Ambipolarity and Dimensionality of Charge Transport in Crystalline

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Ambipolarity and Dimensionality of Charge Transport in Crystalline Group 14 Phthalocyanines: A Computational Study Sai Manoj Gali, Micaela Matta, Benoît H. Lessard, Frédéric Castet, and Luca Muccioli J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b11588 • Publication Date (Web): 17 Jan 2018 Downloaded from http://pubs.acs.org on January 18, 2018

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The Journal of Physical Chemistry

Ambipolarity and Dimensionality of Charge Transport in Crystalline Group 14 Phthalocyanines: A Computational Study Sai Manoj Gali,† Micaela Matta,† Benoît Lessard,‡ Fréderic Castet,∗,† and Luca Muccioli∗,¶ †Institut des Sciences Moléculaires (ISM, UMR CNRS 5255), University of Bordeaux, 351 Cours de la Libération, 33405 Talence, France ‡University of Ottawa, Department of Chemical and Biological Engineering, 161 Louis Pasteur, Ottawa, Ontario K1N 6N5 ¶Department of Industrial Chemistry "Toso Montanari", University of Bologna, Viale Risorgimento 4, 40136 Bologna, Italy E-mail: [email protected]; [email protected]

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Abstract The charge transport properties of ten group 14 phthalocyanine crystals are investigated by means of Kinetic Monte Carlo simulations based on experimental X-Ray structures. All investigated materials are predicted to exhibit ambipolar semiconducting behavior, with hole and electron mobilities lying in the range 0.1-1 cm2 V−1 s−1 , showing their potential for organic electronics devices. The simulations also evidence that the dimensionality of charge transport in these materials can be finely tuned by substituting the phenoxy axial groups with fluorine atoms and by varying their number and positions; a complete substitution gives rise to two-dimensional transport for both electrons and holes. Most remarkably, one of the investigated compounds, incorporating iodine-phenoxy groups as axial substituents, exhibits the largest mobilities both for electrons and holes, owing to large intermolecular couplings and low reorganization energies, and thus emerges as a highly promising one-dimensional semiconductor.

Introduction Metal phthalocyanines (MPc) are conjugated macrocycles with a metal or metalloid atom inclusion. Their ease of synthesis and high chemical stability make them ideal for a variety of low cost applications. 1 In fact, MPcs are among the oldest dyes used commercially as colorants, pigments and even xerographic materials. Silicon phthalocyanines (SiPc) are an emerging class of MPcs with exceptional photoelectronic properties. Recently, SiPcs have found application in organic photovoltaics (OPVs) as both the acceptor or donor molecules in planar heterojunction devices, 2,3 as well as solution processable ternary additives in bulk heterojunction devices. 4–6 SiPcs have also been utilized as hole transport molecules in organic light emitting diodes (OLEDs), 7 and as the emitting molecule in solution processable redNIR emitting devices. 8,9 However, the mobility of charge carriers, which characterizes the efficiency of charge transport through the materials and thus their applicability for organic electronics devices, 2


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is difficult to access experimentally, since measured values can show large variations depending on the experimental technique. 10 In this regard, numerical simulations offer an alternative for establishing relationships linking the chemical nature of the molecules and their supramolecular arrangement to the bulk charge transport properties. It is worth stressing that, although the predominance of a charge-hopping mechanism over a band-like mechanism is well established in disordered organic semiconductors, the degree of localization of the charge carriers over molecular units, 11 as well as the appropriateness of the hopping model in high mobility organic crystals, are still matter of debate. 12,13 In particular, different ways in describing lattice vibrations entering the charge transfer rate expressions, 14 or in incorporating energetic disorder, 15 might result in large variations in the simulated charge carrier mobilities. However, independently on the adopted transport model, simulations showed to provide reliable insights on mobility trends when comparing materials based on similar molecular structures. 15–17 Besides, since predicting crystal structures is still a formidable task, 18 implementing a computational screening procedure based on experimental crystallographic data 19 still appears the only reliable approach at present. F

N

N N

N Y X

F O

N

N Y N

N

F a F

X = Si, Y = F X = Sn, Y = Cl X = Ge, Y = Cl X = Si, Y = a X = Si, Y = b

F

F F F b F

6: X = Si, Y = c 7: X = Si, Y = d 8: X = Si, Y = e 9: X = Si, Y = f 10: X = Si, Y = g

O

F O

d

F

O 1: 2: 3: 4: 5:

F

F

F

O I

F

O F e

f

O g

cF

Figure 1: Chemical structure of the investigated phthalocyanine derivatives.

In this work, we investigate the charge transport properties of ten group 14 phthalocyanine crystals (see Figure 1) by means of Kinetic Monte Carlo (KMC) simulations based on experimental X-Ray structures. Compounds 1-3 incorporate halogen atoms as axial sub3



stituents and differ by the metalloid group 14 core atom, from Sn (1) to Si (2) and Ge (3). All other compounds involve phenoxy axial groups together with a silicon core. In the 4-8 series, the phenoxy groups are substituted with fluorine atoms, by varying their number and position. Compounds 9 and 10 incorporate respectively a methyl group and a iodine atom in meta position of the phenoxy substituents. The electronic parameters governing hole and electron migration at the molecular and supramolecular levels are first calculated for all materials using density functional theory (DFT). Subsequently, KMC simulations in the framework of the Marcus-Levich-Jortner model are carried out to evaluate charge mobilities. The impact of thermal structural fluctuations on the magnitude and dimensionality of the charge transport is then investigated, by introducing fluctuations of the diagonal and off-diagonal charge transport parameters and evaluating their effect on predicted mobilities. In the final process, structure-charge transport relationships are deduced and discussed within this series of materials.

