The Mesoscopic Electrochemistry of Molecular Junctions

Within the context of an electron dynamic (time-dependent) perspective and a voltage driving force acting to redistribute electrons between metallic and addressable molecular states, we define here the associated electron admittance and conductance. We specifically present a mesoscopic approach to resolving the electron transfer rate associated with the electrochemistry of a redox active film tethered to metallic leads and immersed in electrolyte. The methodology is centred on aligning the lifetime of the process of electron exchange with associated resistance and capacitance quantities. Notably, however, these are no longer those empirically known as charge transfer resistance and pseudo-capacitance, but are those derived instead from a consideration of the quantum states contained in molecular films and their accessibility through a scattering region existing between them and the metallic probe. The averaged lifetime (τr) associated with the redox site occupancy is specifically dependent on scattering associated with the quantum channels linking them to the underlying metallic continuum and associated with both a quantum resistance (Rq) and an electrochemical (redox) capacitance (Cr). These are related to electron transfer rate through k = 1/τr = (RqCr)−1. The proposed mesoscopic approach is consistent with Marcus’s (electron transfer rate) theory and experimental measurements obtained by capacitance spectroscopy.


SI. 1.2. Electrochemical Capacitance Spectroscopy Measurements and Summarized Theory
All electrochemical measurements were undertaken on an AUTOLAB PGSTAT fitted with an FRA2 module. A three electrode cell setup was used with a gold (METROHM) working electrode, a platinum wire auxiliary electrode and a Ag|AgCl as reference electrode, providing a half wave potential of ferrocene molecule at about 0.45 V with respect to the reference, where electrochemical capacitive response is expected to maximize. [2][3][4] Impedance spectra were collected between 1 MHz and 0.01 Hz with amplitude of 10 mV (peak to peak). All the impedance spectra were subsequently verified for compliance with linear systems theory by Kramers-Kronig by employing the FRA AUTOLAB software. The ECS analysis of these interfaces can be sensitively analysed measuring complex * ( ) (impedance) function and conversion into * ( ) (capacitance) by * ( ) = 1/ * ( ), where is the angular frequency and = √−1 (i.e., complex number). Herein we then indicate that an asterisk refers to complex conjugate functions, for instance, * ( ) = ′ + ′′ where ′ and ′′ are, respectively, the real and imaginary parts of the complex capacitance function. Therefore, practically ECS procedures involves taking the data resolved in a standard impedance analysis ( * ( )), sampled across a range of frequencies at any steady-state potential, and converting it phasorially into complex capacitance [ * ( )] with its real and imaginary components. In processing * ( ) datasets in this way one obtains the imaginary part of the capacitance by noting that ′′ = ′ and real part as ′ = ′′, where = ( | | 2 ) −1 and | | = √( ′′ ) 2 + ( ′ ) 2 is the modulus of * . If one carries out this analysis inside of the surface potential window where redox activity is observed and then outside the potential window, the difference (between the two impedance measurements) is obtained as the pure electrochemical response constituting of a series resistance and capacitance 1,4 , as fitted in Figure 2a and 2b. It should be noted that, in the absence of a redox film, charging capacitance is comparatively very small for higher molecular coverage and can, alternatively, be estimated by a simple subtraction of the former. 1,4 Plotted data points (in Figure 3) represent mean and standard deviations values across (at least) three different measured junctions (molecular film/electrode).
In more detail Electrochemical Capacitance Spectroscopy (ECS) 1 is a methodology based on the resolved ratio between a time-dependent current response and an imposed small amplitude voltage applied to molecular scale films. We have previously shown that this enables a quantification of an electron chemical (redox) capacitance and its associated quantum components within energetically addressable molecular films, 2,3,5 and that these charging characteristics, in the past loosely ascribed to "pseudo capacitance" (or faradaic capacitance), 5  consideration of quantum capacitance in devices comprising an interphase boundary of metal with a two-dimensional electron gas system. It is also resolved free of non-faradaic capacitive components, i.e. those not associated with redox state occupancy 2,5 . We have shown that is comprised of two distinct series contributions thus 1/ = 1/ + 1/ , where is the electrostatic or geometrical capacitance arising from charge separation in the normal (classical) sense and the quantum capacitance, arising very specifically from the chemical potential changes associated when charging a nanoscale (atomic/molecular scale) entity. 2,5 As the electrochemical capacitance , as an additional chemical interfacial charging, can be spectrally resolved, as demonstrated previously here, its applications into molecular films are useful in sensorial applications and devices [8][9][10] . Since this interfacial charging depends on the quantum mechanical coupling of the electrochemical states and electrode/metallic states, the former spanning a Gaussian in distribution (at finite temperature), one can quantify by integrating over all contributing energy levels thus where, ( ), the redox density of states can be written as     Figure 4a of ( ̅ ) from where was obtained by taking the resonant frequency of 20 Hz. Figure 4b was then calculated as = (2 20 Hz) ( ̅ ). The behavior of the natural logarithm of the electron transfer rate as a function of potential is shown in Figure 4c and was calculated by observing that ( ̅ ) = / ( ̅ ) at 20 Hz.

