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. Rossky Festschrift”. Natalya A. Zimbovskaya and Abraham Nitzan*

Cite This: J. Phys. Chem. B 2020, 124, 2632−2642 Read Online

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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

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. Diverse state probabilities Pa and Pb and the steady state electron reaction geometries may be taken into account by varying ϵ current Iss (positive when electrons go from left to right) are − d and the force constants.62 64 More sophisticated models65 given by make the mathematics more involved but are not expected to 00kba→ kab→ change the essential physics. The reorganization energy Er Pa = Pb = associated with the electron transfer process kkab→→+ ba kkab→→+ ba (8)

2633 https://dx.doi.org/10.1021/acs.jpcb.0c00059 J. Phys. Chem. B 2020, 124, 2632−2642 The Journal of Physical Chemistry B pubs.acs.org/JPCB Article I ss =−kPL0 kP L0 ||e bab→→ aba R0 R0 =−()kPbab→→ − kP aba kkRL− kk RL = abba→→ baab →→ ()kkab→→+ ba (9) The coupling Γof the molecular bridge to electrodes affects the electron transfer rates (and consequently the SMJ transport properties) in two ways. First, as indicated above, it controls the transfer rates between the electrodes and the molecule. Figure 1. Left panel: Current−voltage characteristics computed using This effect is accounted for within the standard Marcus theory. eqs 10−12 (solid lines). The electrode Fermi energy in the unbiased μ ±| | junction is set to 0; the bias is applied symmetrically ( L,R = e V/2) Second, it is manifested in the lifetime broadening of molecular − ff relative to this origin and TL = TR = Ts. The Landauer Buttiker limit levels, an e ect disregarded by this theory. It has been ff suggested in recent work38 that the Marcus expressions for the is represented by the red line (Er = 0). The di erence between the results obtained using eqs 10−12 and the Marcus limit is transfer rates may be generalized to include the broadening demonstrated by comparison of the solid black line with the dashed effect. For a symmetrically coupled system the transfer rates 38,39 line plotted using the Marcus equations for the electron transfer rates may be approximated as follows: at the same value of Er (Er = 0.4 eV). Right panel: Electron current as ff Γ ∞ a function of the temperature di erence symmetrically distributed L,R between the electrodes ((T + T )/2 = T ) in an unbiased SMJ in the kfKab→ = ∫ d1ϵ[ − (βL,R ,) ϵ ]− () ϵ L R s −∞ L,R (10) ℏΓ π Marcus limit. Curves are plotted assuming that kTs = 0.026 eV, = ϵ ϵ − ∞ 0.01 eV, d = 0.1 eV (left panel), and d = 0.02 eV (right panel). L,R Γ kfKba→ = ∫ d(,)()ϵϵϵβL,R + π −∞ L,R (11) works that investigate the implication of Marcus kinetics for where the steady state conduction properties of molecular junctions in the limit of hopping conduction. Here we focus on the πβ β s s 2 energy balance associated with such processes, and the KRe±()ϵ= exp −(( ℏΓ∓ϵ±iEdr −ϵ )) 4EEr 4 r implication of Marcus kinetics on heat transfer. Each electron hopping event between the molecule and an electrode is ÅÄ Ä É Å β Å Ñ accompanied by solvent and metal relaxation and, therefore, by Å s Å Ñ ×ℏÅerfc Å((Γ∓ϵ±−ϵiEdr )) Ñ Å 4E Å Ñ heat production in these environments. We denote these heats Å r ÇÅ ÖÑ (12) ÇÅ Qs and Qe for the solvent and the electrode, respectively. ÑÉ fi L,R i yÑ Speci cally, Qs,a→b denotes the heat change in the solvent when and erfc(x) is the complementaryj error function. AlthoughzÑ the j − zÑ an electron hops from the molecule into the left (L) or right derivation of eqsj 10 12 involves some fairyzÑ strong L,R assumptions,38,39 thek result is attractive for its ability{ÖÑ to yield (R) electrode, and similarly, Qs,b→a is heat change in the the Landauer cotunneling expression in the strong molecule− solvent in the opposite process of an electron moving from the electrode to the molecule. For symmetrically coupled electro- electrode coupling limit EkTrs≪ℏΓ and the Marcus Γ Γ Γ des ( L = R = ) considered in the Marcus limit, these terms expression in the opposite limit. This is shown in the left have the form panel of Figure 1 where three current−voltage curves are Γ β plotted at the same value of and several values of Er. The K Γ s − Q → = ∫ d1ϵ[ −f (β ,)( ϵ ] ϵd − ϵ ) Landauer Buttiker behavior is demonstrated for Er =0.AsEr s,ab kK 4πE K K enhances, the current−voltage curve behavior becomes more ab→ r similar to the Marcus behavior represented by the dashed line. β ×exp −s ()E −ϵ +ϵ2 In the case of the Marcus limit, one observes a very rd 4ER (13) pronounced plateau in the I−V profile around V = 0, as seen Ä É in Figure 1 (dashed line). This plateau develops gradually as and Å Ñ Å Ñ the electron−phonon coupling increases and is a manifestation Å Ñ − Γ Å β Ñ of a Franck Condon blockade similar to that resulting from K = ÇÅ s ϵϵϵ−ϵβ ÖÑ Q s,ba→ K ∫ d(,)()fK K d interactions between electrons and individual molecular kba→ 4πEr vibrational modes.19,20 μ μ μ βs 2 When the two electrodes in an unbiased SMJ ( L = R = ) ×−exp () ϵ+−ϵE ff dr are kept at di erent temperatures, a thermally induced charge 4ER (14) current emerges, as shown in Figure 1 (right panel). The K K 66 where K ={L, R}.