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Irreversibility and hysteresis in redox

molecular conduction junctions.

Agostino Migliore*,†,‡ and Abraham Nitzan*,†

† School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel. Phone: +972-3-6407634.

Fax: +972-3-6409293.

‡ Present address: Department of Chemistry, Duke University, Durham, NC 27708, USA.

Phone: +1-919-6601633.

* E-mails: migliore@post.tau.ac.il and nitzan@post.tau.ac.il

CORRESPONDING AUTHOR: Agostino Migliore. School of Chemistry, Tel Aviv University, Tel

Aviv 69978 Israel. Current address: Department of Chemistry, Duke University, Durham, NC 27708,

USA. Phone: +1-919-6601633. E-mails: migliore@post.tau.ac.il, agostino.migliore@duke.edu.

ABSTRACT

In this work we present and discuss theoretical models of redox molecular junctions that account for

recent observations of nonlinear charge transport phenomena, such as hysteresis and hysteretic negative

differential resistance (NDR). A defining feature in such models is the involvement of at least two

conduction channels - a slow channel that determines transitions between charge states of the bridge and

a fast channel that dominates its conduction. Using Marcus’ theory of heterogeneous electron transfer

(ET) at metal-molecule interfaces we identify and describe different regimes of nonlinear conduction

through redox molecular bridges, where the transferring charge can be highly localized around the

redox moiety. This localization and its stabilization by polarization of the surrounding medium and/or

conformational changes can lead to decoupling of the current response dynamics from the timescale of

the voltage sweep (that is, the current does not adiabatically follow the voltage), hence to the

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appearance of memory (thermodynamic irreversibility) in this response that is manifested by hysteresis

in current-voltage cycles. In standard voltammetry such irreversibility leads to relative shift of the

current peaks along the forward and backward voltage sweeps. The common origin of these behaviors is

pointed out and expressions of the threshold voltage sweep rates are provided. In addition, the theory is

extended (a) to analyze the different ways by which such phenomena are manifested in single sweep

cycles and in ensemble averages of such cycles, and (b) to examine quantum effects in the fast transport

channel.

KEYWORDS: molecular electronics · redox molecular junctions · Marcus theory · hysteresis ·

hysteretic NDR.

INTRODUCTION

Redox molecular junctions, that is junctions whose operation involves two or more oxidation states of

the molecular bridge, have attracted great interest because of their ability to manifest nonlinear effects

in the current-voltage response1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 that are relevant to nanoelectronics, and to provide

control mechanisms based on the connection between the charging state of the molecule and its

conduction properties.1, 2, 3, 5, 9, 11

In a redox molecular conduction junction, the localization of the transferring charge around the redox

center and its stabilization by suitable polarization of the nuclear environment can lead to weak

coupling strengths to the contacts and, as a consequence, to switching between different molecular

charging states by means of sequential ET processes.13 As noted in ref 13, the existence of two (or

more) locally stable charge states is not sufficient to characterize a molecular junction as redox type.

Switching between them by repeated oxidation-reduction processes simply leads to current that depends

on this switching rate. A prerequisite for redox junction behavior, often manifested by the appearance of

negative differential resistance (NDR), hysteresis and hysteretic NDR, is the presence of a second

transport channel whose conduction is large enough to determine the observed current on the one hand,

and is appreciably affected by changes in the redox state of the molecule (caused by relatively slow

electron exchange through the first channel) on the other. Such a mechanism characterizes recent single

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electron counting measurements in quantum point contacts14, 15, 16 and has also been proposed17 as the

physical basis of NDR in spin-blockaded transport through weakly coupled-double quantum dots. While

NDR and its dependence on the temperature and the nuclear reorganization after ET were the focus of

the work in ref 13, the present work also considers the occurrence of hysteresis and hysteretic NDR in

weakly-coupled redox junctions.

The paper is organized as follows. In next section we analyze the common underlying mechanism of

irreversible effects that appear in standard voltammetry employed at single metal-molecule interfaces

and hysteresis in the current-voltage response of metal-molecule-metal junctions. This analysis is then

extended to redox molecular junctions characterized by two interacting, fast and slow, charge-transport

channels, described by three or four molecular states models. Charge transfer kinetics in the slow

channel can be safely described by sequential Marcus rate processes.18, 19, 20 Charge transfer through the

fast channel that dominates the junction current is described either using Marcus rates or as resonant

tunneling according to the Landauer-Büttiker formalism.21, 22

RESULTS AND DISCUSSION

Irreversible voltammetry and hysteretic conduction in a two-state model.

In what follows we refer as irreversible current-voltage response the evolution of a junction that does

not reverse itself when the voltage sweep is reversed. Such irreversible evolution occurs when the

intrinsic charge transfer timescale (measured, e.g., by

1

ρ , eq 3b below) is slow relative to the voltage

sweep rate, so that the current cannot adiabatically follow the instantaneous voltage. Obviously,

irreversibility in a solvated molecular junction (a double molecule-metal interface) and in cyclic

voltammetry under diffusionless conditions23 must have a similar underlying mechanism, still such

studies have progressed separately so far. Several comparative observations such as (i) the behavior of

single molecule conductance against the need for a molecular layer to obtain appreciable current from a

volammogram24, 25 and (ii) the appearance of irreversibility in voltammetry involving diffusionless

molecules at sweep rates lower than those required for observable hysteresis in redox junctions, can be

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explained by addressing them together. One aim of the following analysis is to relate and explain such

phenomena, by affording a common language for their description.

We start by considering the simplest molecular model: a two-state system, an oxidized molecular

form A and a reduced form B, where transitions between them take place by simple rate processes. The

transition rates A → B and B → A (electron injection into and removal from the molecule, respectively)

are denoted by

BA AB

RR

and

AB BA

RR

, respectively. We denote by

A

P and

B PP

the

probabilities to find the molecule in state A and B, respectively, and by

eq,A

P

and

eq

P their equilibrium

values. Obviously,

1

BA

PP

and

BAABA

RPRP

eq eq,

(detailed balance). Under a time dependent

voltage

)(tV

these probabilities can be written as26

),(

)()(

)(

),()(),(

eq,

tVQ

VRVR

VR

tVQVPtVP

BAAB

BA

AA

(1a)

and

),(

)()(

)(

),()(),(

eq

tVQ

VRVR

VR

tVQVPtVP

BAAB

AB

, (1b)

where the departure Q of P from

eq

P depends explicitly on the time t. All the memory effects in the

response of the system to the external bias V can be encapsulated in the function Q, which is obtained as

follows: the master equation

Q)RRQPRQPR

V(PRV(RP

dt

dQ

dV

dP

u

dt

ttV(dP

BAABBBAAAB

BAAB

()()(

))1

)),(

eq,eq,

eq

(2)

where

dt dVu

is the rate of the voltage sweep, is rewritten as

dV

dP

ρ

u

Qρ

dt

dQ

eq

, (3a)

where

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BA AB

RRρ

(3b)

is the effective rate that characterizes the system relaxation after changing the external voltage. For

constant u or over a time interval in which u does not change appreciably this leads to

'

eq

) '

t

(

)(

0

0

0

')()(

t

t

t

tρ

ttρ

dV

dP

edtuetQtQ

(

0tt

) (4)

The transient associated with

)(0tQ

can be disregarded at long time. If u is small enough so that

dVdPeq

remains essentially constant during a time interval comparable with ρ1

, eq 4 results in

dV

dP

ρ

u

tV

(

Q

eq

),

(5)

Eq 5 describes a steady-state value of Q: the difference

eq

PPQ

remains very close to zero while V

is slowly changed.27 Then, at a single molecule-metal interface under reversible conditions the current I

between the metal and the molecule is proportional to u and is given by

)(

eq

eq

PPρρQ

dV

dP

u

e

I

J

(6)

where e is the magnitude of the electron charge.

