J. Phys. Chem. C 2008, 112, 15537–15542
15537
How Important Is the Position of the Molecule between the Electrodes in Tuning Negative Differential Resistance Behavior? Sabyasachi Sen and Swapan Chakrabarti* Department of Chemistry, UniVersity of Calcutta, 92, A.P.C.Ray Road, Kolkata 700 009, India ReceiVed: May 29, 2008; ReVised Manuscript ReceiVed: July 21, 2008
We observed that the negative differential resistance feature of four different arrangements of a thiol-ended trimer unit of cis-polyacetylene is greatly influenced by the orientation of the molecule with respect to the electrodes, and surprisingly, the feature has completely been lost in one of the arrangements. We provide convincing theoretical evidence for the nonexistence of the negative differential resistance feature by monitoring the shift in the transmission peak across the bias window combined with molecular projected self-consistent Hamiltonian states analyses. 1. Introduction
I(V) )
The decrease in current due to an increase in voltage is unequivocally delineated as negative differential resistance (NDR). This NDR feature is of supreme interest in the fabrication of a number of modern electronic devices and finds remarkable applications in low-power memory devices, highfrequency oscillators, analog-to-digital converters, and logic circuits.1-4 Experimentally, NDR behavior has been examined in nanotube devices, nanoscale molecular junctions connected between a pair of metallic electrodes, 2′-amino-4-ethynylphenyl4′-ethynylphenyl-5′-nitro-1-benzenethiolate, diodes based on 1,4dibenzyl-C60 and zinc phthalocyanine hybrid material, etc.5-8 These innovations have greatly enhanced the inquisitiveness for exploring the NDR feature at the theoretical level also. In most of the theoretical as well as experimental analyses, the emergence of the NDR feature has been attributed to the effect of charging, polaron formation, conformational changes in molecules, twisting of the ring structure leading to conformational changes, bond fluctuation, development of charge density wave, intermolecular rotation in organic molecular systems, metal filaments or impurities within a molecular layer, and so forth.9-14 Moreover, it has also been noticed that current through a molecular rod is significantly affected depending upon the position of the anchor group with respect to the electrodes. This fundamental discovery inspired us to face a challenging question; i.e., how important is the position of a molecule between two electrodes in designing NDR material at the molecular level? In this paper, we present the results of a series of computer simulations, which clearly demonstrate that the orientations of the anchor group and that of the molecule have profound effect on the NDR behavior of the system. The current-voltage characteristics have been evaluated using the celebrated quantum transport formulation of Landauer and Bu¨ttiker.15,16 In the present study we have used density functional theory (DFT) combined with nonequilibrium Green’s function (NEGF) approach.17-19 The transport properties are estimated for the region comprising of two semi-infinite electrodes (L and R) coupled via the contact (C) region. With the use of NEGF formalism the current (I) through this system becomes * Corresponding author. E-mail:
[email protected]. Fax: 9133-23519755.
2e h
∫-∞∞ dε[nF(ε - µL) nF(ε - µR)]Tr[ΓL(ε)G†(ε)ΓR(ε)G(ε)](1)
In eq 1, G is the retarded Green’s function of the coupled system, ΣR and ΣL are the self-energies describing the coupling of the right and left electrode of the two-probe system to the rest of the semi-infinite electrodes.
ΓL(ε) ) i[ΣL(ε) - ΣL†(ε)]
(2)
ΓR(ε) ) i[ΣR(ε) - ΣR†(ε)]
(3)
and
V represents the applied bias with eV ) µL - µR. µL and µR are the electrochemical potentials of the left and right electrodes.
