Exploring Bridges between Quantum Transport and Electrochemistry. I

Article pubs.acs.org/JPCC

Exploring Bridges between Quantum Transport and Electrochemistry. I. Kirk H. Bevan,* Md. Sazzad Hossain, Asif Iqbal, and Zi Wang Materials Engineering, McGill University, Montréal, Québec H3A 0C5, Canada ABSTRACT: In this work we present a theoretical connection between the Landauer picture utilized in quantum transport and the Gerischer picture utilized in electrochemistry. A comprehensive analysis of the single-particle picture and total energy picture in electrochemistry is presented, followed by derivation of electron transfer rates utilizing the nonequilibrium Green’s function formalism. Correlations are also made with the Marcus−Hush theoretical approach more often utilized in electrochemistry. The analysis is limited to tunneling (also called outer-sphere) electrochemical reactions. In general, it is expected that this work will serve to further bridge the diverse condensed matter and chemistry foundations inherent to interfacial electrochemistry.

1. INTRODUCTION Electrochemistry is a ubiquitous aspect of our lives: from lithium ion batteries, to biomedical sensors, through to artificial photosynthesis. It quietly permeates every aspect of our modern world.1−3 However, despite its importance, the complex physics and chemistry present during electrochemical reactions have thus far hampered the development of fully quantitative and predictive theories (e.g., from first-principles).1,2 Though an abundance of important and pioneering theoretical approaches exist, most are rooted firmly in the chemistry community extending often from the study of thermodynamics and kinetics.1,2 However, the chemical aspects of electrochemistry constitute only half of the physical picture present in these reactions, which invariably includes a solidstate region of conduction. Thus, electrochemistry is an interfacial process, typically constituting a charge transfer event between a solid and a “liquid”.1 Although the reactant medium need not necessarily be a “liquid”, exemplary systems include organic radical polymer batteries.4 On the other hand, on the solid-state side we have a material, often a metal, understood through concepts firmly rooted in condensed matter physics, whereas, on the “liquid” side, we have reactants, understood through concepts deeply rooted in chemistry. Thus, an enormous difficulty emerges when communicating and quantifying electrochemical processes occurring at solid−“liquid” interfaces to both the physics and chemistry communities. From the perspective of a physicist, one prefers to understand how physical (band) conduction in a material is transformed into a chemical electron transfer event at a solid− liquid interface,1,5 whereas from the perspective of the chemist, one prefers to start from the kinetics present in the “liquid” and subsequently evaluate how charge transfer can occur with a solid.1,6 © XXXX American Chemical Society

The communication breakdown is further complicated by the distinct languages utilized by the chemistry and physics communities. The solid side of the junction is usually understood in terms of what is termed the “single-particle” picture in physics,7 whereas “liquid”-phase reactions are usually quantified by chemists in terms of the “total energy” of the reactants (hence the link to thermodynamics).1,2 Therefore, a conceptual difficulty naturally arises when linking the two phases and arriving at overall interfacial reaction rates. Gerischer’s best known theoretical contribution was to provide a link between the “total energy” and “single-particle” languages.1,2,8−10 However, Gerischer’s approach does not in itself provide a systematic method for arriving at electrochemical rate constants that can be directly compared with experiment. In order to arrive at rate estimates one needs to include the pioneering contributions by Marcus and Hush (or other associated electron transfer approaches).1,2,11,12 On the other hand, in recent years, short-range electron transfer and transport processes have been increasingly captured by physicists and chemists alike utilizing the “single-particle” Landuaer picture for “quantum transport” (that grew out of the condensed matter community).7,13−15 In particular, the Landauer picture has become the de facto method of choice when analyzing electron transport from solid-state regions through molecular (“chemical”) conductors, in the field of molecular electronics.14,15 A possible extension of this Landuaer-based bridge between physics and chemistry is more fully to extend the “single-particle” language present in the physics community by capturing electrochemical rate constants. Important contributions in this regard have been made by Received: October 2, 2015 Revised: December 11, 2015

A

DOI: 10.1021/acs.jpcc.5b09653 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C Nitzan and others.5,16,17 In this work we attempt to extend the Landauer picture more fully to tunneling outer-sphere electrochemical reactions by utilizing the nonequilibrium Green’s function (NEGF) formalism.7,13−15 The primary contribution of this paper is to demonstrate how coherent electrochemical electron transfer events can be captured within Landauer’s picture and the NEGF formalism, by directly building upon the seminal work by Gerischer.1,2,7−10,13 Before proceeding, it is important to note that electrochemistry can be divided into two reaction classifications: inner-sphere and outer-sphere reactions.1 Outer-sphere reactions refer to those in which electron transfer is accompanied only by a tunneling event: the reactant remains separated from the substrate (solid region) by a tunneling barrier through the entirety of the reaction process (as shown in Figure 1a). This

science,19 thus in this work we limit our theoretical study to the more tangible and malleable domain of outer-sphere reactions. Nevertheless, a wide range of exciting work remains to be done in understanding inner-sphere electrochemical reactions.1 To further extend the bridge between physics and chemistry in the domain of electrochemistry our discussion is divided into two main parts. First, in section 2 we connect the “singleparticle” picture utilized by physicists and the “total energy” picture utilized by chemists. This is done to ensure that the connection between Landauer transport and Gerischer’s electrochemical picture, which are primarily contained in section 3.2, is presented in a format familiar to both communities. We begin by outlining Gerischer’s picture in section 2.1, followed by an overview of the more traditional Marcus−Hush electron transfer picture in section 2.2.8−12 Subsequently, in section 3 we develop rate expressions based on the “single-particle” picture. Expressions for the Marcus− Hush electron transfer rates are first overviewed in section 3.1, where important tunneling concepts regarding coherent and incoherent processes are outlined.1,2,7,13 Lastly, in section 3.2 we utilize the NEGF formalism to develop Landauer-based expressions for tunneling (outer-sphere) electrochemical reaction ratesthe key contribution of this paper. Importantly, the Landauer approach is shown to yield near identical outersphere rate expressions to those currently found in the electrochemistry literature.1,17 Throughout this work a deliberate measured attempt is made to clearly connect the condensed matter (“single-particle”) and chemistry (“total energy”) perspectives. In part II of this series, we will extend the derived rates to capture current−voltage characteristics in a model electrochemical system.20

