Conduction in molecular junctions: inelastic effects

Thermoelectricity in Molecular Tunnel Junctions

The growing interest in thermoelectric energy conversion technologies has recently extended to the molecular scale, with molecular tunnel junctions emerging as promising platforms for energy harvesting from heat in a quantum-tunneling regime. This Review explores the advances in thermoelectricity within molecular junctions, highlighting the unique ability of these junctions to exploit charge tunneling and controlled molecular structure to enhance thermoelectric performance. Molecular thermoelectrics, which bridge nanoscale material design and thermoelectric applications, utilize tunneling mechanisms, such as coherent tunneling and hopping processes, including coherent and incoherent pathways, to facilitate energy conversion. Complementing these mechanisms is an array of high-precision fabrication techniques for molecular junctions, from single-molecule break junctions to large-area liquid metal-based systems, each tailored to optimize heat and charge transfer properties. With novel design strategies such as the incorporation of electron-dense ligands, customizable anchor groups, and advanced junction architectures, molecular tunnel junctions hold promise for addressing challenging targets in thermoelectricity. This Review focuses on theoretical models, experimental methodologies, and design principles aimed at understanding the thermoelectric function in molecular junctions and enhancing the performance.

Protein charge transfer far from equilibrium: a theoretical perspective

Potential differences for protein-assisted electron transfer across lipid bilayers or in bio-nano setups can amount to several 100 mV; they lie far outside the range of linear response theory. We describe these situations by Pauli-master equations that are based on Marcus theory of charge transfer between self-trapped electrons and that obey Kirchhoff's current law. In addition, we take on-site blockade effects and a full non-linear response of the local potentials into account. We present analytical and numerical current-potential curves and electron populations for multi-site model systems and biological electron transfer chains. Based on these, we provide empirical rules for electron populations and chemical potentials along the chain. The Pauli-master mean-field results are validated by kinetic Monte Carlo simulations. We briefly discuss the biochemical and evolutionary aspects of our findings.

Electron-Vibration Entanglement of Resonating Dimers in Quantum Transport

Electron transport in a single molecule resulting from the superposition of its vibronic states depends on the coupling strength with the metallic leads. However, dynamical coherence and Fermionic correlation in molecule-molecule and molecule-lead coupling necessitates a critical approach to treat the current and its noise level, especially in the presence of a variable external bias for temperature-dependent conduction. Primarily, this work is a generalization of the theoretical approach of the atomic dimers to incorporate the effect of vibrational modes in current and conductance characteristics. The variation of current and differential conductance due to the external bias reveals a vibrational Coulomb blockade structure corresponding to the functioning vibrational mode in the system. The numerical demonstration for a diverse class of molecules generically shows that electron-vibration interaction can quantitatively predict the nature of coherent electron transport and current noise. Secondly, an attempt has been made to illustrate the effect of magnitude of coherence-induced noise suppression of current as a signature of electron-vibration entanglement. Finally, temperature-dependent conductance of the molecular junction in dimer structure has been estimated along with the peak shifts due to the applied gate voltage.

Generalized Voigt broadening due to thermal fluctuations of electromechanical nanosensors and molecular electronic junctions

Graphene and other 2D materials give a platform for electromechanical sensing of biomolecules in aqueous, room temperature environments. The electronic current changes in response to mechanical deflection, indicating the presence of forces due to interactions with, e.g., molecular species. We develop illustrative models of these sensors in order to give explicit, compact expressions for the current and signal-to-noise ratio. Electromechanical structures have an electron transmission function that follows a generalized Voigt profile, with thermal fluctuations giving a Gaussian smearing analogous to thermal Doppler broadening in solution/gas-phase spectroscopic applications. The Lorentzian component of the profile comes from the contact to the electrodes. After providing an accurate approximate form of this profile, we calculate the mechanical susceptibility for a representative two-level bridge and the current fluctuations for electromechanical detection. These results give the underlying mechanics of electromechanical sensing in more complex scenarios, such as graphene deflectometry.

