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Molecular Sensing Using Armchair Graphene Nanoribbon Mohammad Reza Rezapour,[a,c]y Arunkumar Chitteth Rajan,[b,c]y and Kwang S. Kim*[c] In molecular electronics, the conductance strongly depends on the frontier energy levels and spatial orientations of molecules. Utilizing these features, we investigate the electron transport characteristics of conjugated molecules attached on an armchair graphene nanoribbon. The resulting sharp reduction in the transmission which represents molecular fingerprints and the change of the transmission depending on the molecular
orientation, are examined in accordance with a unified picture of the Fano–Anderson model. These characteristics, being unique for each molecule, would be applicable to molecular C 2014 Wiley Periodirecognition and configurational analysis. V cals, Inc.
Introduction
unique for each molecule, would be effective tools to molecular recognition and conformational analysis. We apply an external perturbation to a 10-AGNR transport system which causes resonances in the corresponding electron transmission spectrum (For simplicity, we denote 10-AGNR as AGNR). An attached molecule on the AGNR surface provides new paths for the ballistic electrons and consequently resonances can be observed in the electron transmission curve due to quantum destructive interference at certain characteristic energies.[31]
Advances in manipulation of a single molecule have led to the fabrication of molecular devices because of their utility as electronically active elements.[1–7] This approach has been applied even to the DNA sequencing using nanopores and nanochannels.[8–13] Conversely, it has been shown that graphene, a honeycomb lattice made of carbon atoms, has extraordinary electronic capabilities which make the transport properties of this material extremely attractive for fabricating future nanoelectronics devices.[14–19] Graphene offers ballistic conductance for charge carriers[20–22] and a narrow graphene nanoribbon (GNR) provides few channels for electron transport, and therefore, it is possible to consider GNR as a quasi one-dimensional (1D) system.[23,24] Even though the production of sub-10 nanometer GNR is an experimental hurdle, anew advancements led to the synthesis of atomically precise GNRs of width less than 1 nm.[25,26] Recently, Kim and coworkers presented two dimensional (2D) molecular electronics spectroscopy by applying pconjugated molecules as external perturbations to a narrow armchair graphene nanoribbon (AGNR) which causes reductions in the corresponding electron transmission spectrum.[12,27] It is worth mentioning that molecules bound to 1D metallic nanowires have been studied to some extent.[28,29] The transport characteristics of molecules attached on AGNR surface strongly depend on the frontier orbital energy levels and spatial orientations of molecules. A sharp dip in the transmission due to resonance with molecular orbital energy levels represents molecular fingerprints, and the position and width of the dip shows the dependency on the molecular configuration. In a realistic case, the stacked molecule over the AGNR surface experiences slight fluctuations due to thermal noise and environment effects which would disturb the characteristics of the resonance dip. In this work, first, we show that the molecule-AGNR transport structure acts as a Fano–Anderson system.[30] Then, the effects of configuration interactions on the specifications of transmission dips are investigated. In this way, the influence of the seperation distance between molecule and AGNR and the rotation and translation of molecule on the characteristics of transmission reduction is studied. We demonstrate that the changes in the width and the position of the transmission reductions, being 1916
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DOI: 10.1002/jcc.23705
Methods We implement the transport calculations by using the density functional theory combined with the nonequilibrium Green’s function method.[32–34] The transmission is given by, †
T5 Tr½GL GGR G ;
(1)
where Tr stands for the trace, CL/R 5 i[RL/R–RL/R†] with RL/R as the self-energy of the left/right electrode and G signifies the Green’s function. The conductance is given in terms of the quantum of conductance as Go 5 (2e2/h). The Ceperley–Alder method for the local density approximation[35] is incorporated as the exchange correlation function. The double-f polarization basis set and the Troullier–Martins pseudo-potential[36] and 500 Ry cutoff energy for the grid-mesh are used in the calculations. The transport calculations are accomplished with POSTRANS code[37,38]plugged into [a] M. R. Rezapour Department of Physics, Pohang University of Science and Technology, Pohang, 790-784, Korea [b] A. C. Rajan Department of Chemistry, Pohang University of Science and Technology, Pohang, 790-784, Korea [c] M. R. Rezapour, A. C. Rajan, K. S. Kim Department of Chemistry, School of Natural Science, Ulsan National Institute of Science and Technology (UNIST), Ulsan, 689-798, Korea. E-mail: [email protected] †
These authors contributed equally to this work. Contract grant sponsor: NRF (National Honor Scientist Program); Contract grant number: 2010-0020414; Contract grant sponsor: KISTI; Contract grant number: KSC-2011-C3-019
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Figure 1. An electron transport system consisting of narrow width 10AGNR with a ring-matched (h 5 0 ) naphthalene molecule stacked above its surface. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
the Siesta package.[39] These computations are performed with a narrow AGNR of width 1 nm. The dangling bonds are hydrogenterminated. Geometry optimization is performed for AGNR using the conjugate gradient minimization and the conjugated mole˚ distance. cule is stacked on the AGNR surface at 3 A
Transmission Reduction of Naphthalene-AGNR System We start applying the perturbation by introducing a naphthalene molecule onto the AGNR surface (Fig. 1). The molecule is
stacked in a way to directly match the benzene-moiety of AGNR. Due to an effective orbital overlap between the molecule and AGNR, a reduction occurs in electron transmission of AGNR from its step-like nature. Figure 2a displays a nearly symmetric minimum transmission at the energy 21.2 eV. As p-p stacking[40] is responsible for the interaction between the molecule and AGNR, their coupling results in a large width of the transmission reduction by one quanta, indicating strong coupling. By increasing the distance between the molecule and AGNR or rotating the molecule about its center, the width of the transmission reduction diminishes. The coupling strength decreases exponentially if the distance between the molecule and AGNR increases.[41] Inset of Figure 2a shows the transmission reductions at different distances between naphthalene and AGNR. Here, we observe a slight shift in the position and a slight change in the width of the dip. Figure 2b shows changes of position and width of the resonance by rotating the naphthalene molecule about its center for three cases of parallel stacking, the ringmatched case (h 5 0 ) and the ring-mismatched cases (h 5 45 and h 5 90 ). The reason behind the shift of the resonance position is that the energy level of naphthalene molecule, which offers a new path for electron transmission, is slightly modified due to the AGNR-molecule coupling.[31] However, the coupling strength does not change significantly for the rotation angles less than 10 and translations within 0.25 A˚, which is useful for molecular recognition. Otherwise, the shift becomes larger so that the configurational differences can be noted. Density of states (DOS)[37] for these cases is presented in Figure 2c. The narrower the DOS, the narrower the width of the transmission reduction would be.