ÅÄ Similarly, Q → andÑÉ Q → are heats current does not appear if ϵ = μ =0. However, when ϵ ≠ μ Å e,a b Ñ e,b a d d generated in electrodeÅ K when an electronÑ leaves (enters) the the current emerges. The current changes its direction at T = Å Ñ L molecule into (from)ÇÅ that electrode: ÖÑ TR. Its magnitude strongly depends on the reorganization energy. Indeed, the thermally induced current takes on Γ β noticeable values only provided that the effects of nuclear Q K = s d1ϵ[ −f (βμ ,)( ϵ ] ϵ − ) e,ab→ K π ∫ K KK reorganization are weak and becomes suppressed when the kab→ 4 Er interaction with the solvent environment increases. βs 2 II.B. Heat Currents and Energy Conservation. The ×exp −()Erd −ϵ +ϵ results summarized above were mostly obtained before in 4Er (15) ÅÄ ÑÉ 2634 Å https://dx.doi.org/10.1021/acs.jpcb.0c00059Ñ Å J. Phys.Ñ Chem. B 2020, 124, 2632−2642 Å Ñ ÇÅ ÖÑ The Journal of Physical Chemistry B pubs.acs.org/JPCB Article and 1 Γ L,R = ϵϵϵ−ϵϵβ Q s,ba→ L,R ∫ d(,)()()fKL,R L,R d + π kba→ (25) K Γ βs Q → = ∫ d(,)()ϵϵ−ϵf βμ eb, a K π K KK L,R 1 Γ kba→ 4 Er Q = d1ϵ[ −f (βμ ,)( ϵ ] ϵ − ) e,ab→ π L,R ∫ L,R L,R L,R β kab→ ×−exp s () ϵ+−ϵE 2 dr K ()ϵ (26) 4ER (16) − Ä É 1 Γ Equations 15 andÅ 16 are analogues ofÑ the corresponding Q L,R = d(,)()()ϵϵ−ϵϵfKβμ Å Ñ e,ba→ L,R ∫ L,R L,R L,R + results reported in refÅ 37. Ñ π kba→ Å Ñ Equations 13−16ÇÅ are expressions for theÖÑ heat changes per (27) specific hopping events. The corresponding heat change rates L,R L,R ϵ − where ka→b, kb→a, and K±( ) are given by eqs 10 12. It should (heat per unit time) in the solvent and the electrodes are be noted that the procedure leading to eq 19 that demonstrates obtained from the energy conservation remains the same when these expressions are used. JQ≡ ̇ s s Results based on eqs 20−23 obtained using Marcus theory 0L LRR rates given by eqs 5 and 6 are displayed in Figure 2. The left =+Pkaab()→ Qs,ab→ kab→ Qs,ab→ 0L LRR ++Pkbba()→ Qs,ba→ kba→ Qs,ba→ (17) and K ̇ K K 0 K K 0 K JQe ≡ e =+ kPQkPQab→ a e,ab→ ba→ b e,ba→ (18) Using eqs 5 and 6 and eqs 13−16, it can be easily established that eqs 17 and 18 imply L R μμ JJJe ++=e s ()LR − Iss (19) L Figure 2. Heat currents Js (left panel), Je (solid lines, right panel), and showing the balance between heat change rates in the solvent R and the electrodes and the heat generated by the current flow Je (dashed lines, right panel) shown as functions of the bias voltage V for several values of the reorganization energy. Curves are plotted across the voltage bias. In the absence of solvent reorganiza- ff omitting the e ect of molecular level broadening and assuming kTs = tion, Js = 0 and eq 19 is reduced to the standard junction ℏΓ ϵ L R μ − μ | | kTL = kTR = 0.026 eV, = 0.01 eV, d = 0.1 eV. The inset in the left energy balance relation Je + Je =( L R)Iss/ e . From eqs − panel focuses on a segment of the main plot that emphasizes the local 13 18 we obtain after some algebra (see Appendix A) cooling of the solvent in the corresponding voltage range. JL =−ϵ−()(μ I Pk0L + Pk 0L ) E − Y e L dss aa→→ b bb a r L (20) panel shows the heat deposited in the solvent environment plotted against the bias voltage for different values of the R μ 0R 0R Je =ϵ−()(d R Iss − Pkaa→→ b + Pk bb a ) ErR − Y (21) reorganization energy. The right panel shows similar results for the left and right electrodes. The following observations can be kk made: J = 2 abba→→EYY++ s rLR fl kkab→→+ ba (22) (a) Re ecting the behavior of the electronic current, energy exchange processes are very weak at low bias due to the where Franck−Condon blockade that hinders electron trans- | | ∂ port. Noticeable heat currents appear when e V exceeds Er fK YK =Γ∫ d ϵ the reorganization energy Er thus lifting the blockade. πβs ∂ϵ (b) The heat deposited into the electrodes shows an asymmetry between positive and negative biases (or β ×−ϵ+−ϵP0 exp s ()E 2 equivalently between left and right electrodes). This b dr fl ff 4ER asymmetry re ects the di erent positioning of the Ä É energy of the transferred electron relative to the left i Å β Ñ 65 +jP0 exp Å −s ()E −ϵ +ϵ2 Ñ and right Fermi energies and was observed exper- j a Å rd Ñ 67 j Å 4ER Ñ (23) imentally. k ÇÅ ÖÑ (c) The heat deposited into the solvent environment (left Ä É The analysis presentedÅ in this sectionÑy remains valid panel) is symmetric with respect to bias inversion Å Ñz regardless of theÅ specific form of theÑ expressionsz for the because it reflects the energy balance relative to both Å Ñz relevant heat flows.ÇÅ These may be defiÖÑned{ within Marcus electrodes. theory by eqs 13−16 or by the generalized expressions derived (d) Note that the heat exchanged with the solvent (nuclear) using the approximation of ref 38: environment can become negative; namely, heat may be pulled out of this environment at some range of bias and 1 Γ L,R = ϵ[ −β ϵ ] ϵ − ϵ ϵ reorganization energy. In the present case of a Q s,ab→ L,R ∫ d1fKL,R (L,R ,)(d )− () π kab→ symmetrically coupled SMJ with a symmetrically | | ≈ (24) distributed bias voltage, this happens at eV 2Er,

