Bias-induced conductance switching in single molecule junctions containing a redox-active transition metal complex
The paper provides a comprehensive theoretical description of electron transport through transition metal complexes in single molecule junctions, where the main focus is on an analysis of the structural parameters responsible for bias-induced conductance switching as found in recent experiments, where an interpretation was provided by our simulations. The switching could be theoretically explained by a two-channel model combining coherent electron transport and electron hopping, where the underlying mechanism could be identified as a charging of the molecule in the junction made possible by the presence of a localized electronic state on the transition metal center. In this article, we present a framework for the description of an electron hopping-based switching process within the semi-classical Marcus–Hush theory, where all relevant quantities are calculated on the basis of density functional theory (DFT). Additionally, structural aspects of the junction and their respective importance for the occurrence of irreversible switching are discussed.
KeywordsElectron transfer Single molecule electronics Density functional theory Marcus theory Redox reactions Nanostructures
Single molecule electronics (SME) provides a promising alternative to conventional semiconductor electronics, where it is envisioned that single, or small ensembles of molecules could be applied as active or passive building blocks in electronic circuits [2, 3].
A variety of possible applications for molecular components in electronic circuits could be identified in the past decades. Single molecules were proposed to function as both passive (wires) [4, 5, 6, 7, 8, 9, 10, 11, 12, 13] and active (diodes, transistors, switches) [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31] devices in electronic components, where their most significant benefit is that their intrinsic functionality can be designed reliably by means of chemical synthesis.
Single molecule switching mechanisms are based on either conformational changes triggered by photons [14, 15, 16, 17] or bias [18, 19, 20, 21, 22, 31, 32], spin crossover [24, 25, 26] or a redox reaction, which is performed via the introduction of oxidizing or reducing agents  or a gate electrode in an electrochemical cell [28, 29, 30].
The possible applicability of transition metal complexes for single molecule switches was first investigated by the group of Jens Ulstrup [33, 34, 35, 36] showing significant redox switching potential of such compounds within the junction supported by a good alignment of the molecular eigenstates with the electrodes Fermi energy. An electron transfer kinetics model was derived for the explanation of the trends found in these measurements [34, 35, 36, 37], which described the electron transport in such junctions as a two-step process of subsequent resonant tunneling events aided by the vibrational relaxation of the molecular orbitals. An adaptation of this scheme based on electron hopping in terms of Marcus theory for single molecule junctions has been addressed by Ulstrup and Kuznetsov [38, 39] and has been further developed by Nitzan and co-workers recently . Both groups, however, did not address the determination of the key parameters in a junction environment from density functional theory (DFT) calculations, but rather used model systems, such as rigid spheres between two metallic electrodes.
Migliore and Nitzan have recently proposed an explanation for hysteresis in single molecule I/V measurements based on the interplay of coherent tunneling, defining the conductance of the junction, and electron hopping causing a time delay or hysteresis in the I/V curves [41, 42]. The most important ingredient of this model is a localized state on the compound exhibiting a low degree of electronic coupling to the electrodes. Based on this model in combination with DFT calculations, hysteresis effects found in mechanically controlled break junction (MCBJ) experiments performed by Schwarz et al.  have been analyzed by us theoretically. In this work, three transition metal complexes with a Fe, Ru, and Mo-center, respectively, have been studied regarding their electronic ground state and switching properties, where the derivation of the structure-dependent key parameters for Migliore and Nitzan’s 2-channel scheme for these structures from DFT has been achieved.
This paper tries to move further in this direction with a special emphasis set on a more detailed analysis of the key quantities relevant for the occurrence of conductance switching in transition metal complex-based single molecule junctions and their relation to the structural properties of the respective compound.
Results and discussion
While the description of coherent electron transport in single molecule junctions is already well established, a treatment of incoherent sequential electron hopping in the literature is still mostly limited to intramolecular charge transfer in push–pull molecules or the charging of adsorbed molecules on a single surface as proposed in a series of articles by Rudolph Marcus [43, 44, 45].
The measurements which will be interpreted theoretically later in this paper, however, have been performed in UHV, where λ solv = 0; therefore, we would like to refer to an earlier paper  for our definition of λ solv in an electrochemical environment.
The calculation of λ in is straightforward, since it is the energy, which is required to relax the nuclei of the reactants from their energetic minimum at the systems equilibrium geometry, namely the uncharged molecule in the junction setup, to their optimal configuration in its final state, i.e., the charged compound between the surfaces after the charge has been transferred. Because during the reaction no significant structural rearrangement of the infinitely large metal electrodes takes place, only changes in the molecular geometry have to be considered for λ in. Therefore, λ in for a redox reaction in a single molecule junction has been calculated from the difference of the neutral (initial, i) molecule’s total energies in the equilibrium structure of its charged (final, f) state E 0(R f ) and its initial geometry E 0(R i ).
In panel b of Fig. 2 the transition into this barrierless bias regime is visible as an increase in k ox,K (black curves in Fig. 2) until the inflection point of the error function is reached at V crit, where k ox,K reaches half of its maximum value.
