Common Principles of Molecular Electronics and Nanoscale

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Common Principles of Molecular Electronics and Nanoscale Electrochemistry Paulo Roberto Bueno Anal. Chem., Just Accepted Manuscript • DOI: 10.1021/acs.analchem.7b04533 • Publication Date (Web): 24 May 2018 Downloaded from http://pubs.acs.org on May 25, 2018

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Analytical Chemistry

Common Principles of Molecular Electronics and Nanoscale Electrochemistry Paulo R. Buenoa,b* a

b

Instituto de Química, Universidade Estadual Paulista, Araraquara, São Paulo, Brazil Department of Chemistry, University of Oxford, South Parks Road, Oxford OX1 3QZ, UK (temporary address as visiting professor)

*[email protected], tel: +55 16 3301 9642, fax: +55 16 3322 2308, Instituto de Química, Universidade Estadual Paulista, CP 355, 14800-900 Araraquara, São Paulo, Brazil

Abstract The merging of nanoscale electronics and electrochemistry can potentially modernize the way electronic devices are currently engineered or constructed. This paper offers a conceptual discussion of this central topic, with particular focus on the impact that uniting physical and chemical concepts at a nanoscale could have on the future development of electroanalytical devices. Bibliography Professor Paulo R. Bueno has a B.Sc. degree in Materials Science and Engineering, an M.B.A. degree, and a Ph.D. in theoretical Physical Chemistry. He is currently a research director at São Paulo State University and Head of the Physical Chemistry Department at that university. His main academic interest focuses on applications of electric and electrochemical spectroscopic methods, aiming to gain an in-depth understanding of the physical and chemical fundamentals of electron transfer and energy storage at the nanoscale. He has authored more than 170 papers, holds six licensed patents, and is one of the founders of Osler Diagnostics, a spin-off company from the University of Oxford, UK. Dr. Bueno has worked as an academic and researcher visitor on various occasions at several European universities, including the University of Paris and the University of Oxford. In the UK, some of his research projects have received awards from the Royal Society (including the Brian Mercer Feasibility and Newton Advanced Fellowship awards). He was endorsed as an exceptional talent in Physical Chemistry by the Royal Society and UK government. Currently, he is a Research Fellow Director of the Royal Society, a Fellow of Royal Society of Chemistry, and an invited member of the American Chemical Society. Dr. Bueno is also a member of other scientific societies, among them the Electrochemical Society, the International Society of Electrochemistry, and the Materials Research Society. Introduction Our understanding of the physics of nanoscale electronics considers an urgent task that affects the future, as laid down in the roadmap for the semiconductor industry. Gate widths for CMOS transistors, which today are smaller than 40 nm, are predicted to soon decrease to between 10 to 5 nm1. The latter nanometric size2-5 is close to the limit whereas yet incompletely understood quantum effects are dominant, preventing semiconductor manufacturing companies from further narrowing CMOS channels to below 10 nm and thus from squeezing more computing power onto chips. This means that Moore’s law – which states that computing power doubles every two years at the same cost – is running out of steam; beyond it, the next cycle – of shrinking the width of the channels etched into silicon chips – is uncertain.

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Nonetheless, from the theoretical standpoint, devising an electronic device using molecules has been the ultimate goal in nanotechnology6,7; hence, in conceptual terms, molecular electronics has been proposed as an alternative to silicon post-CMOS devices. The natural question1 that arises in such a scenario is whether molecular-based quantum devices will supplant traditional CMOS technology. The answer depends on researchers’ ability to understand, control and fabricate physical devices at such a diminutive scale. It has been proven that the impedance of nanoscale devices, such as coherent quantum resistance-capacitance (RC) circuits, violates Kirchhoff’s law8. Undoubtedly, the predicted addition of capacitors and resistors stops at the nanoscale. In other words, at this diminutive scale, the predictability of their functioning and consequently the control of circuit devices in general is inherently challenging because guidelines to build electronic circuits at the nanoscale have not yet been established. Accordingly, the resistance associated with charge relaxation differs from the usual quantum transport and consequently the coherent quantum RC circuit does not comply with quantum conductance9,10 (where conductance is directly proportional to electron transmittance),8,11 although the incoherent one does8. Regardless of the charge transport regime (coherent or incoherent), quantum RC ircuits are thus referential models for the operative dynamic regime of all quantum devices12, although, surprisingly, this has been ignored in the fields of both molecular electronics and electrochemistry.7 Therefore, this is the subject we will focus on throughout this paper. When discussing electronics and electrochemistry comparatively from a historical perspective, it is important to recall Michael Faraday’s belief that electronics and electrochemistry are not different disciplines. Nonetheless, physicists appropriated Faraday’s law of induction, which was important to construct electrically powered apparatuses as a separate branch. In contrast, the different fields of chemistry began to develop the discipline of electrochemistry, and here Faraday’s constant13 and Faraday’s law of electrolysis13 were fundamental for the progress achieved in this field. Thereafter, two different approaches to the phenomenon of capacitance, also introduced by Faraday (the unit of capacitance, farad, was a tribute to him), were applied in electronics and electrochemistry. Distinct interpretations of capacitance and capacitors, as elements of circuits in electronics and electrochemistry, gave rise to separate terminologies, i.e., electrostatic and electrochemical capacitors (double layer, for instance). In the field of electrochemistry, also in tribute to Faraday’s law of electrolysis, the processes occurring in the presence or absence of charge transfer, respectively, were termed faradaic and non-faradaic14-16. In other words, faradaic and non-faradaic are interfacial processes that comply or not with Faraday’s law of electrolysis. Consequently, according to physics and chemistry derived technologies and the interpretation of physical processes associated with charge separation and polarization, it is possible to find technological components engineered from both scientific branches in the same device. For instance, when buying a new smartphone we are invited to check out its performance governed by the processor and by the battery within the mobile device. The former and latter are examples of components based on concepts of electronics and electrochemistry, respectively. Both are equally important in a device, regardless of the scientific branch they originated from. From the perspective of the end-user’s mobile device, why have a fast process or if it works standalone for only few hours? In other words, the performance of the processor is limited to the amount of energy that can be contained in the electrochemical battery component within the device. The industry is struggling to determine the correct balance while physicists and chemists are debating how these disciplines merge. As shall be demonstrated here, there are no physical conceptual differences between electronics and electrochemistry at the nanoscale perspective. Electrochemical devices have the additional contribution of ions in the charging of their capacitive components, but this can be integrated in the theory. Accordingly, although the electrolyte is additional to electrochemical devices, it can be integrated with the theory that underpins mesoscopic electronic devices.