2635 https://dx.doi.org/10.1021/acs.jpcb.0c00059 J. Phys. Chem. B 2020, 124, 2632−2642 The Journal of Physical Chemistry B pubs.acs.org/JPCB Article

namely, when the driving force associated with the bias JQexcess = ̇ excess voltage is nearly counterbalanced by driving for the s s − L LLL solvent reorganization originated from electron phonon =−Gk( ba→ Qs,ba→ kab→ Qs,ab→ interactions. This cooling is reminiscent of similar effects +−R RRR discussed in the low electron−phonon coupling kQba→ s,ba→ kQab→ s,ab→ ) (33) − regime.59 61 K ,excess ̇ K ,excess K K K K JQe = e =− GkQkQ()ba→ e,ba→ ab→ e,ab→ III. DRIVEN JUNCTION (34) Next we consider charge and energy currents in driven biased Using these expressions with eqs 13−16 for the heat junctions, where driving is modeled by an externally controlled currents and eqs 5 and 6,or10 and 11,wefind time dependent parameter in the system Hamiltonian. In the present study we limit our consideration to time dependence JJJexcess ≡+excess L,excess + JR,excess “ ” ϵ tot s e e of the single electron level d of the molecular bridge that LL may in principle be achieved by varying the gate potential. =[Gkk()()μL −ϵd ab→→ + ba Similar studies in the absence of electron−phonon inter- +−ϵ+()()μ kkRR] actions, focusing on a consistent quantum thermodynamic R d ab→→ ba (35) 68−83 description of such systems, were recently published. The Equations 31−35 are exact relations. In particular, eq 31 can model considered below includes strong coupling to the be used as a basis for expansions in powers of ϵ̇ . We start by phonon environment at the cost of treating this coupling d writing G as such a power series, G = G(1) + G(2) + ... with G(n) semiclassically and assuming weak coupling between the representing order n in ϵ̇ , and use this expansion in eq 31 molecule and the electrodes. This model is similar to that d while further assuming that G depends on time only through used to analyze cyclic voltammetry observations when ϵ (n) extended to consider two metal interfaces.84 its dependence on d. Therefore, dG /dt is of the order n +1. In further analysis we assume that the electron level ϵ is a We note that our results are consistent with this assumption. d III.A. Quasistatic Limit: First Order Corrections. To the slowly varying and construct an expansion of the solution of in fi 85 rst order in ϵḋ , the left-hand side of eq 31 vanishes, leading to powers of ϵḋ . To this end we start by separating the time dependent populations into their steady state components ∂P0 ϵ (1) . a 1 (which implicitly depend on time through d(t)) and G =ϵd corrections defined by29 ∂ϵd ()kkab→→ + ba (36)

0 where ka→b and kb→a depend on time through their Ptaa()=ϵ−ϵ P (dd ) G ( , t ) ϵ dependence on d. Note that eqs 30 and 36 imply that =ϵ+ϵ0 (1) (1) 0 Ptbb() P (dd ) G ( , t ) (28) d/dd/dPta =− Ptb =ϵdȧ ∂∂ϵ P /d; namely, this order of the 0 calculation corresponds to the quasistatic limit where all where the steady state populations P satisfy P k → − P k → a,b b b a a a b dynamics is derived from the time dependence of ϵ .Atthe =0,P + P = 1, and are given by eq 8. Then d a b same time, it should be emphasized that this limit is not a 0 reflection of the instantaneous steady state, as is evident from dPaa. ∂P dG =ϵd − eqs 32. dt ∂ϵd dt Consider first the electronic current. Using eqs 32 and 36, 0 the first order correction to the electron exchange rates with dPbb. ∂P dG =ϵd + the left and right electrodes is obtained in the form dt ∂ϵd dt (29) 0 (1) K K (1) ∂Pa From eqs 28 it follows that IKba=+()kkG→→ab =ϵḋ νK ∂ϵd (37) dPabdP =− =Gk()ab→→ + k ba with dt dt (30) K K Comparing eqs 29 and 30, we obtain kkab→→+ ba νK = 0 kkab→→+ ba (38) dG . ∂Pa =ϵd −+Gk()ab→→ k ba fi dt ∂ϵ (31) namely, a product of the ( rst order) change in the electronic d (1) ν population on the molecule dPa /dt and the fraction K of this The electronic currents can be written in terms of G in the change associated with the electrode K. form Next, consider the heat currents. Using eq 36 for G in eq 35 LL excess leads to IkPkPIIL =−=+bab→→ aba ss L ∂PP0 ∂ 0 =−=+RR excess (1) ̇ aȧ μν μν IkPkPIIR bab→→ aba ss R (32) Jtot =−ϵϵdd +ϵd ()L L + R R ∂ϵd ∂ϵd (39) excess K K (where IK =(kb→a + ka→b)G) expressing the fact that the left and right excess particle (electron) currents due to driving are To better elucidate the physical meaning of this result we fi generally not the same. Equations 32 together with analogues rearrange the rst term on the right according to 0 0 00 of eqs 17 and 18 in which Pa,b are replaced by Pa,b which can be −ϵdd ϵ̇ ∂∂ϵ=ϵPPPtaad/ddḋ −ϵ()a/dand use eq 37 to cast eq used to obtain the excess heat currents caused by driving 39 in the form

