Enhanced rectifying performance by asymmetrical gate voltage for BDC₂₀ molecular devices

Abstract

By applying the asymmetrical gate voltage on the 1,4-bis (fullero[c]pyrrolidin-1-yl) benzene BDC₂₀ molecule, we investigate theoretically its electronic transport properties using the density functional theory and nonequilibrium Green's function formalism for a unimolecule device with metal electrodes. Interestingly, the rectifying characteristic with very high rectification ratio, 91.7 and 24.0, can be obtained when the gate voltage is asymmetrically applied on the BDC₂₀ molecular device. The rectification direction can be tuned by the different gate voltage applying regions. The rectification behavior is understood in terms of the evolution of the transmission spectrum and projected density of states spectrum with applied bias combined with molecular projected self-consistent Hamiltonian states analyses. Our finding implies that to realize and greatly promote rectifying performance of the BDC₂₀ molecule the variable gate voltage applying position might be a key issue.

---

1 Introduction

In recent years, various molecular devices^1–14 with interesting physical properties such as single-electron characteristic, negative differential resistance, molecular rectification, field-effect characteristics, and electronic switching have attracted considerable attention owing to their great potential practical applications in atomic-scale circuits. In particular, rectifying behavior^7–14 is the fascinating aspect of the modern quantum transport phenomena because it is one of the most important electronic elements used for logic circuits and memory elements. So far, five aspects for a molecular rectification have been intensely studied: (i) an asymmetry in the coupling of a molecule to two identical electrodes;^9,15 (ii) an asymmetry for a central electroactive unit placed inside a junction, for example, the central electroactive unit connected with alkyl chains of different lengths;¹⁰ (iii) an asymmetry of the electron density of the relevant molecular orbitals in the conjugated core, or D–σ–A molecular rectifier which was introduced by Aviram and Ratner based on a single organic molecule¹¹ and has been widely studied;¹² (iv) a conformational molecular diode, which depends on the molecule's orientation and displacement responding asymmetrically to the interelectrode applied field;¹³ and (v) asymmetrical Schottky barriers at both molecule–electrode contacts due to the difference in band structures of different electrode materials.¹⁶ Though rectification behavior has been found in a variety of molecular devices, unfortunately, the rectification efficiency of them is generally extremely low. Therefore, how to enhance the rectification efficiency of the molecular rectifier and obtain high-performance nanoscale rectifiers has been a focus of study in molecular electronics because the rectification ratio is a crucial parameter for the technical usefulness. Here, we suggest that another type of rectification can also be realized if the gate voltage is asymmetrically applied on the BDC₂₀ molecular device. Yet, as far as we know, it is the first time for the realization in theory of the new molecular rectifier with high rectification ratio.

Manufacture of stable devices with wires fixed between electrodes requires “molecular alligator clips”. Much work has focused on the preparation of molecular wires incorporating thiol end-groups because these groups adhere strongly to gold.¹⁷ However, thiols are known to have many different binding sites on gold, which leads to fluctuations in the electronic properties of the molecular junction.¹⁸ We have turned our attention to using Buckminsterfullerene, as an anchoring group for obtaining a more well-defined contact region to gold. Recently, it has been investigated both experimentally¹⁹ and theoretically.^20–22 Martin et al.¹⁹ have investigated the electrical characteristics of the C₆₀ end-capped compound incorporated between two gold electrodes in a break junction. The studies revealed that in comparison to thiol anchoring, fullerene anchoring leads to a considerably lower spread in low-bias conductance. Furthermore, relevant experiments show that fullerene-anchored benzenes exhibited an increased stretching length before breaking.

