Conductance of a single flexible molecular wire composed of alternating donor and acceptor units

Molecular-scale electronics is mainly concerned by understanding charge transport through individual molecules. A key issue here is the charge transport capability through a single—typically linear—molecule, characterized by the current decay with increasing length. To improve the conductance of individual polymers, molecular design often either involves the use of rigid ribbon/ladder-type structures, thereby sacrificing for flexibility of the molecular wire, or a zero band gap, typically associated with chemical instability. Here we show that a conjugated polymer composed of alternating donor and acceptor repeat units, synthesized directly by an on-surface polymerization, exhibits a very high conductance while maintaining both its flexible structure and a finite band gap. Importantly, electronic delocalization along the wire does not seem to be necessary as proven by spatial mapping of the electronic states along individual molecular wires. Our approach should facilitate the realization of flexible ‘soft' molecular-scale circuitry, for example, on bendable substrates.


Determination of the polymer conformation
The atomic positions in a molecule cannot be easily inferred from STM images and a precise determination requires to combine STM imaging and ESQC calculations 1 . However, some general informations on the molecular conformations can be obtained from measured distances and symmetry arguments when analyzing an STM image, as for instance for a DAD polymer: While the positions of the S atoms in the outer thiophene of a wire (endchains) remain ambiguous, the thiophene arrangement in the inner donor sites (bithiophene connections) can be determined by superimposing various chemical structures to the STM image ( Supplementary Fig.2a-d).
As an example, a straight segment of DAD chain is shown in Supplementary Fig.2a and the relative chemical structure with bithiophene in trans-configuration adapts fairly well to the STM image. Importantly, the other possible trans-arrangement of inner thiophenes (mirror-reflection of (a)) match less with the STM image (see the relative orientation of the acceptor sites compared to the bright lobes in the STM image). For completeness two other possibilities are plotted in Supplementary Fig.2, having chemical structures with the inner bithiophene connections in a cis-configuration (c-d). Notably, these cis arrangements do not allow to reproduce the imaged segment of wire.

Calculated electronic structure of a DAD monomer and (DAD) 2 dimer
The molecular orbitals of a DAD monomer, the electron transmission spectrum (with the tip apex located on a lateral thiadiazole group) and four characteristic ESQC images were calculated for the different electronic resonances as presented in Supplementary Fig.3 (the Fermi level is located around -10.5 eV for this semi-empirical calculation with only the valence electrons considered in ESQC). The HOMO resonance is extremely weak, indicating a small coupling of this electronic state with the Au(111) surface and explaining the difficulty to experimentally detect it by STM (see Fig.2 in the main text).
The A 1 and A 2 resonances derive from the central benzobis(thiadiazole) unit with a large STM junction conductance through a given thiadiazole group. The A 1 to A 2 resonance separation of 1.6 eV results from the large electronic interactions between the two thiadiazole  systems through the central phenyl  system. The two resonances above A 2 are characteristics of the LUMO+2 and LUMO+3 electronic states of the DAD monomer described in a mono-electronic approximation.
Calculations were also performed for a (DAD) 2 dimer ( Supplementary Fig.4). The HOMO resonance is still very weak as compared to the LUMO ones. The A 1 and A 2 resonances are almost at the same energy position as compared with the DAD monomer indicative of a weak electronic interaction through the central thiophene dimer. A new D 1 resonance is now nicely appearing in between. It results from the bonding states of the two thiophenes composing now the central thiophene dimers. The calculated STM image at the A 1 state is very characteristic with its four large lobes. The D 1 molecular state is now very well located on the central thiophene dimer part of the (DAD) 2 molecule. It will give rise to the D 1 second conduction band of the (DAD) n polymer. Notice the nice doublet structure of the A 1 states with a splitting of about 0.1 eV. There is also a doublet structure for the A 2 states (see below) but corresponding to a degenerate LUMO+1 mono-electronic state with its two corresponding molecular orbitals.

Flexibility of the DAD polymers
The presence of bent open structures and closed DAD rings testify the high degree of intermolecular flexibility of DAD-based structures. The most abundant closed structure is Supplementary Information C. Nacci et al.

Tunneling spectroscopy of a DAD ring
We identified donor-like and acceptor-like electronic states on closed structures too. As reported for a DAD 6 ring in Supplementary Fig.6a, dI/dV spectroscopy reveals again the electronic states A 1,2 and D 1 roughly at the same energetic position as found for its linear counterpart (Fig.2 in the main text). dI/dV conductance mapping clearly shows that acceptor states A 1 (Supplementary Fig.6d) and A 2 (Fig.2q in the main text) have the same pattern with the signal maximum at the acceptor groups positions, while the donor state D 1 ( Supplementary Fig.6e) reveal an inverted pattern as expected and as already seen for linear chains (Fig.2 in the main text). At negative bias voltages, the dI/dV spectroscopy does not reveal any molecular-related feature, and accordingly the conductance maps ( Supplementary  Fig.6b-c) are featureless.