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ϵ 0 dissipation caused by driving the level. The last term on the d(dPa ) ̇ (1) 0 (1) ≡ EPJnnM =ϵḋ a −+μμL̇ + Ṙ right-hand side of eq 45 represents the second order dt tot L R (40) contribution to the rate of chemical work; thus, eq 45 is an This equation is a statement of the first law of expression for the first law of thermodynamics at the second ̇(1) thermodynamics, where EM represents, to order 1, the rate order of our expansion. Following refs 68 and 86, the of change of energy in the molecule, and the terms on the right coefficient in front of ϵ̇2 in eq 45 0 d stand for the work per unit time (ϵdȧ P ), rate of heat developing 0 − (1) ≡− L(1) R(1) (1) ∂Pa 1 in the environment ( Jtot (Je + Je + Js )), and rate of γ =− μ (1) μ (1) chemical work ( LIL + RIR ) to the same order. All terms ∂ϵd kkab→→ + ba (48) included in eq 40 are of the form ϵdḋ r()ϵ where r is an arbitrary fi ffi ϵ may be identi ed with the friction coe cient. A similar function and are therefore the same except of sign when d interpretation was suggested for different models for SMJs.68,87 goes up or down, as this should be within the quasistatic γ ϵ Dependencies of on d are shown in Figure 3.Inan regime. unbiased junction the friction coefficient reaches its maximum III.B. Beyond the Quasistatic Regime: Second Order Corrections. Using the expansion for G in eq 31 and keeping only second order terms leads to 1dG(1) G(2) =− kkab→→+ ba dt (41) which, using eq 36, gives ϵ̇2 ∂ ∂P0 1 G(2) =− d a ()kkab→→+ ba∂ϵd ∂ϵd ()kkab →→ + ba (42) Ä É The second order correction toÅ the electron currentÑ is (K = Å Ñ {L, R}): Å Ñ Å Ñ ffi ÇÅ ÖÑ Figure 3. Friction coe cient characterizing dissipation in the system (2) K K (2) caused by driving of the energy level in a symmetrically coupled IKab=+()kkG→→ba junction with a symmetrically applied bias as a function of ϵ . Main 0 d 2 ∂ ∂Pa 1 figure: Dashed lines represent friction in the absence of molecule- =−ϵ̇d νK ∂ϵ ∂ϵ()kk + solvent coupling (Er = 0); solid lines are plotted at Er = 0.05 eV for d d ab→→ ba (43) three values of the bias voltage (indicated by different colors). Inset: Ä É The second orderÅ excess heat is obtainedÑ from eq 35 by Friction at zero bias for the indicated two values of reorganization Å (2) Ñ replacing G with ÅG . The sum of secondÑ order corrections to energy. In all lines, the friction is normalized by its value at zero bias Å Ñ and zero reorganization energy. Other parameters are kTL = kTR = kTs the heat currentsÇÅ then takes the formÖÑ ℏΓ γ = 0.026 eV, and = 0.01 eV. 0(0) is the maximum value of the (2) L(2) R(2) (2) friction coefficient in the junction where the molecule is detached JJtot ≡++e Je Js from the solvent (Er = 0). (2) (2) LL =−ϵdGk()(()ab→→ + k ba + GμL kab →→+ k ba RR γ(0) at ϵ = μ = 0 and falls approaching zero as ϵ moves away ++μR ())kkab→→ ba (44) d d from this position. In this case, γ appears to increase with the − Using eqs 42 and 44, we can present the work energy increasing voltage. This results from the fact that at low bias balance equation at this order as follows: the Franck−Condon blockade discussed above makes the molecule−electrode coupling small, and the friction increases EWJ̇ (2) = ̇ (2) −+(2) ()μμ I(2) + I(2) M tot L L R R (45) upon removing this blockade at higher bias voltage. Also, at where higher voltage the peak splits: Two peaks appear due to electron−electrode exchange near the two Fermi energies (2) d Ė =− [ϵG(1) ] characterizing the biased junction. Note that coupling to the M dt d solvent shifts the positions of these peaks, in correspondence with the eqs 5 and 6 for transfer rates. ∂P0 ∂P0 2 a 11d a III.C. Evolution of the System (Dot) Entropy. fi =−ϵ̇d +ϵd De ne the ∂ϵd kkab→→ + ba dϵd ∂ϵd kkab→→ + ba system entropy by the Gibbs formula for our binary system Ä (46)É Å Ñ S =−kP(aa ln P + (1 − P a )ln(1 − P a )) (49) Å ji zyÑ is the secondÅ order change rate in thej total system energyzÑ Å fi j zÑ Using eqs 28,wefind expressed as theÇÅ time derivative of the rstk order contribution{ÖÑ ϵ fi to this energy (product of d and the rst order correction to S=− kPGPG((00 − )ln( − ) the population G(1)) and aa 00 +−+(1PG )ln(1 −+ PG )) (50) ∂ 0 aa ̇ (2) (1) 2 Pa 1 WG=−ϵḋ ≡−ϵ̇d which can be used to find again an expansion in powers of ϵ̇ : S ∂ϵkk + (47) d d ab→→ ba = S(0) + S(1) + ... In what follows we limit ourselves to the case 0 0 is the second order excess work per unit time (power) which of an unbiased junction in the wide-band limit for which Pa/Pb −βϵ corresponds to the lowest order irreversible work expressing = exp( d). In the absence of driving

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(0) (0) (0) (0) (0) left to the right electrode which may be appreciable if ϵ is S =−kP(aa ln P + (1 − P a )ln(1 − P a )) (51) 0 suﬃciently larger than kT = kT = kT ≡ kT. Mathematically, ≡ L R s and (assuming that TL = TR = Ts T) we treat this deviation from equilibrium dot population (eq 59 and beyond) using our expansions in powers of ϵ̇ for the (1) =−β ϵ (1) d S kGd (52) relevant quantities. The current is given by the average over a The ﬁrst and second order variations in the dot’s entropy period due to driving are obtained as (recall that the sign of Jtot was 1 ττ1 chosen so that the heat current into the environment is It==∫∫d()IttL d()ItR ττ0 0 positive) τ (58)

(0) (0) (1) where IK(t) values are given by eq 32. Further analytical (1) ∂S 1 ∂P J Ṡ =ϵ̇ =ϵϵ̇ a =− tot progress can be made by expanding IK in powers of ϵḋ . d dd δ ∂ϵd T ∂ϵd T (53) However, using this expansion implies that in eq 56 is large enough for the inequality kTΓ(ϵ) ≫ ϵ̇ to be satisfied for all ϵ. and d The lowest nonvanishing contribution to eq 58 is then (1) (1) (2) ∂S (1) ∂G (2) ττ Ṡ =ϵ̇ =−kGββ ϵ̇ −ϵϵ k ̇ 1 (2) 1 (2) d d dd It==d()Ittd()It ∂ϵd ∂ϵd ∫∫L R τ ττ0 0 (59) Ẇ (2) J(2) = − where the second order contributions to the currents are given T T (54) by eq 43. Note that this is the excess current produced by Equation 54 may be rewritten as driving which persists also in the absence of imposed bias. When a voltage bias is imposed so as to drive a current in the (2) (2) (2) (2) J Ẇ opposite direction to ⟨I⟩τ , the total current Ṡ += T T (55) (2) IV()=+ IVss () I () V The left side of eq 55 is the sum of the rate of total entropy ττ (60) ̇(2) fl change in the system (dot/molecule) S and the entropy ux can be used to define the power produced by the engine into the electrodes and solvent environment. Together these terms give the total entropy production due to the irreversible (2) Π=()VVIVIV (()ss + ()) nature of the process at this order. This result is identical to τ (61) that obtained in fully quantum mechanical treatments of The device efficiency is defined as the ratio similar processes evaluated in the absence of coupling to 67,78,80 ff Π()V solvent, except for a sign di erence in the heat η = fi ()V 1 τ de nition. Here, the heat which is going out of the system is ∫ d(,)tẆ (2) V t defined as positive. τ 0 (62) Figure 4 shows the voltage dependence of these engine IV. MARCUS JUNCTION ENGINE characteristics. Obviously, both vanish in the absence of load In this section, we extend the above analysis to discuss a simple model that simulates an atomic scale engine. This can be achieved by imposing asymmetry on the coupling of the molecular bridge to the electrodes that enables to convert the ϵ motion of d to electron current between the electrodes. A simple choice is δ2 Γϵ=ΓL() 22 ()ϵ+ϵ0 +δ δ2 Γϵ=ΓR() 22 ()ϵ−ϵ0 +δ (56) This represents a situation where the moving level is coupled Figure 4. Averaged over the period thermodynamic efficiency η (blue to wide-band electrodes via single-level gateway sites with line) and power Π (red line) produced in the junction by periodically ± ϵ driving the bridge level. Curves are plotted at kT = 0.026 eV, ℏΓ = energies 0 attached to the left/right electrode. The ϵ δ τ parameter δ characterizes widths of the density of states 0.01 eV, E0 =0,E1 = 0.1 eV, Er = 0.1 eV, 0 = 0.1 eV, = 6 meV, = 10 μs. The inset shows the dependence of the power Π on the width peaks associated with these states. The electron transfer rates parameter δ for a given bias voltage V = 0.1 V with all other are calculated from eqs 5 and 6. In further calculations we ϵ parameters taking the same values. assume that d varies according to ϵ=−()tEE cos(2πτ t / ) (57) d01 (V = 0), as well as at the stopping voltage when the current It is intuitively obvious that driving fast enough (small τ)to vanishes, and go through their maxima at different “optimal” induce deviation of the dot population from instantaneous voltages (which in turns depend on the choices of E0 and E1). equilibrium with a choice of origin E0 and amplitude E1 that Note that because of the intrinsic friction in this model, the −ϵ ϵ ffi encompass the interval ( 0, 0) will produce current from the e ciency vanishes rather than maximizes at the stopping

