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C: Energy Conversion and Storage; Energy and Charge Transport

Charge Mobility Prediction in Organic Semiconductors: Comparison of Second-Order Cumulant Approximation and Transient Localization Theory Alessandro Landi J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b04252 • Publication Date (Web): 11 Jul 2019 Downloaded from pubs.acs.org on July 18, 2019

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The Journal of Physical Chemistry

Charge Mobility Prediction in Organic Semiconductors: Comparison of Second-Order Cumulant Approximation and Transient Localization Theory Alessandro Landi∗ Dipartimento di Chimica e Biologia Adolfo Zambelli, Università di Salerno, Via Giovanni Paolo II, I-84084 Fisciano (SA), Italy E-mail: [email protected]

Abstract In the last years, several theoretical models for the simulation of charge transfer in organic semiconductors have been developed. Two particularly interesting approaches, because of their low computational cost and high accuracy, are the transient localization theory (TLT) and the second order cumulant expansion of the density matrix (SOC). In this work, we apply these models for the evaluation of charge carrier mobility of four prototypical molecules, i.e. tetracene, pentacene, picene and rubrene. Their relative performances in reproducing experimental data are discussed and their predictions are compared with the outcomes of semiclassical Marcus theory. We focus on a simplified framework for SOC model, resorting to constant transfer integrals and time-averaged kinetic constants, showing that, in spite of the rather severe approximations introduced, the results are in good agreement with TLT approach and experimental data. The temperature dependence of the mobility is also analyzed for pentacene and rubrene,

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showing that SOC and TLT models lead to very similar results in good agreement with experimental evidences, while the Marcus approach leads to incorrect trends.

Introduction The application of organic semiconductors in electronic devices, also known as organic electronics, has attracted much attention both from the industry and the research community because of the advantages of these materials. Indeed, they have better mechanical properties than their inorganic counterparts, their electronic and chemical properties can be finely tuned using different functional groups, and, above all, their processability is economically advantageous. In fact, they can be produced with a variety of solution-processing techniques or vacuum deposition methods, showing also good compatibility with large-area flexible substrates like paper or plastic. 1,2 In light of these benefits, crystalline and polimeric organic materials have been applied in organic field-effect transistors (OFET), 3 organic light emitting diodes (OLED), 4 and organic solar cells (OSC). 5 The most important quantity characterizing the performances of organic semiconductors is the charge carrier mobility µ, where values around 102 cm2 V−1 s−1 are required for practical applications. 6 In the last years, significant improvements have been achieved in this respect, leading to materials displaying record mobilities 7–9 of almost 50 cm2 V−1 s−1 . Unfortunately, even if organic synthesis provides the possibility to modulate the properties of these materials, their mobility is quite difficult to predict a priori, so that the discovery of new organic semiconductors with better performances has often relied on a trial and error methodology. 10 However, in light of the high costs of synthesis and subsequent electronic characterization, a theory-guided approach could be highly beneficial to rationalize and speed up the pace of the design of new materials. 10–13 A complete de novo design would require crystal structure prediction, which, in spite of some recent successes, 14 still represents a major challenge. 15–17 This is further complicated

