First-Principles Investigation of Anistropic Hole Mobilities in Organic

Jun 9, 2009 - Division of Chemistry and Biological Chemistry, School of Physical and Mathematical Sciences, Nanyang Technological University, 637616 S...
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J. Phys. Chem. B 2009, 113, 8813–8819

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ARTICLES First-Principles Investigation of Anistropic Hole Mobilities in Organic Semiconductors Shu-Hao Wen,†,‡ An Li,† Junling Song,† Wei-Qiao Deng,*,†,‡ Ke-Li Han,*,‡ and William A. Goddard III*,§ DiVision of Chemistry and Biological Chemistry, School of Physical and Mathematical Sciences, Nanyang Technological UniVersity, 637616 Singapore, State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, People’s Republic of China, and Materials and Process Simulation Center, California Institute of Technology, Pasadena, California 91125 ReceiVed: January 18, 2009; ReVised Manuscript ReceiVed: April 20, 2009

We report a simple first-principles-based simulation model (combining quantum mechanics with Marcus-Hush theory) that provides the quantitative structural relationships between angular resolution anisotropic hole mobility and molecular structures and packing. We validate that this model correctly predicts the anisotropic hole mobilities of ruberene, pentacene, tetracene, 5,11-dichlorotetracene (DCT), and hexathiapentacene (HTP), leading to results in good agreement with experiment. 1. Introduction

2. Theoretical Methodology

Although organic materials promise enormous advantages in cost and processability,1-3 technological progress has been slow due to the limited understanding of the fundamental electronic properties of organic semiconductor materials.4,5 The inconsistency of basic performance parameters, such as surface treatment, temperature, material purity, device structure, and variations in morphology, and the intrinsic electrical anisotropy of organic materials make comparisons of experimental data within the literature very difficult. Recently, the development of singlecrystal organic field effect transistors (SCOFETs) provided opportunities to explore intrinsic properties of these promising organics.6-8 These intrinsic properties included angular resolution mobility anisotropy6,9,10 and the Hall Effect.7 Among these properties, the electrical anisotropy of organic materials has attracted much attention.6,9-19 For example, the anisotropic effects were first found by Sundar et al. in rubrene crystals in 2004.6 Then scientists found that the anisotropic field effect is common in various organic crystals such as pentacene9 and dicyclohexyl-R-quaterthiophene.11 Although there are several theoretical studies about anisotropic hole mobilities,15-20 a systemic investigation of anisotropic hole mobilites is still lacking.

Our simulation model is based on first-principles quantum mechanics (QM) calculations combined with Marcus-Hush theory,21,22 which we validate by predicting anisotropic hole mobilities of several p-type organic compounds (no adjustable parameters). Using the hopping mechanism for an organic crystal at room temperature, the electronic hopping rate (W) is given by the Marcus-Hush equation21,22

Here, we developed a mobility orientation function µΦ(V,λ,r,θ,γ;Φ), establishing the quantitative relationship between angular resolution anisotropic mobilities and molecular packing architecture parameters (r, θ, and γ) and underlying electronic properties (electronic coupling V and reorganization energies λ) of organic materials. * To whom correspondence should be addressed. E-mail: wqdeng@ ntu.edu.sg (W.-Q.D.); [email protected] (K.-L.H.); [email protected] (W.A.G.). † Nanyang Technological University. ‡ Chinese Academy of Sciences. § California Institute of Technology.

W)

( ) (

V2 π p λkBT

1/2

exp -

λ 4kBT

)

(1)

where V is the electronic couplings23-27 between neighboring molecules in the organic single crystal, λ is the reorganization energy,5,28 T is the temperature, and kB is the Boltzmann constant. Assuming no correlation between hopping events and that charge motion is a homogeneous random walk,29-31 the hopping rate between neighboring molecules in the organic single crystal leads to the diffusion coefficient in eq 2

D ) lim tf∞

1 〈x(t)2〉 1 ≈ 2n t 2n

∑ ri2WiPi

(2)

i

where n is the spatial dimensionality, i represents a specific hopping pathway with hopping distance ri (the intermolecular center-to-center distances of different dimer types), and P is the hopping probability, which is calculated as by

10.1021/jp900512s CCC: $40.75  2009 American Chemical Society Published on Web 06/09/2009

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Pi )

Wi

∑ Wi

Wen et al.