2638 https://dx.doi.org/10.1021/acs.jpcb.0c00059 J. Phys. Chem. B 2020, 124, 2632−2642 The Journal of Physical Chemistry B pubs.acs.org/JPCB Article voltage point. The device operation depends on the character- produced only in the irreversible regime, and we could istics of the molecule−metal coupling, expressed in our model determine the points of optimal performance of such an engine Γ δ ϵ ffi by the parameters , , and 0. An example is shown in the with respect to power and e ciency. inset to Figure 4. The produced power vanishes as the While our calculations are based on Marcus electron transfer molecule−metal coupling goes to zero (a slip situation). It also kinetics in which level broadening due to molecule−metal vanishes for large δ, namely, when the coupling asymmetry coupling is disregarded, we have shown that the extension to disappears as Γ(ϵ) becomes independent of ϵ. the more general kinetics suggested in ref 38 which The definition of efficiency, eq 62, as the ratio between the (approximately) bridges between Marcus sequential hopping generated power and the invested work, is limited to the and Landauer cotunneling limit is possible. regime where V is smaller than the stopping voltage and the Energy conversion on the nanoscale continues to be a focus current flows against the voltage bias. For higher counter of intense interest. The present calculation provides a first voltage, current will flow in the opposite direction, and the simple step in evaluating such phenomena in a system efficiency as defined will become negative. In fact, the pump involving electron transport, electron−solvent interaction, operation described above changes into a possible motor and mechanical driving. operation that can translate into conversion of electrical bias into mechanical work if a proper load is put on the molecular ■ APPENDIX A 88,89 carrier. Note that the gain of the machine may be defined ̇ L Here, we derive eq 20 for Qe. Starting from eq 15, we present as the total amount of charge transferred against the bias, so L Qe,a→b in the form that at stopping voltage the current Iss disappears. Mathemati- cally, this means that the term proportional to I disappears L ss QEe,ab→ =ϵdr − −μL from eq 61. Physically, this means that the power vanishes when we stop the leakage current. Γ β + s ∫ d1ϵ[ −f (β ,) ϵ ] kL 4πE L L V. CONCLUSIONS ab→ r β In the present work we have analyzed the energy balance in ×( ϵ+E −ϵ )exp −s ()E −ϵ +ϵ2 − rd rd single-molecule junctions characterized by strong electron 4Er phonon interactions, modeled by a single-level molecule (dot) Ä É (A.1) Å Ñ connecting free electron metal electrodes, where charge Å Ñ Similarly Å Ñ transfer kinetics is described by Marcus electron transfer Å Ñ theory. The standard steady state transport theory was ÇÅ ÖÑ Γ β extended to include also slow driving of the molecular level L =−ϵ−+μ s ϵϵβ QEe,ba→ L dr L ∫ d(,)fL L that may be achieved by employing a time dependent gate kba→ 4πEr potential. A consistent description of the energetics of this β process was developed leading to the following observations: ×( ϵ +E −ϵ )exp −s ()E +ϵ −ϵ2 dr 4E rd (a) Accounting for the total energy and its heat, work, and r chemical components shows that energy conservation ÅÄ ÑÉ (A.2) Å Ñ (first law of thermodynamics) is satisfied by this model Integrating by parts, we obtainÅ Ñ Å Ñ at all examined orders of driving. ÇÅ ÖÑ (b) Heat is obviously produced by moving charge across a L Γ Er QEe,ab→ =ϵdr − −μL − L potential bias. In addition, when charge transfer involves kab→ πβs solvent reorganization, the current flowing in a biased ∂ϵf (,)β β junction can bring about heat transfer between the metal L L s 2 ×ϵ∫ d exp −−ϵ+ϵ()Erd and the solvent environments and may even produce ∂ϵ 4Er solvent cooling in some voltage ranges. Ä (A.3)É Å Ñ (c) In the presence of solvent reorganization, the friction Å Ñ ϵ and Å Ñ experienced by the driven coordinate d which expresses Å Ñ − ÇÅ ÖÑ energy loss (heat production) due to the molecule L Γ Er metal electron exchange is strongly affected by the QE→ =−ϵ−−μ dr e,ba L k L πβ presence of solvent reorganization. ba→ s ∂ϵβ (d) Beyond the reversible (driving at vanishingly small rate) fL (,)L βs 2 ×ϵ∫ d exp −+ϵ−ϵ()Erd limit, entropy is produced and is determined, at least to ∂ϵ 4E the second order in the driving speed, by the excess r work associated with the friction affected by the ÅÄ (A.4)ÑÉ Å Ñ molecule coupling to the electrodes and solvent Substituting these expressions intoÅ eq 18, we getÑ the ̇ Å ̇ Ñ ̇ environments. expression for QL given by eq 20. ExpressionsÅ for QR andÑQR ̇ e ÇÅ e ÖÑ e We have also used this model to study a molecular junction and Qs may be derived in the same way. with a periodically modulated dot energy. We have considered a model engine in which such periodic driving with a properly ■ AUTHOR INFORMATION chosen energy dependent molecule−electrode coupling can Corresponding Author move charge against a voltage bias and calculated the power Abraham Nitzan − Department of Chemistry, University of and efficiency of such a device. In the parameter range Pennsylvania, Philadelphia, Pennsylvania 19104, United States; consistent with our mathematical modeling, useful work can be School of Chemistry, Tel Aviv University, Tel Aviv 6997801,

