organic transistors. Mater Today 2007, 10:28–37. 8. Xu Y, Liu C, Khim D, Noh Y-Y. Development of high-performance printed organic field-effect transistors and integrated circuits. Phys Chem Chem Phys, 2014, 17:26553–26574. 9. Sirringhaus H. Organic Field-Effect Transistors: The Path Beyond Amorphous Silicon. Adv Mater 2014, 26:1319–1335. 10. Subramanian V, Chang P, Lee J, Molesa S, Volkman S. Printed organic transistors for ultra-lowcost {RFID} applications. IEEE Trans Compon Packag Technol 2005, 28:742–747.

5. Dimitrakopoulos C, Malenfant P. Organic Thin Film Transistors for Large Area Electronics. Adv Mater 2002, 14:99–117.

11. Steudel S, De Vusser S, Myny K, Lenes M, Genoe J, Heremans P. Comparison of organic diode structures regarding high-frequency rectification behavior in radio-frequency identification tags. J Appl Phys 2006, 99:114519.

6. Newman CR, Frisbie CD, da Silva Filho DA, Brédas J-L, Ewbank PC, Mann KR. Introduction to Organic Thin Film Transistors and Design of n-Channel Organic Semiconductors. Chem Mater 2004, 16:4436–4451.

12. Myny K, Steudel S, Smout S, Vicca P, Furthner F, van der Putten B, Tripathi A, Gelinck G, Genoe J, Dehaene W, et al. Organic RFID transponder chip with data rate compatible with electronic product coding. Org Electron 2010, 11:1176–1179.




13. Kearns D, Calvin M. Photovoltaic Effect and Photoconductivity in Laminated Organic Systems. J Chem Phys 1958, 29:950–951. 14. Tang CW. Two-layer organic photovoltaic cell. Appl Phys Lett 1986, 48:183–185. 15. Riede M, Mueller T, Tress W, Schueppel R, Leo K. Small-molecule solar cells—status and perspectives. Nanotechnology 2008, 19:424001. 16. Kippelen B, Brédas J-L. Organic photovoltaics. Energy Environ Sci 2009, 2:251–261. 17. Dennler G, Scharber MC, Brabec CJ. PolymerFullerene Bulk-Heterojunction Solar Cells. Adv Mater 2009, 21:1323–1338. 18. Chidichimo G, Filippelli L. Organic Solar Cells: Problems and Perspectives. Int J Photoenergy 2010, 2010:123534. 19. Deibel C, Dyakonov V. Polymer–fullerene bulk heterojunction solar cells. Rep Prog Phys 2010, 73:096401. 20. Service RF. Outlook Brightens for Plastic Solar Cells. Science 2011, 332:293. 21. Cao W, Xue J. Recent progress in organic photovoltaics: device architecture and optical design. Energy Environ Sci 2014, 7:2123–2144. 22. Brabec C, Scherf U, Dyakonov V. Organic Photovoltaics – Materials, Device Physics, and Manufacturing technologies. 2nd ed. Weinheim: Wiley-VCH; 2014. 23. Shirakawa H, Louis EJ, MacDiarmid AG, Chiang CK, Heeger AJ, Chem J. Synthesis of electrically conducting organic polymers: halogen derivatives of polyacetylene, (CH)x. J Chem Soc, Chem Commun 1977, 16:578–580. 24. Brédas J-L, Norton JE, Cornil J, Coropceanu V. Molecular Understanding of Organic Solar Cells: The Challenges. Acc Chem Res 2009, 42:1691–1699. 25. Holstein T. Studies of polaron motion: Part I. The molecular-crystal model. Ann Phys 1959a, 8:325–342. 26. Holstein T. Studies of polaron motion : Part {II}. The “small” polaron. Ann. Phys. 1959b, 8:343–389. 27. Peierls RE. Quantum Theory of Solids. 1st ed. Oxford: Oxford University Press; 1955. 28. Hannewald K, Stojanović VM, Schellekens JMT, Bobbert PA, Kresse G, Hafner J. Theory of polaron bandwidth narrowing in organic molecular crystals. Phys Rev B 2004, 69:075211. 29. Hannewald K, Bobbert PA. Ab initio theory of charge-carrier conduction in ultrapure organic crystals. Appl Phys Lett 2004, 85:1535. 30. Ortmann F, Bechstedt F, Hannewald K. Charge transport in organic crystals: interplay of band transport, hopping and electron–phonon scattering. New J Phys 2010a, 12:023011.