(3)

i

to be µ′(Φ) ) 0, while the second derivatives of Φ dependent mobility 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

The drift mobility from charge hopping µ is then evaluated from the Einstein relation, leading to the bulk (isotropic) mobility of the material

Φexterma ) µ)

e D kBT

(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 of the hopping jumps (γi) between adjacent molecules relative to the plane of interest (Vri cos γi). In most instances, 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 γi are 0°. Using the basal plane as the reference, Φ is the orientation angle of the transistor channel relative to the reference axis (crystallographic axis a or b) and {θi} 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 θi-Φ, as shown in Figure 1a for a rubrene crystal. We then project the hopping paths onto the different transistor channel (Wiri cos γi cos(θi - Φ)). In these organic crystals, neighboring molecules can be characterized as transverse dimers T, parallel dimers P, and longitudinal dimers L, as illustrated in Figures 1a and 2a.29 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 molecules surrounding each molecule are identical, so that eqs 2-4 lead to the orientation function describing the mobility in a specific conducting direction on a specific surface in the organic crystal

µΦ )

e 2kBT

∑ Wiri2Pi cos2 γi cos2(θi - Φ)

(

nπ 1 + arctan 2 2

(5)

∑ PiVi2ri2 cos2 γi sin 2θi i

∑ PiVi2ri2 cos2 γi cos 2θi i

)

n ) 0, (1, (2, (3, ... (6)

We first validate this theory (eqs 5 and 6) for some of the highest performing p-type organic semiconductors, ruberene, pentacene, tetracene, 5,11-dichlorotetracene (DCT),32 and hexathiapentacene (HTP),33,34 by comparing to available experimental results. Just two parameters (electron coupling V and reorganization energy λ) determine the relations in eqs 1-6, both of which can be derived from first-principles calculations. We use the adiabatic potential energy surfaces method to calculate λ.28 The geometries for the isolated molecules in the neutral and cationic states are optimized using the B3LYP flavor of DFT with the 6-311G** basis set, as implemented in the Jaguar 4.0 program.35

λ ) λ0 + λ+ ) (E*0 - E0) + (E*+ - E+)

(7)

where E0 and E+ represent the energies of the neutral and cation species in their lowest-energy geometries, respectively; E*0 and E*+ are the energies of the neutral and cation states with the geometries of the cation and neutral species, respectively. We choose the method in refs 26 and 27 to calculate the V of each dimer, and the monomer orbitals with proper orthogonalization are used as the basis set for the Hamiltonian of the dimer system. The geometries of dimer pairs are selected from X-ray crystal structure. The electronic coupling calculations of dimers are performed by the local density functional VWN in the conjunction with the PW91 gradient corrections with the TZ2P basis set, as implemented in the ADF27 program. The V can be calculated from the spatial overlap (SRP), charge transfer integral (JRP), and site energies (HRR,HPP)26

i

Here, Pi cos2 γi cos2 (θi - Φ) describes the relative hopping probability of various dimer types to the specific transistor channel, while ri, γi, and θi are determined by the molecular 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 the combined effects of electronic couplings (V) from different hopping pathways for the mobility anisotropy in organic materials. In eq 5, a specific Φ corresponds to a specific conducting direction, which means that it is a onedimensional model; thus, the spatial dimensionality n in eq 2 is taken to be 1 for the derivation of eq 5. Here, we use n ) 1 approximately to extract the mobility along a specific direction. Equation 5 provides an analytic function to determine the angular resolution 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 the first derivatives of the Φ-dependent mobility

V)

JRP - SRP(HRR + HPP)/2 2 1 - SRP

(8)