Electronic destructive interference
Supplementary Fig.7 shows the electronic properties of the DAD polymer. The small dispersion of the A 1 , D 1 bands and the zero dispersion of the A 2 band can be clearly seen. The calculated dispersion of the A 1 states is 0.4 eV for long (DAD) n oligomers. In the D 1 , A 2 molecular orbital set, the D 1 states are dispersed on about a 0.25 eV range and the A 2 states not dispersed at all. There is a 1.3 eV calculated large separation between these two sets. Therefore the conduction band of a DAD polymer is composed of two not very dispersive sub-bands separated by a large gap.
The small 0.4 eV band dispersion of the A 1 conduction band correspond well to the 0.1 eV separation of the A 1 resonance doublet calculated for (DAD) 2 . The D 1 band is much more dispersive than the A 2 one because of the through acceptor electronic interaction between two dithiophene units along the (DAD) n backbone. The HOMO state dispersion is about 0.8 eV which will give rise to a moderate 1 eV valence band for the DAD polymer. To confirm the measured dispersion of the (DAD) n unoccupied states presented in Fig.3 (main text) and above, band structure calculations of the (DAD) n polymer were performed, compared with the (DAD) n oligomer electronic states dispersion. Then, the ESQC electronic transmission coefficient through a (DAD) n oligomer with n =10 was calculated and compared with a (Dphenyl-D) n oligomer with n = 10 to identify the destructive interference through the (DAD) n oligomer between the A 1 and the D 1 states while tunneling in the energy range between those two conduction bands.

Fitting I(z) curves
The measured I(z) curves are not flat (in a semi-logarithmic plot), but exhibit characteristic oscillations (as presented in the main text). Such a behavior has already been observed for polyfluorene chains and has therein combination with calculationsbeen assigned to the step-by-step detachment of the individual polymer units from the surface, thus the mechanical motion of the polymer 2 .
Following this interpretation, fitting of the I(z) curve is done by using only the minima (or maxima) of the oscillation (see Supplementary Fig.8) and the fitting function I = I o exp(-βz) (with I o being the contact current and β the inverse decay length of the current through the wire). Note that irregular or noisy pulling curves, where the current oscillations cannot be clearly identified, have not been considered in the data analysis. When doing a statistical analysis with various I(z) curves of different oligomer pulling experiments, all done at similar small bias voltages (between -100 mV and +100 mV), we obtain an average value of β = 0.21 Å -1 with ±0.06 Å -1 statistical error.

Calculated (E) and (V) curves
Supplementary Fig.9 shows the calculated transmission spectra for the DAD chain in a planar junction between two electrodes and in a STM configuration where the molecule is connected between the tip and the surface.
Due to the bending of the molecule in the STM configuration, the gap of the chain is decreased in the STM junction and one state is pushed out of the valence band to the middle of the gap. This distortion of the electronic structure due to the bending of the molecule affects to the exponential decay of the transmission and the decay of the current as shown in Supplementary Fig.11. In the planar configuration between two electrodes, the exponential decay as a function of the energy, in the case of the transmission, and as a function of the voltage, in the case of the current, is almost planar in the middle of the gap. The decay decreases when reaching the valence and conduction band and it is zero in the band where the transmission is pseudo ballistic. The state in the middle of the gap affects drastically the decay curves because a pseudo ballistic transmission is produced at the resonance energy.

Inverse decay length β for different oligomer lengths
The β decay factors determined within the investigated range of bias voltages (from -0.8 up to +1.1 V) scatter around the average value and do not reveal a clear tendency. Thus, they seem to be independent from the oligomer lengths and Z drop values (as shown in Supplementary  Fig.10).

Comparison with homogeneous molecular wires
To support our choice of alternating donor and acceptor groups further, we present in Supplementary Fig.12 a comparison with long poly(dithiophene-phenyl) wires, reflecting simply the backbone of DAD molecular wires. Adding lateral electron-acceptor groups (thiadiazole groups) to the poly(dithiophene-phenyl) (i.e. changing from Supplementary   Fig.12d to h) reveals a drastic change in terms of electronic structure.
A substantial shrinking of the HOMO-LUMO gap is predicted from calculations in favour of DAD wires (from 1.56 eV to 0.78 eV) with a clear benefit in terms of reduction of the inverse decay length  from 0.27 Å -1 (poly(dithiophene-phenyl)) down to 0.21 Å -1 (DAD, all in a planar configuration). A large dispersive conduction band is calculated in the case of the poly(dithiophene-phenyl) wire ( Supplementary Fig.12a), while this band splits for DAD wires, resulting in the presence of two not very dispersive sub-bands A 1 and D 1 as shown in Supplementary Fig.12e-f separated by a large gap of 1.3 eV. Calculated electron transmission coefficients in a planar configuration through both wires (Supplementary Fig.12c and g) clearly reveal the destructive interference between the A 1 and D 1 states while tunneling in the energy range between those two conduction sub-bands. This peculiar electronic states dispersion with a large gap between the A 1 and the D 1 states affects (E) via the m*(E) effective mass of tunneling electrons 3 confirming that (E) is not only controlled by the HOMO-LUMO gap but also by the curvature of the (E) inverse parabola.