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by polymorphism (i.e. the ability of organic semiconductors to access different crystalline packing states), since even the slightest change in molecular packing can have a huge impact on electronic properties. 18–23 This should be kept in mind in particular when thin-films are used in measurements, because single-crystals and thin film transistors may have different solid-state structures and moreover the substrate where the semiconductor is deposited on usually induces a molecular packing different from bulk, the so-called substrate-induced polymorph. 24–26 At the moment, computational models can only predict the mobility given the crystalline structure of the materials. This information is however of great importance, since electronic characterization is both a money and time-consuming task, while the crystalline structures of the materials are routinely achievable with virtually no effort. Indeed, several crystallographic databases, such as the Cambridge Structural Database (CSD), 27 contain a huge number of materials whose structure is known but whose electronic properties have never been investigated. Unfortunately, progresses in computer-aided material discovery have been slowed down by the fact that currently there is an active debate 28,29 about the most appropriate theoretical approach to model charge transport in organic semiconductors. On the one hand, the maximum mobilities measured in organic semiconductors (of the order of 10 cm2 V−1 s−1 ) suggest a short electronic mean-free path, on the order of the intermolecular distances, which undermines the basic assumptions of band transport theory. Moreover, deviations from the perfect crystalline arrangement, due to thermal motions, indicates that a periodic Blochstate cannot be assumed. This eventually suggests the inadequacy of the band theory for the description of charge transport in organic semiconductors. 29–33 On the other hand, the Marcus formula, an incoherent hopping model, provides results in line with experimental observations 8 and with semiclassical dynamics models, 34 leading however to wrong temperature dependences. 10,30,35 The inclusion of quantum mechanical effects in the hopping rates, e.g. using Fermi Golden Rule (FGR), leads to mobilities decreasing at higher temperatures 16,36 in agreement with experimental observations. However, FGR strongly overestimates the tran-

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sition rates, in particular for high-mobility organic semiconductors (with µ higher than ≈ 1 cm2 V−1 s−1 ), 35,37 which is the most promising class for practical applications in electronic devices. This shortcoming is probably related to the fact that for such ultrafast transitions coherence effects, neglected in the FGR framework, should be considered. A more suitable approach could be the second-order cumulant expansion of the density matrix of the system (SOC), 38–41 since it allows to take into account the change of the reaction rate with time, describing the population P (t) of the state of interest with an equation of the form: dP (t) = kSOC (t)P (t). dt

(1)

The SOC model is a well-known technique 38,42 only recently applied to realistic systems, 43–45 and preliminary results indicate that SOC predictions are in better agreement with experimental data than FGR ones when dealing with high-mobility semiconductors. 35,37 More recently, a different theoretical approach for the description of CT in organic materials has emerged, based on the observation that thermal motions occurring in organic crystals lead to large fluctuations in the transfer integral values, 30,32,46 an effect known as “dynamic disorder”. This effect is the manifestation of the non-local electron-phonon coupling; in other words, it is due to the interactions between molecular phonons and the charge carriers. Several numerical models have been developed in this framework; 46–55 among these, one of the most appealing approaches, because of its low computational cost and excellent agreement with experimental results 10,56 is the transient localization theory (TLT). 30,50,57 In that model, the dynamic disorder causes a transient localization of the charge carrier over a length Lτ on a timescale τ over which the intermolecular oscillations causes the disorder to fluctuate. Being developed relatively recently, both SOC and TLT models have been applied to the description of CT in organic semiconductors only a few times 10,30,35,43,56,57 and a comparison of their relative performances in reproducing experimental data is still missing. Such an analysis could be however of great importance, since, because of their relatively low com4


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putational cost, both models are promising candidates for the analysis of crystallographic databases, such as the Cambridge Structural Database, to identify new semiconductors. For that reason, we here apply these models for the evaluation of charge carrier mobilities in four among the most studied organic semiconductors: tetracene, pentacene, rubrene and picene, chosen because of their relatively high experimental mobilities. 9 The results are compared with experimental data and also with the predictions of Marcus theory, still the most popular choice for the simulation of charge transport in organic materials. Moreover, we also evaluate the mobility at different temperatures for pentacene and rubrene, whose experimental temperature dependence is known. 8,58

Methods The Hamiltonian used for the description of CT in organic semiconductors is the Peierls Hamiltonian, 32,59 consisting of 3 contributions: the electronic part, the phonon part and their interaction: H=

X i

+

X

X Ei |ii i + Vij |ii j + ~ωM M

i,j6=i

XX i

M

1 2 1 ∂2 + QM + − 2 ∂Q2M 2

(2)

XX

gi,M QM |ii i + gij,M QM |ii j . i,j6=i M

Here, |ii is the electronic state with the charge fully localized on the i-th molecular unit. Only one state per site is considered, e.g. the molecular HOMO for hole transport; Ei is the electronic energy of the site, Vij is the transfer integral between nearest neighbour pairs and QM is the dimensionless coordinate of the normal mode with frequency ωM . The total electron-phonon coupling has been splitted in its local (gi,M ) and non-local (gij,M ) contributions; the former describes the modulation of the site energies, while the latter concerns the modulation of the transfer integrals due to the normal modes M .