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We propose to bring together electronics and electrochemistry by discussing the time-scale of processes, i.e., by debating about the time-dependent response, starting from the analysis of an RC circuit at the nanoscale. In so doing, we are going to end up realizing an intriguing fact – that the fundamentals that connect electronics and electrochemistry are comparable. The impact of the electrolyte environment on the nanoscale system, by attenuating the electron dynamics, results in a larger time-scale of the charging events, which is an inevitable price to be paid. Accordingly, bioelectronics (or bio-electrochemistry) is fated to be slower than electronics, albeit not necessarily less efficient. Evidence in favor of the latter argument can be obtained by looking at the brain’s synapses compared to a computer’s processing capability. Accordingly, we revisit both theoretical and experimental analyses of time-dependent phenomena on a nanoscale, disclosing that quantumRC dynamics is shielded by the electrolyte environment, influencing the use and interpretation of electron transfer theory in both molecular electronics and electrochemistry.17 The implications are that the rate of electrochemical reactions is governed by = / , where is the electrochemical capacitance and is a quantized conductance9. adequately incorporates the required equilibrium electrostatic potential intrinsically associated with the meaning of . Our ultimate goal is to illustrate how the concepts associated with the astonishingly simple formulation = / apply in the design of molecular redox-switchable junctions17-20 as nanoscale energy transducers and actuators for molecular diagnostic applications21-27. Also, how these concepts explain pseudo-capacitive phenomena on the nanoscale, generally affecting our interpretation and the future development of batteries at the nanoscale levels, for instance. The usefulness of the = / formula introduced here is also demonstrated in calculating the quantum of conductance in DNA acting28 as nanowires (which influences our understanding of molecular electronics) and in explaining the intrinsic supercapacitance existing in reduced graphene layers, which is important for the development of molecular supercapacitors.29 We begin our considerations and our introduction to the concepts by considering the simplest model in which electronics and electrochemistry can be brought together, i.e., a junction between a metal and a molecule (see Figure 1), which is thus modeled as a type of quantum RC circuit. This model applies to zero-, one- and two-dimensional electroactive nanostructures, where the charge transfer or electron transport is not accompanied by ionic diffusion. Charge relaxation and chemical capacitance The properties of molecular junctions follow the principles of nanoscale electronics, so a brief introduction to the physical fundamentals governing molecular electronics and the associated time scales is undoubtedly useful, in combination with electrochemistry, in which case the nanoscale electronic circuit operates embedded in an electrolyte environment. For the sake of simplification, the required physical fundamentals are initially introduced at the zero-temperature limit, and an ideal quantum RC time-scale response is predicted at this limit. In subsequent sections, we discuss how these fundamentals are beneficial to understanding the time-scale response of mesoscopic electrochemical structures operating at finite temperatures, placing the assumption discussed here into a more day-to-day context. It should be stated that, at the nanoscale, neither resistance nor capacitance can be considered classical circuit elements and the time-scale of a quantum or nanoscale process will be modeled by a quantum RC circuit. Indeed, at the mesoscopic scale, where practically all nanoelectronic and nanoscale electrochemical devices are situated, there is no prevalence of the classical or quantum mechanics. Mesoscopic systems are those containing a quantity of atoms that are large enough not to follow purely quantum mechanics, but sufficiently small not to follow the laws of classical mechanics. This is the case of quantum RC electronic circuits, which can be modeled as quantum dots ina single contact with a macroscopic electrode, as


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illustrated in Figure 1a. This is the case of an ideal quantum electronic RC circuit (at the zero-temperature limit), where the time-scale to charge a single quantum capacitive state from an electrode(Figure 1a)is8,12,30

= +

=

(1)

where is the elementary charge of the electron, ℎ is the Plank constant, is the electrostatic capacitance and = is the quantum capacitance proportional to the density of electronic states (DOS) of the quantum dot, where one can see that is the series association of the electrostatic capacitance and , following mesoscopic reasoning. = (ℎ/2 !) is a quantized resistance9, where ! is the electron transmission of an individual point contact (essentially, this is associated with the electron transmission probability of the quantum channel, as illustrated in Figure 1), which separates the accessible electronic state in from the electronic reservoir (the electrode of Figure 1). In fact, Figure 1a shows an individual quantum point contact structure (or a quantum dot), which can be used to represent a molecule in physical or chemical contact with an electrode. The quantum point contact comprises a nanoscale chemical capacitor, ,the series association of electrostatic ( ) and quantum capacitors ( ), which, in a mesoscopic approach, takes into account both classical and quantum contributions. The reason for the latter is that, in addition to the electrostatic separation of charges, there are other contributions due to the electronic structure of the chemical constituents of the matter. Accordingly, in Figure 1a, the quantum point contact is assembled over an electrode and thus electronically coupled to it through idealized quantum channels within unitary transmittance, as previously stated. On the other hand, Figure 1billustratesthe behavior of a quantum point contact coupled to a metallic structure when immersed in an electrolyte environment. In the presence of an electrolyte, the associated potential of the gate is screened by the environment, which is illustrated by solvent molecules now surrounding the electrochemical capacitance, , instead of . Thus, nowreplaces as the series association of ionic31 (double-layer as a typical exemplar) and quantum capacitors32,33 (see more details below), thus constituting the missing physical structure of interest to interpret the time-dependent (or the time-scale) response of molecular electronics14, 64-69(attended in an electrolyte environment) and molecular electrochemistry17,34. When an electrolyte is present34, the electrical field screening is a critical aspect29,31 because the solvent chemical environment is essential to dictate the electron dynamics, according to electron transfer models20,35. It should also be noted that in an ideal point contact, ! must equate with the unit so that ℎ/2 (∼12.9 kΩ)is a universal constant, whose inverse is known as the quantum of conductance9,10. Essentially, each molecule in physical or chemical contact with an electrode would be charged (especially if electrons are transferred to the molecule) on a time-scale that follows Eqn. (1). In the absence of an electrolyte, Eqn. (1) summarizes the properties of a coherent (at gigahertz frequencies) ideal quantum electronic RC.


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Figure 1. (a) An individual quantum point contact structure comprising a molecular chemical capacitor, , assembled on an electrode and electronically coupled through quantum channels; (b) the same as in (a), but within the point contacts immersed in an electrolyte environment. In the inset, note how this pictorial approach couples with perturbation theory, ($) = %&($)/%'($), as 36 detailed in reference .

