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Kinetic Schemes in Open Interacting Systems Abraham Nitzan†,‡ and Michael Galperin*,¶

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Kinetic Schemes in Open Interacting Systems Abraham Nitzan†,‡ and Michael Galperin*,¶ Letter Cite This: J. Phys. Chem. Lett. 2018, 9, 4886−4892 pubs.acs.org/JPCL Kinetic Schemes in Open Interacting Systems Abraham Nitzan†,‡ and Michael Galperin*,¶ † Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States ‡ School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel ¶ Department of Chemistry and Biochemistry, University of California at San Diego, La Jolla, California 92093, United States ABSTRACT: We discuss utilization of kinetic schemes for description of open interacting systems, focusing on vibrational energy relaxation for an oscillator coupled to a nonequilibirum electronic bath. Standard kinetic equations with constant rate coefficients are obtained under the assumption of time scale separation between the system and bath, with the bath dynamics much faster than that of the system of interest. This assumption may break down in certain limits, and we show that ignoring this may lead to qualitatively wrong predictions. Connection with more general, nonequilibrium Green’s function (NEGF) analysis is demonstrated. Our considerations are illustrated within generic molecular junction models with electron−vibration coupling. Development of experimental techniques on the nanoscale has conditions for time scale separations are established in a given made studies of single-molecule junctions possible. These range, they may become invalid in other domains of operation. experiments yield unique possibilities to explore physical and Such situations are well-known in classical dynamics. For chemical properties of molecules by measuring their responses example, transition state theory (TST) of molecular rate to external perturbations. Following experimental advances, processes assumes that molecular degrees of freedom except there was rapid development of theoretical approaches. Today the reaction coordinate are at thermal equilibrium (and a variety of techniques, ranging from diagrammatic expansions therefore can be taken to be part of the thermal environment). 1,2 (such as, e.g., the nonequilibrium Green function (NEGF), TST breaks down when the observed rate is on the order of, or 3,4 5,6 quantum master equation, and Hubbard NEGF )to faster than, the rate of thermal relaxation in the molecule, as approximate treatments of strongly correlated systems (e.g., demonstrated in the Kramers theory of activated rate dynamical mean field theory7 and beyond8) and to numerically 24 − processes. exact methods (e.g., renormalization group techniques9 12 and 13−15 Importantly, even if the assumed time scale separation holds continuous time quantum Monte Carlo ), are available. near equilibrium, it might fail far from it. The reason is that Implementation of such schemes, particularly the numerically systems interacting with their equilibrium surrounding remain exact approaches, is often expensive, and their applications for within the energetic domain of thermal energy, while systems Downloaded via UNIV OF PENNSYLVANIA on September 21, 2020 at 14:14:07 (UTC). simulations of realistic systems is limited. coupled to nonequilibrium environments, e.g., under optical At the same time, simple kinetic schemes have been widely See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. illumination or voltage bias, may be driven to energy domains and successfully utilized in the description of rate phenomena where time scale separation does not hold. Thus, while in open molecular systems (for example, donor−bridge− − situations of the first kind (such as activated barrier crossing) acceptor (DBA) molecular complexes).16 23 Such schemes are well understood and documented, mathematically equiv- lead to a description of system states connected by rate alent circumstances in nonequilibrium systems have been often processes whose Markov limit description provides “rate overlooked. One such case is the vibrational dynamics of coefficients” that enter into the kinetic description (master molecules adsorbed at or bridging between metallic interfaces equation) of the system evolution. Such Markov limit descriptions rely on time scale separation between the due to coupling to the thermal electronic baths. Molecular vibrational motion is sensitive to the electronic occupation of observed system evolution and dynamic processes that ff determine the rates; the latter usually involves the dynamics the molecule, which in turn is a ected by the molecule and of relaxation in the bath. Obviously, the details of such kinetic metal electronic structure, their mutual coupling, and the schemes depend on the way system−bath separation is defined junction voltage bias. A common approximation, equivalent