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The Vij can be evaluated as: 60

Vij = φ0i F φ0j ,

(3)

where φ0i and φ0j are the localized HOMOs of states i and j, respectively, and F is the Fock operator of the system. The suffix 0 indicates that the orbitals are unperturbed. The charge mobility µ can be evaluated resorting to the Einstein-Smoluchowsky equation: 35,61 µ=

eD , 2nkB T

(4)

where e is the elementary charge, kB is the Boltzmann constant, T is the absolute temperature, n is the dimensionality of the system and D is the diffusion coefficient, defined as: ∆X 2 . t→∞ t

D = lim

(5)

Here ∆X 2 is the mean-square displacement evaluated at t → ∞, i.e. for long times, when the diffusion regime is established. 30 The models discussed in this work evaluate this quantity resorting to different procedures, which we will discuss in the following.

Marcus Formula Marcus theory assumes that the vibrations can be treated classically, 62,63 expressing the CT rate in condensed media between two electronic states |ii and |ji as:

kMarcus

|Vij |2 = ~

r

# " 2 π (∆G0 + λ) , exp − λkB T 4λkB T

(6)

where ∆G0 is the free energy change of the CT reaction and λ is the reorganization energy. The first one is zero in our case, since we are considering hopping between two equal molecules in the crystal. The reorganization energy consists of an intramolecular and an intermolecular contribution, due to equilibrium position changes of the system and of the surrounding medium upon hole transfer. However, computations for naphthalene in its molecular crystal 6


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structure resorting to a QM/MM approach with a polarizable force field showed that the intermolecular reorganization energy is around a few meV; 64 further studies showed that acenes tipically have intermolecular reorganization energies lower than 0.2 kcal/mol, 65 much smaller than the intramolecular one. Moreover, the reorganization energy of rubrene is almost independent of the molecular packing structure. 66 For these reasons, throughout this work we will consider only the intramolecular part of the reorganization energy.

Second order cumulant approximation Resorting to the SOC model, it is possible to explicitly include in the calculations all the normal modes of the system, expressing the population of the initial state |ii as: 39,43

Pi (t) = exp [K2 (t)] ,

(7)

where: −2

K2 (t) = −2~ Re

Z

t

dτ1 0

Z

0

τ1

i [VI (τ1 ), [VI (τ2 ), ρ(0)]] |ii dτ2 .

(8)

Here VI is the interaction representation of the coupling potential Vij and ρ(0) describes the initial state of the system, usually assumed to be the equilibrium population of the unperturbed initial state 35,43 (see Supporting Information). A better insight into the electronic properties of the materials can be obtained by writing the differential equation governing the electronic population (obtained by simply differentiating eq. 8 with respect to t): dPi (t) = kSOC (t)Pi (t), dt

(9)

where kSOC (t) is dK2 kSOC (t) ≡ = −2~−2 Re dt

Z

0

t

i [VI (τ1 ), [VI (τ2 ), ρ(0)]] |ii dτ.