Formulation of Long-Range Transport Rates through Molecular Bridges: From Unfurling to Hopping

Weak fluctuations about the rigid equilibrium structure of ordered molecular bridges drive charge transfer in donor-bridge-acceptor systems via quantum unfurling, which differs from both hopping and ballistic transfer, yet static disorder (low frequency motions) in the bridge is shown to induce a change of mechanism from unfurling to hopping when local fluctuations along the molecular bridge are uncorrelated. Remarkably, these two different transport mechanisms manifest in similar charge-transfer rates, which are nearly independent of the molecular bridge length. We propose an experimental test for distinguishing unfurling from hopping in DNA models with different helix directionality. A unified formulation explains the apparent similarity in the length dependence of the transfer rate despite the difference in the underlying transport mechanisms.

Single molecule electronic devices

Single molecule electronic devices in which individual molecules are utilized as active electronic components constitute a promising approach for the ultimate miniaturization and integration of electronic devices in nanotechnology through the bottom-up strategy. Thus, the ability to understand, control, and exploit charge transport at the level of single molecules has become a long-standing desire of scientists and engineers from different disciplines for various potential device applications. Indeed, a study on charge transport through single molecules attached to metallic electrodes is a very challenging task, but rapid advances have been made in recent years. This review article focuses on experimental aspects of electronic devices made with single molecules, with a primary focus on the characterization and manipulation of charge transport in this regime.

Heat transfer between elastic solids with randomly rough surfaces

We study the heat transfer between elastic solids with randomly rough surfaces.We include both the heat transfer from the area of real contact, and the heat transfer between the surfaces in the non-contact regions.We apply a recently developed contact mechanics theory, which accounts for the hierarchical nature of the contact between solids with roughness on many different length scales. For elastic contact, at the highest (atomic) resolution the area of real contact typically consists of atomic (nanometer) sized regions, and we discuss the implications of this for the heat transfer. For solids with very smooth surfaces, as is typical in many modern engineering applications, the interfacial separation in the non-contact regions will be very small, and for this case we show the importance of the radiative heat transfer associated with the evanescent electromagnetic waves which exist outside of all bodies.

Progress with molecular electronic junctions: meeting experimental challenges in design and fabrication

Molecular electronics seeks to incorporate molecular components as functional elements in electronic devices. There are numerous strategies reported to date for the fabrication, design, and characterization of such devices, but a broadly accepted example showing structure-dependent conductance behavior has not yet emerged. This progress report focuses on experimental methods for making both single-molecule and ensemble molecular junctions, and highlights key results from these efforts. Based on some general objectives of the field, particular experiments are presented to show progress in several important areas, and also to define those areas that still need attention. Some of the variable behavior of ostensibly similar junctions reported in the literature is attributable to differences in the way the junctions are fabricated. These differences are due, in part, to the multitude of methods for supporting the molecular layer on the substrate, including methods that utilize physical adsorption and covalent bonds, and to the numerous strategies for making top contacts. After discussing recent experimental progress in molecular electronics, an assessment of the current state of the field is presented, along with a proposed road map that can be used to assess progress in the future.

Magnetoresistance of nanoscale molecular devices based on Aharonov-Bohm interferometry

Control of conductance in molecular junctions is of key importance in the growing field of molecular electronics. The current in these junctions is often controlled by an electric gate designed to shift conductance peaks into the low bias regime. Magnetic fields, on the other hand, have rarely been used due to the small magnetic flux captured by molecular conductors (an exception is the Kondo effect in single-molecule transistors). This is in contrast to a related field, electronic transport through mesoscopic devices, where considerable activity with magnetic fields has led to a rich description of transport. The scarcity of experimental activity is due to the belief that significant magnetic response is obtained only when the magnetic flux is of the order of the quantum flux, while attaining such a flux for molecular and nanoscale devices requires unrealistic magnetic fields. Here we review recent theoretical work regarding the essential physical requirements necessary for the construction of nanometer-scale magnetoresistance devices based on an Aharonov-Bohm molecular interferometer. We show that control of the conductance properties using small fractions of a magnetic flux can be achieved by carefully adjusting the lifetime of the conducting electrons through a pre-selected single state that is well separated from other states due to quantum confinement effects. Using a simple analytical model and more elaborate atomistic calculations we demonstrate that magnetic fields which give rise to a magnetic flux comparable to 10(-3) of the quantum flux can be used to switch a class of different molecular and nanometer rings, ranging from quantum corrals, carbon nanotubes and even a molecular ring composed of polyconjugated aromatic materials. The unique characteristics of the magnetic field as a gate is further discussed and demonstrated in two different directions. First, a three-terminal molecular router devices that can function as a parallel logic gate, processing two logic operations simultaneously, is presented. Second, the role of inelastic effects arising from electron-phonon couplings on the magnetoresistance properties is analyzed. We show that a remarkable difference between electric and magnetic gating is also revealed when inelastic effects become significant. The inelastic broadening of response curves to electric gates is replaced by a narrowing of magnetoconductance peaks, thereby enhancing the sensitivity of the device.