As we all know, there are much more atoms in C₆₀ than that in C₂₀, which makes the self-consistent DFT calculations of electronic states calculation difficult and expensive in time. At the same time, considering that the electronic transport of C₂₀ molecular device is similar with that of C₆₀ molecular device.^23,24 In order to reduce the computational cost, in this work, we study the transport through BDC₂₀ molecule (see Fig. 1), in which a phenyl ring is connected to two C₂₀ fullerenes²⁵ on two opposite sides via a pyrrolidine group (in a so-called“dumbbell” fashion). Here, we consider applying a gate voltage on the different positions of BDC₂₀ molecule. It should be possible to boost rectifying efficient because a gate voltage might greatly modify the electronic structure of such a molecule. Indeed, the currently reported very high rectification ratio, 91.7 and 24.0, employing the nonequilibrium Green's function method combined with the density functional theory (NEGF + DFT) calculations for a unimolecule connected with metal electrodes can be predicted when gate voltage is asymmetrically applied on the right C₂₀ of the BDC₂₀ molecule. Our findings highlight that asymmetrically gate voltage is an efficient way to realize a rectification behavior and the suitable gate voltage position is more important for greatly promoting rectifying characteristics and changing rectifying direction.

Fig. 1 Structures of molecular devices in our simulation. The gray, white, blue and yellow spheres represent C, H, N and Au atoms, respectively.

2 Model and method

The molecular devices under study, as illustrated schematically in Fig. 1, are formed by the BDC₂₀ molecule adsorbing onto two semi-infinite 3 × 3 Au (111) electrode surfaces directly, where bias voltage V is applied and the current is collect. Furthermore, an additional gate voltage is applied on the different regions of the molecule. Note that the gate electrode plotted in Fig. 1 is not a real physical electrode, and there is no current running from the source or drain electrode to the gate electrode. The electrostatic effect of the gate electrode in the experiment is simulated by shifting the molecular projected self-consistent Hamiltonian (MPSH) part of the Hamiltonian with the gate voltage. This shows that the gate electrode induces an external electrostatic potential on the molecular region.^4,26 Systems with no gate voltage or only with gate voltage applied on the right C₂₀, central phenyl ring (including pyrrolidine groups), central phenyl ring (including pyrrolidine groups) and right C₂₀ molecule, and the whole BDC₂₀ molecule are referred to as models M₁ or M₂, M₃, M₄ and M₅, respectively. Model M₁, M₃ and M₅ are the models with symmetrical gate voltage. Model M₂ and M₄ are the models with asymmetrical gate voltage. The purpose of forming five such different devices is to demonstrate asymmetrical gate voltage-modified effects on rectifying. To construct these molecular devices, each optimized molecular core by a separate calculation based on DFT is positioned between two Au (111) electrodes and is initially chosen to chemisorb to Au surface through bond contact. The distance between Au surface and C–C is 2.185 Å, which is a typical distance for the Au–C system.²³ Each model is composed of three regions: the left electrode, the scattering region (the device region), and the right electrode. The scattering region contains molecular core and two layers of Au surfaces (18 atoms) on each side of molecule for screening the effects of the electrodes on molecule. The geometries are optimized until all residual forces on each atom are smaller than 0.05 eV Å⁻¹.

The geometrical optimizations and the electronic transport properties are all calculated by the ATK package,^4,27 which is based on the NEGF + DFT technique. In our calculations, the exchange-correlation potential is described by the generalized gradient approximation (GGA) in Perdew, Burke, and Ernzerhof (PBE) form.²⁸ Valence electrons are expanded in a single-ζ plus polarization basis set (SZP) for Au atoms and a double-ζ plus polarization basis set (DZP) for C, N and H atoms. Core electrons are modeled with the Troullier–Martins nonlocal pseudopotential.²⁹ The electrode calculations are performed under periodical boundary conditions, and the Brillouin zone has been sampled with 3 × 3 × 100 points within the Monkhorst–Pack k-point sampling scheme. We set 150 Ry as the cutoff energy for the grid integration and select 10⁻⁵ as the convergence criterion for the total energy. The nonlinear current through the device is calculated using the Landauer–Büttiker formula

I(V) = (2e/h)∫[f(E − μ_L) − f(E − μ_R)]T(E, V)dE, (1)

where h is the Planck's constant, e is the electron charge, f is the Fermi function, V is the applied bias voltage, μ_L and μ_R are the electrochemical potential of the left and right electrodes, respectively,

V = (μ_L − μ_R)/e (2)

μ_L = E + eV/2 (3)

μ_R = E − eV/2 (4)

E_f is the Fermi level of the system, which is set to be zero in our calculations, and T(E, V) is the transmission function. The energy region contributing to the total current integral is called the bias window. In the approach, the molecular vibration effects have not been considered. Some deep discussions on these aspects in quantum transport can be found in ref. 30.

