Incoherent Hopping Research Articles

Electrochemical gating for single-molecule electronics with hybrid Au|graphene contacts.

The single-molecular conductance of a redox active viologen molecular bridge between Au|graphene electrodes has been studied in an electrochemical gating configuration in an ionic liquid medium. A clear "off-on-off" conductance switching behaviour has been achieved through gating of the redox state when the electrochemical potential is swept. The Au|viologen|graphene junctions show single-molecule conductance maxima centred close to the equilibrium redox potentials for both reduction steps. The peak conductance of Au|viologen|graphene junctions during the first reduction is significantly higher than that of previously measured Au|viologen|Au junctions. This shows that even though the central viologen moiety is not directly linked to the enclosing electrodes, substituting one gold contact for a graphene one nevertheless has a significant impact on junction conductance values. The experimental data was compared against two theoretical models, namely a phase coherent tunnelling and an incoherent "hopping" model. The former is a simple gating monoelectronic model within density functional theory (DFT) which discloses the charge state evolution of the molecule with electrode potential. The latter model is the collective Kuznetsov Ulstrup model for 2-step sequential charge transport through the redox centre in the adiabatic limit. The comparison of both models to the experimental data is discussed for the first time. This work opens perspectives for graphene-based molecular transistors with more effective gating and fundamental understanding of electrochemical electron transfer at the single molecular level.

Size effect on optical conductivity of Bi2Te3 nanoparticles due to softening of phonon frequencies

The frequency-dependent optical conductivity σ(w) strongly depends on the particle size in nanostructures. Phonon frequencies are believed to be decrease on a decrease in particle size due to the softening of phonon frequencies around grain boundaries in nanomaterials. The optical conductivity of Bi2Te3 nanoparticles is studied using the two components model: First is the coherent motion of Drude electrons within the Bi2Te3 nanoparticles, and second is the incoherent hopping of charge carriers from one Bi2Te3 nanoparticle to another. The model has the phonon frequency and relaxation rates as the independent free parameters, which depend on nanoparticle size. The relaxation rates that are a function of frequency are incorporated in optical conductivity through memory functions. The coherent motion of Drude carriers exhibited a peak in intensity near zero frequency regime and decreased to a minimum value in the far-infrared region. The incoherent hopping of electrons from one Bi2Te3 nanoparticle to another produces a broad peak in the frequency-dependent optical conductivity at around the mid-infrared region. On decreases in nanoparticle size, softening of phonon frequencies occurs at interfaces due to an increase in interface volume ratio; as a result, the peak in optical conductivity at the mid-infrared region shifts towards lower frequencies.

Effects of localization length and spatial disorder on a charge carrier mobility in organic disordered semiconductors

We investigate the transport properties of charge carrier disordered organic semiconductors with a focus on the determination and analysis of charge carrier mobility. By understanding that charge transport is due to incoherent hopping of charge carriers across localized states, we use a model that relates mobility to charge carrier (not small polarons) hopping by thermal activation. We consider the Miller–Abrahams expression to describe the hopping rate of charge carriers and employ kinetic Monte Carlo simulation methods to generate data from which we can analyze charge carrier mobility as a function of applied electric field, temperature, localization length, and spatial and energetic disorder parameters. Based on our results, we discuss the effects of these parameters on charge carrier mobility. Our results show the importance of the spatial disorder parameter and localization length in the effects of electric field dependence on charge carrier mobility, and we also evaluate the value of localization length that has been mostly considered as 0.1b, where b is the lattice parameter.

Origin of Chirality Induced Spin Selectivity in Photoinduced Electron Transfer.

