Jin-Dou Huang, Shuo Chai, Shu-Hao Wen, Wei-Qiao Deng, and Ke-Li Han*
State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics,
Chinese Academy of Sciences, Dalian 116023, P. R .China
As one of the most important physical properties of the organic material, the anisotropic charge-carrier mobility directly determines device performance to a large extent. In this protocol, we mainly describe efforts in our group over the last few years to establish the quantitative relationship between angular resolution anisotropic mobilities and organic crystal packing architecture parameters (r, θ, and γ). Based on first-principles quantum mechanics (QM) calculations and Marcus-Hush theory in an effective one-dimensional diffusion equations model, the first analytical expressions of organic semiconductor crystal anisotropic mobility is proposed and applied in some typical p-/n-type organic semiconductor materials. The surprisingly agreements between our intuitive model and the available experiments indicate that the proposed anisotropic mobility analytical expressions in terms of fundamental molecular and packing properties can be applied in aiding synthetic design of organic semiconductor materials.
1

Organic semiconductors have attracted comprehensive interest among researchers all over the world, in particular as active components in organic electronic devices and molecular electronics.
Compared with inorganic semiconductors, organic semiconductor materials possess advantages such as low cost, versatility of chemical synthesis, ease of processing, low weight, and flexibility.
However, the performances of most current organic electronic devices are still limited by the relatively low carrier mobility of the organic material. Investigating the structure/property relationships for the ultimate design of new organic semiconductor materials with higher mobility has been a subject of great interest for many years.
1-2 Nevertheless, the successful relations of microscopic molecular and structural characteristics to their macroscopic mobility properties, especially for the intrinsic anisotropic carrier mobility, are yet to be fully clarified. Only recently, with the development of single-crystal organic field-effect transistors (SCOFETs), the direct measurement of carrier mobility as an explicit function of intermolecular proximity and orientation within an organic crystal structure is allowed, 3-5 which facilitate the direct comparison between theory and experiment, and provide the opportunity for theoretician to establish the quantitative relationship even the explicit analytical expression of the intrinsic anisotropic mobility of organic semiconductor crystals. For the final “functionality by design” of organic materials, a successful analytical expression of anisotropic mobility is very helpful and important since most organic crystals show the pronounced anisotropy which has to be taken into account for device design in the commercial application.
From the 1950s, significant progress has been made toward improved understanding of intrinsic charge-transport phenomena in organic materials, and several models, such as band model, tight-binding model and hopping model, have been proposed for the analysis and simulation of low-density intrinsic transport behavior in the organic crystals observed in OFET experiments.
6 In most cases, the hopping model is one of the most appropriate method to describe carrier transport in organic semiconductor materials, especially at room temperature, due to the fact that organic molecules are usually aggregated by weak van der Waals forces and thus the intermolecular
2

electronic couplings (electron transfer integral) are much weaker than the electron-vibration couplings (reorganization energy) for the majority of conjugated organic oligomers. In the hopping model, the intrinsic charge-transport rates for electron and hole transport mainly rely on two contributions. The first is the geometric relaxation of the molecule (inner reorganization energy) and its surroundings (outer reorganization energy) on movement of the charge carriers.
The second is the magnitude of the intermolecular electronic coupling, which is intimately related to crystal packing. The former is mostly the energy change of a single molecule on charge addition/removal (inner reorganization energy), because contributions from the electronic and nuclear polarization/relaxation of the surrounding medium are significantly smaller.
7 The latter can be approximated as nearest-neighbor contributions, as the electronic couplings fall off rapidly with intermolecular separation. In addition, electronic couplings between adjacent molecules in crystals are also highly sensitive to the molecular packing motif, such as the relative positions of the interacting molecules and intermolecular orientations.
8
Another significant aspect of organic crystal transport properties is the anisotropy of charge transport on organic surfaces, which mainly originates from the sensitivity of electronic couplings to the mode of packing.
9 In the last few years, the mobility anisotropy has received more and more attention as the continuous development of organic field-effect transistors (OFETs). In 2004,
Sundar et al. investigated the dependence of the field-effect mobility on the orientation of the transistor channel relative to the crystallographic axes, and observed for the first time a strong anisotropy of the intrinsic hole mobility within the a-b plane of single crystals of rubrene in field-effect experiment.
5 Later on, scientists found that the anisotropic field effect is common in various organic crystals, such as linear acenes and their derivatives/analogues.
4, 9 These experimental results measured through single-crystal devices not only give us an opportunity to completely understand the charge transport mechanisms, but also provide references for the further theoretical study on relationships between the microscopic molecular packing and macroscopic charge transport of the materials since the crystal packing and molecular orientation are clearly fixed. On the theoretical side, there are several theoretical studies about anisotropic hole/electron mobilities and these investigations gave a detailed analysis and qualitative simulation of the anisotropy of charge transport behavior.
10-12 However it should be noted that
3

even though the Holstein–Peierls model can well describe the temperature-dependence and anisotropy of charge-carrier mobilities in organic molecular crystals, it does not give quantitatively predictive values, and in some cases the calculation results from the
Holstein–Peierls model is about one to two orders of magnitude larger than that of the single-crystal experimental measurements.
7 Moreover, the master equation method coupled with the Marcus–Hush electron transfer theory provides with an efficient method to numerically solve the anisotropic charge-carrier mobility from the molecular packing structure, 11 nevertheless, master-equation method is always complex and does not present the inherent relationship between molecular packing architecture parameters and the mobility anisotropy in organic materials, which is very important for the design of new molecular materials and the improvement of the device performance. It will be very helpful for “design” if we can use a simpler and more intuitive model to offer clear physical insight. For this purpose, by ignoring diffusion pathways interaction in the one-dimensional diffusion equations, we developed a first-principles-based simulation model predicting anisotropic hole/electron mobility of organic crystals with only crystal structures needed.
13 The model leads to the first analytical expression for predicting angular resolution anisotropic mobility in the organic crystal, and the mobility orientation function µ
Φ explicitly shows how the hole/electron mobility correlates with the molecular packing and the underlying atomistic electronic properties, which is very useful in aiding synthetic design of organic semiconductors. In spite of the approximations and the simple one-dimensional diffusion model, the analytical expression of our angular resolution anisotropic mobility function made surprising good predictions for the anisotropic mobility distributions in many organic molecular semiconductors such as linear acene, acene derivates, perylene bisimide derivatives, and oligothiophenes as well as their derivatives/analogues. Recently, it has been more and more widely used in the description of charge transfer behaviour and provides a guideline for “tailoring” new organic compounds for organic electronics.
13-28
In this procotol, we mainly describe efforts in our team over the past few years to propose a theoretical model to establish the quantitative relationship between angular resolution anisotropic mobilities and molecular packing architecture parameters (r, θ, and γ) as well as underlying electronic properties of organic materials, and to simulate the anisotropic hole/electron mobilities
4

