Control of electron transport through Fano resonances in molecular wires

发布时间:2021-10-19 23:42:22

Control of electron transport through Fano resonances in molecular wires
T. A. Papadopoulos, I. M. Grace and C. J. Lambert1
1

Department of Physics, Lancaster University, Lancaster, United Kingdom

arXiv:cond-mat/0609109v1 [cond-mat.mtrl-sci] 6 Sep 2006

Using a ?rst principles approach, we study the electron transport properties of two molecules of length 2.5nm which are the building blocks for a new class of molecular wires containing ?uorenone units. We show that the presence of side groups attached to these units leads to Fano resonances close to the Fermi energy. As a consequence electron transport through the molecule can be controlled either by chemically modifying the side group, or by changing the conformation of the side group. This sensitivity, which is not present in Breit–Wigner resonances, opens up new possibilities for novel single-molecule sensors.

The ability to position single molecules in electrical junctions [1, 2, 3, 4, 5, 6] and demonstrations that molecular and nanoscale structures are capable of basic electronic functions such as current recti?cation, negative di?erential resistance and single electron transistor behaviour [7, 8, 9, 10, 11, 12] suggest that single-molecule electronics may play a key role in the design of future nanoelectronic circuits. However, the goal of developing a reliable molecular-electronics technology is still over the horizon and many key problems, such as device stability, reproducibility and the control of single-molecule transport need to be solved. In addition, since contacting molecules via break junction and SPMs is not a scalable technology, the question of wiring large numbers of molecules on a single chip still needs to be addressed. One approach to developing a scalable technology involves placing molecules between arrays of pre-formed, lithographically-grown contacts. Such an approach requires the length of the molecules to match the spacing between the contacts and since in practice, the spacing between such contacts can only be reliably controlled on the scale of 1–10nm, it is desirable to synthesize families of molecular wires with this range of lengths. Recently [13, 14, 15] a family of rigid molecules has been synthesized with lengths up to 10nm. These are π–conjugated oligomers based on rigid-rod-like aryleneethynylene backbones, containing ?uorenone units. The presence of terminal thiol groups allow assembly onto gold surfaces and therefore make them ideal for use in single-molecule device fabrication. In this Letter we investigate the properties of the smallest of these molecular wires, of length 2.5nm, since these form the building blocks of the longer wires. The central part consists of a single ?uorenone unit, which could be chemically modi?ed, e.g. by replacing the oxygen with pyridine or bipyridine rings, as shown in Fig. 1. In many molecular devices, electron transport is dominated by conduction through broadened HOMO or LUMO states, leading to Breit–Wigner resonances [16]. In contrast, for this family of molecules, we ?nd that transport is dominated by Fano resonances [17] associated with the presence of side groups, such as the oxygen atom or bi-pyridine. Generic properties of Fano reso-

nances, have been discussed in several contexts recently. Ref [18] deals with Fano resonances in 1-d waveguides, applied to GaAS heterojunctions. Ref. [19] analyses a generic model of a double quantum dot. Refs. [20, 21] address inelastic scattering, which can be an issue at ?nite bias, but is not the focus of attention in our paper. Ref. [22] is concerned with low temperature transport below the Kondo temperature. ’Conductance cancellation’ in 1-d tight-binding chains has also been noted [23] and a H¨ ckel model of resonant transport recently considered u [24]. None of the above papers contain ab initio, materialspeci?c calculations, which treat the metallic electrodes in a realistic manner. Recently, this limitation has been removed by state-of-the-art calculations on di-thiol benzene [25], which contain many examples of Fano resonances. Depending on speci?c conditions, these can take on a wide variety of shapes, ranging from Breit-Wigner type line shapes to strongly asymmetric pro?les. However the Fano resonances are neither controlled nor engineered into the molecule. They arise from a complicated interaction between the molecule and contacts and cannot be identi?ed with a particular section of the molecule. The aim of this Letter is to demonstrate that there are advantages in engineering molecules to possess Fano resonances associated with speci?c parts of the molecule, which can be modi?ed externally to achieve control over transport. Through an ab initio simulation of the molecules in Fig. 1, we demonstrate that Fano resonances near the Fermi energy can be controlled by altering the properties of the attached side group. This is in marked contrast with the behaviour of Breit-Wigner resonances, which are relatively insensitive to the state of the side group. These results suggest that the control of Fano resonances opens intriguing possibilities for single-molecule sensing. To compute electron transport properties, we use a combination of the DFT code SIESTA [26] and a Green’s function scattering approach, as encapsulated in the molecular electronics code SMEAGOL [27]. Initially the isolated molecule is relaxed to ?nd the optimum geometry and the molecule is then extended to include surface layers of the gold leads, so that charge transfer at the