Assuming that hks is the Kohn-Sham Hamiltonian of the dimer C2 system which consists of two monomers, and φC1 HOMO and φHOMO are the highest occupied molecular orbitals (HOMO) of two monomers, SRP, JRP, HRR, and HPP needed for the calculation of V for p-type organic materials can be obtained from C1 C2 JRP ) 〈φHOMO |hks |φHOMO 〉 C1 C2 SRP ) 〈φHOMO |φHOMO 〉 C1 C1 HRR ) 〈φHOMO |hks |φHOMO 〉

(9)

C2 C2 HPP ) 〈φHOMO |hks |φHOMO 〉

3. Results and Discussion Calculated Angular Resolution Anisotropic Mobility of

Anistropic Hole Mobilities in Organic Semiconductors

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Figure 1. (a) Illustration of projecting different hopping paths to a transistor channel in the a-b plane of a rubrene crystal; θP, θT1, and θT2 are the angles of P, T1, and T2 dimers relative to the reference crystallographic axis b (L dimers are also shown); Φ is the angle of a transistor channel relative to the reference crystallographic axis b. (b) Comparisons of the calculated mobility anisotropy curve with experiments.6,10,36

Rubrene and Pentacene. Figure 1a illustrates the projecting 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 1b compares our theoretical results with experiments.6,10,36 The reference axis is set as the crystallographic b axis, and the orientation angle of the conducting channel relative to the reference b axis is Φ. The hopping paths of T1, T2, and P dimer types are exactly on the a-b plane. The angles between the hopping paths of T1, T2, and P dimer types and reference axis b are θT1, θT2, and θP values of 63.3°, 180°-63.3°, and 0°, and the angles between the hopping paths of T1, T2, and P dimer types and the conducting channel are θT1-Φ, θT2-Φ, and θP-Φ, respectively. Using eq 5 with the V electronic coupling and λ reorganization energy in Table 1 (derived from ab initio calculations) leads to the mobility orientation function in the a-b plane for rubrene.

µΦ ) 0.018 cos2(63.3° - Φ) + 0.018 cos2(63.3° + Φ) + 7.12 cos2Φ (10) Figure 1b shows that the predicted mobility anisotropic curve for rubrene agrees reasonably with most reported experimental data,3,6,10,36,37 as summarized in Table 1. For example, Sunder et al. obtained hole mobilities of 2 and 5 cm2 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 FET device.6 Zeis et al. reported hole mobilities along the a (1.8 cm2 V-1 s-1) and b (5.3 cm2 V-1 s-1) axes based on single-crystal FET with colloidal graphite electrodes and parylene as dielectrics.36 Moreover, Ling et al.10 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 cm2 V-1 s-1 for the a axis and 5.0 cm2 V-1 s-1 for the b axis) that also match our predictions. Although our calculation results are in reasonable agreement with most experimental results, our predictions overestimate the hole mobility at the b axis by 2.12 cm2 V-1 s-1. This overestimation may come from the approximation by using Marcus theory at its first-order perturbative nature.38,39 The hopping paths of T1, T2, and P dimer types are on the basal stacked layer’s a-b plane; thus, the γT1, γT2, and γP are 0°. The between layer dimers L are out of the a-b plane, and they have a weak electronic coupling with a VL of 0. From the rubrene crystal, VT1 ) VT2, rT1 ) rT2, PT1 ) PT2, θT1 ) 63.3°, θT2 ) 180-63.3°, and θP ) 0°. On the basis of eq 6 and the calculated λ and V for rubrene (Table 1), this leads to 2Φexterma ) nπ + arctan 0, indicating that Φexterma ) 0, 180° or 90, 270° directions lead to mobility extrema. Since the second derivative is µ′′(Φ) > 0 for Φ ) 0 and 180° while µ′′(Φ) < 0 for Φ ) 90 and 270°, we expect Φ of 0 and 180° to correspond to the highest mobility and Φ of 90 and 270° to correspond to the lowest mobility. Indeed, this agrees well with experiments by most experiments3,6,10,36,37,40 that find the highest mobility along the b crystallographic axis and the lowest mobility along the a crystallographic axis. Figure 2 compares our predictions for pentacene with experiments. Equation 5 leads to the angular resolution anisotropic mobility orientation function of pentacene in the a-b plane as

