Robust Molecular Anchoring to Graphene Electrodes - ACS Publications

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Robust molecular anchoring to graphene electrodes Hatef Sadeghi, Sara Sangtarash, and Colin J. Lambert Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b01001 • Publication Date (Web): 12 Jul 2017 Downloaded from http://pubs.acs.org on July 13, 2017

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Robust molecular anchoring to graphene electrodes Hatef Sadeghi*, Sara Sangtarash, and Colin Lambert* Quantum Technology Centre, Department of Physics, Lancaster University, Lancaster LA1 4YB, UK * [email protected]; [email protected]

Recent advances in the engineering of pico-scale gaps between electroburnt graphene electrodes provide new opportunities for studying electron transport through electrostatically-gated single molecules. But first we need to understand and develop strategies for anchoring single molecules to such electrodes. Here, for the first time, we present a systematic theoretical study of transport properties using four different modes of anchoring zinc-porphyrin monomer, dimer and trimer molecular wires to graphene electrodes. These involve either amine anchor groups, covalent C-C bonds to the edges of the graphene, or coupling via π-π stacking of planar polyaromatic hydrocarbons formed from pyrene or tetrabenzofluorene (TBF). π-π stacked pyrene anchors are particularly stable, which may be advantageous for forming robust single-molecule transistors. Despite their planar, multi-atom coupling to the electrodes, pyrene anchors can exhibit both destructive interference and different degrees of constructive interference, depending on their connectivity to the porphyrin wire, which makes them attractive also for thermoelectricity. TBF anchors are more weakly coupled to both the graphene and the porphyrin wires, and induce negative differential conductance at finite source-drain voltages. Furthermore, although direct C-C covalent bonding to the edges of graphene electrodes yields the highest electrical conductance, electron transport is significantly affected by the shape and size of the graphene electrodes, because the local density of states at the carbon atoms connecting the electrode edges to the molecule is sensitive to the electrode surface shape. This sensitivity suggests that direct C-C bonding may be the most desirable for sensing applications. The ordering of the low-bias electrical conductances with different anchors is as follows: direct C-C coupling > π-π stacking with the pyrene anchors > direct coupling via amine anchors > π-π stacking with TBF anchors. Despite this dependency of conductances on the mode of anchoring, the decay of conductance with the length of the zinc-porphyrin wires is relatively insensitive, with the associated attenuation factor β lying between 0.9 and 0.11 Å-1. Keywords: Single molecule electronics, graphene electrode, covalent/π-π anchor, porphyrin, attenuation factor Studies of electron transport through metal/molecule/metal junctions utilise a variety of techniques for contacting single-molecules to metallic electrodes, including scanning-tunnelling-microscopy-based break junctions, conducting probe atomic force microscopy break junctions, mechanically controllable break junctions and electromigration1–8. For the purpose of attaching molecules to such metallic electrodes, a variety of anchor groups have been explored2,9. These fundamental studies have demonstrated clear correlations between molecular structure and function, but are not scalable, not CMOS compatible and in many cases do not allow gating by a nearby electrostatic gate. Silicon-based platforms have been proposed10,11 but so far such technologies remain in their infancy. To overcome some of these limitations, strategies for contacting single molecules based on carbon nanotubes12 and graphene13–15 have been developed. In particular electroburnt graphene junctions have been shown to deliver stable electrode gaps below 5nm16–26,

which allow electrostatic gating through buried or side gates. Recently graphene-based molecular junctions were used to realise a stable and reversible photoswitch20 and to study quantum interference16,27,28, electron transport18,19, vibrational properties17,29 and molecular magnetism30 in single molecules. However compared with metal/molecule/metal junctions, graphene-based molecular electronics is still in an early phase of development. In particular to date only relatively few anchor groups have been utilized16–18,21,29,30 and a comparative study of their effect on junction performance is lacking. It took several years to obtain agreement between different research groups working on transport measurements of single molecules using gold electrodes. This was made possible only after comparative systematic theoretical and experimental studies9,31–34 were carried out on homologous series of the molecules with a range of anchor groups. In this Letter, we investigate the effect of different anchoring modalities on transport through