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FIG. 1: The ground state geometry of the 2.5nm molecules extended to self-consitently include 8 layers of gold on the (111) surface, each containing 9 Au atoms. Colour codes: C(grey), H(white), N(dark grey) and O(black). a) The top molecule contains a single oxygen atom attached to the central ?uorenone unit and b) the oxygen has been replaced by a bipyridine unit.

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gold-molecule interface is included self-consistently. The number Ng of incorporated gold layers is increased until computed transport properties no longer change with increasing Ng . We ?nd that this occurs for Ng = 8 layers of gold, each consisting of 9 atoms on the (111) plain. Using a double–ζ basis plus polarization orbitals, Troullier– Martins pseudopotentials [28] and the Ceperley–Alder LDA method to describe the exchange correlation [29], an e?ective tight-binding Hamiltonian of the extended molecule is obtained, from which a scattering matrix and electron transmission coe?cient T (E) are computed. The zero-bias electrical conductance is then given by the Landauer formula G = (2e2 /h)T (EF ), where EF is the Fermi energy [30]. For the molecule shown in Fig. 1a, which has an oxygen atom attached to the ?uorenone unit, the transmission coe?cient T(E) is shown in Fig. 2 (black line). For E ≈ 1.2eV , this exhibits a typical Breit–Wigner resonance. As expected, since the molecule and contacts are symmetric, the maximum value of this peak is unity. However in the vicinity of the Fermi energy (EF = 0) transport is dominated by the presence of an asymmetric Fano resonance [17], comprising a resonant peak followed by an anti-resonance. We now demonstrate that anti-resonances arise from quasi-bound states associated with the side groups of the molecules in Fig. 1. For the molecule of Fig. 1a, this is demonstrated by the simple theoretical trick of arti?cially setting the hopping matrix elements connecting the oxygen atom to the ?uorenone unit to zero. This yields the transmission coe?cient shown by the red line in Fig. 2, which shows that arti?cial removal of the chemical bonds to the oxygen destroys the Fano resonance, whereas the remainder of the transmission spectrum remains largely unaltered. The ability to manipulate Fano resonances, by chemical means or otherwise, opens up new possibilities for

FIG. 2: Black line: Electron transmission coe?cient versus energy for the molecule having attached an oxygen atom as a side group. Dashed line: Oxygen bonds removed from the Hamiltonian.

controlling single-molecule transport. To explore this possibility in greater detail and to demonstrate that Fano resonances are a generic feature of molecular wires with attached side groups, we examine the molecule shown in Fig. 1b, in which the oxygen has been replaced by a bipyridine unit. Recent STM experiments on related wires have shown that it is possible to change the rotational conformation of attached side groups [31] and therefore we examine transport properties as a function of the angle of rotation θ of the bi-pyridine side group. (We de?ne the angle θ = 90? when the two rings lie parallel to the molecule axis and θ = 0? when they lie perpendicular to it.) The computed transmission through this molecule is shown in Fig. 3 for ?ve values of θ. This demonstrates that Fano resonances persist when one side group (namely the oxygen atom) is replaced by another and furthermore, the position of the Fano resonance is sensitive to the conformation of the side group. In contrast, the Breit–Wigner peak at 1.25eV is almost una?ected by such changes. To demonstrate that the Fano resonance is associated with localized states on the side group, we examine the energy spectrum of the isolated molecule. Fig. 4 shows how the energy levels of the isolated molecule depend on the rotation angle θ. This shows that while most of the levels remain una?ected, one of them is sensitive to changes in θ, varying from 1.57eV at θ = 90? to 0.0eV at θ = 0? . To demonstrate that this level belongs to the ?uorenone unit and bi-pyridine, Fig. 5 shows the local density of states (LDOS) for two di?erent energy values when θ = 30? (ground state of the molecule) and θ = 75? . Energies E = 2.0eV clearly correspond to states delocalized along the backbone, with almost no weight on the bi-pyridine. This state is responsible for the Breit–

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FIG. 5: Constant LDOS surface for a) 30? rotation and energy values of 1.46eV and 2.00eV . b) 75? rotation and energy values of 0.54eV and 2.00eV .
B A ... 0
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FIG. 3: Transmission against energy for the bi-pyridine attached molecule for rotation angles of 0? to 90?