Inelastic effects in Aharonov-Bohm molecular interferometers

Inelastic effects arising from electron-phonon coupling in molecular Aharonov-Bohm (AB) interferometers are studied using the nonequilibrium Green's function method. Results for the magnetoconductance are compared for different values of the electron-phonon coupling strength. At low-bias voltages, the coupling to the phonons does not change the lifetime and leads mainly to scattering phase shifts of the conducting electrons. As a result of these dephasing processes, the magnetoconductance of the molecular AB interferometer becomes more sensitive to the threading magnetic flux as the electron-phonon coupling is increased, opposite to the behavior of an electric gate.

Extended spin-boson model for nonadiabatic hydrogen tunneling in the condensed phase

A nonadiabatic rate expression for hydrogen tunneling reactions in the condensed phase is derived for a model system described by a modified spin-boson Hamiltonian with a tunneling matrix element exponentially dependent on the hydrogen donor-acceptor distance. In this model, the two-level system representing the localized hydrogen vibrational states is linearly coupled to the donor-acceptor vibrational mode and the harmonic bath. The Hamiltonian also includes bilinear coupling between the donor-acceptor mode and the bath oscillators. This coupling provides a mechanism for energy exchange between the two-level system and the bath through the donor-acceptor mode, thereby facilitating convergence of the time integral of the probability flux correlation function for the case of weak coupling between the two-level system and the bath. The dependence of the rate constant on the model parameters and the temperature is analyzed in various regimes. Anomalous behavior of the rate constant is observed in the weak solvation regime for model systems that lack an effective mechanism for energy exchange between the two-level system and the bath. This theoretical formulation is applicable to a wide range of chemical and biological processes, including neutral hydrogen transfer reactions with small solvent reorganization energies.

Thermal gating of the single molecule conductance of alkanedithiols

The temperature dependence of the single molecule conductance (SMC) of alpha,omega-alkanedithiols has been investigated using a scanning tunnelling microscopy (STM) method. This is based on trapping molecules between a gold STM tip and a gold substrate and measuring directly the current across the molecule under different applied potentials. A pronounced temperature dependence of the conductance, which scales logarithmically with T(1), is observed in the temperature range between 293 and 353 K. It is proposed the origin of this dependence is the change in distribution between molecular conformers rather than changes in either the conduction mechanism or the electronic structure of molecule. For alkanedithiols the time averaged conformer distribution shifts to less elongated conformers at higher temperatures thus giving rise to higher conductance across the molecular bridges. This is analysed by first calculating energy differences between different conformers and then calculating their partition distribution. A simple tunnelling model is then used to calculate the temperature dependent conductance based on the conformer distribution. These findings demonstrate that charge transport through single organic molecules at ambient temperatures is a subtle and highly dynamic process that cannot be described by analysing only one molecular conformation corresponding to the lowest energy geometry of the molecule.