3 Results and discussion

In Fig. 2, we present the dependence of current I on bias voltage V (I–V characteristics) in a bias region [−1.5 V, 1.5 V] for all five models we studied. In this figure, we can see obvious differences for these models due to the different applying position effects of gate voltage. Several important features in the I–V curves are clearly visible: For M₁, M₃ and M₅, a symmetrical I–V curve arises as expected. The currents in M₁ and M₅ sustain small values in the low bias region and increase quickly at high bias. For M₃, the current almost increases gradually with the increasing of the bias. For M₄, the current rises rapidly and displays a linear behavior, representing an approximate Ohmic behavior under very small bias. And then the current drops dramatically with the increase of the bias and the current can be neglected in the [−0.7 V, −1.5 V] region, showing obvious asymmetry in current under positive and negative biases. This decrease in current due to an increase in bias is a manifestation of the negative differential resistance (NDR) behavior. Very interestingly, the I–V curve for M₂ shows a highly asymmetric characteristic: the current rises rapidly from −0.3 to −0.6 V under negative bias and is almost quenched to zero in the region [−0.3 V, 0.0 V] and whole region of positive bias, which is very analogous to the I–V curve for a traditional diode, and thus a rather large rectification ratio can be expected.

Fig. 2 I–V curves for all models. The I–V curve for M₂ and M₄ shows a highly asymmetric characteristic.

Fig. 3 shows the rectification ratio of five models. The rectification ratio, a ratio of the currents under negative and positive bias with the same voltage magnitude, can be calculated by R(V) = |I(−V)|/I(V). We define that the rectification ratio R(V) < 1[R(V) > 1] is the forward (reversed) rectification. Very interesting, two important physical phenomena can be observed from Fig. 3: (1) rectification directions are intimately related to the applying region of asymmetric gate voltage. We can observe significant reverse rectifying performances in M₂ over our whole calculation region, and M₄ shows a forward rectifying performance. (2) The rectification phenomenon depends strongly on the symmetry of gate voltage. There is no rectification phenomenon in M₁, M₃ and M₅. Obviously, the electronic transport character of M₂ presents the better rectifying effect than that of M₄. The maximum values of rectification ratios are 24.0 at V = 0.8 V, and are far less than that for M₂, 91.7 at V = 0.6 V. This is a currently reported very high rectification ratio with DFT calculations for a unimolecule device with metal electrodes and more than 30 times higher an original A-R diode.¹¹ Other merits for M₂ are its lower threshold bias, larger inverse current, and wider bias range for rectification. The gate voltage is symmetrical for M₁, M₃ and M₅, and asymmetrical for M₂ and M₄. Therefore, it is reasonable to conclude that the asymmetry of gate voltage may be an effective way to promote rectifying performance and modulate the rectifying direction.

Fig. 3 Rectification ratios change with bias for all models.