Irreversibility manifests itself in accumulation of Q during part of a voltage sweep and inversion in

the sign of Q in the backward sweep, with consequent hysteresis over a cycle. The general requirement

for reversible behavior at any V is obtained from eqs 1 and 6 as

1

eq

eq

eq

P

Q

dV

dP

ρP

u

(7)

and can be extended to models of single or double metal-molecule interfaces that include more than two

system states (see next section).

For a molecule stably adsorbed on a single metal electrode that can be characterized as a semi-junction,

eq 2 or 6 can be used to describe on a “per molecule” basis28, 29, 30 the current at a molecule-electrode

interface, as it appears in typical linear scan (cyclic) voltammograms of diffusionless redox systems. In

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such a system, the applied overpotential V operates as a gate voltage, effectively changing the position

of the molecular level relative to the metal Fermi energy. The surface concentration of the electroactive

species in the reduced (oxidized) state is replaced by the occupation probability

PPB

(

PPA

1

) of

the molecular redox site, and its time derivative yields the charge flow originating from the change in

the oxidation state of the molecular system.31 A fast enough voltage sweep leads to the hysteretic

behavior of P shown in Figure 1a, which was obtained by implementing eq 2 in a finite difference

simulation and describing the interfacial ET according to the Gurney32-Marcus model, as in ref 29, but

with the ET rates in the analytical form reported in the Appendix. We assign V as positive when the

electrostatic potential on the molecule is higher than that in the metal,13 so that electrons flow from the

metal, making it identical with the negative of the traditional definition of the overpotential

Figure 1. (a) The occupation probability P of the molecular site plotted against the interfacial voltage V

over a cycle, with a maximum voltage

V5 . 0

axm

V

. The forward and backward sweeps are in black and

red, respectively. The system is modeled by the parameters

K 298

T

,

eV 15. 0

μEAB

, and

eV25. 0

λ

. The initial condition is

) 0 (

eq

P ) 0 (P

. The scan rate is given by

V 10

3

u

. For

example,

sV 10

u

for

14s10

or

sV 100

u

for

15s10

(b) The dimensionless current at the

molecule-metal interface,

)( dt

dPJ

, plotted against V using eq 2. The solid lines are obtained

with the same parameters as in panel a, in particular

eV25. 0

λ

. The dashed lines correspond to the

same parameters, except that

0

λ

.

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Figure 1 relates the splitting of the peak potentials to the hysteresis in P. Such a connection, which is

not explicitly considered in theoretical analyses of electrochemical redox reactions and voltammetry, is

used here to link the peak splitting observed in single irreversible cyclic voltammograms (see Figure 1b,

where the average response of the single adsorbed molecule over many sweeps can be compared to the

voltammogram for a molecular layer) and the hysteretic I-V characteristics of redox junctions.

As investigated both theoretically33, 34 and experimentally,35 the time derivative of

eq

P in eq 6 departs

from the ideal behavior (characterized by a peak of size

Tkue

B

4

that occurs at equal potentials in the

upward and downward scans) when the voltage scan rate is comparable with the ET rate20 as quantified

by the dimensionless kinetic parameter28

euTkkm

B

0

(8)

where

) 0( ) 0(

0

VRVRk

BAAB

with

μEAB

. For the purpose of this work it important to

describe this distortion (and the associated hysteresis in the redox state of the molecule) in terms of a

kinetic parameter that lends itself to generalization and use within the context of redox molecular

junctions. In particular, the explicit dependence on the reorganization energy λ and on the voltage V

needs to be expressed. Our aim is to provide a criterion for the first appearance of hysteresis. To this

aim, we consider that P experiences the largest rate of change, hence the first appearance of hysteresis

(see Figure 1a), about the voltage

eμEV

AB

)(

0

at which

BAAB

RR

(see eq 34 and Figure 2). At

this voltage, the effective rate ρ can be written as (see Appendix and ref 36)37

)(

4

exp

) 0(

),,;(

0

Tkλ

Tk

λ

λ

Tkπ

λ

TλVρ

B

B

B

(9)

Near

0

V , ρ decreases with the reorganization energy (for example, see Figure 5 in ref 38), so that eq 6

can become invalid at any feasible scan rate and irreversible behavior is observed. In contrast, the

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condition for reversibility can be much more easily satisfied at usual sweep rates (up to ~

sV 100

) for

0

λ

, namely, for

ρ

. This is exemplified in Figure 2: for

eV25 . 0

λ

, the evolution of P over a

cycle of the applied voltage is characterized by hysteresis with a corresponding splitting of the peak

voltages in Figure 1b, whereas no hysteresis occurs for

0

λ

. This can be quantified: inserting the

expression of the maximum current (in reversible regime),

Tkue

B

4

, and eq 9 into eq 6, and imposing

the condition

21

Q

for reversibility, we arrive at the limiting sweep rate

),,;(2),,;(

00

TλVρ

e

Tk

TλVu

B

l

, (10)

such that reversible behavior is obtained if

luu

, while hysteresis in the evolution of P over a voltage

cycle, hence distortion of the voltammogram, takes place if

luu~

.

Figure 2. (a)

BA

RY

(red),

AB

R

(blue) and

eq

P (black) plotted against V, and P along the

forward (gray) and backward (pink) voltage sweeps, by using the same model parameters as in Figure

1a. (b) The same quantities as in panel a are shown for the case

0

λ

(as in Figure 1b).

While eq 10 yields a delimiter between the sweep rate ranges with reversible and irreversible behavior

based on the first appearance of hysteresis at

0

VV

, its extension to all voltages is obtained by

application of eq 7 and use of the rate expressions derived in the Appendix , as:

),;(

~

ρ2),,;(TλV

e

Tk

TλVu

B

l

, (11a)

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with

. ) 0(

4

22

exp

exp1

8

),,(

) 0(1

),;(

~

ρ

2

λ

Tλk

αλαλ αλαλ

Tk

ααTλS

λ

TλV

B

B

(11b)

where α is defined by eq 34b. Eqs 10 and 11 provide generalizations of the dimensionless rate constant

m given by eq 8 and employed in standard voltammetry studies. Eq 11 defines the lower boundary of

the irreversible region in the V-u plane as the locus

1)()(

~

ρ2)(

VueTkVVm

B

, while eq 10 yields

an approximation to the maximum of such a curve. These equations can be used in future analyses for

full theoretical characterization of the intermediate behaviors between the reversible (

m

) and

totally irreversible (

0

m

) limits (set in the fundamental work by Laviron28, 34, 39 by using Butler-

Volmer equations) with use of Marcus ET rates and thus consideration of reorganization energy and

temperature effects. In particular, according to above interpretation of eqs 10 and 11 in the V-u plane,

the condition

e

Tk

uTλV

(

u

B

l

2

;

~

),,

0

(12)

defines a regime of sweep rates where irreversibility is seen exclusively in the presence of suitably large

reorganization energy. For example, for

eV25. 0

λ

eq 10 gives

4 . 0

uum

l

, and in fact

irreversible response is found in Figures 1-2, while for

0

λ

it is

20

uul

and no irreversibility

occurs. Eqs 10-12 can be applied, e.g., to the range from ~10 s–1 to ~

16s10

deduced in the

Supporting Information from experimental data, and can be used to explore and predict the effects on

irreversible behaviors of using solvents with diverse polarization properties, hence different resultant

reorganization energy, in distinct experiments.

