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Energy, Work, Entropy, and Heat Balance in Marcus Molecular Junctions Published As Part of the Journal of Physical Chemistry Virtual Special Issue “Peter J

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Energy, Work, Entropy, and Heat Balance in Marcus Molecular Junctions Published As Part of the Journal of Physical Chemistry Virtual Special Issue “Peter J pubs.acs.org/JPCB Article Energy, Work, Entropy, and Heat Balance in Marcus Molecular Junctions Published as part of The Journal of Physical Chemistry virtual special issue “Peter J. Rossky Festschrift”. Natalya A. Zimbovskaya and Abraham Nitzan* Cite This: J. Phys. Chem. B 2020, 124, 2632−2642 Read Online ACCESS Metrics & More Article Recommendations ABSTRACT: We present a consistent theory of energy balance and conversion in a single-molecule junction with strong interactions between electrons on the molecular linker (dot) and phonons in the nuclear environment where the Marcus-type electron hopping processes predominate in the electron transport. It is shown that the environmental reorganization and relaxation that accompany electron hopping energy exchange between the electrodes and the nuclear (molecular and solvent) environment may bring a moderate local cooling of the latter in biased systems. The effect of a periodically driven dot level on the heat transport and power generated in the system is analyzed, and energy conservation is demonstrated both within and beyond the quasistatic regime. Finally, a simple model of atomic scale engine based on a Marcus single-molecule junction with a driven electron level is suggested and discussed. I. INTRODUCTION the bridge sites and/or among the bridge sites subjected to local thermalization. This dynamics is usually described by In the past two decades molecular electronics became a well- − − 3,23 27 established and quickly developing field.1 5 Presently, it successive Marcus-type electron transfer processes. Indeed, Marcus theory has been repeatedly and successfully provides a general platform which can be used to consider 28−35 diverse nanoscale electronic and energy conversion devices. used to study charge transport through redox molecules. Nuclear motions and reorganization are at the core of this The basic building block of such devices is a single-molecule fi junction (SMJ). This system consists of a couple of metallic/ transport mechanism. The theory may be modi ed to include effects of temperature gradients across an SMJ36,37 as well as a semiconducting electrodes linked with a molecular bridge. fi Electron transport through SMJs is controlled by electric nite relaxation time of the solvent environment. Further − generalization of Marcus theory accounting for finite lifetime Downloaded via UNIV OF PENNSYLVANIA on September 4, 2020 at 17:19:33 (UTC). forces, thermal gradients, and electron phonon interactions. In addition, in SMJs operating inside dielectric solvents, broadening of the molecule electron levels was recently suggested.38,39 See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. transport properties may be strongly affected by the solvent − − response to the molecule’s charge states.6 8 Overall, electrons Besides charge transport, electron phonon interactions may strongly affect heat generation and transport through on the molecule may interact with a collection of thermalized 9,10,20 phonon modes associated with the solvent nuclear environ- SMJs. There is an increasing interest in studies of vibrational heat transport in atomic scale systems and in their ment as well as with individual modes associated with − interfaces with bulk substrates.40 45 Electron transfer induced molecular vibrations. Such interactions lead to the energy 36,37 ff exchange between traveling electrons and the environment, heat transport was also suggested and analyzed. E ects of ff structure−transport correlations on heat transport character- thus giving rise to inelastic e ects in the electron transport. In 46−48 − istics of such systems are being studied as well as effects the weak electron phonon coupling limit, inelastic contribu- fi tions may be treated as perturbations of basically elastic originating from speci c features of coupling between the − transport.9 16 Stronger coupling of electrons to phonon modes can result in several interesting phenomena including negative Received: January 3, 2020 differential conductance, rectification, and Franck−Condon Revised: March 6, 2020 − blockade.17 22 Published: March 12, 2020 In the present work, we consider a limit of very strong electron−phonon interactions when electron transport may be described as a sequence of hops between the electrodes and © 2020 American Chemical Society https://dx.doi.org/10.1021/acs.jpcb.0c00059 2632 J. Phys. Chem. B 2020, 124, 2632−2642 The Journal of Physical Chemistry B pubs.acs.org/JPCB Article 49−52 molecular linker and electrodes and the heat currents 1 2 53−55 Ek= λ rectification. Correlations between structure and heat r 2 (3) transfer in SMJs and similar systems may be