To make an analysis on rectifying mechanism, we calculate the spatial distribution of the MPSH¹² for the frontier orbital levels at zero bias, which gives a visual description of the electronic structure and can be employed to predict qualitatively the small-bias rectifying behaviors, as shown in Fig. 4. MPSH state is the self-consistent eigenstate of the molecule in the presence of Au electrodes and thus it contains the molecule–electrode coupling effects during the self-consistent calculations. The highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) are defined as the level with a lower and higher energy position than the Fermi level at zero bias and zero gate voltage. The molecular orbitals shift when applying the gate voltage. For M₁ and M₅, HOMO and LUMO are closer to the E_f. For M₂ and M₄, LUMO and LUMO+1 are closer to the E_f. For M₃, LUMO+1 and LUMO+2 are closer to the E_f. At zero bias, for M₁ and M₅, M₂ and M₄, and M₃, HOMO and LUMO, LUMO and LUMO+1, LUMO+1 and LUMO+2 are the most important molecular orbitals for electronic transport, respectively, therefore their spatial distribution is a good indicator of molecular electronic structures. The spatial distribution of the MPSH for the frontier molecular orbitals in our five models is drawn in Fig. 4. The spatial distribution of these frontier orbitals is a good indicator of molecular electronic transportation.³¹ As can been seen, the frontier molecular orbitals for M₂ and M₄ tend to be localized, which contributes to the most asymmetry of I–V characteristics.³² For M₁ and M₅, the HOMO and LUMO state locate at the left and right C₂₀ molecule, and pyrrolidine group area of the molecular device. For M₃, the LUMO+1 state locate at the pyrrolidine group and central phenyl ring area. The LUMO+2 state mostly locates at the pyrrolidine group area. However, M₂ and M₄ are significantly different because the LUMO and LUMO+1 states locate at the left region (left C₂₀ and pyrrolidine group) for M₂ and at the left region (left C₂₀ and pyrrolidine group), and the left region (left C₂₀ and pyrrolidine group) and right C₂₀, respectively, for M₄. The higher the delocalization extension, the lower the barrier height, however, the less the asymmetry of the potential drop. For M₁, M₃ and M₅, the spatial distribution of the MPSH for the frontier orbital levels is symmetrical, which has no contributions to a rectification under low bias. For M₂ and M₄, the spatial distribution of the MPSH for the frontier orbital levels is asymmetrical. This asymmetric spatial distribution of molecular states is the intrinsic origin for a rectification. The frontier orbital state in M₂ has a stronger localization than that in M₄, leading thus to a better rectifying performance for M₂. Therefore, different transport behaviors, especially for a low bias, in various models can be expected due to different spatial distributions of molecular states.^32–34

Fig. 4 The composition of molecular orbitals for all models at zero bias.

In order to understand the observed rectification behavior over the whole calculated bias region, we plot transmission spectra of all models at various biases in steps of 0.2 V, as shown in Fig. 5, where the Fermi level is set to zero as stated before. The transmission spectra are the most intuitive representation of quantum transport behaviors. According to the Landauer–Büttiker formula, the current is determined by the integral area of transmission curve within the bias window (BW), which is the region [−V/2, V/2], considering the fact that E_f is set to be zero. From Fig. 5, one can see obvious dependence of the transmission spectra on the applied bias. When the bias is applied, the transmission peaks shift relative to the E_f and their strength is changed. For M₁ and M₅, the tunneling peaks move far away from the BW within ±1.2 V and the tunneling peaks under the positive energy shifts into the BW after exceeding around ±1.2 V under positive and negative bias. The case for M₃ is similar to M₁ and M₅ except that a slight less transmission appears within the BW. M₂ is special because the tunneling peak shifts into the BW after exceeding around −0.4 V under negative bias, and moves far away from the BW and disappears under positive bias. The strength of the tunneling peaks reduces drastically when the bias exceeds −0.6 V. Therefore, a large reverse rectification and NDR show up. For M₄, the tunneling peak always stays inside (run into) the BW under positive bias, and stays outside the BW mostly within −0.2 V bias and disappears beyond −0.4 V bias, thus a forward rectification occurs. The strength of the tunneling peaks reduces drastically when the bias exceeds 0.4 V and −0.3 V, leading to a net drop in current and the onset of NDR at 0.4 V and −0.3 V. Here, modified effects on electronic structures by different position of voltage gate can be clearly observed.

Fig. 5 Transmission spectra of all models at various biases in steps of 0.2 V. The red region denotes the bias window (BW).

