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Is the Antiresonance in Meta-Contacted Benzene Due to the Destructive Superposition of Waves Traveling Two Different Routes around the Benzene Ring? Daijiro Nozaki, and Cormac Toher J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 27 Apr 2017 Downloaded from http://pubs.acs.org on May 2, 2017
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The Journal of Physical Chemistry
Is the Antiresonance in Meta-Contacted Benzene Due to the Destructive Superposition of Waves Traveling Two Different Routes around the Benzene Ring? Daijiro Nozaki*,† and Cormac Toher‡
† ‡
Lehrstuhl für Theoretische Materialphysik, Universität Paderborn, 33095 Paderborn, Germany. Department of Mechanical Engineering and Materials Science, Duke University, Durham, North Carolina 27708, USA.
Abstract The well-known antiresonance around the middle of the HOMO-LUMO gap observed in the transmission spectra of the meta-contacted benzene molecular junctions is often explained as being caused by the destructive interference between electronic waves following two different pathways in real space around the phenyl ring. We show one contradictory scenario where this interpretation may break down when one of the bonds in the benzene is attenuated gradually. Interestingly, the dip in the transmission spectra is not attenuated at all even after the complete breaking of the bond. This inconsistency arises from the misinterpretation of the antiresonance observed in energy space as a consequence of the superposition of waves propagating through two independent pathways in real space. We revisit the Landauer model within the Green’s function formalism and propose a different interpretation of the appearance of the antiresonance in energy space in meta-substituted benzene which is compatible with the scenario described above. The quantum interference observed in energy-dependent transmission spectra comes from cancellation between the terms in the Green’s function and is effectively due to interference between the different molecular orbitals.
Introduction The development of measurement techniques and theory has promoted the fundamental understanding of charge transport process in nanostructures.1,2 Although electron tunneling through a molecule embedded between contacts can be roughly estimated from the energy difference between the frontier eigenenergies and the Fermi energy, as well as the molecular length and the coupling strengths with the contacts; sometimes the tunneling is blocked depending on the position of contacts, despite the energetics and coupling with the molecule being the same, due to the destructive quantum interference (DQI) effect. Recently DQI, observed as an abrupt drop in the transmission, has been the focus of signifi-
cant attention due to its unique features and potential applications in nanoelectronics for switching devices and for efficient thermoelectric devices.3–12 DQI is often discussed using the analogy with its classical counterpart. Figs. 1(a)-(b) show the well-known double slit experiments. The two waves, A,B , , coming from different pathways overlap on the screen at position “x” yielding: , = , + , . (1) They destructively interfere at the position where , = − , = , is satisfied, yielding a line of zero amplitude on the screen. On the other hand, the quantum counterpart shown in Figs. 1(c)-(d) is the case when a molecule is bound to metallic contacts. Among the many molecules showing DQI13–45, meta-connected benzene is a well-known chemical building block for this effect6–9,46-83, although there are some arguments that DQI may not be visible or may be elusive for this configuration because of the shift of eigenenergies or because the dominant contribution comes from components.6,8,9,31,73,79,84 The DQI is observed as a dip in the transmission spectra (see Fig. 1(d)). In the quantum case, the interference pattern is projected as a “transmission spectrum” in energy space, while in the classical case it is projected as “maximum absolute amplitude” in real space on the screen. In the literature, the antiresonance in transmission spectra is often explained using the analogy with the classical double slit experiments or Mach-Zehner interferometers (see e.g. reports in Refs. 5-7,9,22-24,3337,44,45,47-49,58-68), i.e. that the destructive interference emerges when the difference of two spatial pathways matches with a phase shift of . The metasubstituted benzene is the most representative case for this scenario as depicted in Fig. 1(e) (see e.g. reports in Refs. 6,7,9,22-24,47-49,58-68). Hereafter we refer to this as the intuitive interpretation and we define it by the following three conditions: (c1) T(E) presents antiresonance behavior.
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(c2) The antiresonance is given by the cancellation of two components having different sign (e.g. –0.5 and 0.5 and –X and X). (c3) Each component in (c2) solely comes from the path A or path B.
Figure 1. Explanatory diagrams for (a)-(b) classical interference phenomena and (c)-(d) quantum interference phenomena. The controversial interpretation of the interference event is shown in (e). is the nearest-neighbor transfer integral and is the homogeneous on-site energy.
