Electrochemical Shot Noise of a Redox Monolayer Simon Grall Shuo Li Laurent Jalabert Soo-Hyeon Kim and Nicolas Cl ement IIS LIMMSCNRS-IIS IRL2820 The Univ. of Tokyo 4-6-1 Komaba Meguro-ku Tokyo 153-8505 Japan

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Electrochemical Shot Noise of a Redox Monolayer

Simon Grall, Shuo Li, Laurent Jalabert, Soo-Hyeon Kim and Nicolas Cl´ement∗

IIS, LIMMS/CNRS-IIS IRL2820, The Univ. of Tokyo; 4-6-1 Komaba, Meguro-ku Tokyo, 153-8505, Japan

Arnaud Chovin, Christophe Demaille†

Universit´e Paris Cit´e, CNRS, Laboratoire d’Electrochimie Mol´eculaire, F-75013 Paris, France

(Dated: March 6, 2023)

Redox monolayers are the base for a wide variety of devices including high-frequency molecular

diodes or biomolecular sensors. We introduce a formalism to describe the electrochemical shot

noise of such monolayer, conﬁrmed experimentally at room temperature in liquid. The proposed

method, carried out at equilibrium, avoids parasitic capacitance, increases the sensitivity and allows

to obtain quantitative information such as the electronic coupling (or standard electron transfer

rates), its dispersion and the number of molecules. Unlike in solid-state physics, the homogeneity in

energy levels and transfer rates in the monolayer yields a Lorentzian spectrum. This ﬁrst step for

shot noise studies in molecular electrochemical systems opens perspectives for quantum transport

studies in liquid environment at room temperature as well as highly sensitive measurements for

bioelectrochemical sensors.

Self-assembled monolayers (SAM) composed of

nanometric-long redox molecules are building blocks

for molecular electronics and electrochemistry. They

can behave as molecular diodes operating at ultra-

high-frequency (potentially as rectenna in the visible

spectrum) [1, 2], with on-oﬀ ratio breaking the Landauer

limit [3, 4], and show interesting features such as signa-

tures of collective quantum interference eﬀects at room

temperature [5–7]. In addition, their operation in liquid

oﬀers a direct link between quantum transport and elec-

trochemistry [8–10] that provides unique opportunities.

For example, the nanoscale measurements of electro-

chemical signals remains extremely challenging while

key to the development of nanobiosensors [11]. Several

approaches have been explored to tackle the challenge,

using redox cycling [12], high frequency measurements

[13] and ﬂuorescence [14]. The underlying challenges

rise from the presence of parasitic capacitances and from

the fact that under typical measurements conditions,

the current scales with the sensor area, leading to

diﬃculties in retrieving the signal with micro- and

nanoscale electrodes. Simultaneously, these systems

oﬀer unique properties as quantum devices. Probably

the most intriguing aspect for the solid-state physics

community is the potential for millions of single-energy

level quantum dots simultaneously operating at room

temperature, with extremely small dispersion, tunable

electronic coupling [15] and Frank Condon eﬀect [16].

We propose here to exploit and formalize the shot

noise induced by reversible single electron transfers of

electroactive molecules attached to an electrode as a

new, very sensitive electrochemical technique and as a

way to characterize the homogeneity in the electronic

∗Nicolas Cl´ement: nclement@iis.u-tokyo.ac.jp

†Christophe Demaille: christophe.demaille@univ-paris-diderot.fr

properties of these assembled molecular quantum dots.

shot noise has been extensively studied in solid-state

physics [17] and more recently in molecular electronics

[18, 19], but not in electrochemistry, except for the shot

noise due to a variation of the number of molecules

in a nanogap [20–23]. Such measurements are usually

challenging because of the ubiquitous 1/f noise (e.g. in

solid-state physics [24], quantum transport [25], molec-

ular electronics [26] or in liquid [27]) which is typically

circumvented by low-temperature measurements and by

measurements at higher relative frequencies.

The 1/f noise is here not dominant thanks to the

well-deﬁned energy level and electron transfer rates

of the redox molecules of the monolayer, allowing to

study its low-frequency shot noise arising from the sum

of single-electrons trapping/detrapping events to each

molecule with a narrow distribution in time constants. A

simple and straight-forward equation of the shot noise is

proposed, giving direct access to the distribution of the

charge transfer rates and the number of charge carriers.