As is well-known, in the transmission spectra, the position of the transmission peak is generally determined by the energy level of the MPSH orbital and the height of the transmission peak is dependent on the delocalization of the molecular orbital. It is worth noting that Fig. 4 is only related to zero-bias molecular states. When the bias is applied, a system is driven out of the equilibrium, and molecular levels should shift due to complicated reasons, such as the enhancement of molecular polarization and variations of the electrode potential, potential drop profile in the whole device region, and the bias-dependent coupling strength between a molecule and electrodes. Therefore, to understand the origins of these transmission peaks within the BW and their changes with the increase of bias, we give an evolution of the molecular orbitals with applied bias for all models, as shown in Fig. 6. For simplicity, the MOs which can enter into or go close to the BW are displayed. In Fig. 6, LUMO, LUMO+1, …, LUMO+6 for M₁, LUMO, LUMO+1, …, LUMO+5 for M₂, LUMO, LUMO+1, …, LUMO+5 for M₃, LUMO, LUMO+1, …, LUMO+7 for M₄, and LUMO, LUMO+1, …, LUMO+8 for M₅, have a contribution to the current integral. As can be seen in Fig. 6(a), (c) and (e), the shift of the molecular orbitals within the BW under applied positive and negative bias is almost the symmetrical, so there are no rectifying performances for M₁, M₂ and M₅. From Fig. 6(b) and (d), we can see the molecular levels within the BW show different shift tendency in the positive and reverse branches of the BW, which should affect the rectification ratio under various bias for M₂ and M₄. The asymmetric shift of frontier orbitals under bias³⁵ is one of the key factors affecting the molecular rectification. To explore the origin of the asymmetric transmission spectra with respect to bias, we calculated the MPSH eigenstates. The isosurfaces of the MPSH eigenstates within the bias window for the M₂ at V = −0.6 V and 0.6 V are plotted in the left and right panels of Fig. 6(b), respectively. At V = −0.6 V, LUMO+1 eigenstates are spatially delocalized throughout the whole molecule region, making a big contribution to the transmission spectrum featured by the big and wide peaks. In contrast, at V = 0.6 V, all of the four eigenstates (LUMO to LUMO+3) are localized in either the left region or the right region, which means that the channel is suppressed so that no transmission peaks pop up in the transmission spectrum curves. This difference accounts for the negative rectifying properties of M₂. The isosurfaces of the MPSH eigenstates within the bias window for M₄ at V = −0.8 V and 0.8 V are plotted in the left and right panels of Fig. 6(d), respectively. At V = −0.8 V, all the six eigenstates (LUMO to LUMO+5) are localized in either the left region or the right region. By contrast, one of the two eigenstates (LUMO+1 and LUMO+2) at V = 0.8 V are delocalized throughout the whole molecule region, causing the big and wide peaks in the transmission spectrum. As a result, the M₄ shows a positive rectification behavior. Obviously, all of these are in agreement with those observed in Fig. 5 for transmission spectra. The above results indicate that the rectifying performance of M₂ and M₄ mainly results from the asymmetry distribution of the frontier MOs and the corresponding asymmetry evolution of the MO eigenenergies under the different direction of the external electric field, which is in agreement with previous predictions.^12,35–38 Interestingly, asymmetrical gate voltage can strengthen this asymmetry (rectifying performance).

Fig. 6 Evolution of the molecular orbitals with applied bias for all models. The MPSH states are plotted as left and right panels for each MO at certain biases.

To understand the rectifying performance with more details, the projected density of states (PDOS) are calculated, as depicted in Fig. 7, for all models at several typical nonzero biases, corresponding to maximum values of rectification ratios. The orbital PDOS here is the density of states projected on the i^th atomic basis orbital |χ_i〉, namely,

where |ψ(E_nk)〉 is a wave function for the device scattering state and could be presented as a linear combination of the pseudoatomic basis orbitals E_nk are energies of incident states of electrodes. More importantly, the bias-polarity dependence of PDOS can be clearly observed, which contributes to the rectification of a nanojunction. For M₁, M₃, and M₅, the PDOS are almost the same at positive and negative biases. In other words, no rectification in M₁, M₃, and M₅ can be expected. Nevertheless, M₂ and M₄ are entirely different; the PDOS peak in the vicinity of the Fermi level at negative bias is remarkably larger than that at positive bias. As a result, it has a best reverse rectification for M₂. There are two large PDOS peaks at positive bias near the Fermi level, which can greatly facilitate the electronic tunneling. As a result, it has a best forward rectification for M₄.

Fig. 7 Device PDOS of all models at several typical nonzero biases.