Further discussion of novelty and significance of eqs 9-12 is afforded in the Supporting Information.

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Next consider the double metal-molecule interface of a redox molecular conduction junction. Here

eq

P

in eqs 1a-b is replaced by

ss

P the (non-equilibrium) steady state probability that the system is in state B.

Eqs 1-5 also apply to the two-state model of such a junction, where both the left (denoted by L) and

right (R) contacts, characterized by coupling the strengths

L

and

R

, respectively, contribute to the

transitions A → B and B → A, so that the respective ET rate constants are given by

R

AB

L

ABAB

RRR

and

R

BA

L

BA BA

RRR

. Considering, for simplicity, a symmetric junction (

RL

) and symmetric

bias drop at the electrodes, the ET rates are still given by eq 34 except that α is replaced by

K ABK

eEμα

(K = L, R), where

LR

V

2

.13 The instantaneous L- and R-terminal currents

are given by

Q)RRJRQPRQPPRRP

e

I

J

L

BA

L

ABAB

L

BA

L

ABA

L

BA

L

AB

L

L

()()()1 (

ssss,

, (13a)

Q)RRJRP PR

e

I

J

R

BA

R

AB AB

R

AB

R

BA

R

R

() 1 (

, (13b)

where

BAAB

L

BA

R

AB

R

BA

L

AB

AB

RR

RRRR

J

(13c)

is the steady-state current. Deviation from steady-state can be expressed by the difference between the

left and right terminal currents:40

dt

dP

Q

)

RRJJ

BAABRL

(

. (14)

While in the single interface case discussed above

dtdP

is the interfacial current, here

dtdP

is a

“leakage” current that vanishes under steady-state conditions. The reversible/irreversible behavior of the

junction, as expressed by hysteresis in the current over a bias cycle, can be described as before:

irreversibility sets in when the bias sweep rate

tVu

is larger than the current relaxation rate

determined by

BA AB

RRρ

, and it can be conveniently described in terms of

ss

PPQ

. Eqs 3 and 5

remain valid also in the present case, because they refer to a generic two-state system, and, together

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with eqs 13a-b, provide the following criterion of reversibility (analogous to eq 7):

ABRL

RρPJJ

ss

(15)

where

Vd

Pd

uρQJJ

RL

ss

(16)

Since

AB

R

determines the order of magnitude of

L J , and

R

J , it follows also that

RLRL

JJJJ,

in

this limit. Eqs 15 and 16 yield the following condition on the sweep rate for the attainment of reversible

current-voltage responses:

dV

Rd

dV

Rd

R

R

ρ

dVPd

ρP

uu

BAAB

AB

BA

2

ss

ss

0

(17)

As detailed in the Supporting Information,

0 u decreases with λ at any voltage. This may suggest the

hysteresis in the current-voltage response of a redox molecular junction is easier to detect with feasible

scan rates when the reorganization energy involved in the electron localization on the redox center is

larger. However, Figure 3 shows that this conclusion is too simplistic because, in contrast with the trend

in the irreversible behavior of a molecule adsorbed on a single electrode, increasing λ not only makes

0 u smaller but also makes the hysteresis cycle narrower.41

Figure 3.

L

J

plotted against V, for

K 298

T

,

eV 15. 0

μEAB

, and a voltage sweep rate that for

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12

1

s 300

is

sV 20

u

. It is

0

λ

(forward and backward sweeps in grey and pink, respectively)

eV 25. 0

λ

(solid black and red lines), and

eV5 . 0

λ

(black and red dashes). A small transient NDR

is seen for

0

λ

due to fast charge accumulation in the molecule, as given by

dtdP

.41

To conclude this section, we consider again the timescale issue. As already stated, irreversibility and

hysteresis occur when the current cannot adiabatically follow the voltage change, which requires that

the characteristic charge transfer rates are slower than the voltage scan rate. On the other hand, in a

single-molecule junction easily observable currents (i.e., currents of the order of 1 nA) require the

transit of ~

s electrons1010

. Thus, the condition for detectable current is clearly incompatible with the

condition for hysteresis with experimentally feasible scan rates. The model considered so far, i.e., a two-

state molecular junction, cannot account for such experimental observations.42 A four-state model (that

becomes a three-state model over voltage ranges where double occupation of the bridge is not allowed)

able to justify the occurrence of significant hysteresis under less restrictive conditions is presented in the

next section.

Redox molecular junctions.

The occurrence of hysteresis, NDR, and hysteretic NDR at sweep rates commonly used in

experiments can be rationalized even in single molecule junctions, provided that charge transport

through the molecule takes place via at least two channels with different characteristics: one (strongly

coupled or "fast") channel dominates the observed current, while the other (weakly coupled, "slow")

channel determines that charging state of the bridge. In ref 13 we have argued that the existence of two

such channels is the hallmark of so called redox molecular junctions.

For definiteness, we consider the neutral molecule (state A) and two single-electron orbitals, b and c,

that can become occupied when the molecule acquires excess electron(s). We assume that orbital c is

strongly localized on the molecule (as would be the case for an orbital localized near a redox center),

therefore weakly coupled to at least one of the electrodes, while orbital b is more delocalized, so more

strongly coupled to both electrodes. In the ensuing kinetics orbital b will provide a relatively fast

channel that determines the magnitude of the observed current, while population and depopulation of

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orbital c takes place on a slow time scale associated with the observed hysteresis and NDR. Figure 4

depicts this model in the molecular state space. Molecular states B and C correspond to the molecule

with an excess electron in orbital b and c, respectively, while the state where both orbitals are occupied

is denoted by D.43 A similar model, excluding population of state D, has been used by Muralidharan and

Datta,17 who proposed a mechanism for NDR in the Coulomb blockade limit, and in works by Flensberg

et al.,44, 45 where it is shown that the blocking state causing NDR can result by breaking of the molecular

symmetries due to image charge interaction. Transport models that comprise interacting fast and slow

channels have been also studied recently in the context of electron counting measurements, where the

current through a point contact is used to monitor the electron occupation in a neighboring weakly

transmitting junction.14, 15, 46 When applied to redox molecular junctions, such models have to take into

account strong electron-phonon coupling and the dynamics of nuclear reorganization, which is done

here by inserting Marcus-type interface ET rates in simple rate (master) equations for both the slow and

fast transport channel.