31. Ortmann F, Bechstedt F, Hannewald K. Characteristics of small- and large-polaron motion in organic crystals. J Phys Condens Matter 2010b, 22:465802. 32. Troisi A, Orlandi G, Anthony JE. Electronic Interactions and Thermal Disorder in Molecular Crystals Containing Cofacial Pentacene Units. Chem Mater 2005, 17:5024–5031. 33. Troisi A, Orlandi G. Charge-Transport Regime of Crystalline Organic Semiconductors: Diffusion Limited by Thermal Off-Diagonal Electronic Disorder. Phys Rev Lett 2006, 96:086601. 34. Troisi A. Prediction of the Absolute Charge Mobility of Molecular Semiconductors: the Case of Rubrene. Adv Mater 2007, 19:2000. 35. Stafström S. Electron localization and the transition from adiabatic to nonadiabatic charge transport in organic conductors. Chem Soc Rev 2010, 39:2484–2499. 36. Dirac PAM. The Quantum Theory of the Emission and Absorption of Radiation. Proc R Soc A 1927, 114:243. 37. Wentzel G. Über strahlungslose Quantensprünge. Z Phys 1927, 43:524–530. 38. Fermi E. Nuclear Physics. Chicago: University of Chicago Press; 1950. 39. Born M, Oppenheimer R. Zur Quantentheorie der Molekeln. Ann Phys 1927, 389:457–484. 40. Condon E. A Theory of Intensity Distribution in Band Systems. Phys Rev 1926, 28:1182–1201. doi:10.1103/PhysRev.28.1182. 41. Condon EU. Nuclear Motions Associated with Electron Transitions in Diatomic Molecules. Phys Rev 1928, 32:858–872. doi:10.1103/PhysRev.32.858. 42. Condon EU. The Theory of Complex Spectra. Phys Rev 1930, 36:1121–1133. 43. Franck J, Dymond EG. Elementary processes of photochemical reactions. Trans Faraday Soc 1926, 21:536–542. doi:10.1039/TF9262100536. 44. Stehr V, Engels B, Deibel C, Fink RF. Anisotropy of singlet exciton diffusion in organic semiconductor crystals from ab initio approaches. J Chem Phys 2014a, 140:024503. 45. Stehr V, Fink RF, Deibel C, Engels B. Charge carrier mobilities in organic semiconductor crystals based on the spectral overlap. J Comput Chem. In press. 46. May V, Kühn O. Charge and Energy Transfer Dynamics in Molecular Systems. 3rd ed. Weinheim: Wiley-VCH; 2011. 47. Scholes GD. Long-Range Resonance Energy Transfer in Molecular Systems. Annu Rev Phys Chem 2003, 54:57–87. 48. Wong KF, Bagchi B, Rossky PJ. Distance and Orientation Dependence of Excitation Transfer Rates in Conjugated Systems: Beyond the Förster Theory. J Phys Chem A 2004, 108:5752–5763.