µΦ ) 0.58 cos2(47.7° + Φ) + 4.80 cos2(54.2° - Φ) + 0.16 cos2 Φ

Φ ) 95° - φ (11)

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Φ and φ are defined as the orientation angle of the conducting channel relative to the reference a and b axis, respectively, and the angle between the a and b axis is 95°. The predicted mobility anisotropy curve for the pentacene crystal (Figure 2b) agrees reasonably with experiment results of Lee et al.,9 and the predicted lowest/highest mobilities (0.66 and 4.88 cm2 V-1 s-1) are close to experiment (0.6 and 2.3 cm2 V-1 s-1). In contrast to rubrene, pentacene has VT1 * VT2, rT1 * rT2, PT1 * PT2, θT2 ) 54.2°, θT3 ) 180-47.7°, and θp ) 0°. Using eq 6 and the calculated λ and V (Table 1) for pentacne, it leads to 2Φexterma ) nπ + arctan(-2.67). To make comparison with Lee et al.’s experiment that the orientation angle of the conducting channel is relative to the reference b axis, we use φexterme ) 95° - Φexterma. We observe that the φexterme ) 39 and 219° clockwise directions relative to the b crystallographic axis of pentacene are the directions for the highest mobilities and that φexterme ) 129 and 309° are the directions for the lowest mobilities. These calculations explain why the highest and lowest mobility channels do not coincide with the texture direction of the pentacene crystal in the Lee et al. experiment.9 The experimental results find the highest mobility at 180° and the lowest mobility at 120°, clockwise relative to the b crystallographic axis. The theory and experiment agree on the direction of lowest mobility channels, but the direction for highest mobility channels is different. We re-evaluated the experimental anisotropic mobility based on their experimental IDS - VG data9 (Figure S2, Supporting Information). We found an alternative interpretation of their experimental results, leading to the red curve in Figure 2b for the anisotropic mobility. With this reinterpretation, the maximum hole mobility is along 30°, in good agreement with our theoretical result of 39°.

Wen et al. TABLE 1: Comparison of Predicted Lowest/Highest Mobility in the a-b Plane of Rubrene (RUBR), Pentacene (PENT), and Tetracene (TETR) Crystals and That in the a-c Plane of DCT, HTP Crystals with Experimental Results Based on Single-Crystal OFETsa predicted µ experimental µ (cm2 V-1 s-1) (cm2 V-1 s-1) 1.2-5.0b 1.8-5.3c 4.4-15.4d 2.4e, 8f 0.6-2.3g 1.9h 2.2i 5j 0.15k 1.3l 2.4m

RUBR

0.03-7.12

PENT

0.66-4.88

TETR

0.01-3.36

DCT

0.006-2.22

1.6n

HTP

0.001-0.16

0.27o

V (eV)

λ (eV)

VT1 ) 0.019 (0.015)p VT2 ) 0.019 (0.015)p VP ) 0.089 (0.083)p VL ) 0.000 (0.000)p VT1 ) 0.085 (0.085)p VT2 ) -0.048 (-0.051)p VP ) 0.032 (0.037)p VL ) 0.000 (0.000)p VT1 ) -0.017 (-0.004)p VT2 ) 0.070 (0.070)p VP ) 0.001 (0.000)p VL ) 0.000 (0.000)p VT1)-0.006 VT2 ) -0.0055 VP ) -0.0863 VL1 ) -0.0017 VT1 ) 0.0035 VT2 ) 0.0031 VT3 ) 0.0026 VT4 ) 0.003 VP ) 0.023 VL ) -0.0006

0.1521 (0.1590)q

0.0976 (0.0992)r

0.1157 (0.1176)r

0.1472

0.1428

a Electronic coupling elements V (compared with Breda`s et. al.’s calculated results5) and reorganization energies λ (compared with experimental results46,37) are listed. b Reference 10. c Reference 36. d Reference 6. e Reference 3. f Refernce 37. g Reference 9. h Reference 40. i Reference 44. j Reference 45. k Reference 41. l Reference 42. m Reference 43. n Reference 32. o Reference 33. p Reference 5. q Reference 46. r Reference 37.