To further rationalize the rectifying nature and profoundly understand the rectifying mechanism in essence, we plot the spatial distribution of the electrostatic difference potential for all models at zero biases, as shown in Fig. 8. Electrostatic difference potential is a solution to Poisson equation for the difference in electron density, which equals to the total density minus the neutral atom electron density. Generally speaking, for bulk hetero-junctions and molecular rectifiers, nonlinear drop of electrostatic potential occurs, and the region with high potential drop plays a dominate role in the rectification performance. Apparently, there exists a asymmetrical potential barrier in M₂ and M₄, which is an intrinsic origin of rectifying effect in our nanojunctions. While in M₁, M₃ and M₅, the electrostatic difference potential distribution is symmetrical, thus there is no rectification. Note that the higher potential barrier appear at the right side of the scattering regions, as can be seen in M₂. When M₂ is positively biased, the electron tunneling is more favorable from the left electrode to right electrode. Consequently, the negative current is larger than the positive current and a reverse rectification occurs. The case M₄ is extremely different from M₂. It is just opposite for M₄. Note that the higher potential occurs at the left side of the scattering regions, as demonstrated in M₄. The electron transfer is more difficult from the left electrode to right electrode. A forward rectification can be expected accordingly.

Fig. 8 The electrostatic difference potential distribution of all models at zero bias.

4 Conclusion

In summary, we have investigated the electronic transport properties of gated BDC₂₀ molecular device by using the first-principle method based on the DFT combined with NEGF. We obtain a rectification ratio as high as 91.7 and 24.0 with a lower threshold bias, and wider bias range for rectification when the gate voltage is symmetrically applied on the right part of the BDC₂₀ molecule. This is an interesting result. The analysis of the molecular projected self-consistent Hamiltonian and the evolution of the frontier molecular orbitals as well as transmission spectrum with various external voltage biases gives an inside view of the observed results, which suggests that rectifying performance are due to the asymmetry distribution of the frontier MOs and the electrostatic potential, and the difference of projected density of states spectrum with applied bias. Our finding implies that to realize rectifying characteristics of molecular device, the suitable applying position of gate voltage might be an effective way, which could be helpful in the field of large rectification ratio molecular devices.

Acknowledgements

This work was supported by the National Basic Research Program of China (Grant no. 2009CB929204), Natural Science Foundation of China (Grant no. 11074146 and 11374183), the Natural Science Foundation of Shandong Province (Grant no. ZR2010AM037 and ZR2012AQ018), the Excellent Youth and Middle Age Scientists Fund of Shandong Province (no. BS2012CL025) and the Independent Innovation Foundation of Shandong University (Grant no. 2009TS097 and 2012TS025).