This approach provides clearly readable signals even

when faradaic currents become unmeasurable, avoids the

parasitic capacitance issue and allows for measurements

without extra excitation other than the thermal noise.

Electroactive redox molecules can be seen as single-

electron quantum dots with extremely small energy dis-

persion, even in liquid and at ambient temperature [5].

The equilibrium reaction of an ideally reversible redox

couple M+/M attached to a metallic electrode and held

at a distance zfrom the electrode (insets Figure 1 (a))

can be written as:

Mkox

−−*

)−−−

kred

M++ e–

Using the Marcus-Hush formalism to describe the elec-

tron transfer rates gives [28]:

arXiv:2210.12943v4 [cond-mat.mes-hall] 3 Mar 2023

FIG. 1. Illustration of current and noise versus time and voltage, considering a slow scan rate compared to the electron transfer

rates (ksum 1/tstep). (a) Ivs E, with the evolution of Ias the time after voltage step is increased. (b) Voltnoisograms

(PSD vs E) taken at low frequency (Eq. 8) corresponding to the same conditions as in (a). (c) Sampled staircase voltammetry

example, with the raw current data (black dots), a double exponential decay ﬁt of the current (red) and the voltage steps

(yellow). In this example, each voltage step is of 2.25 mV, starting at 195 mV. (d) Raw currents subtracted with exponential

ﬁts (blue). (e) PSD spectrum of one current timetrace in (d).

kMH

ox,red =ρH2

¯hrπ

kBT λ

+∞

−∞

1 + ex

kBT

e−(x−λ±η)2

4λkBTdx (1)

with kox the oxidation rate, kred the reduction rate, ρ

the density of state in the metallic electrode, H2the

electronic coupling, ¯hthe reduced Planck constant, λ

the reorganization energy (Frank Condon eﬀect due to

water molecules reorganizing after charging the redox

molecule), Tthe temperature, kBthe Boltzmann con-

stant and η=q(E−E0) with Ethe potential at the

electrode, E0the standard potential of the molecule and

qthe elementary charge. Eq.1 is analogue to the Lan-

dauer formalism in solid-state physics [8]. The speciﬁcity

of the redox molecules is their energy level broadening

due to the large reorganization energy. Eq.1 can be sim-

pliﬁed to Eq. 2 3 (Buttler-Volmer model) when |η|<< λ

which is often the case. It will be used here initially for

its simplicity.

kBV

ox =k0e−βzeαη

kBT(2)

kBV

red =k0e−βz e−(1−α)η

kBT(3)

with βthe tunneling decay coeﬃcient (1 ˚

A−1), αthe

charge transfer coeﬃcient and k0the standard electron

transfer rate at a distance z= 0 (in s−1). The exponential

decay part is formally contained in the electronic coupling

term H2in Eq. 1 but is usually extracted for convenience

to be included in the Butler-Volmer model [28].

Sampled current staircase voltammetry (SCV) is the

electrochemical technique used to interrogate the surface-

attached redox species [29], analogue to the charge pump-

ing technique in semiconductors [30]. The electrode po-

tential is raised in small steps of height Estep, and the

current is recorded as a function of time, up to a time

tstep, corresponding to the steps duration (Figure 1 (c)).

The current I, in the case of slow scan rate and long

sampling time (i.e. ksum =kBV

ox +kBV

red 1/t), Ican be

expressed as (details in SM):

I=Nqν

4kBT

cosh2(η

2kBT)(4)

with Nthe total number of molecules and ν=Estep/tstep

the voltage scan rate. Note that such current represents

the transition of the charges at a certain scan rate,

and not an equilibrium value of the current at a given

potential. Figure 1 (a) shows Iversus applied voltage

Eat a given scan rate and at diﬀerent times tafter the

voltage step, exhibiting a quick decrease of amplitude.

One way to consider the noise of the current versus

time (Figure 1 (b)) is to look at its power spectrum den-

sity (PSD, noted Sin equations). The PSD (Figure 1 (b)

and (e)) can be seen as a description of how the variance

of the measured signal is spread in the frequency domain.