2. SINGLE-PARTICLE PICTURE IN ELECTROCHEMISTRY Various approaches are used to connect the single-particle picture13 with heterogeneous electrochemical reactions at interfaces (again, here we speak of outer-sphere electrochemical reactions involving tunneling).1,2 Those originating from the field of electrochemistry tend to utilize a modified version of Marcus−Hush theory,1,11,12 which we will not discuss. The reader may wish to consult the excellent discussion in ref 1 which covers Marcus−Hush theory. In this section we attempt to provide a model which will generate an intuitive transition from the single-particle picture to the study of tunneling electrochemical reactions at interfaces based on Gerischer’s approach.2,8−10 We hope this model will further the interaction between those coming from a condensed-matter or “band diagram” background and the field of electrochemistry.1,2,13 We divide our discussion on this topic into two parts: first, in section 2.1 Gerischer’s picture is outlined in detail; second, in section 2.2 the relationship between Gerischer’s single-particle picture and homogeneous electron transfer reactions, for which Marcus−Hush theory was originally developed, is explored. 2.1. Gerischer’s Model for Heterogeneous Electrochemical Reactions. First we begin with a brief review of outer-sphere heterogeneous reactions, which involves the tunneling transfer of an electron from a single reactant to a contact (here we use a metal substrate) as shown in Figure 1a. The term “heterogeneous” denotes the fact that the metal contact and reactant are not identical species; we will come to homogeneous reactions later. An oxidized reactant (Ox) has one less electron than a reduced (Red) reactant; hence, oxidized reactants accept an electron from the metal contact,

Figure 1. Connecting the total energy (E) and the single-particle energy (ε) pictures in a heterogeneous reaction between a reactant and metal contact [displayed diagrammatically in (a)]. (b) Depicts the E profile versus nuclear coordinate (qht) for a reactant in a reduced (red) or oxidized (blue) state. Multiples of the heterogeneous reorganization energy (λht) are utilized for the energy scale. The single-particle oxidation and reduction energies (εox and εred) are shown as differences between oxidation (Eox) and reduction (Ered) states in both (b) and more generally across all reaction coordinates as a linear dispersion of single-particle energies in (c). The nuclear probability distribution as a function of qht is shown in (c), for both the reduced (Dred) and oxidized (Dox) states. The linear relationship between E and qht enables a projection of the nuclear probability distribution into the single-particle picture in (d).

tunneling region is usually a few monolayers of solvent molecules, H2O in the case of the canonical outer-sphere reactants [Fe(H2O)6]+3 and [Fe(H2O)6]+2, but can also be a solid insulating tunneling region.1,18 On the other hand, innersphere reactions are invariably accompanied by a reactant chemisorption event whereby the reactant physically bonds with the solid-phase region (e.g., H2 and O2 evolution during artificial photosynthesis).2 Inner-sphere reactions are exceedingly complex and might be classified as a grand challenge in B


Article

The Journal of Physical Chemistry C

contact and the reactant subsequently “reorganized” from qht = −1 to qht = 0. Of course the reverse process can occur, whereby an oxidized state gains an electron from the contact via the reaction Ox + e− → Red. The oxidized species has an electron affinity of εox that represents a bound electron state of negative energy. This electron affinity at qht = 0 is not the same as that at qht = −1 since the nuclear coordinates have changeda point we will come back to shortly. Hence, once the electron is gained by the oxidized state at qht = 0, its total bonding energy lowers by an amount εox to the reduced (red) parabola point at qht = 0 again, electrons move while the nuclei stand still following the Franck−Condon principle. Now the reactant is reduced but sits at a nonequilibrium coordinate configuration qht = 0 (recall that a reduced species has an energetic minimum at qht = −1). Hence, following bonding forces it must rapidly relax to qht = −1, lowering its total bonding energy (E) by an amount of λht (traversing the solid red arrow in Figure 1b). In this process an electron was gained from the contact and the reactant subsequently “reorganized” from qht = 0 to qht = −1. Once again we can repeat the cycle, with the reduced species giving up an electron to the contact, ad infinitum (at least for tunneling electrochemical reactions). Thus, at the reduced (red) nuclear minimum (qht = −1) the ionization/affinity energy for the species is εred, and at the oxidized (blue) nuclear minimum (qht = 0) the ionization/ affinity energy is εox (as shown in Figure 1b). Under the simplified harmonic approximations we have made, these two single-particle energies differ by 2λht such that εox = εred + 2λht. However, we need not limit ourselves to a consideration of ionization/affinity energies at two nuclear coordinates. The total bonding energy (E) for a reactant in each charge state may be expressed as

and reduced reactants give up an electron to the contact (see Figure 1a). It is important to note that the oxidized and reduced species usually correspond to different charge states of the same chemical species (e.g., the canonical outer-sphere reactants [Fe(H2O)6]+3 and [Fe(H2O)6]+2).1 Now, the two different equilibrium charge states (Ox and Red) of a chemical species will correspond to two different nuclear coordinates, simply because the nuclear−electron and electron−electron bonding interactions will change when an electron is added or removed. This is referred to as the “reorganization” of the nuclear coordinates between the two different charge states. This reorganization effect is particular to small reactants (e.g., molecules or solvated ions in liquids):1,2 when a “reactant” is very large (for example our contact in Figure 1a) and the transferred electron is delocalized, there is so little impact from the addition or removal of one electron that its nuclear coordinates do not change. This is not necessarily the case in solids where electrons are localized and polarons can form,3,21,22 but this can be ignored for the moment since metals possess delocalized electrons (which leads to the concept of band conduction).13 In this work we only consider heterogeneous reactions between a metal contact (which does not reorganize upon each charge transfer reaction) and a reactant (which does reorganize). As mentioned, due to the phenomena of nuclear reorganization, upon the addition or removal of an electron, the oxidized and reduced species have different equilibrium bonding configurations. For example, [Fe(H2O)6]+3 (Ox) has a shorter Fe−H2O ligand bond length than [Fe(H2O)6]+2 (Red).1,6 This is shown diagrammatically in Figure 1b, where the oxidized state total bonding energy (Eox) is represented as a blue parabolic dispersion centered at the heterogeneous nuclear coordinate configuration qht = 0, whereas the reduced reactant total bonding energy (Ered) is represented as a red parabolic dispersion centered at qht = −1. This is the often used harmonic approximation for bonding energies.1,2,7 The nuclear coordinate normalization is completely arbitrary, and various representations exist in the literature.1,2 Let us start by considering what occurs when a reduced species loses an electron to the contact via the reaction Red − e− → Ox (as shown in Figure 1b). If we start from the energetic minimum (qht = −1 on the red parabola in Figure 1b) and remove an electron from the reduced species, this is “equivalent” to introducing an ionization energy change to the reduced species (i.e., knocking its highest energy electron off into the vacuum). However, in this case the electron tunnels into the contact at an equivalent bound energyso the total energy of the entire system, reactant plus contact, does not change. Nevertheless, the lost electron is bound at a negative energy εred, and this loss of an electron raises the energy of the reactant by −εred. This brings us to the point of the oxidized species (blue) parabola at qht = −1; following the Franck− Condon principle we have removed an electron from the reactant while keeping its nuclear coordinates fixed. With the electron now departed into the contact, the reactant is now oxidized but at a nonequilibrium nuclear configuration. Hence, bonding forces rapidly push it toward the oxidized state energetic minimum at qht = 0 (following the blue arrow along the blue parabola in Figure 1b), losing λht from the total bonding energywhich we call the heterogeneous reorganization energy. It is assumed to be the same for both charge states in this discussion.1,2 In this process an electron was lost to the