Here we propose a mechanism by which spin-polarization can be generated dynamically in chiral molecular systems undergoing photoinduced electron transfer. The proposed mechanism explains how spin-polarization emerges in systems where charge transport is dominated by incoherent hopping, mediated by spin-orbit and electronic exchange couplings through an intermediate charge transfer state. We derive a simple expression for the spin-polarization that predicts a nonmonotonic temperature dependence, consistent with recent experiments, and a maximum spin-polarization that is independent of the magnitude of the spin-orbit coupling. We validate this theory using approximate quantum master equations and the numerically exact hierarchical equations of motion. The proposed mechanism of chirality induced spin selectivity should apply to many chiral systems, and the ideas presented here have implications for the study of spin transport at temperatures relevant to biology and provide simple principles for the molecular control of spins in fluctuating environments.

Fully Quantum Modeling of Exciton Diffusion in Mesoscale Light Harvesting Systems.

It has long been a challenge to accurately and efficiently simulate exciton–phonon dynamics in mesoscale photosynthetic systems with a fully quantum mechanical treatment due to extensive computational resources required. In this work, we tackle this seemingly intractable problem by combining the Dirac–Frenkel time-dependent variational method with Davydov trial states and implementing the algorithm in graphic processing units. The phonons are treated on the same footing as the exciton. Tested with toy models, which are nanoarrays of the B850 pigments from the light harvesting 2 complexes of purple bacteria, the methodology is adopted to describe exciton diffusion in huge systems containing more than 1600 molecules. The superradiance enhancement factor extracted from the simulations indicates an exciton delocalization over two to three pigments, in agreement with measurements of fluorescence quantum yield and lifetime in B850 systems. With fractal analysis of the exciton dynamics, it is found that exciton transfer in B850 nanoarrays exhibits a superdiffusion component for about 500 fs. Treating the B850 ring as an aggregate and modeling the inter-ring exciton transfer as incoherent hopping, we also apply the method of classical master equations to estimate exciton diffusion properties in one-dimensional (1D) and two-dimensional (2D) B850 nanoarrays using derived analytical expressions of time-dependent excitation probabilities. For both coherent and incoherent propagation, faster energy transfer is uncovered in 2D nanoarrays than 1D chains, owing to availability of more numerous propagating channels in the 2D arrangement.

Modulated-photocurrent spectroscopy of single-crystal organic semiconductor rubrene with pristine and trap-dominated surfaces

High-performance, benchmark organic semiconductor rubrene is investigated by modulated-photocurrent (MPC) spectroscopy. In this technique, periodically modulated and steady-state photocarrier populations are simultaneously generated in the sample by two independent light beams, with the resultant photocurrent measured by a lock-in amplifier. The technique allows identifying a bandlike carrier motion (as opposed to incoherent hopping) as the dominant charge transport mechanism in this highly crystalline molecular semiconductor, in agreement with prior studies, yet without the need for fabrication of complex transistor devices. Moreover, MPC spectroscopy is used to determine the important parameters, such as the density of states for traps, their carrier capture coefficient, and attempt to escape frequency, as well as the quantum efficiency of photocarrier generation and the photocarrier mobility. The MPC spectroscopy is also shown to be able to discriminate charge transport at the surface of the crystal from that occurring in the bulk. The wealth of the thus obtained information is essential for our better understanding of the microscopic mechanisms governing photoconductivity in this class of materials.

Quantum versus classical transport of energy in coupled two-level systems

We consider the problem of energy transport in a chain of coupled quantum systems with the goal of shedding light on how nonclassical resources can affect transport. We study the cases for which either coherent or incoherent energy hopping takes place in the chain. Here, incoherent energy hopping is referred to as the "classical" scenario in allusion to its fully diagonal dynamics in the basis formed by the eigenstates of the decoupled sites. We focus on the case of a linear chain of two-level sites and find a hopping rate threshold above which the coherent quantum case is more efficient than the incoherent counterpart. We then link the quantum hopping rate to the coherence global maximum, which allows us to state that there is a coherence threshold above which the quantum scenario is more efficient. Next, we consider the integrated coherence generated by the dynamics and show how it is related to what is known as the invasiveness of a quantum operation. Our results strongly suggest the significant role played by quantum invasiveness as a resource for quantum transport.