of typical p-/n-type organic semiconductor materials. In Section II, we briefly describe our simulation model based on quantum chemical approaches to compute the molecular parameters that govern the intermolecular charge transfer process, followed by our proposed mobility orientation function μ
Φ
(V, λ, r, θ, γ; Φ) that describes the mobility in a specific conducting direction on a specific surface in the organic crystal. Section III and Section IV focus on the applications of our simulation model to the p-type and n-type organic semiconductor materials, respectively. The conduction mechanism and the resistances at the TTF-TCNQ organic hetero-interface, based on our calculations, are discussed in Section V. A summary and outlook are present in the last section.
Our simulation model is based on first-principles quantum mechanics (QM) calculations combined with Marcus-Hush theory, 29-30 which we have validated by predicting anisotropic hole mobilities of several typical p-type organic compounds.
13 Using the hopping mechanism for an organic crystal at room temperature, the nonadiabatic electronic hopping rate ( W ) is given by
Marcus-Hush equation: 29-30
W

V
2



 k T
B
 1 2 exp




4 k T
B


(1), where V is the electronic coupling between neighboring molecules in the organic single-crystal, λ is the reorganization energy, T is the temperature, and k
B
is the Boltzmann constant.
Assuming no correlation between hopping events, the hopping rate between neighboring molecules in the organic single-crystal leads to the diffusion coefficient
D

1
2 n
 i
2 r W P i i i
(2), where n is the spatial dimensionality and i represents a specific hopping pathway with hopping distance r i
(the intermolecular center-to-center distance of different dimer types). P is the hopping probability which is calculated as
P i

 i
W i
W i
(3).
5

The diffusive mobility from charge hopping μ is then evaluated from the Einstein relation, leading to the bulk (isotropic) mobility of the material:
  e k T
B
D (4).
The magnitude of the field-effect mobility in a particular transistor channel depends on the specific surface of the organic crystal. We analyze the mobility of components for each surface in terms of angles ( γ i
) between the charge hopping pathways and the plane of interest (W i
·r i
·cosγ i
).
13
In most instances, unsubstituted π-conjugated molecules crystallize into a layered herringbone packing, which gives rise to a 2D transport within the basal stacked organic layers while transport between layers is less efficient. For the hopping paths in the basal stacked layers, the γ i
are 0 degrees.
Using the basal plane as the reference, Φ is the orientation angle of the transistor channel relative to the reference axis (such as the crystallographic axes a, b or c ) and { θ i
} are the angles of the projected hopping paths of different dimer types relative to the reference axis. Thus the angles between the hopping paths and the conducting channel are
θ i
– Φ
(see Figures 1a and 1c). We then project the hopping paths onto the different transistor channels ( W i
· r i
· cos γ i
· cos( θ i
– Φ )). In the typical layered herringbone packing, neighboring molecules in the same layers can be characterized as transverse dimers T and parallel dimers P as illustrated in Figures 1a and 1c. For crystals with structural disorder, we use distribution functions to describe the probability density of dimer types. For the ideal high-purity crystals without disorder, the orientations of the surrounding molecules are identical, so that equations 2-4 lead to the orientation function describing the mobility in a specific conducting direction on a specific surface in the organic crystal: 13


 e
2 k T
B
 i
2
W r P i i i
2
 i
2 cos cos (
  i

) (5).
Here P i
·cos 2 γ i
·cos 2 ( θ i
– Φ ) describes the relative hopping probability of various dimer types to the specific transistor channel, while r i
, γ i and θ i
are determined by the molecular packing architecture in the organic crystal; other terms are defined as above. In eq (5), we suggest that the mobility in a special conducting direction is determined by all related hopping pathways and it is a
6

combined effect of electronic couplings ( V ) from different hopping pathways in organic materials.
In eq (5), a specific Φ corresponds to a specific conducting direction, which means that it is a one-dimensional model. Thus, the spatial dimensionality n in eq (2) is taken to be 1 for the derivation of eq (5).
Equation (5) provides an analytic function to determine the angular-resolved anisotropic mobilities for any type of organic semiconductors by relating the crystal packing and electron coupling V to the angle Φ . We describe the mobility as a function of the orientation angle of the transistor channel in a plane, taking μ' ( Φ ) = 0 while μ'' ( Φ ) > 0 defines the direction for the conducting channels with the highest mobility in the plane and μ'' ( Φ )< 0 defines the direction for the conducting channels having the lowest mobility in the plane. Taking μ' ( Φ ) = 0, leads to:
 extrema
 n


1
2 2 arctan


 i
 i
2 2
PV r i i i
2 2
PV r i i i cos
2  i sin 2
 i cos
2
 i cos 2
 i


, n
    
(6).
When the orientation angle of the transistor channel equals to Φ extrema
, the highest/lowest mobility in the plane could be calculated by substitution of the Φ extrema
values into equation (5). We have validated equations (5) and (6) for some of the highest performing p-type organic semiconductors, such as ruberene, pentacene, tetracene, 5,11-dichlorotetracene (DCT) and hexathiapentacene
(HTP), by comparing the calculated results with the available experimental data.
13
Just two parameters, electron coupling V and reorganization energy λ , determine the relations in equations (1) - (6) and both of which can be derived from first principle calculations. We use the adiabatic potential-energy surfaces method to calculate
λ
.
8 The geometries for the isolated molecules in the neutral and cationic/anionic states are optimized using DFT with the B3LYP functional and with the 6-311G** basis set. For the neutral monomer A, the reorganization energies for hole transport and electron transport are as follows:
 hole
 