Figure 2. (a) Illustration of projecting different hopping paths to a transistor channel in the a-b plane of a pentacene crystal; θP, θT1, and θT2 are the angles of P, T1, and T2 dimers relative to the reference crystallographic axis a; Φ is the angle of a transistor channel relative to the reference crystallographic axis a. (b) Comparison of the predicted mobility anisotropy curve with experiment9 and evaluation (aFigure S2, Supporting Information).

Anistropic Hole Mobilities in Organic Semiconductors

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Figure 3. The predicted anisotropic mobility on the a-b plane (relative to the crystallographic axis b) of a tetracene crystal.

Figure 4. The predicted anisotropic mobility on the a-c plane (relative to the crystallographic axis a) of a DCT crystal.

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Wen et al.

Figure 5. The predicted anisotropic mobility on the a-c plane (relative to the crystallographic axis a) of a HTP crystal.

It is generally thought that the hopping pathway with the largest electronic coupling (i.e., the largest hopping rate) will be the direction of the largest mobility. Indeed, for rubrene, the predicted directions of the highest/lowest mobility direction agree with the experiments3,6,10,36,37,40 that the highest mobility direction is along the P dimers with the largest electronic couplings. However, in pentacene, the predicted direction with the highest mobility is not along the T1 dimer with the largest electronic couplings.5 We can explain this phenomenon based on the mobility orientation function. In rubrene, the electronic couplings of the P dimers are much larger than those of other dimers (the combined effects are small), and the packing architectures in rubrene crystals have the high symmetry (rT1 ) rT2, PT1 ) PT2, θT1 ) 63.3°, and θT2 ) 180-63.3°). However, in pentacene, the electronic couplings of T1, T2, and P dimers are close (the combined effects from the electronic couplings of different dimers are significant), and the packing architectures have the low symmetry (rT1 * rT2, PT1 * PT2, θT1 ) 54.2°, and θT2 ) 180-47.7°). Owing to these different parameters of rubrene and pentacene, the mobility orientation function directly leads to the difference of the mobility anisotropic curve (Figures 1b and 2b) between rubrene and pentacene. From these results on rubrene and pentacene, we conclude that the simulation model using eqs 1-9 can predict reasonable angular resolution anisotropic hole mobilities of organic semiconductors. The Sunder et al. experimental results6 highlighted the importance of controlling the orientation of the organic crystals relative to the FET channel. Thus, our mobility orientation function eq 5 combined with QM calculation results provides the simple means for an experimentalist to predict the direction that would lead to the highest mobility. Equation 6 shows that it is the combined effects of electronic couplings (V) of different hopping pathways and molecule architecture parameters in the crystal (ri, γi, and θi) that determine the anisotropy of carrier transfer in organic materials.