Molecule-AGNR System as a Fano–Anderson Structure Considering AGNR as a quasi one-dimensional structure[42] and the molecule as a side-attached defect, we treat this transport system as a Fano–Anderson configuration[43] (Fig. 3). Then, the Hamiltonian of the system can be written as follows[44]: X X † Cun21 un 1Ea jaj2 1 Vj uj 1c:c: H5 n
Figure 2. (a) Ballistic electron transmission (T against energy E–EF; EF: Fermi energy) of AGNR (dashed line) perturbed with a ring-matched (h 5 0 ) naphthalene molecule (solid line). The inset shows that as the molecule is displaced ˚ , dotted line; 3.75 A˚, dash-dot line), away from AGNR (3.25 A˚, dashed line; 3.50 A the width of the transmission reduction decreases with slight energy shift. (b/c) The transmission-reduction/DOS for h 5 0 (solid line) and the rotated configurations of h 5 45 (dotted line) and h 5 90 (dashed line). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
(2)
j
Here, un is the Bloch wave at site n, coupled by nearest neighbor hopping parameter C in a chain. Ea is the modified energy of state a belonging to the molecule and Vj is the coupling strength between site j and state a. The first term in eq. (2) describes the interaction between sites in the chain. Hence, our model is different from the usual tight binding description of graphene-like systems.[19] We consider a local coupling when Vj 5 0 for J 6¼ 0. The resulting transmission probability that describes the transmission reduction can be expressed as, T5rk 2 =ðrk 2 11Þ;
(3)
where rk 5 ck(E–Ea)/V02 and ck 5 2C sin k is the group velocity of the electron wave with a wave number k. The width of the Journal of Computational Chemistry 2014, 35, 1916–1920
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Multiple Fano Resonance and Effect of Configuration Interaction on Transmission Reduction
Figure 3. (a) Lattice structure of AGNR in which a molecule a is stacked. Each box stands for a unit cell that is symbolized by a circle in a chain. The ˚ is the carbon-carbon distance between two adjacent circles is 3r, r 5 1.42 A bond length. (b) Schematic representation of the Fano–Anderson model. The chain of circles denotes a continuum of states of the AGNR structure while a discrete state of the molecule a couples to u0 of the AGNR continuum state. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
resonance is ca 5 V02 /C sin kf, where kf is the wave number at the resonance. Indeed, eq. (3) is a special case of the Fano formula with Fano factor which equals to zero.[31] The above description depicts the transmission reduction caused by molecule stacked over the AGNR surface as a Fano resonance as follows: Side-attaching a p-conjugated molecule to the nanoribbon provides new paths for the ballistic electrons. Due to quantum interference at certain molecular characteristic energy the resonance would be observed in the electron transmission curves.
