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Effect of Applied Voltage on the Geometrical and Electronic Structures of Dipyrimidinyl-Diphenyl Diblock as a Molecular Diode: A DFT Study Morad M. El-Hendawy,† Ahmed M. El-Nahas,*,† and Mohamed K. Awad‡ Chemistry Department, Faculty of Science, El-Menoufia UniVersity, Shebin El-Kom, Egypt, and Chemistry Department, Faculty of Science, Tanta UniVersity, Tanta, Egypt ReceiVed: July 27, 2010; ReVised Manuscript ReceiVed: October 24, 2010
The effect of an external electric field on the geometrical and electronic structures of the nonsymmetrical dipyrimidinyl-diphenyl (DPDPh) molecule has been investigated using B3LYP density functional theory (DFT) in the absence and presence of an electric field in an attempt to rationalize its rectifying behavior. A detailed comparison with the isoelectronic symmetrical tetraphenyl analogue was used to justify the rectification ability of DPDPh. The calculations showed that the insertion of nitrogen atoms in the conjugated backbone modifies the electronic structure of DPDPh compared to its parent hydrocarbon and then affects its electrical properties. Consequently, the HOMO level of DPDPh under positive voltage became closer to the Fermi level of the electrode than it does under negative bias giving rise to a rectifying character. 1. Introduction It is known that direct current (dc) is needed for the functioning of many electronic devices. Dc is derived from the alternating current (ac) using rectifiers composed of one or more diodes. Diodes are components that allow current to pass in one direction. The fabrication of silicon devices down to sub100 nm is becoming undesirable because of expensive cost and high power consumption, and a serious search for cheaper alternatives is needed.1 A promising starting point is to mimic the electronic function of the silicon-based rectifiers. This is first envisioned by Aviram and Ratner in 1974.2 Since their first theoretical proposal of using organic molecules as a molecular diode, many experiments and theoretical calculations have been conducted to design new molecules with the desired electrical properties.3-46 The Aviram-Ratner (AR) molecular diode consists of donor and acceptor π-conjugated segments separated by an insulating σ-bridge in which the forward current flows from the acceptor to the donor. Recently, a new class of molecular diodes called diblock has been synthesized and showed a pronounced rectification.5-8 The investigated molecular diode dipyrimidinyldiphenyl (DPDPh) diblock, Figure 1, consists of an electronrich diphenyl block covalently bonded to an electron-deficient dipyrimidinyl subunit. Due to the electron demand difference at its ends, DPDPh resembles the p-n junction of a conventional semiconductor device which makes it attractive for molecular diode studies.6-8 Dı´ez-Pe´rez et al.8 used the acassisted scanning tunneling microscope (STM) break-junction method9 to measure the current-voltage (I-V) characteristics of the nonsymmetrical DPDPh diblock and its symmetric tetraphenyl (TPh) analogue, each bridged between two gold electrodes.8 They found that the DPDPh has stronger rectification than TPh. In addition, TPh and DPDPh showed symmetric and asymmetric I-V curves, respectively. Very recently,23,24 two separate theoretical studies calculated the I-V curves of terphenyl23 (similar to TPh) and DPDPh24 and showed analogous profiles to the measured ones.8 * Corresponding author,
[email protected] (Ahmed M. El-Nahas). † El-Menoufia University. ‡ Tanta University.