FIG. 6: A backbone of sites labelled 1 to N coupled to left and right leads by matrix elements V, W and to a side group of sites by H1 .

9 0 6 0

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through a single backbone state Φm (k), with a resonant energy ε0 . In this case, the relevant self-energy Σ is Σ= ωω ? E ? ε0 + iΓ

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where ω = Ψ|H1 |Φ . In this expression, Γ = Γ1 + Γ2 is the broadening due to coupling of the backbone to the leads, with Γ1 = |V Φm (1)|2 N0 (E), Γ2 = |W Φm (N )|2 NN +1 (E) and N0 (E) (NN +1 (E)) the local density of states for the left (right) contacts. This yields for the transmission coe?cient, T = 4Γ1 Γ2 E ? ε0 ?
ωω ? E?ε 2

FIG. 4: Rotation angle θ dependence of the energy levels of the isolated molecule. The Fano eigenstates are represented by triangles while the circles represent the Breit–Wigner energy levels.

. + Γ2

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Wigner resonances in ?gures 2 and 3. On the other hand, for E = 1.46eV and E = 0.54eV the orbitals are found to be localized on the central unit. To capture the generic features of this combination of a Fano resonance and nearby Breit–Wigner resonance, we now develop a model of the system sketched in Fig. 6, which consists of a backbone A composed of atomic orbitals numbered i = 1, 2, . . . , N , coupled to a side group B by matrix elements H1 . (The geometric arrangement of the sites is irrelevant, provided A-sites are weakly coupled to B-sites.) Sites i ≤ 0 belong to the left lead and sites i ≥ N +1 to the right lead. These are coupled to the ends of the backbone via weak hopping elements V, W . In the absence of coupling to the side chain, we assume that transmission through the backbone takes place

Equation (1) shows that when ω = 0, no anti-resonance occurs and the transmission coe?cient exhibits a simple Breit–Wigner peak. More generally, eq. (1) shows that transmission is a maximum at energies E = ε± , where ε± are the roots of equation (E ? ε0 )(E ? ε) ? ωω ? = 0 and vanishes when E = ε. For small ω, a Breit–Wigner peak of width Γ occurs in the vicinity of ε+ ≈ ε0 . In addition a Fano peak occurs in the vicinity of ε? ≈ ε with Γωω ? width (ε0 ?ε)2 . A comparison between eq. (1) and the ab initio results of Fig. 3 is shown in ?gure 7. This demonstrates that with an appropriate choice of parameters, eq. (1) captures the essential features of Fano resonances in aryleneethynylene molecular wires. In summary we have shown that transport through a new class of molecular wires, composed of ?uorenone subunits with side groups, is dominated by Fano resonances rather than Breit–Wigner resonances. As a consequence electron transport through the molecule can be

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FIG. 7: a) A plot of eq. (1) for two sets of parameters, chosen to ?t the ab initio results for θ = 0? and θ = 75? . Black line: The values used are ε+ = 1.34eV , ε = 0.86eV and ε? = 0.77eV . Dashed line: ε+ = 1.23eV , ε = ?0.72eV and ε? = ?0.74eV . For both curves Γ = 0.05eV . b) For comparison, the lower ?gure shows the corresponding ab initio results.

controlled either by chemically modifying the side group, or by changing the conformation of the side group. This sensitivity, which is not present in Breit–Wigner resonances, opens up the possibility of novel single-molecule sensors. Acknowledgements: This work was supported by EPSRC under grant GR/S84064/01 (Controlled Electron Transport) (Durham and Lancaster), a LancasterEPSRC Portfolio Partnership and MCRTN Fundamentals of Nanoelectronics.

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