Calculation of the effective mass m*
According to the electronic scattering theory applied to a periodic system (presenting an electronic band structure with gaps separating the bands), the mono-electronic Schrödinger equation can be solved for this periodic system assuming a real k vector or an imaginary k = iq vector (q in Supplementary Fig.13) 4 . For the real k, this gives rise to the standard E(k) band structure calculations, well-characterized in terms of scattering through the periodic system by a perfect T(E) transmission coefficient (for each real k vector E(k) band energy range).
For an imaginary k vector and for example between two bands, conduction band (CB) and valence band (VB), where in the corresponding energy range T(E) = 1, the dispersion relation has a parabolic form (as identified first by L. Brillouin for a genuine periodic system 5 ). Generalization from semiconductor quantum transport calculations (in the valence-conduction band gap) to molecular wires has been reported 6 and can now also be calculated by DFT methods for simple molecular wires with a minimum of hetero-atoms 7 .
For molecular wires, the number of calculated complex valued band structures remains limited as compared to semiconductor and magnetic materials since up to now there was no experimental way to measure the tunneling decay in the HOMO-LUMO gap through the molecular wire. In all these calculations, the effective m* mass is given by calculating the curvature of the imaginary paraboloid at its maximal extension 6 . In general, m* depends on the energy of the tunneling electrons 6 . But in many cases and for molecular wires with not too many hetero-atoms per unit cell, m* calculated exactly at the saddle point of the tunneling paraboloid (and kept constant) is enough to reproduce the entire parabola using the universal law for the tunneling decay:

Supplementary Equation 1
This is the case for example in the Supplementary Fig.13b, c and d band structures where in each case, the blue parabola was calculated using the above universal formula for (E) with a constant m*. This approximation is generally working for quite symmetric valence and conduction band structure molecular wires 6 . The DAD molecular wire is an interesting exception with hetero-atoms because it presents an asymmetry between the CB and the VB leading to a small deviation between the calculated tunneling paraboloid and the universal decay law with an assumed constant m*. This is presented in Supplementary Fig.13a by superimposing the blue universal law on the calculated complex valued band structure. The direct consequence is that the paraboloid of the complex-valued part of the band structure is flatter at its saddle point as compared to the universal law, leading as a consequence to a smaller  value and therefore to a smaller m* for the same HOMO-LUMO gap. This is a direct consequence of the A 1 , D 1 structure of the conduction band which is not present at the valence band side.
The fitting of the exact calculated paraboloid by the above (E) universal law is possible by finding a good m*(E) function. But this type of work is out of the scope of this work and enters now the theory of complex valued band structure as explored already 6,7 .

Calculation of the pulling mechanics
The mechanics of the pulling of a (DAD) 5 oligomer was studied theoretically by means of the semi-empirical ASED+ method 8 . The interaction of the (DAD) 5 oligomer with the Au(111) surface was described by means of van der Waals interactions 8 . As shown in previous works 2,3 , when the tip is approached to the dangling bond of the molecule then a chemical bond is created between the molecule and the tip. After, the molecule is pulled up in steps of 0.05Å. For each step, the geometry of the molecule is optimized until a threshold of 0.01eV/Å S17 is reached for the force on each atom. For each step in the pulling, the electronic transmission between the tip and the electrode was computed by the ESQC technique 9 .
The mechanics of the pulling reveals a rotation of the benzobis(thiadiazole) unit when this group is lifted up from the surface due to the van der Waals attraction. This rotation changes slightly the contact of the molecule with the surface making oscillations in the I(z) curve. This is in agreement with the experiments where oscillations of one unit size are observed. In the theoretical side, besides the 12 Å oscillation of a DAD unit, we obtain oscillations also for the benzobis(thiadiazole) and thiophene groups ( Supplementary Fig.11). Maximum points in the curve correspond to configurations where the oligomer is completely parallel to the surface while minimums correspond to geometries where the part of the molecule lifted up is bent respect to the surface.

Atomic coordinates for DAD molecules
The atomic coordinates (x,y,z where z is the distance from the surface) used in the calculations are presented in separated files: STM image calculations of DAD (Supplementary Data 1) presented in Supplementary Fig.8, of the dimer (DAD) 2 (Supplementary Data 2) presented in Supplementary Fig.9, for an oligomer (DAD) 10 (Supplementary Data 3) and the electronic transport calculations through (DAD) 10 ( Supplementary Fig.11). For DAD and (DAD) 2 , their molecular electronic structure on Au(111) were optimized using the semi-empirical ASED+ code.Furthermore, the Slater atomic orbitals parameters used in ESQC are presented (Supplementary Data 4).