Ered = λht(qht + 1)2

(1)

Eox = λhtqht2 + |εox | + λht

(2)

where the separation of the parabolic curves in Figure 1b may be set to any oxidation energy εox. Here we have used εox = −6λht purely for illustrative purposes. The single-particle ionization/affinity energies ε of the reactant across all nuclear coordinates (qht) can be arrived at by subtracting these two expressions ε = Ered − Eox = 2λhtqht − |εox |

(3)

which gives the linear dispersion shown in black in Figure 1c. This linear dispersion defines the single-particle energies for the reactant in either the reduced or oxidized state across all nuclear coordinates with respect to the reference vacuum energy of 0 (not merely at the bonding minima where it takes on the values εred and εox, respectively).2,8−10,16 Approximately, in the single-particle nomenclature,7 the linear dispersion in Figure 1c represents both the “LUMO” level (electron affinity energy) of the oxidized state and the “HOMO” level (electron ionization energy) of the reduced state. Strictly speaking, ionization and affinity energies subjected to either electronic or nuclear reorganization should be compared via total energy differences rather than single-particle Hamiltonian eigenenergies.7,23 This, however, is not the end of our conversion to the singleparticle picture. Though the full range of ionization/affinity energies (plotted in black in Figure 1c) remains accessible to C


Article


the meantime, it is important to understand the fundamental relationship between Gerischer’s model and the Marcus−Hush homogeneous reaction model that is more commonly discussed in the literature, as it will form the basis of a fundamental assumption on how we model heterogeneous tunneling rates. 2.2. Relation to Homogenous Electron Transfer Reactions. To those entering the electrochemical community from a “band diagram” background, the concept of a “reorganization energy” often leads to a confusion between heterogeneous reactions and homogeneous reactionsand the interpretation of electron transfer processes. Homogeneous reactions involve an electron transfer reaction between two identical species (e.g., [Fe(H2O)6]+3 + [Fe(H2O)6]+2 ⇌ [Fe(H2O)6]+2 + [Fe(H2O)6]+3), and no contact or substrate is involved in this process. Typically they occur between species dissolved in solution, though this is not a fundamental restriction,4 and in the nonadiabatic limit can be described in the original Marcus−Hush approach sketched in Figure 2a.1,2 Here, in Figure 2a, we label two identical species (apart from their charge state) by the nomenclature α and β and examine the reaction αox + βred ⇌ αred + βox (e.g., [Fe(H2O)6]+3 + [Fe(H2O)6]+2 ⇌ [Fe(H2O) 6]+2 + [Fe(H2O)6]+3). The exchange of an electron between the two species is driven by

both reduced and oxidized states, the probability of an oxidized or reduced species possessing a given affinity/ionization energy is very different. This is because they have two different total energy minima as a function of the nuclear coordinate qht (as shown in Figure 1b). Following transition state theory,24 the probability of an oxidized species residing at a given nuclear coordinate may be approximated as a Boltzmann distribution of the form 2

Dox (qht) ∝ e−λhtqht / kB ;

(4)

where kB is Boltzmann’s constant and ; is the temperature. Likewise, the probability of a reduced species residing at a given nuclear coordinate is 2

Dred (qht) ∝ e−λht(qht + 1)

/ kB ;

(5)

Both probability distributions are sketched in Figure 1c, where it can be seen that an oxidized reactant (blue) is strongly localized around qht = 0 and a reduced (red) reactant is strongly localized around qht = −1. This probability of nuclear distributions can be transformed into an affinity/ionization energy “occupation” probability by rearranging eq 3 into the form qht = (ε + |εox|)/2λht and inserting it into eqs 4 and 5 to obtain Dox (ε) =

⎛ −(ε − ε )2 ⎞ 1 ox ⎟ exp⎜ 4πλhtkB ; ⎝ 4λhtkB ; ⎠

(6)

Dred (ε) =

⎛ −(ε − ε )2 ⎞ 1 red ⎟ exp⎜ 4πλhtkB ; ⎝ 4λhtkB ; ⎠

(7)

where we have normalized the single-particle occupation distribution of each state to 1. This is Gerischer’s description of oxidation and reduction states in the single-particle picture as displayed in Figure 1d.2,8−10 Here we have juxtaposed the oxidation and reduction single-particle distributions against the distribution of single-particle states typified by a metal,7 which contains a continuum of single-particle states separated by an electrochemical potential (μeq, also called a Fermi level) that divides the filled states (shaded in Figure 1d) from the empty states (above the shaded substrate region in Figure 1d). The filled states in the metal substrate are occupied according to the Fermi function f (ε) = 1/(1 + exp[(E − μeq )/kB ;]), and the empty states are similarly distributed according to [1 − f(ε)]. In this combined single-particle picture a reactant may exchange an electron with the contact/substrate through the isoenergetic transfer of an electron (speaking in terms of single-particle energies). In short, Gerischer’s picture is simply a transformation from nuclear coordinates to single-particle ionization/affinity coordinates, as sketched in Figure 1, made possible by the linear relationship between the two “coordinates” in the harmonic approximation limit. In the single-particle picture, we can view Dox(ε) as the broadened energy distribution [or density of states (DOS)] of the affinity level εox and likewise Dred(ε) as the broadened DOS of the ionization level εred.7 However, it is important to recognize that Gerischer’s interpretation is likely only valid in the weak coupling limit where an electron tunnels to/from the metal contact to the reactant (what is called an outer-sphere reaction in electrochemistry).1,16,20 We will further explore Gerischer’s model when we develop tunneling rate expressions based on the Landauer picture in section 3. In

