Dielectric properties of crystalline organic molecular ...

G. D’Avino, D. Vanzo, Z. G. Soos, The Journal of Chemical Physics, accepted (2016)

Dielectric properties of crystalline organic molecular lms in the limit of zero overlap Gabriele D'Avino∗ Laboratory for Chemistry of Novel Materials, University of Mons, Place du Parc 20, BE-7000 Mons, Belgium; Department of Physics, University of Liège, Allée du 6 Août 17, BE-4000 Liège, Belgium

Davide Vanzo and Zoltán G. Soos† Department of Chemistry, Princeton University, Princeton New Jersey 08544, USA

(Dated: January 5, 2016)

Abstract We present the calculation of the static dielectric susceptibility tensor and dipole eld sums in thin molecular lms in the well-dened limit of zero intermolecular overlap. Microelectrostatic and charge redistribution approaches are applied to study the evolution of dielectric properties from one to a few molecular layers in lms of dierent conjugated molecules with organic electronics applications. Because of the conditional convergence of dipolar interactions, dipole elds depend on the shape of the sample and dierent values are found in the middle layer of a thick lm and in the bulk. The shape dependence is eliminated when depolarization is taken into account, and the dielectric tensor of molecular lms converges to the bulk limit within a few molecular layers. We quantify the magnitude of surface eects and interpret general trends among dierent systems in terms of molecular properties, such as shape, polarizability anisotropy, and supramolecular organization. A connection between atomistic models for molecular dielectrics and simpler theories for polarizable atomic lattices is also provided.

∗ †

Electronic address: Electronic address:

[email protected] [email protected]

1


I. INTRODUCTION The dielectric susceptibility is a key physical property of solids with localized electrons. In molecular crystals with applications in organic electronics, the dielectric constant (or better its anisotropic counterpart, the dielectric tensor) governs fundamental phenomena such as the response to internal and applied elds, the energy levels of charge carriers, and the electron-hole binding energy of excitons. On the other hand, devices based on organic semiconductors feature planar architectures in most cases, with active components consisting of lms whose thickness ranges from macroscopic to one molecular layer. Scaling the size of materials down to the molecular level often results in altered physical properties that can hardly be anticipated with concepts borrowed from the bulk or from continuous systems, although such concepts are typically used to interpret experiment and in theoretical models. In this paper, we discuss the dielectric properties of organic molecular lms with variable thickness down to one monolayer (ML). We seek to clarify the evolution of dielectric properties with thickness in two-dimensional (2D) systems, to quantify surface eects and to disclose specic 2D features. The Clausius-Mossotti equation (CME) [1, 2] relates the relative permittivity κ = ε/ε0 of a macroscopic gas or dilute liquid to the microscopic polarizability α and the number density n = N/V , the inverse of the volume per particle:

αn κ−1 . = κ+2 3ε0

(1)

Here α is the isotropic (scalar) polarizability of atoms or the average (trace) of the molecular polarizability tensor and ε0 is the vacuum permittivity. As shown by Lorentz, the CME holds in cubic crystals of atoms where interactions between induced dipoles at lattice points vanish by symmetry. [3, 4] More generally, a uniform applied electric eld E induces dipoles µ = αF in simple atomic lattices and the total eld F at lattice points is

F = fE =

E . 1 − αnL/ε0

(2)

The isotropic Lorentz factor L = 1/3 in cubic crystals becomes anisotropic in lattices of lower symmetry, as shown by Mueller [5] long ago for simple tetragonal lattices and more recently by Purvis and Taylor [6] for orthorhombic lattices. The factors Lj are the three principal components of the Lorentz tensor L(r) with unit trace evaluated at lattice points. 2


The eld F is strongly enhanced over E when αnLj /ε0 → 1, and large Lj clearly increases

f in Eq. 2. Colpa has carefully related L(r) = D + Λ(r) in uniformly polarized atomic lattices to the depolarization tensor D of a continuous dielectric and the dipole eld Λ(r) of discrete induced dipoles at lattice points. [7] Depolarization is governed by the shape of the sample and the distant

dipoles may be viewed as a continuous dipole density. The dipole eld Λ(r) depends

on the lattice structure and leads to innite 3D sums whose conditional convergence and evaluation have been discussed separately. [6, 810] The dielectric properties of crystalline molecular lms require additional extensions of Eq. 2, starting with thickness down to one molecule. We present in this paper the dielectric properties of molecular lms with organic electronics applications by generalizing Eq. 1 to anisotropic κ tensors in crystalline lms of molecules with polarizability tensor α. Our approach explictly target the electronic response at zero frequency and retains the assumption of zero intermolecular overlap, which is rigorous for a gas or dilute liquid or even a noble-gas crystal. Finite overlap is mandatory for charge transport that is essential for electronic applications, although the limited mobility and hopping transport in organic molecular crystals or lms point to small intermolecular overlap. Zero overlap is a practical and accurate approximation for obtaining the collective response to an applied electric eld using molecular and structural inputs. The large basis sets needed for accurate molecular α are prohibitive for direct calculations of lms. Density functional perturbation theory allows the calculation of the full dielectric susceptibility tensor [11, 12] accounting for intermolecular overlap, but to the best of our knowledge this approach has never been applied to organic crystals or lms. Heitzer

et al.

recently reported the out-

of-plane dielectric response of molecular lms using plane wave density functional theory (DFT) in the presence of an applied electric eld. [13, 14] The important point for the present development is that zero-overlap approaches allow the calculation of microscopic induced dipoles and the polarization P per unit cell for a given molecular α and lattice. Zero overlap is tacitly assumed in all versions of microelectrostatic (ME) theory [1518] in which molecular α tensors are partitioned among atoms in order to compute electric elds in molecular crystals, as well as in intramolecular charge redistribution (CR), which describes classical electrostatic interactions between quantum mechanical molecules. [19, 20] Intramolecular charge ow actually accounts for most of the 3