The number of Fano sites can be increased by adding another naphthalene molecule to the system. We examine the effect of configuration on the transmission reduction due to rotation of naphthalene molecules by fixing one of them and rotating the other about its center without altering their parallel orientation with AGNR. Two cases are considered. First, both molecules are placed on the same side of AGNR at a distance 6r ˚ ) is the carbon-carbon bond between them, where r (5 1.42 A distance in AGNR. Figure 4a represents the transmission curves of these configurations. The wider resonance corresponds to the ring-matched case, h1 5 h2 5 0 . As compared with the resonance due to a single naphthalene molecule, this is a double width resonance case because of the contributions from two degenerate energy levels of the molecules. This equal contribution is shown in Figure 4b by means of DOS (solid line). Rotating one of the naphthalene molecules by angle h1 5 30 results in splitting of the resonance into two resonances with different widths. The wider transmission reduction arises from the ring-matched molecule due to the larger p-orbital overlap. The rotated configuration of the other molecule produces the resonance curve of lesser width and is shifted due to the variation in coupling strength. Double Fano resonance is found to shrink to a single resonance case at h1 5 90 . But in fact, it is still a combination of two resonances caused by the ringmatched and mismatched cases. One might note that the positions of the transmission reductions of the cases h 5 0 and h 5 90 are the same in Figure 2b. Equation (3) is still valid for the case of two noninteracting Fano sites a and b where rk 5 (E–Ea)(E–Eb)/c(E–Et) with c 5 ca 1cb and Et 5 (caEb1cbEa)/c.[45] Here, ca and cb are the widths of the resonances. In the second case, two naphthalene molecules are stacked on AGNR from its opposite sides (Fig. 5). In the first trial, we place two naphthalene molecules facing each other from the opposite sides of AGNR. This configuration allows naphthalene molecules to have interaction through AGNR. In this case, the modified Hamiltonian of the system can be written as H 5 H 1 H0 , where H is given by eq. (2) with the local coupling condition H0 5 Eb|b|[2] 1 b†aVab 1 b†Vbu0 1 c.c. Here, Va/b is the individual coupling strength of state a/b with site J 5 0, and Vab describes the interaction between two naphthalene molecules. Hence, rk 5
2 ðE2Ea ÞðE2Eb Þ2Vab V
V2
(4)
ab ca ½ðE2Ea Þ1Vab Vba 1cb ½ðE2Ea Þ2 ðE2E bÞ
Figure 4. (a) Transmission curve due to two naphthalene molecules placed on the same side of AGNR at a distance 6r between them. The first molecule is rotated from its ring-matched configuration to h1 5 30 and h1 5 90 , and the second one is made fixed (h2 5 0 ). (b) Corresponding DOS and a schematic diagram of the system. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
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produces the transmission coefficient in accordance with eq.(3). Figures 5a–5b illustrate the transmission curves and DOS, respectively. Figure 5a represents multiple Fano resonances.[46] In the ring-matched case, h1 5 h2 5 0 ; we observe two major resonances at the energies 20.95 and 21.47 eV with a small asymmetric transmission reduction with high DOS, at 21.41 eV (inset of Fig. 5b). These two wider resonances indicate higher coupling between molecules and AGNR due to the WWW.CHEMISTRYVIEWS.COM
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Figure 5. Transmission and DOS of two naphthalene molecules on rotation where each ones are side coupled from the opposite surfaces of AGNR. (a,b) correspond to directly interacting molecules facing each other from the opposite sides of AGNR; whereas, in (c,d) the upper naphthalene molecule is moved by a distance 6r along the AGNR length to avoid the interaction between the molecules.
maximum p-orbital overlap. We disturb the ring-matched case slightly by rotating the upper naphthalene molecule about its center. The positions of two dips come closer to each other at h1 5 30 and h1 5 60 (not shown here) and they become narrow. Because in these cases, the overlap between p-orbitals is less, the coupling strength is changed and a displacement in the resonance energy position is observed. Finally, when h1 5 90 , the wider resonance is located at the same resonance position of h1 5 0 . In the second trial, to avoid the interaction between two naphthalene molecules, one of the molecules is moved along the AGNR length by a distance 6r. Now, we expect to observe Fano resonances caused by two noninteracting Fano sites. Figure 5c illustrates the Fano dips similar to the transmission curves given in Figure 4a. By rotating one of the naphthalene molecules, the double-width Fano resonance in the case of h1 5 h2 5 0 turns into two Fano dips at h1 5 30 . The position of the dips caused by the ring-matched naphthalene molecule at 21.14 eV remains unaffected, while the resonance at 21.30 eV is evolved from the rotated naphthalene configuration.
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Finally, at h1 5 90 , we observe the recombination of the two resonances into a double-width single resonance dip. The DOS corresponding to the noninteracting naphthalene molecules is given in Figure 5d. We also discuss the effect of configuration on Fano resonance profile due to translation. Figures 6a–6c illustrate the transmission profiles of three cases on translation (by dTz) along the AGNR length–a single naphthalene molecule, two naphthalene molecules on the same side of AGNR and two naphthalene molecules facing each other through opposite sides of AGNR. Here, we consider the configurations corresponding to the ringmatched cases (solid line) of Figures 2b, 4a, and 5a. In Figure 6a, the single Fano resonance at the initial position ˚ ) gets weakened and shifted to a less-wider single (dTz 5 0 A ˚ . A similar effect Fano resonance on translation by dTz 5 2.4 A is observed in Figure 6b. The narrow resonance at dTz 52.4 A˚ for this configuration still shows the double-width behavior. The double Fano resonance (Fig. 6c) with almost equal widths (solid line) goes to a nearly single Fano resonance (dotted line) by moving naphthalene molecules by dTz 5 1.2 A˚, and at ˚ the transmission profile is observed with double dTz 5 2.4 A Fano resonance of different widths (dashed line). As AGNR is periodic, translation by dTz 5 3r gives the same resonance profile as that of the initial position. Cytosine, a DNA base, on its parallel-stacked configuration above AGNR produces overlapped double Fano resonance due to its nearly degenerate levels of highest occupied molecular orbitals. Figure 6d illustrates its Fano resonance profile (solid line) consisting of two dips with different widths (ca > cb). The overlapped double Fano resonance profile of the initial configuration is transformed into two almost separated Fano ˚ with ca>>cb. A tilted configuration resonances at dTz 5 2.6 A of cytosine molecule with respect to the width of AGNR by an
Figure 6. (a–c) Schematic diagrams and transmission profiles of single and double naphthalene molecules on translation along the length of AGNR. (d–e) Transmission profile of a parallel-stacked (u 5 0 ) or tilted (u 5 5 ) configuration of a cytosine molecule above AGNR on translation.