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Finally we demonstrate that the deviation from planarity of TBF anchors, combined with their weak coupling to the porphyrin wires leads to negative differential conductance at finite source-drain voltages.

graphene-based molecular junctions. We study the electronic properties of zinc-porphyrin monomers, dimers and trimers coupled to the graphene electrodes using a series of π-π or covalent anchoring groups. Zinc-porphyrins are interesting, because due to their highly conjugated backbone, the electrical conductance of 10-4.7 – 10-4.5 G0 is relatively high compared with other single molecule junctions of similar length9. Furthermore, zinc-porphyrins are attractive candidates for molecular-scale wires, because their measured conductance G decays exponentially with molecular length L, as G α exp(-βL), with an unusually low attenuation factor35,36 β ~ 0.04 Å-1. In addition, electronic properties of zinc-porphyrins as central molecular cores have been studied experimentally and theoretically in junctions formed from gold electrodes35–38, which allows us to make a comparative study of their properties within graphene junctions. We shall demonstrate that different modes of anchoring have a greater effect on graphene-based device properties compared with equivalent molecular junctions formed using gold electrodes, which suggests that the choice of anchor group should be targeted at a particular application. Depending on the choice of anchoring, the conductance could be higher or lower than measured conductance values of zinc-porphyrin wires using gold electrodes. For example, the conductance of zinc-porphyrin monomers (measured as 10-4.3 G0 on gold35) varies from 101.7 G0 with direct C-C anchoring to 10-5.8 G0 with pyrene anchors, 10-7.1 G0 with amine anchors and 10-8.1 G0 with TBF anchors. Despite these variations in conductance, the decay of conductance with the length of zinc-porphyrin wires connected to graphene electrodes is only slightly affected by the choice of anchor and the associated attenuation factor β varies only over a small range from 0.9 to 0.11 Å-1. Furthermore, we show that the sensitivity of electrical conductance to electrode shape and edge termination is more pronounced in junctions with direct CC covalent bonding to the graphene edges compared with robust π-π stacked anchoring. This suggests that C-C covalent anchoring is more desirable for sensing applications, whereas π-π stacking may be more advantageous for robust single-molecule transistors and thermoelectricity. Surprisingly, we find that quantum interference plays a significant role in controlling the electrical conductance with planar anchor groups, which despite their multi-π-orbital coupling to the electrodes, can exhibit both constructive and destructive interference39, depending on their connectivity to the porphyrin wire.

Figure 1. Graphene junctions bridged by porphyrin wires formed from one, two and three zinc porphyrin units (n = 1, 2, 3). (a) The porphyrin wires are contacted to the graphene by pyrene anchors. Other modes of anchoring studied include covalent anchoring to the edge of graphene electrodes via (b) a direct carbon-carbon bond and (c) amine anchors or through π-π interactions with (d) pyrene-1 anchors (e) pyrene-2 anchors and (f) tetrabenzofluorene (TBF) anchors. Details of these junctions are shown in fig. S1 of the supporting information (SI).

We study electron transport through the graphene nanojunctions shown in figure 1a, bridged by porphyrin nanowires with one, two and three (n=1, 2, 3) zinc porphyrin units (see fig. S2 in the SI for molecular structure of porphyrin nanowires and fig. S1 for details of the junctions) using density functional theory DFT combined with phase coherent, Green’s function-based scattering theory (see methods), because measurements35 of porphyrin molecular wires with different lengths show that room-temperature electron transport remains phase coherent up to length scales of order 3nm. In the device structure of figure 1a, monomer, dimer and trimer zinc porphyrin wires are linked to the graphene electrodes through either covalent C-C bonds to the edge of the graphene ribbons (fig. 1b), through amine anchors (fig. 1c) or π-π stacking through pyrene with two different connectivities (figs. 1d,e) or a tetrabenzofluorene TBF anchor group (fig. 1f). All molecular wires studied here are


