Electron transfer at different electrode materials: Metals, semiconductors, and graphene

Journal Pre-proof Electron transfer at different electrode materials: metals, semiconductors and graphene Elizabeth Santos, Renat Nazmutdinov, Wolfgang Schmickler PII:

S2451-9103(19)30167-X

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https://doi.org/10.1016/j.coelec.2019.11.003

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COELEC 479

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Current Opinion in Electrochemistry

Received Date: 9 September 2019 Revised Date:

18 November 2019

Accepted Date: 19 November 2019

Please cite this article as: Santos E, Nazmutdinov R, Schmickler W, Electron transfer at different electrode materials: metals, semiconductors and graphene, Current Opinion in Electrochemistry, https:// doi.org/10.1016/j.coelec.2019.11.003. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Elsevier B.V. All rights reserved.

Electron transfer at different electrode materials: metals, semiconductors and graphene Elizabeth Santos1,*, Renat Nazmutdinov2 and Wolfgang Schmickler1

Abstract Electron transfer reactions are the most important processes at electrochemical interfaces. They are determined by the interplay between the interaction of the reactant with the solvent and the electronic levels of the electrode surface. Theoretical treatments only based on Density Functional Theory calculations are not sufficient. This review emphasizes mainly the effect of the electronic structure of the electrode material on electron transfer under different kinetic regimes. Our goal is to understand experimental results in the framework of a theory valid for arbitrary strengths of electronic coupling. Adresses 1 Institute of Theoretical Chemistry, Ulm University, D-89069 Ulm, Germany. 2 Kazan National Research Technological University, 420015 Kazan, Russian Federation. * Corresponding author: Santos, Elizabeth, [email protected] 1. Introduction Electron transfer reactions in an electrochemical environment are governed by both the electronic interactions of the reactant with the electrode material and with the solvent. The first process is determined by the coupling parameter Veff or chemisorption function ∆ (ε), as it is called in the model of Anderson and Newns [1,2]. This quantity reflects the electronic structure of the electrode material and is proportional to its density of states: 2

(1) ∆(ε ) ≈ π Veff ρelec (ε ) When an electron transfer takes place at the interface, it is accompanied by changes in intramolecular degrees of freedom and the reorganization of the solvent. In the case of chemisorption (strong electronic interactions), the reactant can totally lose its solvation shell, and bond-breaking can also happen (inner sphere reaction). If the interaction with the electrode is comparatively weak, the reactant preserves its whole solvation shell. This is an outer sphere electron transfer reaction; the reactant is not in direct contact with the electrode surface. At least one layer of solvent or some other ligand separates reactant and electrode. Although the reactant preserves its inner shell, the solvent in the vicinity of the reactant must reorient during the reaction because reactant and product carry different charges. The reaction rate is mainly determined by this reorganization of the solvation shell characterized by the concomitant energy λ, and the theoretical basis is given by extensions of the Marcus – Hush model [3,4]. Figure 1a illustrates these two types of electron transfer reactions [5-8]. In the case of inner sphere reactions, a strong dependence on the electrode material is observed, while in the case of outer sphere reactions the electrode material does not play such an important role. In order to emphasize the complexity of inner sphere reactions, the paradigmatic hydrogen evolution reaction is shown as an example. This reaction occurs at least through two elementary steps, and the limiting step on platinum group metals and on mercury-like metals is different. A typical mistake for numerous publications of "volcano plots" and other correlations is to consider the exchange current without taking into account the different mechanisms, as has been stressed in

other publications [9-10]. It is not easy to obtain the components of the total current corresponding to the different steps of the reaction. Nevertheless, according to the strength of the coupling, an electron transfer can occur within an adiabatic or nonadiabatic regime. The interactions between the reactant and the metal by adiabatic reactions are sufficiently strong such that the electron (a) outer sphere reactions e e

Metal

Z

Metal

e

inner sphere reactions e

A

Metal e

B

Metal

(b)