In summary, small atomic or molecular entities attached to an electrode immersed in a vacuum or ambient air environment can be modeled by using Eqn. (1), although this experimentally one-terminal contact mode (between molecules and electrode) cannot be probed electrically in the traditional direct current (DC) configuration. The reason for this is that a one-terminal contact does not allow electrons to flow continually. Hence, a DC measurement needs two-terminal contacts to induce a potential difference between the terminals for electric current to flow across a molecule that separates the terminals. The latter is the typical experimental configuration adopted in molecular electronics experiments, and is also the architecture used in the operation of classical transistors. However, electrical measurements in one-terminal contact mode are possible by using alternate current (AC) approaches, such as impedance methods28,33. Accordingly, a single molecule or a molecular film can be studied based on the linear response of sinusoidal perturbations provided by impedance spectroscopy. These perturbations allow electrons to be transferred to the molecule and reflected from the molecule back to the electrode (so there is an electron relaxation and dynamics), and the response is predicted to follow Eqn. (1). Thanks to these dynamics, molecular electronics can be probed using time-dependent electrochemical methods and well-established experimental approaches. Electrochemical capacitance An equivalent nanoscale or quantum RC circuit exists if the molecular junction is inan electrochemical environment, as shown in Figure 1b. This is only a matter of evaluating the charge relaxation in the presence of an existing ionic environment and the hypothetically quantum point contact, as discussed in the preceding section, is embedded in a dielectric continuum (Figure 1b) containing ions (the electrolyte itself), so that is now replaced by (the electrochemical capacitance), such as12,33

= + (

(2)


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where ) is the ionic capacitance of the interface37(normally modeled by the double layer phenomenon, see below). A difference between the and depiction of the charging events should be highlighted simply because ) differsfrom , as illustrated in Figure 2. The ionic capacitance per unit of area is ) = *+ *, -, where - is ionic-strength-dependent, i.e.,- is the inverse of the Debye length31, thus differing from the geometric factor 1// contained in .Figure 2 clarifies the differences between electrostatic plate capacitor (as an example of ) and an ionic plate capacitor (where the double layer is an example of ) ). In the electrostatic plate capacitor, the capacitance per unit of area is inversely dependent only on the distance between the plates. In the ionic plate capacitor, the capacitance per unit of area is dependent on -, i.e., the inverse of the Debye length. In the latter, when an electrode plate is in contact with an electrolyte in the absence of electroactive species, an electrochemical element analogous to the plate electrostatic capacitance exists (Figure 2b), but charge separation is modulated by ionic strength, in the case of ) . The simplest capacitive model for the junction shown in Figure 2b is known as a double-layer structure, which is negatively charged in this illustration. The structure closest to the electrode is known as the Helmholtz inner plane of the double-layer model.13,31 Numerous examples of applications are found using this simple double layer capacitive model14,38-43.

Figure 2. (a) Electrostatic plate capacitor in which the capacitance per unit of area depends solely on the geometry. (b) When a plate electrode is in contact with an electrolyte in the absence of an electroactive species, an electrochemical element analogous to the electrostatic plate capacitor exists, and is now dependent on the inverse of Debye length, -, which is modulated by the ionic strength. The inset in (b) demonstrates that the atomic structure of this junction is manifold. Reprinted (adapted) with permission from (Joshua Lehr, Justin R. Weeks, Adriano Santos, Gustavo T. Feliciano, Melany, I. G. Nicholson, Jason J. Davis and Paulo R. Bueno; Mapping the ionic fingerprints of molecular monolayers, Physical Chemistry and Chemical Physics. Copyright (2017) Royal Society of Chemistry).

Nevertheless, additional components associated with exist when electrons are transferred from the electrode to accessible electronic states at the electrolyte side of the junction, where the energy alignment between the electrode and the molecular states is critical to quantify the charge contribution of . For instance, in the absence of Faradaic activity, 1/ is depreciable in Eqn. (2), since the electronic density of states is inaccessible to charging by a potential difference established with the electrode, corresponding to the situation where polarization occurs without charge transfer (referred to as non-Faradaic events).31 Eqn. (2) predicts a double layer capacitive model and also contemplates other capacitive effects associated with the charging of molecular ensembles. If accessible electronic states exist, there is aninherentpresence ofquantizedcapacitive effectsusually engrained within what is described as “pseudo capacitance.”17Under


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such circumstances, then, both ionic and electronic contributions must be considered in a combined electrochemical capacitance which includes a consideration of Thomas-Fermi electric field screening29,31,33

[where ionic () ) and quantic ( ) terms contribute], and - = ( /*0 *, )/ is the Thomas-Fermi wave-vector that governs the potential decay associated with the electric field screening.29,31 The meaning of - is therefore important, since it is dependent on and dielectric environments, *0 *, . The meaning of faradaic and non-faradaic contributions

Both ionic (distinguishable) and electron (indistinguishable) particles are engrained in experimentally resolved of an electrochemical interface. If a Boltzmann approximation is assumed, considering the screening is controlled by ions in a pure non-faradaic charging of the interface, then ∝ exp[−'/7 8], where 7 is the Boltzmann constant and 8 is the absolute temperature. In this Boltzmann approximation and dynamics dominated by ions, fluctuations in the electric potential obviously follow classical dynamics. On the other hand, in assuming a pure faradaic charging of the interface, governing by electron dynamics only so ∝ (1 + exp[−'/7 8]), and fluctuations of the electric potential distribute according to Fermi-Dirac statistics. The Boltzmann approximation leads to a wave vector of the form - = [(2 )/ (*, *0 : 8)]/ , which is engrained within the Debye-Hückel model44,45. The Debye length can be retrieved from the Thomas-Fermi wave vector, demonstrating that the double layer phenomenon is only an approximation of Eqn. (2), i.e., when we accept that only ions control the electric field screening in an electrochemical junction.31 In summary, both faradaic and non-faradaic situations are approximations of the same mesoscopic event governed by electronic and ionic dynamics. As expected, double-layer (or Debye electric field screening) phenomena prevail in the absence of accessible electronic states of molecular layers assembled over an electrode.31 This is the case of graphene oxide (low density of electronic states) contrasting with reduced graphene oxide (high density of electronic states);see further discussion about super-capacitance phenomena in graphene sheets, Figure 8.29 Henceforth, Thomas-Fermi screening governs reduced graphene oxide over Debye screening in graphene oxide. In summary, in situations where there is a significant accessible nanoscale electron density of states(often resulting in faradaic events), is dominated by and, accordingly, the electrical field screening is ruled by electron dynamics attenuated by the electrochemical environment. In short, the purpose of faradaic and non-faradaic terminologies used in describing electrochemical charging events, from the point of view of mesoscopic physics, is directly associated with the type of electrical screening that governs the electrochemical junction. The faradaic and non-faradaic phenomena are thus only different quantitative approximations of a common physical event operating at nanoscale electrochemical junctions. Electron transfer rate and electrochemical capacitance Now let us demonstrate what the time-scale of an electrochemical process would be, according to Eqn. (1),but whose capacitance is . The latter is the case where the charge of the mesoscopic (or molecular) scale capacitance is embedded in an electrolyte environment, like that depicted in Figure 1b.17 This obviously corresponds to the time-scale of the electrochemical reaction conforming to a single electron transfer such as17,35

=

+ = (


(3)

