Charge transport through extended molecular wires with strongly correlated electrons

Electron–electron interactions are at the heart of chemistry and understanding how to control them is crucial for the development of molecular-scale electronic devices. Here, we investigate single-electron tunneling through a redox-active edge-fused porphyrin trimer and demonstrate that its transport behavior is well described by the Hubbard dimer model, providing insights into the role of electron–electron interactions in charge transport. In particular, we empirically determine the molecule's on-site and inter-site electron–electron repulsion energies, which are in good agreement with density functional calculations, and establish the molecular electronic structure within various oxidation states. The gate-dependent rectification behavior confirms the selection rules and state degeneracies deduced from the Hubbard model. We demonstrate that current flow through the molecule is governed by a non-trivial set of vibrationally coupled electronic transitions between various many-body ground and excited states, and experimentally confirm the importance of electron–electron interactions in single-molecule devices.


Synthetic Procedure and Characterisation
FP3 was synthesized by Sonogashira coupling, as shown in Scheme 1. The Br2FP3 and TDP building blocks were made according to literature procedures. 1,2 All reagents were obtained from commercial sources and used as received without further purification. Toluene and DIPA were taken from a solvent drying system (MBraun MB-SPS-5-Bench Top) under nitrogen. Petroleum ether (PE) with a 40-60 °C boiling point range was used. Column chromatography was carried out using Merck Geduran silica gel 60 under N2 pressure. TLC was carried out on Merck silica gel 60 F254 Al plates. MALDI-TOF-MS was carried out in positive reflectron mode using a Bruker MALDI microflex instrument with dithranol as a matrix. NMR spectroscopy measurements were recorded using a Bruker AVII400 instrument. All peaks were referenced to the residual solvent peak. Scheme 1. Synthesis of fused porphyrin trimer with tridodecyloxypyrene anchoring groups. Ar = 3,5bis(trihexylsilyl)phenyl.

FP3
Br2FP3 (2.35 mg, 0.46 μmol) and TDP (1.79 mg, 2.3 μmol) were dissolved in toluene (0.8 mL) and DIPA (0.2 mL). The solution was degassed by freeze-pump-thaw three times before [Pd(PPh3)4] (0.1 mg, 0.08 μmol) and CuI (0.02 mg, 0.1 μmol) were added under a flow of argon. After another freezepump-thaw cycle, the mixture was heated to 50 °C for 3 hours. The solvents were removed and the product purified by chromatography (SiO2, PE:   The spectrum was recorded in a 0.1 M tetrabutylammonium hexafluorophosphate DCM solution as the electrolyte. Glassy carbon, platinum wire, and Ag|AgNO3 (10 mM) were used as the working, counter and reference electrodes respectively. Ferrocene was added at the end of the spectrum and the peaks reported are reference to the Fc+|Fc peak.

Device Fabrication
Device fabrication followed previously reported procedures. 2,3 Devices were fabricated on n-doped silicon wafers with 300 nm of thermally grown SiO2. For device A and device B the underlying doped silicon was used as a global gate for all devices on each chip. For device C a local gate electrode was patterned. The local gates were fabricated by optical lithography and e-beam evaporation of titanium (10 nm) and gold (30 nm). A dielectric layer of HfO2 (10 nm) was subsequently deposited by atomic layer deposition. Source and drain contact pads were patterned onto the SiO2 (device A and B) or HfO2 (device C) by optical lithography and e-beam evaporation of titanium and gold (10 nm / 60 nm).
CVD-grown monolayer graphene was transferred onto the devices by Graphenea. The graphene was patterned into bow-tie shapes with a width of approximately 100 nm at the narrowest point. First the devices were spin-coated with the negative tone resist ma-N 2403 and patterned using e-beam lithography with a dose of 120 µC cm −2 and an accelerating voltage of 50 kV. The pattern was developed with ma-D 525 to remove the unexposed resist, and unprotected regions of graphene were etched by O2 plasma. The developed resist was removed with an NMP-based remover REM660 to give the bowtie-shaped graphene. Finally, the patterned graphene was formed into a nanometerspaced graphene tunnel junctions, graphene nano-gaps, by feedback-controlled electroburning 4, 5 with a threshold resistance of 600 MΩ. The IV curves after electroburning were fitted with the Simmons model to estimate the spacing between the graphene source and drain electrodes to be around 1.5 nm.
The gate-dependence of the source-drain current was measured at room temperature before deposition of the molecular solution on an automated probe station. The molecules were deposited onto the graphene nano-gaps from a 2 µM toluene solution, and the devices were measured again.
In total 950 devices were fabricated by feedback-controlled electroburning, and were measured before and after FP3 deposition at room temperature. 101 devices were wire-bonded and cooled down to either 77 K or 4 K. 37 of these devices broke during wire-bonding or cool-down. 11 of the remaining 64 devices showed no sign of SET/Coulomb blockade, and 29 had gate coupling too weak or conductance too low to allow for further investigation. 10 showed a similar pattern of addition energies. 3 (device A, device B and device C) of these displayed N-4/N-3 transitions within the experimental window, and had sufficiently clean stability diagrams prior to molecular deposition to warrant modelling with the Hubbard framework.