Figure 2. Connecting the single-particle (ε) and total energy (E) pictures for reactants of the same species (labeled α and β) participating in a homogeneous reaction. (a) Displays E of both species as a function of the homogeneous reaction coordinate qhm, with solid black corresponding to α oxidized (blue) and β reduced (red) and vice versa for the dashed black line. Reduced species are represented diagrammatically by a red sphere and oxidized species by a blue sphere, with sphere size corresponding to the reaction coordinate. (b) Simplified separate total reaction energy contributions by each species providing the plots in (a). (c) The single-particle (ε) energy dispersion of both species as a function of qhm intersecting when both species (α and β) have the same reaction coordinates, thereby enabling an electron transfer reaction at rate khmcombined dashed-red/solidblue corresponds to ε for α, combined dashed-blue/solid-red corresponds to ε for β. D


Article

The Journal of Physical Chemistry C the change in their homogeneous nuclear coordinate qhm (see Figure 2), which is related to but not quite the same as the heterogeneous reaction coordinate explored earlier in Figure 1. There are two parabolic minima corresponding to one species being reduced and the other oxidized, and a reduced species is represented by red and an oxidized species by blue. In Figure 2a we ascribe the charge state αox and βred to the solid left harmonic total energy dispersion with a minimum at qhm = −0.5 and the charge state αred and βox to the dashed right parabolic total energy dispersion with a minimum at qhm = +0.5. At the nuclear coordinate qhm = −0.5 species α has “contracted” nuclear coordinates (e.g., [Fe(H2O)6]+3) [sketched as a small blue circle in Figure 2a] relative to species β which has “expanded” nuclear coordinates (e.g., [Fe(H2O)6]+2) [sketched as a larger red circle in Figure 2a]. The electron prefers to be in the state which “gives it more room” and hence tends to reside on species β at qhm = −0.5. The reverse scenario is sketched at qhm = +0.5 in Figure 2a, where α has “expanded” nuclear coordinates (e.g., [Fe(H2O)6]+2) and β “contracted” nuclear coordinates (e.g., [Fe(H2O)6]+3), such that the electron now prefers to reside on species α. Just as with the heterogeneous reaction discussed in the previous section, contraction or extension of the nuclear coordinates corresponds (in the simplest terms) to the extension and contraction of the Fe− H2O ligand bond.1,2,6 However, there are also polarization/ dielectric contributions which are discussed at length in ref 1. Now, kinetic activation is the typical mechanism by which an electron is exchanged between α and β, whereby the nuclear coordinates are traversedwe will explore this shortly.1 However, there is another mechanism, whereby the species stay fixed at a given qhm coordinate and an electron is knocked off the “expanded” species and placed on the “contracted” species. This is sketched in Figure 2a at qhm = ±0.5, whereby we traverse from the solid/dashed parabola to a point directly above on the dashed/solid parabola. At qhm = ±0.5 the cost of exchanging an electron in this manner is the affinity energy of the oxidized species (εox) less the ionization energy of the reduced species (εred), which we call the homogeneous reorganization energy λhm = εox − εred = 2λht, and it is twice the heterogeneous reorganization energy (see the earlier discussion pertaining to Figure 1).2 To see why this is so, we need to plot the individual total energy contributions by the reduced (red) and oxidized (blue) species as shown in Figure 2b. In a simplified description, the two solid lines in Figure 2b sum to give the total energy solid dispersion in Figure 2a; likewise the two dashed lines in Figure 2b sum to give the total energy dashed dispersion in Figure 2a. What can be seen in Figure 2b is that species α has an ionization/affinity energy of εox, and species β has an ionization/affinity energy of εred at qhm = −0.5; the converse holds at qhm = +0.5. But again, as described earlier in the context of heterogeneous reactions, we need not limit ourselves to considering the reactant ionization/affinity energies at just two coordinates. By subtracting the individual total energy contributions of each species in Figure 2b at all nuclear coordinates, we arrive at the two linear single-particle dispersions in Figure 2c. The dashed-red/solid-blue line in Figure 2c represents the single-particle ionization/affinity energy for species α across all qhm. Likewise, the dashedblue/solid-red line in Figure 2c represents the single-particle affinity/ionization energy for species β across all qhm. Importantly, it is “only” at coordinate qhm = 0 that both

species have the same ionization/affinity energy such that a radiationless (coherent) electron tunneling event can occur, that is, the reaction αox + βred ⇌ αred + βox. In Figure 2c we represent the “special” point at which both species have the same ionization/affinity energy (and thus the same coordinates) by red and blue “coordinate circles” of equal magnitude. This picture is very different from the heterogeneous reaction process in Figure 1d, where the substrate/ contact always has a “continuum” of single-particle states that a reactant electron can tunnel to/from regardless of its nuclear coordinate (as sketched in Figure 1d). Though the probability of a reactant being at a given coordinate is localized around εox and εred, the only caveat is that the “continuum” of states is determined by the substrate’s bandstructure and the availability of occupied/empty states as dictated by Fermi−Dirac statistics (we will discuss this in more detail shortly). Quite the opposite holds in a homogeneous reaction: according to the singleparticle picture in Figure 2c, there is only “one” nuclear coordinate at a which a radiationless reaction can take place. This “one” nuclear coordinate restriction can be “lifted” for homogeneous reactions if an external energy is imparted into the system, such that the oxidation or reduction total energy is raised or lowered for a given reactant1 or an electron is “knocked off” as described earlier.