0
 



E
0
A
* 
E
0
A
 
E
A

* 
E
A


(7),
 electron
 
0
 



E
0
A
* 
E
0
A
 
E
A

* 
E
A


(8), where E
0
A
and E

A
/ E

A
denote the energies of neutral and cation/anion monomers in their respective optimized geometries, and
A
*
E and
0
E
A
*

/ E
A
*
 denote the energies of neutral and
7

cation/anion monomers with the cation/anion and neutral geometries, respectively. The electronic coupling calculations of dimers, whose geometries are selected from X-ray crystal structures, are performed by the local density functional VWN in conjunction with the PW91 gradient corrections, as implemented in the ADF program.
31 The electronic coupling V can be calculated from the spatial overlap ( S ), charge transfer integral (
RP
J ), and site energies (
RP
H ,
RR
H ): 8
PP
V

J
RP

S
RP
( H
RR

H
PP
) / 2
1
 2
S
RP
(9).
Assuming h ks
is the dimer system Kohn-Sham Hamiltonian which consists of two monomers, and

C 1 ,

C 2 are the highest occupied molecular orbitals (HOMO) or the lowest unoccupied molecular orbitals (LUMO) of two monomers , thus S ,
RP
J ,
RP
H and
RR
H
PP for the V calculation of p -type organic materials can be obtained from:
J
RP
 
C 1
S
RP
 
C 1
H
RR
 
C 1
H
PP
 
C 2 h
/ ks

C 2

C 2
/ h ks

C 1
/ h ks

C 2
(10)
1.1 Linear Acenes and Their Derivatives
Linear acenes and their functional derivatives have recently attracted much attention as a new class of promising materials for use in OFETs. Of these molecules, pentacene is among the first conjugated organic oligomers to be used as a p-type semiconductor and is still used as the standard for all newly developed organic semiconductors.
1, 9 Since metal-insulator semiconductor field-effect transistor (MISFET) devices based on pentacene films were first reported by
Dimitrakopoulos et al., 32 numerous research groups have studied and optimized the performance of these devices.
4, 33-34 In 1997, Gundlach and co-workers reported that pentacene OTFTs could
8

achieve a field-effect mobility as large as 0.7 cm 2 /(V·s), 34 which is much higher than the maximum field-effect mobility of 0.038 cm 2 /(V·s) reported for the pentacene MISFET. Then
Butko and co-workers reported on the fabrication and characterization of field-effect transistors of single-crystal pentacene, and the device exhibited hole conductivity with room-temperature effective mobility up to 0.30 cm 2 /(Vs).
33 In 2006, Lee et al. designed fan-shaped electrodes on a
Si/SiO
2
substrate to investigate the anisotropic field effect mobility in freestanding single crystal pentacene. They found that the field effect mobility showed remarkably anisotropic behaviour (i.e., angular dependence) and the highest mobility value was estimated to be 2.3 cm 2 /(Vs) at room temperature.
4 In an effort to provide an assessment of the possible range of intrinsic and anisotropic charge-transfer rates for hole transport, without considering the complications caused by structural disorder and defects, we described the intrinsic mobility as a function of the orientation angle of the transistor channel in a plane (see Figure 1a). The angular resolution anisotropic mobility orientation function of pentacene in the a-b plane estimated from equation 5 is shown as follows: 13
μ
Ф
= 0.58cos
2 (47.7° + Ф) + 4.80cos
2 (54.2° - Ф) + 0.16cos
2 Ф Ф = 95° - φ.
Here, Φ and φ are defined as the orientation angle of the conducting channel relative to the reference a and b axes, respectively, and the angle between the a and b axes is 95°. As shown in
Figure 1a, we compared our predictions for pentacene with experiments. The predicted mobility anisotropy curve for the pentacene crystal (Figure 1b) agrees reasonably well with the experiment results of Lee et al., 4 and the predicted lowest/highest mobilities (0.66 and 4.88 cm 2 V -1 s -1 ) are also very close to experiment (0.6 and 2.3 cm 2 V -1 s -1 ).
Another typical p-type organic semiconductor material is rubrene, which is a tetraphenyl derivative of teracene. Besides air stability and ease crystal growth, the most crucial characteristic of rubrene is its high field-effect charge mobility. The rubrene SCFETs first fabricated by
Podzorov et al. exhibited mobility in the range 0.1 – 1 cm 2 /(V·s) at room temperature.
35
Subsequently, the mobility of the SCFETs was significantly improved to 8 cm 2 /(V·s) after optimization of the fabrication process.
36 Based on pure rubrene single crystals with colloidal graphite electrodes and parylene as a dielectric, the SCFETs prepared by Zeis et al. exhibited a maximal mobility of 13 cm 2 /(V·s) with strong anisotropy.
37 It is also worthwhile to mention that
9