large polarizability and hyperpolarizabilities of planar conjugated molecules, a feature that is qualitatively captured also by Hückel or Pariser-Parr-Pople theories. CR and ME results for the dielectric tensor and polarization energy of charge carriers in organic molecular crystals have recently been presented [21] and compared with each other and with experiment. In that work, we found that the two methods reliably describe both magnitude and anisotropy of the dielectric tensor κ, though ME systematically overestimates its largest component in highly polarizable molecules. CR has been instead found to give very accurate results for κ, attesting to the soundness of the zero overlap approximation. Zero overlap is also assumed implicitly in the charge response kernel approach to the static permittivity tensors of organic crystals. [22] The kernel indicates how the charge at one atom responds to a potential change at another atom in the same molecule[23] which is precisely the role of the atom-atom polarizability tensor in CR. [19] The kernel is DFT-based with virtual sites for planar molecules, while the CR tensor is based on INDO/S and atomic dipoles for planar molecules. The permittivity tensors of representative crystals computed with the two methods agree within a few percent with CR results or with experiment,[22] as expected for very similar approaches. In this work we rely primarily on CR to discuss the dielectric properties of representative molecular lms. The development is general, however, for CR, ME or other methods that neglect intermolecular overlap. Molecular lms are modeled as free-standing lms whose structure is taken directly from the bulk crystal, thereby neglecting structural changes associated with thin lm formation, the possible presence of defects, the polarizability of the substrate, as well as structural deformations due to the applied eld. The paper is organized as follows. In Section II we present the dielectric properties of molecular lms that, as sketched in Fig.1, consist of a nite number of innite 2D MLs. The polarization per unit cell P is obtained self consistently and contributions to internal electric elds are computed layer by layer. In Section III we consider representative molecular lms such as, among others, C60 , oligoacenes, oligothiophenes and perylenetetracarboxylic dianhydride (PTCDA). The CME holds rigorously for C60 crystals in the atomic limit of point molecules, but molecular size is shown to increase the Lorentz factor to L > 1/3. Since α is isotropic by symmetry, C60 lms illustrate anisotropic κ tensors due entirely to the sample's shape. Acene lms ranging from naphthalene to hexacene consist of standing molecules with increasingly anisotropic α with largest principal component along the long 4


molecular axis. The dielectric properties of lms evolve from ML to multilayers and the middle layer of 5ML lms is already close to the bulk. Interactions between induced dipoles normal to the lms are depolarizing. A similar trend is found in oligothiophene lms. PTCDA is a taken as a representative of lms of lying down molecules and, unlike other systems, presents polarizing interlayer interactions. In Section IV we discuss general trends and conclude.

II. DIELECTRIC PROPERTIES OF MOLECULAR FILMS We present in this Section a general approach to the relative permittivity κ of molecular lms based on the molecular α tensor, a specied structure, and zero intermolecular overlap. The lm's surfaces are dened by Miller indices (hkl) and the thickness is a multiple of the separation d between adjacent planes. We consider free standing lms and neglect structural dierences, in many cases small, between crystals and lms. Films and crystals are therefore built from the same (bulk) unit cell and translation symmetry holds for innite 3D crystals or 2D lms. In multilayer lms, the translationally invariant unit is represented by the composite cell made by crystal unit cells stacked along the plane normal. Figure 1 illustrates anthracene (A3) lms of 1 and 3 MLs with z normal to the surface. Identical inputs, namely the molecular polarizability tensor α and the crystal structure, ensure the full comparability of lm and bulk results. In Section II A we obtain general relations between the permittivity tensor κ of 2D lms, the unit cell polarization P and the depolarization tensor D. The evaluation of P in ML and multilayer lms is discussed in Section II B.

Throughout the paper, we resort to CR and

ME schemes as implemented in the MESCal code presented in Ref. [21], to which we defer the reader for further information.

Molecular α tensors are evaluated at the B3LYP/6-

311++G** level, which provides a good compromise between accuracy and cost. [24] In ME, α is distributed over atoms proportionally to the atomic polarizability and retains the anisotropy of the molecular tensor. CR is governed by an atom-atom polarizability tensor that is evaluated at the INDO/S level and accounts for 80-90% of the molecular polarizability. The remainder is distributed over atoms as in ME.

5


FIG. 1: Sketch of anthracene (A3) thin lms of 1 and 3 ML built from the bulk crystal structure cut along the (001) facet, the ab crystallographic plane featuring herringbone packed molecules. z is the normal to the lm plane, the choice of the origin and in-plane axes are left open. The blue frames show the 2D unit cell, which for a multilayer lm consists of bulk cells stacked along the crystallographic c axis (composite cell). The red dashed frame illustrates the crystal unit cell in the middle layer of the 3 ML lm.

A. Relative permittivity

κ

In a uniformly polarized dielectric, the polarization P per unit cell is related to the applied electric eld E or to the total macroscopic eld F inside the material,

P = ε0 χF = ε0 ζE.

(3)

By denition, the tensor χ = κ − 1 is the electric susceptibility. The tensor ζ that we compute in Section II B is the response to a uniform eld E that is applied normal to the lm and along two orthogonal in-plane directions. Uniform polarization requires a sample whose macroscopic shape is an ellipsoid or a limiting case of a generalized ellipsoid such as a thin lm. [4] In a continuous dielectric, F and E are uniform and related by

F=E−

P·D ε0

(4)

Here P · D is the depolarization eld and D is the depolarization tensor, with principal components (1/3, 1/3, 1/3) for a sphere and (0, 0, 1) for a lm in the xy plane. [3, 4] Equation 4 is strictly valid for continuous dielectrics. At the microscopic level of discrete 6


atoms and molecules, however, the internal elds that contribute to P are periodic functions, the same in all unit cells, but not uniform within a cell. When α is a molecular tensor, induced dipoles µ are vectors that in ME are conventionally located at atoms. The internal eld that generates µ is the applied eld E plus the eld of other polarized molecules, the so-called dipole eld Fdip that requires a self-consistent evaluation in order to account for mutual intermolecular interactions. Since dipole eld sums converge conditionally in 3D, the tensor ζ depends on the sample's shape and is quite dierent at the center of a large spherical crystal and at the middle of a thick crystalline slab. But the bulk κ or χ does not depend on the shape of the sample when depolarization is taken into account. The depolarization eld in a sphere is uniform and

. When ζ

isotropic

is obtained in the limit of an innite sphere, the tensors χ, κ and ζ have the same principal axes but dierent principal components. [19, 20] Depolarization in 2D lms is uniform but . When ζ is obtained in the limit of a thick slab, its principal axes do not coincide

anisotropic

with those of κ or χ. The tensor ζ for a lm is well dened: it depends on the lm's thickness and converges to the thick-slab limit, which is dierent from the bulk value. The tensors κ and χ that include depolarization also depend on the lm thickness and do converge to the bulk value. The induced dipole per unit cell is the polarization P that in ME is computed as

P=

1 X 1 X µm = αm (E + Fdip m ) = ε0 ζE. Vc m Vc m

(5)

The sum is over the M atoms in the unit cell of volume Vc . For the general development, we suppose that Fdip is known at all atoms of the unit cell, determined as shown in Section II B. The eld normal to the lm returns the elements ζxz , ζyz and ζzz . The remaining elements of the real symmetric tensor are found with E in the plane. Moreover, we can quantify surface eects by evaluating P in the dierent layers of a thick lm, or by comparing results for composite 2D cells and the middle layer. We then use Eq. 4 to obtain the relationship between ζ and κ or χ:

χ = κ − 1 = (1 − D · ζ)−1 ζ.