DC

BA

CD

DC

DBBD CA AC

AB

BA

R

R

RRRR

R

R

II

I

(a)

BA

BA

II

I

(b)

Figure 4. (a) The four-state model described in the text. The transition rate for the process

XW

(W,

X = A, B, C, D) is denoted by

WX

R

. I and II denote the conduction modes of the "fast" transport channel

in the two oxidation states. The vertical arrows depict the changes in the molecular oxidation state by

electron transfer via the "slow" channel that often involves transient localization of excess charge in the

redox group. (b) Same as (a), in a reduced picture where A and B are obtained from states A and B,

respectively, by charging the redox site.

It should be noted that the applicability of such a kinetic description using Marcus ET rates is not at

all obvious. It is certainly justified for the slow channel, where the timescale for molecular charging and

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discharging is slow relative to that of nuclear rearrangement, however it may be questionable for the

charge transfer transitions associated with the fast channel. In next section, we will consider a novel

conduction model where the hopping mechanism is assumed for the slow channel while the fast channel

is described in the Landauer-Büttiker limit (coherent transport), as it may be appropriate depending on

the metal-molecule coupling strengths. Here we continue to assume that both channels can be described

with Marcus hopping kinetics. The corresponding master equation is

C CDB BDD DC DB

D

D DCA ACC CD CA

C

D DBA ABB BDBA

B

C CAB BAA ACAB

A

PRPRPRR

dt

dP

PRPRPRR

dt

dP

PRPRPRR

dt

dP

PRPRPRR

dt

dP

)(

)(

)(

)(

(19)

where

X P is the probability of the molecular state X (X = A, B, C and D) and

1

DCBA

PPPP

.

The total probability of charge localization in the redox site will be denoted by P, namely,

DC

PPP

.. The transport process given by eq 19 can be described in different ways. First, in a single-

electron picture, orbitals b and c describe two different distributions of a transferring electron on the

molecule, which correspond to the two transport channels discussed above. These channels will be

denoted by 1 and 2, respectively. Alternatively, because of the vastly different timescales associated

with these channels, and because channel 2 contributes negligibly to conduction, we can consider two

conduction modes of channel 1 that correspond to the different occupation states of orbital c. In the

molecular state language, these conduction modes, denoted by I and II in Figure 4, correspond to the

BA

and

DC

processes, respectively. We will sometime simplify the notation further, denoting

these processes by

BA

and

BA

, where A and B represent the molecular states of the

“charged” (in the sense that c is occupied) molecule in which orbital b is empty or occupied,

respectively, as seen in Figure 4b. Note that in most calculations reported below we also take into

account the small contribution to the current from channel 2.

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15

The ET rates in eq 19 are given by expressions similar to eq 34, except that the state energies and

molecule-electrode coupling parameters are adjusted to take into account electron-electron interaction

as expressed in the properties of the molecular states B, C and D. Specifically, the state energies satisfy

)()(

ACABAD

EEEEEE

, where is the energy of interaction between the two excess

charges in state D. In general, ≠ 0 and the

DC

and

DB

transition rates depend on the energy

differences

ABCD CD

EEEE

and

ACBD

EE

, respectively. Furthermore, we consider the

possible effects of charging one channel on the properties of the other channel. Denoting by

iλ and

K

i

(i = 1, 2) the reorganization energy47 and the coupling strength (expressed by the corresponding electron

loss rate) to the K (= L, R) contact in channel i in the case where = 0, we neglect the effect of

occupying the (relatively delocalized) b orbital on the localized c wave function, hence on the

parameters

L

2 ,

R

2 and

2λ associated with this orbital, as suggested by recent studies based on the

Density-Functional theory.48, 49 In contrast, charge localization in c causes significantly inhomogeneous

spatial changes in the effective potential seen by the other transferring charge, with non-negligible

effects on orbital b. In particular, a change in the wave function tails on the two electrodes may lead to

significant changes in the metal-molecule electronic couplings. This is modeled by assigning the

coupling strengths

K

1

K

1

κ

(with κ a constant and K = L, R) to channel 1 in the conduction mode II.

On the other hand, we disregard a possible effect of charging orbital c on

1λ (see the inclusion of this

effect in the Supporting Information).

To summarize, the essential features of the above four-state model for a redox molecular junction are:

(a) Such junctions are characterized by two conduction channels: a fast channel, 1, and a slow

channel, 2. The charging transitions in the latter are dominated by electron localization at a

molecular redox center.

(b) The transitions between the charged and uncharged states of the redox group (slow channel)

contribute negligibly to the junction current but can affect significantly the conductance via

channel 1.

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(c) The timescale separation between molecular charging (transitions in channel 2) and conduction

through channel 1 results in transitions between junction states characterized by different

steady- states currents.

The timescale separation between the two channels is the essential attribute of the redox junction

property, which is amplified by the solvent reorganization about the redox site. It has the important

consequence that the decoupling of the timescale associated with charging/discharging of the molecular

redox site and that of the voltage scan occurs at scan rates far slower than the charge transfer through

the fast channel that dominates the junction current. Therefore, in contrast to the two-state case,

irreversibility and memory effects in the population kinetics of the redox center will be expressed

visibly, sometimes prominently, in the observed conduction. In particular, hysteresis and hysteretic

NDR will be dominated by the slow channel, and telegraphic noise associated with transitions in this

channel is expected in some voltage range. Note that because of the large reorganization energy

associated with channel 2, its effect on the molecular conduction begins at voltages higher than the

threshold for significant current through channel 1.13

Next consider the junction transport properties as described by eqs 19. The solution of these equations

is greatly simplified by exploiting the timescale separation between the two channels. Assuming (see

point (c) above) that channel 1 is at steady-state, so that

BAB ABA

RPRP

(20a)

DCD CDC

RPRP

(20b)

eqs 19 reduces to

D DBB BD

DB

C CAAAC

C

A

PRPR

dt

dP

dt

dP

PRPR

dt

dP

dt

dP

(21)

The left terminal current, normalized to e, is given by

L

DBD

L

CAC

L

DCD

L

BAB

L

BDB

L

ACA

L

CDC

L

ABAL

RPRPRPRPRPRPRPRPV(J

)

;

(22)

Page 17

17

Using eqs 20, 22, 34 and the relation

DC

PPP

allows us to write this current as a function of the

voltage V and the interaction parameter in the form (see Supporting Information)

);();()

;

(

VVJVJ

L

P

L

, (23a)

where

)();();()( )];( 1 [);(VJVrVPVJVPVJ

ABAB

P

(23b)

with

T

T

2

R

BA

B

R

k

B

L

L

L

L

AB

B

L

k

B

R

R

R

R

BA

B

R

T

L

AB

B

L

T

AB

BA

R

α

Tkλ

λα

αTλS

αT

,

λ

,

S

(

R

α

Tkλ

λα

αTλS

αT

,

λ

,

S

(

R

k

α

R

k

α

κ

J

J

V

(

r

2

exp1

2

exp

)

),,(

exp1

2

exp

)

),,(

exp1exp1);

1

1

1

1

1

1

1

1

(23c)

is the contribution to the current by channel 1. In eq 23c,

BA

J

(or, equivalently,

CD

J

; see Figure 4)

denotes the steady-state current carried by channel 1 through the reduced molecule (a molecule with an

excess electron localized at the redox center). It is given, in analogy to eq 13c, by