A deeper understanding of transport properties through the suggested molecules is needed before synthesizing new candidates. Quantum chemical calculations are very helpful tool in this respect. Since the first proposal of a molecular rectifier by Aviram and Ratner,2 many theoretical studies have been done and contributed to the progress of molecular electronics.10-21 Although the electrical properties of molecular devices with a finite number of molecules between metallic electrodes have been measured experimentally, it is often hard to give a full interpretation of the observed properties. A variety of theoretical approaches have been applied to model the electron transport in molecular diodes.10-21 Most of these studies did not take into account the molecular interactions such as electrostatic and van der Waals interactions, solvent, and external electrical field effect. The latter is a very important factor because the application of small volts on the functional molecule may greatly affect its geometrical and electronic structures and modulate the passing current. This paper aims at the investigation of the effect of external electric field on geometrical parameters and electronic structures of the asymmetric DPDPh diblock molecular diode and the symmetric TPh. This effect will reflect their electrical properties. The studied parameters and properties include bond length, twist angle, Mulliken charges, dipole moment, and energies and spatial distributions of the frontier molecular orbitals (FMO: HOMO and LUMO). Some of these properties are used to interpret the stronger rectification of DPDPh compared to TPh. Moreover, symmetric and asymmetric current-voltage (I-V) curves of TPh and DPDPh, respectively, were explained. 2. Computational Details All the calculations were performed using the Gaussian 03 program.47 The investigated molecules were fully optimized using the hybrid B3LYP functional48-50 with 6-31G(d) basis set. To study the effect of external electric field on geometrical and electronic structures of DPDPh and TPh, the terminal sulfur atoms of the fully optimized structure were fixed in space to simulate the connection with the electrodes.17 Then the structure was optimized under the effect of uniform electric field of different strengths (0-0.005 au). These values correspond to
10.1021/jp107014g 2010 American Chemical Society Published on Web 11/19/2010
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Figure 1. B3LYP/6-31G(d) optimized structures of (a) DPDPh and (b) TPh. The numbering scheme for atoms is shown on the structure. A, B, C, and D symbols are labels for the four rings. The presented symbols Φ1, Φ2, and Φ3 in the figure are the torsion angles between each two adjacent rings. The uniform external electrical field aligned along the two terminal sulfur atoms in the positive x-axis direction is presented. All labels are the same for the TPh molecule.
TABLE 1: Spatial Distruibtion of HOMO-1, HOMO, LUMO, and LUMO+1 of TPh and DPDPh Molecules at Zero Bias
0.0-0.26 V/Å or to 0.0-0.4.90 V (1 au ) 51.42 V/Å), where the effective molecular lengths (S25 · · · S26 distance) for DPDPh and TPh are 19.07 and 19.37 Å, respectively. The positive and negative bias voltages are applied along both positive and negative x molecular axis, respectively, of TPh and DPDPh molecules; see Figure 1. Frequency calculations of the optimized geometries have been carried out at the same level of theory to ensure that the optimized structures are minima on the potential energy surface of the respective molecules. The calculated frequencies are very close to the previously reported values.51,52 For example, the multiple peaks found in the region 3030-3170 cm-1 agree with the experimental finding of 3035 and 3061 cm-1. Moreover, the aromatic C-C ring stretching frequencies located at 1010, 1405, and 1465 cm-1 also match well with experiment (1003, 1408, and 1477 cm-1). 3. Results and Discussion It has been reported that electron transport characteristics of given molecular systems depend on the intrinsic properties of these molecules.20,22 These properties include length, conformation, dipole moment, HOMO-LUMO energy gap (Eg), and the alignment of this gap with the metal Fermi level. Therefore, investigation of geometrical parameters and electronic structures of DPDPh and TPh under external electric field will be helpful in understanding the different rectification behaviors of these two molecules and in designing novel molecular devices. 3.1. Geometrical and Electronic Structures at Zero Bias Voltage. The optimized structures of DPDPh and TPh are displayed in Figure 1. The calculations show that both DPDPh and TPh adopted twisted structures. The phenyl rings of TPh are mutually twisted by 38°. The pyrimidinyl rings of DPDPh
are coplanar and twisted by 36°, with respect to the diphenyl block. The average Mulliken atomic charges (MAC) at the nitrogen atoms of terminal pyrimidinyl moiety amounts to -0.460 e, while those of the inner pyrimidinyl ring bear an average charge of -0.410 e. Moreover, the sulfur atom linked to pyrimidinyl block carries a charge of 0.017 e, while that connected to a diphenyl block acquires a charge of -0.017 e. On the contrary, sulfur atoms connected to TPh carry a charge of -0.023 e. This reflects different electronic demand of the segments of DPDPh. On the basis of these charge separations, the DPDPh molecule gained a strong dipole moment (5.4 D) at zero bias directed toward dipyrimidinyl block. On the other hand, TPh has a dipole moment of 1.7 D. This indicates that DPDPh exhibits a significant polarization compared to TPh. As depicted in Table 1, the HOMO of DPDPh slightly tends to be localized on the diphenyl segment, while the LUMO tends to localized on the dipyrimidinyl segment. This property differs notably from that of the AR rectifier in which the HOMO and LUMO are almost entirely localized on the donor and acceptor parts, respectively.13 Contrarily, HOMO-1, HOMO, LUMO, and LUMO+1 of the TPh block exhibit highly symmetric delocalization over the whole molecular skeleton. Accordingly, TPh behaves like a molecular wire. The most pronounced difference between the two molecular systems is LUMO+1, where it is highly delocalized over TPh and highly localized onto the dipyrimidinyl segment of DPDPh. 3.2. Effect of Voltage on Geometrical and Electronic Structures. The variations in MAC of non-hydrogen atoms of TPh and DPDPh, with respect to zero bias voltage, against different voltages are illustrated in Figures 2 and 3, respectively. Figure 2 shows that the most affected atoms in TPh are those
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Figure 2. Deviation of Mulliken atomic charge (MAC) on each non-hydrogen atom along the conjugation backbone of TPh under (a) positive bias and (b) negative bias with respect to those of zero bias. Atomic indices are given in Figure 1.