7

(6)


For a thin lm in the xy plane, the components of the symmetric tensor κ are

1 1 − ζzz 2 κzz = 1 + ζxx + ζxz

κzz = κxx

(7)

2 κyy = 1 + ζyy + ζyz κzz

κxz = ζxz κzz κyz = ζyz κzz κxy = ζxy + ζxz ζyz κzz . The eigenvalues and eigenvectors of the 3 × 3 matrix dened by Eq. 7 are the principal components κX , κY , κZ , and principal axes (XY Z ) of the static permittivity tensor. All components of κ contain the factor κzz that increases sharply as ζzz approaches unity. It is instructive to consider the eect of depolarization in the special case of cubic structures whose Cartesian axes are principal axes of both κ and ζ . Figure 2 shows the relationship between ζ and κ (Eqs. 6 and 7) for a spherical crystal and for the in- and out-of-plane components of a thin lm. The κ(ζ) curve of the 3D crystal increases monotonically and diverges at ζ = 3 when the depolarization eld in Eq. 4 counterbalances E exactly. Depolarization in 2D is limited to E normal to the lm and leads to divergent κZ (ζZ ) at ζZ = 1, while the in-plane components κX,Y = 1 + ζX,Y are not depolarized. Since the depolarization eld is related to the sample's shape, the qualitative implications of Fig. 2 are not limited to cubic lattices or indeed to atomic lattices. The principal axes of κ and ζ coincide in tetragonal or orthorhombic atomic lattices but not in systems of lower symmetry. Nevertheless, the divergence of the out-of-plane κ component as ζzz → 1 is a general feature of thin lms. Molecular lms in Section III with tensor α and dierent principal axes for κ and ζ also show strong amplication as ζzz approaches unity. Macroscopically thick slabs, on the other hand, return the same bulk κ as spherical crystals. We comment that depolarization in atomic lattices with isotropic α is usually included at the beginning by using Eq. 4 to eliminate E in Eq. 5. Polarization in atomic lattices with one atom per unit cell is simply P = nαF, where F is the total eld and n is the number density. Depolarization in molecular lattices could also be introduced at the beginning. Since the depolarization eld is uniform in appropriately shaped systems, it does not aect the calculation of dipole eld sums. In molecular lms it is convenient to compute the tensor

ζ in Eq. 3 and then obtain the χ and κ tensors as shown above. 8


FIG. 2: Relationship between the external eld susceptibility ζ and the dielectric tensor κ for uniformly polarized dielectrics in the shape of a 3D sphere and a thin 2D lm, in the special case of ζxy = ζxz = ζyz = 0. The dielectric constant of the crystal or the out-of-plane of the lm is amplied when ζ or ζZ approaches the corresponding asymptotic value (vertical lines).

B. Internal elds We now address the evaluation of internal elds, starting with ME methods where each atom bears a fraction of the total molecular polarizability, αm = gm α, with weights gm that depend on the chosen scheme. As shown in Figure 1, we consider lms in the xy plane dened by Miller indices (hkl) whose thickness depends on the number of MLs. The unit cell contains several molecules with a total of M atoms at rm and dened polarizability tensors αm . We evaluate the internal eld Fm = E + Fdip m at atom m and generalize later to

F(r) with r 6= rm . The applied and dipole elds at atom m of the reference cell are Fm = E + Fdip m = E+

X0

Tms αs Fs .

(8)

s

The sum is over all atoms not in the same molecules as m. The dipole eld tensor is αβ Tms =

α β 2 3rms rms − rms δms 5 rms

(9)

where α, β = x, y, z are Cartesian coordinates and rms = rs − rm is the distance between atoms m and s. The tensor Tms is symmetric and traceless. Sums over atoms are geometrical 9


quantities that are fully specied by the structure and that appear in many contexts for electric or magnetic dipoles. In the case of lms, the dipole eld at atom m always includes an innite 2D sum over the atoms of all other molecules in the same ML. Explicit expressions for 2D sums are given in Ref. [25]. In multilayer lms, Fm also has contributions from induced dipoles that we evaluate separately as a 2D sum for each ML. Intermolecular dipole-dipole interactions can be polarizing or depolarizing, i.e. Fdip m can enhance or decrease the value of the applied eld

E at atom m. When E is applied perpendicular to the lm, interactions among molecules in the same ML are depolarizing as expected for parallel dipoles in the same plane, while the contribution from other layers can be either polarizing or depolarizing. The latter are typically small and decrease rapidly with the separation between MLs. The left-hand equality of Eq. 5 holds also for the CR model, where the total induced dipole of a molecule is

µ=

X

em ρm rm + µ

(10)

m

where ρm is the induced charge at atom m at rm . Since we are considering neutral molecules, the total induced charge is zero and the sum over ρm rm does not depend on the origin. Small

e m are scaled atomic polarizabilities chosen to ensure constant µ = αE. [19] induced dipoles µ The calculation of internal elds is more involved in CR since one has to take into account the contribution from induced charges and dipoles. Unit cells are equivalent by symmetry in 3D crystals or 2D MLs. In either case, Eq. 8 is a system of 3M linear equations for the three components of Fm at M atoms for a given orientation of E. In CR the number of linear equations is 4M due to ρm . In multilayer lms, we have to consider the 2D (or composite) unit cell, i.e. the translationally invariant unit - see bottom panel of Figure 1. Induced dipoles (and charges in CR) in the reference unit cell are evaluated in the self-consistent eld of its periodic replica within a radius R, and the induced polarization is computed by summing up molecular dipoles in the cell. The computational eort goes as M : more than 100 layers are readily accessible in atomic lms with a few atoms per unit cell; a few layers of a molecular lm with hundreds of atoms in the 2D cell is more demanding. Dipole eld sums converge absolutely as R−1 in 2D, allowing us to obtain the ζ tensor for innite lms by extrapolating the results on nite disk-shaped molecular assemblies. The

10


extrapolation procedure is illustrated in Fig. 3 for CR calculations on anthracene (001) lms of 1 and 3 ML. These results are representative for all lms discussed below, for which accurate R−1 trend have been obtained upon increasing the disc's radius. In the case of monoclinic A3 lms, the y axis, taken parallel to the crystal's b axis, is a principal axis of ζ , ensuring ζxy = ζzy = 0. It is worth noting that the polarization converges to dierent values in dierent layers, and that dierent results are obtained when ζ is computed in the middle or outer layers of the 2D cell. Surface eects and convergence of ζ to the thick-slab value are discussed in the coming sections.

FIG. 3: Extrapolation of the ζ tensor for A3 (001) lms of 1 and 3 MLs. The panels show the non-zero components of the tensor obtained with CR calculations. Closed symbols correspond to

ζ components determined for molecular discs of radius R. Lines are linear ts on the data for R > 100 nm. Open symbols mark the extrapolated values in innite discs. For the 3 ML lm, ζ is calculated for the composite unit cell as well as for unit cells in the middle and outer layer (see Figure 1).