3. ELECTRON TRANSFER RATES IN ELECTROCHEMISTRY With the single-particle energy picture in both heterogeneous and homogeneous reactions elucidated, we can now begin to develop expressions for the electron transfer rate in both scenarios. We will begin by briefly exploring the well-known Marcus−Hush rate expression for homogeneous electron transfer in section 3.1, followed in section 3.2 by a detailed derivation of heterogeneous electron transfer rates utilizing the single-particle Landauer picture and the nonequilibrium Green’s function formalism. 3.1. Marcus−Hush Homogeneous Electron Transfer Rate. As discussed toward the end of section 2.2, an electron transfer event in a homogeneous electrochemical reaction occurs when both reactants have the same nuclear coordinates (qhm = 0 in Figure 2). This precondition corresponds to both reactants possessing the same single-particle ionization/affinity energy, such that an electron may transfer/tunnel from the reduced reactant to the oxidized reactant (as shown in Figure 2c). Within the Marcus−Hush approach, electron tunneling/ transfer at this intersection coordinate has a rate proportional to 2π|M|2/ℏ, where M represents the electronic coupling between species α and β in Figure 2c, and ℏ is Planck’s constant divided by 2π.2,5 However, to arrive at a total rate we must multiply the electron transfer prefactor by the “density of nuclear states”, which for this symmetric example works out to exp( −λhm /4kB ;)/ 4πλhmkB ; . 5 The total homogeneous reaction rate for our nonadiabatic homogeneous system is therefore given by1,5,11,12 k hm =

|M |2 ℏ

⎛ λ ⎞ π exp⎜ − hm ⎟ λhmkB ; ⎝ 4kB ; ⎠

(8)

where the “density of nuclear states” in our rate expression essentially tells us the probability of a reaction pair being at the electron transfer nuclear coordinate (qhm = 0 in Figure 2) and follows from the Franck−Condon principle.1,5 We note that this “density of nuclear states” can be interpreted as a E


Article

The Journal of Physical Chemistry C probability distribution and is temperature dependent unlike a true “electronic density of states”. This nomenclature is found in the chemistry literature.5 In addition to limiting the reaction rate, our adherence to the Franck−Condon principle has another fundamental consequence: it means we are assuming that each electron transfer/ tunneling event is a coherent process. That is, we are assuming that no phonons (quantized nuclear vibrations) are emitted or adsorbed during each electron transfer event. If phonons were emitted or absorbed, then we would be simultaneously traversing nuclear coordinates (qhm) while transferring an electron (e.g., an electron transfer event at qht = 0 would propel the nuclear coordinates from qhm = 0 to some other end coordinate in Figure 2). In general, we are also assuming that no photons are emitted or adsorbedthough this is an interesting topic that lies outside the scope of our present discussion.2 Nevertheless, when addressing the problem of quantifying heterogeneous electron tunneling/transfer rates, it is logical to carry forward the Franck−Condon approximation and assume that each reactant−substrate tunneling event might be treated as a coherent processafter all a “contact” is merely a macroscopic “reactant” with a continuum of single-particle ionization/affinity energies as sketched in Figure 1. This is the first and foremost approximation which we shall use in our detailed analysis of heterogeneous electron transfer rates. We leave a detailed analysis of incoherent interactions to future work.7,13 3.2. Arriving at Heterogenous Electron Transfer Rates by Combining Gerischer’s Picture with Landauer’s Approach. The Landauer picture and the supporting machinery of the nonequilibrium Green’s function formalism provide a robust method for analyzing coherent tunneling.25−29 It is also the approach of choice for theoretical studies of electron transport processes in a wide range of systems from spintronics through to molecular electronics.7,15 To enhance our overall understanding of electron transport and transfer processes at the atomic scale, it would be immensely beneficial if heterogeneous electrochemical tunneling reactions could be brought under this same “theoretical umbrella”. In the below analysis we attempt to provide such a connection, which through reductions to the transfer Hamiltonian approach is shown to produce near identical expressions to those often found in the electrochemistry literature (as it pertains to outersphere electrochemical reactions).1 However, to begin this process we first need to clearly define the quantities of interest as it pertains to the NEGF formalism, and this is done in section 3.2.1; subsequently, in section 3.2.2 a full derivation connecting Gerischer’s picture with Landauer’s approach is provided. 3.2.1. NEGF Partitioning of the Heterogeneous Tunneling Problem. To utilize the machinery of the NEGF formalism, we must first divide up our reaction system into manageable quantities as illustrated in Figure 3a. An excellent discussion on the NEGF method is provided in refs 7 and 13. To start, let us define the spectral function of the substrate and reactant as AS and AR, respectively, as shown in Figure 3a; here we use AR to describe the reactant in either the oxidized (blue) or reduced (red) state since the same continuous single-particle dispersion is available to both reactant configurations as discussed earlier with respect to Figure 1c. The spectral function is a concept commonly utilized in solid-state physics. The more familiar density of states concept is related to the spectral function through the trace of the matrix (assuming a linear combination

Figure 3. Combining the Landauer and Gerischer pictures to describe heterogeneous electron transfer rates. (a) Depicts the NEGF formalism pertaining to oxidation and reduction reactions: the metal substrate (gray) and reactants (blue for oxidized and red for reduced) are partitioned by a green tunneling medium described by a Green’s function (G). (b) Through various deductions we arrive at the forward electron transfer rate (kf) and the backward electron transfer rate (kb) within the Landuaer picture, utilizing the NEGF formalism, incorporating Gerischer’s DOS description.

of atomic orbitals matrix representation, as is common in quantum chemistry). Hence, for our substrate and reactant regions we may define the density of states as DS = tr[AS]/2π and DR = tr[AR]/2π.7,13 Note, in this entire discussion we are assuming that all quantities depend on nuclear coordinates (qht), particularly those of the reactant, and are expressed in the single-particle energy representation (ε). However, for the sake of brevity we do not explicitly write this dependence in each variable. Within our approach sketched in Figure 3a, the substrate and reactant are partitioned by a tunneling medium which we describe by a “device” Green’s function (G). This tunneling medium (green in Figure 3a) can take several forms but is typically a solvent; for example, it can be a monolayer or so of water molecules separating [Fe(H2O)6]+2/+3 from the substrate (for further discussion see ref 1). In an orthogonal basis representation, the device Green’s function is given by G = [(ε + i0+)I − H − ΣS − ΣR]−1.7,13 Here, H is the Hamiltonian of the tunneling region, and ΣS,R represents the substrate/reactant self-energies interacting with the tunneling medium; I is simply the identity matrix. An important point to underscore is that we can likely only partition the reactant and substrate in this manner in the tunneling limit. If the reactant and substrate were to come in close proximity, for example through a chemisorption or an inner-sphere reaction process, then we would need to reassess our approach. Only when the reactant and substrate are weakly interacting can we confidently apply this Green’s function partitioning approach at this time. Further work is needed to assess the degree to which this formalism might be applied to inner-sphere electrochemical reactions. F