t(ε) ) [ΓR(ε)]1⁄2G(ε)[ΓL(ε)]1⁄2
(4)
is called left to right transmission amplitude. By using eqs 1 and 4 we obtain the Landauer-Bu¨ttiker formula for the conductance. 2. Computational Details We have chosen cis-polyacetylene as our model system since it is the simplest organic nanowire. It is well-known that polyacetylene has two geometric isomers, namely, cis- and trans-polyacetylene. Of the two conformations, the structure of trans-polyacetylene is thermodynamically more stable. Interestingly, the NDR effect has already been observed in the trimer unit of cis-polyacetylene at the theoretical level.19 To elucidate the effect of orientation/placing of the molecule as well as its anchor group we have chosen four different self-assembled arrangements (on the Au(111) surface) of a thiol-ended trimer unit of cis-polyacetylene. Corresponding two-probe configurations are presented in Figure 1a-d. The geometries of all the thiol-ended molecules are optimized with the inclusion of two Au(111) triangles at the end. Optimizations have been performed at the B3LYP20,21 level of theory using the Gaussian 03 suite of programs.22 The basis set used here is 6-31+G(d,p) for C, S, H atoms and LANL2DZ for Au atoms. In the optimized structure, C-S (thiol end group) distance is measured as 1.76 Å and Au-S distance is 2.39 Å. For the gold electrodes of the
10.1021/jp804754h CCC: $40.75 2008 American Chemical Society Published on Web 09/06/2008
15538 J. Phys. Chem. C, Vol. 112, No. 39, 2008
Sen and Chakrabarti
Figure 2. Zero-bias transmission spectra of the arrangement 1a at different levels of k-point sampling (perpendicular to the direction of transport).
Figure 1. (a-d) Four different arrangements of the two-probe system of the thiol-ended trimer unit of cis-polyacetylene self-assembled on Au(111) surface. (e) Selected view of the arrangement in (a).
two-probe systems a separate calculation has been performed in ATK 2.0.4.23 In all the two-probe configurations we have used the same Au-S distances as obtained from Gaussian 03. It is quite evident from Figure 1 that the 1a and 1c configurations are different from the other two configurations as the molecule is placed diagonally between the Au electrodes in configurations 1b and 1d. Moreover, a closer inspection of Figure 1, parts a and c, further reveals that they differ in the placement of the anchor group. A comparison of Figure 1, parts a and c, demonstrates that in the 1c configuration, sulfur atoms
are placed up and down with respect to the end carbon atoms of the cis-polyacetylene backbone. Similarly, in the 1b and 1d arrangements also the anchor groups are not placed in the identical manner. In the 1d configuration the sulfur atoms are positioned up and down with respect to the end carbon atoms of the trimer unit of cis-polyacetylene, whereas in 1b both the sulfur atoms are placed in the same side. In Figure 1e, we present a selected view of the arrangement shown in Figure 1a. It is also to be noted that Au(111) has five different adsorption sites and quantum transport phenomena are strongly affected by the adsorption sites at which molecule-electrode coupling takes place. To make our calculations consistent, we have chosen only the on-top sites in all the four configurations. As a consequence the role of adsorption sites has been ruled out, and this in turn will certainly help us to understand the importance of molecular/anchor group orientations in designing efficient NDR materials. All quantum transport calculations have been performed using a double-ζ polarized basis function in combination with the Perdew-Zungerfunctional24,25 andnorm-conservingTroullier-Martins pseudopotentials26 as implemented in ATK 2.0.4 software.23 Transport properties are estimated by employing 1 × 1 × 500 k-point sampling. We also analyzed the zero-bias transmission spectra at 4 × 4 × 500 k-point sampling and 6 × 6 × 500 k-point sampling and compared those results with 1 × 1 × 500 k-point sampling. It has been noticed that k-point sampling has little impact on the transmission spectra of the chosen system. Related variation is presented in Figure 2. In addition, we also checked the dependency of the transmission spectra over the k-point sampling in the direction of transport (z-direction) and found no appreciable change in the transmission spectra when the level of k-point sampling is changed from 1 × 1 × 100 to 1 × 1 × 500 and 1 × 1 × 1000. Figure 3 demonstrates the relevant variation. Moreover, to justify that our findings are not due to an artifact of analysis we calculated the band structure of the electrode with 1 × 1 × 500 k-point sampling and found
Orientation Effects in NDR
J. Phys. Chem. C, Vol. 112, No. 39, 2008 15539
Figure 5. Differential conductance (dI/dV) vs voltage (V) plot for all four arrangements of the two-probe system of the trimer unit of cispolyacetylene self-assembled on Au(111) surface.
Figure 3. Zero-bias transmission spectra of the arrangement 1a at different levels of k-point sampling along the z-direction (direction of transport).