The estimated numerical accuracy of ζ elements is better than 10−4 . The estimate is based on (i) the calculated values of o-diagonal elements ζα,β that vanish by symmetry; (ii) 11


the calculated dierence between nite α, β and β, α o-diagonal elements; and (iii) on R−1 extrapolations such as shown in Fig. 3 to innite layers. Corrections to the R−1 behavior increase with the lm thickness, requiring larger R for extrapolated results. Once the elements of ζ are determined for an innite lm of given thickness, the dielectric tensor κ is computed via Eq. 7. We also followed the convergence of κ in the middle layer with increasing thickness to the bulk value in spherical crystals. The agreement between thick-lms and bulk (sphere) κ tensors of molecular lms is within 10−2 or better, quite sucient for our purposes, but distinctly less than agreement within numerical accuracy in atomic lattices. [25] Since dipole elds of adjacent layers decrease rapidly with interlayer separation, a few layers are typically sucient to converge internal elds to the thick-slab value. Internal elds due to induced dipoles are equal by symmetry at equivalent points in unit cells but are far from constant within unit cells. Eqs. 8 and 9 give the internal elds at atomic position rm generated by the induced dipoles of all atoms that are not in the same molecule. The internal eld can be however evaluated at any point r within a molecule (e.g. at the molecular centroid) once we dene the molecular volume, for example using van der Waals surfaces or Wigner-Seitz cells, by writing Eqs. 8 and 9 with r instead of rm . To conclude this Section, we note that the long-range of dipole-dipole interactions complicates the evaluation of dipole eld sums, but is well understood. The problem in crystalline free-standing lms is resolved by 2D translational symmetry with M atoms per (composite) unit cell that makes possible the self-consistent evaluation of Fm as discussed above. A single defect, either a vacancy or an impurity, could be modeled with supercells that are large enough to neglect interactions between defects, although this implies an increase of the computational cost due to the larger M . Multiple defects, dislocations or grain boundaries call for other approaches. Since Fm is the sum of many small contributions, however, rst-order perturbation theory should be viable: the contribution from defects is taken into account without recalculating Fm self-consistently. A low density of point defects, vacancies or interstitials mainly changes the dipole eld in the immediate vicinity of defects from the defect-free value Fm .

12


III. MOLECULAR FILMS A. C60 fullerene C60 is a rare molecule with isotropic α by symmetry. C60 crystals are therefore ideal for illustrating the role of molecular size and, to be specic, the dierence between a polarizable point at the center and 60 atoms with α/60. The crystal is face centered cubic with molecules centered on lattice points. [26] The lattice constant a = 14.052 Å accurately xes the number density n = 4/a3 . The polarizability is α = 81.60 Å3 at the B3LYP/6-311++G** level. The important point here is that α is a molecular quantity rather than its exact magnitude. We start with the dielectric properties of the bulk crystal, take the origin at a lattice point and consider the dipole eld Fdip (r) generated by induced dipoles of molecules whose centers are within a sphere of radius R. We immediately have Fdip (0) = 0 by symmetry in the approximation of one polarizable point per molecule and obtain the textbook result that the Lorentz tensor is isotropic, L = 1/3, in cubic lattices. The CME, Eq. 1, directly relates isotropic κ = 3.914 of the crystal to the molecular α = 81.60 Å3 . The enhancement factor in Eq. 2 is f = 1.914 for the crystal. In an atomistic ME picture each atom bears a fractional polarizability and induced dipoles are oset from the molecular centroid. As far as we know, nite-size eects to the CME have not been recognized previously. Induced dipoles and dipole elds at the atomic sites of the reference molecule, computed for a eld E along z , are shown in the left and right panels of Figure 4, respectively. Dipole elds are substantial, reaching at some atoms 10% of E, but are much smaller on average. Both Fdip (0) and the average hFdip m i over atoms are nite, directed along the applied eld and polarizing. The x and y components of Fdip m are also nite but cancel on summing over all atoms. Dipole elds therefore represent a small but non-negligible correction to E that originates from molecular size. The dielectric constant is isotropic but increases to 4.024 in the ME calculation with induced dipoles at atoms. The CR result for the same α is κ = 3.948, a smaller increase over point dipoles. Although beyond direct verication at present, we note that Antoine et al. [27] pointed out that measured κ in C60 crystals indicate a slightly (∼ 2 Å3 or 2%) smaller α than molecular calculations. Molecular size increases κ by 1% (CR) or 2% (ME) at constant α.

13


Atomic lattices with scalar α illustrate the evolution of dielectric properties from a ML to thick lms. For this purpose we discard the atomistic picture and model C60 lms as atomic lattices with an induced dipole at the center. The Miller indices (001) or (111) yield a square planar or hexagonal ML, respectively. Interactions between induced dipoles normal to the ML are depolarizing. Topping [28] evaluated long ago the dipole-eld sums in the two lattices K001 = −9.033 and K111 = −11.034 in a dierent context. The principal component of the Lorentz tensor normal to the ML is

LZ = 1 −

Khkl √ . 4π 2

(11)

We obtain LZ = 0.4917 for (001) and 0.3791 for (111). Axial symmetry leads to LX =

LY = (1 − LZ )/2 in either case. Strong depolarizing interactions between dipoles normal to the plane are also conrmed by the large dierence between the molecular response

α/(ε0 Vmol ) = 1.478 (Vmol is the molecular volume) and the much smaller ζZ = 0.844 and 0.771 for (001) and (111) MLs, respectively. The relative permittivity κ is also axial and diers from the CME. The shape dependence

FIG. 4: Induced dipoles and dipole elds in C60 crystals computed for an applied electric eld of 0.1 V/Å along z . Left panel: Atomic dipoles of the reference molecule induced by the external eld and the dipole eld of other molecules. Right panel: Dipole eld at atoms of the C60 molecule (z component, dots) plotted as a function of the atomic position. The dashed line shows the dip average value over atoms hFm iz = 0.002 V/Å. The pattern reects the identical Fdip m components

at symmetry-equivalent sites. Fdip m components normal to the applied eld have similar magnitude but average to zero.

14


in systems with isotropic α is [6, 25]

κj − 1 =

αnfj αn = ε0 ε0 − αnLj

(12)

where fj is dened in Eq. 2 and the CME is regained for Lj = L = 1/3. The principal axis

Z of κ and ζ are normal to the lm. Both MLs have LZ > 1/3 leading to κZ = 6.412 and 4.393, signicantly larger than κ = 3.914 in the bulk. Multilayer lms with (001) or (111) facets also have principal components κZ and ζZ , with

Z normal to the lm. The left and right panels of Figure 5 show the thickness dependence of ζZ and κZ , respectively. Decreasing ζZ with thickness is a direct consequence of depolarizing inter-layer dipole elds. Convergence to the crystal value of κ = 3.914 for point dipoles at the molecular centroid is slower for the (001) lm. Depolarizing interactions also occur in other atomic lattices with staggered layers, namely (001) lms of body centered cubic or tetragonal structures. [25] Surface eects in multilayers are quantied by the dierence between ζZ computed for the composite 2D cell and for the middle layer. ζZ values in the middle layer are very close to the thick slab limit already at 3 ML, conrming the rapid decay of dipole elds with inter-layer spacing. Large ML values of κ are clearly related to small

1 − ζZ (see Eq. 6 and Fig. 2). Surface eects limited to one or two atomic layers have also be discussed at the interface of semi-innite crystals and in nanoscopic systems described by simple cubic lattices of polarizable points. [2931] Our results quantify the evolution of surface eects with thickness in C60 , a face centered cubic crystal. More importantly, the method holds for crystals of less than cubic symmetry as we discuss in the following.