ABBA

L

BA

R

AB

R

BA

L

AB

BA

RR

RRRR

J

(23d)

where the ET rates are given by the analogue of eq 34 for the conduction mode II of channel 1 (i.e.,

using

CD

E

rather than

AB

E

in eq 34 of the Appendix). The second term in the right side of eq 23a is the

small contribution to the current by channel 2, given by

Page 18

18

L

CA

R

BA

R

AB

L

BA

L

AB

B

R

AB

R

AB

L

AB

L

AB

R

BA

R

AB

L

BA

L

AB

R

AB

R

AB

L

AB

L

AB

L

AC

L

AC

BAAB

BAAB

L

AC

R

DCCD

L

CA DC

L

DBCD

BAAB

L

AC BA

L

BD AB

L

R

RηRη

Tk

RηRη

RηRηR

ηRηη

PR

R

RRη

P

RR

RRRR

P

RR

RRRR

PV

(

exp

) 1 (

) 1 ();

(23e)

with

,

2

) 2(

exp

),,(

),,

λ

(

)(

);(

);(

1

1

1

1

T

λ

k

λα

αTS

αTλS

κ

VR

VR

Vη

B

K

K

K

K

AB

K

CD

K

AB

(K = L, R) (23f)

and

L

AC

η

expressed similarly to

L

AB

η by replacing κ ,

1λ and

AB

E

with 1,

2λ and

AC

E

, respectively. For

= 0,

L

takes the simple form

L

CA

L

AC

PRRP

1

and under steady-state conditions it is given by eq 13c

with B replaced by C. This small contribution to the current is disregarded in Figure 4b.

Eq 23b expresses

P

J , the dominant contribution to the current

L J at the left electrode, as a weighted

average of the currents carried by channel 1 in its conduction modes I and II. When the interaction

between the two excess electron charges in state D is neglected, the ratio r of the steady-state currents

BA

J

and

AB

J

is voltage independent,

κV

(

r

) 0 ;

, as seen from eqs 13c and 23f. An analogous

expression can be written for the R-terminal current,

R

J :

);();();(

VVJVJ

R

P

R

(24a)

with

R

CA

R

BA

R

AB

L

BA

L

AB

B

R

AB

R

AB

L

AB

L

AB

R

BA

R

AB

L

BA

L

AB

R

AB

R

AB

L

AB

L

AB

R

AC

R

AC

BAAB

BAAB

R

AC

R

R

R

RηRη

Tk

RηRη

RηRηR

ηRηη

P

R

R

RRη

PV

(

exp

)1 (),

(24b)

The memory effects that appear in fast sweeps are embodied into eqs 23-24 through the deviation Q

of P from its steady-state value

ss

P . The evolution of P is derived from eq 21 (see Supporting

Information) as

Page 19

19

QA

dt

dQ

dV

dP

u

dt

dP

2

ss

(25)

(the dependence of P on

utV

and is not explicitly shown here), where

ACBA BDAB

CADCDB CD

DCCD

BAAB

RRRR

RRRR

RR

RR

A

A

P

1

1

2

1

ss

, (26a)

with

AC

R

AC

R

AC

L

AC

L

AC

ABBA

BA AB

AC

BD

BAAB

AB

AC

BA AB

BA

R

RηRη

RR

RR

R

R

RR

R

R

RR

R

A1

(26b)

R

TkR

ηRη

RηRηRηRη

Tk

RηRηRηRη

R

A

R

RR

R

R

RR

R

R

RR

R

R

RR

R

A

BAC

R

CA

R

AC

L

CA

L

AC

R

AB

R

AB

L

AB

L

AB

R

BA

R

AB

L

BA

L

AB

B

R

BA

R

AB

L

BA

L

AB

R

AB

R

AB

L

AB

L

AB

AC

DB

DC CD

CD

CA

DCCD

DC

BD

BAAB

AB

AC

BAAB

BA

exp

exp

1

2

(26c)

and

R

AC

η

obtained from

R

AB

η , eq 23f, by replacing κ ,

1λ and

AB

E

with 1,

2λ and

AC

E

, respectively.

From eqs 23-26 it follows that if the sweep rate is significantly smaller than

dVPd

PA

u

ss

ss2

0

. (27)

channel 2 also works under steady-state conditions, so that

ss

PP

and the I-V characteristics of the

junction does not exhibit any hysteretic behavior . In this case,

RL

and consequently

RL

JJ

. If,

instead, u is similar to or smaller than

0 u in some bias range, Q is an appreciable fraction of

ss

P and

irreversibility appears in

L J and

R

J through both the main contribution

P

J and the residual terms

L

and

R

. Still, since

LR

P

J

,

,

dt dPJJ

RLRL

is much smaller than

L J and

R

J , so

that

P

RL

JJJ

. It is worth noting that the kinetic parameter m introduced in eq 8 is extended to the

Page 20

20

present model by replacing

eTkk

B

0

with the limiting voltage sweep rate

0 u of eq 27. Then,

0 uu

amounts to the condition

1)(

Vm

that was discussed for the two-state case.

Figures 5-9 show some characteristic behaviors resulting from Eqs 23, 25 and 26 (for specificity, only

the current

L J is shown). In these examples we assume equal potential drops across the two molecule-

lead interfaces.50 Figure 5 shows the occurrence of hysteresis, NDR, and hysteretic NDR for a given set

(see caption) of junction parameters. As seen in Figures 5c and 5f, high enough scan rates lead to

hysteresis irrespective of the value of the electron-electron interaction parameter . On the other hand,

large enough causes NDR irrespective of the scan rate (see Figures 5d-f). Thus, hysteretic NDR

occurs for sufficiently high values of both u and .

Figure 5.

L

L

J

1 plotted against V over a voltage cycle, eq 23. The forward and backward sweeps are

in black and red, respectively (they are on top of each other in panels a, b, d, e). The following model

parameters are used:

R

2

L

2

R

1

L

1

100

,

K298

T

,

eV15. 0

μEAB

,

eV 3 . 0

μEAC

,

eV25 . 0

1

λ

,

eV 5 . 0

2

λ

,

4

κ

. is zero in panels a-c and is 0.5 eV in panels d-f.

L

u

1 is

V 102

5

(left panels),

V 102

4

(center panels), and

V102

3

(right panels).

Page 21

21

Figure 6. (a) The Y axis represents the left-terminal current,

L

L

J

1 (black solid line), the steady-state

current through channel 1 for empty redox site,

L

AB

J

1 (gray dashed line), and the redox site

occupation P (blue dashed line), plotted as functions of V during a forward voltage sweep, for the same

parameters as in Figure 5c. (b) Same observables during the backward sweep.

L

L

J

1 and P are

represented by red solid and pink dashed lines, respectively. (c-d) Same as a-b with the parameters of

Figure 5f. Note that

L

AB

J

1 is the same in all figures.

To understand these behaviors, we consider in Figure 6 the voltage dependences of P and

AB

J

that

appear in the main contribution

P

J to the current, eq 23b, for the situations of Figures 5c and 5f.

