Measuring Quantum Capacitance in Energetically Addressable

Letter pubs.acs.org/ac

Measuring Quantum Capacitance in Energetically Addressable Molecular Layers Paulo R. Bueno*,† and Jason J. Davis*,‡ †

Physical Chemistry Department, Institute of Chemistry, Univ. Estadual Paulista (São Paulo State University), CP 355, 14800-900 Araraquara, São Paulo, Brazil ‡ Department of Chemistry, University of Oxford, South Parks Road, Oxford OX1 3QZ, U.K. S Supporting Information *

ABSTRACT: The Fermi level or electrochemical signature of a molecular film containing accessible orbital states is ultimately governed by two measurable series energetic components, an energy loss term related to the charging of appropriately addressable molecular orbitals (resonant or charge transfer resistance), and an energy storage or electrochemical capacitance component. The latter conservative term is further divisible into two series contributions, one being a classic electrostatic term and the other arising from the involvement and charging of quantized molecular orbital states. These can be tuned in and out of resonance with underlying electrode states with an efficiency that governs electron transfer kinetics and an energetic spread dependent on solution dielectric. These features are experimentally resolved by an impedance derived capacitance analysis, a methodology which ultimately enables a convenient spectroscopic mapping of electron transfer efficacy, and of density of states within molecular films.

T

the resonant communication of these with confined orbital states. It is specifically the case that the redox process is resolvable as a series combination of charge transfer resistance (energy loss) Rct and electrochemical (or redox) capacitance (energy storage), Cr, elements (see Figure 1a).15 We show herein that the latter energy storage term is directly derived from a combination of the electrostatic field associated with charge separation and a quantized term resulting from resonant access to specific confined molecular states. Such a directly and conveniently resolved mapping of interfacial electron transfer is unprecedented and provides a powerful means of studying the fundamental electrochemistry of molecular and biological systems generally.15,16,20,24 The main purpose of the current work is to demonstrate that the charging of a film constraining electrochemical (or redox) accessible orbitals is fundamentally and resolvably associated with a communication between a metal density of states, DOS (i.e., considered to be infinite in magnitude) and individual quantized energy levels (associated with a small finite DOS) that, en masse, constitute a population DOS that can be energetically mapped by capacitance spectroscopy.15,16,22,24 It will be additionally demonstrated that electrochemical capacitance obtained from these spectroscopic analyses is divisible into electrostatic and quantum components (see

he measured electron transport or capacitive characteristics of mesoscopic systems are dependent on both atomic scale phenomena and are interfacing with this at, often, microscopic levels.1−6 An extrapolation of mesoscopic physics to analyzable molecular scale quantized characteristics is, though, rare.7 One example of such has been the work of Murray in which the quantized charging of metallic and nonmetallic nanoparticles has been shown to be relevant to the kinetics of electron transfer from the underlying electrode to the particle-tethered redox sites.8−14 In the present work, we demonstrate that the capacitive signature of a molecular film attached to a metallic substrate (or electron reservoir), as analyzed by impedance derived capacitance,15,16 is equivalent to the quantum capacitance features associated with nanoscale devices such as field effect transistors (FETs), carbon nanotubes, graphene, and two-dimensional (2D) topological insulators.5,6,17 The electrochemical interrogation of redox-accessible thin films at engineered interfaces, more generally, constitutes a potentially powerful means of studying fundamental aspects of interfacial electron transfer.18−23 In seeking to address the commonly distorting and low signal-to-noise characteristics of standard electrochemical methodologies, we have recently introduced impedance-derived capacitance spectroscopy15 as a means of accessing, in a frequency-resolved manner, faradaic (redox) features free from the (otherwise inherent) distorting and distracting resistive and capacitive effects caused by ionic polarization and other nonfaradaic phenomena (see Figure 1c).24 Herein, the redox centers and faradaic activity should be treated as synonymous with metallic states (Fermi levels) and © 2014 American Chemical Society

Received: September 30, 2013 Accepted: January 9, 2014 Published: January 9, 2014 1337

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sums of edVl and edVr for the left and right electrodes, respectively. The associated capacitance, Ce, is purely electrostatic in origin and constitutes that classically associated with a parallel plate configuration.26 If the metal plates are replaced with finite DOS [i.e., when the chemical (or chemical potential energy) of the electron in the reservoir is important], the field induced electronic redistribution is followed by a secondary energy perturbation (red hashed line in Figure 2b), related to the chemical potential energy of electrons such that the measured potential in the plates deviates from its equilibrium value by an additional amount dμi where the index i represents the left and/or right plate in Figure 2b. The net charge developed (through both the primary and secondary energetic effects) on the plate i, dqi, is then given by dqi = gi(E)e(dμi − edVi )