15


FIG. 5: Out-of-plane principal components ζZ and κZ of C60 lms from 1 to 12 MLs. Circles and squares refer to (001) and (111) faces, respectively. Open symbols show results for the middle cell (see Figure 1). The dashed line at κ = 3.914 is the isotropic crystal value. Results from ME calculations with C60 molecules modeled as one polarizable point.

16


B. Acenes Acene lms illustrate the general case of anisotropic (tensor) α, several molecules per unit cell, and permittivity tensors κ whose principal components (κX ,κY ,κZ ) and axes (X ,Y ,Z ) vary with thickness. We consider linear oligoacenes (abbreviated as An for n rings) from naphthalene (A2) to hexacene (A6). The optical and solid-state properties of A2 and A3 crystals were extensively studied [15, 32] long before A5 lms became a prototypical organic semiconductor. Acene crystals have herringbone packing with two molecules per unit cell. As sketched in Fig. 1, acene lms typically have ab or (001) facets and standing molecules with approximately equal footprints. The long molecular axis is tilted about θ ∼ 30 degrees from the normal to the ab plane. The tensors α are anisotropic, increasingly so for large n, and their principal directions are xed by symmetry. The largest principal component

αL is along the long molecular axis, the smallest αN is normal to the molecular plane and the medium αM is in the plane, normal to the long axis. The κ tensors of An crystals have recently been obtained using ME and CR methods, and compared with experiment. [21] Dipole-eld sums were evaluated in spherical crystals with increasing radii. Here we report CR calculations on An lms from 1 to 5 MLs with ab faces for crystal structures reported in Refs. [3337] for A2 to A6. The elements of the ζ tensor are computed as illustrated in Fig. 3 for A3 lms and then used to obtain the κ matrix in Eq. 7. The representative Fig. 6 for A3 lms shows the convergence of the principal components of κ to the crystal value with increasing thickness. The estimated accuracy of κ components is ±0.01 for the reasons mentioned in Section II. Table I lists the principal components of κ of A3 and A5 lms from ML to crystals. The 5 ML entries refer to the composite 2D unit cell and the unit cell in the middle layer. Overall, the dielectric tensors of An lms follow the pattern of C60 lms in Fig. 5, suggesting that the main features of dipole elds are only quantitatively aected by the anisotropy of α. More specically, our results show that: (i) strong intra-layer interactions are depolarizing for induced dipoles normal to plane and (ii) polarizing for induced dipoles in the plane; (iii) interlayer interactions between dipoles normal to the lm are weakly depolarizing and responsible for the trends of κ components with the lm thickness. The principal components of κ are essentially the same in the crystal and in the middle layer of 3 ML lms. Convergence to the crystal with increasing thickness is faster than in C60 . 17


FIG. 6: Principal components of the κ tensor of A3 from 1 to 5 ML, for induced polarization computed on the 2D composite cell (full circles) and on the middle cell (open circles). Y is along the crystallographic b axis, Z is tilted by 33 degrees from z , the lm normal (see Table II). The dashed line shows the κ principal components of bulk A3. Results from CR calculations.

A3 κZ

κY

A5 κX

κZ

κY

κX

ML

4.239 2.853 2.235

5.353 3.114 2.384

5 ML

4.101 2.869 2.234

5.203 3.110 2.387

5 ML†

4.072 2.874 2.234

5.180 3.111 2.389

Crystal

4.073 2.874 2.234

5.188 3.113 2.389

TABLE I: Principal components of the dielectric tensor for A3 and A5 lms and bulk. In both cases κ for the middle layer of a 5 ML lm and in the crystal are nearly the same. Results from CR calculations. † Polarization computed for the middle cell of the lm (see Fig. 1.)

18


Figure 7 shows the principal components of κ tensors of An lms with ab faces up to 5 ML. Results for A2, A4 and A6 lms conrm and complete the picture found for A3 and A5, characterized by weak depolarizing interlayer interactions and rapid convergence to the bulk. Systematic variations of π -electronic structure remain a major theme in theoretical and computational chemistry, and acenes are second only to linear polyenes in this respect. The αN component is due to the distortion of atomic orbitals in the eld, while intramolecular ow of π -electrons dominates the larger αL and αM components, the former growing superlinearly with molecular size.

FIG. 7: Principal components of κ for An bulk and lms with ab faces up to 5ML. For lms, full symbols refer to the composite cell unit cell, open symbols to the middle cell, and lines are a guide for eyes. Results from CR calculations.

The orientation of the κ principal axes is approximately thickness independent. The unit cell of A2 and A3 are monoclinic with two molecules that are equivalent by symmetry. For these systems Y is along the crystallographic b axis and Z and X are tilted in the ac plane by the angle φ (see Table II). A4, A5 and A6 are triclinic and the XY Z frame orientation can be given in terms of Euler axes. [38] The X principal axis of κ has the largest projection along the normal to the molecular planes for all An; the small variation of κX − 1 from 1.17 (A2) to 1.38 (A6) in Fig. 7 is mainly due to αN . The κY component is largely dominated by αM , while the Z principal axis is always close to the long molecular axis with large αL , and κZ − 1 increases from 2.4 (A2 ML) to 5.1 (A6 ML). Instead of principal components, the component κzz normal to the lm is probed by experiment or obtained directly in DFT calculations. [13, 31, 39, 40] The tilts θ of the 19


ML

θ

κzz

κZ

5 ML

φ

κzz

κZ

φ

A2

30.7

3.158 3.429 27.8

3.056 3.331 29.7

A3

31.4

3.697 4.239 31.3

3.564 4.103 32.8

A4

22.4†

4.256 4.656 26.3

4.166 4.567 27.2

A5

24.8†

4.748 5.353 29.3

4.619 5.215 30.2

A6

25.6†

5.329 6.096 29.1

5.200 5.950 29.7

A5F 13.9

4.343 4.529 20.5

4.258 4.439 20.9

T4

31.9

4.640 5.497 32.0

4.427 5.278 33.1

T6

33.6

6.012 7.545 33.4

5.635 7.157 34.3

TABLE II: Out-of-plane (κzz ) and largest principal (κZ ) component of the dielectric tensor of 1 and 5 ML lms of standing molecules, obtained with CR calculations. φ is the angle between the plane normal z and the principal direction Z ; θ is the tilt angle between the long molecular axis and z . Angles are given in degrees. † θ averaged over the two symmetry inequivalent molecules.