Theoretical methodology Electron and hole mobilities were calculated by means of KMC simulations using X-Ray crystal structures taken from references 2,3,20,21 , whose unit cell parameters are reported in Table 1. The supramolecular arrangements viewed along the (a, b), (a, c) and (b, c) crystallographic planes, together with representative intermolecular distances, are provided in the supplementary material for all investigated systems. KMC simulations were conducted on supercells obtained by replicating the X-ray unit cell 10-15 times along each crystallographic direction, such that the supercells are approximately cubic with all sides larger than 100 Å and contain a total number of molecules (hopping sites) ranging between 2000 and 3500. The simulation conditions are identical to the ones employed in reference 22 , with random initial position of the charge, propagation of the charge for d = 4 µm along the applied electric field, and mobility calculated as averages over 500 propagations.

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The charge transfer rates between adjacent molecules i and j were calculated according to the Marcus-Levich-Jortner (MLJ) rate expression. 23,24 MLJ assumes a hopping mechanism for charge transport and treats high energy intramolecular vibrations quantum mechanically, through a single effective vibrational mode with frequency ωeff and Huang-Rhys (HR) factor Seff . 10 As thoroughly discussed by Engels and coworkers 14 , even if sometimes the simpler Marcus model gives mobility closer to experiments, the MLJ model is physically superior to the Marcus one, since the latter treats all the vibration classically and consequently should be strictly valid only for kB T h ¯ ωeff . In the MLJ formalism, the charge transfer rates are expressed as: ∞ n X Seff Jij2 (∆Gij + λS + n¯ hωeff )2 2π √ exp(−Seff ) × exp − kij = h ¯ 4πλS kB T n=0 n! 4λS kB T

(1)

where T is the temperature (set to 300 K in all simulations), and Jij is the electronic coupling between the hopping sites i and j. λS is the classical contribution to the reorganization energy, associated to the dynamic polarization of the molecules surrounding the charged one. 11,25,26 Unless otherwise specified, λS was set to 0.2 eV both for holes and electrons, neglecting the small electron-hole asymmetry in dynamic polarization energy which can be registered in presence of large molecular multipoles. 27 The choice of using the same λS value for all compounds is however justified owing to the similarity of compounds 1-10 in chemical structure, flexibility, and absence of dipole moment. ∆Gij is the free energy associated to the electron (hole) transfer reaction, resulting from the application of an external ~ and from the difference in electron affinity (ionization potential) between the electric field E molecules involved in the transfer, j and i : ~ · R~ij ∆Gij = j − i + q E

(2)

~ ij the distance vector between the with q = +e (−e) for hole (electron) transport and R hopping sites. Seff is the HR factor associated to a single effective intramolecular vibrational 5



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mode of energy h ¯ ωeff , and describes the strength of the electron-vibration coupling. This term is directly related to the internal reorganization energy which quantifies the changes in geometry of the two molecules involved in the charge transfer process, by λi = Seff h ¯ ωeff . HR parameters for hole and electron transport were computed following reference 28 using molecular geometries and vibrational frequencies calculated at the B3LYP/6-31G(d) level (since the 6-31G(d) is not available for I, Ge and Sn, the double-zeta-split-valence + polarization (DZVP) basis set 29 was used for these atoms). Internal reorganization energies were calculated at the same level of theory using the four point adiabatic potential approach 30–33 (see supporting information for details). Transfer integrals Jij for hole (electron) transport were obtained by employing the projection method, based on a Löwdin’s orthogonalization scheme involving the HOMOs (LUMOs) of monomers. Since all molecules possess degenerate or nearly degenerate LUMO (L) levels, the effective squared transfer integrals for electron 2 2 2 2 + JL+1,L + JL,L+1 + JL+1,L+1 , in line with pretransport were computed as (J e )2 = 21 JL,L vious works. 11,34 Table 1: Crystal parameters and number of formula units (Z) in the unit cell, as well as space group (SG) of the investigated phthalocyanine crystals. M and T stand for monoclinic and triclinic symmetries, respectively. Mol. 1 2 3 4 5 6 7 8 9 10

a (Å) b (Å) c (Å) α (deg.) β (deg.) 8.61 13.34 10.38 90.0 93.5 7.36 8.67 11.04 74.2 80.3 7.36 8.70 10.96 73.8 80.6 8.34 10.31 11.52 72.5 70.1 8.41 10.16 11.51 72.5 69.5 8.44 13.60 15.23 90.0 97.4 10.20 17.69 10.79 90.0 117.6 9.63 19.77 10.04 90.0 118.6 10.25 16.56 11.51 90.0 115.8 12.64 19.58 7.54 90.0 103.2

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γ (deg.) Z SG 90.0 2 M:P21/n 85.5 1 T:P¯1 86.3 1 T:P¯1 83.8 1 T:P¯1 81.3 1 T:P¯1 90.0 2 M:P21/c 90.0 2 M:P21/n 90.0 2 M:P21/c 90.0 2 M:P21 90.0 2 M:P21/c