(10)

kSOC (t) can be interpreted as a time-dependent rate of the electronic transition 43 (see also 7



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Supporting Information). The time-dependent nature of the SOC transition rate makes this approach particularly useful for ultra-fast transitions, where FGR often fails. Indeed, it has been shown that for slower charge transfer processes, FGR and SOC lead to similar results, also in good agreement with experimental data, thus indicating the adequacy of both models when dealing with materials displaying low to moderate charge mobilities. 35,37 We remark that, even though it is possible to take into account the oscillations of the transfer integral in the SOC framework, 43 we will here neglect this effect, as we are here interested in assessing the performances of this model while keeping as low as possible its computational cost (see Results section). When a rate constant for charge transport can be defined, such as in the Marcus and SOC framework, the diffusion coefficient for charge transport can be computed as: 67–69

D=

N X

d2a ka pa ,

(11)

a=1

where the sum runs over all neighbouring molecules N . Here, da is the hopping distance determined from the CIF files, ka the relative rate constant (given by eq. 6 or 10 depending P on the model used), and pa the hopping probability (pa = ka / N b kb ).

Transient Localization Theory The TLT model is based on the idea that disorder in organic crystals leads to a “transient localization” of the charge carrier over a length L(τ ) within a fluctuation time τ . After this time, the disorder fluctuates because of thermal intermolecular oscillations and charge diffusion can ultimately take place. In this framework, the diffusion coefficient Dx along the x direction is related to the transient localization length L2x (τ ) by: 50 Dx =

L2x (τ ) . 2τ

8


(12)

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As shown in ref. 57, L2x (τ ) is computed as L2x (τ ) =

where Z =

P

n

2 2 1 X −βEn , e n [H, X ] m 1 2 Z n,m ( τ ) + (Em − En )2

(13)

e−En /(KB T ) is the partition function and X is the the position operator.

|mi and |ni are two generic eigenvectors, with corresponding eigenvalues Em and En , of the Hamiltonian in eq. 2. In particular, the disorder in organic crystal due to thermal motion, which causes fluctuations in the transfer integral value, is taken into account as a off-diagonal disorder in the Hamiltonian. This disorder is of the order of the variance of

the transfer integral σ 2 = (V − hV i)2 , which can be evaluated from the electron-phonon

coupling gij,M as: 70

σij2

= (Vij − hVij i)

2

=

X |gij,M |2 M

2

coth

~ωM , 2kB T

(14)

When considering the influence of the displacements QM on the transfer integral Vij , the gij,M appear in the second term of the Taylor series expansion of Vij around the equilibrium position: Vij ({QM }) ∼ = Vij ({QM } = 0) +

X

gij,M QM +

M

X

M,N

∂ 2 Vij QM QN + . . . ∂QM ∂QN

(15)

that is, the gij,M can be evaluated as

gij,M

∂Vij ({QM }) = ∂QM {QM }=0

(16)

In other words, the electron-phonon coupling describes the change in the transfer integral value due to displacements along normal modes, as a result of thermal motions. Unfortunately, evaluation of gij,M is a computationally expensive step, 71 thus severely hindering the application of this model to material discovery. For that reason, in our previous work 72 we

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have developed a methodology that ensures a significant shortening in the computational time without affecting the accuracy of the results. , We represent the normal mode M as a vector of Cartesian displacements QM = xM k

which spans the entire supercell, so gij,M can be written as:

gij,M = ∇Vij · QM .

(17)

Here, ∇Vij is the Cartesian gradient of the transfer integral, that is the derivative of the transfer integral with respect to the Cartesian displacement of an atom k:

∇Vij =

∂Vij ∂xk

,

(18)

where the elements of the vector ∇Vij are zero if displacements xk do not belong to molecule i or j. In other words, instead of evaluating gij,M by displacing the atomic position along normal modes (eq. 16), we first evaluate the change in Vij when the atoms are moved along cartesian displacements (i.e. ∇Vij , eq. 18) and then we project that value over the actual normal modes (eq. 17). Eq. 17 is particularly suited for the development of a fast protocol because the first term of the product (∇Vij ) only includes 6NA differentiations (where NA is the number of atoms in one molecule) and the second term does not depend on the transfer integral, so it needs to be evaluated only once. It is important to notice that, even if we have presented the one-dimensional (1-D) case, generalization to 2-D or 3-D is straightforward, 10 since the total localization length is the sum of the contributions along different directions, say x and y for the 2-D case: L2 (τ ) = L2x (τ ) + L2y (τ ).