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(figs. 2a,b,c) through π-π stacking with pyrene (for two different connectivities denoted pyrene-1 and pyrene-2) and TBF anchors. For comparison, we have also calculated the transmission 𝑇𝑇(𝐸𝐸) of MZnP, DZnP and TZnP (fig. 2d) anchored to gold electrodes using pyridine anchors (see SI for the relaxed junction geometries), which shows good agreement with measured conductances obtained elsewhere35. Upon increasing the number of ZnP units, the HOMO-LUMO gap shrinks and the conductance decreases in the co-tunnelling regime (ie when 𝐸𝐸𝐹𝐹 is in the vicinity of the middle of HOMO-LUMO gap41). As demonstrated in figures 2a,b,c, the anchor groups have strong effect on the electrical conductance in the vicinity of the middle of HOMO-LUMO gap. In particular, the conductance is significantly reduced (by ~ 2 orders of magnitude) when TBF anchors are used, due to a combination of two effects. First, whereas the pyrene anchor is planar, the TBF is twisted due to the interactions between the inner hydrogen atoms (see fig. S3 in the SI). This creates a non-uniform ππ overlap between π orbitals of TBF and the graphene electrodes, which reduces the electronic coupling. Secondly, the conjugation is broken at the cyclopentadiene connection point, which further reduces the conductance. As we show below, these features lead to negative differential conductance at finite source-drain voltages. For pyrene anchors, the conductance is sensitive to the choice of connection point to the pyrene (figs. 2a,b), which induces constructive or destructive interference near the gap centre. In the high conductance case (pyrene-1, fig. 2a), the conductance is in the range of 10-5 – 10-7 G0 in the vicinity of gap centre, which is close to the measured conductance of MZnP, DZnP and TZnP with gold electrodes35,36 and close to our calculation with gold electrodes shown in figure 2d. In addition, in contrast with the junctions formed from gold electrodes, where transport is LUMO (HOMO) dominated with pyridine35 (thiol36) anchors, the centre of the HOMO-LUMO gap is close to the DFT-predicted Fermi energy (EF = 0eV) when graphene electrodes are used and the slope of the transmission function at E=EF is low. In contrast the transmission functions of pyrene-2 junctions exhibit lower conductance near the gap centre (fig. 2b). It is worth to mention that this is less affected by the edge termination of graphene nano-gap i.e. the transmission function of pyrene2 junction with –OH graphene nano-gap edges (fig. S4 in the SI) shows lower conductance than pyrene-1 comparable to the junctions with H-termination. As shown in shown in figure 3c, figures S5 and S6 of the SI, the

synthetically feasible and the junctions are experimentally realisable. To study transport properties of the structures of figure 1, we first obtained their ground state geometry and mean field Hamiltonians using DFT and then used our transport code Gollum40 to calculate the transmission coefficient T(E) of electrons with energy E passing from one graphene electrode to other (see methods). Using the Landauer formula, the low-temperature, low-bias electrical conductance is given by 𝐺𝐺 = 𝐺𝐺0 𝑇𝑇(𝐸𝐸𝐹𝐹 ), where 𝐺𝐺0 is the conductance quantum and 𝐸𝐸𝐹𝐹 the Fermi energy of the electrodes, whereas the room-temperature conductance is obtained from an integral of T(E), weighted by the derivative of the Fermi function (see methods).

Figure 2. Transmission coefficients for electrons of energy E passing through the junctions with graphene electrodes possessing periodic boundary conditions. Monomer, dimer and trimer porphyrin wires connected to the graphene electrodes through π-π stacking (a) pyrene-1 anchor (b) pyrene-2 anchor, (c) TBF anchor or (d) gold electrode with pyridine anchor (fig. S1) or to the edges of the graphene electrodes with (e) amine, (f) a direct C-C bond. Further details of these junctions are shown in fig. S1 of the SI.

Figure 2 shows the electrical conductance 𝐺𝐺/𝐺𝐺0 (electron transmission coefficient 𝑇𝑇(𝐸𝐸𝐹𝐹 )) for the zinc porphyrin monomer (MZnP) (blue, n=1), dimer (DZnP) (red, n=2) and trimer (TZnP) (orange, n=3) bonded to the graphene