Figure 1. (a) experimental values of the standard exchange current for an outer sphere reaction (filled symbols); data obtained from references [5,6] and for an inner sphere reaction, such as the hydrogen evolution reaction; data obtained from references [7,8]. At right: schematic representations of outer and inner sphere reactions. (b) dependence of the rate constant on the interaction strength ∆ based on a computer simulation [15] [13]. The black line shows the prediction from first order perturbation theory. exchange takes place every time the system reaches the transition state. Thus the system is in electronic equilibrium for all solvent configurations. In the case that the electronic interactions are weaker, the system can pass the saddle point of the reaction

coordinate without an electron transfer, so that it subsequently returns to its initial state. These reactions are called nonadiabatic, and the interaction strength will enter into the pre-exponential factor of the expression for the velocity. In summary, there are three different kinetic regimes depending on the strength of the electronic interactions between the reactant and the electrode material as illustrated in Figure 1b: I. If the interactions are sufficiently weak, the reaction proceeds non-adiabatiacally and first order perturbation theory can be applied [11-14][9-12]. This corresponds to the linear behavior observed in Figure 1b, where the pre-exponential factor is proportional to ∆. II. If the interactions are of intermediate strength, solvent dynamics starts to influence the rate through the friction factor, which can be described in terms of Kramer's theory [15] [13]. Therefore, in a certain region the rate constant is almost independent of ∆. This regime is obeyed by outer sphere reactions, and the behavior can be described in the framework of Marcus and Hush model [3,4], where the reaction barrier is not affected by the electronic coupling. III. If the interactions are very large, a catalytic effect is observed, and the rate constant increases again, due to the lowering of the activation energy by the electronic coupling (more details about our theory of electrocatalysis can be found in [16-17] [14-15] and previous review articles [18-21] [16-19]). In the case of metallic electrodes, electron transfer reactions proceed mainly adiabatically, while on semiconducting and carbonaceous electrodes is still controversial if they take place in an adiabatic or non-adiabatic regime. 2. Electronic structure of electrodes from different materials The electronic structure of different electrode materials can be quite complex. Therefore, our strategy is to use idealized band shapes for calculations, in order to understand the influence of the electronic structure on the mechanism of electron transfer at different materials. Figure 2 illustrates different models assumed for the density of states (DOS) for diverse electrode materials. The simplest possible model is the wide band approximation with constant density of states [22] [20]. In the case of metals, the Fermi level lies within this band. In the case of semiconductors, a step function can be used assuming that the DOS is constant within two bands and vanishes in the gap with the Fermi level within this band gap [23,24**] [21,22**]. However, this approximation is certainly not realistic. A better model considers semielliptic bands [1,2]. In Figure 2 it was assumed that the center of the band coincides with the Fermi level for the metal, while two semiellipses represent the valence and conductance bands for the semiconductor with the Fermi level at center of the band gap [24**][22**]. Obviously, the center, the width and the separation (band gap) of the bands can be shifted according to the specific system to be investigated. Ideal intrinsic semiconductors show sharp band edges. However, doping and defects introduce localized electronic states in the band gap. In order to analyze this effect, we consider an amorphous semiconductor with a density of states in the band gap that takes the form of a negative semiellipsis (see green line in Figure 2). The electronic structure of graphene is particularly interesting. We have calculated the density of states using the tight binding model [25] [23]. The presence of topological defects or doping introduces mid gap states on the density of states [26-28] [24-26]. We have accounted for such distortions in the electronic structure introducing a

Lorentzian function at three different positions relative to the Fermi level (see Figure 2) [29**] [27**]. 3. Electrochemical response for the different electrode materials In the case of a non-adiabatic regime, the electronic interactions between electrode and reactant are very weak, in consequence the electron transfer occurs from (or into) a very sharp electronic level of the reactant. The expression for the currentoverpotential dependency j(η) for the reduction of a species A is given by [13,16] [11,14]:

jred = P

Veff h

2

( 4πλ kBT )

−1/2

+∞

∫ ρ (ε ) f (ε,T ) W (ε, λ, η) dε elec

FD

elec

(2)

ox

−∞

Figure 2. Normalized density of electronic states (DOS) of all the model systems considered in this work (more details in the text).