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Charge transport parameters The energies of the three orbitals considered for charge transport, namely HOMO, LUMO and LUMO+1, as well as ionization potentials (IP) and electron affinities (EA) obtained from differences in the total energies of the charged and neutral molecules in their optimized geometries, are reported in Table 2 for all compounds. Vertical transition energies (∆E), calculated using time-dependent DFT, are also reported. Although not directly comparable, optical band gaps as well as experimental IP and EA values deduced from thin film measurements 2,3 are also provided. The frontier molecular orbitals of compound 1 are illustrated Figure 2, while those of the remaining compounds are given in the supplementary material. In all compounds, both HOMO and LUMOs are spread over the phthalocyanine backbone, with no delocalization on core and axial substituents X and Y. However, non-linear axial substituents determine a reduction of symmetry of the electron density, which is reflected in the removal of the degeneration between LUMO and LUMO+1 for compounds 4-10, albeit the difference in energy is not substantial. The IPs calculated in gas phase lie in the range 6.0-6.4 eV with a weak dependence on the chemical nature of X and Y. As expected, theoretical values are systematically larger than the experimental ones, since these latter were obtained from measurements on thin films, in which local charges are stabilized by the polarizable environment. 27,35 For the same reason, the calculated EAs are underestimated with respect to experiments. 27 As also expected from DFT calculations, 36 the absolute value of the HOMO energy is smaller than the IP calculated from total energy differences, while the negative of the LUMO energy is larger than the calculated EA. Consistently with the localization of the frontier orbitals on the conjugated phthalocyanine core (Figure 2), and with the available experimental results shown in Table 2, the axial substitutions hardly affect IPs, EAs, and optical bandgaps of the studied compounds. As a consequence, the calculations indicate that compounds 1, 2, 9, and 10, for which measurements are not available, should exhibit values very similar to the ones reported for compounds 3-8. 7



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Table 2: Energy of the HOMO, LUMO and LUMO+1, ionization potential (IP), electron affinity (EA), and optical band gap (∆E) calculated with ORCA 37 at the B3LYP/6-31G(d) level (using the DZVP basis set for I, Ge and Sn). Experimental values measured on thin films are also listed. All values are in eV. Molecule IPexp 1 – 2 – 3 5.8a 4 5.7a 5 5.7b 6 5.8b 7 5.9b 8 5.4b 9 – 10 –

IPcalc 6.075 6.357 6.252 6.161 6.067 6.174 6.269 6.120 6.005 6.141

EAexp – – 4.0a 3.8a 3.8b 3.9b 4.0b 3.5b – – a from

EAcalc HOMO 1.829 -5.012 2.216 -5.325 2.075 -5.201 2.111 -5.168 2.013 -5.077 2.085 -5.178 2.201 -5.270 1.912 -5.022 1.908 -4.998 2.091 -5.155 2 b reference ; from

LUMO LUMO+1 -2.857 -2.856 -3.210 -3.209 -3.078 -3.075 -3.072 -3.036 -2.977 -2.945 -3.067 -3.005 -3.158 -3.105 -2.895 -2.871 -2.860 -2.810 -3.029 -2.992 3 reference .

∆Eexp – – 1.56a 1.62a 1.66b 1.65b 1.61b 1.61b – –

∆Ecalc 2.27 2.22 2.22 2.20 2.20 2.22 2.20 2.22 2.02 2.22

Figure 2: Isodensity plots of frontier molecular orbitals of compound 1. From left to right: HOMO, and degenerate LUMO and LUMO+1.

Intramolecular reorganization energies and effective Huang-Rhys parameters for holes and electrons are reported in Table 3. Consistently with record low values for λhi reported previously for zinc phtalocyanines, 38 internal reorganization energies for hole transport assume values in the range 30-70 meV, which are considerably smaller than that reported for other crystalline hole-transporting organic semiconductors such as metal-free phthalocyanines (114 meV 39 ), rubrene (146 meV 40 ) or pentacene (95 meV 14,40,41 ). Not surprisingly, compounds 1-3 bearing a simple halogen substituent in Y position exhibit smaller reorganization energies than derivatives 4-10 wherein Y is a bulkier phenoxy-based substituent, the smallest λhi value being obtained for compound 1 with Y=F. Moreover, compounds 2 and 3 exhibit very 8


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similar λhi values, evidencing the weak impact of the nature of the central metalloid core. Hole reorganization energies of compounds 4-10 also show weak variations, lying between 51 and 64 meV. Compounds 9 and 10, in which the phenoxy group is substituted respectively by a methyl or an iodine, show slightly smaller λhi than those in which the phenoxy h group is fluorinated. Compared to reorganization energies, Huang-Rhys factors Seff are more

sensitive to subtle structural changes and display larger variations, going from 0.137 (1) to 0.644 (4). Hole reorganization energies and Huang-Rhys factors for electrons are consistently larger than the ones calculated for holes, suggesting that group 14 phthalocyanines should favor hole transport over electron transport, as these two parameters must be minimized to achieve the highest transfer rates (see Eq. 1). Given that in typical conditions the n = 0 term in the summation in Eq. 1 is the predominant one, for a given value of λi , the smaller h the Seff the fastest the transfer rate. From this perspective, from table 3 it emerges that

compounds 1-3, 9-10 possess optimal parameters for hole transport, and only 1, 9 and 10 for electrons.