10


(19)

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Results Crystal structures for pentacene, tetracene, rubrene and picene, have been derived from the Cambridge Structural Database. 27 Like the majority of molecular semiconductors, these molecules have a crystal structure with a high-mobility plane, whereas the mobility perpendicular to this plane is 1-2 orders of magnitude smaller, 61,73–75 so here we refer to a standard 2D lattice where there can be up to three nearest-neighbour transfer integrals (Fig. 1). Indeed, allowing for some of the transfer integrals to be zero, or for pairs of parameters to be identical, such a lattice can describe almost all high-mobility organic molecular semiconductors. 10

Figure 1: Definition of a standard 2D lattice used to present the parameters needed for the calculation of the mobility. a and b identify any two crystallographic axes and γ is the angle between them. In this respect, in Fig. 2 the crystal structures for the molecules under study are shown and the fastest hole paths 69,76 are labeled according to the scheme in fig. 1. Using this layout, we have evaluated the transfer integral (V ), the norm of the transfer integral gradient (|∇V |), and the fluctuation of the transfer integral (σ) for each pair in the high-mobility plane of the molecules in Fig. 2 as well as their reorganization energy (λ). These quantities are reported in Table 1. In order to work on the same structure used in experimental measurements, we have evaluated the ∇V starting from the molecular arrangements as reported in the CIF files from the CSD database, unlike our previous work, 72 where a geometry optimization step was performed on the experimental structures. Nevertheless, 11



Figure 2: Hole hopping paths in single crystal structure for tetracene (top left), pentacene (top right), picene (bottom left) and rubrene (bottom right).

a comparison between the values reported in this work and data in ref. 72 shows that the results are quite similar, in particular for charge transfer paths displaying high transfer integral values (within 20%). Moreover, the σ reported in Table 1 are very similar to those computed by other groups with different methods, 71,77,78 ensuring the reliability of this procedure. Using data in Table 1, we have applied the Marcus formula, the second order cumulant approach and the transient localization theory to evaluate the charge carrier mobilities along each charge transfer path for the molecules under study, see Table 2, where we have also reported the kinetic constants and the transient localization lengths. Analysis of Table 2 shows that the SOC approximation and TLT model lead to very similar results, with few interesting exceptions. The most striking case is channel A of tetracene, where TLT predicts a mobility two orders of magnitude higher than SOC or Marcus model. That discrepancy is due to the quite high σ/V displayed by this path; indeed, a high degree of disorder is indicative of large amplitude fluctuations of the molecular positions, which could lead to more advantageous molecular arrangements (i.e. with higher transfer integral) than equilibrium geometry for charge transport. An interesting consequence is that, in TLT

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Table 1: λ, V , |∇V | and σ at T = 300 K for the interacting couples of molecules in List of pairs in the high-mobility plane for molecules in Fig. 2

λ (meV) V (meV)

|∇V | σ (meV) (meV/Å)