However, the intuitive interpretation of the antiresonance in as depicted in Fig. 1(e) raises the following questions: (1) What exactly travels through the two different pathways? And what kinds of quantity destructively interfere at site 3? (2) How does the cancellation of something in question (1) at site 3 in real space create the dip at = in the transmission spectra in energy space? How does one mathematically derive = = 0 from this intuitive picture? (3) How is the difference of 2a (where a is the bond distance) in the length of the pathways between Path A and B in real space associated with the energy at which the antiresonance in the transmission spectra occurs? Why does the wave having a de Broglie wavelength = 4 result in an antiresonance at energy = , instead of at an energy which is a function of this distance such as = !" = !#/ ? (4) The standing wave description is often used for the explanation of the DQI. However, this stand-
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ing wave appears at the resonance energies (eigenenergies) in the isolated systems. Is it relevant to use this standing wave description at the off-resonance energy where the antiresonance is observed? (5) How is the splitting of the antiresonance explained from the intuitive picture when the bond is attenuated (see %& ' 0.5 in Fig. 2)? Considering these questions, the critical concern arises: is it relevant to interpret the destructive interference in energy space in Fig. 1(d) as being due to the overlap of two pathways in real space as depicted in Fig. 1(e)? Recently, similar concerns were also pointed out as a commentary85 to the research on the DQI of biphenyl wires comparing photo-induced electron transfer with electron transport in molecular junctions.86 In order to demonstrate this concern explicitly, we consider one scenario where the strength of one of the bonds in a benzene ring is attenuated gradually (see inset of Fig. 2). According to the intuitive interpretation given by Fig. 1(e), the antiresonance due to the destructive interference should be attenuated since the flow through the path A in Fig. 1(e) is suspended. Interestingly, as we will see later, the antiresonance, often discussed as being due to the destructive overlap of two waves (or electrons) traveling through different pathways, is not attenuated at all even after the complete breaking of the bond as shown in Fig. 2. This inconsistency arises from the misinterpretation of the antiresonance observed in energy space as a consequence of the superposition of waves (or electrons) propagating through two pathways in real space. In this work, we revisit the transmission function within the Green’s function formalism. Then we reformulate the consistent interpretation of the appearance of the antiresonance in energy space in meta-substituted benzene so as to be compatible with the beforementioned scenario of the “scissor-cutting” of the bond. Methods Before proceeding we briefly summarize the theoretical framework for this research. The transmission is given by the standard Landauer model87,88 with the Fisher-Lee relation as = Trace.Γ0 1 EΓ3 1 4 5, where 1 /4 is the retarded (advanced) Green’s function, defined as 1 = . + 678 − 9 − Σ0 − Σ3 5;< and 1 4 = .1 5= . 8, 7, and Σ0/3 are the identity matrix, infinitesimal imaginary value, and self-energy for the left/right contact, respectively. For simplicity, we consider only the pz orbitals of the carbon atoms forming the electron molecular orbitals and use a noninteracting Hückel-like Hamiltonian H with nearestneighbor transfer integral and homogeneous on-site energy . Since we assess the origin of the antiresonance which appears even in the noninteracting description, the influence of the electron-electron interactions is
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outside of the scope of this research. Since our focus is not on the DQI arising from the correction by the Σ0/3 but on the DQI originating from the molecule itself, we use the wide-band limit approximation for the contacts. Each contact is attached to a single site of the device with broadening > . Then, the transmission between site i and j can be written as: = % 4> % ?1,@ ? , where 1,@ is the off-diagonal element of 1 . Thus, the behavior of the transmission profile and the (anti-)resonance can be analyzed only by 1 = 0.01 and the transfer integral %& between site 2 and 3 is tuned. Results & Discussion Figure 2 presents the transmission of metaconnected benzene with different transfer integral values %& . With the reduction of the transfer integrals %& , the degeneracy of the HOMO and LUMO of benzene breaks and new resonances appear. As mentioned earlier, it is interesting to note that the antiresonance around the midgap = remains intact during the attenuation even if one of the pathways is fully blocked. Note that in the fully attenuated limit (%& = 0.0), the system can be alternatively interpreted as a T-shaped motif where a side group attached to main conduction channel disturbs conduction at the eigenenergy of the side-group, causing the DQI.38,45,50,52 The strong coupling limit (%& = 1.0) can also be alternatively interpreted from the analytic expression for Green’s function for a molecule with homogeneous inter-site coupling .58 Although the intermediate regime (0.0 B %& B 1.0) could be speculatively interpreted to be zero due to a combination of the processes involved in these two limits, this requires further assessment. 89 The intermediate regime shows the presence of an intact antiresonance in spite of the suppression of the wave propagation through path A. The reduction of the transfer integral %& also corresponds to the elongation of the bond between site 2 and 3. If the intuitive interpretation holds, the antiresonance at the mid-gap should be affected because of the change of the phase shift accompanying the elongation of the upper pathway. However, such an attenuation or shift of the antiresonance is not seen from the result in Fig. 2. From another point of view, one may also say that the back-scattering of the electron wave function caused by the reduction of the bond may contribute to the destructive interference with another electron wave propagating through path B, thus retaining the presence of the antiresonance. In this work, we explore a consistent interpretation for the intact antiresonance using the MO approach. Another notable feature is the phase transition of DQI where a single antiresonance splits into three branches when %& ' 0.5. Although this splitting is known to happen in tight-binding models when second and third nearest neighbor interactions are included26,28,30–32,56, it
is noteworthy that the antiresonance can split even when only nearest neighbor interactions are considered, as demonstrated here.
Figure 2. Transmission spectra for benzene molecules contacted at the meta-position with different transfer integral values %& between sites 2 and 3. Interestingly, the antiresonance remains present during the breaking of the bond.
In order to reinterpret the counter-intuitive behavior of the antiresonance at the mid-gap energy, we use the Lehmann representation of the retarded Green’s function to get its spectral decomposition in terms of the molecular orbitals (MOs) ΨD of the benzene molecule19,20,26,27,29,50,52,56: 1
