Irreversibility and hysteresis in redox molecular conduction junctions.
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 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.
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ABSTRACT: Switching events in the current flowing through individual redox proteins, (azurin) spontaneously wired between two electrodes, are studied using an electrochemical scanning tunneling microscope (ECSTM). These switching events in the current-time trace are characterized using conductance histograms, and reflect the intrinsic redox thermodynamic dispersion in the azurin population. This conductance switching may pose limitations to miniaturizing redox protein-based devices.Small 03/2014; · 7.82 Impact Factor
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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
BAAB
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
BA AB
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
BAABB BAA AB
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
e dtuetQtQ
(
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