The system under study has two-states related to the ox-

idized/reduced states of the molecules, here attached to a

single electrode. Each molecule is expected to lead to the

so-called Random Telegraph Signal (RTS) which is a shot

noise due to individual transfer of electrons in and out of

the single-electron boxes. To avoid confusion, in such a

single-electrode system, the shot noise is not expected to

be compared to 2qI because at equilibrium, where both

oxidation and reduction reactions compensate each other,

I= 0 while S > 0 (discussion in SM) [31]. In general,

RTS is typically associated with 1/f noise due to the wide

range of energy levels and electron transfer rates[32, 33].

However, an ensemble of reversible redox couples, like

those found in a redox SAM in liquid, can be thought

of as an ensemble of quantum dots with very similar en-

ergy levels because the molecules that make up the SAM

have strictly identical atomic structures and may diﬀer

only in their orientation relative to the surface[5]. As-

suming ﬁrst that all Nmolecules have identical energy

levels E0and charge transfer rates kox/kred for oxida-

tion/reduction, respectively, the PSD can be expressed

as [32]:

S(f, η, N)=4N∆I2koxkred

kox +kred

(kox +kred)2+ (2πf)2

(5)

with fthe frequency and ∆Ithe current corresponding

to the oxidation (or reduction) of one molecule. If we

consider ∆Ias the transfer of one electron of charge q

per the average time taken for transferring one electron

(i.e., ∆I=q

kox +1

kred

), Scan be rewritten as:

S(f, η, N)=4Nq2(koxkred)3

(kox +kred)3

(kox +kred)2+ (2πf)2

(6)

which becomes at low frequency (assuming α= 0.5):

lim

f→0S(η, N)=4N q2(koxkred)3

(kox +kred)5(7)

8Nq2k0e−βz

cosh5(η

2kbT)(8)

0.1 0.2 0.3 0.4 0.5 0.6

-4

-3

-2

-1

j (mA/cm2)

E (V)

9 mV/s

4.7 mV/s

2.4 mV/s

1 mV/s

FIG. 2. Example of current CVs obtained at diﬀerent ν(elec-

trode area ≈45 mm2,Evs Ag/AgCl (3 M NaCl), electrolyte:

[NaClO4]=0.5 M).

This equation expresses the dependence of the low fre-

quency electrochemical shot noise of the redox SAM ver-

sus the electrode potential. The corresponding curve is

plotted in Figure 1 (b). Similarly to current SCV signals,

it presents a peak at E0, but narrower than that of the

SCV peak, with a full width at half-maximum (FWHM):

F W HM = 4 acosh( 5

√2)kBT

q≈56 mV (9)

Note that unlike the current, Sdoes not depend on ν

as the PSD is considered for a system at equilibrium. S

is also independent of the potential scan direction. In-

terestingly, the limiting cases of η→0 and f→0 give

access to the electron transfer rate k0and the total num-

ber of molecules N.

lim

η→0S(f, N) = 1

2Nq2k0e−βz 1

4+( 2πf

k0e−βz )2(10)

with the corner frequency fc=1

2πk0e−βz:

lim

η→0,f→0S(N) = 1

8Nq2k0e−βz (11)

The main result of the present work is Eq. 11, linking

directly and simply k0and Nto the noise measured

at low frequency for E=E0. Provided the corner

frequency of the PSD fccan be measured (Figure S1

(b)), the individual values of k0and Nare obtained

from Eq. 10 and Eq. 11. Alternatively, if Nis known

independently, k0can be straightforwardly derived from

Sat E=E0(Eq. 11).

To demonstrate the validity of the previous analy-

sis, an experiment is set using ferrocene undecanethiol

Fc(CH2)11SH self-assembled on a gold microelectrode. A

two-electrode electrochemical cell setup is used in a Fara-

day cage, using a [NaClO4]=0.5 M aqueous electrolyte

and a Ag/AgCl electrode (3 M NaCl) acting as both ref-

erence and counter electrode. Details about the sample

preparation and the measurement setup can be found in

Supplementary Materials (Figure S2 and S3). The sys-

tem is interrogated using staircase voltammetry (Figure

1 (c)), which is equivalent to linear cyclic voltammetry

(CV) at slow scan rates [34]. Our motivation is to oﬀer a

comparison of the well-known technique of cyclic voltam-

metry with the results obtained looking at the shot noise

of the system.