Figure 3. Deviation of Mulliken atomic charge (MAC) on each non-hydrogen atom along the conjugation backbone of DPDPh under (a) positive bias and (b) negative bias with respect to those of zero bias. The atom indices of the acceptor part range from 1 to 12 plus 25, while those of the donor part range from 13 to 24 plus 26.
near the high potential end. Under a positive bias, the MAC of atoms incorporated in the terminal phenyl ring (D) exhibit the highest variations. Upon reversing the applied voltage, the highest change is shifted toward the terminal pyrimidinyl ring (A). This explains a symmetrical effect of electric field on the charge distribution of TPh. Contrarily, applying either positive or negative voltage on DPDPh shows that the highest variations in MAC appear on ring D. This indicates the asymmetrical effect of the electric field on the charge distribution of DPDPh. The midpeak observed at atom C13 in Figure 3b can be ascribed to the fact that it is the closest atom of the donor part to the electron-deficient acceptor part. Therefore, the response of C13
to electron demand is expected to be higher than that for other neighboring atoms. The different intrinsic character of TPh and DPDPh, which appears during the response to the electric field in both directions, could explain their different rectification behavior.8 It is expected that these changes in the charge distribution under the bias effect will cause a variation in the bond lengths. Figures 4 and 5 display these variations for some selected bonds of TPh and DPDPh molecules, respectively. The change in bond lengths is an important parameter for understanding the optical and electrical prosperities of molecules.53,54 The bond length variation in both molecules shows a zigzag path over
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Figure 4. Deviation of the bond length (DBL) of some selected bonds along the conjugated backbone of TPh under (a) positive bias and (b) negative bias with respect to those of zero bias.
Figure 5. Deviation of the bond length (DBL) of some selected bonds along the conjugated backbone of DPDPh under (a) positive bias and (b) negative bias with respect to those of zero bias.
the range of the applied biases with remarkable changes in DPDPh compared to TPh. For example under positive bias, the bonds of the acceptor part (dipyrimidinyl block) in the case of DPDPh exhibit larger variation compared to the donor part. Only the C22-S26 bond in the donor part shows a significant change. Upon reversal of the applied voltage, all bonds in the conjugation backbone are significantly changed. On the other hand, the variation in bond lengths of TPh seems relatively small due to the weak polarization in the molecular backbone. However, the variation in bond lengths under positive voltages is somehow comparable to that under negative biases due to the symmetry of the molecule. From these results we can conclude that DPDPh is more sensitive to the electric field due to its asymmetry.