References

1. S. N. Rashkeev, M. D. Ventra and S. T. Pantelides, Phys. Rev. B: Condens. Matter Mater. Phys., 2002, 66, 033301.
2. Z. H. Zhang, X. Q. Deng, X. Q. Tan, M. Qiu and J. B. Pan, Appl. Phys. Lett., 2010, 97, 183105.
3. Z. Q. Fan and K. Q. Chen, Appl. Phys. Lett., 2010, 96, 053509.
4. M. Brandbyge, J. L. Mozos, P. Ordejon, J. Taylor and K. Stokbro, Phys. Rev. B: Condens. Matter Mater. Phys., 2002, 65, 165401.
5. S. J. Tans, A. R. M. Verschueren and C. Dekker, Nature, 1998, 393, 49.
6. T. M. Perrine, R. G. Smith, C. Marsh and B. D. Dunietz, J. Chem. Phys., 2008, 128, 154706.
7. I. I. Oleynik, M. A. Kozhushner, V. S. Posvyanskii and L. Yu, Phys. Rev. Lett., 2006, 96, 096803.
8. X. Q. Deng, J. C. Zhou, Z. H. Zhang, G. P. Tang and M. Qiu, Appl. Phys. Lett., 2009, 95, 103113.
9. Z. Zhang, Z. Yang, J. Yuan and M. Qiu, J. Chem. Phys., 2008, 128, 044711.
10. P. E. Kornilovitch, A. M. Bratkovsky and R. S. Williams, Phys. Rev. B: Condens. Matter Mater. Phys., 2002, 66, 165436.
11. A. Aviram and M. A. Ratner, Chem. Phys. Lett., 1974, 29, 277.
12. J. B. Pan, Z. H. Zhang, X. Q. Deng, M. Qiu and C. Guo, Appl. Phys. Lett., 2010, 97, 203104.
13. A. Troisi and M. A. Ratner, Nano Lett., 2004, 4, 591.
14. M. Bütiker, Phys. Rev. Lett., 1986, 57, 1761.
15. J. Taylor, M. Brandbyge and K. Stokbro, Phys. Rev. Lett., 2002, 89, 138301.
16. J. B. Pan, Z. H. Zhang, K. H. Ding, X. Q. Deng and C. Guo, Appl. Phys. Lett., 2011, 98, 092102.
17. K. Nørgaard, M. B. Nielsen and T. Bjørnholm, Functional Organic Materials, VCH-Wiley, Weinheim, Germany, 2007.
18. C. Li, I. Pobelov, T. Wandlowski, A. Bagrets, A. Arnold and F. Evers, J. Am. Chem. Soc., 2008, 130, 318.
19. C. A. Martin, D. Ding, J. K. Sørensen, T. Bjørnholm, J. M. van Ruitenbeek and H. S. J. van der Zant, J. Am. Chem. Soc., 2008, 130, 13198.
20. J. K. Sørensen, J. Fock, A. H. Pedersen, A. B. Petersen, K. Jennum, K. Bechgaard, K. Kilsa, V. Geskin, J. Cornil, T. Bjørnholm and M. B. Nielsen, J. Org. Chem., 2011, 76, 245.
21. T. Markussen, M. Settnes and K. S. Thygesen, J. Chem. Phys., 2011, 135, 144104.
22. S. Bilan, L. A. Zotti, F. Pauly and J. C. Cuevas, Phys. Rev. B: Condens. Matter Mater. Phys., 2012, 85, 205403.
23. Y. P. An, C. L. Yang, M. S. Wang, X. G. Ma and D. H. Wang, J. Chem. Phys., 2009, 131, 024311.
24. Z. Q. Fan, K. Q. Chen, Q. Wan, B. S. Zou, W. H. Duan and Z. Shuai, Appl. Phys. Lett., 2008, 92, 263304.
25. K. D. Sattler, Handbook of Nanophysics: Clusters and Fullerenes, CRC Press, 2011.
26. X. Q. Shi, Z. X. Dai, X. H. Zheng and Z. Zeng, J. Phys. Chem. B, 2006, 110, 16902.
27. J. Taylor, H. Guo and J. Wang, Phys. Rev. B: Condens. Matter Mater. Phys., 2001, 63, 245407.
28. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865.
29. N. Troullier and J. Martins, Phys. Rev. B: Condens. Matter Mater. Phys., 1991, 43, 1993.
30. S. Alavi, B. Larade, J. Taylor, H. Guo and T. Seideman, Chem. Phys., 2002, 281, 293.
31. J. M. Seminario, A. G. Zacarias and P. A. Derosa, J. Phys. Chem. A, 2001, 105, 791.
32. J. W. Zhao, C. Yu, N. Wang and H. M. Liu, J. Phys. Chem. C, 2010, 114, 4135.
33. Z. H. Zhang, Z. Yang, J. H. Yuan, H. Zhang, X. Q. Deng and M. Qiu, J. Chem. Phys., 2008, 129, 094702.
34. H. M. Liu, N. Wang, P. Li, X. Yin, C. Yu, N. Gao and J. W. Zhao, Phys. Chem. Chem. Phys., 2011, 13, 1301.
35. X. Yin, H. Liu and J. Zhao, J. Chem. Phys., 2006, 125, 094711.
36. R. Stadler, V. Geskin and J. Cornil, Adv. Funct. Mater., 2008, 18, 1119.
37. R. Stadler, V. Geskin and J. Cornil, J. Phys.: Condens. Matter, 2008, 20, 374105.
38. A. Staykov, D. Nozaki and K. Yoshizawa, J. Phys. Chem. C, 2007, 111, 11699.

---