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angle u 5 5 produces overlapped Fano resonances (Fig. 6e). Interesting forms of the resonances are seen on translating this ˚, particular configuration above the AGNR surface. At dTz 5 0.6 A the overlapped Fano resonances are transformed into two separated single resonances with different widths. A wider reduction ˚ , where the resoin the transmission is reported at dTz 5 3.0 A nance profile is a single Fano dip; that is, cb 0. These resonance profiles correspond to the coupled-resonator induced reflection in the Fano–Feshbach resonance.[45]
Conclusions In summary, we demonstrate that an electron transport system comprising a molecule as an external perturbation to AGNR can be treated as a Fano–Anderson model. Characteristic Fano resonances in the transmission profile of AGNR caused by a perturbing molecule are analyzed. As the asymmetric Fano resonances in electron transmission, irrespective of edgeroughness and width of AGNR,[27] are characterized by the molecular orbitals, identifying molecules of p-conjugated systems can be made reality. Such signals can be measured with an atomic force microscopy (AFM)-like tip on which short narrow GNR is attached with a gate voltage scanning capability for molecules on the surface of solid, liquid, or immersed solid. Thus, 2D conductance measurement would be possible for molecular characteristics and conformation orientation on the surface. Therefore, based on our analysis of the p-conjugated systems on GNR which can be extended to carbon nanotubes, the 2D conductance spectroscopy would be realized, as in 2D nuclear magnetic resonance (2D NMR) or 2D infra red (2D IR). This would open a new avenue for molecular spectroscopy. Keywords: electron transport Fano resonance molecular electronics single molecule sensing
How to cite this article: M. R. Rezapour, A. C. Rajan, K. S. Kim J. Comput. Chem. 2014, 35, 1916–1920. DOI: 10.1002/jcc.23705
[1] J. R. Reimers, Computational Methods for Large Systems: Electronic Structure Approaches for Biotechnology and Nanotechnology; Wiley: Hoboken, 2010. [2] J. Cuevas, E. Scheer, Molecular Electronics: An Introduction to Theory and Experiment; World Scientific Publishing Company Incorporated: Singapore, 2010. [3] A. Nitzan, M. A. Ratner, Science 2003, 300, 1384. [4] W. Y. Kim, Y. C. Choi, S. K. Min, Y. Cho, K. S. Kim, Chem. Soc. Rev. 2009, 38, 2319. [5] L. Shen, M. Zeng, S.-W. Yang, C. Zhang, X. Wang, Y. Feng, J. Am. Chem. Soc. 2010, 132, 11481. [6] E. Kan, W. Hu, C. Xiao, R. Lu, K. Deng, J. Yang, H. Su, J. Am. Chem. Soc. 2012, 134, 5718. [7] W. J. Cho, Y. Cho, S. K. Min, W. Y. Kim, K. S. Kim, J. Am. Chem. Soc. 2011, 133, 9364. [8] J. Prasongkit, A. Grigoriev, B. Pathak, R Ahuja, R. H. Scheicher, Nano Lett. 2011, 11, 1941.
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[9] R. H. Scheicher, A. Grigoriev, R. Ahuja, J. Mater. Sci. 2012, 47, 7439. [10] T. Ahmed, S. Kilina, T. Das, J. T. Haraldsen, J. J. Rehr, A. V. Balatsky, Nano Lett. 2012, 12, 927. [11] S. K. Min, Y. Cho, D. R. Mason, J. Y. Lee, K. S. Kim, J. Phys. Chem. C 2011, 115, 16247. [12] S. K. Min, W. Y., Kim Y. Cho, K. S. Kim, Nat. Nanotechnol. 2011, 6, 162. [13] S. Thomas, A. C. Rajan, M. R. Rezapour, K. S. Kim, J. Phys. Chem. C 2014, 118, 10855. [14] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, A. A. Firsov, Science 2004, 306, 666. [15] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, A. K. Geim, Rev. Mod. Phys. 2009, 81, 109. [16] V. Georgakilas, M. Otyepka, A. B. Bourlinos, V. Chandra, N. Kim, K. C. Kemp, P. Hobza, R. Zboril, K. S. Kim, Chem. Rev. 2012, 112, 6156. [17] K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J.