the Sundar group reported a strong anisotropy of the field-effect mobility within the a-b plane of single crystals of rubrene, which was observed in field-effect experiments for the first time.
5 Later, the Bao group reported anisotropic field-effect mobility with higher angular resolution, and these results offered an improved data set for further studying the correlation between crystal structure and charge carrier mobility.
3
Based on our mobility orientation function, equation 5, combined with first-principles quantum mechanics (QM) calculations, we predicted the possible range of charge-transfer rates in rubrene single crystal and simulated the mobility anisotropy curve for rubrene. Figure 1c illustrates the projection of various hopping paths onto a transistor channel in the a-b plane of a rubrene crystal
(the herringbone layer of a rubrene crystal), and Figure 1d compares our theoretical results with experimental measurements. The predicted mobility anisotropic curve for rubrene agrees reasonably with most reported experimental data (see Figure 1d). For example, Sundar et al. obtained hole mobilities of 2 and 5 cm 2 V -1 s -1 along the a and b axes, respectively, by using a two-probe technique to measure the rubrene with a single-crystal OFET device.
5 Zeis et al. reported hole mobilities along the a (1.8 cm 2 V -1 s -1 ) and b (5.3 cm 2 V -1 s -1 ) axes based on a single-crystal FET with colloidal graphite electrodes and parylene as dielectrics.
37 Moreover, Ling et al. employed a novel bottom-contact oxide architecture and fan-shaped electrodes to achieve
30° resolution for the field effect mobility in the rubrene single crystal, leading to results (1.2 cm 2 V -1 s -1 for the a axis and 5.0 cm 2 V -1 s -1 for the b axis) that also match our predictions.
3
Besides pentacene and rubrene, our orientation functions are also used to describe the anisotropic hole mobilities of tetracene, 5,11-dichlorotetracene (DCT), hexathiapentacene
(HTP), 13 and pentacene derivatives with different electron-withdrawing substituents (such as F,
Cl, Br, CN), 17 respectively. Comparisons of predicted mobility values to available experimental data are summarized in Table 1, where we see reasonable agreement with most experiments.
Besides, Liu et al. have theoretically investigated the transport property of the cross stacking crystal of trans-2,5-diphenyl-1,4-distyrylbenzene (trans-DPDSB) based on the Marcus electron transfer theory and our angular resolution mobility function.
22 They found the unique anisotropic hole and electron transport with nearly balanced mobility in the trans-DPDSB crystal, and based on these findings they proposed a possible novel device structure for 3D-bulk carrier
10

recombination in the theoretical stage. Li et al. calculated the hole and electron mobilities of tetrathiafulvalene (TTF) derivative crystals.
23 Their results indicated that a better agreement between the prediction and experimental was found in TTF crystal when the anisotropic transfer properties was taken into account in calculations.
1.2 Oligothiophenes and Their Derivatives/Analogues
Oligothiophenes with well-defined structures, which are highly crystalline in nature, constitute another representative class of organic π-electron systems for use in electronic and optoelectronic devices.
1 The interest in this class of materials has been intense ever since the first organic transistor, built with sexithienyl as the active semiconducting materials, was reported.
38 To date, a number of unsubstituted oligothiophenes with varying conjugation lengths have been synthesized and characterized. X-ray diffraction studies of the oligothiophenes with four (4T), five (5T), six
(6T), and eight (8T) thiophene rings showed that they all display similar solid-state ordering in the bulk and in thin films: planar conformations, herringbone-type packing motifs and well-defined monolayers.
1 Figure 2c shows the layered herringbone packing of compound 6T, which is also the typical molecular arrangement for the majority of unsubstituted conjugated organic oligomers. In order to obtain anisotropic mobility, the electronic hopping rates in various hopping paths are projected to the transistor channel in the b-c plane of a sexithiophene crystal (see Figure 2c).
15
From the calculated mobility anisotropy curve shown in Figure 2d, we can see that there exists a small difference between the mobility values in different directions. The highest value is 0.09 cm 2 /
(V·s) at 90°/270° transport directions and the lowest is 0.06 cm 2 /(V·s) at 0°/180° directions. The highest/lowest mobility axes do not completely coincide with the texture direction. These observations are mainly derived from the anisotropy of the crystal structure and direction-dependent overlap of the π electrons, as has been explained in Ref 15. Although there is not yet reported angularly resolved anisotropic mobility measurements for 6T, the ranges of mobility values estimated in the same layer agrees well with the corresponding experimental values (0.006~0.009 cm 2 V -1 ·s -1 ).
15 More recently, Duan et al. also used our simulation model described in Section II to predict the anisotropic mobility of cyanovinyl-substituted oligothiophenes and to further probe the relationship between the intermolecular interaction and
11

the anisotropic mobility in order to gain insight into the influence of cyanovinyl groups.
39 Though the angular resolution anisotropic mobilities analysis, they found that enriching intermolecular interactions induced by introduction of cyanovinyl-substituents to the thiophene units could be favourable for controlling the transport channel and thus get high mobility.
Moreover, a series of α-oligofurans, which are close oligothiophene analogues, have been synthesized and characterized by Gidron et al.
40 An interesting discovery was that long oligofurans exhibit tighter herringbone solid-state packing, greater rigidity, and greater solubility than the corresponding oligothiophenes, which means that oligofurans fulfill the most important requirements for a wide range of applications. However, much less attention has been devoted to the oligofurans due to the poor chemical stability of the long-chain oligofurans. Based on our first-principles simulation model, we investigated the intrinsic charge-transfer properties of unsubstituted α-oligofurans, and provided an assessment of the possible range of charge-transfer rates in oligofuran materials.
15 Figure 2b shows the predicted anisotropic hole mobilities of sexifuran (6F), for which there are not yet reported angularly resolved anisotropic mobility measurements. The anisotropic mobility curve of 6F is quite different from that of 6T due to the difference in their packing arrangements. The angle dependence of mobility in the 6F single crystal shows remarkable anisotropic behaviour and the highest mobility value appears when the value of Φ is near 0°/180°, which is in sharp contrast with 6T. Moreover, it is worth noting that the maximum hole mobility value for 6F is nearly 17 times larger than that for the 6T single crystal, which indicates that holes in the 6F crystal are intrinsically far more mobile than those in the 6T crystal. These results are consistent with Bunz’s conjecture that charge-carrier mobility in the 6F crystal might be larger than that in the 6T crystal because of its tighter packing and more electron-rich nature.
41
Besides these typical organic semiconductor materials, the conducting properties of oligothienoacenes and their derivatives that exhibit extended π-conjugation and more rigid structures are also systematically investigated in our theoretical calculations.
18 The calculated hole-transfer mobilities matched reasonably well with the experimentally reported values. Our theoretical results also showed that the chemical oxidation of the thiophene ring led to optimal intermolecular orbital π-overlap and enhanced intermolecular electronic coupling, more
12