Article

The Journal of Physical Chemistry C 1 h 1 = h 1 = h

The self-energies of the substrate and reactant are given by ΣS = τ†SGSτS and ΣR = τRGRτ†R, where the substrate/reactant Green’s functions GS,R are related to their spectral functions via AS,R = i(GS,R − G†S,R).7,13 The tunneling medium’s interaction with the substrate and reactant is determined by the coupling matrices τS and τR, respectively, as sketched in Figure 3a. The imaginary part of the substrate and reactant self-energy leads to the concept of a broadening matrix ΓS,R = i(ΣS,R − Σ†S,R). With further deductions, utilizing our earlier expressions, we can write ΓS = τ†SASτS and ΓR = τRARτ†R, which we shall make extensive use of shortly. Moreover, we can write the spectral function of the tunneling medium as A = i(G − G†) = G(ΓS + ΓR)G†, which leads to an analogous concept of an electron correlation function Gn = G(fΓS + θΓR)G† whose diagonal elements represent the electron density at a given energy.7 Here θ represents the occupation factor of the reactant in either the reduced (θ = 1) or oxidized (θ = 0) state, and f(ε) is the Fermi function distribution of the substrate [as sketched in Figure 3b and discussed earlier in the context of eqs 6 and 7]. The concept of broadening can be extended to capture the lifetime of an electron by dividing by Planck’s constant (h); for the substrate the lifetime of an electron escaping to or from the tunneling reaction is given by ΓS/h. This leads to two final important concepts sketched in the center of Figure 3a: first, the “out-flow” of electrons per unit energy across the partitioning boundary between the substrate (gray) and tunneling (green) regions may be expressed as fΓSA/h; conversely, the “in-flow” of electrons per unit energy into the substrate across the partitioning boundary region may be expressed as ΓSGn/h.7,13 By utilizing these “out-flow” and “inflow” concepts we will be able to arrive at heterogeneous rate constants consistent with Landauer’s picture. 3.2.2. Connecting Gerischer’s Picture with Landauer’s Approach. By the principle of current conservation, we can determine the forward (kf) and backward (kb) rate of electron flow from/to the substrate via the reactants by counting those electrons which cross the boundary between the substrate and the tunneling region (gray and green regions, respectively, in Figure 3a). For electron flow from the substrate to oxidized reactants, the nuclear coordinate (qht) resolved transfer rate is given by the substrate “out-flow” less the “in-flow” summed across all single-particle energies (ε) 1 h 1 = h 1 = h 1 = h

k f (qht) =

k b(qht) =

∫ tr[ΓSGn − f ΓSA]θ=1dε ∫ (1 − f )tr[ΓSGΓRG†]dε ∫ [1 − f (ε)]T(qht , ε)dε

(10)

which also reduces to include Landauer’s transmission probability. Both eqs 9 and 10 have a simple intuitive interpretation. The probability of an electron transferring into an oxidized state is the probability of finding an electron at a given energy ( f) multiplied by the probability of a transfer event (T). Likewise, the probability of an electron transferring from a reduced state to the substrate is the probability of finding an empty substrate state at a given energy (1 − f) multiplied by the probability of a transfer event (T). Similar intuitive explanations exist in the electrochemistry literature.1,17 Note, here we have assumed that reduced states are always occupied, and oxidized states are always empty. This is a common procedure in the electrochemistry literature.1,17 A detailed discussion on the electrochemical potential and statistical occupation distribution in the liquid can be found in ref 30. It is important to emphasize that eqs 9 and 10 assume the coherent transfer of electrons at each nuclear coordinate (qht). This is a direct extension of the coherent tunneling assumptions in homogeneous electron transfer theories discussed at length earlier in section 3.1. Following the Franck−Condon principle, we are assuming the electrons transition while the nuclei “stand still”.1 This implies that an electron transition between the substrate and a reactant will not simultaneously facilitate a change in the nuclear coordinates of the reactant. Incoherent electron tunneling processes do existinelastic electron tunneling spectroscopy is a canonical example7but are usually associated with nuclear excitations or relaxations within the tunneling barrier itself and are typically included through the use of additional self-energy terms in the NEGF formalism.7,13 We will not explore this avenue but choose to treat heterogeneous electron transfer on the same coherent footing as homogeneous electron transfer. This is an approximation that can be further explored in future work. However, as we shall see shortly, within the Landauer approach this coherent approximation leads to the rate expressions commonly found in the electrochemistry literature (although through a very different route) that have explained observed heterogeneous experimental phenomena quite well.17,18 To our knowledge there are currently no reported outer-sphere electrochemical measurements which have thus far demonstrated incoherent tunneling phenomena (i.e., inelastic electron scattering).1,7,13,17,18 Yet to arrive at a full rate expression, we need to go beyond considering one nuclear coordinate (qht) and consider the total rate averaged across all nuclear coordinates. This can be accomplished by weighting the transfer rate at each nuclear coordinate by the probability of the reactant residing at a given coordinate and summing across all coordinates. Returning to our Boltzmann-based nuclear distributions in eqs 4 and 5, discussed when we developed our single-particle picture in section 2.1, the nuclear coordinate averaged electron transfer rates may then be expressed as

∫ tr[f ΓSA − ΓSGn]θ=0 dε ∫ tr[f ΓSG[ΓR + ΓS]G† − ΓSG[θ ΓR + f ΓS]G†]θ=0 dε ∫ f (ε)tr[ΓSGΓRG†]dε ∫ f (ε)T(qht , ε)dε (9)

where T(qht, ε) is the Landauer transmission probability.13 We have explicitly written the nuclear coordinate dependence (qht) in the Landauer transmission probability to underscore that this expression corresponds to the electron transfer rate for a given reactant nuclear coordinate. Likewise, the rate of electron flow from the reduced reactants to the substrate can be expressed as the difference between the substrate “in-flow” less the “outflow” G