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HOMO and LUMO wavefunction of isolated pyrene are suppressed at the pyrene-2 connection point, leading to a low conductance. This demonstration of anchor-dependent quantum interference leading to destructive or constructive interference in the vicinity of middle of HOMO-LUMO gap (co-tunnelling regime) in the junctions with π-π anchoring is unusual, because electrons are injected into the planar anchor through multiple π orbitals. This effect can be understood using ‘magic ratio theory’41,42 combined with a quantum circuit rule43, which starts by noting that if the Fermi energy of the electrodes lies close to the centre of the HOMO-LUMO gap and provided the two anchor groups are weakly coupled to the central moiety and to the external electrodes, then the electrical conductance GA-B-A of a molecule of the form AB-A (comprising two anchor groups ‘A’ and a central moiety ‘B’ ) is proportional to the product GA × GB× GA , where GA (GB) are contributions to the electrical conductance from the separate entities ‘A’ and ‘B’43. For a polyaromatic-hydrocarbon (PAH) moiety ‘A’ such as pyrene, in which electrons enter the anchor group at site i and leave the anchor group at site j (or vice versa), GA is proportional to the square of the corresponding “magic integer” M α (Ct)-1 at the centre of the HOMO-LUMO gap where C is connectivity table41,42 obtained by inserting -1 for all connected sites in figure 3a and zero otherwise. For molecules such as pyrene (fig. 3a), which can be represented by bipartite lattices in which even numbered sites are connected to odd numbered sites only, Mij vanishes when i,j are both odd or both even41. When i is odd and j is even (or vice versa) the table of magic integers for pyrene is shown in figure 3b (note that Mij = Mji). For pyrene-1, the non-zero connectivities correspond to i=1 (green arrow in fig. 3a) and j even (or j=1 and i even), while for pyrene-2, j=2 (purple arrow in fig. 3a) and i is odd or vice versa41. The numbers in this table are a measure of the strength of the constructive interference associated with the different connectivities and since the numbers in row 1 (associated with pyrene-1 shaded green) are typically larger than those in column 2 (associated with pyrene-2 shaded purple), we anticipate that the electrical conductance of a molecule with pyrene-1 anchors will exceed that of the same molecule, but with pyrene-2 anchors. We now consider two scenarios. On the one hand, since the average squared magic number for pyrene-1 is (1/8) ∑𝑗𝑗 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒(𝑀𝑀1,𝑗𝑗 )2 = 32 × 7/8, whereas for pyrene-2 it is (1/8) ∑𝑖𝑖 𝑜𝑜𝑜𝑜𝑜𝑜 (𝑀𝑀𝑖𝑖,2 )2 = 27/8, we conclude that if electrons are only injected at a single atom, which is

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selected at random, then on average [GA for pyrene-1] / [GA for pyrene-2] = 7/3. Since GA-B-A is proportional to (GA)2GB, this suggests that [GA-B-A for pyrene-1] / [GA-B-A for pyrene-2] = (7/3)2 = 49/9 ≈ 5. On the other hand, we are interested in the case where the pyrene is π-stacked onto graphene and therefore electrons are injected in a phasecoherent manner, simultaneously into all sites of the pyrene, but with unknown amplitudes. This means that GA is proportional to the squared modulus of the sum of magic numbers41, weighted by their incoming amplitudes (rather than the sum of the squares), which for pyrene-1 is | ∑𝑗𝑗 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑟𝑟𝑗𝑗 𝑀𝑀1,𝑗𝑗 |2 . Ie if electrons are injected with uncorrelated random injection amplitudes 𝑟𝑟𝑖𝑖 , which satisfy < 𝑟𝑟𝑖𝑖 >= 0, < 𝑟𝑟𝑖𝑖∗ 𝑟𝑟𝑗𝑗 >= 𝜎𝜎 2 for 𝑖𝑖 = 𝑗𝑗 and < 𝑟𝑟𝑖𝑖∗ 𝑟𝑟𝑗𝑗 >= 0 for 𝑖𝑖 ≠ 𝑗𝑗, then for pyrene-1, < 𝐺𝐺𝐴𝐴 > is proportional to < | ∑𝑗𝑗 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑟𝑟𝑗𝑗 𝑀𝑀1,𝑗𝑗 |2 > = (∑𝑖𝑖,𝑗𝑗 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 < 𝑟𝑟𝑖𝑖∗ 𝑟𝑟𝑗𝑗 > 𝑀𝑀1,𝑗𝑗 𝑀𝑀1,𝑖𝑖 ) = 𝜎𝜎 2 × 32 × 7 and similarly for pyrene-2. Consequently their

conductance ratio remains at 49/9 ≈ 5.