Insights into Photosynthetic Energy Transfer Gained from Free-Energy Structure: Coherent Transport, Incoherent Hopping, and Vibrational Assistance Revisited.

Giant strides in ultrashort laser pulse technology have enabled real-time observation of dynamical processes in complex molecular systems. Specifically, the discovery of oscillatory transients in the two-dimensional electronic spectra of photosynthetic systems stimulated a number of theoretical investigations exploring the possible physical mechanisms of the remarkable quantum efficiency of light harvesting processes. In this work, we revisit the elementary aspects of environment-induced fluctuations in the involved electronic energies and present a simple way to understand energy flow with the intuitive picture of relaxation in a funnel-type free-energy landscape. The presented free-energy description of energy transfer reveals that typical photosynthetic systems operate in an almost barrierless regime. The approach also provides insights into the distinction between coherent and incoherent energy transfer and the criteria by which the necessity of the vibrational assistance is considered.

Temperature-Dependent Tunneling in Furan Oligomer Single-Molecule Junctions.

Two commonly observed charge transport mechanisms in single-molecule junctions are coherent tunneling and incoherent hopping. It has been generally believed that tunneling processes yield temperature-independent conductance behavior and hopping processes exhibit increasing conductance with increasing temperature. However, it has recently been proposed that tunneling can also yield temperature-dependent transport due to the thermal broadening of the Fermi energy of the contacts. In this work, we examine a series of rigid, planar furan oligomers that are free from a rotational internal degree of freedom to examine the temperature dependence of tunneling transport directly over a wide temperature range (78-300 K). Our results demonstrate conductance transition from a temperature-independent regime to a temperature-dependent regime. By examining various hopping and tunneling models and the correlation between the temperature dependence of conductance and molecular orbital energy offset from the Fermi level, we conclude thermally assisted tunneling is the dominant cause for the onset of temperature-dependent conductance in these systems.

Charge Transfer and Spin Dynamics in a Zinc Porphyrin Donor Covalently Linked to One or Two Naphthalenediimide Acceptors.

Quantum coherence effects on charge transfer and spin dynamics in a system having two degenerate electron acceptors are studied using a zinc 5,10,15-tri(n-pentyl)-20-phenylporphyrin (ZnP) electron donor covalently linked to either one or two naphthalene-1,8:4,5-bis(dicarboximide) (NDI) electron acceptors using an anthracene (An) spacer, ZnP-An-NDI (1) and ZnP-An-NDI2 (2), respectively. Following photoexcitation of 1 and 2 in toluene at 295 K, femtosecond transient absorption spectroscopy shows that the electron transfer (ET) rate constant for 2 is about three times larger than that of 1, which can be accounted for by the statistical nature of incoherent ET as well as the electron couplings for the charge separation reactions. In contrast, the rate constant for charge recombination (CR) of 1 is about 25% faster than that of 2. Using femtosecond transient infrared spectroscopy and theoretical analysis, we find that the electron on NDI2•- in 2 localizes onto one of the two NDIs prior to CR, thus precluding electronically coherent CR from NDI2•-. Conversely, CR in both 1 and 2 is spin coherent as indicated by the observation of a resonance in the 3*ZnP yield following CR as a function of applied magnetic field, giving spin-spin exchange interaction energies of 2J = 210 and 236 mT, respectively, where the line width of the resonance for 2 is greater than 1. These data show that while CR is a spin-coherent process, incoherent hopping of the electron between the two NDIs in 2, consistent with the lack of delocalization noted above, results in greater spin decoherence in 2 relative to 1.