importantly, it provided a promising way to convert p-type semiconductors to n-type or ambipolar materials.
In n-type semiconductors the majority of carriers are electrons. The development of n-type materials has largely lagged behind that of p-channel materials due to the facts that the transport in n-channel conductors is degraded easily by air, which acts as an electron trap together with the dielectric surface trapping sites, and that most known organic materials tend to conduct holes better than electrons. Besides the susceptibility of organic anions to atmospheric oxidants, such as
O
2
and H
2
O at the surface of organic semiconductors or the hydroxyl groups at the surface of gate insulators, finding suitable metals for contacts to n-channel organic materials is also a difficulty that limits n-channel organic semiconductors use in OFET applications. The metals typically used for making contact to organic materials (Au, Ag, and other noble metals primarily) have work functions better suited for injection of holes into the HOMO than of electrons into the LUMO for typical organic semiconductors. Although low-work-function metals such as Al, Mg, or Ca, are expected to have lower electron injection barriers, they tend to oxidize easily and readily form reactive complexes with the organic semiconductor. From a practical standpoint, the electron affinity of n-type organic semiconductors needs to be at least 3.0 eV, but should not be much greater than 4.0 eV since the high electron affinity facilitates efficient injection of electrons into the empty LUMO of the semiconductor molecules. However, if the molecule is too electrophilic, its stability in ambient conditions would be overly compromised.
42 Therefore, it has been proposed in previous studies that electron transfer materials for organic electronic applications should satisfy a variety of design requirements such as high electron affinities, large intermolecular electronic couplings, and small reorganization energies.
7
Electronically, electron-transport materials should be electron deficient since electron deficient molecules with relatively low-energy LUMOs can be more readily reduced than more electron rich molecules. The reduced molecules will be thermodynamically less reducing and will therefore be potentially less reactive with various impurities in the systems. How does one tailor molecules to make them electron deficient? It is natural to consider the introduction of side or peripheral
13

electron-withdrawing groups to a larger π-conjugated moiety. Indeed, both experimental and theoretical studies showed that functionalizing p-type semiconductors with electron-withdrawing groups is a promising way to convert them to n-type materials.
2 Recently, the electron transport properties of derivatives of organic semiconductors with electron-withdrawing substituents have been investigated with the angular-resolution anisotropic mobility orientation function and density functional theory methods in our group 16-18 . The theoretical simulations mainly focus on the anisotropic transport behaviors of electrons, the effect of molecular structure on mobility and search for the effective methods to improve the electron transport ability.
2.1
Perylene Bisimide Derivatives
Perylene bisimides (PBIs) have been demonstrated to be n-type semiconductors that have large conjugated structures and strong electron affinities. Because of the representative geometric, electronic and transport properties, PBIs and their derivatives show promise in optical and electronic devices. Extensive experiments illustrated that attaching substituents, such as perfluoroalkyl, fluorine, chlorine, cyanide, etc. to the conjugated perylene bisimide core could further stabilize molecular orbitals and facilitate electron transport successfully.
2, 43 Based on first-principles QM calculations combined with the mobility orientation function, we provided the theoretical study on these n-type perylene bisimide (PBI) derivatives with electron-withdrawing substituents at both bay and imide nitrogen positions (see Figure 3), and systematically discussed the detailed influences of these electron-withdrawing substituents on molecular orbitals, air stability, electronic properties and charge transport behaviours.
16
Figure 4 shows the molecular packings of compound A
2
, B
3
and C
1
as well as the torsion angles of perylene bisimide plane. When there is no substituent at the bay position, the conjugated perylene bisimide part is almost planar, like in A
1
, A
2
and A
3
. The torsion angle from the optimized molecular geometries (0.9°) and experimented measurement (1.5°) 43 of compound A
2 indicated the near planarity of this kind of molecule. Electron-withdrawing substituents (i.e. halogen atoms) at the bay positions can result in the severe distortion in the conjugated PBI skeletons due to the steric hindrance. The calculated torsion angle of the B
3
molecule is 37.3° which is in good agreement with the angle of 37.2° from experiment.
43 These geometric
14

conformation changes induced by attaching electron-withdrawing groups do not facilitate the intermolecular electronic coupling. As shown in Figure 4, the molecular packing of the A
2
crystal keeps a slipped stack while the C
1
crystal changes to a herringbone one with an increased intermolecular center-to-center distance. The centroid distance is 4.91 Å of the closest conjugated molecules for A
2
while it is 7.19 Å for B
3
and 5.28 Å for C
1
. The planar torsion usually hinders the approach of the molecules and obstructs the molecule from packing densely. Table 2 lists the maximum electron mobilities of some PBI derivatives. In these PBI derivatives, A
2
exhibits good transport properties for electrons, with the electron mobility as large as 0.123 cm 2 V -1 s -1 . The close packing of the molecules in the solid state allows the strong electronic coupling of the p-conjugated orbitals which determines the high mobility. It is noteworthy that when attaching the two fluorine atoms at the centrosymmetric bay positions of the perylene core, electronic coupling between LUMOs has been enhanced significantly, about two orders of magnitude larger than that for HOMOs. The compound C
1
shows favorable electron transport properties. The LUMOs of conjugated molecules show a strong coupling interaction and the mobility for electrons can come to 0.514 cm 2 V -1 s -1 . For compound C
1
the experimentally measured electron mobility is 0.35 cm 2 V -1 s -1 by Schmidt 43 and our theoretical simulation provides a reliable result. For the PBI derivatives with four electron-withdrawing atoms, i.e., B
2
, B
3
and B
4
, the charge transport mobilities are not satisfying. Due to the repulsion of the introduced electron-withdrawing atoms, the regular molecular packing of the PBI compounds has been distorted and the distance between the conjugated centers has been lengthened largely.
The highly anisotropic character of molecular packing of organic single crystals gives rise to the highly anisotropic mobility. Understanding the anisotropic mobility can assist in controlling the directions of transistor channels relative to the reference directions of the molecular crystal to obtain the highest charge mobility.
5 Figure 5 describes the angular-resolution anisotropic mobilities of electron transport of compounds A
2
and C
1
. Here, we choose the direction of strongest electronic couplings of LUMOs as the reference direction (the 0° direction in the angular-resolution figures). For the electron transport of compound A
2
, since the electronic coupling of LUMOs for dimer D
2
is comparable to the one for D
1
, the mobility maximum of electron transport is along about 5° in the angular-resolution figure. For compound C
1
the electron
15