-H. Ahn, P. Kim, J.-Y. Choi, B. H. Hong, Nature 2009, 457, 706. [18] Y.-W. Son, M. L. Cohen, S. G. Louie, Nature 2006, 444, 347. [19] A. Cresti, N. Nemec, B. Biel, G. Niebler, F. Triozon, G. Cuniberti, S. Roche, Nano Res. 2008, 1, 361. [20] J. Tworzydło, B. Trauzettel, M. Titov, A. Rycerz, C. W. J. Beenakker, Phys. Rev. Lett. 2006, 96, 246802. [21] E. Munoz, J. Lu, B. Yakobson, Nano Lett. 2010, 10, 1652. [22] K. I. Bolotin, K. J. Sikes, J. Hone, H. L. Stormer, P. Kim, Phys. Rev. Lett. 2008, 101, 096802. [23] W. Y. Kim, K. S. Kim, Nat. Nanotechnol. 2008, 3, 408. [24] L. Yang, C.-H. Park, Y.-W. Son, M. L. Cohen, S. G. Louie, Phys. Rev. Lett. 2007, 99, 186801. [25] J. Cai, R. Pascal, R. Jaafar, M. Bieri, T. Barun, S. Blankenburg, M. Muoth, A. P. Seitsonen, M. Saleh, X. Feng, K. Mullen, R. Fassel, Nature 2010, 466, 470. [26] M. Koch, F. Ample, C. Joachim, L. Grill, Nat. Nanotechnol. 2012, 7, 713. [27] A. C. Rajan, M. R. Rezapour, J. Yun, Y. Cho, W. J. Cho, S. K. Min, G. Lee, K. S. Kim, ACS Nano 2014, 8, 1827. [28] A. Kumar, R. Heimbuch, B. Poelsema, H. J. W. Zandvliet, J. Phys. Condens. Matter 2012, 24, 082201. [29] D. E. P. Vanpoucke, J. Phys. Condens. Matter 2014, 26, 133001. [30] A. E. Miroshnichenko, S. Flach, Y. S. Kivshar, Rev. Mod. Phys. 2010, 82, 2257. [31] U. Fano, Phys. Rev. 1961, 124, 1866. [32] J. Taylor, H. Guo, J. Wang, Phys. Rev. B 2001, 63, 245407. on, J. Taylor, K. Stokbro, Phys. Rev. B [33] M. Brandbyge, J.-L. Mozos, P. Ordej 2002, 65, 165401. [34] S. Datta, Electronic Transport in Mesoscopic Systems; Cambridge University Press: Cambridge, 1997. [35] D. M. Ceperley, B. J. Alder, Phys. Rev. Lett. 1980, 45, 566. [36] N. Troullier, J. L. Martins, Phys. Rev. B 1991, 43, 1993. [37] W. Y. Kim, K. S. Kim, J. Comput. Chem. 2008, 29, 1073. [38] W. Y. Kim, K. S. Kim, Acc. Chem. Res. 2010, 43, 111. [39] J. Soler, E. Artacho, J. Gale, A. Garcıa, J. Junquera, P. Ordej on, D. Sanchez-Portal, J. Phys. Condens. Matter 2002, 14, 2745. [40] K. S. Kim, P. Tarakeshwar, J. Y. Lee, Chem. Rev. 2000, 100, 4145. [41] Y. Cho, S. K. Min, W. Y. Kim, K. S. Kim, Phys. Chem. Chem. Phys. 2011, 13, 14293. [42] D. H. Lee, J. D. Joannopoulos, Phys. Rev. B 1981, 23, 4988. [43] A. E. Miroshnichenko, S. Flach, Y. S. Kivshar, Rev. Mod. Phys. 2010, 82, 2257. [44] A. E. Miroshnichenko, S. F. Mingaleev, S. Flach, Y. S. Kivshar, Phys. Rev. E 2005, 71, 036626. [45] S. F. Mingaleev, A. E. Miroshnichenko, Y. S. Kivshar, Opt. Express 2008, 16, 11647. [46] A. E. Miroshnichenko, Y. S. Kivshar, Phys. Rev. E 2005, 72, 056611.
Received: 13 July 2014 Revised: 21 July 2014 Accepted: 27 July 2014 Published online on 13 August 2014
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Molecular Sensing Using Armchair Graphene Nanoribbon Mohammad Reza Rezapour,[a,c]y Arunkumar Chitteth Rajan,[b,c]y and Kwang S. Kim*[c] In molecular electronics, the conductance strongly depends on the frontier energy levels and spatial orientations of molecules. Utilizing these features, we investigate the electron transport characteristics of conjugated molecules attached on an armchair graphene nanoribbon. The resulting sharp reduction in the transmission which represents molecular fingerprints and the change of the transmission depending on the molecular
orientation, are examined in accordance with a unified picture of the Fano–Anderson model. These characteristics, being unique for each molecule, would be applicable to molecular C 2014 Wiley Periodirecognition and configurational analysis. V cals, Inc.