Theoretical studies of electron transfer through dendrimeric architecture

We have analyzed the steady-state electron transfer rate through a bridge of dendrimeric architecture. The difference between the linear chain and the dendrimeric architecture has also been demonstrated with steady-state rate as a main observable in the coherent and incoherent regimes of interactions. It is shown that generally the rate of electron transfer in dendrimeric architecture is faster than the rate associated with their linear chain counterpart with similar kind of bonding connectivities. The rate depends upon the size of the molecule, core branching, and the nature of the coupling among the different nodes on the dendrimer molecule. Depending upon the nature of the donor and acceptor, phenomenological dephasing coefficient due to environment and the geometry of the dendrimeric architecture, the modification of electron transfer rate has been studied. In the regime of fully coherent interactions where all quantum effects are considered the rate shows a multiple inversion due to the dendrimer architecture which is neither available in the regime of incoherent interaction nor in the linear chain case in similar condition. We have discussed about the applicability of our model in metal-molecule-metal junction, photoinduced electron transfer process, and molecular conductor.

Spin-boson thermal rectifier

Rectification of heat transfer in nanodevices can be realized by combining the system inherent anharmonicity with structural asymmetry. We analyze this phenomenon within the simplest anharmonic system-a spin-boson nanojunction model. We consider two variants of the model that yield, for the first time, analytical solutions: a linear separable model in which the heat reservoirs contribute additively, and a nonseparable model suitable for a stronger system-bath interaction. Both models show asymmetric (rectifying) heat conduction when the couplings to the heat reservoirs are different.

Effect of electron-phonon coupling on the conductance of a one-dimensional molecular wire

The effect of inelastic scattering, particularly that of the electron-phonon interactions, on the current-voltage characteristics of a one-dimensional tight-binding molecular wire has been investigated. The wire has been modeled using the Su-Schreiffer-Heeger Hamiltonian and we compute the current using the Landauer's scattering formalism. Our calculations show that the presence of strong electron-lattice coupling in the wire can induce regions of negative differential resistance (NDR) in the I-V curves. The reasons for this can be traced back to the quasidegeneracy in few of the low-energy molecular levels in the presence of electron-phonon coupling and an external applied bias. The molecular levels become highly delocalized at the critical bias at which the NDR is seen, corresponding to the vanishing of the electron-phonon coupling with equal bond lengths.

Bridge mediated two-electron transfer reactions: On the influence of intersite Coulomb interactions

Donor-acceptor two-electron transfer (TET) mediated by a linear molecular bridge is described theoretically. The particular case is considered where the TET takes place in the presence of a strong electronic intersite coupling within the bridge and against the background of fast vibrational relaxation processes. For such a situation the coarse-grained description of bridge-assisted electron transfer in molecular systems can be utilized [Petrov et al., J. Phys. Chem. B 106, 3092 (2002)]. In the present case it leads to kinetic equations and rate expression for TET reactions. Our recent treatment of completely nonadiabtic TET reactions [Petrov et al., J. Chem. Phys. 120, 4441 (2004)] including a reduction to single-exponential kinetics (with overall transfer rate K(TET)) is generalized here to the case of strong intrabridge coupling and the presence of intersite Coulomb interactions. The dependence of K(TET) on the bridge length which is determined by a separate stepwise and concerted contribution is discussed in detail. It is found that the intersite Coulomb interaction favors the TET if the donor and the acceptor are uncharged in their completely reduced states (with two excess electrons present).

Vibrational effects in laser-driven molecular wires

The influence of an electron-vibrational coupling on the laser control of electron transport through a molecular wire that is attached to several electronic leads is investigated. These molecular vibrational modes induce an effective electron-electron interaction. In the regime where the wire electrons couple weakly to both the external leads and the vibrational modes, we derive within a Hartree-Fock approximation a nonlinear set of quantum kinetic equations. The quantum kinetic theory is then used to evaluate the laser driven, time-averaged electron current through the wire-leads contacts. This formalism is applied to two archetypical situations in the presence of electron-vibrational effects, namely, (i) the generation of a ratchet or pump current in a symmetrical molecule by a harmonic mixing field and (ii) the laser switching of the current through the molecule.

Thermally Activated Conduction in Molecular Junctions

We report temperature dependence measurements on the conductance of individual molecular wires. The results show for the first time in a molecular junction the theoretically predicted transition from coherent superexchange tunneling conductance to an activated hopping mechanism as temperature is increased.