Figure 3. Pyrene M-table and HOMO and LUMO isosurfaces. (a) pyrene numbering where green and purple arrows denote pyrene-1 and pyrene-2 connectivities respectively, (b) pyrene Mtable41 for given numbering (note that all odd to odd and even to even elements are zero). (c) HOMO and LUMO isosurfaces. Wavefunction amplitude is much lower in pyrene-2 conection point compared to pyrene-1 conection point in both HOMO and LUMO.

These scenarios correspond to two extremes of single-point entry and uncorrelated multi-point entry, but in both cases pyrene-1 has the higher conductance, because of the stronger constructive interference experienced by pyrene1. When a pyrene anchor is π-stacked to graphene, but the position of the pyrene varies randomly in time, but slowly on the timescale needed for an electron to traverse the


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amine anchors is due to the presence of the carbonyl group. This is illustrated by the local density of states shown in figure S10 in the SI, which is suppressed in the vicinity of the carbonyl group due to the broken conjugation.

molecule, the latter scenario is the more reasonable, although the random amplitudes could be more correlated. The above argument provides an estimate of the ratio of transmission coefficients at the centre of the HOMOLUMO gap. A detailed discussion of this effect at all energies in the presence of multi-point contacts using both simple tight binding model and material specific mean field Hamiltonian is presented in section 7 of the SI, figs S7-S9. It is interesting to note that since transmission functions of pyrene-2 junctions decreases with high slope around the middle of HOMO-LUMO gap, these junctions are attractive for thermoelectricity, because the Seebeck coefficient is proportional to the slope of ln T(E), dlnT(E)/dE at E=EF 44 and the slope of ln T(E) in the vicinity of gap centre is much higher with destructive pyrene-2 connectivity (fig. 2b) compared to the pyrene-1 connectivity (fig. 2a). We now consider the effect of coupling to the edge of graphene electrodes through either direct C-C bonds to carbon atoms on the edge of the graphene (fig. 1b) or direct covalent bonds to carbonyl groups on the edge of graphene via amine anchors (fig. 1c). Amine contacts have been tested experimentally with carbon nanotube and graphene electrodes45. Amine anchors may be experimentally relevant when electroburnt graphene junctions are formed in air, which could lead to the edges terminated with carboxylic groups created during the electroburning process16. Figures 2e,f show the conductance of MZnP, DZnP and TZnP junctions with amine-mediated (fig. 2e) and direct C-C bonds (fig. 2f). Clearly the conductance is significantly higher in the junctions with direct C-C bonds compared with all other junctions in figure 2 and furthermore the HOMO-LUMO gap is quite symmetrically positioned around the DFT Fermi energy. To demonstrate that the high conductance of junctions with direct C-C bonds is due to the higher coupling to the electrodes, we present the conductances of MZnP with different anchors on a linear scale in the figure 4a, which reveals that the width of the HOMO resonance is greatest in the junctions with direct C-C bonds. Since the widths of transmission resonances are determined by the coupling to the electrodes46, the higher width demonstrates that junctions with direct C-C bonds are coupled to the electrodes more strongly than all other junctions. The conductance order is as follows (see table S1): direct C-C coupling > π-π stacking with the pyrene-1 anchor > π-π stacking with the pyrene-2 anchor > direct coupling via amine anchors > π-π stacking with TBF anchors. The lower conductance for

Figure 4. Coupling strengthen to graphene with different anchors and attenuation β factor. (a) Conductance as a function of Fermi energy on a linear scale. These curves are re-plotted on linear scale from the blue curves in figure 2. The widths of the resonances plotted on a linear scale are an indication of the strength of the coupling to the electrodes. The highest coupling is obtained in the junctions with direct C-C coupling. The orders are as follows: direct C-C coupling > π-π stacking with pyrene-1 anchor > π-π stacking with pyrene-2 anchor > direct coupling through amine anchor > π-π stacking with TBF anchor. (b) The length dependent conductance of the porphyrin wires connected to graphene electrodes with different anchors and through the direct covalent bond or π-π stacking. Also shown is the length dependent conductance of the porphyrin wires connected to two gold electrodes through the pyridine anchor. All values are obtained using the DFT-predicted Fermi energy. The values of attenuation factor β are shown in the legend.

