Page 1

1

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

Page 2

2

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

Page 3

3

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

Page 4

4

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

Page 5

5

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