Molecule

Label

Tetracene

A B C

112

+5.59 +18.9 −86.4

235 456 318

33.5 23.8 21.1

5.99 1.26 0.244

Pentacene

A B C

95.9

+40.2 −77.5 +44.6

273 265 394

19.4 17.9 31.2

0.483 0.231 0.700

Rubrene

A B C

146

+140 −21.3 −21.3

289 124 124

42.3 18.7 18.7

0.302 0.878 0.878

Picene

A B C

181

−115 +89.8 +81.8

217 527 526

27.5 20.0 25.1

0.239 0.223 0.307

σ/V

framework, channel A of tetracene is predicted to have a higher mobility than channel B, in spite of a lower V and a higher σ. This is an intriguing results, since it points out that a high σ can have a beneficial impact on the charge transport properties of a molecule if the transfer integral V has a rather low value. It is worth noting that, despite this difference, the average mobilities obtained with Marcus, TLT and SOC are quite similar to each other, being also in good agreement with reported experimental values 79 (µ ≈ 2.4 cm2 V−1 s−1 ). This is probably due to the fact that all these models lead to similar µ along the high-mobility direction (path C), which obviously has the highest impact on the total electronic properties of the whole crystal. We remark that the selection of experimental mobility values is of critical importance, since different research groups have often obtained different results for the same material. 10,87 As discussed in several papers, 10,35,72 the most reasonable procedure would consist in considering only experimental mobilities showing good reproducibility among different groups, as these are as close to the ideal intrinsic behavior, as can be achieved today. 10,72 We also point 13



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Table 2: Rate constants k (s−1 ×1013 ), square of transient localization length L2 (Å2 ) and mobilities µ

(cm2 V−1 s−1 ) along different hole transfer paths for the molecules under study. Average mobility is calculated taking into account the contributions by all the paths (eq.s 11 and 19). The error in the computed mobilities is of the order of the last digit. In the last column we report experimental values taken from the literature.

Molecule

Label

Tetracene

Pentacene

Rubrene

Picene

a

ref. 79;

b

Marcus k µ

A B C Average

0.053 0.604 12.6

A B C Average

3.46 12.9 4.26

A B C Average

20.9 0.471 0.471

A B C Average

9.02 5.50 4.56

ref. 80–84;

c

ref. 58,85;

SOC

TLT 2

k

µ

L

0.0237 0.307 5.66 2.70

0.011 1.89 15.0

0.005 0.959 6.72 3.03

97 48 405

2.38 5.81 1.91 2.21

4.39 12.3 5.36

3.14 5.87 2.77 2.15

222 358 108

20.8 0.577 0.577 9.97

22.2 0.521 0.521

22.1 0.637 0.637 10.6

1256 61 61

4.97 4.02 2.33 2.03

20.6 12.6 10.4

11.3 9.23 5.31 4.65

907 520 360

d

µ

µexp

1.44 0.715 6.02 2.77

2.4a

3.31 5.33 1.62 2.98

1.0-3.5b

18.7 1.82 1.82 10.8

12.0c

13.5 7.74 5.36 8.68

9.0d

ref. 86

out that we will limit to consider disordered systems; indeed, the discussed models applies if charge carriers are localized over spatially and energetically distributed transport sites. For that reason, the models used cannot reproduce some experimental results such as those in ref.s 8 and 88, where the authors deal with highly purified samples where coherent effects play a role. Also pentacene and rubrene predicted mobilities are very similar to experimental values (µ in the range 1 − 3.5 cm2 V−1 s−1 for pentacene and µ in the range 8.5 − 12 cm2 V−1 s−1 for rubrene have been reported 10,58,80–85 ) regardless of the model used. In other words, these semiconductors represent a particularly favorable case where all the models here investi14


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gated are adequate in reproducing their charge transfer properties, probably because these molecules display intermediate values of V , σ/V and λ. When these conditions do not hold, the results obtained resorting to Marcus, SOC or TLT can significantly diverge, as in the case of picene, whose predicted average as well as 1-D mobilities are quite different resorting to different models. Indeed, Marcus formula leads to significantly lower mobilities than the other methods, being also smaller than experimental data (µ up to 9.0 cm2 V−1 s−1 have been reported 86 ), a drawback probably related to its rather large reorganization energy, the highest among the molecules under study. On the other hand, 1-D SOC and TLT mobilities are significantly larger and in satisfying agreement between each other. Nevertheless, the average TLT mobility is almost twice times higher than SOC ones. This is probably related to the approximated formula used for the evaluation of average mobilities in the framework of Marcus and SOC models (eq. 11). Indeed, it resorts on the combination of kinetic constants evaluated along the single charge transfer path, while in real crystals different charge migration channels are simultaneously available. 16 In the TLT framework, where all the possible charge migration directions are considered at once, computed average mobility is higher, especially when two or more path have similar transfer integral value, the most interesting kind of semiconductors, since it ensures the possibility of circumventing a defect in the material, or of transferring the charge through percolation paths 89 in bulk heterojunction solar cells. It is worth remarking that this whole analysis has been performed in a simplified framework for SOC model, i.e. using constant transfer integrals and time-averaged kinetic constants (see Computational Details), since we are here interested in assessing its performances while keeping as low as possible the computational cost. Further studies taking into account the time dependence of the rates and modifying the methodology to handle the oscillations of the transfer integrals are ongoing in our group and will be the subject of future works. Until now, we have not focused on Marcus formula results since, with the notable exception of picene, they are quite similar to the SOC ones. However, Marcus theory fails