In the tunnelling regime, the conductance is expected to decay exponentially with length G =A exp(-βL), where L is the tip to tip length, A is a pre-factor and β is the attenuation factor. To demonstrate how the attenuation factor of ZnP depends on the electrodes and the anchors, figure 4b shows the length-dependent electrical conductance at room temperature, evaluated using the DFT-predicted Fermi


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energy (see table S1 in the SI). The β factor is of order ~0.9 to 0.11 Å-1 in the junctions with graphene electrodes, which is higher than that of gold-electrode junctions. Clearly the exponent β is only slightly affected by the choice of anchor group, whereas the overall magnitude of the conductance (ie the pre-factor A) strongly depends on the anchor. In the experiments, at large distances from the nanogap, the electrodes are of course much wider, but after successive electroburning steps, the electrodes in the vicinity of the gap is much narrower, with only a few scattering channels. This is why we now consider finite electrode sizes. In the above calculations, to focus attention on the role of the anchor groups and to minimise the effect of the transverse edges of the graphene ribbons, periodic boundary conditions were employed in the transverse direction (ie in the direction perpendicular to the current flow). To probe the effect of changing the shape and edges of the electrodes, we have repeated the above calculations using graphene-nanoribbon electrodes with hydrogenterminated edges. In agreement with other studies47, for graphene leads with periodic boundary conditions, two conduction channels are open around the Fermi energy, compared with one open channel in the hydrogenterminated case followed by three open channels for a small range of energies due to the edge states48,49 (see fig. S11 in the SI for the band structures, density of states and number of open conduction channels). Figure 5 shows the transmission coefficient T(E) for junctions with amine-mediated and direct C-C covalent bonds to the edges of the hydrogen-terminated graphene nanoribbons electrodes as well as π-π stacked configurations with pyrene-1 anchors. The main differences between periodic boundary conditions and Hterminated ribbons are associated with features in the local density of state LDOS near the tip surface in the latter, which are not present in the former. The H-terminated edge near the triangular ends of the ribbons creates a bound state and an associated resonance in the local density of states28,50, which couples to the continuum and creates a Fano resonance, which is visible when there is one open channel (ie in the hydrogen-terminated graphene nanoribbon lead), but hidden when there are two open channels (ie with periodic boundary conditions). Furthermore the coupling of the bound state to the transmitting channel depends on the type of anchor. In hydrogen-terminated graphene nanoribbon electrodes, since there is only one open channel, the transmission vanishes at the Fano resonance. However, in the electrode

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with periodic boundary conditions, there are two open channels and therefore when transmission is suppressed by a Fano resonance in one channel, finite transmission in the second channel prevents the overall transmission from vanishing.

Figure 5. Transmission coefficients for electrons of energy E passing through junctions with two hydrogen terminated graphene-nanoribbon electrodes. Monomer, dimer and trimer porphyrin wires connected to the electrodes with (a) amine, (b) pyrene-1and (c) direct C-C covalent bond connected to graphene electrodes. Further details of these junctions are shown in fig. S1 of the SI.

To investigate the sensitivity of different anchoring modalities to the shape and width of the graphene electrodes, we have calculated the transmission coefficient for the four different junction geometries connected to the graphene leads with direct C-C bonds. When we vary the electrode width or rotate the molecule 90 degrees with respect to the plane of the electrodes, or use rectangular lead surfaces (figs. S12b-d in the SI) compared with the circular (fig. S12a in the SI), the transmission coefficient is significantly affected by the shape and width of the electrodes (fig. S13 in the SI). This sensitivity arises because the local density of states at the carbon atoms connecting the leads to the molecule is affected by the interference patterns on the surface of the lead, which is sensitive to the electrode surface shape. Surprisingly, this could even lead to a negative β factor (the gap-centre behaviour in fig. S13a in the SI). It is interesting to note that in the junctions with direct C-C anchoring, molecule orientation has a significant effect on the conductance. For example, the conductance of a planar junction (fig. S13b,d) decreases by orders of magnitude if