Table 3: Internal reorganization energies (λi , eV), effective Huang-Rhys factors (Seff ), and Effective frequencies (¯hωeff , eV) for holes (h) and electrons (e) transport. Mol. 1 2 3 4 5 6 7 8 9 10

λhi 0.036 0.046 0.040 0.064 0.061 0.058 0.064 0.063 0.054 0.051

h Seff 0.137 0.211 0.174 0.644 0.529 0.268 0.544 0.495 0.197 0.206

h h ¯ ωeff 0.261 0.217 0.229 0.099 0.115 0.215 0.117 0.127 0.272 0.246

λei 0.206 0.216 0.220 0.240 0.239 0.253 0.244 0.236 0.221 0.228

e Seff 1.559 1.917 1.924 2.319 2.265 1.963 1.996 2.121 1.135 1.010

e h ¯ ωeff 0.132 0.112 0.114 0.103 0.105 0.128 0.122 0.111 0.195 0.225

Transfer integral values for holes and electrons were calculated with the B3LYP functional and the 6-31G(d) basis set between one reference molecule and all neighboring ones. It

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should be mentioned that DFT values are functional-dependent and that a scaling factor of up to two can be expected from one functional to the other. Nevertheless, this deviation is systematic and consequently transfer integrals obtained with a given functional for similar molecules can be safely compared. 42 For the set of pththalocyanines investigated here, the largest couplings, which correspond to the crystalline direction along which the π-overlap between molecules is maximal, lie in the range 19-121 meV for holes and 23-86 meV for electrons (Table 4). Compound 10, in which Y = OPhI, displays the maximum couplings for both electrons and holes along the π-stacking direction (parallel to the c crystallographic axis), due to a relative orientation favoring a constructive overlap between the frontier MOs of the interacting molecules. This is illustrated in Figure 3, which compares the supramolecular arrangement along the π-stacking direction in compound 9 (in which Y = OPhMe) and 10. As a consequence of a smaller lateral shift between the interacting molecules, the intermolecular overlap is much larger in compound 10, resulting in high transfer integrals (see Table 4). As we shall investigate in the following, charge transport in the latter system is however expected to be one-dimensional, the coupling terms along the other crystallographic directions being at least one order of magnitude smaller for both electrons and holes. On the contrary, significant secondary coupling terms exist in other derivatives, in particular for compounds 1-5 (e.g. e e in crystal 1 where J(1/2,1/2,1/2) is more than half the main transfer integral J(1,0,0) ), opening

pathways for multidimensional charge transport. Finally, the MLJ charge transfer rates between a reference molecule and its first neighbors, calculated using Eq. 1 for holes and electrons, are collected in Table 4. Although the rate amplitudes roughly follow those of transfer integrals, the reported values also highlight the impact of the intramolecular HR parameters on the efficiency of the hopping events. As a representative example, compounds 2 and 7 exhibit the same transfer integrals for holes along the π-stacking direction (J h = 36 meV), while the CT rate of 2 is significantly larger h due to a smaller value of Seff . Similarly, although in compound 2 J h ≈ J e along the ±(0, 1,

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0) direction , the transfer rate for holes is more than 5 times larger than the one for electrons, e h . is one order of magnitude smaller than Seff since Seff

Table 4: Transfer integrals (absolute value of J ≥ 2, in units of meV) and charge transfer rates (given in units of 1010 s−1 , calculated using Eq. 1 with ∆Gij = 0 and at T=300 K) between a reference molecule and its first neighbors for holes (h) and electrons (e) in the investigated phthalocyanine crystals. The crystallographic directions corresponding to intermolecular vectors joining one central molecule with its neighbours are given in the basis of direct lattice vectors. # 1

direction ±(1, 0, 0) ±( 12 , ± 12 , 12 ) ±(0, 1, 0) 2 ±(0, 1, 0) ±(1, 1, 0) ±(1, 0, -1) ±(0, 0, 1) ±(1, 0, 0) 3 ±(0, 1, 0) ±(1, 1, 0) ±(1, 0, -1) ±(0, 0, 1) ±(1, 0, 0) 4 ±(1, 0, 0) ±(1, 0, -1) ±(0, 1, -1) 5 ±(1, 0, 0) ±(0, 1, -1) ±(1, 0, -1) 6 ±(1, 0, 0) ±( 12 , ± 12 , 12 ) 7 ±(1, 0 0) ±( 12 , ± 12 , 12 ) ±(1, 0, 1) 8 ±(1, 0, 1) ±(0, 0, 1) ±(0, ± 12 , 12 ) 9 ±(1, 0 0) ( 12 , ± 12 , 12 ) (- 12 , ± 12 , - 12 ) ±(1, 0, 1) 10 ±(0, 0, 1) ±(1, 0, 0)

Jh Je kh ke 42 31 825 114 8 18 30 38 3 2 4 0 36 34 563 102 12 11 63 11 6 8 16 6 2 19 2 32 0 4 0 1 33 30 491 78 13 11 76 11 5 6 11 3 3 19 4 31 0 4 0 1 29 43 251 116 11 12 36 9 9 5 24 2 40 42 520 116 11 6 39 2 10 12 33 9 18 23 133 43 7 4 20 1 36 37 415 109 8 10 20 8 2 14 1 16 54 40 971 117 2 17 1 21 1 5 0 2 19 25 159 108 8 11 28 21 7 10 22 17 3 8 4 11 121 86 6389 1446 5 4 11 3

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Figure 3: Supramolecular arrangement (top and side views) along the π-stacking direction in compound 9 (left) and 10 (right), highlighting the higher spatial overlap between the aromatic planes for derivative 10.