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Israel; orcid.org/0000-0002-8431-0967; Email: anitzan@ (15) Koch, T.; Loos, J.; Fehske, H. Thermoelectric effects in sas.upenn.edu molecular quantum dots with contacts. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 89, 155133. Author (16) Zimbovskaya, N. A. The effect of dephasing on the Natalya A. Zimbovskaya − Department of Physics and thermoelectric efficiency of molecular junctions. J. Phys.: Condens. Electronics, University of Puerto RicoHumacao, Humacao, Matter 2014, 26, 275303. Puerto Rico 00791, United States (17) Koch, J.; von Oppen, F. Franck-Condon blockade and giant Fano factors in transport through single molecules. Phys. Rev. Lett. Complete contact information is available at: 2005, 94, 206804. https://pubs.acs.org/10.1021/acs.jpcb.0c00059 (18) Koch, J.; von Oppen, F.; Andreev, A. Theory of the Franck- Condon blockade regime. Phys. Rev. B: Condens. Matter Mater. Phys. Notes 2006, 74, 205438. The authors declare no competing ﬁnancial interest. (19) Hartle, R.; Thoss, M. Vibrational instabilities in resonant electron transport through single-molecule junctions. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 83, 125419. ■ ACKNOWLEDGMENTS (20) Hartle, R.; Thoss, M. Resonant electron transport in single- The present work was supported by the U.S. National Science molecule junctions: Vibrational excitation, rectification, negative differential resistance, and local cooling. Phys. Rev. B: Condens. Matter Foundation (DMR-PREM 1523463). Also, the research of Mater. Phys. 2011, 83, 115414. A.N. is supported by the Israel-U.S Binational Science (21) Galperin, M.; Ratner, M. A.; Nitzan, A. Hysteresis, switching, Foundation, the German Research Foundation (DFG TH and negative differential resistance in molecular junctions. A polaron 820/11-1), the U.S National Science Foundation (Grant model. Nano Lett. 2005, 5, 125−130. CHE1665291), and the University of Pennsylvania. N.A.Z. (22) Zazunov, A.; Feinberg, D.; Martin, T. Phonon-mediated acknowledges support of the Sackler Visiting Professor Chair negative differential conductance in molecular quantum dots. Phys. at Tel Aviv University, Israel. Rev. B: Condens. Matter Mater. Phys. 2006, 73, 115405. (23) Marcus, R. A. On the Theory of Oxidation-Reduction ■ REFERENCES Reactions Involving Electron Transfer. I. J. Chem. Phys. 1956, 24, 966−982. (1) Nitzan, A.; Ratner, M. A. Electron transport in molecular wire (24) Marcus, R. A. Chemical and Electrochemical Electron-Transfer junctions. Science 2003, 300, 1384−1389. Theory. Annu. Rev. Phys. Chem. 1964, 15, 155−196. (2) Agrait, N.; Yeati, A. L.; Van Ruitenbeek, J. M. Quantum (25) Marcus, R. A.; Sutin, N. Electron transfers in chemistry and properties of atomic-sized conductors. Phys. Rep. 2003, 377,81−279. biology. Biochim. Biophys. Acta, Rev. Bioenerg. 1985, 811, 265−308. (3) Coropceanu, V.; Cornil, J.; da Silva Filho, D. A.; Olivier, Y.; (26) Marcus, R. A. Electron transfer reactions in chemistry. Theory Silbey, R.; Bredas, J.-L. Charge transport in organic semiconductors. and experiment. Rev. Mod. Phys. 1993, 65, 599−653. Chem. Rev. 2007, 107, 926−952. (27) Nitzan, A. Chemical Dynamics in Condensed Phases: Relaxation, (4) Cuevas, J. C.; Sheer, E. Molecular Electronics: An Introduction to Transfer and Reactions in Condensed Molecular Systems; Oxford Theory and Experiment; World Scientiﬁc: Singapore, 2010. University Press, 2006. (5) Venkataraman, L.; Klare, J. E.; Nuckolls, C.; Hybertsen, M. S.; (28) Jia, C.; Migliore, A.; Xin, N.; Huang, S.; Wang, J.; Yang, Q.; Steigerwald, M. L. Dependence of single-molecule junction Wang, S.; Chen, H.; Wang, D.; Feng, B.; Liu, Z.; Zhang, G.; Qu, D. conductance on molecular conformation. Nature 2006, 442, 904− H.; Tian, H.; Ratner, M. A.; Xu, H. Q.; Nitzan, A.; Guo, X. Covalently 907. bonded single-molecule junctions with stable and reversible photo- (6) Li, X.; Hihath, J.; Chen, F.; Masuda, T.; Zang, L.; Tao, N. G. switched conductivity. Science 2016, 352, 1443−1445. Thermally activated electron transport in single redox molecules. J. (29) Migliore, A.; Nitzan, A. Irreversibility and hysteresis in redox Am. Chem. Soc. 2007, 129, 11535−11542. molecular conduction junctions. J. Am. Chem. Soc. 2013, 135, 9420− (7) Li, C.; Mishchenko, A.; Li, Z.; Pobelov, I.; Wandlowski, Th.; Li, 9432. X. Q.; Wurthner, F.; Bagrets, A.; Evers, F. Electrochemical gate- (30) Migliore, A.; Schiff, P.; Nitzan, A. On the relationship between controlled electron transport of redox-active single perylene bisimide molecular state and single electron pictures in simple electrochemical molecular junctions. J. Phys.: Condens. Matter 2008, 20, 374122. junctions. Phys. Chem. Chem. Phys. 2012, 14, 13746. (8) Li, Z.; Liu, Y.; Mertens, S. F. L.; Pobelov, I. V.; Wandlowski, Th. (31) Kuznetsov, A. M.; Medvedev, I. G. Coulomb repulsion effect on From redox gating to quantized charging. J. Am. Chem. Soc. 2010, 132, the tunnel current through the redox molecule in the weak tunneling 8187−8193. limit. Phys. Rev. B: Condens. Matter Mater. Phys. 2008, 78, 153403. (9) Galperin, M.; Ratner, M. A.; Nitzan, A. Molecular transport (32) Kuznetsov, A. M.; Medvedev, I. G.; Ulstrup, J. Coulomb junctions: vibrational effects. J. Phys.: Condens. Matter 2007, 19, repulsion effect in two-electron nonadiabatic tunneling through a one- 103201. level redox molecule. J. Chem. Phys. 2009, 131, 164703. (10) Dubi, Y.; Di Ventra, M. Heat flow and thermoelectricity in (33) Yuan, L.; Wang, L.; Garrigues, A. R.; Jiang, L.; Annadata, H. V.; atomic and molecular junctions. Rev. Mod. Phys. 2011, 83, 131−154. Antonana, M. A.; Barco, E.; Nijhuis, C. A. Transition from direct to (11) Zimbovskaya, N. A.; Pederson, M. R. Electron transport inverted charge transport Marcus regions in molecular junctions via through molecular junctions. Phys. Rep. 2011, 509,1−87. molecular orbital gating. Nat. Nanotechnol. 2018, 13, 322−329. (12) Cizek, M.; Thoss, M.; Domcke, W. Theory of vibrationally (34) Bueno, P. R.; Benites, T. A.; Davis, J. J. The mesoscopic inelastic electron transport through molecular bridges. Phys. Rev. B: electrochemistry of molecular junctions. Sci. Rep. 2016, 6, 18400. Condens. Matter Mater. Phys. 2004, 70, 125406. (35) Migliore, A.; Nitzan, A. Nonlinear charge transport in redox (13) Ren, J.; Zhu, J.-X.; Gubernatis, J. E.; Wang, C.; Li, B. molecular junctions: A Marcus perspective. ACS Nano 2011, 5, Thermoelectric transport with electron-phonon coupling and 6669−6685. electron-electron interaction in molecular junctions. Phys. Rev. B: (36) Craven, G. T.; Nitzan, A. Electron transfer through a thermal Condens. Matter Mater. Phys. 2012, 85, 155443. gradient. Proc. Natl. Acad. Sci. U. S. A. 2016, 113, 9421−9432. (14) Walczak, K. Thermoelectric properties of vibrating molecule (37) Craven, G. T.; Nitzan, A. Electron transfer at thermally asymmetrically connected to the electrodes. Phys. B 2007, 392, 173− heterogeneous molecule-metal interfaces. J. Chem. Phys. 2017, 146, 179. 092305.