The twist angles (Φ1, Φ2, and Φ3) also exhibit a bias dependence; see Table 2. In the case of TPh, a general slight decrease in Φ1, Φ2, and Φ3 angles are observed by about 2.66, 1.91, and 1.40°, respectively, under either positive or negative voltage. On the other hand, DPDPh shows a different behavior. Sweeping applied voltage from -4.90 to 4.90 V, the Φ1 angle increases steadily from -0.31 to 1.30° except for the point at zero voltage. Meanwhile, Φ2 and Φ3 angles increase until they reach a maximum at 0.98 and 1.96 V by 5.39 and 4.34°, respectively, then decrease with raising the voltage. It is apparent that replacement of the diphenyl block of TPh by dipyrimidinyl reduces all torsion angles between rings especially the Φ1 angle which makes rings A and B coplanar. These variations can be attributed to two factors, steric repulsion and π-electron con-
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TABLE 2: External Bias Dependence of the Torsion Angles (deg), Φ1, Φ2, and Φ3, between Two Adjacent Rings of TPh and DPDPh TPh
DPDPh
bias voltage (V)
Φ1
Φ2
Φ3
Φ1
Φ2
Φ3
4.90 3.92 2.94 1.96 0.98 0.00 -0.98 -1.96 -2.94 -3.92 -4.90
34.84 35.93 36.78 37.59 37.85 37.70 37.75 37.58 36.64 36.12 35.24
-36.12 -36.72 -37.27 -37.44 -37.76 -38.18 -38.15 -38.11 -37.69 -36.96 -36.42
-37.53 -37.88 -38.15 -38.01 -38.14 -38.49 -38.32 -38.21 -37.75 -37.02 -36.65
-1.30 -0.93 -0.72 -0.79 -0.78 -0.41 -0.70 -0.48 -0.42 -0.38 -0.31
-34.60 -35.32 -35.86 -36.22 -36.27 -35.64 -35.64 -34.83 -33.87 -32.69 -30.88
35.54 36.09 36.36 36.40 36.37 35.77 35.76 35.27 34.49 33.43 32.06
jugation.17 If the latter factor dominates, the molecule shifts toward coplanarity. However, the predominance of the steric factor gives a twisted configuration. The response of molecular configurations to external electric field is beneficial in modeling the electrical behavior of molecular electronic devices.55,56 Introducing asymmetry to the TPh molecule by replacement of a diphenyl segment with a dipyrimidinyl one generates a relatively strong built-in electric field (dipole) outbound from diphenyl to dipyrimidinyl.8 Due to different electron affinities of the two segments, DPDPh resembles a traditional p-n junction. In order to quantify the built-in electric field under the action of electric field, we calculated the dipole moments of DPDPh and TPh when they integrated into the electric circuit. Figure 6 depicts the bias dependence of the change in dipole moments of DPDPh and TPh. The dipole moment change is symmetric around zero bias for TPh. The dipole moment of TPh amounts to 1.70 D, and its direction is almost perpendicular to the molecular axis. However, the molecule is symmetrical with respect to the two electrodes and the effect of the applied field on the dipole moment on moving from zero bias in the negative direction should be the same as moving to the positive direction. On the other hand, DPDPh behaves differently where a monotonous increase is observed with raising negative bias, while first decrease and then increase in the dipole moment variation is recorded in the positive direction of voltage. This observation exactly agrees with the experimental findings reported by Dı´ez-Pe´rez et al. for DPDPh.8 The applied electric
Figure 6. External bias dependence of the deviation in dipole moment of DPDPh and TPh.
Figure 7. External bias dependence of (a) HOMO and LUMO energies, and (b) HOMO-LUMO energy gap (Eg) of DPDPh and TPh molecules.
field in the positive direction forces electrons to move from dipyrimidinyl to diphenyl moiety (i.e., the opposite direction of the built-in electric field), and thus the dipole moment reduces initially. Further increase in positive bias increases the dipole moment again. Therefore, the theoretical calculations could justify the pronounced rectification behavior of the nonsymmetric DPDPh diblock compared to the symmetric TPh block. The changes in the electrical behavior of DPDPh and TPh shed some light onto the importance of insertion of nitrogen atoms in changing the electronic function of molecular devices. These results agree with He et al.23 who reported that the electrical rectification from introducing asymmetry is attributed to the intrinsic dipole moment which makes the molecule responses different when the direction of electric field changes. 3.3. Effect of Voltage on FMOs. It is known that energies of FMOs and Eg are good indices for measuring the response of molecules to external electric field.20,22 Figure 7 displays variation of FMO energies and Eg versus voltages. In the case of TPh, upon increase in the applied bias in both directions the HOMO and LUMO energies as well as Eg change symmetrically. However, the asymmetric DPDPh diblock shows unsymmetrical behavior. Under negative bias, LUMO energy decreases linearly, while increasing gradually in the case of positive bias. The HOMO energy shows a little response until (0.98 V and then increases in both directions with rapid increase in the positive voltage. The asymmetrical electrical properties of DPDPh, with respect to TPh, reflect the p-n
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Figure 8. Schematic representation for the electron transport process through a diblock molecular diode wired in the circuit. Where the left electrode is the STM tip as ground terminal and the right electrode is the substrate as positively (i) or negatively (ii) biased terminal according to the technique used in ref 8. (i) Applying positive bias facilitates the electron transport from the cathode (the tip electrode) into the anode (the sample electrode). (ii) Applying negative voltage impedes the current flow in the opposite direction (i.e., from the sample electrode to the tip).
