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Magic Ratios for Connectivity-Driven Electrical Conductance of Graphene-Like Molecules

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Magic Ratios for Connectivity-Driven Electrical Conductance of Graphene-Like Molecules Magic ratios for connectivity-driven electrical conduct- ance of graphene-like molecules Yan Geng,1‡ Sara Sangtarash,*,2‡ Cancan Huang,1‡ Hatef Sadeghi,2‡ Yongchun Fu,1 Wenjing Hong,*,1 Thomas Wandlowski,1 Silvio Decurtins,1 Colin J. Lambert,*,2 and Shi-Xia Liu*,1 1 Department of Chemistry and Biochemistry, University of Bern, Freiestrasse 3, CH-3012 Bern, Switzerland 2 Lancaster Quantum Technology Centre, Department of Physics, Lancaster University, LA14YB Lancaster, UK centre of the H-L gap. For the purpose of identifying conditions ABSTRACT: Experiments using a mechanically-controlled break for destructive QI within the delocalised π-system, a useful con- junction and calculations based on density functional theory ceptual approach is to represent molecules by lattices of con- demonstrate a new magic ratio rule (MRR), which captures the 2 contribution of connectivity to the electrical conductance of gra- nected sites (C(sp ) atoms), such as those shown in Fig. 1, in phene-like aromatic molecules. When one electrode is connected which 1a represents a benzene ring, 1b represents naphthalene, to a site i and the other is connected to a site i' of a particular mol- 1c represents anthracene and 1d represents anthanthrene. Such ecule, we assign the molecule a “magic integer” Mii'. Two mole- abstractions highlight the role of connectivity in determining the cules with the same aromatic core, but different pairs of electrode presence or absence of destructive QI. For example, the lattices connection sites (i,i' and j,j' respectively) possess different magic of Fig. 1 are bipartite, being composed of equal numbers of integers Mii' and Mjj'. Based on connectivity alone, we predict that ‘primed’ and ‘unprimed’ sites, such that primed sites (labelled when the coupling to electrodes is weak and the Fermi energy of by primed integers such as 1′,2′,3′) are connected to unprimed the electrodes lies close to the centre of the HOMO-LUMO gap, sites only (labelled by non-primed integers such as 1,2,3) and 2 33-38 the ratio of their conductances is equal to (Mii' /Mjj') . The MRR is vice versa. It is well known , (see also the Mathematical exact for a tight binding representation of a molecule and a quali- Methods below) that if electrodes are connected to two sites tative guide for real molecules. which are both primed or both unprimed, then destructive inter- ference occurs and the contribution from π-orbitals to the elec- INTRODUCTION Charge transport through polycyclic aro- trical conductance G vanishes. For a phenyl ring this corre- matic hydrocarbons (PAHs) has attracted intensive attention in 31 1,2 sponds to the well-known case of meta-coupled electrodes , but recent years , partly due to their role in the design and devel- more generally it holds for any bipartite lattice. opment of molecular electronic devices3-6. Since PAHs are well-defined and defect free, they also provide model systems Studies of such lattices have yielded a variety of simple rules for understanding transport in graphene, treated as an infinite for the appearance of destructive QI2132-37, for which the π- 7-9 alternant PAH, and graphene-based nanostructures . When a orbital contribution to G vanishes. The aim of the present pa- single molecule is connected to metallic electrodes, electrons per is to elucidate a simple rule for determining the non-zero passing through the molecule from one electrode to the other values of electrical conductance arising from constructive QI 10,11 can remain phase coherent, even at room temperature . This in aromatic molecules. At first sight, this task seems rather has led to a great deal of discussion about the role of quantum daunting, because there is only one conductance (i.e. G = 0) interference (QI) in determining the electrical conductance of when QI is destructive, whereas there are many possible non- 12-21 single molecules , culminating in a series of recent experi- zero values of G when QI is constructive. Furthermore, the 22-30 ments revealing room-temperature signatures of QI . non-zero values of conductances in the presence of construc- tive QI depend on the strength and detailed nature of the con- Both experiment and theory have focused primarily on elucidat- tacts to electrodes. ing the conditions for the appearance of constructive