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49. Hennebicq E, Pourtois G, Scholes GD, Herz LM, Russell DM, Silva C, Setayesh S, Grimsdale AC, Müllen K, Brédas J-L, et al. Exciton Migration in Rigid-Rod Conjugated Polymers: An Improved Förster Model. J Am Chem Soc 2005, 127:4744–4762. 50. Malagoli M, Coropceanu V, da Silva Filho DA, Brédas JL. A multimode analysis of the gas-phase photoelectron spectra in oligoacenes. J Chem Phys 2004, 120:7490–7496. 51. Medvedev ES, Osherov VI. Radiationless Transitions in Polyatomic Molecules. Springer Series in Chemical Physics, vol. 57. Berlin/Heidelberg: Springer; 1995. 52. Huang K, Rhys A. Theory of Light Absorption and Non-Radiative Transitions in F-Centres. Proc Roy Soc Lond A 1950, 204:406–423. doi:10.1098/ rspa.1950.0184. 53. Laguerre E. Sur la transformation des fonctions elliptiques. Bull Soc Math 1878, 6:72–78. 54. Bronstein IN, Semendjajew KA, Musiol G, Mühlig H. Taschenbuch der Mathematik. 5th ed. Thun and Frankfurt am Main: Verlag Harri Deutsch; 2000. 55. Spano F, Siddiqui S. Exciton–vibrational coupling in pinwheel aggregates of π-conjugated molecules. Chem Phys Lett 1999, 314:481–487. 56. Duschinsky F. On the interpretation of electronic spectra of polyatomic molecules. I. Concerning the Franck-Condon principle. Acta Phisicochim 1937, 7:551–566. 57. J. Burkhardt. Electron–phonon coupling in large chromophores, Master’s thesis. Lund: School Lund University, 2008. 58. Sánchez-Carrera RS, Coropceanu V, da Silva Filho DA, Friedlein R, Osikowicz W, Murdey R, Suess C, Salaneck WR, Brédas J-L. Vibronic Coupling in the Ground and Excited States of Oligoacene Cations. J Phys Chem B 2006, 110:18904–18911. 59. Marcus RA. On the Theory of Oxidation-Reduction Reactions Involving Electron Transfer. J Chem Phys 1956, 24:966. 60. Marcus RA. Electron transfer reactions in chemistry. Theory and experiment. Rev Mod Phys 1993, 65:599–610. 61. Levich V, Dogonadze R. Theory of non-radiation electron transitions from ion to ion in solutions. Dok Akad Nauk SSSR 1959, 124:123–126. 62. Levich VG. Present state of the theory of oxidationreduction in solution (bulk and electrode reactions). Adv Electrochem Electrochem Eng 1966, 4:249–371. 63. Jortner J. Temperature dependent activation energy for electron transfer between biological molecules. J Chem Phys 1976, 64:4860. 64. Förster T. Zwischenmolekulare Energiewanderung und Fluoreszenz. Ann Phys 1948, 437:55–75.

65. Förster T. 10th Spiers Memorial Lecture. Transfer mechanisms of electronic excitation. Discuss Faraday Soc 1959, 27:7–17. 66. Fink RF, Pfister J, Zhao HM, Engels B. Assessment of quantum chemical methods and basis sets for excitation energy transfer. Chem Phys 2008, 346:275–285. 67. Dexter DL. A Theory of Sensitized Luminescence in Solids. J Chem Phys 1953, 21:836. 68. Nolting W. Grundkurs Theoretische Physik 3: Elektrodynamik. 6th ed. Heidelberg: Springer; 2003. 69. Davydov AS. The Theory of Molecular Excitons. Sov Phys Usp 1964, 7:145. 70. Newton MD. Quantum chemical probes of electrontransfer kinetics: the nature of donor-acceptor interactions. Chem Rev 1991, 91:767–792. 71. Coropceanu V, Cornil J, da Silva Filho DA, Olivier Y, Silbey R, Brédas J-L. Charge Transport in Organic Semiconductors. Chem Rev 2007, 107:926–952. 72. Koopmans T. Über die Zuordnung von Wellenfunktionen und Eigenwerten zu den Einzelnen Elektronen Eines Atoms. Physica 1934, 1:104–113. ∂E = ε in density-functional the∂ni ory. Phys Rev B 1978, 18:7165–7168. doi:10.1103/ PhysRevB.18.7165.

73. Janak JF. Proof that

74. Valeev EF, Coropceanu V, da Silva Filho DA, Salman S, Brédas J-L. Effect of Electronic Polarization on Charge-Transport Parameters in Molecular Organic Semiconductors. J Am Chem Soc 2006, 128:9882–9886. 75. Li H, Brédas J-L, Lennartz C. First-principles theoretical investigation of the electronic couplings in single crystals of phenanthroline-based organic semiconductors. J Chem Phys 2007, 126:164704. 76. Rosso K, Dupuis M. Electron transfer in environmental systems: a frontier for theoretical chemistry. Theor Chem Acc 2006, 116:124–136. 77. Coropceanu V, Malagoli M, da Silva Filho DA, Gruhn NE, Bill TG, Brédas JL. Hole- and ElectronVibrational Couplings in Oligoacene Crystals: Intramolecular Contributions. Phys Rev Lett 2002, 89:275503. 78. Norton JE, Brédas J-L. Polarization Energies in Oligoacene Semiconductor Crystals. J Am Chem Soc 2008, 130:12377–12384. 79. McMahon DP, Troisi A. Evaluation of the External Reorganization Energy of Polyacenes. J Phys Chem Lett 2010, 1:941–946. 80. Ponder JW, Wu C, Ren P, Pande VS, Chodera JD, Schnieders MJ, Haque I, Mobley DL, Lambrecht DS, DiStasio RA, et al. Current Status of the AMOEBA Polarizable Force Field. J Phys Chem B 2010, 114:2549–2564.