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where is the well-known electron transfer rate constant. Eqn. (3) is validated in reference 36 and thus represents the quantum RC circuit model for electrochemistry at room temperature. The temperature obviously introduces broadening effects that can be explained using statistical thermodynamics. This topic will be introduced later herein. As a result, the predicted thermal broadening of the density of electrochemical states that governs ,as part of Eqn. (3) (with a Gaussian shape), is experimentally measured46,47, as shown in Figure 5b. The Gaussian shape directly associated with the electrochemical DOS shape can be used to investigate how spreads48in response to changes in the external capacitance, ;< , i.e., a capacitance generically representing the non-quantized contribution of the chemical or electrochemical capacitance. Broadly speaking, ;< takes into account the electrostatic or ionic contribution of the capacitive component (see Figure 3), which predictably affects according to = *0 *, - . An electrolytic environment can be changed by keeping the same concentration of ions and altering only the solvent (see Figure 5). This has different impacts on the density of states of the system, but the length scale follows - = ( /*0 *, )/ . Further details are discussed later. In summary, ;< is associated with environmental effects only [in a vacuum environment this is (electronics)and in an electrolyte it is equivalent to ) (electrochemistry)], as discussed previously. The broadening of has two main sources, one coming from temperature, as predicted by statistical thermodynamics, and the other associated with the dielectric characteristics of the environment. This affects how we interpret electron transfer theory, when applied realistically in modeling molecular junctions. The aforementioned assumption is in agreement with the fact that electrochemical reactions inherently occur at finite temperatures and are known to be explicitly dependent on the dielectrics of the environment. Electron transfer rate, conductance and electrochemical capacitance Note that Eqn. (3) can be rewritten in a simpler form: = /, where = 1/ is the quantum of conductance and is the quantum limit of the charge transfer resistance (=< ) obtained in these junctions. Briefly, =< is defined in electrochemistry as resistance to the transfer of electrons from one phase (metallic electrode, for instance) to another (for example, a molecular gate, as illustrated in Figure 3a). In reference 36 , note that = / is the expected response of an ensemble of molecules assembled in an electrode, comprising a molecular film, as displayed in Figure 3a.17,35 The response that confirms the interdependence of electrochemical capacitance, quantum of conductance and electron transfer through the definition of = /is shown in Figure 4 for alkane molecular films17. Also interesting is the correspondence of with the first-principles density functional theory that was recently postulated,33,49 allowing electrochemical events to be rationalized within an approach of conceptual quantum chemistry and first principle quantum mechanics.50 Eqn. (3) thus connects the electron transfer rate and the quantum of conductance concepts in such a way that renders electron transfer and transport totally interchangeable through the microscopic accessing of . Figure 3 shows (see reference 36 for a demonstration) that an ensemble of individual quantum point contacts exemplified by ideally individual ℎ/2 and ,) components, associated in series, provides an average response that is a series association encompassing the individual elements to provide and as universal elements representing, for instance, the experimental response of electroactive molecular films. This is the response obtained experimentally by impedance spectroscopy measurements.17-19,33,36,37 In other words, the quantum equivalent RC circuit of the ensemble is depicted as the series combination of and elements, which globally provides an electron transfer rate of = ( ) , as shown in Figure 3c. Figure 3dshows how the quantum RC representation (red box) of a molecular film, as illustrated in Figure 3a,


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is expressed as the sum of ideal series quantum resistors (ℎ/2 ) and capacitors ( ) coupled with an external capacitance ( ;< ), the latter representing the non-faradaic contributions. The conductance = 1/ (Figure 4b and 4d) and (Figure 4a) are experimentally accessible by impedance measurement17 in alkane molecular films wherein tunneling and hopping mechanisms can be studied, as commonly reported through other methods.28,46 The independence of as a function oftemperature variations (see Figure 4c) is suggestive of a tunneling mechanism, although separately, and are temperature dependent within a particular interval of frequencies. An ultimate electron transfer rate depends on , and hence, on the associated nature of the electronic density of states. has been an ignored component in molecular electronics or nanoscale electrochemistry.7,51 Nonetheless, , as an experimentally accessible parameter, is a quantitative measurement of HOMO-LUMO states.33,36

Figure 3. (a) Molecules assembled on an electrode, forming a molecular film embedded in an electrolyte. This situation can be 36 modeled by an ensemble of individual quantum point contacts(see more details in reference ). (b) This is the expected thermal broadening given by %〈〉/%(') ∝ 〈〉(1 − 〈〉) [see also Eqn. (4)]. /corresponds to the thickness of the electron path through the molecular layer.Note that resistors were depicted as ideal conductor channels, where ! equates with unit. (c) Indicated the related to the equivalent circuit response of a quantum RC ensemble as shown in (d).

In summary, in treating the molecular states electronically coupled with electrodes as point contacts, and further, in electrochemically measuring and interpreting the properties of an ensemble from a timedependent standpoint, a correspondence between quantum of conductance9 and electrochemical capacitance is found33,34 through the electron transfer rate.17,52 In this theoretical framework, the electron transfer rate depends explicitly on within its quantum mechanics components, which differ from , where electrical field screening is different, as discussed previously. At this point, it is important to differentiate unequivocally between two-terminal contacts, as generally used in molecular electronics, and the one-terminal contact configuration, as used in electrochemistry. The dissimilarities determine the differences in electric potential the molecule is subjected to in each of the configurations. This allows us to explicitly distinguish an electrochemical %@̅ = −%' potential difference attained for a one-terminal contact in electrochemistry from the chemical potential (non-equilibrium)


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differences (%@ = −%') obtained in a molecular electronics two-terminal configuration, i.e., whether or not they are electrochemically gated, the one and two-terminal experimental setups clearly differ. The differences are associated not only with issues involving chemical bonds to the electrodes (one or two contacts) but also and mainly with electrical field screening, which is altered. Experimental evidence of the alignment between conductance and electrochemical capacitance The temperature dependence of and is consistent with the fact that an increase in capacitance causes a concomitant increase in conductance (see Figure 4), as predicted by = /. The temperature independence of resonant frequency is additional evidence demonstrating the predominance of a tunneling mechanism accompanied with electron transfer, as indicated in Figure 4c. Figure 4dshows how can be measured equivalently as a function of potential at a fixed frequency, and as predicted, maximizes when the electrode energy is aligned with redox energy levels, B+ [according to Eqn. (4), below], which corresponds to the Fermi level of these quantum, electrochemically active molecular junctions. The vertical bars in Figure 4a, 4b and 4c correspond to the value of the resonant frequency directly quantified as = / .

Figure 4. (a) Corresponds to the Bode representation of the real component of complex capacitance. (b) Shows the Bode diagram of AC conductance [($) = $′′], meaning the admittance of electrons to molecular redox switch ensembles. Note that(b) is obtained at the Fermi level (associated with the formal potential) of the junction. (c) Shows the temperature independence of resonant frequency, = / . (d) as a function of potential measured at a fixed frequency (at20 Hz).Reprinted (adapted) with permission from (Paulo R. Bueno, Tiago A. Benites and Jason J. Davis; The Mesoscopic Electrochemistry of Molecular Junctions, Scientific Reports. Copyright (2016) Nature).