Figures 6a-b focus on the case 0, while the case 0.5 eV is shown in Figures 6c-d. The following

points are notable:

(a) The comparison of Figures 6a and 6b displays the hysteresis in the redox state of the molecule

during a bias cycle, which results from the delay in the evolution of P with respect to

ss

P . In fact

Page 22

22

during the forward sweep P remains negligible over a bias range wider than that predicted by eq

26 for

ss

P . Similarly, during the backward sweep, P takes the plateau value

1)(

ss

VP

over

a voltage range wider than that pertaining to

ss

P . Therefore, the switch of the transport channel

1 from the conduction mode I (empty redox site and coupling strengths

R

1

L

1

to the

electrodes) to the more conductive mode II (occupied redox site and coupling strength

K

1

K

1

κ

to the K = L, R lead, with

4

κ

) during the forward sweep occurs at higher bias

voltages than the reverse switch in the backward sweep. Consequently, counterclockwise

hysteresis (current is smaller in the forward voltage sweep than in the backward direction) is

observed in Figure 5c.

(b) Clockwise hysteresis can be obtained if the system starts from state C (P = 1). Moreover, if P is

not zero at the end of a voltage cycle (e.g., see Figure S1a in the Supporting Information), the

system can end a single realization of the voltage cycle and start the next one in the conduction

mode II. This kind of behavior is observed, e.g., in the experiments of ref 51 (see I-V curves in

Figure 3 therein). Clockwise hysteresis loops are also found if

1

κ

, namely, if the conductance

of the reduced bridge is smaller than that of the uncharged molecule (examples are given in

Supporting Information). This prompts future tests of our model against experimental data4, 7, 51

that show occurrence of clockwise and/or counterclockwise hysteresis loops, as well as the

possibility to predict similar behaviors in redox junctions manufactured so to fit within suitable

parameter ranges.

(c) The following physical interpretation of NDR emerges from Figures 6c-d. Starting with the

molecule in state A and focusing for example on the current at the left interface,

ABL

JJ

at

sufficiently low biases where P is negligible, as predicted by eq 23b. As V increases, P becomes

appreciable and consequently channel 1 can switch with probability P to the conduction mode

II. At higher bias voltages this switch will lead to a current

AB

BA

JJ

, but, because the

threshold voltage13 of mode II is higher by

eV12

than that of mode I, the current will

Page 23

23

decrease (NDR) before starting to rise again around the threshold bias voltage of mode II,

eμEλ

AB

) ( 2

1

, and finally reaching the high-voltage plateau as

BA

L

JJ

. During the

backward sweep, because of the memory effects in the evolution of P, conduction mode II

remains significantly populated over the voltage range in which the current is appreciable, and

L

J is accordingly closer to

BA

J

than in the forward sweep, with little NDR (or no NDR for

sufficiently high scan rate).

(d) For zero or small enough ,

eμEλ

AB

) ( 2

1

is smaller than the threshold bias voltage for

molecular charging. Thereby, conduction mode II is accessible where the redox site begins to be

populated, which means that the current rises from

AB

J

to

BA

J

without NDR (Figures 5a-c).

The connection between the sweep rate and the appearance of hysteresis is investigated in Figure 7.

Figure 7a shows the threshold sweep rate for hysteresis,

)(

0Vu

, together with the voltage sweep rates

V 102

4

11

L

u

and

V 102

3

12

L

u

used in Figures 5b and 5c, respectively.

1u is smaller than

)(

0Vu

at each bias, but

1u is an appreciable fraction of

)(

0Vu

over the voltage range in which the rate

of electron injection

L

AB

R

starts not to be negligible compared to the rate of electron delivery

R

BA

R (see

Figure S2 in the Supporting Information) and thus, according to eqs 13c and 23a-b, the current is

appreciable. Consequently, hysteresis appears in Figure 5b, although it is barely visible. In the same

voltage range

2 u is larger than

)(

0Vu

so that considerable hysteresis occurs in the case of Figure 5c.

Further insight into the hysteretic behavior of the molecular system is gained by the analysis in Figures

7b-c, where the appearance of hysteresis in the current-voltage response is related to the irreversible

evolution of P, as described by eq 25 for large enough u values. In these two panels we report the

evolutions of molecular reduction,

dt dP

, its reversible component,

dV dPu dt dP

ssss

, and its

irreversible part,

dtdQ

, for

1uu

and

2 u . Since

dtdP

is of the order of

dV dPu

ss

, the maximum of

this rate in Figure 7c is about a factor

10

12

uu

larger than that in Figure 7b. Furthermore, the relative

deviation

ss

PQ

increases considerably with the sweep rate u. Since the memory effects cause a delay

Page 24

24

in the evolution of P compared to

ss

P , the delay in charging the molecule corresponds to accumulation

of the negative deviation

dt dQQ

, whereas the delay in the achievement of full reduction (i.e., the

high-V plateau of P) is responsible for decrease in Q. Significant memory effects occur mainly over the

bias range in which u is larger than or of the same order as

)(

0Vu

. Nevertheless, the accumulation of

memory in the response of the molecular system to the changing voltage can begin before it becomes

actually observable, where the current and P are both negligible but u is of the order of magnitude of

0 u , so that no hysteresis can be seen although Q is an appreciable fraction of P (see, for example, the

low-voltage range where the sweep rate and the threshold rate

0 u are comparable in Figure 7a, but the

current in Figure 5c and Q in Figure 7c are still negligible). In addition to this, the accumulated memory

affects the current-voltage characteristic over a V range where

0 uu

but Q has not been fully

dissipated yet (see the high-voltage tail of the positive Q peak in Figure 7c and compare with Figure

7a).

Figure 7. (a) The threshold voltage sweep rate, expressed as

uu0 10

log

with

V1

1

L

u

(black solid

line) plotted against the voltage, using the same model parameters as in panels 6b-c. The sweep rates

V 102

4

11

L

u

and

V 102

3

12

L

u

are also displayed as

uu110

log

and

uu2 10

log

(horizontal gray and black dashed lines, respectively). (b-c)

dtdPY

(black),

dtdPss

(red), and

dtdQ

(blue) versus V during the forward bias sweep, for

1uu

and

2 u , respectively. The time unit is

L

1

100

. Y = 0 is marked by the dashed line.

The above discussion has been focused on the average response of the observed system over many

Page 25

25

similar bias sweeps, yet it allows direct comparison with experiments consisting in single or few voltage

sweeps.1, 4, 7, 52, 39, 51 Of particular interest is the comparison between the ensemble average and the

individual sweeps. Such a comparison is shown in Figure 8 (see computational details in the Supporting

Information). Figure 8a shows the average current-voltage characteristics for reversible and irreversible

behaviors (as determined by the sweep rate), while Fig 8b shows the corresponding results for a single

realization, where the transition probabilities

tRAC and

tRCA , with

t chosen as a suitable

simulation time step, are used to generate a single trajectory according to the stochastic simulation

procedure53 detailed in the Supporting Information. Of particular interest are the qualitatively different

behaviors of single realizations depending on the scan rate. At fast scan rates (red and black curves in

Figure 8b, whose averages over many realizations yield the red and black curves in Figure 8a),

stochastic hysteresis is seen, with a single jump to the high-conductance mode in the upward run and

persistence of this mode in the downward run. The single jump takes place at different biases in

different sweeps, leading to the average hysteresis cycle of Figure 8a. In contrast, in the slower bias

scan that on the average yields the grey curve of Figure 8a, the single realization is characterized by

multiple switching between the two conductions modes of channel 1 leading to the appearance of

telegraphic current noise.