Impact of energetic disorder Two sources of energetic disorder were introduced in the KMC simulations. Diagonal disorder, which is associated to variations in the geometrical structure of the molecular sites as well as to the change in the electrostatic landscape experienced by charge carriers during their migration, was incorporated through the variations of the site energies i that contribute to the total free energy of the charge transfer reaction (eq. 2). The amount of diagonal disorder was defined by the standard deviation of a normal Gaussian distribution, in the spirit of the Gaussian disorder model introduced by Bässler. 43,44 Since thermal fluctuations induce variations in the relative positions of the molecules that can strongly affect the electronic

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couplings, 45–47 off-diagonal disorder was also considered by introducing a modulation the transfer integrals with respect to the values reported in table 4. To address the effect of diagonal and off-diagonal disorder on charge transport, KMC simulations were initially performed using different simulation conditions, choosing at this stage compound 1 as representative of the whole family. Diagonal disorder was investigated by considering three limiting cases: 1. No diagonal disorder (denoted 0 ): this case corresponds to a perfectly ordered crystal, devoid of any chemical or physical defects, for which in all cases i =j . 2. Static diagonal disorder (S ): energetic disorder is introduced by initializing the i values at the beginning of every KMC by random sampling from a Gaussian distribution with standard deviation σ . This case corresponds to the situation where the fluctuations in the molecular geometries are slow in comparison to the charge transport, or where site energy differences are static in time, for instance because of physical defects in the crystal or of the presence of impurities acting as weak dopants. The chosen amplitude σ = 2 kB T = 51.4 meV is typical of mild disorder conditions, in line with experimental data: for instance Qiao et al. reported values of about 90 meV for amorphous thin films of compound 2. 48 3. Dynamic diagonal disorder (D ): site energies are re-initialized at each hopping event, using a Gaussian distribution with standard deviation σ = 2 kB T , describing fluctuations occuring on time scales that are smaller than the charge hopping time. Note that this approach is not fully justified since intramolecular and intermolecular vibrational modes are actually already accounted for by the respective reorganization energy terms in MLJ kinetic models. 26 However, it could be seen as an effective way to consider the dynamic effect of the transit of other charges in high charge density conditions. Additionally, to take into account the dynamic fluctuations in the relative positions of electronically coupled hopping sites, off-diagonal energetic disorder was modeled with three 13



different schemes: 1. No off-diagonal disorder (J0 ): transfer integrals values Jij reported in Table 4 are used in equation 1 without any variation. 2. "Thermalized" off-diagonal disorder (JT ): disorder is introduced by replacing Jij valq ues in equation 1 by Jij2 + σJ2 , with σJ = 0.5 Jij , a typical standard deviation for acenes. 46,49 This approach implicitly accounts for a Gaussian distribution of transfer integrals, by replacing the square of the mean transfer integral Jij2 by the mean squared transfer integral hJij2 i, which for a Gaussian distribution of standard deviation σJ corresponds in fact to Jij2 + σJ2 . 22,46,47 This approximation has the neat effect of increasing charge mobility with respect to the J0 case, for any value of σJ . 3. Dynamic off-diagonal disorder (JD ): disorder is introduced by randomly sampling transfer integrals at every KMC step from a Gaussian distribution centered at Jij and with a standard deviation σJ = 0.5 Jij . In this case we expect the hopping time, proportional to 1/hJi2 , to increase with respect to the two previous cases. Indeed, the average hopping time obtained from the probability density function of a normal R∞ exp[(J−Jij )2 /(2σJ2 )] √ dJ, is always higher than 1/Jij2 (J0 case) distribution, 1/hJi2 = −∞ J12 2πσJ and than 1/(Jij2 + σJ2 ) (JT case): mobility is then awaited to decrease accordingly. This situation is equivalent to consider thermal fluctuations of molecular positions and orientations affecting the associated instantaneous Jij values at each hopping step. 50

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Figure 4: Supramolecular arrangement along the π-stacking direction (a axis) in compound 1 (left), and hole (middle) and electron (right) mobility (log scale) along this direction, computed as a function of electric field using different energetic disorder configurations and λS = 0.2 eV.