TABLE 3: Spatial Distribution of HOMO and LUMO of TPh and DPDPh under Different Biases
character of DPDPh as a molecular diode. As displayed in Figure 7b, Eg of TPh shows a symmetrical variation with respect to zero bias compared to DPDPh which gives an asymmetric one. In the positive bias region, the Eg for DPDPh decreases slowly compared to negative bias. The difference in the response of FMO energies and Eg of the two molecules to the external electric field reflects the effect of insertion of nitrogen atoms in the conjugated backbone on the electrical properties of molecular electronic devices. This result does not match with He et al.23
who observed narrower Eg under positive bias than under negative bias. This may be attributed to the differences in the nature of the asymmetric molecules, where their molecule has the form D-π-A (i.e., the donor and the acceptor part is separated by π bridge) and ended with two acetylenic groups which extends the conjugation length, while in our molecule there is no π bridge. As an approximation, the Fermi level is defined as the midway between the HOMO and LUMO energy levels.57 Previous
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studies reported that the electron tunneling through the HOMO of a molecular system is the key factor for electron transport, because the Fermi level of metallic electrode is closer to HOMO than LUMO.6,7,58-62 It is worth mentioning that the rapid increase in the HOMO energy of DPDPh under a positive bias brings HOMO more closer to the Fermi level. Therefore, it is expected that the current that passes through DPDPh under positive biases should be more than that which passes through under negative ones. Then we could justify the pronounced rectification function of DPDPh and agree with the experimental observation.8 On the other hand, the HOMO of TPh shows a symmetrical soaring up under both negative and positive biases, and consequently it has a symmetric I-V curve.8 This also explains why DPDPh shows a pronounced rectification compared to TPh. Although one can guess that the generated built-in electric field may impede the current flow in the positive direction. However, the closeness of the HOMO level to the Fermi level would facilitate current flow in this direction. Hence, we could conclude that the passing current due to tunneling process dominates easily over the opposite current due to the intramolecular charge transfer. The aforementioned interpretations could be clearly illustrated in a schematic representation of the electron transport process through the diblock molecular diode wired in the circuit under the effect of positive and negative biases; see Figure 8. Applying a certain voltage on the circuit will create some perturbation on the energy of the Fermi level (EFermi) of both anode and cathode, which develops a potential difference between them and, hence, affects the rate of the electron tunneling process. For example, upon applying a positive voltage +υ V, the EFermi of the sample electrode goes down by υ V (i.e., the energy becomes EFermi - υ) and then acts as an anode relative to the tip electrode (EFermi) which acts as a cathode. Then the former can resonate with the HOMO level of the molecule, Figure 8(i). Hence, the current flow in the forward direction is favored. On the contrary, applying a negative voltage -υ V causes the EFermi of the sample electrode to go up by υ V (i.e., the energy becomes EFermi + υ) and then acts as a cathode relative to the tip which acts as an anode. In such cases, neither the HOMO nor the LUMO level is aligned with the cathode, and hence the backward current is impeded in this direction, Figure 8(ii). The electron transport through a molecular system can be better seen from the change in spatial orientation of frontier molecular orbitals across the molecule.17,56,63,64 Table 3 displays the spatial distributions of FMOs of TPh and DPDPh under different biases. At first glance, the HOMO and LUMO surfaces of DPDPh are highly affected by the applied voltage compared to TPh. Applying 0.98 V bias leads to full delocalization of the HOMO with a little change in the LUMO. Further increase in bias shifts the HOMO and LUMO more toward dipyrimidinyl and diphenyl segments, respectively. This is due to the applied field at 0.98 V overcoming first the obstacle due to the intrinsic charge transfer from diphenyl to dipyrimidinyl segments. Upon reversing the applied voltage, the distribution of FMOs also reversed, but strongly. It was observed that at low positive or negative voltages, the spatial distributions of the HOMO and LUMO are greatly changed, while at higher voltages the change is very limited. This could be used to explain the I-V curve of the studied molecules. It is assumed that the unaltered electronic structure of the molecule under applied voltage is the reason behind the fixed rate current flow (i.e., saturation) at higher voltage. For example, by sweeping voltage from 0 to 2.98 V, both the HOMO and LUMO of the DPDPh molecule are affected