long axis of An molecules from the ab plane normal are listed in Table II for An crystals, as well as other systems with similar crystal packing, namely peruoropentacene (A5F), quaterthiophene (T4) and sexithiophene (T6). The angle θ is retained by hypothesis in lms derived from the crystal structure. Since the largest κZ component is always parallel to the long molecular axis, a comparable angle φ is obtained between z and Z . Table II lists

φ for 1 and 5 ML lms of An along with the κzz and κZ values. The dierences between φ for 1 and 5 MLs are small. We underscore that κ depends on α and structure, but not on molecular dipole, quadrupole or higher permanent multipoles. Both A5 and A5F have 22 π -conjugated electrons that lead to large and nearly equal α. The quadrupole tensors of A5 and A5F have principal components of similar magnitude but with reversed signs. [41] The ionization potential (IP) and the electron anity (EA) strongly depend on molecular multipoles, primarily the quadrupole moment in planar centrosymmetric molecules. This eect is responsible for an increase of the IP and a decrease of EA with respect to the gas phase in A5, and an opposite trend for A5F. [41] Opposite IP shifts in photoelectron spectra have been highlighted by the Koch group for the two molecules. [4244] The electrostatic energy of the lm depends 20


on both α and multipoles, but κ depends only on dipoles induced by the applied eld. In terms of dielectric properties, the relevant dierence is the 15% increase of the unit cell volume, 685 Å3 for A5 compared to 797 Å3 for A5F. The induced polarization scales with the α/Vc ratio (see Eq. 5), which accounts for the κ dierences in Table II. We further notice that the composite cell of A5F entails weaker intermolecular interaction within each layer, though the leading eect is the change in the polarizability density.

21


C. Thiophenes and PTCDA The notation T4 and T6 refers to linear quaterthiophene and sexithiophene, respectively, while PTCDA abbreviates perylenetetracarboxylic dianhydride. Thin lm transistors based oligothiophenes, polythiophenes, PTCDA and substituted perylenes have been widely studied. In the context of dielectrics, T4 and T6 resemble acenes: they form lms of standing and herringbone-packed molecules with αL along the long molecular axis, which is generally larger than in acenes of similar size. By contrast, lms of PTCDA and substituted perylenes have lying down molecules with large αL and αM in the plane and the small αN normal to the lm. Films of metal-phthalocyanines or metal-porphyrins also have planar molecules at on the surface with the smallest αN normal to the lm. Roughly speaking, acenes and thiophenes are representative of prolate spheroids, while PTCDA illustrates an oblate spheroid; C60 and Alq3 = tris-(8-hydroyquinoline)aluminum are isotropic or slightly anisotropic both in shape and in polarizability. Since T4 and T6 resemble An, we list CR results for 1 and 5 ML lms in Table II. Also in this case we build (001) lms starting from the crystal structure of the so-called high temperature polymorphs with two molecules in the unit cell. [45, 46] The tilt angle between the long molecular axis and the ab plane is again θ ∼ 30◦ , and similarities and dierences with respect to An can be explained in terms of the αL /Vc ratio. The unit cell volume of T4 is 731 Å3 , almost 7% larger than in A5, while αL = 106 Å3 is 10% larger than in A5. These opposite variations roughly compensate each other and account for the comparable

κzz and κZ of T4 and A5 lms without taking into account the slightly dierent molecular tilt or other structural details. On the other hand, the larger T6 has Vc = 1064 Å3 , some 36% larger than for A6, and αL = 222 Å3 compared to 127 Å3 for A6. The greater increase of αL compared to Vc is reected in larger κzz and κZ for T6 lms than for A6 lms. We model PTCDA (100) lms taken from the crystal structure of the so-called α polymorph. [47] PTCDA lms present at lying molecules in the bc plane and eclipsed molecular stacks along the plane normal in multilayer lms. The lm normal is the Z principal axis of both the κ and ζ tensors. The evolution with the lm thickness of κZ and ζZ up to 7 ML is shown in Figure 8. As in the other cases, we obtain convergence of ζ and κ to the thick-slab and bulk limits, respectively. The convergence of κZ computed for the composite cell is considerably slower than in An and Tn and comparable to that of C60 (111) lms. 22


Nevertheless, the middle layer of a 3 ML lm is again already close to the crystal value.

FIG. 8: Thickness dependence of the out-of-plane principal components of ζ and κ in PTCDA (100) lms calculated for the 2D composite cell (lled circles) and for the middle layer (open circles). The dashed line shows the κZ bulk value. Unlike in the other lms considered in this work, ζZ and κZ increase with the lm thickness due to polarizing interlayer interactions. Results from CR calculations.

Apart from these similarities, PTCDA (100) lms present the following two important dierences with respect to the other systems considered in this work: (i) κZ increases with thickness in PTCDA lms and converges to the bulk limit from below. This is an indication of polarizing interactions between eclipsed and closely stacked molecular layers. (ii) κZ is the smallest component of the dielectric tensor and results from the small out-of-plane polarizability αN that in CR calculations is entirely described by induced dipoles. The value of the other two components, due to charge ow in the π -conjugated system, are κX = 3.57 and κY = 4.31 in the crystal.

23


IV. DISCUSSION The Clausius-Mossotti equation is a cornerstone of our microscopic understanding of dielectric properties of materials. The polarizability per unit volume αn is the only parameter entering this famous equation which, derived for gases and liquids, is also valid in cubic crystals. Organic crystals or lms are more complex even in the limit of zero overlap, due to molecular tensors α whose principal axes dier in general from the crystal or lm axes, and because there is usually more than one molecule per unit cell. The dielectric susceptibility of lms, either atomic or molecular, depend on the crystal facet and converge to the bulk value with increasing thickness, with surface eects mostly limited to the rst monolayer. Thin lms of organic conjugated molecules are very diverse. Nevertheless some general, yet qualitative, statements about their dielectric properties can be made. Classical continuous dielectrics support uniform polarization in samples whose shape is a general ellipsoid or one of its limiting cases, such as thin lms. The actual thickness is not relevant for an innite continuous lm featuring a uniform areal density of induced dipoles in an applied eld normal to its plane. This induced dipoles density is well dened and for zero thickness generates a step function in the potential. [3, 4] The dipole eld Fdip is rigorously zero above and below the lm. Discrete induced dipoles induced by a eld normal to the plane lead to dipole-dipole interactions within a ML and to a nite eld Fdip (r) above and below the lm, which in turn governs the thickness dependence of the dielectric properties. The decay of dipole elds with z is exponential and depends on the periodicity of the lattice of polarizable points, as was recognized all along in a simple cubic lattice. [48] Dipole elds at lattice sites are zero by symmetry in cubic systems, but are polarizing in simple tetragonal lattices featuring eclipsed layers. In the case of adjacent layers that are oset, or staggered in the xy plane, interlayer interaction can be either polarizing or depolarizing, and dipole elds vanish for a specied oset. [25] Molecular shape and orientation must also be taken into account for interactions between induced dipoles in dierent layers. The simplest approximation is to take the molecular volume Vmol as a fraction of the unit cell volume and the molecular footprint A = a2 for the in-plane lattice constant a (for simplicity we assume a square lattice with one molecule per cell). The ML thickness is then d = Vmol /a2 . Interlayer dipole-dipole interactions are 24