Перенос заряда в молекуле ДНК в рамках простой модели открытой квантовой системы

The quantum-mechanical model proposed earlier by Skourtis and Nitzan (J. of Chem. Phys. 119, (2003) 6271) to describe a charge transfer in a fragment of artificial DNA molecule has been numerically investigated. The current rationale for the model is carried out and values of its parameters are indicated. Within this model, the description of the transport of a hole carrier in DNA is based on solutions to the time-dependent Schrödinger equation including damping effects. The non-unitary dynamics of the hole carrier is treated by us within the framework of a theory of continuous quantum measurements by the environment in an open quantum system. Results of numerical analysis of the model are in a good agreement with experimental observations and demonstrate two different types of the charged carrier motion, presumably tunneling and incoherent hopping. The main concepts of the theory of decoherence and superselection for open quantum systems and the prospects for their application for further study of various mechanisms of motion of a charged carrier in DNA are briefly discussed.

Observation of Intramolecular Interaction in Fluorescent Star-Shaped Polymers: Evidence for Energy Hopping between Branch Chains.

Time-resolved fluorescence anisotropy measurements were performed on three-branched star-shaped polymers, based on precisely synthesized poly(9,9-di-n-octyl-fluorene vinylene)s containing C6F5 end groups. The star-shaped polymers showed identical fluorescence spectra, fluorescence lifetimes, and quantum yields to those of the reference single-chain oligomer. However, a rapid fluorescence anisotropy decay was observed in two kinds of star-shaped polymers, while such decay was not seen in the corresponding single-chain oligomer. On the basis of the analysis using an incoherent hopping model, the observed rapid anisotropy decay is attributable to energy hopping processes between branches within a single polymer species, and its rate was deduced to be ca.100 ps depending upon the core part.

Kinetic Monte Carlo Study of Triplet-Triplet Annihilation in Conjugated Luminescent Materials

It is well known that in organic solids the collision of two excitons can give rise to delayed fluorescence (DF). Revived interest in this topic is stimulated by the current endeavor towards the development of efficient organic optoelectronic devices such as organic light-emitting diodes (OLEDs) and solar cells, or sensitizers used in photodynamic therapy. In such devices, triplet excitations are ubiquitously present but their annihilations can be either detrimental, e.g., giving rise to a roll-off of intensity in an OLED, or mandatory, e.g., if the sensitizer relies on up-conversion of long-lived low-energy triplet excitations. Since the employed materials are usually noncrystalline, optical excitations migrate via incoherent hopping. Here, we employ kinetic Monte Carlo simulations (KMC) to study the complex interplay of triplet-triplet annihilation (TTA) and quenching of the triplet excitations by impurities in a single-component system featuring a Gaussian energy landscape and variable system parameters such as the length of the hopping sites, i.e., a conjugated oligomer, the morphology of the system, the degree of disorder (\ensuremath{\sigma}), the concentration of triplet excitations, and temperature. We also explore the effect of polaronic contributions to the hopping rates. A key conclusion is that the DF features a maximum at a temperature that scales with $\ensuremath{\sigma}/{k}_{B}T$. This is related to disorder-induced filamentary currents and thus locally enhanced triplet densities. We predict that a maximum for the TTA process near room temperature or above requires typically a disorder parameter of at least 70 meV.

Observing Donor-to-Acceptor Electron-Transfer Rates and the Marcus Inverted Parabola in Molecular Junctions.

A recurring theme in molecular electronics is the relationship between the intramolecular electron transfer rate in a donor-bridge-acceptor system and the conductance at low bias in the corresponding electrode-bridge-electrode junction. The similarities between through-bridge donor-to-acceptor electron tunneling and through-bridge electrode-to-electrode Landauer transport led to the suggestion of an approximate linear relationship between the rate and the conductance for any given bridge. A large body of work indicates that the two quantities are not necessarily linearly related, showing different behaviors as a function of temperature, voltage and bridge length. Apart from Landauer tunneling, incoherent hopping can be an important mechanism in molecular junctions. We propose a donor-bridge-acceptor molecular junction, functioning in the incoherent hopping regime, that is suited for establishing direct correlations between the electrode-to-electrode current and the intramolecular donor-to-acceptor electron transfer rate. We suggest that this type of junction may be used to observe the Marcus-inverted-parabola dependence of the intramolecular rate on energy gap by varying the bias voltage. The realization of such an experiment would enable meaningful comparisons between solution-phase electron transfer rates and molecular-junction currents for the same molecule.