Figure 4. Band structure of the Au electrode with 1 × 1 × 500 k-point sampling.
that continuity of the energy band across the Fermi level is restored. Relevant band structure is depicted in Figure 4. 3. Results and Discussion In Figure 5, we present a variation of the differential conductance (dI/dV) against voltage (V) for the two-probe configurations presented in Figure 1a-d. It is quite evident from Figure 5 that in the bias range of (2.1 to (2.45 V, dI/dV values are negative in case of arrangements 1a, 1b, and 1c, and the same is not perceptible in arrangement 1d. Therefore, Figure 5 quite clearly indicates that the arrangement 1d does not exhibit any NDR character, whereas this feature is strongly present in other three configurations, i.e., 1a, 1b, and 1c. Another interesting aspect of the dI/dV versus V curve is the emergence of a
peak at zero-bias voltage in case of 1a, 1b, and 1c. Surprisingly, this peak is absent in case of arrangement 1d. An explanation of this observation could be attained if we compare the zerobias transmission spectra of four different arrangements (see Figure S11 of the Supporting Information). From the zero-bias transmission spectra, we notice that in the 1d configuration there is no resonance peak at the Fermi energy level. This certainly denotes reduction in conductivity. Again in order to provide a more fundamental origin of the observed feature we performed an analysis of symmetrized fragment orbitals (SFOs) of different irreducible representations. SFO analysis suggests that in case of the 1d arrangement the eigenstate corresponding to the Fermi energy is resulted by the s orbital of Au, p orbitals of Au, C, S, and dz2, dxz orbitals of Au. In the 1a arrangement, contribution of d orbital is absent and the resonance peak at the Fermi energy level corresponds to the s orbital of Au and p orbitals of Au, C, S. The d orbital contribution is also absent in case of 1c, and the resonance peak is resulted by the s orbital of Au and p orbitals of C and S. In the 1b arrangement the resonance peak corresponds to the s orbital of Au, p orbitals of C, S, Au, and dz2 orbital of Au. A closer look at Figure 5 reveals that the dI/dV versus V characteristics of all the two-probe configurations are asymmetric in nature. Earlier, a mechanically controllable break junction experiment of Reichert et al.27 pointed out that contact asymmetry might cause asymmetry in I-V characteristics even in a perfectly symmetric molecule. Later on Zahid et al.28 gave a theoretical justification of the observed asymmetry in I-V characteristics of the spatially symmetric 9,10-bis((2′-p-mercaptophenyl)-ethinyl)-anthracene molecule. In the present investigation, the asymmetric nature of dI/dV versus V curves of 1b, 1c, and 1d is due to the inherent asymmetry imposed on the molecular structure. Although asymmetry in the molecular structure is not fairly noticeable in the arrangement 1a, the presence of a slight asymmetry can be captured if we look at the coordinates of the molecules, surface atoms, and electrodes (see the Supporting Information). In addition, in Figure 1e we present a selected view of the arrangement 1a. It is quite evident from Figure 1e that in the arrangement of 1a one of the sulfur atoms is slightly upward with respect to the nearby end carbon atom (numbered 9 and 1 in Figure 1e) and the other sulfur atom is slightly downward with respect to the nearby end carbon atom (numbered 10 and 8 in Figure 1e). Hence, a slight asymmetry is present in the optimized structure of 1a, and as a result of
15540 J. Phys. Chem. C, Vol. 112, No. 39, 2008
Figure 6. Transmission spectra of the arrangement 1a, at bias voltages of 0.0, (1.0, (2.0, (2.3, and (2.4 V, respectively. Dashed lines indicate the bias window. A positive bias corresponds to electron current from the left to the right electrode.