Introduction
unique for each molecule, would be effective tools to molecular recognition and conformational analysis. We apply an external perturbation to a 10-AGNR transport system which causes resonances in the corresponding electron transmission spectrum (For simplicity, we denote 10-AGNR as AGNR). An attached molecule on the AGNR surface provides new paths for the ballistic electrons and consequently resonances can be observed in the electron transmission curve due to quantum destructive interference at certain characteristic energies.[31]
Advances in manipulation of a single molecule have led to the fabrication of molecular devices because of their utility as electronically active elements.[1–7] This approach has been applied even to the DNA sequencing using nanopores and nanochannels.[8–13] Conversely, it has been shown that graphene, a honeycomb lattice made of carbon atoms, has extraordinary electronic capabilities which make the transport properties of this material extremely attractive for fabricating future nanoelectronics devices.[14–19] Graphene offers ballistic conductance for charge carriers[20–22] and a narrow graphene nanoribbon (GNR) provides few channels for electron transport, and therefore, it is possible to consider GNR as a quasi one-dimensional (1D) system.[23,24] Even though the production of sub-10 nanometer GNR is an experimental hurdle, anew advancements led to the synthesis of atomically precise GNRs of width less than 1 nm.[25,26] Recently, Kim and coworkers presented two dimensional (2D) molecular electronics spectroscopy by applying pconjugated molecules as external perturbations to a narrow armchair graphene nanoribbon (AGNR) which causes reductions in the corresponding electron transmission spectrum.[12,27] It is worth mentioning that molecules bound to 1D metallic nanowires have been studied to some extent.[28,29] The transport characteristics of molecules attached on AGNR surface strongly depend on the frontier orbital energy levels and spatial orientations of molecules. A sharp dip in the transmission due to resonance with molecular orbital energy levels represents molecular fingerprints, and the position and width of the dip shows the dependency on the molecular configuration. In a realistic case, the stacked molecule over the AGNR surface experiences slight fluctuations due to thermal noise and environment effects which would disturb the characteristics of the resonance dip. In this work, first, we show that the molecule-AGNR transport structure acts as a Fano–Anderson system.[30] Then, the effects of configuration interactions on the specifications of transmission dips are investigated. In this way, the influence of the seperation distance between molecule and AGNR and the rotation and translation of molecule on the characteristics of transmission reduction is studied. We demonstrate that the changes in the width and the position of the transmission reductions, being 1916
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DOI: 10.1002/jcc.23705
Methods We implement the transport calculations by using the density functional theory combined with the nonequilibrium Green’s function method.[32–34] The transmission is given by, †
T5 Tr½GL GGR G ;
(1)
where Tr stands for the trace, CL/R 5 i[RL/R–RL/R†] with RL/R as the self-energy of the left/right electrode and G signifies the Green’s function. The conductance is given in terms of the quantum of conductance as Go 5 (2e2/h). The Ceperley–Alder method for the local density approximation[35] is incorporated as the exchange correlation function. The double-f polarization basis set and the Troullier–Martins pseudo-potential[36] and 500 Ry cutoff energy for the grid-mesh are used in the calculations. The transport calculations are accomplished with POSTRANS code[37,38]plugged into [a] M. R. Rezapour Department of Physics, Pohang University of Science and Technology, Pohang, 790-784, Korea [b] A. C. Rajan Department of Chemistry, Pohang University of Science and Technology, Pohang, 790-784, Korea [c] M. R. Rezapour, A. C. Rajan, K. S. Kim Department of Chemistry, School of Natural Science, Ulsan National Institute of Science and Technology (UNIST), Ulsan, 689-798, Korea. E-mail: [email protected] †
These authors contributed equally to this work. Contract grant sponsor: NRF (National Honor Scientist Program); Contract grant number: 2010-0020414; Contract grant sponsor: KISTI; Contract grant number: KSC-2011-C3-019
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Figure 1. An electron transport system consisting of narrow width 10AGNR with a ring-matched (h 5 0 ) naphthalene molecule stacked above its surface. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
the Siesta package.[39] These computations are performed with a narrow AGNR of width 1 nm. The dangling bonds are hydrogenterminated. Geometry optimization is performed for AGNR using the conjugate gradient minimization and the conjugated mole˚ distance. cule is stacked on the AGNR surface at 3 A
Transmission Reduction of Naphthalene-AGNR System We start applying the perturbation by introducing a naphthalene molecule onto the AGNR surface (Fig. 1). The molecule is
stacked in a way to directly match the benzene-moiety of AGNR. Due to an effective orbital overlap between the molecule and AGNR, a reduction occurs in electron transmission of AGNR from its step-like nature. Figure 2a displays a nearly symmetric minimum transmission at the energy 21.2 eV. As p-p stacking[40] is responsible for the interaction between the molecule and AGNR, their coupling results in a large width of the transmission reduction by one quanta, indicating strong coupling. By increasing the distance between the molecule and AGNR or rotating the molecule about its center, the width of the transmission reduction diminishes. The coupling strength decreases exponentially if the distance between the molecule and AGNR increases.[41] Inset of Figure 2a shows the transmission reductions at different distances between naphthalene and AGNR. Here, we observe a slight shift in the position and a slight change in the width of the dip. Figure 2b shows changes of position and width of the resonance by rotating the naphthalene molecule about its center for three cases of parallel stacking, the ringmatched case (h 5 0 ) and the ring-mismatched cases (h 5 45 and h 5 90 ). The reason behind the shift of the resonance position is that the energy level of naphthalene molecule, which offers a new path for electron transmission, is slightly modified due to the AGNR-molecule coupling.[31] However, the coupling strength does not change significantly for the rotation angles less than 10 and translations within 0.25 A˚, which is useful for molecular recognition. Otherwise, the shift becomes larger so that the configurational differences can be noted. Density of states (DOS)[37] for these cases is presented in Figure 2c. The narrower the DOS, the narrower the width of the transmission reduction would be.