(1)

where gi(E) represents the DOS (i.e., the number of states per interval of energy) in plate i where its occupancy is a function of the potential energy developed, E = −eV, and Vi is the potential in “plate” i. Note that there must, of course, be charge equivalence (since any charge removed from one DOS is introduced to the other; i.e., dql = dqr, where l and r denote left and right plates, respectively. Using eq 1 and the fact that dVdc = dVl + dVr, one resolves that the overall capacitance, the electron chemical or redox capacitance is Cr = edq/dμ, where

Figure 1. (a) The equivalent circuit depiction of faradaic activity at an electroactive molecular film modified metallic electrode, comprising charge transfer resistance, Rct, and redox capacitance, Cr.16 As shown in (b), the latter is divisible into electrostatic Ce and localized DOS (or quantum)-based contributions, Cq. (c) Represents a depiction of the ionic polarization contributions that can be subtracted to provide the faradaic branch shown in (a).15,16,24 (d) A schematic representation depicting the equivalence of a surface confined redox addressable state to a “quantum cavity” with its associated “quantum capacitance”.25 In reality, a population of such states is defineable by a Gaussian distribution about a mean state energy Er, with respect to a reference potential, Eref, as depicted in (e).

1 1 1 = + Cr Ce Cq

(2)

and Ce = dq/dVr. Note here that Ce arises physically from the electrostatics associated with charge separation from metal to confined molecular redox states. At molecular scales this term is large such that 1/Ce ∼ 0 and 1/Ce ≪ 1/Cq, where Cq is the quantum capacitance. The latter term is fundamentally associated with the electronic redistribution of electrons we associate with a density of states or redox density of a given redox process and defined as25

Figure 1b), with the latter dominant for molecular systems attached to an electron reservoir (metallic electrode). Let us start with a brief review of the energy storage associated with the application of a DC voltage (Vdc) across parallel metallic plates spanning a dielectric. As depicted in Figure 2a, the initially equalized Fermi energies diverge by the

1 1⎛ 1 1 ⎞ ⎟ = 2 ⎜⎜ + Cq gr (E) ⎟⎠ e ⎝ gl (E)

(3)

Since the infinite DOS associated with a metallic plate (Figure 2a) is high, gl(E) and gr(E) ≫ 1 such that Cq tends to null and Cr is equivalent to Cq. Equation 2 is thus representative of a general capacitive analysis that extends beyond that taught by classical electrostatics. In applying this framework to an experimental configuration of more direct electrochemical relevance, we can initially note that eq 2 represents a case where a voltage is applied between two DOS (Ce will always be present; Cq will be an additional term if one or both of the sets of DOS are energetically finite and confined−situations analogous to heterogeneous and homogeneous electrochemistry, respectively). If we are specifically considering the voltammetry of a metallic electrode confined molecular film (Figure 1, panels d and e), then we have one metallic probe reservoir with an infinite DOS ρ(E) and a second “plate” now represented by a molecular film containing finite energetically addressable orbital levels with a DOS gr(E) (Figure 1d). In the expected limit of ρ(E) ≫ gr(E) (an infinite density of state in contact with limited surface state density), eq 3 becomes

Figure 2. Schematic energetic diagrams of a capacitive element comprising two equivalent plate materials. (a) The conduction bands (shaded yellow) of a macroscopic capacitor shift under an applied voltage by an amount of edV. (b) If the infinite DOS characteristics of a macroscopic conductor are replaced with the confined DOS features of a nanoscopic system this same applied voltage generates an additional perturbation in chemical potential (by dμ from an equilibrium value, μe, i.e., the black dots). The hashed red line in (b) depicts chemical potential levels of electrons after their redistribution between the plates. This field-induced energetic and spatial redistribution of electrons between coupled metallic and confined DOS (or indeed two finite confined DOS) is thus responsible for the generation of an additional capacitive term in addition to that arising from simple electrostatics. 1338

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Figure 3. (a) Nyquist impedimetric diagram of an 11-ferrocenyl-undecanethiol film in three different solvent dielectric environments [water εs ∼ 80 (yellow), acetonitrile εs ∼ 40 (green), and dichloromethane εs ∼ 10 (red)]. As can be noted, there is no apparent changes so that there is little suitable information in impedance representation as discussed in the previous work.16,22 (b) The analogous capacitative Nyquist response (obtained by using the Z* = 1/jωC* relationship to convert the complex impedance function into capacitance; see the Supporting Information and refs 16 and 22 for more detail) where the impact of solvent static dielectric constant on experimentally resolved Cr spread is clear. Ce (and therefore experimental Cr) is intrinsically associated with a dielectric constant, according to dielectric theory.26 Note that, predictably, the integrated charging (i.e., the value of redox molecular DOS) is constant. The electron transfer rate, kr, is affected as shown in (c), and the inset where an expected linear relationship of logarithmic kr and εs is clear.27 In (d), the dispersion of quantum conductance as a consequence of dielectric environment is observed. Note that the spread of DOS (quantified by σ) in (c) is consistent with Marcus theory trends,15 but its magnitude does not directly report this as will be addressed in future work. The frequency axis in Figure 3d is reported here as the base ten exponents only for clarity. In all cases, plots are of mean values from three independent measurements.