The results in Figures 5 and 8 compare well with experiments such those in ref 7 (in particular,

compare Figure 8b with Figure 4 in ref 7) and refs 1, 51. In ref 7 the voltage change is implemented in

steps of duration t referred to as current measurement integration time), so that our voltage sweep rate

u is proportional to

1

t

. The observation7 of hysteresis at small t (0.64 ms) and telegraphic noise

that averages to no hysteresis at large

t (320 ms) corresponds to the results shown in Figure 8.54

Obviously, this agreement with observation does not provide a detailed description of the particular

experiment, but, rather, shows the generic nature of the phenomenon and the ability to reproduce the

experimental data with a generic model.

Page 26

26

Figure 8.

L

L

J

1 plotted against V, using the parameters of Figure 5 with = 0, except that

V 102

2

1

L

u

for the black and red lines while

V 102

5

1

L

u

for the grey and pink lines (forward

and backward sweeps, respectively). (a) Average over many realizations. The grey and pink lines are on

top of each other (reversible behavior). (b) A single realization of the system behavior over a single

voltage cycle. The small direct contribution from channel 2 is neglected in this calculation.

In summary, the presence of a redox center on the molecular bridge leads to system response on a

slow timescale that gives rise to hysteresis and NDR phenomena when the voltage changes on this

timescale or faster. As seen in Figure 5 (see also examples reported in Figures S6-7 of the Supporting

Information), the reorganization energies involved in the interfacial ET processes play an important role

in determining and shaping this behavior. On the timescale considered, the system displays a bistable

behavior which is enhanced by the additional stabilization provided by this reorganization. Another

manifestation of this bistability is the appearance of telegraphic noise in single, slow, potential sweeps

as seen in Figure 8.

A Landauer-Büttiker-Marcus model of redox junctions.

The two conduction channels model used in the previous section can be seen as a simplification of a

quantum transport problem described by a model of two interacting transport channels (e.g. a bridge

comprising two single-electron levels, each of them coupled to the leads, with Coulomb interactions

between the electronic populations on these two levels) characterized by given couplings to the leads

Page 27

27

and to the phonon environment. As already noted, such models were discussed in conjunction with

single electron counting measurements using point contact detectors.14, 15, 16 In the previous section we

have assumed that the molecule-lead couplings in both channels are small enough to allow treatment by

classical kinetic equations and, furthermore, that electron-phonon coupling is large and temperature is

high enough so that Marcus rates can be used in these kinetic equations. Here we examine another limit,

where transport in the "slow" channel is assumed to be described by Marcus kinetic equations as before,

however in the fast channel molecule-lead coupling is assumed to be large enough so that transport in

this channel is a coherent co-tunneling process that can be described by the standard Landauer- Büttiker

theory.21, 22 As in any mixed quantum-classical dynamics, this level of description has its own

intricacies and is treated in what follows with further approximations. A comparison with a numerical

calculation based on the pseudo-particle Green function formalism55 will be presented in a subsequent

publication.

An approximate kinetic description of this limit can be obtained by assuming that on the timescale of

interest the system can be in two states: one, denoted as state S1, where the slow channel 2 - the

molecular redox site - is occupied, and the other where it is not (state S0). In terms of the probabilities

of the four states in eq 19, the probabilities that the system is in states S1 and S0 are

DCS

PPPP

1

(28a)

BAS

PPPP

1

0

(28b)

In each of these states, the current

1I through the fast channel 1, as well as the average bridge

population

1n

in this channel, are assumed to be given by the standard Landauer theory,

disregarding the effect of electron-phonon interaction,21, 22

I

2

VεfVεf

εε

dε

π

e

εV

RL

RL

;;

2

;

2

11

11

11

(29)

2

2

1

2

1

11

11

;;

2

1

π

;

εε

VεfVεf

dεεVn

R

R

L

L

(30)

Page 28

28

where

Kf (K = L, R) denotes the Fermi-Dirac function of the K electrode (K = L, R) and

RL

111

,

and where

1ε and

K

1 take the values

AB

Eεε

) 0 (

11

,

K

1

KK

0

11

in state S0 and

) 0 (

1

ε

) 1 (

1

ε

1

ε

,

0

11

KK

κ

in state S1. The switching kinetics in channels 2 is described by

1

010

) 1 (

SSSS

kPkP

dt

dP

(31)

where for the average switching rates we invoke one of the following models:

Model A. The rates are written as weighted averages over the populations 0 and 1 of channel 1,46 with

respective weights

1

1n

and

1n

:

BDSACSSS

RnRnk

010110

1

(32a)

DBSACSSS

RnRnk

111101

1

(32b)

where

AC

R

,

AC

R ,

BD

R

,

DB

R

are the Marcus rates defined in the above section (see discussion of eq 19

and Figure 4).

Model B. The rates are written as Marcus ET rates between the two system states S0 and S1, whose

energy difference is taken to be

11

)0(

201

)0(

1

)0(

211

) 1 (

101

SSSSS

nεnεεnεEE

, where

AC AC

EEEε

)0(

2

.

These two models are associated with different physical pictures. Model A assumes that the switching

rates see the instantaneous population in channel 1, while model B assumes that these switching rates

are sensitive only to the average population

1n

. Model B suffers from an additional ambiguity: the

apparent change in the number of electrons on the molecular bridge between the two states S0 and S1 is

11

0111

SS

nnn

. Nevertheless, Marcus-type rates for transferring one electron between

the metal and the molecule are calculated. A discussion of these models and their validity in comparison

to a quasi-exact calculation will be given elsewhere.

For both models we assume that the potential sweep is slow enough so that the above rates follow it

adiabatically. As before, the time evolution of P, eq 31, over a voltage sweep can lead to hysteresis in

Page 29

29

the current response over a bias voltage cycle if the scan rate u is fast enough. The average (over many

sweep cycles with similar initial conditions) current is given by

V

]V

V

) 1 (

1

ε

1

) 0 (

1

ε

1

;; 1 [VIPVIP eJVI

(33)

where the contribution of channel 2 to the observed current is neglected. Results based on eqs 29, 31

and 33, using models A and B for the redox reaction rates, are shown in Figure 9 (see the

implementation of these equations in the Supporting Information), while a single realization of the bias

sweep will show telegraphic noise similar to that seen in Figure 8. The I-V responses predicted by both

models A and B, similarly to those arising from the kinetic model in the previous section (see Figure 5),

show hysteresis and hysteretic NDR. However, the considerable quantitative differences (for example,

unlike in the full hopping model, no NDR occurs unless is sufficiently large and/or κ is small

enough) offer the possibility to discriminate between the conduction mechanisms corresponding to the

two classes of models in their application to experiments. This may have relevant implications not only

for the study of specific systems, but also for a more general classification of the redox molecular

systems currently used in nanoelectronic experiments, based on the few global parameters

characterizing the above models.