El-Hendawy et al. greatly, after that no change. On the other hand, under negative voltage they are unaffected from -1.96 V. This could explain qualitatively the higher flow of forward current through DPDPh under positive voltage than that under negative voltage. 4. Conclusions We have studied the effect of a homogeneous electric field on the geometrical and electronic structures of the asymmetric dipyrimidinyl diblock (DPDPh) at the B3LYP/6-31G(d) level of theory. A detailed comparison with its symmetric isoelectronic tetraphenyl (TPh) block was used to justify the pronounced rectification behavior of DPDPh. The results obtained can be summarized as follows: 1. Compared to TPh, the introduced asymmetry affected charge distribution, and therefore DPDPh became more sensitive to the electric field. The response of DPDPh to different voltage is beneficial in modeling the electrical behavior of molecular electronic devices. 2. We quantified the built-in electric field under the applied EF through the dipole moment. For the DPDPh diblock, the closeness of the HOMO level to the Fermi level of the electrode under positive biases compared to negative ones explains why the forward current passes easier than the reverse one which agrees with the experimental observation. 3. Finally, the results shed some light onto the importance of nitrogen atoms in modification of the electrical properties of conjugated molecular electronic devices which is useful for the design of new molecular electronic devices, particularly the p-n molecular diode. References and Notes (1) Kumar, M. J. Recent Pat. Nanotechnol. 2007, 1, 51. (2) Aviram, A.; Ratner, M. A. Chem. Phys. Lett. 1974, 29, 277. (3) Introducing Molecular Electronics; Lecture Notes in Physics; Cuniberti, G., Fagas, G., Richter, K., Eds.; Springer: Berlin, 2005; Vol. 680. (4) Metzger, R. M. Chem. ReV. 2003, 103, 3803. (5) Jiang, P.; Morales, G. M.; You, W.; Yu, L. Angew. Chem., Int. Ed. 2004, 43, 4471. (6) Morales, G. M.; Jiang, P.; Yuan, S.; Lee, Y.; Sanchez, A.; You, W.; Yu, L. J. Am. Chem. Soc. 2005, 127, 10456. (7) Lee, Y.; Carsten, B.; Yu, L. Langmuir 2009, 25, 1495. (8) Diez-Perez, I.; Hihath, J.; Lee, Y.; Yu, L. P.; Adamska, L.; Kozhushner, M. A.; Oleynik, I. I.; Tao, N. J. Nat. Chem. 2009, 1, 635. (9) Xu, B.; Tao, N. J. Science 2003, 301, 1221. (10) Dutta, S.; Lakshmi, S.; Pati, S. K. Bull. Mater. Sci. 2008, 31, 353. (11) Zhenyu, L.; Jing, H.; Qunxiang, L.; Jinlong, Y. Sci. China, Ser. B: Chem. 2008, 51, 1159. (12) Brady, A. C.; Hodder, B.; Martin, A. S.; Sambles, J. R.; Ewels, C. P.; Jones, R.; Briddon, P. R.; Musa, A. M.; Panettaf, C. A.; Mattern, D. L. J. Mater. Chem. 1999, 9, 2271. (13) Majumder, C.; Mizuseki, H.; Kawazoe, Y. J. Phys. Chem. A 2001, 105, 9454. (14) Stokbro, K.; Taylor, J.; Brandbyge, M. J. Am. Chem. Soc. 2003, 125, 3674. (15) Lu, J. Q.; Wu, J.; Chen, H.; Duan, W.; Gu, B. L.; Kawazoe, Y. Phys. Lett. A 2004, 323, 154. (16) Karzazi, Y.; Cornil, J.; Bredas, J. L. Nanotechnology 2003, 14, 165. (17) Li, Y. W.; Yao, J. H.; Yang, C. L.; Zhong, S. K.; Yin, G. P. Mol. Simul. 2009, 35, 301. (18) Mizuseki, H.; Niimura, K.; Majumder, C.; Kawazoe, Y. Comput. Mater. Sci. 2003, 27, 161. (19) Parr, R. G.; Yang, W. Density Functional Theory of Atoms and Molecules; Oxford University Press: New York, 1989. (20) Salomon, A.; Cahen, D.; Lindsay, S.; Tomfohr, J.; Engelkes, V. B.; Frisbie, C. D. AdV. Mater. 2003, 15, 1881. (21) Majumder, C.; Briere, T. M.; Mizuseki, H.; Kawazoe, Y. J. Chem. Phys. 2002, 117, 2819. (22) Oleynik, I. I.; Kozhushner, M. A.; Posvyanskii, V. S.; Yu, L. Phys. ReV. Lett. 2006, 96, 096803. (23) He, H.; Randey, R.; Mallick, G.; Karna, S. P. J. Phys. Chem. C 2009, 113, 1575.
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