(We note that for λ − >0 V crit,ox,K = V crit,rd,K, which is the case depicted in Fig. 2.)
When the participation of both electrodes is taken into account, the respective reaction rates describing the redox reaction with one electrode and the molecule simply add up. This is due to the fact that no matter in which direction the electron(hole) exchange happens, it always results in a reduction or oxidation of the molecular species. These summed up reaction rates k rd and k ox are shown in panel c of Fig. 2. The total oxidation rate in the described case is still zero at small biases, since no oxidation reaction of the molecule, with an electron moving to any of the two electrodes happens. For V < V crit,ox,L and V > V crit,ox,R, however, a reaction involving one of the two leads happens with a frequency of γ K, while the opposite electrode does not participate to the same reaction. This is due to the fact that at V crit,L/R the energy barrier for the electron (or hole) transfer from the molecule to L/R is fully compensated by the applied bias, while on the other electrode (R/L) it is even increased due to the relation Φ L = −Φ R.
The total reduction rate k rd for our model system, in panel c of Fig. 2, on the other hand, is γ L + γ R, (or 2γ in a symmetric junction) for zero bias, which is due to the systems total energy reduction, when an electron occupying an electrode surface state at μ is transferred onto α with its eigenenergy ε 0. When no external potential is applied, the energy gain is the same on both electrodes, because μ L = μ R.
For V < V crit,rd,L or V > V crit,rd,R, however, the reaction with one of the two respective electrodes becomes unfavorable, since its Fermi energy is lowered by such an amount that the occupation of α with an electron arriving from the respective electrode does not lead to a total energy reduction anymore, therefore, leading to k rd,K → 0. However, k rd does never reach a value of zero since the reaction between the molecule and the other of the two electrodes is still barrierless and this electrode is, therefore, able to provide the electron for the reduction reaction.
Recently, Migliore and Nitzan  proposed a model mechanism causing hysteresis in I/V curves based on two different types of electron transfer reactions occurring simultaneously but on different time scales. While the faster one of the two reactions in this two-channel model is defining the measured conductance, the slower one is the reason for the hysteresis or switching observed in I/V measurements. In a single molecule junction setup, this means that coherent electron transport is mainly responsible for the conductance and defines the “fast channel”. For the switching in conductance for such compounds as described in this article, the most plausible mechanism is a change in the compounds redox state via electron hopping from one of the electrodes onto a localized eigenstate close to the electrodes’ Fermi level. This process can be quantitatively described in terms of electron transfer rates according to Marcus theory, as described above, where the key parameters are derived from DFT calculations.
Based on Migliore and Nitzan’s model, an algorithm for the simulation of such hysteresis effects and switching has been used for the theoretical analysis of the experimentally found bias driven switching found in Ref. , which we recapitulate in more detail in the following:
In a next step, the hopping reaction involving a weakly coupled MO has to be analyzed regarding its time scale, to determine if and how often the corresponding redox reaction happens within the time span for the measurement of one individual current value in the experiment.
Electronic coupling H α,K/eV of the molecular frontier orbitals to the electrodes as determined from Eq. (14)
4.9 × 10−2
2.1 × 10−2
2.0 × 10−2
5.0 × 10−2
2.6 × 10−3
1.6 × 10−3
1.2 × 10−5
In contrast to the Fe containing compounds 1 and 2, a triplet state has been determined as the ground state for compound 3, which contains Mo. As a consequence only for 3, a splitting of the eigenenergies is found for different spins, changing the energetic sequence of its MOs close to μ and only the MO containing the metal d xy AO is occupied for both spins for this compound, making it now its HOMO.
For all compounds, the spatial distributions of the frontier orbitals, which are situated near the Fermi Level of the electrodes in a junction setup, are shown as insets in Fig. 4. It can be seen that the d xz and d yz metal AOs hybridize rather strongly with the respective ligands leading to a delocalization of the resulting MOs in the transport direction, whose contribution to the phase coherent conductance is dominant. The d xy orbital, on the other side, is not oriented along the transport direction and is, therefore, not contributing to the coherent tunneling conductance. Its very low (but still finite) coupling to the metallic bands combined with its energetic proximity to μ in 3, however, makes this MO accessible for electron hopping, which can cause reversible, but now also irreversible switching events in I/V measurements, as we discuss in the following.
By applying the two-channel model described earlier in this article, we were able to reproduce the key characteristics of the experimentally determined I/V curves by Schwarz et al. , namely pocket-like hysteresis features for 2 and both reversible and irreversible switching for 3. In the following analysis, we would like to focus our attention on the irreversible switching events found for compound 3.