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We will now discuss how Eqn. (3) complies with electrical field effects and thermal broadening.17,20 Electrical field effects and thermal broadening It is essential to highlight how previously introduced nanoscale circuit elements ( , and ) and the relationship between them governs electrochemistry at the nanoscale, particularly the electrochemistry of molecular junctions. Considered a useful theory for nanoscale electrochemistry, the concepts underpinning these circuit elements can now be applied to envision applications, particularly those associated with electroanalytical methodologies. Moreover, to help us design electrochemical devices as electronics engineers do, it is useful to make an analogy with field-effect transistors operating under equilibrium conditions (more details can be found in reference 36), thus leading to novel approaches in the design of electrochemical field effect transistors, where mesoscopic effects prevail. Let us starting by noting that the differences between the potential in the channel/bridge and in the gateare dependent on (see Figure 3). The gate capacitance is thus governed by and Eqn. (2) can be deduced equivalently only by analyzing field-effect transistors at equilibrium, which clearly demonstrates that, in experimentally accessing , the thermodynamic properties of the gates are traceable, so that sensitive molecular changes in the electric potential of the gates are quantifiable. Consequently, the associated energy scale49 of the gate is B = & /2 , where & is the electric charge. Variations in this energy due to the occupancy of the states associated with variations in are reported through derivatives of B with respect to the charge, resulting in %B/%& = &/ , which is an electric potential difference32. In conclusion, the inverse of reports on the variations of the energy of the interface. Additionally to the manifestation of field effects associated with is the fact that, realistically, electrochemical reactions operate at room temperature. Hence, the zero temperature approximation [of Eqn. (2) and (3)] is impracticable, and our discussion now cannot proceed without taking into account thermal broadening. Since is proportional to the DOS, the easiest way to experimentally access the thermal broadening is by measuring as a function of potential. The experimental shape thus obtained can be explained by treating the ensemble of parallel quantum capacitors (see Figure 3b) using statistical mechanics, i.e., by applying the grand canonical ensemble, as stated briefly in the preceding sections and in more detail in reference 36. The variance in capacitor numbers to the thermal broadening (at constant temperature and volume)53, following evidence of Tomas-Fermi screening, is governed by the Fermi-Dirac statistics of occupation, where 〈〉 = (1 + exp[−'/7 8]) is the average number of quantum capacitors and −' = B+ − @̅ , whereB+ is the formal potential of the electrochemical accessible states. In equilibrium conditions, the electrochemical overpotential is null and@̅ = B+ . In other words, in electrochemical equilibrium conditions, the occupation of the states associated with is half, i.e., 〈〉 = 1/2, which means that half of the states are oxidized and half of them are reduced. Considering a molecular coverage number, D, which defines the number of quantum capacitors (or redox molecules) covering the surface of the electrode, is finally obtained as

F

= EG IJ 〈〉(1 − 〈〉),

(4)

H

Since /7 = K/,where is the ideal gas constant and K is the Faraday constant, a relationship between this mesoscopic model and classical electrochemistry is then obviously settled. Classical electrochemistry is


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retrieved from the quantum derived model (see reference 36 for further details on this). It has been proven in reference 36 that a collection of individual molecules form an ensemble assembled on the surface of an electrode with a capacitive macroscopic response accessible experimentally as by impedance or potential sweep methods. In electrochemical equilibrium conditions (in which, as previously stated,〈〉 = 1/2),the term〈〉(1 − 〈〉) = 1/4 and thus = D/48. Eqn. (4) thus conforms to the classical electrochemical analysis (as demonstrated in detail in reference 36). Briefly, in classical electrochemical transient-based methods, the exchange electrochemical current density is obtained, according to Laviron’s formalism,20,54,55 as M = ( D/48)N, where N = %'/%O is the scan rate in the case of transient potential scan perturbing methods20,54,55.The electrochemical capacitance is thus the proportional term between Mand N. Note that M = ( D/48)N = N is equivalent to Eqn. 2.2 of reference 20. Nonetheless, traditional electrochemistry literature or textbooks do not explicitly correlate Laviron’s kinetic equation with the electrochemical capacitance of the interface. Finally, it should be noted that, whether or not Eqn. (4) is taken into account, the conductance of the system is directly associated with quantized amounts of 2 /ℎ and that, overall, the total conductance is proportional to , which is = , as validated previously. In summary, countless applications and electrochemical devices can be envisaged following the reasoning previously introduced by Eqn. (4). For instance, Eqn. (4) must participate not only in molecular diagnostic devices37 but also in the design of molecular super-capacitors29,56-58, batteries and solar cells59-63, etc., to cite but a few. Eqn. (4) theoretically governs the nanoscale electrochemistry of molecularly modified electrodes and applies to those molecular systems introduced in reference20. For instance, with few modifications and considerations, Eqn. (4) was used to explain the capacitance characteristics of Geobacter biofilm64, which controls the performance of this microbial biofilm respiration64 and thus is directly related to the production of biochemical electricity. The implications of the electrochemical or redox capacitance in biology are of tremendous importance, and there is still a large field open for exploration. Several applications of this previously mentioned mesoscopic electrochemical theory in different molecular devices will be introduced in the following sections, obviously aiming to generalize the applicability of the concepts described herein. We start by demonstrating the effect of the solvent on the shape of electrochemical capacitance as a function of the potential of the electrode and how this complies with (or affects) the contemporary use of the semi-classical electron transfer model to interpret (sometimes incorrectly) the electrochemical behavior of molecularly modified electrodes. Effect of the solvent environment and spreading of the electronic density of states The semi-classical electron transfer theory is underpinned by the transition state theory, in which the electron transfer assumes the form of = , exp [−B ∗ /7 8] and where , is the maximum rate constant and B ∗ is the activation energy.65,66 B ∗ generally has two contributions, one coming from mechanical elastic distortions from internal force fields and another from charge injections65,66 that occur in the presence of an electrostatic external potential accounted for in ;< (see Figure 3), which involves the role of polarization properties of the solvent.52 None of these energetic components (contributing to B ∗ ), alone or combined, can adequately describe the quantum electrochemical RC model. The reason for this is that, in the DebyeHückel theory, fluctuations of the screened Coulomb potential operate over46,47 an exponential damping term ∝ -Q , wherein Q is the spatial position and - is the inverse of Debye length within a Boltzmann approximation of the electric field screening. Nonetheless, - following classical approximations is unsuitable according to previous reasoning validating Thomas-Fermi against Debye screening. In conclusion, the meaning of capacitance in Debye electric field screening is totally distinct from the Thomas-Fermi31 case.