Figure 4 reports the variation of the hole and electron mobilities along the π-stacking direction (a axis) of compound 1 with respect to the electric field amplitude, and illustrates the impact of including diagonal and off-diagonal energetic disorder into KMC simulations. At low fields, almost irrespectively of the simulation conditions, both hole and electron mobilities display a Poole-Frenkel behavior, in which the logarithm of the mobility is proportional to the square root of the electric field:

µ(E) = µ0 exp(β

p |E|)

(3)

Zero-field mobilities µ0 and Poole-Frenkel factors β, extracted by fitting the linear part of the mobility curves (between 0.1 and 1.5×106 V/cm) using equation 3, show a rather strong dependence on the disorder model (see Table 5). The largest µ0 values are obtained for trapfree structures, i.e. when no diagonal disorder is included (0 conditions). Predictably, 43 including static disorder in the hopping site energies (S conditions) considerably lowers the charge migration efficiency, as a consequence of the introduction of static traps (sites with low i ). Considering the role of positional disorder, KMC simulations conducted using the thermalized JT conditions provide mobility curves with the same shape but translated to

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higher values with respect to those obtained without any disorder (J0 ). On the other hand, introducing positive or negative fluctuations within the transfer integrals through the JD conditions lowers the charge mobilities, in particular at high fields, through an attenuation of the electric field dependence, summarized by the lower values of β in Table 5. To gain further insight on the impact of energetic disorder on mobility values and on the Table 5: Zero field mobility (µ0 , cm2 V−1 s−1 ) and Poole-Frenkel factor (β, 10−3 cm1/2 V−1/2 ) for holes (h) and electrons (e) in compound 1, obtained by fitting simulated mobilities with equation 3 using different simulation conditions. Disorder 0 |J0 0 |JD 0 |JT S |J0 S |JD S |JT D |J0 D |JD D |JT

µh0 2.29 1.39 2.88 0.16 0.13 0.19 1.01 0.73 1.27

βh 0.10 0.10 1.40 0.90 1.44 0.25 0.04 0.25

µe0 0.37 0.32 0.46 0.03 0.03 0.04 0.28 0.23 0.35

βe 0.18 0.04 0.18 1.25 0.98 1.28 0.19 0.11 0.19

Table 6: Maximum hole (h) and electron (e) mobilities (cm2 V−1 s−1 ) along the a,b,c crystallographic directions of compound 1, mobility ratios and dimensionality ∆ of the transport, calculated using different simulation conditions for energetic disorder. disorder

µha

µhb

µhc

0 |J0 0 |JD 0 |JT S |J0 S |JD S |JT

2.56 1.14 3.20 0.68 0.33 0.87

0.14 0.12 0.15 0.05 0.04 0.05

0.07 0.06 0.07 0.01 0.01 0.02

µh a µh b

µh a µh c

∆h

µea

µeb

µec

18.3 9.5 21.3 15.1 8.0 18.1

36.6 19.0 45.7 52.3 30.0 58.0

1D 1D 1D 1D 1D 1D

0.43 0.33 0.54 0.11 0.08 0.14

0.16 0.16 0.20 0.04 0.04 0.05

0.08 0.08 0.11 0.01 0.01 0.02

µea µeb

µea µec

∆e

2.7 2.1 2.7 2.6 2.0 2.8

5.4 4.1 4.9 7.9 7.3 7.0

2D 2D 2D 2D 2D 2D

anisotropy of transport properties, hole and electron mobilities were computed in the three crystallographic planes of compound 1, using an applied electric field of 1 · 106 V/cm, typical of time-of-flights experiments. 48 Mobility values along the three crystal axes are reported in Table 6, together with the dimensionality ∆ of charge transport, arbitrarily defined as follows: i) 1D mobility: µi /µj ≥ 3 and µi /µk ≥ 3, ii) 2D: µi /µj < 3 and µi /µk ≥ 3 and iii) 16


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Figure 5: Polar plots of hole (left, filled circles) and electron (right, empty circles) mobilities (in units of cm2 V−1 s−1 ) within the ab plane of compound 1, computed using 0 |JT (black symbols) and S |JT conditions (blue symbols). The electric field amplitude was set to 1 · 106 V/cm in all simulations. 3D: µi /µj < 3 and µi /µk < 3, where i, j and k correspond to the crystallographic directions with µi > µj > µk . The values in Table 6 evidence that, whatever the simulation conditions, mobility ratios for holes are much larger than for electrons, as a result of a less marked difference in the e e transfer integrals (see Table 4). Hence, hole transport exhibits a more and J(1/2,1/2,1/2) J(1,0,0)

pronounced anisotropy and lower dimensionality than electron transport. It is also important to note that, although mobility ratios can be sensitive to the simulation conditions (especially for holes where µa /µb and µa /µc values show large dispersion), the disorder model has no impact on the overall dimensionality of charge transport. As a representative example, Figure 5 displays the anisotropy of hole and electron mobilities in the ab plane of compound 1 for two different models, 0 |JT and S |JT . The different scales in the respective black and blue plots evidence once again that varying the simulation conditions changes the absolute magnitude of the mobility, although the range of values spanned is not larger than the one typically found in experiments when comparing different techniques for measurements and sample preparation. The polar plots are also useful for identifying the dimensionality of mobility in a given plane: for instance it can be easily noticed how for compound 1 hole mobility (left panels) is one dimensional in the ab plane, with a typical "swimming goggles" shape of the plot hinting to a ratio µa /µb very different from unity. Conversely mobility is quite balanced, and then two-dimensional for electrons 17



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(right panels), giving rise to a "peanut" or "hourglass" shape of the plot. Compound 1

Compound 2

Figure 6: Polar plots of hole (red) and electron (blue) mobilities (units: cm2 V−1 s−1 ) in the crystal of 1 (left) and 2 (right) within the corresponding crystallographic planes (from top to bottom: ab, ac, bc for molecule 1, ba, bc, ac for molecule 2, respectively).