Predicted Angular Resolution Anisotropic Mobility Curve of TETR, DCT, and HTP. Our first-principles simulation model focuses on predicting the intrinsic properties of organic semiconductors. The comparisons to experiment are all to the OFET measurement of hole mobilities at room temperature for high-purity, high-quality single crystals. The room-temperature mobilities of the highest-quality single-crystal OFETs are expected to approach their intrinsic values. This is because shallow traps can be “invisible” at room temperature since the characteristic time of charge release by shallow traps decreases exponentially with the temperature. For high temperatures where bandwidth and kBT become similar, mobilities become small, the mean free paths (ri in our model) reduce finally to one lattice constant, and the coherent description of band transport finally breaks down. Figures 3-5 show the predicted anisotropic hole mobilities of tetracene, 5,11-dichlorotetracene32 (DCT), and hexathiapentacene33,34 (HTP), respectively, for which there are not yet reported angular resolution anisotropic mobility measurements. The anisotropic mobility curves of tetracene are close to those of pentacene due to the similar crystal structure between them. In DCT and HTP crystals, the electronic couplings of P dimers are much larger than those of other dimers, and the combined effects from different dimers are small; thus, the anisotropic mobility curves are similar to those of rubrene. Comparisons of predicted mobility values to available experimental data are summarized in Table 1, where we see reasonable agreement with most experiments. The predicted lowest/highest hole mobilities of tetracene, 0.01/3.36 cm2 V-1 s-1, span the range of the experiments in particular directions, 0.15,41 1.3,42 and 2.443 cm2 V-1 s-1. We also predict hole mobilities for two new semiconductor compounds, DCT (0.006-2.22 cm2 V-1 s-1) and HTP (0.001-0.16 cm2 V-1 s-1), which are in reasonable agreement with experimental results (1.6 cm2 V-1 s-1 for DCT26 and 0.27 cm2 V-1 s-1 for HTP,27 respectively). The agreement

Anistropic Hole Mobilities in Organic Semiconductors between theory and the SCOFETs experimental results supports the use of the hopping-type mechanism for investigating charge transport at room temperature. By carefully looking into the experimental and theoretical results of anisotropic mobility, we can find the differences in the values and directions. The differences can be ascribed to these possible reasons: the influence of carrier traps and energy traps in experiments (perfect and ideal crystal in calculation), the neglected influences of lattice vibration on the electronic coupling,5 and the approximation5,38,39 of using eq 1 for hopping rate. However, we also find that the profiles of the mobility orientation function curves agree well with the profiles of most experimental measurements of anisotropic mobility.6,10,12,13 We suggest that cosine square shape is the intrinsic characteristic of the electrical anisotropy of organic materials. 4. Conclusion Summarizing, we develop a first-principles-based simulation model predicting anisotropic hole mobility of organic crystals with only crystal structures needed. These first-principles simulations based on a hopping-type transport mechanism lead to the first model µΦ(V,λ,r,θ,γ;Φ) for predicting angular resolution anisotropic mobility in the organic crystal. This quantitative function shows how the hole mobility correlates with the molecular packing (r, θ, and γ, the molecular architecture parameters of the crystal) and the underlying atomistic electronic properties (V, electronic coupling, and λ, reorganization energies), providing a guideline for “tailoring” new organic compounds for organic electronics. Acknowledgment. This work was partially supported by RG54/07 (Grant M52110068); K.-L.H. acknowledges partial support from NKBRSF (Grant 2007CB815202) and NSFC (20833008); W.A.G. acknowledges partial support from the FENA MARCO project. Supporting Information Available: Illustrations of the identical surrounding of random selected molecules in the rubrene crystals (Figure S1); re-evaluation of the experimental anisotropic mobility of pentacene in ref 9 (Figure S2); the derivation of eq 6; and computation details of internal reorganization energy and electronic coupling elements (Tables S3 and S4). This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Rogers, J. A. Science 2001, 291, 1502. (2) Forrest, S. R. Nature 2004, 428, 911. (3) Briseno, A. L.; Mannsfeld, S. C. B.; Ling, M. M.; Liu, S. H.; Tseng, R. J.; Reese, C.; Roberts, M. E.; Yang, Y.; Wudl, F.; Bao, Z. N. Nature 2006, 444, 913. (4) Karl, N. Synth. Met. 2003, 133, 649. (5) Coropceanu, V.; Cornil, J.; da Silva, D. A.; Olivier, Y.; Silbey, R.; Bredas, J. L. Chem. ReV. 2007, 107, 926. (6) Sundar, V. C.; Zaumseil, J.; Podzorov, V.; Menard, E.; Willett, R. L.; Someya, T.; Gershenson, M. E.; Rogers, J. A. Science 2004, 303, 1644. (7) Podzorov, V.; Menard, E.; Rogers, J. A.; Gershenson, M. E. Phys. ReV. Lett. 2005, 95, 226601.

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