accompanied by 9,10,56−61 reflects the strength of interactions between electrons on the heating/cooling of the molecular bridge environment. bridge and the solvent environment. For E = 0, electron Nevertheless, the analysis of heat transfer in SMJs is far from r transport becomes elastic. being completed, especially in molecular junctions dominated The overall kinetic process is determined by the transfer by Marcus-type electron transfer processes. In the present L,R L,R rates k → and k → which correspond to transitions at the left work, we theoretically analyze energy balance and conversion a b b a (L) and right (R) electrodes between the occupied (a) and in such systems. For the molecular bridge, we use the standard unoccupied (b) molecular states (namely, a → b corresponds single-level model that describes two molecular electronic to electron transfer from molecule to metal and b → a denotes states in which the level is either occupied or unoccupied. The the opposite process). Because of the time scale separation electrodes are treated as free electron reservoirs with respective between electron and nuclear motions, these transfer processes chemical potentials and temperatures. We assume that the level have to satisfy electronic energy conservation: may be slowly driven by an external agent (such as a gate voltage) which moves it over a certain energy range. Also, we g(,xExExϵ= )ba () − () +ϵ=0 (4) assume that the temperature of the solvent environment of the ff where ϵ is the energy of electron in the metal. This leads to the bridge may di er from the electrode temperatures. Despite its 27 simplicity, the adopted model captures essential physics of Marcus electron transfer rates given by energy conversion in such a junction. β ∞ K =ϵΓϵ[−ϵs β ] The paper is organized as follows. In Section II, we review kab→ ∫ d()1(,)K fK K the application of Marcus rates to the evaluation of steady state 4πEr −∞ currents resulting from voltage and temperature bias across the βs 2 junction. We study the relationship between heat currents ×−ϵ+−ϵexp ()Erd flowing into the electrodes and into the solvent environment of 4Er (5) the molecular bridge and demonstrate overall energy ÅÄ ÑÉ β Å ∞ Ñ conservation. In Section III, we analyze the energy balance K sÅ Ñ kba→ =ϵÅ ∫ d()(,)ΓK ϵϵf βKÑ in a system where a bridge electronic level is driven by an Å −∞ K Ñ 4πEÇÅr ÖÑ external force. We discuss the irreversible work thus done on β the system and the corresponding dissipated power and the s 2 ×−ϵ+−ϵexp ()drE entropy change. In Section IV, we describe a simple model for 4Er (6) a Marcus junctione engine and estimate its efficiency. Our where K = {L, R}ÅÄ stands for the left andÑÉ right electrodes. In conclusions are given in Section V. Å Ñ Å Γ π 2 ρ ℏ Ñ ρ these expressions,Å K =2VK K/ whereÑ K is the single electron densityÅ of states for the electrodeÑ K and V the II. STEADY STATE CURRENTS ÇÅ ÖÑ K tunneling coupling for electron transfer between this electrode β −1 β −1 II.A. Electron Transfer Rates and Electronic Currents. and the molecule. L,R =(kTL,R) and s =(kTs) indicate We consider a molecular junction where electrons move the temperatures of the electrodes and the molecule environ- between electrodes through a molecular bridge (or dot) that ment. k is the Boltzmann constant, and f L,R are Fermi can be occupied (state a) or unoccupied (state b). Adopting distribution functions for the electrodes with chemical μ the Marcus formalism, we assume that each state corresponds potentials L,R. Expressions in eqs 5 and 6 assume that to a free energy surface which is assumed to be parabolic in the electron transfer takes place from an equilibrium solvent and collective solvent coordinate x. Here and below, we take metal configurations, namely, that thermal relaxation in the “solvent” to include also the intramolecular nuclear motion metal and solvent environments are fast relative to the metal− which contributes to the electronic charge accumulation. We molecule electron exchange processes. When TL = TR = Ts, eqs use the simplest shifted surfaces model: In terms of x, the 5 and 6 are reduced to the standard Marcus−Hush−Chidsey energy surfaces associated with the two electronic states are expressions for electron−electrode transfer rates.27,62 In further assumed to take the forms of identical harmonic surfaces that analysis, we assume that the molecule is symmetrically coupled Γ Γ Γ are shifted relative to each other. to the electrodes ( L = R = ), and unless stated otherwise, we take Γ as a constant independent of energy. 1 2 Given these rates, the probabilities that the dot is in the Exa()= kx 2 (1) states a or b at time t, Pa and Pb, are determined by the kinetic equations 1 2 dPa dPb Exb()=−+ϵ kx()λ d =−Pk Pk =−Pk Pk 2 (2) dt bb→→ a aa b dt aa→→ b bb a (7) Here, λ represents a shift in the equilibrium value of the ϵ ff =+LR =+ LR reaction coordinate, and d is the di erence between the where kkkkkkab→→→→→→ ab ab; ba ba ba. The steady 0 0 equilibrium energies of the two electronic states.
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