Since the main focus of the following discussion is the comparison between the different

18


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compounds, rather than the prediction of the exact values of their mobility, and considering that the results obtained for compound 1 suggest only a weak impact of the disorder model on the resulting dimensionality of charge transport, from now on we will discuss only mobilities obtained with S |JT conditions, coinciding with a moderate amount of static energetic disorder (σ = 2kB T ), and with transfer integrals slightly increased by thermal fluctuations of molecular positions.

Charge carrier mobilities in phthalocyanine crystals Before comparing the charge mobility in crystal phase between the ten different group 14 phthalocyanines subject of this study (Figure 1), we discuss in detail the mobility anisotropy of compounds 1 and 2, shown in Figure 6, because it is illustrative of the intrinsic difficulties in establishing design rules for maximizing or more in general tuning charge transport. In fact, the chemical structures of these two compounds are so similar, and their crystal packing as well, that one would envisage the two systems to show approximately the same charge mobility and anisotropy. Actually it is well known that the chemical similarity is not a sufficient requisite, since different polymorphs of the same material can exhibhit very different semiconducting properties only arising from the different disposition of molecules in space. 51,52 The case of compounds 1 and 2 can be then also representative of the situation of a compound exhibiting two very similar polymorphs, like e.g. the case of pentacene. 53 Phthalocyanine 1 possesses a monoclinic cell with Z=2 and π-stacking along the b axis, while 2 has a triclinic cell with Z=1 and π-stacking along the a axis (see cell parameters in Table 1). To properly compare the two compounds, the mobility anisotropy in Figure 6 is then plotted in the ab, ac, bc planes for molecule 1 versus ba, bc, ac planes for molecule 2. First of all it can be noticed that the range of explored mobilities is the same for the two compounds, and that electron mobilities for compound 2 are in line with available experimental data (0.1-0.3 cm2 V−1 s−1 for the best performing devices 54 ). This agreement validates

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to some extent our computational procedure. By observing mobilities in the ab planes, it is apparent how hole transport is faster than electron transport (red vs. blue curves), and how mobility is strongly one-dimensional and dominated by the transport along the π-stacking direction, with the exception of electron mobility for compound 1. It can be also noticed that the direction of maximum mobility does not correspond to one of the crystal axes for 2, while it does for 1, probably because of the triclinic shape of the crystal cell of the former compound. Moving to ac vs. bc planes, the picture is similar to the one of the ab plane, with the difference that here higher mobility directions are slightly tilted with respect to the a or b axes (in this case both the β angle for 1 and the α angle for 2 deviate from 90◦ ), and that the electron mobility is this time more isotropic for compound 1. The situation emerging from the bottom panels of Figure 6 is the most striking one: in this case electron and hole mobilities are comparable, small but not negligible, and rather isotropic, but while for compound 1 the plot for holes is almost identical to the one for electrons, for compound 2 the directions of higher mobility are reversed for hole and electrons, revealing not only how subtle changes on the molecular packing can give rise to large and unpredictable anisotropic effects, but also that design rules obtained for hole transport may not apply to electron transport, and vice versa.

Finally, the mobility values along the three crystallographic axes (a, b, c) computed using the S |JT are reported in Table 7 for all compounds, along with the dimensionality of their transport properties. For all systems, the computed mobilities are the largest along the crystalline direction maximizing the π-stacking interactions (and thus the electronic couplings and charge transfer rates), namely the a axis for compound 1, 4-7 and 9, b axis for 2 and 3, and c axis for 8 and 10 (compare bold values in Table 7). Consistently with MLJ rates (Table 4), the marked difference in the internal reorganization energies for a given molecule, evidenced in Table 3, is reflected in maximum hole mobilities being always higher than electron ones.

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Table 7: Mobilities (cm2 V−1 s−1 ) computed for electron (µh ) and holes (µe ) along the a, b, c crystallographic directions in the ten investigated phthalocyanine crystals (with maximum values in bold), and dimensionality ∆ of the charge transport, as obtained with S |JT KMC simulation conditions). Molecule 1 2 3 4 5 6 7 8 9 10

µha µhb µhc 0.87 0.04 0.01 0.06 0.76 0.03 0.06 0.68 0.02 0.22 0.03 0.12 0.43 0.05 0.18 0.20 0.05 0.05 0.71 0.06 0.02 0.11 0.01 0.78 0.28 0.06 0.03 0.07 0.02 1.72

µea µeb µec 0.14 0.05 0.02 0.02 0.13 0.08 0.01 0.10 0.07 0.09 0.03 0.01 0.10 0.01 0.03 0.05 0.01 0.01 0.20 0.02 0.03 0.02 0.03 0.19 0.19 0.04 0.03 0.06 0.01 0.50