SI. 2.1. Quantum Transport (in time-independent regime)
The physics associated with molecular scale electronic 11 Figure 1a of the main text. Therefore, the experimentally studied devices contain narrow quasi-onedimensional obstructions to conduction and so that the conductance of such systems is generally quantised. The electrons in such quasi-one-dimensional conductor occupy a set of sub-bands (or quantum wells) due to a quantisation of the electronic states in the transverse direction of the quantum wells. In assuming there is no scattering of electrons between sub-bands or quantized wells, the electrons in each sub-band (or quantized energies) can be considered as a one-dimension electron gas. Therefore, in a one-dimensional DC conductive regime the density of states at energy, ̅ , for electrons propagating in one direction is where is the length of the conductor and is the wavenumber of the electronic states. The group velocity of the electrons that have energy is In considering that all the electronic states propagating to the right in the energy range between ̅ and ̅ + Δ ̅ are occupied and all states propagating to the left are unoccupied the current carried by the electrons in the energy ranges of Δ ̅ is given by which is independent of ̅ and now considering that the energies over which all the right-propagating (from lower to high potential in Figure 1a and Figure 1c) electronic states are occupied and the left-propagating states are unoccupied were equal to , where is the potential (or voltage difference) across the conductor (see Figure 1a and Figure 1c), the conductance of the system would be  The problem with Eqn. (SI. 6) is that it does not take into account self-consistency in a way that a more appropriate formulation of the problem that take into account the self-consistency was proposed by Buttiker 13 and today is known as Landauer-Buttiker formalism where the sum is over all incident channels and , and are the longitudinal velocity, reflection coefficient and transmission coefficient for channel .

SI. 2.2. Lifetime in Collision Theory and Electrochemistry
The time-dependent analysis of the capacitance in redox molecular films in the present work and in other from us [1][2][3][4] has been focused on the mesoscopic characteristics of organic films. Herein specifically we emphasize the use of time-dependent transport to gain information on the charge transfer and transport characteristics of the redox film and its collective (ensemble) dynamics, i.e. multiple redox states connected to or resonating with metallic states of the electrode (which implies conformational as well). It means the treatment is not concerned with stationary time-independent regime (as shown in Figure 1a) of the atomic or molecular structures involved, but in driving them from one state to another (for instance, in linearly perturbing the redox couple, the Fe +2 /Fe +3 in the molecular monolayer) and doing this in different frequencies in a single probe format as indicated in Figure 2b.
Therefore, the duration of the redox process, characterized by perturbing the electrons from metallic probe to redox states in the molecule, can be treated equivalently as the duration of a collision in the collision theory (the mathematical treatment is equivalent since the collision theory is the precursor theory for the collision frequency in the chemical reactions). This approach was first introduced by Markus Buttiker to describe the concept of mesoscopic capacitance. 6 In following the seminal works of Eisenbud 15 , Wigner 16 and Smith 17 it has been noted that in collision theory the lifetime matrix ( ) is directly correlated with scattering matrix ( ). The problem is treated quantum mechanically, using steady-state wave functions; the average time of residence in a region is the integrated density divided by the total flux in (or out), and the lifetime is defined as the difference between these residence times with and without interaction and the transformation properties require construction of the lifetime matrix which was proved by Smith to be exactly the same as the lifetime . This proof provides the clue to a general relationship between the scattering matrix and the lifetime matrix . If is written as = then This is a generalized expression and is the scattering matrix for elastic or inelastic collision and is the lifetime matrix no matter the process is elastic or inelastic and was proved by Smith 17 in the space (in the one-dimension case). The case we are working on is exactly the one-dimension accordingly as presented in Figure 2b and as shown in the main text, in the context of molecular electrochemistry, we have considered that this situations is governed by an RC circuit where = and then where is the relaxation resistance as discussed and introduced in the main text and is the redox capacitance as discussed in previous works. [1][2][3][4]

SI. 2.3. Electron Density and Capacitance
Very recently we have shown the relationship between capacitance spectroscopy (CS) and density function theory (DFT). 5 In DFT, the fundamental variable associated with all observables is, of course, the electron density, i.e.
= ∑ 2 , where can be expressed as a linear combination of Kohn-Sham orbitals. Therefore, in considering the mathematical methodology associated with Kohn-Sham formulation of finite-temperature density function theory, 18 the energy defining equations arise as 19 where i.e., effectively by considering the shape of redox capacitance per unit of volume of the molecular film, . This electrochemical/redox capacitance is a function of ̅ , i.e. the electrochemical of electrons in the metallic states. Note also that ̅ is related to a change in electrochemical potential ( ) of the electrode by ̅ = − as mentioned in section 1.2.
Remembering that (− ) = − / (see section 1.2). The corresponding version of Eqn. (S1. 14) to calculate the number of electrochemistry states, , directly from electrochemical capacitance as a function of potential at finite temperatures is Therefore, practically is a quantity easily obtained by the integral of redox capacitance as a function of electrochemical potential of the electrode as indicated in Figure 4a of the main text. The obtained pattern is a Gaussian shape indicated as discussed in previous works. [1][2][3][4][5] Note that in the zero temperature limit ( ̅ ) = 2 ( ̅ ) so that = ∫ ( ̅ ) ̅ = ∫ 2 ( ̅ ) ̅ as indicated in the main text.

SI. 2.4. Electrochemical States (or Density) and the Quantum Transport
In considering the redox states coupled to metallic single probe are continuum it can be observed that the redox charge is coupled to the redox capacitance by = (SI. 16) where is the charge associated with the redox process (redox reaction) only. In considering infinitesimal variation in time (at steady-state condition) of Eqn. (SI. 16) we obtain / = / which is equivalent to / = / , where is the redox current (faradaic only 2,5,20 ). In adopting ̅ = − and with some rearrangement And by comparison with Eqn. (2) and Eqn. (4) it can be noted that in an ensemble of continuum quantum channels = ∫ ̅ the conductance is proportional to the density of quantum/redox channels. Precisely, for a continuum redox density of states (as is the case of the molecular junctions studied in this work 5 ), this is equivalent to assume that each redox state is associated with a quantum conducting channel (bridging the redox molecular states and those of the metallic probe).
In summary, the redox/electron density obtained in the electrochemistry of molecular (single probe) junctions is equivalent to the conductance through by considering the charge transfer rate involved and connecting the metallic states to those of redox states.