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Although the semi-classical electron transfer theory is generally adequate to predict the impact of the environment’s dielectric properties on the electron transfer rate52, especially in a solvent bulk environment, because of Debye vs. Thomas-Fermi electrical field screening, it fails to predict how solvent reorganizes and thus governs the charge transfer in electroactive molecular scale junctions. At these interfaces, the HOMO and LUMO differences in the molecular ensemble (whose inverse is directly associated with )33 govern the electron transfer rate, as demonstrated previously, according to Eqn.(3). Consequently, it is the electronic structure of the molecules at the interface that governs the electron transfer, i.e., it is - = ( /*0 *, )/ that regulates the way states attained between the HOMO and LUMO levels couple with an electrode.33 The consideration of the Thomas-Fermi screening affects the reorganization energy – the latter, which is a key parameter to be considered in the electron transfer theory, still remains to be evaluated in depth. Figure 5b shows, as exemplified in the 11-ferrocenyl-undecanethiol monolayer assembled on gold electrodes, how the Gaussian shape obtained experimentally from impedance measurements thus spreads with the dielectric environment, complying with Eqn. (3). Note that the electronic DOS (Figure 5b) associated with Eqn. (3) (right side) is not the nuclear Franck-Condon weighted DOS (the left side has been developed semi-classically).17,35 Both DOS functions affect the electron transfer rate; however, from a detailed quantum statistical mechanics standpoint, it is not completely understood how they are fundamentally correlated, an issue to be evaluated in greater detail elsewhere. Moreover, quantum mechanics effects are remarkable and the differences in the electronic structure of the molecular film (Figure 5) as a function of variations in the dielectric environments cannot be explained classically or semi-classically. The electronic structure is wave or electron density-dependent and the spatial position of the atoms, by itself, cannot determine the physical properties of the system. For instance, naively misinterpreting Eqn. (4) for this particular situation, and by following the previous reasoning, one would be able to determine the molecular coverage as D = (48)/ at the formal potential of the junction. for the latter expression maximizes, as shown by the peaks in Figure 5b, as ∼380 µF cm-2, ∼170 µF cm-2and ∼80 µF cm-2 for dichloromethane (DCM), acetonitrile (MeCN) and aqueous solvent (AS, H2O), respectively, as inferred for the three different positions of the maximum value of in Figure 5b. Therefore, different average values of capacitance or molecular coverage would be obtained for the same11-ferrocenylundecanethiol molecular film when placed indifferent solvents. The Gaussian DOS responses in different dielectric environments are reversible,18,19,49 so the molecular coverage and the film atomic structure are kept steady (there is neither loss of ferrocene electrochemical centers to the environment nor a decrease in electrochemical activity). Why, then, does ferrocene molecular coverage vary in different dielectric environments? This is not an inaccuracy resulting from theory, but is based on a common classical interpretative sense of quantum effects (sometimes not even recognized as quantum). Therefore, what is actually incorrect is our classical mindset, because what governs now is the integrated DOS and not the local, as stated in Eqn. (4), from which D = (48)/ was derived considering single molecular states. In integrating the overall energy levels, the total electronic DOS is obtained and is thus constant as 1.37 ± 0.4 x 1014 states cm-2. This demonstrates that the correct interpretation is enabled by the theory, i.e., the electron density remains constant over all levels of charging energy. The impact of the dielectric environment is distributed over the electronic structure of the junction. The structure therefore only reorganizes its electron density49 differently following the polarization associated with ;< , although the total electron density at the molecular junction remains constant, regardless of the dielectric environment.


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Figure 5. The impact of solvent static dielectric constant (water *S ∼ 80 [yellow], acetonitrile *S ∼ 40 [green], dichloromethane *S ∼ 10 [red]) on experimentally resolved and for an 11-ferrocenyl-undecanethiol film showing mean values of three independent measurements. is intrinsically associated with the dielectric constant, according to Eqn. (2). Note that, predictably, the charged states, i.e., the total DOS (computed by integrating the Gaussian functions to obtain the area of the curves), is constant. The electron transfer rate, , is affected, as shown in (a) and the inset in (b). In (b), the dispersion of and DOS Gaussian function is observed as a consequence of effects of the dielectric environment on the quantum RC ensemble. The inset in (b) demonstrates the linear relationship of logarithmic and *S , ultimately related to Marcus theory, but with reservations, as discussed in the main text. Also, in (b)the Fermi level associated with the formal electrochemical potential of the electrode was zeroed simply for convenience. Reprinted (adapted) with permission from (Paulo R. Bueno and Jason J. Davis; Measuring Quantum Capacitance in Energetically Addressable Molecular Layers, Analytical Chemistry. Copyright (2014) American Chemical Society).

In summary, the intrinsic existence of an electronic DOS beyond the nuclear Franck-Condon weighted DOS per se is not predicted by semi-classical electron transfer rate theory. Therefore, the effect of the solvent on the electronic DOS is not predicted by this theory but is envisioned by those accounted for by the mesoscopic theory underpinning the quantum RC circuit. Energy Transducer and Sensing Now we will discuss how previous concepts can be used to design molecular diagnostic assays, based on a unique example of quantum RC electrochemical dynamics as applied to sensing devices. If an interface containing an ensemble of individual electrochemical capacitors (representing the capacitance associated with redox active molecules)is arranged with receptive biological receptors, a sensorial interface is constructed (Figure 6a),because the electronic occupancy of the ensemble is sensitive to the biological binding event (Figure 6b). Such a biosensing interface has been demonstrated to be useful in developing molecular diagnostic devices using redox-active alkanes25,26 and redox-tagged peptide monolayers27 or even graphene layers37. The variation in energy of the interface per number of electrons occupying the electrochemical capacitive ensemble is modulated by a chemical potential difference such as %B/%〈〉 = 〈〉( / ), where ( / ) is the energy required to individually charge the molecular capacitors, and such a change in interfacial energy can be appropriately used as a transducer signal of the biological binding event. An example is given in Figure 6a for a molecular redox-tagged interface, where two important characteristics should be noted. The first has to do with the difference in the effect of the solvent (previously shown in Figure 5) and the second with the environment effect due to the occupancy of receptor sites at the interface (Figure 6b and 6c).


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As shown in Figure 5b, a change in the local dielectric constant (which is achieved by changing the solvent) translates into a resolved DOS energetic redistribution without a change in total state occupancy, as expressed3 by the total area of the normal DOS distribution function. In other words, upon integrating the curves in Figure 5b, the total electrochemical DOS is obtained for the same molecular junction placed in different solvents, and the value is constant, as discussed in the preceding section. The shape of the DOS changes, but the area is the same, meaning the states reorganize energetically according to different dielectric environments. On the other hand, as is the case in Figure 6b, the dispersion of the DOS does not change (there is no change in the standard deviation of the Gaussian DOS shape), but the intensity of the average value (and therefore, the area of the DOS function) changes. Accordingly, a molecular binding event (such as that occurring at a neighboring receptive site) triggers a resolved change in the intensity of the average value of the DOS, and hence, in the occupation of the states (Figure 6b). We can try to distinguish between these two environmental triggers by looking at the DOS shape and its associated energetic spread as expressed through T[%〈〉/%B] = ln WX Y2Z[ (the entropy function of the normal DOS distribution), where [ is the basis of natural logarithms and WX is the standard deviation. In the case of changes in the dielectric environment, there is a clearly resolved change in dispersion (Figure 5b) and Fermi level displacement (not shown)37. The effects of a local binding event are different insofar as there is an associated change in chemical potential, resolvable through DOS occupation (or measured ) without a change in electronic dispersion (Figure 6b) or in the Fermi level.

