The conductance of porphyrin-based molecular nanowires increases with length

High electrical conductance molecular nanowires are highly desirable components for future molecular-scale circuitry, but typically molecular wires act as tunnel barriers and their conductance decays exponentially with length. Here we demonstrate that the conductance of fused-oligo-porphyrin nanowires can be either length independent or increase with length at room temperature. We show that this negative attenuation is an intrinsic property of fused-oligo-porphyrin nanowires, but its manifestation depends on the electrode material or anchor groups. This highly-desirable, non-classical behaviour signals the quantum nature of transport through such wires. It arises, because with increasing length, the tendency for electrical conductance to decay is compensated by a decrease in their HOMO-LUMO gap. Our study reveals the potential of these molecular wires as interconnects in future molecular-scale circuitry.

: A schematic of a generic molecular junction and fused-oligo-porphyrin (FOP) monomer, dimer and trimer molecular wires. (a) Shows the schematic of a generic molecular junction containing a fused porphyrin trimer. (b) a porphyrin monomer connected to electrodes from m and m' connection points (c) A fused porphyrin dimer, comprising two monomers connected to each other through three single bonds (red bonds) and connected to electrodes from d and d' connection points and (d) A fused porphyrin trimer connected to electrodes from t and t' connection points. Figure 1 shows the molecular structure of a porphyrin monomer, a fused dimer and a fused trimer, in which two or three porphyrins are connected to each other through three single bonds (shown by red lines in fig. 1c,   1d). We first consider molecular junctions in which the carbon atoms labelled (m,m'), (d,d') and (t,t') respectively are connected to electrodes via acetylene linkers (see SI for the molecular structure of junctions). Figure 2a shows an example of the junction with graphene electrodes (see fig. S1a-c in the SI for the detailed molecular structure) where the porphyrin wires are connected to the edges of rectangular shaped graphene electrodes with periodic boundary conditions in the transverse direction. To calculate the room temperature electrical conductance G, we calculate the electron transmission coefficient T(E) using the Gollum transport code 27 combined with the material specific mean field Hamiltonian obtained from SIESTA implementation of density functional theory (DFT) 28 and then evaluate G using the Landauer formula (see methods). Results for the monomer, dimer and trimer attached to graphene electrodes (see figure 2a) are shown in figure 2b. Fig. 2: Transport through monomer, dimer and trimer molecular wires attached to two graphene electrodes. (a) A fused porphyrin molecular wire connected to graphene electrodes via acetylene linkers. (b) the room temperature electrical conductance for the porphyrin monomer (blue curve), porphyrin dimer (red curve) and porphyrin trimer (green curve) as a function of the electrode Fermi energy E F , in units of the conductance quantum G 0 = 77 micro siemens.
For these highly-conjugated wires, the energy level spacing decreases as their size increases. Therefore, the energy gap between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) of the dimer is smaller than that of the monomer and in turn, the HOMO-LUMO (HL) gap of the trimer is smaller than that of the dimer. This behaviour is reflected in the conductance resonances of figure 2b, which are furthest apart for the monomer (blue curve) and closest together for the trimer (green curve). This can be understood by starting from a chain of N isolated monomers. Since each monomer has a HOMO energy and a LUMO energy , the isolated chain has N-fold degenerate HOMO and N-fold degenerate LUMO. When the monomers are coupled together to form a fused wire, the degeneracies are lifted, to yield a HOMO, N-tuplet with molecular orbital energies < < ⋯ < … < and a LUMO, N-tuplet < < ⋯ < … < . Consequently the new HL gap ( ) = − is lower in energy than that of the monomer.  figure 3b shows that for the thiol-anchored wires, if − is lower than the mid-gap (0.18 eV) of the trimer, β is zero or slightly positive, otherwise β is negative. show that for a value α = -0.65γ, the curves overlap and for more negative values of α, the transmission coefficient increases with length for energies within the HL gap of the trimer ( fig. 4a), in agreement to the above DFT results. To demonstrate that the decrease in the HL gap is due to a splitting of the HOMO and LUMO degeneracies, figure 4b shows the transmission curves of the trimer over a larger range of energy, for a series of values of the coupling α. For small α, the HOMO and LUMO are each almost triply degenerate and as the magnitude of α increases, the degeneracy is increasingly lifted, leading to a reduction in the HL gap. semiconductors, meaning that eventually the conductance will begin to decrease with length. In practice, this decrease is likely to be slower than exponential, because at room temperature and large enough length scales, inelastic scattering will become significant and a cross-over from phase-coherent tunnelling to incoherent hopping will occur 10 . For comparison, figure S6 of the SI shows the transmission curves for butadiyne-linked porphyrin monomer, dimer and trimer molecular wires, for which the attenuation factor β is clearly positive for a wide range of energies within the HL gap of the trimer in agreement with the reported measured values 21 . The fact that fused porphyrin ribbons are narrow-gap semiconductors means that for a finite oligomer, when electrons tunnel through the gap there will be contributions to the transmission coefficient from both the HOMO and the LUMO bands. Figure S9 of the SI shows that the qualitative features of figure   4a and figure 2 can be obtained by summing these two contributions.
The tight-binding results of figure 4 and the DFT results with a non-specific anchor (figure 3) suggest that a negative beta factor is a generic feature of the fused porphyrin core, provided the centres of the HOMO-LUMO gaps of the monomer, dimer and trimer are coincident. However whether or not it is measured experimentally depends on level shifts of molecular orbitals after attaching to the electrodes. This is illustrated by the calculations shown in figure S10 in the SI using direct C-Au covalent anchoring to gold electrodes, where the HOMOs of the monomer, dimer and trimer coincide and therefore the centres of their HOMO-LUMO gaps are not coincident. This spoils the generic trend and leads to a positive beta factor.
In summary, we have demonstrated that the electrical conductance of fused oligo porphyrin molecular wires can either increase with increasing length or be length independent in junctions formed with graphene electrodes. This is due to alignment of the middle of the HOMO-LUMO gap of the molecules with the Fermi energy of the graphene electrodes. In addition, we show that in junctions formed with gold electrodes, this generic feature is anchor group dependent. This negative attenuation factor is due to the quantum nature of electron transport through such wires and arises from the narrowing of the HOMO-LUMO gap as the length of the oligomers increases.

Computational Methods
The Hamiltonian of the structures described in this paper was obtained using DFT (as described below) or constructed

Supporting Information
The conductance of porphyrin-based molecular nanowires increases with length    Then if we assume no interference between the HOMO and LUMO, the total transmission coefficient is