where: 2 Wox (ε, λ, η ) = exp − ( λ − ε − η ) / ( 4 λ kBT )

fFD (ε,Telec ) = 1/ 1+ exp (ε / kBTelec )

fFD(ε,Telec) is the Fermi-Dirac distribution, i.e. the probability to find an occupied state in the electrode and η is the overpotential. In equation (2) the Fermi level is taken as zero. P is an extra factor, which depends on the reactant work term (i.e. the Boltzmann factor that converts the bulk to the surface concentration) and on the reaction volume resulting from its integration over the distance (of the order of 10-10 – 10-9 m) [13,16] [11,14]:. According to Gerischer’s interpretation [30,31] [28,29], Wox(ε,λ,η) is the probability that an electron from state ε in the metal is transferred to an unoccupied (oxidized) state in the solution. It accounts for the fluctuations in the solvent shell, and therefore it is independent of the electrode material. This must not be confused with the electronic density of states of the reactant, these are sharply localized at a given energy. An equivalent expression for the oxidation can be easily found. In the case of an adiabatic regime, where the electronic interactions between reactant and electrode are stronger, a broadening of the electronic level of the reactant takes place. Therefore, the total energy of the system, also taking into account the interactions with the solvent is given by:

Etot (λ, q, η ) = Esolv (λ, q) + Eelec (λ, q, η ) = λ q + 2zqλ + 2

EF =0

∫

ε ρ A (ε, q, η ) dε

(3)

−∞

Here, q is the normalized solvent coordinate, which describes the reaction path [3,4, 16 14], and z is the charge number. The latter term in equation (3) is the electronic energy, and it is this term that contributes to the decrease of the activation barrier when strong interactions with the electrode causes the broadening of the density of states of the reactant ρA. 4. Results of model calculations Figure 3 shows the results of numerical calculations for a non-adiabatic regime of the integral in equation (2) J(ρelec, λ, η) as a function of the overpotential η, using the density of states ρelec for the different electrode materials of Figure 2. The absolute values of the current are determined by the pre-factor containing the coupling constant |Veff|2 and we shall return to this later. Now, it is interesting to analyze the features of these curves, which are determined by the features of the density of states of the electrode material. The usual near-activationless regime is attained at η of about 0.7 eV for the metal, while it is still not observed for the other materials even at higher overpotentials. It can be explained taking into account that while the product fFD . ρelec for the metal is almost constant below the Fermi level, that product for the other materials increases monotonically when the electronic energy becomes more negative. An interesting behaviour is observed when the graphene contains electronic states in the mid-gap. In some cases, a maximum is observed, which could be interpreted as a vestige of the inverse region in Marcus theory. However, a clear explanation based on the Gerischer’s approach can be found. The contribution of the mid-gap states is stronger at low overpotentials, because the overlap of the probability Wox(ε,λ,η) with these states is larger in this case. At larger overpotentials this overlap occurs with the deeper electronic states, which are similar for all graphene materials. Therefore a maximum appears at intermediate intervals.

Figure 3. Effect of the electronic structure of the electrode material on the current overpotential dependencies for the non adiabatic limit. In contrast to the non-adiabatic regime in the adiabatic case, the reactant and the electrode share their electrons and a broadening of the electronic level of the reactant is produced. Herein lies the key effect of the electronic structure on the kinetics of the reaction. These effects can be better understood by analyzing the density of states of the reactant (for a detailed discussion see reference [29**27**]). Pre-exponential factor of the rate constant Finally, we shall briefly discuss the pre-exponential factor, which has recently received renewed attention [32** 30**]. For adiabatic outer-sphere electron transfer there is a fairly well developed theory, which in agreement with experiment gives preexponential factors of the order of 104 cm s-1; tunneling effects of inner-sphere modes can reduce this typically by one order of magnitude. For inner sphere reactions, both theory and some experimental data suggest a lower pre-exponential factor than for outer-sphere transfer, but the theory is not well developed, and experimental data are sometimes contradictory and affected by anion adsorption. For details we refer to the cited article. For non-adiabatic reactions, this factor depends strongly on the overlap between the reactants and thus on their electronic properties. Hence there can be no general rule for this class of reactions. For specific cases, the pre-exponential factor can be estimated from quantum-chemical calculations. These can be based on first order perturbations theory following eq. (2) by evaluating the overlap matrix element |Veff |2 and placing it directly into this equation. Alternatively, a Landau-Zener factor γe given by [10, 11, 33 31]: (4) κ = 1− exp− (2πγ ) e