Figure 6.(a) Illustration of electroactive speciesarranged with receptive biological receptors, forming a sensorial electroactive (fieldeffect sensitive)interface that can be used in molecular diagnostics. (b) The responsiveness of electrochemical DOS to molecular recognition within a mixed redox switchable and antibody-constrained film as shown in (a). (b) Shows the invariance in both Fermi energy of the junction (∼ 0.49 V versus Ag|AgCl chemical reference) and dispersion of electronic states as a function of C Reactive Protein target concentration. (c) Analytical curves obtained by plotting variations of the energy of the accessible interfacial electrochemical states (∆B) at the B+ for CRP and PSA biomarkers. Reprinted (adapted) with permission from (Paulo R. Bueno and Flavio C. B. Fernandes and Jason J. Davis; Quantum Capacitance as a Reagentless Molecular Sensing Element, Nanoscale. Copyright (2017) Royal Society of Chemistry).

In summary, the electronic DOS, presented by a redox molecular film, responds to a neighboring molecular recognition, such that the resolved electronic charge is proportional to the target/analytic concentration (Figure 6c; the target here being C-reactive protein, CRP, and prostatic acid phosphatase, PAP). The energy-related signal directly reports on the electron occupation of quantized states instead of the DOS shape dispersion. In other words, when a target biomarker is recruited, the occupation of the electronic states changes and this is experimentally measured as the variation in interfacial energy (∆B). Indeed, ∆B = ( / ) corresponds to the amount of energy per electrons modified in the electrode junction


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as a result of binding equilibrium during recognition of the target, which is a function of the concentration of the target in the electrolytic environment. The sensor design is thus based on the response of a confined and resolved electronic density of states for target binding and the associated change in interfacial chemical potential.37 This concept has been demonstrated with a number of clinically important markers, representing a new, potent and ultrasensitive molecular detection system24,27,67-73 (femtomolar to picomolar) enabling an energy transducer principle that can, in a single step and without reagents, quantify markers within biological fluid.24-27,37,67,68,70,74 The mechanism of operation of a quantum electrochemical field-effect transistor, as introduced above, differs from the traditional three-terminal field-effect transistors (FET), in which the modulation of conductance (flow of electrons or holes) is controlled by electrostatic potential variations (in the gate environment) that induce charge density variations in the channel. The mechanism operating in quantum FETs is based on the detection of changes in the electrochemical charge density, or the Thomas-Fermi “field effect” [see considerations based on Eqn. (4) and, for more details, reference36], which changes the electrochemical density of states as a function of chemical variations induced by a biological recognition event, for instance, between a target molecule and the receptor. Consequently, it is the molecular quantum capacitors that work as the gate terminal. These capacitors are surrounded by a chemical environment that determines the sensitivity of the device. In molecular electronics, it is the flow of DC current through the channel that governs, as normally assumed in three-terminal traditional FET configuration. Therefore, nanoscale electrochemical FETs based on molecular quantum capacitors do not follow the traditional DC transistor architecture. To conclude, we exemplified herein that the occupancy of quantized states responds sensitively to changes in the local electrochemical potential caused by the chemical binding reaction taking place in the receptive layer. By properly introducing receptors, this entirely reagentless sensing method becomes highly specific and very sensitive. We recently stated37 that the combination of this approach with standard microfluidic and micro-fabrication methods would, in the near future, offer a great deal to both diagnostics and chemical sensing in general. Quantum Conductance of DNA wires We are now interested in pointing out that “wires” made of molecules can connect the electrochemical capacitive resonant switchers (for instance, ferrocene) to the electrode (see Figure7a); these molecular wires (sometimes called molecular nanowires)act as the bridge whereby electrons can be transmitted or transported along the molecular electronic structure by hopping, enabling transport over larger distances28,75. DNA strands are used as nanowires (with length ∼ 7 nm) and the propensity they acquire to transmit or transport charge can be evaluated by using the dynamics of a quantum electrochemical RC circuit, as introduced above, by simply applying the = / relationship.35 Regardless of whether the process is tunneling or hopping, the = / relationship may be attended and the influence of the electronic structure is achieved implicitly by (and the contained DOS shape). The hypothesis that charge transport in ds-DNA is a function of the distance between the GC base pair, with the electronic transfer rate decreasing by about one order of magnitude with each intervening AT base pair, is confirmed28. According to this hypothesis, increasing the amount of adenine and thymine causes an exponential decrease in the electrical response; thus, AT sites are known to act as conduction barriers. Charge flows through poly(dG)poly(dC) (ds-DNAc) are therefore expected to be comparatively easier than through poly(dA)-poly(dT) (dsDNAi).


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The simplest analysis can be made at the Fermi level and low frequencies, in which = = ( D/47 8) holds. By using the latter relationship, the conductance of DNA strands (composed of different sequences of nucleobases) was proven to be measurable.28 The conductance levels obtained were 4.6 ± 0.4 S m-1and 0.48 ± 0.05 S m-1 (in resistivity values, ~0.2 Ω m and ~2.1 Ω m) for ds-DNAc and ds-DNAi, respectively28, in agreement with other methodologies such as STM.76,77

Figure 7.(a) Illustration of the covalent attachment and electronic coupling of redox-tagged (ferrocene) double-stranded DNA (about 7 nm in length) on a gold electrode.(b) Keeping astationary environment (electrolyte) and constant length, an electronic DOS can be constructed for different sequences of nucleobases, in which charge transport is therefore expected to be different78-81. Reprinted (adapted) with permission from (William W. C. Ribeiro, Luís M. Gonçalves, Susana Liébana, Maria I. Pividori, Paulo R. Bueno; Molecular Conductance of Double-stranded DNA Evaluated by Electrochemical Capacitance Spectroscopy, Nanoscale. Copyright (2016) Royal Society of Chemistry).