∆h 1D 1D 1D 2D 2D 1D 1D 1D 1D 1D

∆e 2D 2D 2D 2D 1D 1D 1D 1D 1D 1D

Compounds 1-3 show very similar trends with rather high hole and electron mobilities, with an interesting two dimensional character in the case of electron transport. Varying the metalloid group 14 core atom from Si (1) to Sn (2) and Ge (3), appears to lead to a small but detectable decrease of mobility values along the π-stacking direction. Larger variations in the charge transport efficiency are observed when considering the series based on fluorophenoxy-substituted phthalocyanines (4-8), with maximal µh and µe values lying respectively in the 0.22-0.78 and 0.05-0.20 cm2 V−1 s−1 ranges. Within this series, maximal hole mobilities evolve in the order 8 ' 7 > 5 > 4 ' 6, and electron mobilities follow a very similar trend but with 4 > 6. This result cannot be completely explained neither just from the evolution of the transfer integrals J h along the π-stacking axis (given in the same order, 54, 36, 40, 29 and 19 meV for holes, and 40, 37, 42, 43, 23 meV for electrons), nor from the evolution of the MLJ rates (in 1010 s−1 : 971, 415, 520, 251 and 133 for holes, and 117, 107, 116, 116, 43 for electrons). This result highlights how surmising trends from transfer integral values, or even from charge transfer rates, could lead to wrong interpretations, and the importance of performing KMC simulations for obtaining reliable design rules. From the latter, relationships linking the molecular structure to the charge transport properties of the material can be established: the incorporation of three fluorine atoms in meta-para 21



or ortho-para positions of the phenoxy groups, like in derivatives 7 and 8, gives rise to large charge carrier mobilities, while substituting only the meta positions has the opposite effect. Moreover, it appears that the dimensionality of charge transport is affected by the number of fluorine atoms in the phenyl ring, with 2D hole transport predicted for compound 5 (four fluorines), and 2D transport both for holes and electrons for 4 (five fluorines). It is not surprising that the largest hole and electron mobilities along the π-stacking crystalline direction are obtained for compound 10, since the efficient charge transport in this material is directly related to the magnitude of the electronic couplings (121 and 86 meV for holes and electrons, respectively), which are more than double of the other systems. The ambipolar charge transport is however highly anisotropic and limited to the c axis, the transfer integrals along the two other crystal axes being very weak (4-5 meV), giving rise to negligible charge transfer rates. It is however striking that substituting methyl groups (in compound 9) by iodine atoms (in compound 10) in meta position of the phenoxy substituents, induces such a large increase of the electronic couplings for both holes and electrons and of the associated mobilities, since these substituents do not differ significantly in term of size and polarizability (see SI). The different packing of compound 10 could be possibly ascribed to intermolecular halogen bondings between iodine atoms, and nitrogens and/or hydrogens of the phtalocyanine core. 55

Conclusions Kinetic Monte Carlo simulations based on experimental X-ray structures have been carried out in order to determine the mobility of charge carriers in ten phthalocyanine crystals. From a methodological point of view, this computational screening evidenced that the only knowledge of prominent descriptors for the mobility of charges, such as transfer integrals and reorganization energies, is not sufficient to fully rationalize the charge transport properties in this series of materials. Since subtle changes in the spatial arrangement of the molecules can

22


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give rise to large and unpredictable effects on the magnitude and dimensionality of mobilities, reliable structure-property relationships have to be based on full-featured charge transport simulations. These latter predicted that all investigated systems should exhibit ambipolar semiconducting behaviour at room temperature, with hole and electron mobilities of the order of 0.1-1 cm2 V−1 s−1 , thus confirming their potential for applications in organic electronics. 2,3 Our mobility calculations also indicate that the good semiconducting properties of group 14 phthalocyanines arise from remarkably small reorganization energies and sizable electronic couplings. The dimensionality of charge transport in these structure-related materials can be finely tuned by chemical substitutions, as illustrated by the changes predicted in the mobility of charge carriers when varying the number and positions of fluorine atoms in compounds 4-8. Among others, compound 10 is especially promising because it possesses both high transfer integrals and low reorganization energies, in particular for holes, even though its application as an electroactive material in crystal phase might be limited by the one dimensional character of the charge transport. These one- or two-dimensional charge transport characteristics in the crystal phase can be either exploited in specific single crystal applications, or more simply overcome by utilizing these materials in amorphous morphologies. 56

Acknowledgement This work has been financed by the French national grant ANR-10-LABX-0042-AMADEus managed by the French National Research Agency under the initiative of excellence (IdEx) Bordeaux program (reference ANR-10-IDEX-0003-02), and by the University of Ottawa for funding through the International Research Acceleration Program (IRAP). Computer time was provided by the Mésocentre de Calcul Intensif Aquitain (MCIA) of the University of Bordeaux, financed by the Conseil Régional d’Aquitaine and the French Ministry of Research and Technology. The authors thank Prof. Timothy Bender (University of Toronto) for

23



providing the crystal structure of all investigated phthalocyanines. L. M. acknowledges Veaceslav Coropceanu (Georgia Institute of Technology, USA) for enlightening discussions about disorder models in charge transport.

Supporting Information Available Supramolecular arrangements and intermolecular distances; calculation of charge transport parameters (internal reorganization energies, Huang-Rhys parameters, transfer integrals) for holes and electrons; polar plots of charge mobility. This material is available free of charge via the Internet at http://pubs.acs.org/.

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Graphical TOC Entry CH3 O

I

Mobility Holes x 6 Electrons x 3

O

32


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Ambipolarity and Dimensionality of Charge Transport in Crystalline

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