In the left panel of Fig. 5, the I/V sweeps resulting in irreversible switching in our simulations are shown for the positive bias range. Such irreversible switching has been found in 16 out of 100 independent simulation runs. As can be seen from the figure, in which the system resides in the lower conducting (reduced) state at the start, an oxidation reaction can happen once V crit,ox,r is reached, leading to a substantial increase in the conductance of the junction. In the selected curves, the reduction back into the ground state does not happen during the timespan of the simulation run, therefore, leaving the system in its charged state even when the bias is turned off again. As a consequence, on/off ratios of up to 200 can be achieved in these sweeps at small voltages. For 61 out of the 100 runs, on the other hand, after the oxidation of the compound into its charged state, a reduction back into its ground state happens during the respective simulation runs. This latter finding can be rationalized in terms of k ox and k rd, as shown in Fig. 2. For the oxidation reaction to occur, the system needs an applied voltage which reduces the energetic barrier defined by ΔG 0 and λ; therefore, this reaction is very unlikely before V crit,ox,K is reached. For the reduction reaction, on the other side, k rd does never fall below the preexponential γ defined in Eq. (3), since (at least) one of the electrodes always enables the reaction. Additionally, k rd even reaches its maximum of 2γ at biases in the range V crit,rd,L < V < V crit,rd,R, making the reduction of the system into its ground system even more probable.
Since the microscopic structure in experimental MCBJ junctions is unknown, structural information regarding their symmetry can only be deduced from individual I/V traces, which are rarely found to be symmetric with respect to the current direction. Therefore, we also studied asymmetry in our simulated junctions by introducing a factor H α,L/H α,R. In terms of Fig. 2, such an asymmetry factor changes the relation between k ox and k rd in the way, that in the bias range studied in Fig. 5, k ox is only dependent on H α,R, with k ox,L negligible in the whole positive bias range. For k rd, however, the situation is different in the sense that at V < V crit,rd,R both k rd,L and k rd,R are maximal, therefore, leading to k rd(V < V crit,rd,R) = γ L + γ R. For biases above V crit,rd,R, on the other side, k rd,R → 0, leading to k rd, (V > V crit,rd,R) = γ L. In other words, this means that reducing the ratio γ L/γ R. = (H α,L/H α,R ) 2 , while keeping H DA,R constant does not influence the rate of the oxidation reaction, while the reduction probability is strongly reduced. This finding indicates that the probability for irreversible switching events to occur is systematically enhanced by structural asymmetry in the junction.
Statistics of the switching behavior for 3, where the ratio between the coupling to the left and right electrodes is varied systematically
H DA,L/H DA,R
In summary, we gave a detailed account of the theory behind the measured irreversible switching events reported recently. These events can be explained in terms of electron hopping onto a localized state of the compound near the electrodes Fermi level. The bias dependence of the reaction rates for both oxidation and reduction has been discussed in a junction environment applying a model based on DFT results with coherent electron tunneling for the conductance and electron hopping for the switching, which enables us to qualitatively reproduce the experimentally found behavior. Statistics over 100 simulation runs show that irreversible switching happens in around 16 % of the cases, while reversible switching due to a reduction of the system back into its ground state is dominant. The ratio between irreversible and reversible switching events can, however, be increased by introducing asymmetry in the junction, which is also likely to be encountered in the MCBJ experiments the simulations are mimicking.
All electronic structure calculations in this paper were performed with the GPAW code [63, 64], in which the core electrons are described by the projector augmented wave (PAW) method and the basis set for the Kohn–Sham wave functions has been chosen to be a linear combination of atomic orbitals (LCAO)  on a double-zeta level with polarization functions (DZP) for all electronic structure calculations. The sampling of the potential energy term in the Hamiltonian is done on a real-space grid when using GPAW, for which we chose 0.18 Ǻ for its spacing and a Perdew–Burke–Ernzerhof (PBE) parametrization for the exchange–correlation (XC) functional throughout this paper. The scattering region for the NEGF-DFT scheme was defined by the molecular compound between two Au-fcc electrodes with 6 × 6 atoms in the surface plane in (111) orientation and one or three Au ad atoms for modeling top and hollow adsorption configurations, respectively. These rather large surfaces have been chosen for the purpose of excluding possible interactions of the molecule with its images in neighboring cells. All DFT calculations for such defined scattering regions were performed allowing for spin polarization and applying periodic boundary conditions, where seven layers of gold were used to reach Au bulk potential as required for the matching with the leads. The electronic structure for the lead regions has been obtained from Au bulk calculations with 6 × 6 × 3 Au atoms in the unit cell with a 1 × 1 × 15 k-point mesh, where the z direction was defining the transport direction.
Open access funding provided by TU Wien (TUW). We would like to thank Florian Schwarz, Emanuel Lörtscher, Koushik Venkatesan, and Heinz Berke our experimental partners in Ref.  for helpful discussions. GK gratefully received the grant “Stipendium der Monatshefte für Chemie” cofunded by the Austrian Academy of Sciences (ÖAW), the Austrian Chemical Society (GÖCH) and the Springer Media company. RS is currently supported by the Austrian Science fund FWF (project No. P27272). All DFT calculations have been carried out on the computing facilities of the Vienna Scientific Cluster VSC (project No. 70671).
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