The Figure 2 shows an example of current CVs at dif-

ferent (low) scan rates ν. The signal is centered around

a potential value of E0= 0.35 ±0.02 V vs Ag/AgCl,

which corresponds to the expected standard potential

for such surface-attached Fc molecules [35–37]. The den-

sity is estimated here at 4.2×10−10 mol/cm2, close to

the values reported in the literature for packed SAMs

(4.4∼4.9×10−10 mol/cm2) [1, 38]. The peak current of

the CV exhibit the usual behavior for a surface-conﬁned

reversible couple, with a linear dependency of the current

versus ν(example data Figure S11).

PSD signals were measured at several scan rates (see

details in SM), their magnitude at 20 Hz versus E

(called “voltnoisogram” for concision) shown in Figure

3 (a), (full set Figure S12). Similar data without the

Fc molecules can be found in Figure S10. The PSD volt-

noisograms behave as expected with a peak-shaped curve

centered around E0≈0.35 V, close to the standard po-

tential of Fc. As predicted from Eq. 8, the peak value

of the PSD voltnoisograms (Figure 3 (b)) remains quasi-

constant for ν < 3 mV/s (see details in SM Figure 3).

Figure 4 (a) shows a cyclic voltammetry (CV) scan at

ν= 0.014 mV/s where no faradaic current signal can

be identiﬁed. Figure 4 (b) shows power spectral density

(PSD) values at E < E0,E≈E0and E > E0on the

forward scan. Figure 4 (c) shows the variation of PSD at

10 Hz as a function of the voltage, showing a clear peak.

These results demonstrate the ability to detect an elec-

trochemical reaction at an electrode through shot noise

measurements, even when the average current signal

from the CV does not show any reaction. The number of

molecules N= 7.5×1010 is calculated from the CV data

at higher scan rates (Figure S11), and using this value

and Eq. 11, the peak amplitude of PSD data shown in

Figure 3 yields k0= 6.3×107s−1(z= 1 nm) is in good

agreement with literature values for this molecule[15, 20].

The FWHM of the PSD peaks on Figure 4 is ≈90 mV,

broader than the 56 mV predicted by Eq. 9. Taking

η= 0 in the Eq. 1 gives an expression for k0, notably

showing dependencies with Hand λ[28]. Previous work

10-5 10-4 10-3 10-2

10-25

10-24

10-23

10-22

PSD peak @ 20 Hz (A2/Hz)

(V/s)

0.2 0.25 0.3 0.35 0.4 0.45 0.5

10-25

10-24

PSD @ 20 Hz (A2/Hz)

E (V)

2.7 mV/s

0.12 mV/s

0.014 mV/s

FIG. 3. (a) PSD of the current versus Eobtained at diﬀerent

ν, at f≈20 Hz. (b) PSD at f≈20 Hz and E= 0.35 V

versus ν.

[15] showed that variation of lambda within physically

reasonable limits do not signiﬁcantly impact the electron

transfer rates. However, the electronic coupling term H

typically varies following lognormal distributions [5] and

can impact signiﬁcantly the resulting value of k0with

variations of just a few percent of its average value (see

Figure S13 & S14). If a lognormal distribution of the

ﬂuctuation of H(and thus, of k0as well) is assumed, a

standard deviation of σ≈4%, comparable to what was

reported in ref [5] (≈2%), can explain the broadening

observed in PSD (Figure 4 (c) dashed line).

There is a signiﬁcant diﬀerence between molecular

monolayers in liquid and solid-state devices in terms of

electrostatic forces. In the ﬁrst case, the electrostatic in-

teractions between neighboring molecules are greatly re-

duced thanks to the high permittivity of water. Previous

research on π−πcoulomb repulsion ϕ(Eq. 12) within

similar Fc SAMs showed negligible impact on current CV

[5, 39].