or destruc- tive interference. In the simplest case, where electrons are in- jected at the Fermi energy EF of the electrodes, constructive QI arises when EF coincides with a delocalized energy level En of the molecule. Similarly a simple form of destructive QI occurs when EF coincides with the energy Eb of a bound state located on a pendant moiety31,32. In practice, unless energy levels are tuned by electrostatic, electrochemical or mechanical gating, molecules located within a junction rarely exhibit these types of QI, because EF is usually located in the HOMO-LUMO (H-L) gap. For this reason, discussions have often focussed on condi- tions for destructive or constructive QI when EF is located at the 1’ 2’ 3’ 1’ 3 1 1 4 2 9’ 10 11 4’ 3’ 2’ 9 10’ 11’ 5 3 5’ 8 (a) |M|=1 (b) |M|= (1,2) (c) |M|= (1,2,3,4) (d) |M|= (1,2,3,4,6,7,8,9) Figure 1 Four examples of bipartite lattices, with the magnitude of their magic numbers shown underneath each lattice. (a) repre- sents benzene, (b) naphthalene, (c) anthracene and (d) anthanthrene. Remarkably, in what follows, we demonstrate a “magic ratio 1′ 2′ 3′ 4′ 5′ rule” based on tables of quantum numbers Mii’, which capture the contribution of connectivity to the electrical conductance 1 -2 1 -1 1 -1 of graphene-like aromatic molecules, or molecules with gra- 2 -1 -1 1 -1 1 phene-like cores, when one electrode is connected to an ‘un- 3 1 -2 -1 1 -1 primed’ site i and the other is connected to a ‘primed’ site i′. In 4 -1 2 -2 -1 1 particle physics, quantum numbers such as ‘charm’ and ‘col- 5 2 -1 1 -1 -2 our’ are assigned to elementary particles. In the case of lattices such as those in Fig. 1, we refer to these new quantum num- bers as ‘magic integers M ’. For each of the molecules shown ii’ Table 1 The M-table of MIs M for the naphthalene lattice of in Fig. 1, the allowed values of |M | are shown beneath each ii’ ii’ Fig. 1b. lattice. Clearly the spectrum of magic integers increases with size of the aromatic core. The precise values of Mii’ are not 1′ 2′ 3′ 4′ 5′ 6′ 7′ 8′ 9′ 10′ 11′ trivial, since for example Mii’ = 5 is missing from the set of anthanthrene integers. 1 - 7 - 4 - 1 - 1 - 2 -3 9 4 1 1 1 RESULTS AND DISCUSSION 2 - - 4 - 1 - 1 - 1 -2 3 1 7 4 1 1 Magic integers (MIs) capture the complexity of interference 3 1 - - 4 - 1 - 1 - 2 -3 patterns created by electrons at the centre of HOMO-LUMO 3 4 1 1 1 gap and allow the prediction of conductance ratios via the 4 - 3 - - 1 - 1 - 1 -2 3 following ‘magic ratio rule’ (MRR), which states that “the 1 6 4 1 1 ratio of conductances of two molecules is equal to the squares 5 1 - 6 - - 1 - 1 - 2 -3 of the ratios of their magic integers.” Clearly, when comparing 3 6 1 1 1 conductances of the same aromatic core, but different contacts, the signs of the MIs are irrelevant. This rule is derived in the 6 - 3 - 6 - - 1 - 1 -2 3 1 6 9 1 1 Mathematical Methods section. To each lattice such as those 7 3 - 8 - 7 - - 3 - 6 1 in Fig. 1, the quantum numbers Mii’ form a table of MIs, which 9 8 7 3 3 we refer to as M-tables. As shown in the SI, for the benzene ring 1a, this is a 3x3 table, with all entries equal to +/-1, so 8 - 8 - 6 - 4 - - 6 -2 -2 6 6 4 4 6 that |Mii’| = 1 is the only possibility and therefore as expected, para (i.e. 3,1′) or ortho (3,2′ or 3,3′) connectivities yield the 9 6 - 6 - 4 - 4 - - 2 2 same electrical conductances. For the naphthalene lattice 1b, 8 6 4 4 6 the 5x5 M-table is shown in Table 1. As expected from sym- 10 3 1 - 2 - 3 - 3 - -4 1 metry, this table shows the conductances associated with con- 2 3 3 3 tact sites 1,1′ and 5,5′ are equal and proportional to (2)2 = 4. It 11 - 6 - 2 2 - 2 - 2 -4 -4 also shows that the conductance with contact sites 4,2′ or 4,3′ 2 2 2 2 would take the same value, which is a less obvious result. Table 2 The M-table for the anthanthrene lattice of Fig. 1d. Note that the first (row) index is non-primed and the second (column) index is primed. The MRR is an exact formula for conductance ratios of tight- cal reactions and photooxidation42-44. The synthetic approach to binding representations of molecules in the weak coupling limit, the two novel pyridine-terminated anthanthrene derivatives (1 when the Fermi energy is located at the centre of the HOMO- and 2, as shown in Scheme 1) is reported in the SI. To measure LUMO (H-L) gap. It does not depend on the size of the H-L gap their single-molecule electrical conductances, we employed a and is independent of asymmetries in the contacts. In what fol- MCBJ setup capable of operating in solution.
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