Cq(E) = e 2gr (E)

kB the Boltzmann constant, T the absolute temperature, and μe the electron chemical potential (related to the potential, V, applied with respect to a reference energy by −ejV = dμe = dE where e is elementary charge). μe is related to the electron free energy of a single electron transfer step, according to ΔG = Er − μe, where the electrical work defined by edV translates into associated change in free energy and ultimately stored chemical energy (i.e., dμe). In taking the derivative of eq 5 with respect to μe (and assuming that q = ne, where n = fΓ), Cr,15,16,24 for a single redox energy state Er, can be given by (noting that the derivative of eq 5 with respect to μe is df/dμe = f(1 − f)/kBT

(4)

Since the physical origin of Cq is the field-induced redistribution of electrons between a confined finite DOS and an underlying infinite DOS, we can examine Cq, and thus Cr, in terms of an occupation function describing the population of redox accessible orbitals (it is exactly this occupation that equates to Cq). This function, f, follows f = F(Er,μe) = n/ Γ,15,16,24 where n is the number of occupied molecular orbital centers (in equilibrium with metallic density of states) and Γ is the surface molecular density/coverage and is based entirely on Fermi−Dirac statistics such that f = F(Er , μe ) =

1 1 + exp[(Er − μe )/kBT ]

Cr(μe ) =

(5)

where Er is the redox accessible orbital energy level (directly related, of course, to the electrochemical half wave potential),

e dq df e2 Γ = e2 Γ = f (1 − f ) dμe dμe kBT

(6)

Note that eq 6 presents a maximum value of Cr when f = 1/2 and f(1 − f) = 1/4. 1339

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ferrocene film if one seeks to associate one electrode with each redox unit confined at the surface).23,29,30 Within Figure 3, the polarizing role of solution (or local) dielectric on DOS Gaussian shape is evident; specifically, we can see that a higher dielectric translates into more polarization and energetic spread.28 This directly reflects the influence of the dielectric environment on polarization of the (or electrostatic interaction with) individual molecular states that constitute the film. The influence of local and solvent electrostatics on the energy of quantized states has been integral in the work of Warshel and Levitt (recent Nobel laureates),31,32 where classical electrostatic effects have been combined with quantum mechanics in chemical reaction analysis.32 The resolved DOS shape reflects inherent thermal dispersion within Fermi energies,28 equivalent in all molecular films at the same temperature, and localized dielectric polarization effects: these are film order/solvent/environmentally sensitive and expected to be specific to a specific film in a specific medium.31,32 Furthermore, the spread of the DOS Gaussian shape can be correlated to a change in the entropy of the molecular energy levels due the dielectric effects (note the entropy of a Gaussian is given by S = ln σ(2πen)1/2, where en is the mathematical constant 2.718 so that standard deviation magnitude directly reflects entropy).33 Furthermore, it is worth noting here that in contrast to a classical electrochemical picture where the number of electrons exchanged through faradaic activity is given by n = fΓ (i.e., directly proportional to the physical coverage of energetically homogeneous redox molecules), the treatment here inherently includes a consideration of energetic spread through n = fgr(μe). Although the integrated redox DOS (i.e., the electron density) is, of course, constant in magnitude across different solutions,28 the associated energetic spread increases with εs (Figure 3a). Note that in terms of electron transfer kinetics this, experimentally supported, picture implies that the true origin of kinetic dispersion in redox-confined monolayers is through the influence of the local environment on a redox quantized DOS shape.28 The previously noted resistive term (Rct) associated with faradaic activity is directly resolvable through the observed timescale of (resonant) electron transfer (rate kr) and the equivalence of this with the reciprocal product of Rct and Cr (i.e., RctCr = 1/kr).15,16,24 The kinetics are of electron transfer being maximized where Rct is minimized (with a concomitant maximization of Cr ) at the half wave potential. The conductivity, reciprocally related to Rct, is proportional to the imaginary part of complex capacitance (Figure 3b) and is thus directly resolved from this capacitance spectroscopy. Note than the magnitude of the resonant C″ peak decreases as the redox DOS spreads in energy (Figure 3a). If one considers the resonant charging of redox levels coupled to underlying metallic states to be associated with an electron wave movement through quantum channels, we can define a single channel quantized resistance Rq as10 Rq = h/2e2 ∼ Rct (where the associated conductance is σq = L/ARq = ωC″). The peak of C″ (Figure 3b) as a function of frequency ultimately represents the resonance frequency from where kr (in reality a range of rate constants clearly exist, with a spread that is directly dependent on solution dielectric, Figure 3a). These timescales can equivalently be written RqCr = 1/kr (i.e., as a direct function of quantum resistance and redox capacitance).25 Indeed, Figure 3b also reports on a distribution of kinetics observed within any