Article

The Journal of Physical Chemistry C kf =

1 h

∫

λht −λhtq 2 / kB ; ht e π kB ;

∫ f (ε)T(qht , ε)dεdqht

kb =

1 h

∫

λht −λht(q + 1)2 / kB; ht e πk B ;

∫ [1 − f (ε)]T(qht , ε)dεdqht (12)

However, these are general expressions which require some work to appear in an intuitive form similar to those often utilized by the electrochemistry community. First, we need to develop a more “flexible” expression for our transmission function utilizing the NEGF machinery outlined in section 3.2.1 T (qht , ε) = tr[ΓSG ΓR G†] = tr[τS†ASτSGτR AR τR†G†]

kf =

= tr[ASτSGτR AR τR†G†τS†] = tr[ASMAR M†] = tr[M†ASMAR ]

2π h

∫ f (ε)tr[M†ASMDox ]dε

(15)

kb =

2π h

∫ [1 − f (ε)]tr[M†ASMDred]dε

(16)

⎛ −(ε − ε )2 ⎞ π ox ⎟d ε exp⎜ λhtkB ; ⎝ 4λhtkB ; ⎠

∫ [DS(1 − f )] |Mℏ|

⎛ −(ε − ε )2 ⎞ π red ⎟d ε exp⎜ λhtkB ; λ ; 4 k ⎝ ⎠ ht B

which includes an expression nearly identical to that given by Marcus−Hush theory (eq 8). That is, for both of the above equations in the interior of the integral, we have the rate of electron transfer expressed as a product of the electronic coupling at a given energy multiplied by the “density of nuclear states” at that same energy just as we do in eq 8. However, to obtain our heterogeneous electron transfer rate we multiply the availability of filled [DS f(ε)] and empty states [DS(1 − f(ε))], respectively, providing and accepting electrons, and integrate over energy to obtain our respective reduction and oxidation rates (kf and kb). Nevertheless, we have not made this deductive journey to merely suggest that one should always apply eqs 17 and 18 to outer-sphere electrochemistry problems. Certainly, there is a richness of chemistry and physics to be explored by further developing rate expressions along the lines of the more general expressions in eqs 11 and 12 [as well as eqs 15 and 16], for example, through the development of first-principles electrochemistry models including electron transfer interactions with the substrate and nuclear coordinate fluctuations.1,25 One particularly interesting avenue of investigation is the possibility of nonconstant coupling matrix elements (such as in Fowler− Nordheim tunneling) which would not be captured via eqs 17 and 18 and thereby necessitate integration of the transmission function over energy in a manner provided by eqs 15 and 16.25 However, due to the present lack of unequivocal experimental data indicating coupling nonlinearity in electrochemical rate measurements,1,17 we shall err on the side of caution and leave a comprehensive investigation of this topic for future work. We have also not fully extended these heterogeneous rate expressions to arrive at experimentally realizable current− voltage characteristics; this shall be the focus of our second paper in this series (part II).20

where the reactant spectral function integrates out via convolution to produce the Gerischer broadened distributions Dox and Dred given by eqs 6 and 7, respectively, in section 2.1. Note here we are expressing the reactants in a single-level eigenbasis (as shown in Figure 1c), This results in a 1 × 1 matrix which allows us to draw a direct correspondence between Gerischer’s DOS expressions and the spectral function expression in the NEGF formalism (see the discussion in section 3.2.1). However, our reduced Landauer rate expressions in eqs 15 and (16) still do not correspond to a form that can be “intuitively” correlated with experiments.17,18 To produce more manageable expressions we can further assume that both the substrate−reactant coupling matrix (M) and substrate spectral function (AS) are independent of energy and can be approximated as 1 × 1 matrices. This leads to the final simplification for the rate expressions sketched diagrammatically in Figure 3b

∫ f (ε)Dox dε

|M |2 ℏ

(20)

(14)

kf =

4π 2 2 |M | DS h

∫

[DSf (ε)]

2

kb =

where ε′(qht) = 2λhtqht − |εox| is the reactant single-particle dispersion as given by eq 3 and derived in section 2.1. Furthermore, by transforming the nuclear coordinate integral present in both eqs 11 and 12 via the substitution qht = (ε′ + |εox|)/2λht we arrive at

kf =

(18)

(19) (13)

where M = τ SGτ R is the substrate−reactant coupling matrix.13,25,26,31 The first line in eq 13 through to the last line in eq 13 can be shown to follow from the Fisher−Lee relation32 as shown in Chapter 3 of ref 13.31 Now if we assume that only the spectral function of the reactant (AR) varies with the nuclear coordinate (qht) of the reactant and further express it in terms of its eigenbasis7 such that AR = 2πδ(ε − ε′(qht)), then the transmission function becomes T (qht , ε) = 2πtr[M†ASMδ(ε − ε′(qht))]

∫ [1 − f (ε)]Dreddε

where we have again used the 1 × 1 matrix relation DS = 2πAS. Expressions of this sort are usually arrived at through the transfer Hamiltonian formalism and are particularly applicable to tunneling systems.13,25−29 They are also nearly identical to those found in the electrochemistry literature.1,17,18 Thus, we have established an intuitive bridge between the Landauer picture and Gerischer’s formulation of tunneling electrochemical reactions.1,7−10,13 To come full circle, it is interesting to make one final comparison between the heterogeneous rates (kf and kb) and the Marcus−Hush homogeneous rate (khm). By expanding and rearranging variables in eqs 17 and 18 we arrive at

(11)

kb =

4π 2 2 |M | DS h

4. CONCLUSION To summarize, we have developed a formal bridge between the Landauer electron transport picture (often utilized in quantum transport) and Gerischer’s formulation of tunneling electrochemical reactions. This was accomplished by first developing