Electrical Measurements
Device A and device B were wire-bonded into a chip carrier and measured in a dip-stick setup at 77 K. The dip-stick was evacuated and immersed in a dewar of liquid nitrogen. A HP33120A function generator was used to apply the source-drain voltage. The gate voltage was applied by a Keithley 2450 SourceMeter. A Stanford Research Systems SR570 low-noise current amplifier was used to measure the source-drain current, and the data collected by a National Instruments BNC-2090A DAQ. Device C was measured in an Oxford Instruments 4K Pucktester. An Adwin Gold II data acquisition system was used to apply the source-drain and gate voltages. An SR570 was used to measure the current which was collected by the Adwin Gold II.       A (a, b), B (d, e), and C (g, h). Conductance zero-bias gate traces before and after zero-bias gate traces for devices A (c) B (f), and C (i).

Electron-Vibration Coupling
Electron-vibration coupling can be incorporated into the energy dependence of the electron-transfer rate constants, → . For single-molecule junctions, the electron-transfer rate constants (assuming thermalized vibrations and the wide-band approximation) are given by: 3 where ρ = +1 for reduction processes or -1 for oxidation and µ is the the chemical potential of the transition, and = 2ℏ/Γ. The phononic correlation function, B(t) which can be thought of as a time-dependent Franck-Condon factor is given by: As mentioned in the main text, the spectral density, ( ), is constructed of two parts, ( ) = ∑ | | 2 ( − ) + ( ), that account for the inner and outer sphere contributions to the rates of electron transfer. The electron-vibration coupling constants, , are calculated from a DFT calculation from the optimized geometries and frequencies of FP3 in the relevant charge states (as shown in the following section). The second term is the outer sphere contribution that corresponds to the reorganization energy of the substrate and wider local environment upon charging the molecule. This is modelled by phenomenologically by a superohmic spectral density function of the form: where is the total outer sphere reorganization energy and is the cut-off frequency, (which is chosen to be 25 meV). 3 For the N-4/N-3 transition, the electron-transfer rates for reduction: → + and → − (and similarly for oxidation, +→ and −→ ) are assumed to be the same except for an offset in energy by spacing between the doublets, 2t. By making this assumption we take the geometric change that occurs upon oxidation to be greater than the geometric differences upon excitation from + −3 to − −3 . Therefore the experimental N-4/N-3 IV traces undergo fitting with three free parameters, , Γ , and Γ .
The N-3/N-2 resonant IV curve can also be fitted using this approach. The geometry of the N-2 state (FP3 2+ ) is optimized by DFT in the singlet or triplet ground state. Therefore the +→ − and +→ are evaluated separately. As with the N-4/N-3 transition we assume the geometry of the doublets, + −3 and − −3 are the same. The transitions to/from the triplet are offset in energy by ( −2 ) − ( -−2 ), as calculated from the Hubbard model for each device. Therefore, as before, the fitting parameters are: , Γ , and Γ , and the electronic coupling to all states for the same charge transition are assumed to be the same.

Calculation of Electron-Phonon Coupling Constants
Gaussian16 6 was used to carry out geometry optimisation and frequency calculations of FP3 in charge (and spin) states: N-4, N-3, and N-2 (singlet, S = 0) and N-2 (triplet, S = 1). A B3LYP/6-31G(d) functional/basis set was used. Frequency calculations confirmed that each optimised geometry was an energy minimum. Electron-vibration coupling constants, Λ , for mode, were calculated for each transition using a curvilinear coordinates system using the DUSHIN code, 7 and are displayed in Figure S13.
The total inner sphere reorganization energy, of FP3 upon electron transfer can be calculated as the sum of contributions from each mode: where = Λ 2 = / is the Huang-Rhys parameter.
Inner-sphere reorganization energies for electron transfer were also calculated using the single-point energy calculations using the equation: Where ( ) and +1 ( +1 ) are the energies of the molecule in charge state N and N+1 at their respective equilibrium geometries. ( +1 ) is the energy of the molecule in the N charge state but at equilibrium geometry of the N+1 charge state. Similarly +1 ( ) is the energy of the molecule in the equilibrium geometry of the N state, with N+1 electrons. 8 The inner sphere reorganization energies calculated from both methods match well, and are displayed in Figure S14.  The two states are in fact degenerate at ~ 30°, therefore the splitting of only a few meV observed experimentally is a reasonable value. At > 30°the wavefunction localises on the anchors and the overall ground state changes from singlet to triplet. (b) The energy of the N state as a function of , and the calculation of t at each value, demonstrating the dependence on the molecular conformation. The distortion along the angle is one of many potential low-energy modes that the molecular can deform along upon adsorption onto the nanogap.