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conjugation is broken. Non-planarity means that the electronic coupling between the TBF and graphene is weaker than that of pyrene, and consequently the level structure of the TBF is preserved. Furthermore the presence of the cyclopentadiene connection point as well as acetylene linker between the TBF anchors and the porphyrin creates a weak link, which at finite source-drain voltages, allows the energy levels of the TBF to be moved out of alignment with the levels of the porphyrin. If the energy levels of the anchor are not broadened by contact to the electrodes and if the levels can be moved out of alignment, then negative differential conductance can occur54. We investigate this possibility using a finitevoltage non-equilibrium Green’s function calculation as discussed in section 8 of the SI, fig. S15 and obtain the current-voltage characteristic shown in figure 6. This clearly shows negative differential conductance at voltages around +/- 1volt.

the zinc-porphyrin is rotated by 90 degrees (fig. S13c in the SI). This sensitivity to changes in the electrodes or ZnP angle in junctions anchored with direct C-C bonds may be desirable for sensing applications and for switching devices. The conductance sensitivity to the electrode surface shape and size with direct C-C bonding is less evident in junctions with π-π anchor groups such as pyrene (see fig. S14 in the SI). In the latter, the coupling to the molecule is through an array of overlapping π orbitals beneath the extended footprint of the planar anchor, in contrast with the direct C-C bond where only one conduction path is available through a single atom. This suggests that π-π anchoring is attractive for molecular electronics, since the conductance is less sensitive to details of the electrodes, provided the sliding of planar anchors along the surface of the graphene electrodes is not detrimental. For example, a study of the stability of planar anchor groups on graphene51 shows that the binding energy of pyrene on graphene is approximately -1.03eV (ie 100kJ per mol), which is 40 times higher than kBT at room temperature and comparable to the sulfur-gold bond. On the other hand, the binding energy to sliding and rotation is only 0.01eV, which is comparable to room temperature, and means that anchors can sample a range of anchoring positions on the surface on nanosecond timescales, unless they are pinned by defects or edges52. Table S2 of the SI shows the change in energy of pyrene anchors as they approach the edges of graphene electrodes and reveals that an energy barrier of up to 0.96eV prevents the pyrenes from sliding across the edge of a graphene ribbon. This means that once a pyreneanchored molecule bridges the gap, it is constrained to remain across the gap, even though the pyrenes can rapidly sample multiple locations within their respective electrodes. It is worth mentioning that at present, little is known about how the graphene edges are terminated in graphene nanojunctions formed by electroburning. However, images of the creation of carbon chains23,53 in vacuum suggest that the edges are not terminated at least in the transient process before the junction stabilized. These very reactive edges therefore would bind to molecules provided they were deposited in a vacuum-controlled environment or could be functionalised to obtain new anchoring modalities. Finally we examine the effect of non-planarity of TBF and the fact that this anchor couples to the porphyrin backbone via a cyclopentadiene connection point where the

Figure 6. The current-voltage characteristic in the zinc-porphyrin monomer with TBF anchors, showing negative differential conductance at V=+/-1V.

In summary, we have studied the electronic properties of zinc-porphyrin monomer, dimer and trimer molecular wires coupled to graphene electrodes through either covalent C-C bonds to the edges of the graphene, through amine anchor groups or via π-π stacking through pyrene and TBF anchors. The ordering of the electrical conductances with different anchors is as follows: direct CC coupling > π-π stacking with the pyrene-1 anchor > π-π stacking with the pyrene-2 anchor > direct coupling via amine anchors > π-π stacking with TBF anchors. The sensitivity of electrical conductance to electrode shape and edge termination is anchor-group dependent; the conductance of π-π stacked pyrene anchors is the least sensitive and direct C-C coupling is the most sensitive. This


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(1 + exp((𝐸𝐸 − 𝐸𝐸𝐹𝐹 )⁄𝑘𝑘𝐵𝐵 𝑇𝑇))−1 is the Fermi-Dirac distribution function, T is the temperature and kB= 8.6×10-5 eV/K is Boltzmann’s constant.

sensitivity suggests that direct C-C bonding may be the most desirable for sensing applications, because the local density of states is affected by analytes binding to the electrode surface. On the other hand, despite their planar, multi-atom coupling to the electrodes, pyrene anchors can exhibit both destructive interference and different degrees of constructive interference, depending on their connectivity to the porphyrin wire, which in view of the Mott formula55, makes them attractive for thermoelectric energy conversion. Finally, we predict that the nonplanarity of TBF anchors, combined with their weaker coupling to the porphyrin wires leads to negative differential conductance at finite source-drain voltages.