15



below room temperature, leading to a thermally activated behaviour in sharp contrast to experimental data, as shown in Fig. 3, where the predicted hole mobilities of rubrene (top) and pentacene (bottom) are reported along with experimental results.

Figure 3: Predicted hole mobilities as a function of the temperature for: rubrene (top) and pentacene (bottom). Blue dashed line: TLT; green full line: SOC; pink dashed line with dots: Marcus formula; black dots: experimental values from ref.s 58 (rubrene) and 8 (pentacene). As concerns rubrene, the observed mobility in the intrinsic regime varies from ≈ 25 cm2 V−1 s−1 at 175 K up to ≈ 10 cm2 V−1 s−1 at 300 K, in good agreement with both TLT and SOC prediction, while Marcus formula leads to a completely wrong trend. In particular, Marcus formula predicts a thermally activated behaviour because of the exponential dependence from temperature in eq. 6. This failure at low temperatures is somewhat expected, since that model treats molecular modes classically, an approximation valid only at relatively high temperatures. On the other hand, the inclusion of quantum mechanical effects in the

16


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hopping rates, such as in the TLT and SOC approaches, where the vibrations are fully taken into account, allows to recover the experimental trend. Marcus formula fails in reproducing experimental trends also when applied to pentacene crystals, see bottom of Fig. 3. It is worth noting that SOC and TLT, while correctly predicting an increase in the mobility with decreasing temperature, lead to values somewhat different from observed ones. This discrepancy can be due to various factors. First of all, probably polymorphism, i.e. the ability for a given molecule to crystallize in different packing structures, plays a role. Indeed, coexistence of two or more polymorphs in the sample has been reported for pentacene, 26,90,91 as well as interconversion between each other, which is strongly influenced by the temperature and by the type and density of defects and impurities. 90 Moreover, delocalization effects can also play a role, since the difference between experimental and theoretical data increases as the temperature decreases, and delocalization effects are known to be larger at lower temperatures.

Conclusions Hole mobilities along single crystal charge transfer paths for tetracene, pentacene, rubrene, and picene have been computed by first principles, resorting to three different methods: the semiclassical Marcus approach, the second order cumulant expansion of the reduced density matrix (SOC) and the transient localization theory (TLT). Predicted mobilities by SOC approach are in good agreement with experimental data, despite the severe approximations used (constant transfer integral and time-averaged rate constants). TLT model leads to mobilities very similar to SOC ones, thus indicating that, at least for simple molecules displaying intermediate values of V and σ/V , the two approaches are both adequate for simulating charge transfer processes. When dealing with molecules with low σ/V such as picene, TLT leads to results in better agreement with experimental evidences, thus indicating that further improvements

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are needed before SOC model can be applied in material discovery for organic electronics. Marcus semiclassical approach can provide qualitatively correct values of mobilities at room temperature, but it is unable to provide reliable temperature dependencies because it cannot account for tunneling effects which dominate at low temperatures.

Acknowledgement There are no funding to declare. The author wishes to thank Mr. Domenico Landi and Mrs. Incoronata Ferracane for fruitful discussion.

Supporting Information Available Computational details and further information about second order cumulant approach and transient localization theory are provided.

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