An improved and more realistic extension of this model is one where the occupation of redox state energies are not homogeneous with discrete energy Er but described by a distribution of states, gr(μe). The redox occupancy is now given by n(μe) = fgr(μe) (instead of n = fΓ), and the redox capacitance obtained by integrating over all the contributions of available energy states, thus Cr(μe ) = e 2

∞

df

2

∫−∞ gr(μe ) dμ dμe = ke T e

B

∞

∫−∞ gr(μe )f (1 − f )dμe

(7)

In now assuming the zero-temperature approximation, this becomes Cr(μe ) = e 2gr (μe )

(8)

Equivalently, this treatment confirms not only that the experimental measured redox capacitance15,16,24 is dominated by quantum capacitance but also that it is proportional in magnitude to the redox DOS, gr(μe), since eq 8 is equivalent to eq 4. In extrapolating this to include distributed states and temperatures above absolute zero, then gr(μe), the molecular confined redox density of states,16 can be obtained as gr (μe ) =

⎡ (E − μ )2 ⎤ 1 r e ⎥ exp⎢ − ⎢⎣ σ 2π 2σ 2 ⎥⎦

(9)

where σ is the standard deviation (and σ the variance). Equation 9 defines the Gaussian format of gr(μe), the states charged by electron flux to/from the electrode. In summary, the electrochemical signature of electrodeconfined molecules is capacitive in origin and a specific case of the more generalized situation as depicted in Figure 2b (see the Supporting Information for detail on experimental procedures). The electronic coupling between an infinite metallic DOS and the energetically constrained and finite DOS of a molecular film generates, under a field induced driving force, both electrostatic and quantized capacitive energy storage terms.25 In redistributing electrons between electrode levels and energetically restricted redox states which are nonadiabatically coupled, one must consider resistive paths that are idealized “quantum channels” associated with a measurable resistive loss. Such channels themselves possess a timescale that can be derived from s/hv, where h is Planck’s constant, s is the effective electron path length, and v is the channel velocity (i.e., ultimately related to electron transfer kinetics in electrochemical nomenclature) at the Fermi energy,25 and enable the charging of the redox site “quantum cavity” (Figure 1d). The latter stores Cq (the energy storage component of the electrochemical process) while the “quantum channels” are responsible for an associated charge transfer resistance, Rct (numerically related to the magnitude of the imaginary component of capacitance C″; see below). In looking at how this model applies to an experimentally well-defined redox film (11-ferrocenyl-undecanethiol), we can immediately see (Figure 3) that Cr, with its Cq and Ce components, is frequency resolved as a Gaussian function (following eq 9) with electrode (potential) energy and that this function depends on the solvent dielectric.28 Integrating the Cr exposes the electron density (or integrated DOS) responsible for the observed faradaic activity (here, 1.37 ± 0.4 × 1014 states cm−2, a number comparable to that obtained for a pure alkyl 2

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given redox confined film, constituting a fingerprint of electron transfer kinetic dispersion.28 In conclusion, we have established herein a theoretical formulation of interfacial electronics, in which the coupling between a metallic electrode DOS and a surface molecular confined finite redox DOS (or molecular Fermi level) and the capacitive charging terms that arise when an electrochemical potential difference is considered. The dominant contribution to this resolved charging is that arising from access to the quantized levels of redox sites, Cq. We have also shown that Cq relates directly and in magnitude to an occupation function and the density of accessible states within this film. In examining this as a function of imposed electrode potential, one then has direct access to the redox site DOS and, in so doing, can directly resolve the influence of the solution dielectric on the associated energetic spread of quantized redox levels. Since these capacitive analyses are inherently frequency resolved, they also report on the kinetics of orbital occupancy change (electron resonance between the metallic and molecular DOS), the associated charge transfer resistance, and the distribution of kinetics observed within any given film. In summary, the faradaic activity of a redox (or electronically) switchable film generates a quantum capacitance that can be resolved by capacitance spectroscopy and whose form maps both electron transfer efficacy and density of states inherent in any real molecular film. We believe this spectrally resolved analysis to be valuable in both the development and understanding of nanometer scale devices and the application of films within, for example, molecular and bioelectronics.

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ASSOCIATED CONTENT

S Supporting Information *

Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. Tel: +55 16 3301 9642. Fax: +55 16 3322 2308. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the São Paulo state research funding agency (FAPESP) and UNESP grants. The authors thank Mr. Joshua Ryall and Mr. Flávio Bedatty Fernandes for supplying raw experimental data.

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REFERENCES

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