Realistic model of charge mobility in π-conjugated polymer systems

Charge transfer processes and charge mobility are investigated in the poly(p-phenylenevinylene) model system. Realistic disordered polymer conformations are created and used in a coarse-grained model. Localized and quasiextended states are obtained using the Holstein Hamiltonian. Charge transport is modeled as an incoherent hopping mechanism in the framework of unimolecular and bimolecular Marcus theory for intramolecular and intermolecular processes, respectively, to account for the electron-phonon coupling present in π-conjugated polymer systems. Static and quasidynamic disorder effects are both considered using the “fluctuating bridges” approach. Charge mobility is calculated using kinetic Monte Carlo simulations for a range of physically relevant parameters. We examine the relative importance of intramolecular and intermolecular mechanisms and the role of localized and extended states in the transport process. We discuss the role of disorder and temperature and show that a log μ∝−F electric field dependence in the high field regime naturally emerges from our model. We show that disorder significantly reduces the mobility at low fields but slightly increases it at high fields. We also show that the mobility is dominated by interchain charge transfer between low energy localized states at low fields, but at higher fields, intrachain transfer to more delocalized higher energy states becomes equally important. This crossover is the cause of anisotropic charge mobility at intermediate field strengths.

Assessing Possible Mechanisms of Micrometer-Scale Electron Transfer in Heme-Free Geobacter sulfurreducens Pili.

The electrically conductive pili of Geobacter sulfurreducens are of both fundamental and practical interest. They facilitate extracellular and interspecies electron transfer (ET) and also provide an electrical interface between living and nonliving systems. We examine the possible mechanisms of G. sulfurreducens electron transfer in regimes ranging from incoherent to coherent transport. For plausible ET parameters, electron transfer in G. sulfurreducens bacterial nanowires mediated only by the protein is predicted to be dominated by incoherent hopping between phenylalanine (Phe) and tyrosine (Tyr) residues that are 3 to 4 Å apart, where Phe residues in the hopping pathways may create delocalized "islands." This mechanism could be accessible in the presence of strong oxidants that are capable of oxidizing Phe and Tyr residues. We also examine the physical requirements needed to sustain biological respiration via nanowires. We find that the hopping regimes with ET rates on the order of 108 s-1 between Phe islands and Tyr residues, and conductivities on the order of mS/cm, can support ET fluxes that are compatible with cellular respiration rates, although sustaining this delocalization in the heterogeneous protein environment may be challenging. Computed values of fully coherent electron fluxes through the pili are orders of magnitude too low to support microbial respiration. We suggest experimental probes of the transport mechanism based on mutant studies to examine the roles of aromatic amino acids and yet to be identified redox cofactors.

Engineering bilinear mode coupling in circuit QED: Theory and experiment

Photonic states of superconducting microwave cavities controlled by transmon ancillas provide a platform for encoding and manipulating quantum information. A key challenge in scaling up the platform is the requirement to communicate on demand the information between the cavities. It has been recently demonstrated that a tunable bilinear interaction between two cavities can be realized by coupling them to a bichromatically-driven transmon ancilla, which allows swapping and interfering the multi-photon states of the cavities [Gao et al., Phys. Rev. X 8, 021073(2018)]. Here, we explore both theoretically and experimentally the regime of relatively strong drives on the ancilla needed to achieve fast SWAP gates but which can also lead to undesired non-perturbative effects that lower the SWAP fidelity. We develop a theoretical formalism based on linear response theory that allows one to calculate the rate of ancilla-induced interaction, decay and frequency shift of the cavities in terms of a susceptibility matrix. We treat the drives non-perturbatively using Floquet theory, and find that the interference of the two drives can strongly alter the system dynamics even in the regime where the rotating wave approximation applies. We identify two major sources of infidelity due to ancilla decoherence. i) Ancilla dissipation and dephasing leads to incoherent hopping among Floquet states which occurs even when the ancilla is at zero temperature, resulting in a sudden change of the SWAP rate. ii) The cavities inherit finite decay from the relatively lossy ancilla through the inverse Purcell effect; the effect can be enhanced when the drive-induced AC Stark shift pushes certain ancilla transition frequencies to the vicinity of the cavity frequencies. The theoretical predictions agree quantitatively with the experimental results, paving the way for using the theory to design future experiments.