81. Ponder JW, Ren P, Pappu RV, Hart RK, Hodsdon ME, Ponder JW, Cistola DP, Kundrot CE, Richards FM. Tinker Software Tools for Molecular Design, Version 6.3, howpublished. St. Louis, MO: Washington University School of Medicine; 2014, http://dasher.wustl.edu/tinker. 82. Stone AJ. Distributed Multipole Analysis: Stability for Large Basis Sets. J Chem Theory Comput 2005, 1:1128–1132. 83. Thole B. Molecular polarizabilities calculated with a modified dipole interaction. Chem Phys 1981, 59:341–350. 84. Honerkamp J. Stochastische dynamische Systeme. Weinheim: VCH-Verlagsgesellschaft; 1990. 85. Gfeller N, Calzaferri G. Energy Migration in DyeLoaded Hexagonal Microporous Crystals. J Phys Chem B 1997, 101:1396–1408. 86. Vega-Durán J, Díaz-Torres L, Barbosa-García O, Meneses-Nava M, Mosiño J. Exact solution to the general non-radiative energy transfer master equations in crystalline materials. J Luminescence 2000, 91:233–241. 87. Yu ZG, Smith DL, Saxena A, Martin RL, Bishop AR. Molecular Geometry Fluctuation Model for the Mobility of Conjugated Polymers. Phys Rev Lett 2000, 84:721–724. 88. Yu ZG, Smith DL, Saxena A, Martin RL, Bishop AR. Molecular geometry fluctuations and field-dependent mobility in conjugated polymers. Phys Rev B 2001, 63:085202. 89. Tutiš E, Batistic I, Berner D. Injection and strong current channeling in organic disordered media. Phys Rev B 2004, 70:161202.

mobilities in organic semiconductor crystals. Phys Rev B 2011, 83:155208. 97. Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes in Fortran 77: The Art of Scientific Computing. 2nd ed. Cambridge: Cambridge University Press; 1992. 98. Einstein A. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann Phys 1905, 322:549–560. 99. von Smoluchowski M. Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen. Ann Phys 1906, 326:756–780. 100. Metropolis N, Ulam S. The Monte Carlo Method. J Am Stat Assoc 1949, 44:335–341. doi:10.1080/ 01621459.1949.10483310. 101. Metropolis N. The beginning of the Monte Carlo method. Los Alamos Science 1987, 15:125–130. 102. Schönherr G, Eiermann R, Bässler H, Silver M. Dispersive exciton transport in a hopping system with gaussian energy distribution. Chem Phys 1980, 52:287–298. 103. Schönherr G, Bässler H, Silver M. Dispersive hopping transport via sites having a Gaussian distribution of energies. Phil Mag B 1981, 44:47–61. 104. Bässler H. Localized states and electronic transport in single component organic solids with diagonal disorder. Phys Stat Sol B 1981, 107:9–54. 105. Richert R, Pautmeier L, Bässler H. Diffusion and drift of charge carriers in a random potential: Deviation from Einstein’s law. Phys Rev Lett 1989, 63:547–550. 106. Pautmeier L, Richert R, Bässler H. Anomalous timeindependent diffusion of charge carriers in a random potential under a bias field. Phil Mag B 1991, 63:587–601.

90. Beljonne D, Hennebicq E, Daniel C, Herz LM, Silva C, Scholes GD, Hoeben FJM, Jonkheijm P, Schenning APHJ, Meskers SCJ, et al. Excitation Migration along Oligophenylenevinylene-Based Chiral Stacks: Delocalization Effects on Transport Dynamics. J Phys Chem B 2005, 109:10594–10604.

107. Abkowitz M, Bässler H, Stolka M. Common features in the transport behaviour of diverse glassy solids: Exploring the role of disorder. Phil Mag B 1991, 63:201.

91. Freire JA, Voss G. Master equation approach to charge injection and transport in organic insulators. J Chem Phys 2005, 122:124705.

108. Borsenberger PM, Pautmeier L, Richert R, Bässler H. Hole transport in 1,1-bis(di-4-tolylaminophenyl) cyclohexane. J Chem Phys 1991, 94:8276–8281.