e

which has the meaning of a transmission coefficient in the sense of activated complex or Kramer’s theory, i.e. the probability of electron transfer after the redox system attains the saddle point. The quantity γe depends on the matrix element, and for small interactions the two approaches are equivalent. Obviously the overlap decreases strongly, typically exponentially, with the distance of the reactant from the electrode. In the non adiabatic regime the electronic transmission coefficient is directly proportional to the electrode DOS near the Fermi level, while in the adiabatic limit this quantity does not depend on the DOS. The DOS dependent κe in the intermediate regime is considered in detail in reference [34** 32*]. As an example, Figure 4 shows the transmission coefficient for Fc/Fc+ couple as a function of the distance from a graphene electrode, Two different orientations of the reactant have been considered – the axis vertical or parallel to the surface – and two different positions relative to the graphene lattice: either above a bridge or above a hollow site. For comparison, the results for a Au(111) surface are also shown. For graphene, the vertical orientation is more favorable, and only weakly depends on the site, while for the horizontal orientation the bridge position is more favorable. For Au(111) κe decays much more slowly and is independent of the orientation and position. Nevertheless, even for graphene the reaction is predicted to be almost adiabatic κ e ≈ 1 for distances x<4.5 Å.

Figure 4. Electronic transmission coefficient (from reference [29**]) calculated for a Fc/Fc+ redox couple at a graphene surface for several orientations of Fc+. The results for the Au(111) surface were obtained in Ref.[33] using the same model and averaging over four different orientations of Fc+ .

Conclusions We have discussed electron transfer reactions in an electrochemical environment for different electrode materials. We have focused on the effect of the electronic structure of the substrate under diverse kinetics regimes: outer- and inner-sphere, adiabatic and

non adiabatic reactions. Idealized electronic structures have been used to predict the electrochemical behavior depending on the strength of the interactions between electrode and reactant. Our goal was to shed light on the controversy found in the literature about the electron transfer mechanism. The related problems and the appeal to theoreticians to develop new tools to understand the fundamental electrochemistry of graphene has been emphasized by Patten et al. [35 33]. Electron transfer reactions measurements at graphene electrochemical interfaces, particularly of outer sphere redox processes have been reported in the literature. Experiments using scanning electrochemical cell microscopy (SECCM) are interesting since they allow local measurements at the nanoscale [36-38]. Frequently, oversimplified models are applied. In the case of metals, density functional theory calculations are extended to electrochemical interfaces. However, charged species and solvent effects are very difficult to model within this computational approach. In the case of semiconductors and graphene, The Marcus-Hush and Gerischer frameworks are the most popular in explaining the experimental data of electron transfer. We have shown that much attention must be paid to the strength of the interactions and the electronic structures of the electrode material in order to correctly analyze experimental results. Our theoretical work is complementary to recent extended reviews [34*39-40*], which focus on electron transfer at graphene from the experimental point of view. The results of our model calculations for pristine graphene agree qualitatively with recent experimental findings [41 35, 36*]. The experimental data from [36*] cannot be fit with extant Marcus−Hush theory. Therefore, the authors assume an empiric linear variation with energy of the DOS of graphene and consequently obtain an improved fit. However, they stress the necessity for a deeper theoretical investigation. Our model can explain these results in a consistent and realistic way. Another promising point is the modification of the electronic structure of semiconductors by intercalation or doping, which might result in a noticeable increase of the rate of electron transfer across such electrodes [4237]. In the case of graphene, an interesting aspect is the role of graphene edges in the electron transfer kinetics [43] Finally, an important advantage of our work is the fact that it is valid for arbitrary interactions strength. This makes it possible to apply it for investigating photocatalytic reactions on semiconductors. An interesting effect is the generation of the so called hot electrons by means of a laser pulse [44* 38*]. This phenomenon can be also well described within our model, as we have shown for graphene [29**27**].

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Declarations of conflict of interest none