(17) H


Article


(16) Migliore, A.; Schiff, P.; Nitzan, A. On the Relationship between Molecular State and Single Electron Pictures in Simple Electrochemical Junctions. Phys. Chem. Chem. Phys. 2012, 14, 13746−13753. (17) Chidsey, C. E. Free Energy and Temperature Dependence of Electron Transfer at the Metal-Electrolyte Interface. Science 1991, 251, 919−922. (18) Miller, C.; Graetzel, M. Electrochemistry at ω-hydroxythiol Coated Electrodes. 2. Measurement of the Density of Electronic States Distributions for Several Outer-Sphere Redox Couples. J. Phys. Chem. 1991, 95, 5225−5233. (19) Fleming, G. R.; Ratner, M. A. Grand Challenges in Basic Energy Sciences. Phys. Today 2008, 61, 28−33. (20) Hossain, M. S.; Bevan, K. H. Exploring Bridges between Quantum Transport and Electrochemistry. Part II. A Theoretical Study of Redox-Active Monolayers. J. Phys. Chem. C 2015, DOI: 10.1021/acs.jpcc.5b09654. (21) Feng, Z.; Timoshevskii, V.; Mauger, A.; Julien, C. M.; Bevan, K. H.; Zaghib, K. Dynamics of Polaron Formation in Li2O2 from Density Functional Perturbation Theory. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 88, 184302. (22) Timoshevskii, V.; Feng, Z.; Bevan, K. H.; Goodenough, J.; Zaghib, K. Improving Li2O2 Conductivity via Polaron Preemption: An Ab Initio Study of Si Doping. Appl. Phys. Lett. 2013, 103, 073901. (23) Szabo, A.; Ostlund, N. S. Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory; Dover Publications, 1996. (24) Sholl, D.; Steckel, J. A. Density Functional Theory: A Practical Introduction; Wiley-Interscience, 2009. (25) Bevan, K. H.; Zahid, F.; Kienle, D.; Guo, H. First-Principles Analysis of the STM Image Heights of Styrene on Si(100). Phys. Rev. B: Condens. Matter Mater. Phys. 2007, 76, 045325(1−10). (26) Bevan, K. H. A. First Principles Scanning Tunneling Potentiometry Study of an Opaque Graphene Grain Boundary in the Ballistic Transport Regime. Nanotechnology 2014, 25, 415701. (27) Cerdá, J.; Van Hove, M. A.; Sautet, P.; Salmeron, M. Efficient Method for the Simulation of STM Images. I. Generalized GreenFunction Formalism. Phys. Rev. B: Condens. Matter Mater. Phys. 1997, 56, 15885−15899. (28) Hofer, W. A.; Foster, A. S.; Shluger, A. L. Theories of Scanning Probe Microscopes at the Atomic Scale. Rev. Mod. Phys. 2003, 75, 1287−1331. (29) Jelínek, P.; Ondrácě k, M.; Flores, F. Relation Between the Chemical Force and the Tunnelling Current in Atomic Point Contacts: A Simple Model. J. Phys.: Condens. Matter 2012, 24, 084001. (30) Reiss, H. The Fermi level and the redox potential. J. Phys. Chem. 1985, 89, 3783−3791. (31) Samanta, M. P.; Tian, W.; Datta, S.; Henderson, J. I.; Kubiak, C. P. Electronic conduction through organic molecules. Phys. Rev. B: Condens. Matter Mater. Phys. 1996, 53, R7626−R7629. (32) Fisher, D. S.; Lee, P. A. Relation between conductivity and transmission matrix. Phys. Rev. B: Condens. Matter Mater. Phys. 1981, 23, 6851−6854.

the single-particle picture within electrochemistry for both heterogeneous and homogeneous reactions. After briefly summarizing the well-known rate constant expression for homogeneous reactions, in particular the assumption of coherent electron transfer processes, the single-particle picture was then employed within the NEGF formalism to arrive at Landauer-type expressions of heterogeneous electron transfer. Through the simplifying assumptions of a constant substrate DOS and constant coupling between the reactants and substrate, reduced Landuaer rate expressions were develop that are nearly identical to those often found in the electrochemistry literature.1,17,18 In general, it is hoped that this work will spur further efforts to connect electrochemistry with quantum transport, by expanding the concepts developed here in the context of advanced atomistic approaches including first-principles methods.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors gratefully acknowledge financial support from NSERC of Canada and FQRNT of Québec.

■

REFERENCES

(1) Schmickler, W. Interfacial Electrochemistry, 1st ed.; Oxford University Press: New York, 1996. (2) Memming, R. Semiconductor Electrochemistry, 1st ed.; WILEYVCH Verlag, 2007. (3) Kuznetsov, A.; Ulstrup, J. Electron Transfer in Chemistry and Biology; John Wiley and Sons: New York, 1999. (4) Nishide, H.; Suga, T. Organic Radical Battery. Interface 2005, 14, 32−36. (5) Nitzan, A. A. Relationship between Electron-Transfer Rates and Molecular Conduction. J. Phys. Chem. A 2001, 105, 2677−2679. (6) Bard, A. J.; Faulkner, L. R. Electrochemical Methods: Fundamentals and Applications, 2nd ed.; Wiley: New York, 2000. (7) Datta, S. Quantum Transport: Atom to Transistor; Cambridge University Press, 2005. (8) Gerischer, H. Ü ber den Ablauf von Redoxreaktionen an Metallen und an Halbleitern. I. Allgemeines zum Elektronenübergang zwischen einem Festkörper und einem Redoxelektrolyten. Z. Phys. Chem. 1960, 26, 223−247. (9) Gerischer, H. Ü ber den Ablauf von Redoxreaktionen an Metallen und an Halbleitern. II. Metall-Elektroden. Z. Phys. Chem. 1960, 26, 325−338. (10) Gerischer, H. Ü ber den Ablauf von Redoxreaktionen an Metallen und an Halbleitern. III. Halbleiterelektroden. Z. Phys. Chem. 1961, 27, 48−79. (11) Marcus, R. A. Electron Transfer Reactions in Chemistry. Theory and Experiment. Rev. Mod. Phys. 1993, 65, 599−610. (12) Hush, N. S. Adiabatic Theory of Outer Sphere ElectronTransfer Reactions in Solution. Trans. Faraday Soc. 1961, 57, 557− 580. (13) Datta, S. Electronic Transport in Mesoscopic Systems; Cambridge University Press, 1995. (14) Damle, P.; Ghosh, A. W.; Datta, S. First-Principles Analysis of Molecular Conduction Using Quantum Chemistry Software. Chem. Phys. 2002, 281, 171−187. (15) Nitzan, A.; Ratner, M. A. Electron Transport in Molecular Wire Junctions. Science 2003, 300, 1384−1389. I


Exploring Bridges between Quantum Transport and Electrochemistry. I

Recommend Documents