transport mobility maximum is 0.514 cm 2 V -1 s -1 along the dimer D
1
vector, mainly because the strongest coupling of LUMOs occurs in D
1
.
Moreover, perylene bisimide derivatives are superior organic semiconductor materials with respect to electron affinity, absorption spectra and so on.
Electron affinities related to the ability of electrons coming across the energy barrier and injecting into empty orbitals are shown in Figure
6a. For these PBI derivatives, the electron affinity ranges from 2.90 eV to 3.51 eV which is desired for n-type material applications.
Therefore, the introduction of side or peripheral electron-withdrawing groups increases the electron affinity and facilitates the efficient injection of electrons from metal electrodes .
Time-dependent DFT calculations are employed to simulate the absorption spectra of organic molecules both in gas and liquid phases. The simulated spectra in tetrahydrofuran solution show the investigated PBI derivatives have broad and strong absorption from 400nm to 700nm, see Figure 6b. They could utilize the UV-Visible light and have potential in organic solar cell applications.
Other researchers after us have paid attention to our angularly resolved anisotropic mobility orientation function and performed similar investigations to discuss the electron mobility of the aromatic diimides, especially the anisotropic mobility.
19-21, 25, 28 Chen et al. investigated and compared the electron injection ability, the stability of radical anions, the carrier mobility as well as their anisotropy for naphthalene, anthracene and perylene diimides with five- or six-membered imide ring.
25 Their results indicated that molecules with a six-membered imide ring should be more suitable for n-type organic semiconductor materials. Geng et al. discussed the anisotropy of mobility for the core-chlorinated naphthalene tetracarboxylic diimides (NDIs ) with fluoroalkyl chains, and the anisotropies and temperature dependences of mobilities suggested that the intrinsic electron mobility of NDI with two chlorine substituents in the crystalline state are lower than those of unsubstituted NDI and tetrachlorinated NDI.
28 According to Zhao et al., two dipyrro-boradiazaindacenes (BODIPY) derivatives are investigated theoretically and the anisotropic mobility has been determined.
19 These two isomeric complexes are demonstrated to have large electron-transfer mobility and be regarded as favorable n-type organic semiconductors.
Their calculation results also showed that the isomerization played an important role in intermolecular interaction and thus the charge transport mobility.
16

2.2
Pentacene Derivatives
Pentacene, as a typical p-type hole-transporting semiconductor, has a large electronic coupling of HOMOs (85.32 meV) which is greater than that of the LUMOs (6.91 meV) evidently in the same channel. However, the electron-withdrawing substituted pentacene molecules showed lower molecular orbital energy levels and stronger LUMO electronic couplings, which means that high electron mobilities and electron affinities can be achieved in these materials. In many cases, the electronic couplings of the LUMOs could approach the same order or even surpass those of the
HOMOs.
17 Figure 7 shows the crystal structures and the angular resolution anisotropic mobilities of PF-PENT, TIPS-PENTF8, TIPS-PENTBr4, and TIPS-THIOCl4. The distributions of their mobilities vary significantly along the different directions and express the representative angular-resolution anisotropic characteristic.
17 As charge transport in organic semiconductors happens preferentially along the p-conjugated orbitals of adjacent molecules, the molecular arrangement mostly determines the transport efficiency. Both the internal molecular conformation and the intermolecular interaction affect the molecular packings of organic semiconductor materials. Many p-conjugated molecular crystals are arrayed in regular edge-to-face packing, e.g., the widely studied pentacene and oligothiophene exhibit the orderly herringbone stacking. In fact, this kind of stacking is not the favorable one for orbital overlap and electronic coupling.
Functionalization by adding TIPS to the parent pentacene or tetracenothiophene could improve the molecular packing from edge-to-face to face-to-face though it may cause the lengthening of the center-to-center distance. This face-to-face stacking could increase the effective p-orbital overlap and enhance the charge transport between conjugated molecules.
Comparing the molecular packing of PF-PENT with that of TIPS-PENTF
8
in the nearest dimer channel of the two crystals (the distance is 4.49A˚ and 7.61A˚ respectively), TIPS-PENTF
8
has a stronger LUMOs electronic coupling of 96.20 meV as compared to 79.36 meV for PE-PENT. The stronger LUMOs electronic coupling will contribute to a high mobility for electron transport of
TIPS-PENTF
8
, see equation 5. The maximum electron mobility of 2.22 cm -2 V -1 s -1 for
TIPS-PENTF
8
is obtained compared to 0.49 cm -2 V -1 s -1 for PE-PENT. This kind of stacking facilitates the electron transport and provides the proof for the design of n-type materials. Our angular resolution anisotropic mobility also shows that the mobility in different directions is a
17

combined effect of all electronic couplings, which suggests that when we design a device we should not just simply choose the face-to-face or face-to-edge packing mode direction to obtain the optimal electron mobility. We should also control the orientation of the materials relative to the device channel to obtain the highest electron mobility performance. Recently, Chen et al. also systematically investigated oligoacenes and their derivatives, and simulated the (anisotropic) electron mobilities. On the basis of the geometrical and electronic structures, molecular stacking motifs, carrier injection and transport properties analyses, they proposed that introducing pyrazine is also an effective approach to obtain the excellent n-type OFET materials.
24 Wang et al. performed theoretical predictions to study the charge-transfer properties of novel diazapentacene derivatives.
26 Their angular resolution anisotropic mobility analyses indicated that 5,7,12,14- tetrachloro-6,13-diazapentacene, 5,7,12,14-tetrachloro-6,13-diaza-6,13-dihydropentacene, and
5,7,12,14-tetrafluoro-6,13-diazapentacene exhibited similar anisotropic mobilities but remarkably different anisotropic behaviors in comparison with the 6,13-dihydro-6,13-diazapentacene.
Interfaces between different materials often exhibit unexpected electrical properties. For example, metallic conductivity and even super-conductivity have been obtained at interfaces formed by insulating transition metal oxides; 44 ultra-low resistances ranging from 1 to 30 kΩ per square have also recently been observed at the interface between two insulating organic materials.
45 These findings have attracted great interest with regard to the properties of the interface. Although the research of Huijben et al. has indicated that the metallic conductivity at the interface between oxides is due to electronic coupling between oxide layers, 46 the metallic conduction mechanism at organic interfaces has yet to be theoretically explained.
Based on a combination of first-principles calculations, Marcus–Hush theory and our anisotropic mobility model, we carried out a theoretical analysis of the metallic conduction mechanism at the experimentally characterized TTF (tetrathiofulvalene)–TCNQ
(7,7,8,8-tetra-cyanoquinodimethane) interface for the first time.
14
For the laminated TTF plane–TCNQ plane interfaces with the TTF crystal a–b, a–c or b–c plane facing the TCNQ crystal a–b, a–c or b–c plane faces respectively, the TTF–TCNQ pairs
18