2640 https://dx.doi.org/10.1021/acs.jpcb.0c00059 J. Phys. Chem. B 2020, 124, 2632−2642 The Journal of Physical Chemistry B pubs.acs.org/JPCB Article

(38) Sowa, J. K.; Mol, J. A.; Briggs, D.; Gauger, E. M. Beyond (62) Marcus, R. A. On the theory of electron-transfer reactions. VI. Marcus theory and the Landauer-Bu ttiker approach in molecular Unified treatment for homogeneous and electrode reactions. J. Chem. junctions: A unified framework. J. Chem. Phys. 2018, 149, 154112. Phys. 1965, 43, 679−694. (39) Sowa, J. K.; Lambert, N.; Seideman, T.; Gauger, E. M. Beyond (63) Stahler, J.; Meyer, M.; Zhu, X. Y.; Bovensiepen, U.; Wolf, M. Marcus theory and the Landauer-Buttiker approach in molecular Dynamics of electron transfer at polar molecule-metal interfaces: the junctions II. A self-consistent Born approach. J. Chem. Phys. 2020, role of thermally activated tunnelling. New J. Phys. 2007, 9, 394. 152, 064103. (64) Zanetti-Polzi, L.; Corni, S. A dynamical approach to non- (40) Cahill, D. G.; Ford, W. K.; Goodson, K.; Mahan, G. D.; adiabatic electron transfers at the bio-inorganic interface. Phys. Chem. Majumdar, A.; Maris, H. J.; Merlin, R.; Phillpot, S. R. Nanoscale Chem. Phys. 2016, 18, 10538. thermal transport. J. Appl. Phys. 2003, 93, 793−818. (65) Zotti, L. A.; Burkle, M.; Pauly, F.; Lee, W.; Kim, K.; Jeong, W.; (41) Leitner, D. M. Thermal boundary conductance and thermal Asai, Y.; Reddy, P.; Cuevas, J. C. Heat dissipation and its relation to rectification in molecules. J. Phys. Chem. B 2013, 117, 12820−12828. thermopower in single-molecule junctions. New J. Phys. 2014, 16, (42) Leitner, D. M. Quantum ergodicity and energy flow in 015004. molecules. Adv. Phys. 2015, 64, 445−517. (66) This happens because the distribution of holes (empty electron (43) Li, N.; Ren, J.; Wang, L.; Zhang, G.; Hanggi, P.; Li, B. states) below the Fermi energy μ and the distribution of electrons Phononics: Manipulating heat flow with electronic analogs and (occupied states) above the latter are the same. Both electrons and beyond. Rev. Mod. Phys. 2012, 84, 1045−1089. holes are coupled to the electrodes through the dot level. Therefore, − ϵ μ (44) Luo, T.; Chen, G. Nanoscale heat transfer from computation at d = , the distributions of electrons and holes are symmetric, and to experiment. Phys. Chem. Chem. Phys. 2013, 15, 3389−3397. the electron current cancels the hole current. (45) Rubtsova, N. I.; Nyby, C. M.; Zhang, H.; Zhang, B.; Zhou, X.; (67) Lee, W.; Kim, K.; Jeong, W.; Zotti, L. A.; Pauly, F.; Cuevas, J. Jayawickramarajah, J.; Burin, A. L.; Rubtsov, I. V. Room-temperature C.; Reddy, P. Heat dissipation in atomic-scale junctions. Nature 2013, ballistic energy transport in molecules with repeating units. J. Chem. 498, 209−212. Phys. 2015, 142, 212412. (68) Bruch, A.; Thomas, M.; Kusminskiy, S. V.; von Oppen, F.; (46) Marconnet, A. M.; Panzer, M. A.; Goodson, K. E. Thermal Nitzan, A. Quantum thermodynamics of the driven resonant level conduction phenomena in carbon nanotubes and related nano- model. Phys. Rev. B: Condens. Matter Mater. Phys. 2016, 93, 115318. structured materials. Rev. Mod. Phys. 2013, 85, 1295−1343. (69) Esposito, M.; Ochoa, M. A.; Galperin, M. Quantum (47) Al-Ghalith, J.; Ni, Y.; Dumitrica, T. Nanowires with dislocations Thermodynamics: A Nonequilibrium Green’s Function Approach. for ultralow lattice thermal conductivity. Phys. Chem. Chem. Phys. Phys. Rev. Lett. 2015, 114, 080602. 2016, 18, 9888. (70) Esposito, M.; Ochoa, M. A.; Galperin, M. Nature of heat in (48) Foti, G.; Vazquez, H. Adsorbate-driven cooling of carbene- strongly coupled open quantum systems. Phys. Rev. B: Condens. Matter based molecular junctions. Beilstein J. Nanotechnol. 2017, 8, 2060− Mater. Phys. 2015, 92, 235440. 2068. (71) Ludovico, M. F.; Arrachea, L.; Moskalets, M.; Sanchez, D. (49) Chen, Y.-C.; Zwolak, M.; Di Ventra, M. Inelastic effects on the Probing the energy reactance with adiabatically driven quantum dots. transport properties of alkanethiols. Nano Lett. 2005, 5, 621−624. Phys. Rev. B: Condens. Matter Mater. Phys. 2018, 97, 041416. (50) Losego, M. D.; Grady, M. E.; Sottos, N. R.; Cahill, D. G.; (72) Ludovico, M. F.; Moskalets, M.; Sanchez, D.; Arrachea, L. Braun, P. V. Effects of chemical bonding on heat transport across Dynamics of energy transport and entropy production in ac-driven interfaces. Nat. Mater. 2012, 11, 502−506. quantum electron systems. Phys. Rev. B: Condens. Matter Mater. Phys. (51) O’Brien, P. J.; Shenogin, S.; Liu, J. P.; Chow, K.; Laurencin, D.; 2016, 94, 035436. Mutin, P. H.; Yamaguchi, M.; Keblinski, P.; Ramanath, G. Bonding- (73) Ludovico, M. F.; Battista, F.; von Oppen, F.; Arrachea, L. induced thermal conductance enhancement at inorganic hetero- Adiabatic response and quantum thermoelectrics for ac-driven interfaces using nanomolecular monolayers. Nat. Mater. 2013, 12, quantum systems. Phys. Rev. B: Condens. Matter Mater. Phys. 2016, 118−122. 93, 075136. (52) Foti, G.; Vazquez, H. Interface tuning of current-induced (74) Ludovico, M. F.; Arrachea, L.; Moskalets, M.; Sanchez, D. cooling in molecular circuits. J. Phys. Chem. C 2017, 121, 1082−1088. Periodic energy transport and entropy production in quantum (53) Segal, D. Thermoelectric effect in molecular junctions: A tool electronics. Entropy 2016, 18, 00419. for revealing transport mechanisms. Phys. Rev. B: Condens. Matter (75) Ludovico, M. F.; Lim, J. S.; Moskalets, M.; Arrachea, L.; Mater. Phys. 2005, 72, 165426. Sanchez, D. Dynamical energy transfer in ac-driven quantum systems. (54) Wu, G.; Li, B. Thermal rectification in carbon nanotube Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 89, 161306. intramolecular junctions: Molecular dynamics calculations. Phys. Rev. (76) Thingna, J.; Barra, F.; Esposito, M. Kinetics and thermody- B: Condens. Matter Mater. Phys. 2007, 76, 085424. namics of a driven open quantum system. Phys. Rev. E: Stat. Phys., (55) Wu, L.-A.; Segal, D. Sufficient conditions for thermal Plasmas, Fluids, Relat. Interdiscip. Top. 2017, 96, 052132. rectification in hybrid quantum structures. Phys. Rev. Lett. 2009, (77)Haughian,P.;Esposito,M.;Schmidt,T.L.Quantum 102, 095503. thermodynamics of the resonant-level model with driven system- (56) Lu, J.-T.; Zhou, H.; Jiang, J.-W.; Wang, J.-S. Phonon thermal bath coupling. Phys. Rev. B: Condens. Matter Mater. Phys. 2018, 97, transport in silicon-based nanomaterials. AIP Adv. 2015, 5, 053204. 085435. (57) Pecchia, A.; Romano, G.; Di Carlo, A. Theory of heat (78) Cavina, V.; Mari, A.; Giovannetti, V. Slow Dynamics and dissipation in molecular electronics. Phys. Rev. B: Condens. Matter Thermodynamics of Open Quantum Systems. Phys. Rev. Lett. 2017, Mater. Phys. 2007, 75, 035401. 119, 050601. (58) Yang, N.; Xu, X.; Zhang, G.; Li, B. Thermal transport in (79) Dou, W.; Ochoa, M. A.; Nitzan, A.; Subotnik, J. E. Universal nanostructures. AIP Adv. 2012, 2, 041410. approach to quantum thermodynamics in the strong coupling regime. (59) Galperin, M.; Saito, K.; Balatsky, A. V.; Nitzan, A. Cooling Phys. Rev. B: Condens. Matter Mater. Phys. 2018, 98, 134306. mechanisms in molecular conduction junctions. Phys. Rev. B: Condens. (80) Ochoa, M. A.; Bruch, A.; Nitzan, A. Energy distribution and Matter Mater. Phys. 2009, 80, 115427. local fluctuations in strongly coupled open quantum systems: The (60) Lykkebo, J.; Romano, G.; Gagliardi, A.; Pecchia, A.; Solomon, extended resonant level model. Phys. Rev. B: Condens. Matter Mater. G. C. Single-molecule electronics: cooling Individual vibrational Phys. 2016, 94, 035420. modes by the tunneling Current. J. Chem. Phys. 2016, 144, 114310. (81) Bruch, A.; Lewenkopf, C.; von Oppen, F. Landauer-Bu ttiker (61) Kilgour, M.; Segal, D. Coherence and decoherence in quantum approach to strongly coupled quantum thermodynamics: Inside- absorption refrigerators. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. outside duality of entropy evolution. Phys. Rev. Lett. 2018, 120, Interdiscip. Top. 2018, 98, 012117. 107701.