Molecule-AGNR System as a Fano–Anderson Structure Considering AGNR as a quasi one-dimensional structure[42] and the molecule as a side-attached defect, we treat this transport system as a Fano–Anderson configuration[43] (Fig. 3). Then, the Hamiltonian of the system can be written as follows[44]: X X † Cun21 un 1Ea jaj2 1 Vj uj 1c:c: H5 n
Figure 2. (a) Ballistic electron transmission (T against energy E–EF; EF: Fermi energy) of AGNR (dashed line) perturbed with a ring-matched (h 5 0 ) naphthalene molecule (solid line). The inset shows that as the molecule is displaced ˚ , dotted line; 3.75 A˚, dash-dot line), away from AGNR (3.25 A˚, dashed line; 3.50 A the width of the transmission reduction decreases with slight energy shift. (b/c) The transmission-reduction/DOS for h 5 0 (solid line) and the rotated configurations of h 5 45 (dotted line) and h 5 90 (dashed line). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
(2)
j
Here, un is the Bloch wave at site n, coupled by nearest neighbor hopping parameter C in a chain. Ea is the modified energy of state a belonging to the molecule and Vj is the coupling strength between site j and state a. The first term in eq. (2) describes the interaction between sites in the chain. Hence, our model is different from the usual tight binding description of graphene-like systems.[19] We consider a local coupling when Vj 5 0 for J 6¼ 0. The resulting transmission probability that describes the transmission reduction can be expressed as, T5rk 2 =ðrk 2 11Þ;
(3)
where rk 5 ck(E–Ea)/V02 and ck 5 2C sin k is the group velocity of the electron wave with a wave number k. The width of the Journal of Computational Chemistry 2014, 35, 1916–1920
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Multiple Fano Resonance and Effect of Configuration Interaction on Transmission Reduction
Figure 3. (a) Lattice structure of AGNR in which a molecule a is stacked. Each box stands for a unit cell that is symbolized by a circle in a chain. The ˚ is the carbon-carbon distance between two adjacent circles is 3r, r 5 1.42 A bond length. (b) Schematic representation of the Fano–Anderson model. The chain of circles denotes a continuum of states of the AGNR structure while a discrete state of the molecule a couples to u0 of the AGNR continuum state. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
resonance is ca 5 V02 /C sin kf, where kf is the wave number at the resonance. Indeed, eq. (3) is a special case of the Fano formula with Fano factor which equals to zero.[31] The above description depicts the transmission reduction caused by molecule stacked over the AGNR surface as a Fano resonance as follows: Side-attaching a p-conjugated molecule to the nanoribbon provides new paths for the ballistic electrons. Due to quantum interference at certain molecular characteristic energy the resonance would be observed in the electron transmission curves.
The number of Fano sites can be increased by adding another naphthalene molecule to the system. We examine the effect of configuration on the transmission reduction due to rotation of naphthalene molecules by fixing one of them and rotating the other about its center without altering their parallel orientation with AGNR. Two cases are considered. First, both molecules are placed on the same side of AGNR at a distance 6r ˚ ) is the carbon-carbon bond between them, where r (5 1.42 A distance in AGNR. Figure 4a represents the transmission curves of these configurations. The wider resonance corresponds to the ring-matched case, h1 5 h2 5 0 . As compared with the resonance due to a single naphthalene molecule, this is a double width resonance case because of the contributions from two degenerate energy levels of the molecules. This equal contribution is shown in Figure 4b by means of DOS (solid line). Rotating one of the naphthalene molecules by angle h1 5 30 results in splitting of the resonance into two resonances with different widths. The wider transmission reduction arises from the ring-matched molecule due to the larger p-orbital overlap. The rotated configuration of the other molecule produces the resonance curve of lesser width and is shifted due to the variation in coupling strength. Double Fano resonance is found to shrink to a single resonance case at h1 5 90 . But in fact, it is still a combination of two resonances caused by the ringmatched and mismatched cases. One might note that the positions of the transmission reductions of the cases h 5 0 and h 5 90 are the same in Figure 2b. Equation (3) is still valid for the case of two noninteracting Fano sites a and b where rk 5 (E–Ea)(E–Eb)/c(E–Et) with c 5 ca 1cb and Et 5 (caEb1cbEa)/c.[45] Here, ca and cb are the widths of the resonances. In the second case, two naphthalene molecules are stacked on AGNR from its opposite sides (Fig. 5). In the first trial, we place two naphthalene molecules facing each other from the opposite sides of AGNR. This configuration allows naphthalene molecules to have interaction through AGNR. In this case, the modified Hamiltonian of the system can be written as H 5 H 1 H0 , where H is given by eq. (2) with the local coupling condition H0 5 Eb|b|[2] 1 b†aVab 1 b†Vbu0 1 c.c. Here, Va/b is the individual coupling strength of state a/b with site J 5 0, and Vab describes the interaction between two naphthalene molecules. Hence, rk 5
2 ðE2Ea ÞðE2Eb Þ2Vab V
V2
(4)
ab ca ½ðE2Ea Þ1Vab Vba 1cb ½ðE2Ea Þ2 ðE2E bÞ
Figure 4. (a) Transmission curve due to two naphthalene molecules placed on the same side of AGNR at a distance 6r between them. The first molecule is rotated from its ring-matched configuration to h1 5 30 and h1 5 90 , and the second one is made fixed (h2 5 0 ). (b) Corresponding DOS and a schematic diagram of the system. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
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produces the transmission coefficient in accordance with eq.(3). Figures 5a–5b illustrate the transmission curves and DOS, respectively. Figure 5a represents multiple Fano resonances.[46] In the ring-matched case, h1 5 h2 5 0 ; we observe two major resonances at the energies 20.95 and 21.47 eV with a small asymmetric transmission reduction with high DOS, at 21.41 eV (inset of Fig. 5b). These two wider resonances indicate higher coupling between molecules and AGNR due to the WWW.CHEMISTRYVIEWS.COM
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Figure 5. Transmission and DOS of two naphthalene molecules on rotation where each ones are side coupled from the opposite surfaces of AGNR. (a,b) correspond to directly interacting molecules facing each other from the opposite sides of AGNR; whereas, in (c,d) the upper naphthalene molecule is moved by a distance 6r along the AGNR length to avoid the interaction between the molecules.