at the interface are all tail-to-tail modes, as shown in Figure 8 which depicts a part of interface configurations with the TTF crystal a-b, a-c and b-c plane on the TCNQ crystal a-b plane, respectively. We calculated the electronic couplings for these tail-to-tail TTF–TCNQ pairs, and all the calculated electronic coupling values are almost zero, which indicated the hybridization between the molecular orbitals of TTF and TCNQ on either side of the interface is very weak. Hence, we concluded that the electronic couplings between TTF and TCNQ at the interface are not responsible for the conduction mechanism. We then calculated the electronic coupling matrix elements in individual TTF and TCNQ crystals, respectively, to evaluate the hole and electron diffusive mobilities in their respective surfaces at the
TTF–TCNQ interface. The results of the calculation showed that the electronic coupling matrix elements of the longitudinal dimers in both of the organic crystals are ~10 -3 eV. Hence, the charge transport between layers in individual TTF or TCNQ crystals is also less efficient,such that the transferred charge will be confined at the TTF–TCNQ interface in decoupled individual TTF and TCNQ layers on either side of the interface in the TTF and
TCNQ crystal. The calculated electronic coupling elements in TTF and TCNQ crystals also showed that in both of these crystals the a–b plane of the molecular stacking layer is favored for charge transport. From the calculated anisotropic mobility in the a–b plane of TTF and
TCNQ single crystals shown in Figure 9, it is apparent that in the TTF crystal the direction with the largest hole mobility is along the b axis (0, b, 0) of the crystal cell. Along this axis the mobility value is 0.799 cm 2 V -1 S -1 , which is close to the experimental value of several cm 2 V -1 S -1 . In the TCNQ crystal, the direction with the largest electron mobility is along the b axis of the primitive cell, i.e. the (a/2, b/2, 0) direction of the TCNQ crystal cell. Along this axis the mobility value is 0.365 cm 2 V -1 S -1 , which agrees well with the experimental value of
0.4 cm 2 V -1 S -1 .
We analyzed the charge transfer at the TTF–TCNQ molecular crystal interface using the molecular orbital approach. The TTF–TCNQ partial charge transfer relies on the critical separation distance between TTF and TCNQ. We used two cofacial neutral TTF and TCNQ molecules with a separation distance between 0.34 and 5 nm and a step -length of 0.2 nm. We performed a full optimization of these TTF–TCNQ complexes and made single-point energy
19

calculations with the orthogonalized TTF and TCNQ monomer orbitals as the basis set. Figure
10a shows the interface electron and hole density versus the separation distance between TTF and TCNQ. It can be seen that the surface carrier density decreases with increasing separation distance, and a critical separation distance of ~1.6 nm was found, beyond which the interface carrier density falls to near zero. Figure 10b shows the resistance per square at the interface versus the separation distance between TTF and TCNQ. From Figure 10b, the resistance of the interface should be between 39 and 64 kΩ per square at 300 K, which is close to the experimental value of 1–30 kΩ per square at room temperature.
Our calculations indicated that the transferred charge from the HOMO of TTF to the
LUMO of TCNQ would be confined at the interface due to the inefficient charge transport across the interface and between layers in individual TTF and TCNQ crystals . This is the origin of the conducting interface and results in the formation of an excitonic insulator. It also suggests that the dimensionality of the interface is the a–b plane of TTF to the a–b plane of
TCNQ crystal, and that the charge behavior is 2D transport (Figure 9).
The 2D conductivity is responsible for the absence of a Peierls transition at the TTF–TCNQ interface, which is one of the most significant differences compared to bulk TTF–TCNQ.
Based on first-principle calculations combined with the hopping description of charge mobilities, we presented an intuitive and desirable simulation model predicting anisotropic hole/electron mobility of organic crystals with only crystal structures needed. The derived analytical expression of angular resolution anisotropic mobility function provides a explicit description of how the hole/electron mobility correlates with the fundamental organic materials properties such as crystal packing (the molecular packing architecture parameters of the crystal r,
θ, and γ) and the underlying atomistic electronic properties (electronic coupling V, and reorganization energies λ). On the basis of our computational model, we have systematically investigated the charge-transfer properties of various p-type and n-type organic semiconductor materials, such as oligothiophenes, oligofurans, perlene bisimide derivatives, and linear acenes as well as their derivatives. The simulation results of charge transport indicated that our
20