Bridge mediated two-electron transfer reactions: Analysis of stepwise and concerted pathways

A theory of nonadiabatic donor (D)-acceptor (A) two-electron transfer (TET) mediated by a single regular bridge (B) is developed. The presence of different intermediate two-electron states connecting the reactant state D-(-)BA with the product state DBA-(-) results in complex multiexponential kinetics. The conditions are discussed at which a reduction to two-exponential as well as single-exponential kinetics becomes possible. For the latter case the rate KTET is calculated, which describes the bridge-mediated reaction as an effective two-electron D-A transfer. In the limit of small populations of the intermediate TET states D-B-A, DB-(-)A, D-BA-, and DB-A-, KTET is obtained as a sum of the rates KTET(step) and KTET(sup). The first rate describes stepwise TET originated by transitions of a single electron. It starts at D-(-)BA and reaches DBA-(-) via the intermediate state D-BA-. These transitions cover contributions from sequential as well as superexchange reactions all including reduced bridge states. In contrast, a specific two-electron superexchange mechanism from D-(-)BA to DBA-(-) defines KTET(sup). An analytic dependence of KTET(step) and KTET(sup) on the number of bridging units is presented and different regimes of D-A TET are studied.

Ab initiostudy of the alternating current impedance of a molecular junction

The small-bias conductance of the C6 molecule, stretched between two metallic leads, is studied using time-dependent density functional theory within the adiabatic local density approximation. The leads are modeled by jellium slabs, the electronic density and the current density are described on a grid, whereas the core electrons and the highly oscillating valence orbitals are approximated using standard norm-conserving pseudopotentials. The jellium leads are supplemented by a complex absorbing potential that serves to absorb charge reaching the edge of the electrodes and hence mimic irreversible flow into the macroscopic metal. The system is rapidly exposed to a ramp potential directed along the C6 axis, which gives rise to the onset of charge and current oscillations. As time progresses, a fast redistribution of the molecular charge is observed, which translates into a direct current response. Accompanying the dc signal, alternating current fluctuations of charge and currents within the molecule and the metallic leads are observed. These form the complex impedance of the molecule and are especially strong at the plasmon frequency of the leads and the lowest excitation peak of C6. We study the molecular conductance in two limits: the strong coupling limit, where the edge atoms of the chain are submerged in the jellium and the weak coupling case, where the carbon atoms and the leads do not overlap spatially.

Two-electron transfer reactions in proteins: bridge-mediated and proton-assisted processes

Nonadiabatic two-electron transfer (TET) reactions through donor-bridge-acceptor (DBA) systems is investigated within the approximation of fast vibrational relaxation. For TET reactions in which the population of bridging states remains small (less than 10(-2)) it is demonstrated that a multiexponential transition process reduces to three-state kinetics. The transfer starts at the state with two excess electrons at the D center (D(2-)BA), goes through the intermediate (transient) state with one electron at the D center and one at the A center (D-BA-), and ends up with the two electrons at the A center (DBA2-). Furthermore, if the population of the intermediate state becomes also small the two-exponential kinetics can be transformed with high accuracy to single-exponential D-A TET kinetics. The related overall transfer rate contains contributions from stepwise and from concerted TET. The latter process is determined by a specific two-electron superexchange coupling incorporating the bridging states (D-B-A and DB-A-) as well as the intermediate state (D-BA-). As an example, the reduction of micothione reductase by nicotinamide adenine dinucleotide phosphate is analyzed. Existing experimental data can be explained if one assumes that the proton-assisted reduction of the enzyme is realized by the concerted TET mechanism.

Redox State Dependence of Single Molecule Conductivity

Spontaneous formation of stable molecular wires between a gold scanning tunneling microscopy (STM) tip and substrate is observed when the sample has a low coverage of alpha,omega-dithiol molecules and the tunneling resistance is made sufficiently small. Current-distance curves taken under these conditions exhibit characteristic current plateaux at large tip-substrate separations from which the conductivity of a single molecule can be obtained. The versatility of this technique is demonstrated using redox-active molecules under potential control, where substantial reversible conductivity changes from 0.5 to 2.8 nS were observed when the molecule was electrochemically switched from the oxidized to the reduced state.