2641 https://dx.doi.org/10.1021/acs.jpcb.0c00059 J. Phys. Chem. B 2020, 124, 2632−2642 The Journal of Physical Chemistry B pubs.acs.org/JPCB Article

(82) Semenov, A.; Nitzan, A. Transport and thermodynamics in quantum junctions: A scattering approach. arXiv 2019, 1912.03773. (83) Migliore, A.; Nitzan, A. Irreversibility in redox molecular conduction: single versus double metal-molecule interfaces. Electro- chim. Acta 2015, 160, 363−375. (84) Oz, A.; Hod, A.; Nitzan, A. Numerical approach to nonequilibrium quantum thermodynamics: Nonperturbative treatment of driven resonant level model based on the driven Liouville von Neimann formalism. J. Chem. Theory Comput. 2020, 16, 1232. (85) The smallness of ϵḋ should be established by comparison to kTΓ and Γ2/ℏ (ref 67). (86) Bode, N.; Kusminskiy, S. V.; Egger, R.; von Oppen, F. Current- induced forces in mesoscopic systems: A scattering-matrix approach. Beilstein J. Nanotechnol. 2012, 3, 144−162. (87) Einax, M.; Solomon, G. C.; Dieterich, W.; Nitzan, A. Unidirectional hopping transport of interacting particles on a finite chain. J. Chem. Phys. 2010, 133, 054102. (88) Bustos-Marun,́ R. A.; Refael, G.; von Oppen, F. Adiabatic quantum motors. Phys. Rev. Lett. 2013, 111, 060802. (89) Fernandez-Alcá zar,́ L. J.; Pastawski, H. M.; Bustos-Marun,́ R. A. Nonequilibrium current-induced forces caused by quantum local- ization: Anderson adiabatic quantum motors. Phys. Rev. B: Condens. Matter Mater. Phys. 2019, 99, 045403.

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