100101102

10-27

10-25

10-23

10-21

10-19

PSD (A2/Hz)

f (Hz)

E<E0 (0.2 V)

E=E0 (0.37 V)

E>E0 (0.5 V)

0.2 0.25 0.3 0.35 0.4 0.45 0.5

I (pA)

E (V)

= 0.014 mV/s

0.2 0.25 0.3 0.35 0.4 0.45 0.5

2x10-25

4x10-25

6x10-25

8x10-25

PSD @ 10 Hz (A2/Hz)

E (V)

FIG. 4. (a) CV of a FcC11SH SAM on gold. (b) PSD mea-

sured at 0.014 mV/s at E < E0,E≈E0and E > E0.

using Eq. 7 considering a lognormal distribution of Hwith

σ= 3.97% of the mean value of H.

ϕ=q1−1+(ra

d)2−0.5

4πε0εrd(12)

with dthe intermolecular distance, rathe distance to

counter ions, ε0the permittivity of the vacuum and

εr=εH2O|εSiO2the permittivity of the medium under

consideration. Taking the same formalism and distri-

bution of das in [5] for ϕbut changing εH2O= 79 to

εSiO2= 3.9 results in variations of ES

F W HM from 1% to

45% respectively (see Figure S15 & S16). As a result

the screening of electrostatic interaction by water avoids

a dispersion of the energy levels, such as the one ob-

served in nanotransistors [33], and thus, avoids the dom-

ination of a 1/f noise resulting from the sum of multiple

Lorentzian spectra with diﬀerent amplitude/corner fre-

quencies.

In conclusion, we demonstrated the measurement of

the shot noise generated by an ensemble of surface-

attached Fc redox molecules, which can be seen as identi-

cal single-electron boxes, in liquid and in ambient condi-

tions. A formalism is proposed to understand it and ex-

hibit dependencies between such noise and electronic cou-

pling. This constitutes a further step toward nanoelec-

trochemistry and single molecule measurements, which

could be practically achieved using our technique com-

bined with a transducer such as a nanotransistor and be

extended to other systems such as quantum dots mono-

layer [40].

Our technique allows for the measurement of electron

transfer rates at low frequencies without the need for

highly time-resolved instrumentation. Although we com-

pared our technique with traditional voltammetry tech-

niques, exhibiting a clear signal in PSD when Itended to

zero, the very concept of “potential scan” is actually not

required to perform noise measurements. As few as two

points at potentials far from E0and one at E0can suﬃce

to resolve the eventual background noise of the experi-

ment and the noise due to the attached molecule, yielding

k0and Nprovided the knowledge of βand z. Concur-

rently, since the measurements is carried out at equi-

librium, capacitive contributions are altogether avoided,

improving the signal and simplifying drastically the in-

terpretation of the data. This opens perspectives in the

ﬁeld of biosensors [11], where the limit of detection of ex-

isting techniques could be further extended by shot noise

analysis, and in high-frequency molecular diodes, where

the electron transfer rate can be estimated through the

low-frequency noise.

ACKNOWLEDGMENTS

This work has been supported by the EU-ATTRACT

project (Unicorn-Dx), the French “Agence Nationale

de la Recherche”(ANR) through the “SIBI” project

(ANR-19-CE42-0011-01) and the JSPS Core-to-Core

Program (JPJSCCA20190006).

S.G designed the acquisition system, conducted the ex-

periments and data analysis and developed the theory,

S.L. fabricated the devices, L.J. designed the acquisition

system, SH. K. and A.C. contributed to the scientiﬁc in-

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ElectrochemicalShotNoiseofaRedoxMonolayerSimonGrall,ShuoLi,LaurentJalabert,Soo-HyeonKimandNicolasClementIIS,LIMMS/CNRS-IISIRL2820,TheUniv.ofTokyo;4-6-1Komaba,Meguro-kuTokyo,153-8505,JapanArnaudChovin,ChristopheDemailleyUniversiteParisCite,CNRS,Laboratoired'ElectrochimieMoleculaire,F-75013Paris,...

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Electrochemical Shot Noise of a Redox Monolayer Simon Grall Shuo Li Laurent Jalabert Soo-Hyeon Kim and Nicolas Cl ement IIS LIMMSCNRS-IIS IRL2820 The Univ. of Tokyo 4-6-1 Komaba Meguro-ku Tokyo 153-8505 Japan

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