92. Rappaport N, Preezant Y, Tessler N. Spatially dispersive transport: A mesoscopic phenomenon in disordered organic semiconductors. Phys Rev B 2007, 76:235323.

109. Bässler H. Charge Transport in Disordered Organic Photoconductors. Phys Stat Sol B 1993, 175:15–56.

93. Seneta E. Markov and the Birth of Chain Dependence Theory. Int Stat Rev 1996, 64:255–263. 94. Pauli W. Festschrift zum 60. A. Sommerfelds. Leipzig: Hirzel; 1928.

Geburtstage

95. Nitzan A. Chemical Dynamics in Condensed Phases. Oxford Graduate Texts. Oxford: Oxford University Press; 2006. 96. Stehr V, Pfister J, Fink RF, Engels B, Deibel C. Firstprinciples calculations of anisotropic charge-carrier

110. Baranovskii SD, Faber T, Hensel F, Thomas P. On the Einstein Relation for Hopping Electrons. Phys Stat Sol B 1998, 205:87–90. 111. Bongiovanni G, Botta C, Silvestro GD, Loi M, Mura A, Tubino R. Energy transfer in nanostructured oligothiophene inclusion compounds. Chem Phys Lett 2001, 345:386–394. 112. Houili H, Tutiš E, Batistic I, Zuppiroli L. Investigation of the charge transport through disordered organic molecular heterojunctions. J Appl Phys 2006, 100:033702.


Advanced Review


113. Novikov SV, Malliaras GG. Transversal and longitudinal diffusion in polar disordered organic materials. Phys Stat Sol B 2006, 243:391–394. 114. Poulsen L, Jazdzyk M, Communal J-E, SanchoGarcía JC, Mura A, Bongiovanni G, Beljonne D, Cornil J, Hanack M, Egelhaaf H-J, et al. ThreeDimensional Energy Transport in Highly Luminescent Host-Guest Crystals: A Quantitative Experimental and Theoretical Study. J Am Chem Soc 2007, 129:8585–8593. 115. Curutchet C, Mennucci B, Scholes GD, Beljonne D. Does Förster Theory Predict the Rate of Electronic Energy Transfer for a Model Dyad at Low Temperature? J Phys Chem B 2008, 112:3759–3766.

Conjugated Polymers. J Phys Chem C 2008, 112:11532–11538. 126. Köse ME, Graf P, Kopidakis N, Shaheen SE, Kim K, Rumbles G. Exciton Migration in Conjugated Dendrimers: A Joint Experimental and Theoretical Study. ChemPhysChem 2009, 10:3285–3294. 127. Papadopoulos TA, Muccioli L, Athanasopoulos S, Walker AB, Zannoni C, Beljonne D. Does supramolecular ordering influence exciton transport in conjugated systems? Insight from atomistic simulations. Chem Sci 2011, 2:1025–1032. 128. Zhang X, Li Z, Lu G. First-principles simulations of exciton diffusion in organic semiconductors. Phys Rev B 2011, 84:235208.

116. Martinelli NG, Olivier Y, Athanasopoulos S, Ruiz Delgado M-C, Pigg KR, da Silva Filho DA, SánchezCarrera RS, Venuti E, Della Valle RG, Brédas J-L, et al. Influence of Intermolecular Vibrations on the Electronic Coupling in Organic Semiconductors: The Case of Anthracene and Perfluoropentacene. ChemPhysChem 2009, 10:2265–2273.

129. Bjorgaard JA, Köse ME. Simulations of Exciton Diffusion and Trapping in Semicrystalline Morphologies of Poly(3-hexylthiophene). J Phys Chem C 2014, 118:5756–5761.

117. Nenashev AV, Jansson F, Baranovskii SD, Österbacka R, Dvurechenskii AV, Gebhard F. Effect of electric field on diffusion in disordered materials. II. Two- and three-dimensional hopping transport. Phys Rev B 2010, 81:115204.

131. Grimme S. Improved second-order Møller-Plesset perturbation theory by separate scaling of parallel- and antiparallel-spin pair correlation energies. J Chem Phys 2003, 118:9095.