this the related I-V characteristics curve becomes slightly asymmetric. Apart from these, we also performed quantum transport calculations on a model perfectly symmetric geometry of a thiol-ended trimer unit of cis-polyacetylene. The symmetry of the molecular structure was verified by performing singlepoint calculation and found to be Cs type. The resulting twoprobe configuration is depicted in Figure S13 of the Supporting Information. From the results of quantum transport calculation we notice that the NDR feature is again present over a certain range of bias voltage and the I-V as well as dI/dV versus V curves are symmetric in nature (Figure S14 of the Supporting Information). Interesting to note is that in the symmetric configuration the magnitude of the NDR feature is smaller than that of 1a, 1b, and 1c. In order to explain the absence of NDR in the 1d arrangement, we initiated a systematic monitoring of the transmission resonance peaks across the bias window over the bias range in which the NDR feature is envisaged. According to the Landauer-Bu¨ttiker formula the current (I) is directly dependent on transmission amplitude. Consequently, the extent of transmission spectra within the bias window will determine the amount of current passing through a particular molecular system. Figures 6 and 7 illustrate transmission spectra corresponding to arrangements 1a and 1d, and the rest of the spectra for the other two arrangements are given in the Supporting Information. In Figure 6, we present the transmission spectra of 1a corresponding to the bias voltages of 0.0, (1.0, (2.0, (2.3, and (2.4 V, respectively. Transmission spectra of 1d, corresponding to the bias voltages of 0.0, (1.0, (2.2, (2.3, and (2.6 V, are presented in Figure 7. Figure 6 recommends that as the applied bias voltage is increased from (2.0 to (2.40 V, transmission resonance peaks contributing to the bias window gradually decrease which clearly is a manifestation of the appearance of the NDR feature in 1a. No such change is detectable in the transmission spectra of 1d over the bias range of (2.2 to (2.6 V and is quite evident from Figure 7. This in turn justifies the absence of NDR behavior in 1d. Furthermore, from the transmission spectra analysis we also notice that the peak at approximately 1.5 eV plays a crucial role in determining NDR behavior. A closer inspection of the transmission spectra presented in Figures 6 and 7 reveals that the resonance peak at 1.5 V (approximately) does not enter into
Sen and Chakrabarti
Figure 7. Transmission spectra of the arrangement 1d, at bias voltages of 0.0, (1.0, (2.2, (2.3, and (2.6 V, respectively. Dashed lines indicate the bias window. A positive bias corresponds to electron current from the left to the right electrode.
Figure 8. MPSH states of the arrangement 1a contributing to the bias window at the bias voltages of 2.3 and 2.4 V. A positive bias corresponds to electron current from the left to the right electrode.
the bias window (shown by the dotted line) at 0 and (1.0 V in the 1a and 1d configurations. In case of the 1d arrangement, this resonance peak contributes to the bias window between (2.2 and (2.6 V (Figure 7). The insertion of this resonance peak into the bias window is an indication of the enhancement of current through the molecular system. Unlike to the 1d configuration, in the 1a arrangement this resonance peak enters into the bias window above (2.4 V. As a result current through the 1a configuration is decreased between (2.0 and (2.4 V due to the gradual reduction of some other transmission peaks present inside the bias window. Therefore, the NDR feature is observed in the 1a arrangement over the bias range of (2.0 to (2.4 V. In order to provide a molecular origin of the absence of NDR behavior in the 1d configuration, we performed eigenstates analyses of molecular projected self-consistent Hamiltonian (MPSH).29 The finite MPSH matrix is expressed as
[
HL + ΣL VL 0 HC VR VL† 0
VR† HR + ΣR
]
(5)
where HL, HR, and HC are Hamiltonian matrices in the left electrode (L), right electrode (R), and contact region (C), respectively. Likewise, VL and VR are the interaction potentials between the L-C and L-R regions, respectively. In Figures 8 and 9, we provide some representative MPSH states of 1a and
Orientation Effects in NDR
J. Phys. Chem. C, Vol. 112, No. 39, 2008 15541 4. Conclusions
Figure 9. MPSH states of the arrangement 1d contributing to the bias window at the bias voltages of 2.3 and 2.4 V. A positive bias corresponds to electron current from the left to the right electrode.