computational model not only made good predictions for the anisotropic mobility distributions in the organic crystals, but also attained near quantitative agreement with experiments. In addition, we also carried out a theoretical analysis of the metallic conduction mechanism at the experimentally characterized TTF–TCNQ interface, and the calculated resistances were in good agreement with the experimental measurements, which further illustrated the validity of our simulation method. Our model supplies the analytical expression of angular resolution anisotropic mobility for the priori design and screening of new organic electronic materials, and aids the device design by aligning the conducting channel along the maximum mobility direction for highest device performance. Despite of the agreement between our model and experiment, we should consider the approximations and the errors cancellation effects in the model. Important approximations, such as weak-coupling limit, Marcus−Hush ET theory in classical forms, tunnelling effects, the neglected influences of lattice vibration on the electronic coupling, the outer reorganization energy, the interactions between hopping pathways, the large electric-field modulation effects, should be investigated carefully and detailed in the future, which is vital to find the source of the errors cancellations and improve the current model for a more concise analytical expression of mobility anisotropy.
Furthermore, it should also be noted that because conventional first-principles methods fail to describe weak intermolecular interactions such as accurate van der Waals, most theoretical calculations and analysis are based on the experimental single-crystal structures.
7-8 The dependence of theoretical predictions for the intrinsic charge mobility on experimental crystal structures makes computations impractical for new molecules where there is no knowledge of the crystal molecular packing structure. Although there has been some recent success for the prediction of crystal structures of small organic rigid molecules when the contents of the asymmetric unit is known, the success of such predictions rely on highly accurate dispersion-corrected Density Functional Theory (DFT) methods that are precluded for the large extended π-conjugated molecules.
47 Therefore, accurate prediction of the molecular packing within the solid has always been an important subject of the future theoretical design of organic materials. Recently, some groups have gained important progress for the accurate and affordable local and global optimization of molecular crystal.
48-50 Moreover, the mechanism of charge
21

transport in organic materials is still controversial from experimental and theoretical perspectives, and the current understanding is still limited.
Despite various models, such as the band model and hopping model, being proposed and widely used in the description of charge transport in organic semiconductors, a systematic approach for predicting charge-transfer rates that is valid for arbitrary strengths of electronic coupling and local electron-phonon interactions and over the full range of temperatures has remained a challenging task confronted by theoreticians. The development of a better theoretical method that is universally applicable still is one of hot spots in present researches. Therefore, while significant progress has been made, there is still a long way to go for the full understanding of charge-transfer mechanisms and the true realization of the computationally led design of organic semiconductor materials.
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25

Table 1 . Simulated hole mobility values for linear acenes and their derivatives; the range of predicted mobility is calculated based on eq (9); and the predicted values are compared with experimental result (come from Ref. 13 and 17).
Molecular crystals
(a)
RUBR
PENT
TETR
HTP (a)
Ref. 13 (b)
(a)
(a)
(a)
DCT (a)
TIPS-THIOCl
PF-PENT (b)
TIPS-PENTF
8
(b)
TIPS-PENTBr
2
(b)
TIPS-PENTBr
4
(b)
TIPS-PENTCN
4
4
(b)
(b)
Ref. 17
Predicted Hole Mobility
(cm 2 V -1 s -1 )
0.03-7.12
0.66-4.88
0.01-3.36
0.006-2.22
0.001-0.16
0-1.36
0.01-0.08
0.26-2.21
0-1.21
0-2.81
0.63-4.79
Experimental
(cm 2 V -1 s -1 )
0.27
0.105
0.0964
0.148
2.2
5
0.15
1.3
2.4
1.6
1.2-5.0
1.8-5.3
4.4-15.4
2.4, 8
0.6-2.3
1.9
Mobility
26

Table 2.
The calculated hole and electron transport mobility (in cm 2 V -1 s -1 ) of some PBI derivatives. (The corresponding experimental values from Ref. 43 are in parentheses).
A
2
B
2
μ hole
0.552 0.005
μ electron
0.123 (0.67) 0.002 (0.0005)
B
3
0.137
B
4
0.014
C
1
0.019
0.024 (0.0003) 0.002 (0.025) 0.514 (0.35)
27

Figure 1.
(a) and (c) Illustration of projecting different hopping paths to a transistor channel in the a-b plane of a pentacene crystal and a rubrene crystal; for pentacene, θ
P
, θ
T1
, and θ
T2
are the angles of P, T
1
, and T
2
dimers relative to the reference crystallographic axis a, Φ is the angle of a transistor channel relative to the reference crystallographic axis a; and for rubrene, θ
P
, θ
T1
, and θ
T2 are the angles of P, T
1
, and T
2
dimers relative to the reference crystallographic axis b, Φ is the angle of a transistor channel relative to the reference crystallographic axis b. (b) and (d)
Comparisons of the calculated mobility anisotropy curves with experiments.
Figure 2 (a) and (c) illustrate different hopping paths projecting to a transistor channel in the a b plane of 6F and 6T single crystals respectively; for 6F (6T), θ
1
and θ
2
are the angles of T
1
and T
2 dimers relative to the reference crystallographic axis b (c) ; Φ is the angle of a conduction channel relative to the reference crystallographic axis b (c) . (b) and (d) show the calculated angle resolved anisotropic hole mobility of 6F and 6T respectively.
Figure 3.
Schematic of the structures of investigated PBI derivatives.
Figure 4.
Molecular packings of compounds (a) A2, (b) B3 and (c) C1 (Ref. 43) with the center-of-mass distance between the conjugated centers and the torsion angles of the perylene bisimide plane by DFT calculation (cal.) as well as the experimental results (exp.). Fluoroalkyl groups have been removed for clarity.
Figure 5.
The angular-resolution anisotropic electron mobility of A
2
crystal (a) and C
1
(b).
Figure 6.
(a) The adiabatic electron affinity of investigated PBI derivatives. (b) UV-Vis absorption spectra of A2, B1, B2, B3 and C1 in tetrahydrofuran (THF) solution.
Figure 7.
Crystal structures and the calculated angular resolution anisotropic mobilities of holes and electrons for a) PF-PENT, b) TIPS-PENTF
8
, c) TIPS-PENTBr
4
, and d) TIPS-THIOCl
4
.
Figure 8.
The schematic of TTF-TCNQ interface configurations with the TTF crystal a-b, a-c and b-c plane on the TCNQ crystal a-b plane, respectively.
Figure 9.
Simulated anisotropic mobility in the a–b plane of a TTF and TCNQ crystal at 300
K.
Figure 10.
(a) Interface carrier density and (b) interface resistance vs. the separation distance between TTF and TCNQ.
28

(a)
47.7
° conducting channel
(b)
(c) conducting channel
(d)
29

Φ
Φ
θ
1
θ
2 c
(a)
Φ
Φ
θ
1
θ
2 b
(c) b c
(b)
(d)
30

31

32

(a)
33

(a) (b)
34

35

36

37

38