maximum p-orbital overlap. We disturb the ring-matched case slightly by rotating the upper naphthalene molecule about its center. The positions of two dips come closer to each other at h1 5 30 and h1 5 60 (not shown here) and they become narrow. Because in these cases, the overlap between p-orbitals is less, the coupling strength is changed and a displacement in the resonance energy position is observed. Finally, when h1 5 90 , the wider resonance is located at the same resonance position of h1 5 0 . In the second trial, to avoid the interaction between two naphthalene molecules, one of the molecules is moved along the AGNR length by a distance 6r. Now, we expect to observe Fano resonances caused by two noninteracting Fano sites. Figure 5c illustrates the Fano dips similar to the transmission curves given in Figure 4a. By rotating one of the naphthalene molecules, the double-width Fano resonance in the case of h1 5 h2 5 0 turns into two Fano dips at h1 5 30 . The position of the dips caused by the ring-matched naphthalene molecule at 21.14 eV remains unaffected, while the resonance at 21.30 eV is evolved from the rotated naphthalene configuration.
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Finally, at h1 5 90 , we observe the recombination of the two resonances into a double-width single resonance dip. The DOS corresponding to the noninteracting naphthalene molecules is given in Figure 5d. We also discuss the effect of configuration on Fano resonance profile due to translation. Figures 6a–6c illustrate the transmission profiles of three cases on translation (by dTz) along the AGNR length–a single naphthalene molecule, two naphthalene molecules on the same side of AGNR and two naphthalene molecules facing each other through opposite sides of AGNR. Here, we consider the configurations corresponding to the ringmatched cases (solid line) of Figures 2b, 4a, and 5a. In Figure 6a, the single Fano resonance at the initial position ˚ ) gets weakened and shifted to a less-wider single (dTz 5 0 A ˚ . A similar effect Fano resonance on translation by dTz 5 2.4 A is observed in Figure 6b. The narrow resonance at dTz 52.4 A˚ for this configuration still shows the double-width behavior. The double Fano resonance (Fig. 6c) with almost equal widths (solid line) goes to a nearly single Fano resonance (dotted line) by moving naphthalene molecules by dTz 5 1.2 A˚, and at ˚ the transmission profile is observed with double dTz 5 2.4 A Fano resonance of different widths (dashed line). As AGNR is periodic, translation by dTz 5 3r gives the same resonance profile as that of the initial position. Cytosine, a DNA base, on its parallel-stacked configuration above AGNR produces overlapped double Fano resonance due to its nearly degenerate levels of highest occupied molecular orbitals. Figure 6d illustrates its Fano resonance profile (solid line) consisting of two dips with different widths (ca > cb). The overlapped double Fano resonance profile of the initial configuration is transformed into two almost separated Fano ˚ with ca>>cb. A tilted configuration resonances at dTz 5 2.6 A of cytosine molecule with respect to the width of AGNR by an
Figure 6. (a–c) Schematic diagrams and transmission profiles of single and double naphthalene molecules on translation along the length of AGNR. (d–e) Transmission profile of a parallel-stacked (u 5 0 ) or tilted (u 5 5 ) configuration of a cytosine molecule above AGNR on translation.
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angle u 5 5 produces overlapped Fano resonances (Fig. 6e). Interesting forms of the resonances are seen on translating this ˚, particular configuration above the AGNR surface. At dTz 5 0.6 A the overlapped Fano resonances are transformed into two separated single resonances with different widths. A wider reduction ˚ , where the resoin the transmission is reported at dTz 5 3.0 A nance profile is a single Fano dip; that is, cb 0. These resonance profiles correspond to the coupled-resonator induced reflection in the Fano–Feshbach resonance.[45]
Conclusions In summary, we demonstrate that an electron transport system comprising a molecule as an external perturbation to AGNR can be treated as a Fano–Anderson model. Characteristic Fano resonances in the transmission profile of AGNR caused by a perturbing molecule are analyzed. As the asymmetric Fano resonances in electron transmission, irrespective of edgeroughness and width of AGNR,[27] are characterized by the molecular orbitals, identifying molecules of p-conjugated systems can be made reality. Such signals can be measured with an atomic force microscopy (AFM)-like tip on which short narrow GNR is attached with a gate voltage scanning capability for molecules on the surface of solid, liquid, or immersed solid. Thus, 2D conductance measurement would be possible for molecular characteristics and conformation orientation on the surface. Therefore, based on our analysis of the p-conjugated systems on GNR which can be extended to carbon nanotubes, the 2D conductance spectroscopy would be realized, as in 2D nuclear magnetic resonance (2D NMR) or 2D infra red (2D IR). This would open a new avenue for molecular spectroscopy. Keywords: electron transport Fano resonance molecular electronics single molecule sensing
How to cite this article: M. R. Rezapour, A. C. Rajan, K. S. Kim J. Comput. Chem. 2014, 35, 1916–1920. DOI: 10.1002/jcc.23705
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Received: 13 July 2014 Revised: 21 July 2014 Accepted: 27 July 2014 Published online on 13 August 2014
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