1d, and other MPSH states are given in the Supporting Information. It is quite evident from these figures that, whereas MPSH states of 1d are completely delocalized, the feature is absent in MPSH states of 1a in the bias range of 2.3 and 2.4 V. As a result, the NDR feature has not been observed in 1d. It is also interesting to note that the magnitude of the current is relatively higher in 1d (also see the I-V curve of the Supporting Information) than that of the rest of the three arrangements. This feature is also attributed to the higher delocalization of MPSH states in 1d. We also investigated the nature of the state associated with the resonance peak at approximately 1.5 eV. For our analysis we choose only the states associated with in the 1a and 1d configurations. At the zero-bias voltage this peak corresponds to the MPSH state 237 in 1a and 235 in 1d. From Figure S12 of Supporting Information, it is evident that both the MPSH states are delocalized and can contribute to additional current if they enter into the bias window. In addition, we also perform population analysis of the respective MPSH states. Our analysis suggests that the MPSH state 237 is mainly contributed by p orbitals of sulfur atoms and that of carbon atoms having delocalized π character. In case of 1d, MPSH state 235 is an admixture of the s orbital of Au and p orbitals of carbon and sulfur. Finally, if we look at the results of SFO analyses on the MPSH states within the bias window then it turns out that in case of 1a MPSH states corresponding to the Fermi energy level and 1 eV less than Fermi energy are linear combinations of s and p orbitals only and all other MPSH states contributing to the bias window are admixture of s, p, dz2, and dyz orbitals. However, for 1d all the MPSH states within the bias window are the linear combinations of s, p, dxz, and dz2 orbitals. When similar analysis is performed in case of arrangement 1b, we notice that relevant MPSH states are contributed by s, p, and dz2 orbitals and in case of 1c contribution of the d orbital is completely absent. Since the molecule between the electrodes lies in the xz-plane and the lobes of dxz also reside in between the x- and z-axes, as a consequence it can make a better overlap with the orbitals of sulfur which can give rise to delocalized conducting states. A closer inspection of SFO results suggests that except in the 1d configuration, the dxz orbital is not participating in the formation of MPSH states within the bias window which in turn plays a decisive role in tuning the NDR feature of cis-polyacetylene in four different configurations.
In summary, we have critically examined the role of orientation of a molecule on its NDR behavior. We calculated current voltage characteristics, bias-dependent transmission spectra, and MPSH states for the trimer unit of cis-polyacetylene at four different configurations. Our investigation suggests that the relative strength of NDR is tremendously affected by the orientation of the molecule with respect to the position of the electrodes and remarkably the NDR feature is totally missing in one of the arrangements (Figure 1d). The exclusion of NDR behavior in arrangement 1d has been explained by scrutinizing the shift in transmission resonance peaks across the bias window at various bias voltages. We also offer an explanation on the molecular origin of the absence of the NDR feature through an analysis of MPSH states contributing to the bias window at specific bias voltages. We conclude that participation of the specific d orbital in the formation of MPSH states over the bias window is very crucial depending upon the plane at which the molecule resides. Finally, to provide a more precise physical picture, we will consider the transfer integral between molecule and electrode30 and will try to justify how the NDR behavior is affected by the transfer integral in our future correspondence. Acknowledgment. We convey