Acknowledgment H.S. thanks UK EPSRC for a post doctorate position funded by Quantum Effects in Electronic Nanodevices “QuEEN” platform grant no. EP/N017188/1. S.S. thanks the European Commission (EC) for a Marie Curie Early Stage Researcher position within EC FP7 ITN Molecular-Scale Electronics “MOLESCO” project no. 606728. This work was also supported by UK EPSRC grant no. EP/M014452/1. We thank H. Anderson and J. Thomas for useful discussions. Supporting Information Available containing details of geometries, electronic structure and transport properties of molecular junctions, and analytics of multi-point contacts.

Computational methods The optimized geometry and ground state Hamiltonian and overlap matrix elements of each structure studied in this paper was self-consistently obtained using the SIESTA56 implementation of density functional theory (DFT). SIESTA employs norm-conserving pseudo-potentials to account for the core electrons and linear combinations of atomic orbitals (LCAO) to construct the valence states. The generalized gradient approximation (GGA) of the exchange and correlation functional is used with the Perdew-Burke-Ernzerhof (PBE) parameterization57 and a double-ζ polarized (DZP) basis set. The real-space grid is defined with an equivalent energy cut-off of 250 Ry. The geometry optimization for each structure is performed to the forces smaller than 20 meV/Å. Where needed, the molecular orbital calculation and visualization of isolated molecules were obtained using Gaussian g0958 package with rb3lyp/6-31++g(d,p) functional and basis set. The mean-field Hamiltonian obtained from the converged SIESTA DFT calculation was combined with our implementation of the non-equilibrium Green’s function method, the Gollum40, to calculate the phase-coherent, elastic scattering properties of the each system consist of left (source) and right (drain) graphene leads connected to the scattering region formed from monomer, dimer and trimer porphyrin wires with different anchor groups. The transmission coefficient T(E) for electrons of energy E (passing from the source to the drain) is calculated via the relation 𝑇𝑇(𝐸𝐸) = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇(𝛤𝛤𝑅𝑅 (𝐸𝐸)𝐺𝐺 𝑅𝑅 (𝐸𝐸)𝛤𝛤𝐿𝐿 (𝐸𝐸)𝐺𝐺𝑅𝑅† (𝐸𝐸)). In this expression, 𝛤𝛤𝐿𝐿,𝑅𝑅 (𝐸𝐸) = 𝑖𝑖 �∑𝐿𝐿,𝑅𝑅 (𝐸𝐸) − ∑𝐿𝐿,𝑅𝑅 † (𝐸𝐸)�

describe

References (1) (2) (3)

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the

level broadening due to the coupling between left (L) and right (R) electrodes and the central scattering region, ∑𝐿𝐿,𝑅𝑅 (𝐸𝐸) are the retarded self-energies associated with this coupling and 𝐺𝐺 𝑅𝑅 = (𝐸𝐸𝐸𝐸 − 𝐻𝐻 − ∑𝐿𝐿 − ∑𝑅𝑅 )−1 is the retarded Green’s function, where H is the Hamiltonian and S is the overlap matrix. Using obtained transmission coefficient 𝑇𝑇(𝐸𝐸), the conductance could be calculated by Landauer formula 𝐺𝐺 = 𝐺𝐺0 ∫ 𝑑𝑑𝑑𝑑 𝑇𝑇(𝐸𝐸)(−𝜕𝜕𝜕𝜕(𝐸𝐸, 𝑇𝑇)/ 𝜕𝜕𝜕𝜕) where 𝐺𝐺0 = 2𝑒𝑒 2 /ℎ is the conductance quantum, 𝑓𝑓(𝐸𝐸) =

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Robust Molecular Anchoring to Graphene Electrodes - ACS Publications

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