The comparison of molecular electronics (through direct conductance measurements) and electrochemistry (indirectly inferred from charge transfer rate measurements) as discussed recently35, in which a power-law relationship82 was empirically demonstrated, can now be theoretically corroborated. Accordingly, we recently demonstrated how = theoretically elucidates this power-law dependence.35 Supercapacitance of graphene layers Since = / is based on first-principle quantum mechanics35,49, its generality is expected to be applicable beyond molecular conductance or biosensing. In this section, we support this statement by demonstrating the applicability of the theory to elucidate capacitance phenomena in carbonaceous compounds. Graphene is one of the most intriguing molecular compounds, and owing to its unique and extraordinary electronic properties, it has been speculated that graphene can carry a supercurrent (i.e., abipolar current).83 One fact that has been neglected is that the outstanding electron transport properties of graphene are intrinsically associated with a large capacitance, which, from the standpoint of classical mechanics, is antagonistic to electron transport. Accordingly, although there are many reviews on the properties and applications of graphene,84-86 only a few studies have connected its intrinsic characteristics with its outstanding capacity for energy storage.87 The origin of the molecular supercapacitance of electrochemically modified graphene mesoscopic layers and its behavior when embedded in an electrolyte


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was only recently elucidated through the use of the quantum electrochemical RC model, as introduced above.29 Figure 8indicatesthat the value of capacitance obtained for reduced graphene oxide is very high, undeniably within mF cm–2 (∼5.1 mF cm–2), which is about 1,000-fold higher than that obtained for graphene oxide (typically between 2 and 10 µF cm–2). The associated increase in the electroactive area(due to the electrochemical reduction process) is approximately 25-fold, much lower than the observed increase in capacitance. Therefore, this increase in capacitance cannot be attributed solely to an increase in the specific surface area. It is worth emphasizing that the only difference between reduced graphene oxide and graphene oxide per se is associated with the electrochemical reduction of graphene oxide29. This experimental observation29 thus enabled a revisitation to the origin of the supercapacitance of graphene, where the electronic density of states of graphene was demonstrated to be crucial in understanding its supercapacitance phenomena, which cannot be explained exclusively by a double-layer model or by nonfaradaic interfacial charging effects. The higher accessible electronic density of states observed in reduced graphene oxide was associated with the dominance of a quantum RC electrochemical effect29 (see Figure 8). This effect can be observed in the diagrams depicted in Figure 8, which demonstrate that the nature of the RC-time constant is associated with charging of capacitive states. In the case of reduced graphene oxide, the contribution of quantum RC-time constant explains the abrupt increase in capacitance values by the electrochemical reduction of graphene oxide, which is equivalent to chemical doping. The capacitance of a glassy carbon electrode (the substrate) and graphene oxide is comparable; nonetheless, the capacitance of reduced graphene oxide (after reduction of graphene oxide in the same electrode) is more than three orders of magnitude higher. The explanation has to do with changes in the density of states of graphene oxide when it is reduced, which clearly demonstrates the quantum effects and their resulting impact on the RCtime constant. The accessible quantum states in reduced graphene oxide contribute to additional charging states and concomitantly increase the electrochemical RC-time constant. Hence, the capacitance of reduced graphene oxide is associated with charging electronic states, exhibiting a behavior predicted by the quantum electrochemical RC circuit in which Thomas-Fermi prevails over Debye screening.

Figure 8.Typical (a) Bode and (b) Nyquist capacitive diagrams obtained for GCE (glassy carbon electrode, as a reference for capacitance values and used as substrate) (▲), GO (graphene oxide) (■), and RGO (reduced graphene oxide) (●) in the aqueous electrolyte of 0.05 M phosphate-buffered saline. The inset in (b) zooms into the high-frequency region. Reprinted (adapted) with permission from (Fabiana A. Gutierrez, Flavio C. B. Fernandes, Gustavo A. Rivas and Paulo R. Bueno; Mesoscopic behavior of multi-


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layered graphene: the meaning of supercapacitance revisited, Physical Chemistry and Chemical Physics. Copyright (2017) Royal Society of Chemistry).

In summary, the RC characteristics associated with mesoscopic-type electronic relaxation, where both and are interdependent, demonstrate that the higher the conductance (lower resistance) associated with accessible states, the higher the supercapacitive effect in graphene compounds.

Final Remarks and Conclusions The covalent attachment of molecules to electrodes and control of the properties of the junction represents the state of the art in molecular electronics and molecular electrochemistry. We demonstrated that the future development of analytical methods will benefit in controlling the properties of molecularly designed junctions. Consequently, we evaluated the time-dependent electronic features of these junctions accessed by electrochemical impedance methods, demonstrating that both capacitive and resistive phenomena involved in electron dynamics throughout molecular-scale junctions embedded in an electrolytic environment (a requisite for electrochemical and electroanalytical applications) are governed by mesoscopic principles. In addition, an overview was made of electrochemical capacitance, , supported by first principle quantum mechanics in the context of its impact on nanoscale electrochemistry and molecular electronics. It was demonstrated that the associated energy (∝ 1/ ) defines the electron transfer rate as = / , in which the electron conductance is the proportional term. cantherefore be defined directly through an electron transfer rate () and a quantized conductance () such as = /. Based on this definition of we thus recognized that double-layer and pseudo capacitances, the latter empirically qualified and poorly understood, are fully incorporated in the definition of . In other words, double-layer and pseudo capacitances were both revealed to be merely particular estimations of , uniquely dependent, at the nanoscale, on the type of electric field screening taking place, namely Debye (non-faradaic type) or ThomasFermi (faradaic type) screenings. Furthermore, we demonstrate that time-dependent electrochemical methods can access nanoscale systems with an unprecedented resolution that strongly influences the future progress of electrochemical nanoscale devices, especially molecular scale transistors and transducers, both extremely useful in developing better electroanalytical methods. Electron dynamics operates in such a way that the electron transfer rate, which governs the rate of electrochemical reactions, is given by = / . Thus, it was confirmed that, in essence, the only difference between nanoelectronics and nanoscale electrochemistry is the presence of an electrolyte in the latter within a shielded electrical field, which governs the equilibrium electrostatic potential of capacitors. In general, it can be assumed that the physics of nanoscale electronics and electrochemistry are fundamentally the same and that they are ruled by conjoint quantum mechanical principles. Surprisingly, the simple = / relationship reconciles molecular electronics and electrochemistry and its usefulness is demonstrated in accessing the energy for charging molecular redox switches, in quantifying the discharge from these switches as the operative transducer signal in molecular diagnostics, in obtaining the conductance of DNA nanowires,and finally, in explaining the phenomenon of supercapacitance of reduced graphene molecular layers. An avenue of scientific and technological possibilities is open for exploration, based on the fundaments driving the common concepts of electronics and electrochemistry at the nanoscale.


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Acknowledgements I gratefully acknowledge the Royal Society and FAPESP (São Paulo Research Foundation) for their financial support. I am also indebted to Jason J. Davis for his invaluable discussions and comments. Last but not least, I would like to warmly acknowledge the assistance of some of my group’s graduate students, who provided raw data that illustrate some sections of this manuscript, and their continuing enthusiasm for this subject. For TOC purpose The merging of nanoscale electronics and electrochemistry can potentially modernize the way electronic devices are currently engineered or constructed. This paper offers a conceptual discussion of this central topic, with particular focus on the impact that uniting physical and chemical concepts at a nanoscale could have on the future development of electroanalytical devices (artwork created by Kieran Tam).

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