our special thanks to Atomistix Inc. for allowing us to use ATK 2.0.4 for electronic transport calculations. The financial support from the DST, Government of India (under the FIST Program) to purchase the GAUSSIAN 03 program is gratefully acknowledged. S. Sen acknowledges Professor A. Guha, Director, JIS College of Engineering. S. Sen also thanks P. Seal and Dr. S. Banerjee for providing necessary help in computational purpose. Supporting Information Available: Optimized Cartesian coordinates of the all four two-probe systems, current-voltage graph of all four arrangements, transmission spectra of all four arrangements (2D view), detailed MPSH analysis of all four configurations, selected view of the arrangement in 1a, zerobias transmission spectra of all four arrangements, MPSH states corresponding to the peak at approximately 1.5 eV in arrangements 1a and 1d, current-voltage characteristic and differential conductance versus voltage plot of the symmetric arrangement. This material is available free of charge via the Internet at http:// pubs.acs.org. References and Notes (1) Xu, B. Q.; Li, X. L.; Xiao, X. Y.; Sakaguchi, H.; Tao, N. J. Nano Lett. 2005, 5, 1491. (2) Csa´thy, G. A.; Tsui, D. C.; Pfeiffer, L. N.; West, K. W. Phys. ReV. Lett. 2007, 98, 066805. (3) Broekaert, T. P. E.; Brar, B.; van derWagt, J. P. A.; Seabaugh, A. C.; Morris, F. J.; Moise, T. S.; Beam, E. A.; Frazier, G. A. IEEE J. Solid-State Circuits 1998, 33, 1342. (4) Liu, R.; Ke, S.-H.; Baranger, H. U.; Yang, W. J. Am. Chem. Soc. 2006, 128, 6274. (5) Xue, Y.; Datta, S.; Hong, S.; Reifenberger, R.; Henderson, J. I.; Kubiak, C. K. Phys. ReV. B 1999, 59, 7852 (R) (6) Khondaker, S. I.; Yao, Z.; Cheng, L.; Henderson, J. C.; Yao, Y.; Tour, J. M. Appl. Phys. Lett. 2004, 85, 645. (7) Gergel, N.; Majumdar, N.; Keyvanfar, K.; Swami, N.; Harriott, L. R.; Bean, J. C.; Pattanaik, G.; Zangari, G.; Yao, Y.; Tour, J. M. J. Vac. Sci. Technol., A 2005, 23, 880. (8) Lin, J.; Zheng, M.; Chen, J.; Gao, X.; Ma, D. Inorg. Chem. 2007, 46, 341. (9) Seminario, J. M.; Zacarias, A. G.; Derosa, P. A. J. Chem. Phys. 2002, 116, 167. (10) Galperin, M.; Ratner, M. A.; Nitzan, A. Nano Lett. 2005, 5, 125. (11) Bauschlicher, C. W., Jr.; Ricca, A. Phys. ReV. B 2005, 71, 205406. (12) Larade, B.; Bratkovsky, A. M. Phys. ReV. B 2005, 72, 035440.
15542 J. Phys. Chem. C, Vol. 112, No. 39, 2008 (13) Franke, K. J.; Schulze, G.; Henningsen, N.; Ferna´ndez-Torrente, I.; Pascual, J. I.; Zarwell, S.; Ru¨ck-Braun, K.; Cobian, M.; Lorente, N. Phys. ReV. Lett. 2008, 100, 036807. (14) Chen, J.; Reed, M. A.; Rawlett, A. M.; Tour, J. M. Science 1999, 286, 1550. (15) Landauer, R. IBM J. Res. DeV. 1957, 1, 223; Phys. Lett. 1981, 85A, 91. (16) Bu¨ttiker, M. Phys. ReV. Lett. 1986, 57, 1761. (17) Zhou, Y.-h.; Zheng, X.-h.; Xu, Y.; Zeng, Z. Y. J. Chem. Phys. 2006, 125, 244701. (18) Lawson, J. W.; Bauschlicher, C. W., Jr. Phys. ReV. B 2006, 74, 125401. (19) Sen, S.; Chakrabarti, S. J. Phys. Chem. C 2007, 112, 1685. (20) Becke, A. D. J. Chem. Phys. 1993, 98, 1372. (21) Lee, C.; Yang, W.; Parr, R. G. Phys. ReV. B 1988, 37, 785. (22) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J. J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.;
Sen and Chakrabarti Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian 03; Gaussian, Inc.: Pittsburgh, PA, 2003. (23) www.atomistix.com. (24) Perdew, J. P.; Zunger, A. Phys.ReV.B 1981, 23, 5048. (25) Ceperley, D. M.; Alder, B. J. Phys. ReV. Lett. 1980, 45, 566. (26) Troullier, N.; Martins, J. L. Phys. ReV. B 1991, 43, 1993. (27) Reichert, J.; Ochs, R.; Beckmann, D.; Weber, H. B.; Mayor, M.; Lo¨hneysen, H. v. Phys. ReV. Lett. 2002, 88, 176804. (28) Zahid, F.; Ghosh, A. W.; Paulsson, M.; Polizzi, E.; Datta, S. Phys. ReV. B 2004, 70, 245317. (29) Staykov, A.; Nozaki, D.; Yoshizawa, K. P. J. Phys. Chem. C 2007, 111, 11699. (30) Datta, S. Electronic Transport in Mesoscopic Systems; Cambridge University Press: Cambridge, 1995.
JP804754H