enhanced in (oblate) systems with a > d and reduced in (prolate) systems with a < d. The anisotropy of α should be taken into account for quantitative results. Interlayer interactions between induced dipoles along z can be either polarizing or depolarizing in molecular lms. Molecular lms of standing molecules have a < d and reduced interactions between layers even before accounting for molecular tilt. This explains why the dielectric properties of oligoacenes and oligothiophenes lms are rather insensitive to thickness and close to the bulk. Decreasing κZ from ML to bulk indicates depolarizing interactions between layers. C60 lms are face-centered cubic lattices with staggered layers: every second (001) layer or third (111) layer is eclipsed. Staggered adjacent layers account for decreasing κZ from ML to bulk in either case. The larger footprint of C60 accounts for a pronounced thickness dependence of dielectric properties in Figure 5, especially for the more open structure of (001) lms. It is worth recalling that while the susceptibility ζ to the applied eld directly probes dipole elds, the dielectric tensor κ includes also the eect of depolarization, which is particularly eective in thin lms. In fact, because of depolarization κzz diverges when ζzz approaches unity, amplifying the dielectric response of molecules with large polarizability (per unit volume) normal to the lm. This accounts for the large κZ in C60 (001) ML, 50% higher than in the bulk, while a smaller increase is obtained for (111) ML. Similarly, among standing molecules the largest variation of κZ between ML and bulk is obtained for T6, i.e. the molecules with largest αL . Divergent behaviors should in principle be possible in thin lms of molecules with higher polarizability/volume ratio, but the actual structure of the lm may be dierent. PTCDA (100) lms are representative of at lying eclipsed molecules with a > d. In this case inter-layer interactions are found to be polarizing as in eclipsed atomic lattices and rather strong due to the large molecular footprint compared to the inter-layer spacing. Accordingly, PTCDA is the only system among those considered in this work whose out-ofplane κ component converges to the bulk limit from below upon increasing the lm thickness. The convergence to the bulk κ is much slower than in standing molecules and comparable to that of C60 (001) lms. However, PTCDA exposes its small αN component normal to the lm, leading to a modest variation of κZ from ML to bulk. We remark that that we cannot conclude that polarizing interactions are a general feature of lms of at lying molecules. Dipole eld sums should be instead evaluated case by case because, due to the size and shape of the molecules, the polarizing or depolarizing nature of interlayer interactions is hardly 25


predictable on simple grounds. In summary, we have presented a general approach to the dielectric properties of crystalline molecular lms in the well-dened limit of no intermolecular overlap, and applied it to a set of pi-conjugated molecules of interest for organic electronics applications. The method is applicable to innite 2D lms with a specied number of MLs and can be combined with any scheme (e.g. microelectostatics or charge redistribution) that relates the best theoretical or experimental molecular polarizability α of molecules to the unit cell polarization P of the lm.

26


Acknowledgments GD gratefully thanks Luca Muccioli for stimulating discussions and support in the early stage of this research, and acknowledges support from EU and the University of Liege trough the FP7-PEOPLE-COFUND-BEIPD and FP7-PEOPLE-2013-IEF programs. ZGS and DV thank the National Science Foundation for support through the Princeton MRSEC (DMR0819860).

[1] R. Clausius,

Die Mechanische Warmetheorie II

(Vieweg Braunschweig, 1879).

[2] O. F. Mossotti, Rendiconti dell'Accademia Nationale della Scienze detta dei XL.

XXIV,

49

(1850). [3] J. Jackson, [4] C. Kittel,

Classical Electrodynamics

(Wiley, New York, 1962).

Introduction to Solid State Physics

(John Wiley & Sons, Inc., New York, 1986), 6th

ed. [5] H. Mueller, Phys. Rev.

47, 947

(1935), URL http://link.aps.org/doi/10.1103/PhysRev.

47.947. [6] C. K. Purvis and P. L. Taylor, Phys. Rev. B

26, 4547

(1982), URL http://link.aps.org/

doi/10.1103/PhysRevB.26.4547. [7] J. Colpa, Physica

56,

185 (1971), ISSN 0031-8914, URL http://www.sciencedirect.com/

science/article/pii/0031891471900784. [8] B. Nijboer and F. D. Wette, Physica

24,

422 (1958), ISSN 0031-8914, URL http://www.

sciencedirect.com/science/article/pii/S0031891458958038. [9] S. W. de Leeuw, J. W. Perram, and E. R. Smith, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences

373, 27 (1980), ISSN 0080-4630.

[10] H. León and E. Estevez-Rams, Journal of Physics D: Applied Physics 42, 185012 (2009), URL

http://stacks.iop.org/0022-3727/42/i=18/a=185012. [11] X. Gonze and C. Lee, Phys. Rev. B

55, 10355 (1997), URL http://link.aps.org/doi/10.

1103/PhysRevB.55.10355. [12] M. Ferrero, M. Rérat, R. Orlando, R. Dovesi, and I. J. Bush, Journal of Physics: Conference Series

117, 012016 (2008), URL http://stacks.iop.org/1742-6596/117/i=1/a=012016. 27


[13] H. M. Heitzer, T. J. Marks, and M. A. Ratner, Journal of the American Chemical Society

135,

9753 (2013), pMID: 23734640, http://dx.doi.org/10.1021/ja401904d, URL http://dx.

doi.org/10.1021/ja401904d. [14] H. M. Heitzer, T. J. Marks, and M. A. Ratner, Journal of the American Chemical Society 137, 7189 (2015), pMID: 25978594, http://dx.doi.org/10.1021/jacs.5b03301, URL http://dx.doi.

org/10.1021/jacs.5b03301. [15] M. Pope and C. Swenberg,

Electronic processes in organic solids

(Clarendon, Oxford, 1982),

and references therein. [16] J. Rohleder and R. Munn,

Magnetism and Optics of Molecular Crystals

(Wiley, New York,

1992), and references therein. [17] J. Applequist, J. R. Carl, and K.-K. Fung, Journal of the American Chemical Society 94, 2952 (1972), http://pubs.acs.org/doi/pdf/10.1021/ja00764a010, URL http://pubs.acs.org/doi/

abs/10.1021/ja00764a010. [18] B. Thole, Chemical Physics

59,

(1981), ISSN 0301-0104, URL http://www.