Page 30

30

Figure 9. Dimensionless current

L

J

1

versus voltage V over a bias cycle, using eq 33 with P evolved

according to eq 31. The forward and backward sweeps are in black and red, respectively. Models A and

B for the average rates of transition between states S0 and S1 are used in a-b and c-d, respectively. The

following model parameters are employed:

K 298

T

,

eV15 . 0

) 0 (

1

ε

μ

,

eV3 . 0

) 0 (

2

ε

μ

,

0

1

λ

,

eV 5 . 0

2

λ

,

eV 1 . 0

11

RL

,

R

2

L

2

100

,

eV 5 . 0

,

V 10

2

3

2

2

L

u

. The dimensionless

electrode coupling parameter κ takes the value 4 in panels a and c, and 1 in panels b and d.

CONCLUSIONS

The irreversible behavior characteristic of sufficiently fast voltammograms, expressed by distortion

and shift39 and the appearance of hysteresis and hysteretic NDR in the current-voltage response of

molecular conduction junctions9 are manifestations of the interplay between two time scales: the

observation time and the characteristic time for switching between different charging states of the

molecular system. The presence, on the molecular system, of redox centers on which electron

localization is stabilized by a polar environment serves to separate the timescale of (slow) charging-

discharging transitions from that associated with the current flow that is relatively fast even for the

smallest observable currents. This localization plays a crucial role in the appearance of irreversibility

effects in the range of commonly used voltage sweep rates.6, 7, 9, 13, 39 This paper analyzes a simple

generic spinless model for this phenomenon that accounts for a broad range of observed behaviors. In

what follows we summarize the key features of the model and its implications:

Page 31

31

(a) The molecular system can exhibit at least two (relatively long-lived) oxidation states characterized

by charge localization in a redox site. Consequently, four distinct molecular states are considered

in the kinetic version of the model.

(b) The transient localization of transferring charge and its stabilization by environmental polarization

corresponds to the presence of a slow charge transport channel characterized by small interfacial

ET rates. We have assumed that these rates are given by the Marcus theory of heterogeneous

electron transfer,18, 19 implying full equilibration of the environmental polarization response on the

timescale of the observed kinetics and highlighting the role played by solvent reorganization about

the molecular bridge.

(c) We have studied the onset of irreversibility, expressed by the appearance of bistability and

hysteresis in the current/voltage response. It should be emphasized that the bistability alluded to in

this paper is a transient phenomenon, characterized by the timescale of the slow channel. Its

observation is determined by the voltage scan rate as compared with the rate of these charging

transitions, while the observed current is determined by the second "fast" channel whose

transmission properties depend on the occupation state of the redox center.

(d) Simple criteria (see eqs 10, 11, 12, 17, and 27) were obtained for the departure of the I-V response

of the junction from steady-state behavior as dependent on the bias sweep rate. Accordingly, the

first appearance of hysteretic behavior and the bias voltage range where it is predominantly

observed can be rationalized and “predicted” from the steady-state response of the system. In

principle, this may provide a route to control and modulate the junction response properties, in

particular, memory effects of interest to nanoelectronics applications.

(e) The effect of solvent reorganization on the appearance of irreversibility at a single metal-molecule

interface and in a molecular redox junction is analyzed and its role in the occurrence of hysteresis,

NDR and hysteretic NDR is explicitly described. This can suggest suitable choices of junction

components for tailoring specific features of the current-voltage response.

(f) While much of our analysis was based on a classical kinetics model with Marcus rates, quantum

Page 32

32

coherent transport in the fast channel has been considered as well. Such model shows qualitatively

similar behavior, with significant quantitative differences. It also raises important issues in the

approximate description of the mixed quantum classical dynamics that will be further discussed in

a future publication.

Finally, we wish to note that the simplicity and generic character of the presented models may provide

a useful framework for further theoretical developments, including the consideration of situations where

neither the hopping nor the fully coherent mechanisms are appropriate to describe the conduction via

the effective transport channels.

ACKNOWLEDGEMENTS. This research was supported by the Israel Science Foundation, the

Israel-US Binational Science Foundation and the European Research Council under the European

Union's Seventh Framework Program (FP7/2007-2013; ERC grant agreement n° 226628). We wish to

thank Michael Galperin for fruitful discussions.

APPENDIX

Marcus ET rates and their approximation near

eμEV

AB

)(

. The interfacial ET rates can

be written as the following sums of analytic functions:13, 30

λk

T

λα

4

αTλSR

B

AB

)(

exp),,(

4

2

,

λk

T

λα

4

αTλSR

B

BA

)(

exp),,(

4

2

, (34a)

where

eVEμα

AB

, (34b)

n

j

N

00

),,(),,( ) 1(

2

1

),,(

n

jj

j

n

αTλαTλ

j

n

αTλS

, (34c)

2 (

T

4

Tλk

αλj

λk

αλj

αTλ

B

B

j

2

) 1 2 (

erfc

) 1

exp),,(

2

, (34d)

and the limit superior N truncates the otherwise infinite sums. In eq 34, is the coupling strength to

Page 33

33

the metal, taken as a constant.

B k is the Boltzamann constant, T is the temperature, λ is the

reorganization energy of the molecular system (including the solvent), and

AB AB

EEE

, where

A

E

and

B

E are the energies of the states A and B, respectively.

For

λμE eV

AB

and

Tkλ

B

, one can write the above ET rates in the Gaussian-like form

proposed my Marcus.56 Therefore, it is

Tk

λ

TλS

e

μE

R

e

μE

R

B

AB

BA

AB

AB

4

exp 0 ,,

4

. (35)

0 ,,TλS

is truncated as in eq 34c for any practical calculation. However, in the exact limit N → ∞, it

can also be recast as30, 57

j

j

k

0

2

0

2

12

erfc

4

) 1

T

2 (

exp ) 1(4 ) 0 ,

T

,( ) 1(4 ) 0 ,

T

,(

BB

j

j

j

Tk

λjλj

λλS

. (36)

For

Tkλ

B

, one can use the asymptotic expansion of the complementary error function and write

.

12

1

j

) 1

(8

12

1

j

) 1

(8

4

) 1

k

2 (

exp

12

2

j

4

) 1

k

2 (

exp ) 1

(4) 0 ,

T

,(

1

1

0

0

22

j

j

j

j

B

j

B

B

B

B

j

πλ

Tk

πλ

Tk

T

λj

πλ

Tk

T

λj

λS

(37)

Then, using the equation (where

n

E2 are Euler numbers)58

j

1

2

22

12

12

1

)! 2 (2

12

1

) 1(

n

n

n

n

j

E

n

π

j

(38)

with n = 0 (hence,

1

0

E

), we obtain

λ

Tkπ

jπλ

Tk

TλS

B

j

j

B

2

12

1

) 1

(8) 0 ,,(

1

1

, (39)

hence the expression of ρ for

0

λ

in eq 9 after substitution in eq 35 and use of eq 3b. This analysis

can clearly be applied to each interface of a junction, with V replaced by

K

(K = L or R).

Page 34

34

ASSOCIATED CONTENT

Supporting Information

Insights into the connection between voltage sweep rate ad reversible/irreversible behavior; typical

lead-molecule coupling strengths deduced from experiments; analytical expressions of the threshold

scan rate for appearance of irreversibility; additional applications of the two and four state redox

junction models; generalizations of the four-state hopping model with Marcus rates; implementation of

four-state Marcus model and Landauer-Büttiker-Marcus model. This material is available free of charge

via the internet at http://pubs.acs.org.

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36

49.

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50.

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