118. Di Donato E, Fornari RP, Di Motta S, Li Y, Wang Z, Negri F. n-Type Charge Transport and Mobility of Fluorinated Perylene Bisimide Semiconductors. J Phys Chem B 2010, 114:5327–5334. 119. Köse ME, Long H, Kim K, Graf P, Ginley D. Charge Transport Simulations in Conjugated Dendrimers. J Phys Chem A 2010, 114:4388–4393. 120. Di Motta S, Siracusa M, Negri F. Structural and Thermal Effects on the Charge Transport of CoreTwisted Chlorinated Perylene Bisimide Semiconductors. J Phys Chem C 2011, 115:20754–20764. 121. Nan G, Li Z. Crystal structure versus charge transport in organic single crystals of [1]benzothieno[3,2b][1]benzothiophene derivatives from a multiscale theoretical study. J Mater Chem C 2014, 2:1447. doi:10.1039/C3TC31711K. 122. Canola S, Negri F. Anisotropy of the n-type charge transport and thermal effects in crystals of a fluoroalkylated naphthalene diimide: a computational investigation. Phys Chem Chem Phys 2014, 16:21550–21558. 123. Canola S, Negri F. Role of the HOMO-1 Orbital on the p-Type Charge Transport of the Fused-Ring Thienoacene DBTDT. J Phys Chem C 2015, 119:11499–11505.

130. Frenkel D, Smit B. Understanding Molecular Simulation. London: Academic Press; 2002.

132. Hellweg A, Grün SA, Hättig C. Benchmarking the performance of spin-component scaled CC2 in ground and electronically excited states. Phys Chem Chem Phys 2008, 10:4119–4127. 133. Schirmer J. Beyond the random-phase approximation: A new approximation scheme for the polarization propagator. Phys Rev A 1982, 26:2395–2416. 134. Dunning TH. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J Chem Phys 1989, 90:1007. 135. Christiansen O, Koch H, Jørgensen P. The secondorder approximate coupled cluster singles and doubles model CC2. Chem Phys Lett 1995, 243:409–418. 136. Hättig C, Weigend F. CC2 excitation energy calculations on large molecules using the resolution of the identity approximation. J Chem Phys 2000, 113:5154. 137. Hättig C. Geometry optimizations with the coupledcluster model CC2 using the resolution-of-the-identity approximation. J Chem Phys 2003, 118:7751. 138. Köhn A, Hättig C. Analytic gradients for excited states in the coupled-cluster model CC2 employing the resolution-of-the-identity approximation. J Chem Phys 2003, 119:5021.

124. Canola S, Pecoraro C, Negri F. Modeling p-type charge transport in thienoacene analogs of pentacene. Theor Chem Acc 2016, 135:1–10.

139. Stehr V, Fink RF, Engels B, Pflaum J, Deibel C. Singlet Exciton diffusion in organic crystals based on Marcus transfer rates. J Chem Theory Comput 2014, 10:1242–1255.

125. Athanasopoulos S, Hennebicq E, Beljonne D, Walker AB. Trap Limited Exciton Transport in

140. Dirac PAM. Quantum Mechanics of Many-Electron Systems. Proc R Soc Lond A 1929, 123:714–733.




141. Slater JC. A Simplification of the Hartree-Fock Method. Phys Rev 1951, 81:385–390. 142. Vosko SH, Wilk L, Nusair M. Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis. Can J Phys 1980, 58:1200–1211. 143. Becke AD. Density-functional exchange-energy approximation with correct asymptotic behavior. Phys Rev A 1988, 38:3098–3100. 144. Lee C, Yang W, Parr RG. Development of the ColleSalvetti correlation-energy formula into a functional of the electron density. Phys Rev B 1988, 37:785–789. 145. Becke AD. Density-functional thermochemistry. III. The role of exact exchange. J Chem Phys 1993, 98:5648. 146. Ahlrichs R, Bär M, Häser M, Horn H, Kölmel C. Electronic structure calculations on workstation computers: The program system Turbomole. Chem Phys Lett 1989, 162:165–169. 147. TURBOMOLE V6.2 2010, a development of University of Karlsruhe and Forschungszentrum Karlsruhe GmbH, 1989-2007, TURBOMOLE GmbH, since 2007; available from http://www.turbomole.com. 148. Furche F, Ahlrichs R, Hättig C, Klopper W, Sierka M, Weigend F. Turbomole. WIREs Comput Mol Sci 2014, 4:91–100. 149. Heisel F, Miehe JA, Schott M, Sipp B. Measurement of Singlet Exciton Diffusion Coefficient in the C’-Direction in Crystalline Naphthalene. Mol Cryst Liq Cryst 1978, 41:251–255. 150. Powell RC, Soos ZG. Singlet exciton energy transfer in organic solids. J Lumin 1975, 11:1–45. 151. Birks JB. The Fluorescence and Scintillation Decay Times of Crystalline Anthracene. Proc Phys Soc 1962, 79:494. 152. Birks JB. The Influence of Reabsorption and Defects on Anthracene Crystal Fluorescence. Mol Cryst Liq Cryst 1974, 28:117–129. 153. Mulder BJ. Anisotropy of light absorption and exciton diffusion in anthracene crystals determined from the externally sensitized fluorescence. Philips Res Rep 1967, 22:142. 154. Donati D, Williams JO. Exciton Diffusion Lengths for Pure and Doped Anthracene Single Crystals from Microscopic Measurements. Mol Cryst Liq Cryst 1978, 44:23–32. 155. Chai J-D, Head-Gordon M. Long-range corrected hybrid density functionals with damped atom-atom dispersion corrections. Phys Chem Chem Phys 2008, 10:6615–6620. 156. Brückner C, Engels B. A theoretical description of charge reorganization energies in molecular organic p-type semiconductors. J Comput Chem 2016, 37:1335–1344.