to and used. The general practice dictated by balance between the fast bath assumption discussed above, is to disregard the simplicity and rigor takes “the system” to be comprised by the observed variables together (when possible) with other Received: June 15, 2018 variables whose inclusion makes the dynamics Markovian. Accepted: August 10, 2018 This practice should be exercised with caution because even if Published: August 10, 2018 © 2018 American Chemical Society 4886 DOI: 10.1021/acs.jpclett.8b01886 J. Phys. Chem. Lett. 2018, 9, 4886−4892 The Journal of Physical Chemistry Letters Letter effect of vibrational dynamics on the electronic subsystem, influence between the system and environment, namely, the representing the latter by a thermal electronic bath or, for a effect of the environment on the system as well as the back- biased (current carrying) junction, by the corresponding action from the system on the environment. We show that the steady-state electronic distribution, assumed unaffected by basis of the zero-order approximation, the assumption of time the vibrational process. This level of description, which scale separation between molecular electronic and vibrational effectively takes the molecular electronic degrees of freedom degrees of freedom (i.e., ability to neglect back-action of − as part of the (generally nonequilibrium) electronic bath,25 27 molecular vibration on the electronic bath), does not hold has been recently used to discuss bias-induced vibrational automatically and in fact has a limited range of validity. − instabilities.28 30 While the limitations of such treatments are Consequently, while regimes of significantly enhanced vibra- sometimes pointed out,30 in other publications, they are tional heating can be found in biased junctions with electron− ignored. Indeed, such instabilities were recently claimed31 to phonon coupling (and heating transient spikes may occur be generic properties of wires whose conduction is dominated following sudden parameter change), instabilities identified as by distinct electronic resonances (or, in the language of ref 31, the appearance of a negative vibrational dissipation rate do not by “separated” electronic states). occur. In our consideration below, the molecular vibration It should be noted that, in general, zero-order treatments of (system) is weakly coupled to the electronic degrees of the kind described above are known to violate conservation freedom (bath), which is the usual setup in considerations of laws.32,33 Thus, notwithstanding their usefulness in many system and bath separation. We stress that even in this applications, such treatment should be regarded with caution, favorable situation kinetic considerations may lead to in particular, when unusual behaviors are observed. For qualitative failures. example, the observation of a negative sign of the vibrational Below, after introducing the model, we discuss its general dissipation rate at an apparent steady state of a molecular treatment using NEGF and its connection to simple kinetic junction should not be regarded as an indicator of a true considerations. We consider the steady state of such a model vibrational instability in the system but (like in linear stability under voltage bias and illustrate failures of standard kinetic analysis of nonlinear differential equations) as an indication of description within numerical examples. failure of the underlying assumptions that lead to such a result. Model and Method. We consider a junction consisting of a It should be emphasized that (again, as in linear stability molecular bridge M coupled to two contacts, L and R (Figure analysis) such analysis can be useful as an indicator that a real 1). Besides electronic degrees of freedom, a molecular ω stable state exists elsewhere (which in a real anharmonic vibration, modeled as a harmonic oscillator of frequency 0 molecule may or may not lie beyond a bond-breaking coupled to the molecular electronic subsystem, is included. threshold). Still, many low-order treatments28,29 of vibrational The contacts are reservoirs of free charge carriers, each at its instabilities in harmonic bridge models of molecular junctions own equilibrium. The model Hamiltonian is leave the reader with the message that the observed “runaway ̂ ̂ ̂ ̂ phonon” describes the full physical behavior. H = HHVM + ∑ ()KKM+ Exact numerical solutions43 areobviouslycapableof KLR= , (1) exploring the correct physical picture. Here we show that an † Ĥ = ∑ HddM ̂ ̂ + ω aa†̂ ̂ approximate self-consistent treatment that does not violate M mm12mm 1 2 0 conservation laws can already avoid the qualitative pitfalls of a mm12, ∈ M linear theory. We consider simple junction models with † + ∑ Uaadd()̂ + †̂ ̂ ̂ electron−vibration coupling (see Figure 1), treated within mm12m 1 m 2 mm12, ∈ M (2) ̂ † HK =ϵ∑ kkcĉ
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