341

sciencedirect.com/science/article/pii/0301010481851762. [19] E. V. Tsiper and Z. G. Soos, Phys. Rev. B

64,

195124 (2001), URL http://link.aps.org/

doi/10.1103/PhysRevB.64.195124. [20] Z. Soos, E. V. Tsiper, and R. Pascal, Chemical Physics Letters

342, 652

(2001), ISSN 0009-

2614, URL http://www.sciencedirect.com/science/article/pii/S0009261401006613. [21] G. D'Avino, L. Muccioli, C. Zannoni, D. Beljonne, and Z. G. Soos, Journal of Chemical Theory and Computation

10, 4959 (2014).

[22] J. Tsutsumi, H. Yoshida, R. Murdey, S. Kato, and N. Sato, The Journal of Physical Chemistry A

113,

9207 (2009), pMID: 19618912, http://dx.doi.org/10.1021/jp903420w, URL http://

dx.doi.org/10.1021/jp903420w. [23] A. Morita and S. Kato, Journal of the American Chemical Society

119,

4021 (1997),

http://dx.doi.org/10.1021/ja9635342, URL http://dx.doi.org/10.1021/ja9635342. [24] A. L. Hickey and C. N. Rowley, The Journal of Physical Chemistry A

118,

3678 (2014),

pMID: 24796376, http://dx.doi.org/10.1021/jp502475e, URL http://dx.doi.org/10.1021/

jp502475e. [25] D. Vanzo, B. J. Topham, and Z. G. Soos, Advanced Functional Materials ISSN 1616-3028, URL http://dx.doi.org/10.1002/adfm.201402405.

28

25,

2004 (2015),


[26] D. Andrè, A. Dworkin, H. Szwarc, R. Cèolin, V. Agafonov, C. Fabre, A. Rassat, L. Straver, P. Bernier, and A. Zahab, Molecular Physics http://www.tandfonline.com/doi/pdf/10.1080/00268979200102101,

76,

URL

1311 (1992),

http://www.

tandfonline.com/doi/abs/10.1080/00268979200102101. [27] R. Antoine, P. Dugourd, D. Rayane, E. Benichou, M. Broyer, F. Chandezon, and C. Guet, The Journal of Chemical Physics

110,

9771 (1999), URL http://scitation.aip.org/content/

aip/journal/jcp/110/19/10.1063/1.478944. [28] J. Topping and S. Chapman, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences

113, 658 (1927), ISSN 0950-1207.

[29] G. D. Mahan and G. Obermair, Phys. Rev.

183,

834 (1969), URL http://link.aps.org/

doi/10.1103/PhysRev.183.834. [30] G. D. Mahan, Phys. Rev. B

74,

033407 (2006), URL http://link.aps.org/doi/10.1103/

PhysRevB.74.033407. [31] A. Natan, N. Kuritz, and L. Kronik, Advanced Functional Materials

20,

2077 (2010), ISSN

1616-3028, URL http://dx.doi.org/10.1002/adfm.200902162. [32] E. A. Silinsh and V. ápek, port Phenomena

Organic Molecular Crystals: Interaction, Localization, and Trans-

(American Institute of Physics, 1994).

[33] C. P. Brock and J. D. Dunitz, Acta Crystallographica Section B

38, 2218 (1982), URL http:

//dx.doi.org/10.1107/S0567740882008358. [34] C. P. Brock and J. D. Dunitz, Acta Crystallographica Section B

46, 795 (1990), URL http:

//dx.doi.org/10.1107/S0108768190008382. [35] D. Holmes, S. Kumaraswamy, A. J. Matzger, and K. P. C. Vollhardt, Chemistry A European Journal 5, 3399 (1999), ISSN 1521-3765, URL http://dx.doi.org/10.1002/(SICI)

1521-3765(19991105)5:113.0.CO;2-V. [36] T. Siegrist, C. Kloc, J. H. Schön, B. Batlogg, R. C. Haddon, S. Berg, and G. A. Thomas, Angewandte Chemie International Edition

40,

1732 (2001), ISSN 1521-3773, URL http://

dx.doi.org/10.1002/1521-3773(20010504)40:93.0.CO;2-7. [37] M. Watanabe, Y. J. Chang, S.-W. Liu, T.-H. Chao, K. Goto, M. M. Islam, C.-H. Yuan, Y.-T. Tao, T. Shinmyozu, and T. J. Chow, Nature Chemistry

4,

574 (2012), URL http:

//dx.doi.org/10.1038/nchem.1381. [38] The principal axes of κ of triclinic systems is dened by the following Euler angles (zyz

29


convention): α = −139.1, β = 27.0 γ = −134.5 for A4; α = −144.5, β = 30.0 γ = −128.4 for A5; α = 144.4, β = 29.5 γ = 129.8 for A6. [39] N. Shi and R. Ramprasad, Phys. Rev. B 74, 045318 (2006), URL http://link.aps.org/doi/

10.1103/PhysRevB.74.045318. [40] L. Romaner, G. Heimel, C. Ambrosch-Draxl, and E. Zojer, Advanced Functional Materials 18, 3999 (2008), ISSN 1616-3028, URL http://dx.doi.org/10.1002/adfm.200800876. [41] B. J. Topham and Z. G. Soos, Phys. Rev. B

84, 165405 (2011), URL http://link.aps.org/

doi/10.1103/PhysRevB.84.165405. [42] I. Salzmann, S. Duhm, G. Heimel, M. Oehzelt, R. Kniprath, R. L. Johnson, J. P. Rabe, and N. Koch, Journal of the American Chemical Society

130,

12870 (2008), pMID:

18771262, http://pubs.acs.org/doi/pdf/10.1021/ja804793a, URL http://pubs.acs.org/doi/

abs/10.1021/ja804793a. [43] S. Duhm, G. Heimel, I. Salzmann, H. Glowatzki, R. L. Johnson, A. Vollmer, J. P. Rabe, and N. Koch, Nature Materials 7, 326 (2008). [44] G. Heimel, I. Salzmann, S. Duhm, and N. Koch, Chemistry of Materials

23,

359 (2011),

http://pubs.acs.org/doi/pdf/10.1021/cm1021257, URL http://pubs.acs.org/doi/abs/10.

1021/cm1021257. [45] T. Siegrist, C. Kloc, R. A. Laudise, H. E. Katz, and R. C. Haddon, Advanced Materials 10, 379 (1998), ISSN 1521-4095, URL http://dx.doi.org/10.1002/(SICI)1521-4095(199803)10:

53.0.CO;2-A. [46] T. Siegrist, R. Fleming, R. Haddon, R. Laudise, A. Lovinger, H. Katz, P. Bridenbaugh, and D. Davis, Journal of Materials Research

10,

2170 (1995), ISSN 2044-5326, URL http://

journals.cambridge.org/article_S0884291400079140. [47] T. Ogawa, K. Kuwamoto, S. Isoda, T. Kobayashi, and N. Karl, Acta Crystallographica Section B

55, 123 (1999), URL http://dx.doi.org/10.1107/S0108768198009872.

[48] J. E. Lennard-Jones and B. M. Dent, Trans. Faraday Soc.

doi.org/10.1039/TF9282400092.

30

24,

92 (1928), URL http://dx.