157. Karl N, Rohrbacher H, Siebert D. Dielectric tensor and relaxation of photoexcited charge carriers in single crystal anthracene in an alternating field without direct contacts. Phys Stat Sol A 1971, 4:105–109. 158. Munn RW, Nicholson JR, Schowob HP, Williams DF. Dielectric tensor of anthracene as a function of temperature and pressure. J Chem Phys 1973, 58:3828. 159. Speight J, Lange NA. Lange’s handbook of chemistry. 16th ed. New York: McGraw-Hill; 2005. 160. Faltermeier D, Gompf B, Dressel M, Tripathi AK, Pflaum J. Optical properties of pentacene thin films and single crystals. Phys Rev B 2006, 74:125416. 161. Schatschneider B, Liang J-J, Reilly AM, Marom N, Zhang G-X, Tkatchenko A. Electrodynamic response and stability of molecular crystals. Phys Rev B 2013, 87:060104. 162. Xia Y, Kalihari V, Frisbie CD, Oh NK, Rogers JA. Tetracene air-gap single-crystal field-effect transistors. Appl Phys Lett 2007, 90:162106. 163. Lee JY, Roth S, Park YW. Anisotropic field effect mobility in single crystal pentacene. Appl Phys Lett 2006, 88:252106. 164. Martinelli NG, Idé J, Sánchez-Carrera RS, Coropceanu V, Brédas J-L, Ducasse L, Castet F, Cornil J, Beljonne D. Influence of Structural Dynamics on Polarization Energies in Anthracene Single Crystals. J Phys Chem C 2010, 114:20678–20685. 165. Galanin MD, Chizhikova ZA. Duration of the Photo and Radioluminescence of Anthracene and Naphthalene Crystals. Opt Spectrosc 1961, 11:143. 166. Birks JB, King TA, Munro IH. The Photoluminescence Decay of Organic Crystals. Proc Phys Soc 1962, 80:355. 167. Cohen MD, Klein E, Ludmer Z. Micromeasurements of exciton diffusion lengths in single crystals of anthracene. Chem Phys Lett 1976, 37:611–613. 168. Brock CP, Dunitz JD. Temperature dependence of thermal motion in crystalline naphthalene. Acta Cryst B 1982, 38:2218–2228. 169. Brock CP, Dunitz JD. Temperature dependence of thermal motion in crystalline anthracene. Acta Cryst B 1990, 46:795–806. 170. Holmes D, Kumaraswamy S, Matzger AJ, Vollhardt KPC. On the Nature of Nonplanarity in the [N]Phenylenes. Chem Eur J 1999, 5:3399–3412. 171. Mattheus CC, Dros AB, Baas J, Meetsma A, de Boer JL, Palstra TTM. Polymorphism in pentacene. Acta Cryst C 2001, 57:939–941. 172. Watanabe M, Chang YJ, Liu S-W, Chao T-H, Goto K, Islam MM, Yuan C-H, Tao Y-T, Shinmyozu T, Chow TJ. The synthesis, crystal structure and charge-transport properties of hexacene. Nat Chem 2012, 4:574–578.