A polaron approach to photorefractivity in Fe : LiNbO3

The thermally activated, incoherent hopping of small electron polarons generated by continuous illumination in iron-doped lithium niobate is simulated by a Marcus-Holstein model for which all the input parameters are known from literature. The results of the calculations are compared with a comprehensive set of data obtained from photorefractive, photogalvanic and photoconductive measurements under green light excitation on samples with different doping levels and stoichiometries in the temperature range between and room temperature. We show that the temperature and composition dependence of the photorefractive observables can be interpreted by a change in the abundance of the different hop types that a polaron performs before being captured by a deep Fe trap. Moreover, by a comparison between experimental and numerical data we obtain new insights on the initial photo-excitation part of the photorefractive process. In particular all results are consistent if a single value of the photogalvanic length is assumed for all the samples and all the temperatures. The photo-generation efficiency ϕ under green light excitation (somewhere denoted as quantum efficiency) is also estimated. It appears to decrease from 10%–15% at room temperature to about 5% at 150 K. This behavior is qualitatively interpreted in terms of a temperature-dependent re-trapping probability of the light-emitted particles from the initial Fe donor center.

Designing Long-Range Charge Delocalization from First-Principles.

Efficient electronic communication over long distances is a desirable property of molecular wires. Charge delocalization in mixed-valence (MV) compounds where two redox centers are linked by a molecular bridge is a particularly well-controlled instance of such electronic communication, thus lending itself to comparisons between theory and experiment. We study how to achieve and control long-range charge delocalization in cationic organic MV systems by means of Kohn-Sham density functional theory (DFT) and show that a captodative substitution approach recently suggested for molecular conductance ( Stuyver et al. J. Phys. Chem. C 2018 , 122 , 3194 ) greatly enhances charge delocalization in p-phenylene-based wires. To ensure the adequacy of our DFT methods, we validate different protocols for organic MV systems of different lengths. The BLYP35 hybrid functional combined with a polarizable continuum model, established by Renz and Kaupp, is indeed capable of correctly describing experimentally observed length-dependent charge delocalization, in contrast to the long-range corrected functionals ω-B97X-D and ω-PBE. We also discuss the implications of these results for a first-principles description of the transition between coherent tunneling and incoherent hopping regimes in molecular conductance.

Using fluorescence detected two-dimensional spectroscopy to investigate initial exciton delocalization between coupled chromophores.

Förster theory describes electronic exciton energy migration in molecular assemblies as an incoherent hopping process between donor and acceptor molecules. The rate is expressed in terms of the overlap integral between donor fluorescence and acceptor absorption spectra. Typical time scales for systems like photosynthetic antennae are on the order of a few picoseconds. Prior to transfer, it is assumed that the initially excited donor molecule has equilibrated with respect to the local environment. However, upon excitation and during the equilibration phase, the state of the system needs to be described by the full density matrix, including coherences between donor and acceptor states. While being intuitively clear, addressing this regime experimentally has been a challenge until the recently reported advances in fluorescence detected two-dimensional spectroscopy. Here, we demonstrate using fourth order perturbation theory the conditions for the presence of donor-acceptor coherence induced cross-peaks at zero waiting time between the first and the second pair of pulses. The approach is illustrated for a heterodimer model which facilitates an analytical solution.