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Phys. Status Solidi B 250, No. 11, 2249–2266 (2013) / DOI 10.1002/pssb.201350048
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basic solid state physics
Forty years of molecular electronics:
Non-equilibrium heat and charge
transport at the nanoscale
Review Article
Justin P. Bergfield* and Mark A. Ratner
Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA
Received 25 June 2013, revised 29 August 2013, accepted 2 September 2013
Published online 19 October 2013
Keywords chemical symmetry, future of molecular electronics, molecular electronics overview, molecular thermoelectric response,
non-equilibrium quantum heat and charge transport theory
∗
Corresponding author: e-mail [email protected], Phone: +1-874-467-4987, Fax: +1-847-491-7713
The “Quo Vadis?” meeting in Bremen (March 2013) was a
spectacular opportunity for people involved in molecular electronics to catch up on the latest, to think back, and to project
into the future. This manuscript is divided into two halves. In
the first half, we address some of the history and where the
field has advanced in the areas of measuring, modeling, making, and understanding materials. We review some big ideas
that have animated the field of molecular electronics since its
beginning, and are at the height of interest and accomplishment
at the moment. Then, we discuss six major areas where the field
is evolving, and in which we expect to see very exciting work in
the years and decades ahead. As a representative of one of the
newer themes, the second half of the paper is devoted to molecular thermoelectrics. It contains some formalism, some results,
some experimental comparison, and some intriguing conceptual
questions, both for pure science and for device applications.
An artist’s rendition of a self-assembled monolayer of
polyphenylether molecules on Au contacted by a Au STM.
© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Where we have been
1.1 Simple transport: Models and measurements The transfer of charge within molecules is generally
referred to as electron transfer, and became an important
element in chemistry in the 1950s, when the availability of
radioisotopes made it possible to measure the rates of electrons transferring from one part of a molecule to another. Very
important work on solid state systems by Hush [1, 2] permitted both classification of these charge transfer reactions and
their understanding in terms of different degrees of freedom,
and different couplings within the electronic Hamiltonian.
Following many beautiful experiments, particularly those of
Taube, Marcus [3] utilized dielectric theory, and what is now
known as the polaron model [4], to develop an equation that
gave the rate of electron transfer between two entities (which
could be parts of a molecule or two different molecules) in
terms of an attempt frequency (for the so-called adiabatic
transfer) or a matrix element of the Hamiltonian operator (for
non-adiabatic processes) and a density-of-states weighted
Franck–Condon factor, that described the process of energy
dissipation into the degrees of freedom of the medium and
of the molecule.
In the 1970s, very creative work from Mann and Kuhn
[5] in organic solids led to the question of how charge could
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J. P. Bergfield and M. A. Ratner: Forty years of molecular electronics
Figure 1 A schematic representation of a Au–C60 –Au singlemolecule junction.
move through molecules, as well as measurements of such
systems and some suggestive insights. In 1974, Aviram’s PhD
thesis was devoted to the topic of a single molecule junction
(that is, a single molecule strung between two electrodes) as
a device (for rectification), and as a model problem. Some of
the calculations there were somewhat naı̈ve, but the general
ideas that one could think of a single molecule as a measurable system, that such a molecule could be synthesized and
placed between electrodes, and that such a molecule could
act as a device (in this case a rectifier) were quite original
[6]. Following the Aviram contribution, there was extensive
interest in the general area of molecular electronics, but the
difficulty of unambiguously creating the structure sketched
in Fig. 1 (i.e. a single molecule bonded to two macroscopic
electrodes) was formidable.
The entire field changed with the invention of scanning
probe microscopy in the early 1980s. First the scanning tunneling microscope (STM) [7] and then the atomic force
microscope (AFM) [8] made it possible to apply a voltage across a single molecule, and to measure the current.
Although these scanning probe techniques were originally
developed to study surface physics and surface structure,
their application to molecular electronics really marked its
emergence as a legitimate and significant area of research.
In the 1980s and 1990s, several attempts were made to
measure current through single molecules between two electrodes [9–11]. The measurements of Reed et al. [12] (utilizing
a creative form of break junction methodology) constitute the
first report claiming to have successfully measured the transport across a single molecular bridge. The Reed paper [12]
was the start of observable molecular electronics, in the sense
that it represented repeatable measurement on a (relatively)
well-defined and stable system. The most accepted measurements for molecular transport junctions are now based on
break junctions. The mechanical break junction was used by
Reed et al. [12], and is still the prime tool for low-temperature
measurements and for those carried out in vacuum
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[13–17]. In solution, the electrochemical break junction
and its STM-based variations are widely used and effective
[18–22].
Within the next decade, the theory involved in modeling
molecular transport (starting from the Landauer approach
which recognized that the dissipation of energy occurred
in the downstream electrode, and extending that to deal
with strongly non-equilibrium situations using the nonequilibrium Green’s function (NEGF) statistical mechanics
methodology [23–37]) permitted a straightforward recipe for
expressing the flow of charge and heat in terms of Green’s
functions and self-energies, which in turn could be calculated
from electronic structure calculations. In this formulation
of transport, based partly on the idea that “transmission is
conductance” [23, 24, 29], a junction’s electronic scattering
function connects the microscopic molecular and junction
properties to the observables quantities measured in the lab.
For instance, in the linear-response regime (i.e., when the
bias voltage ΔV and applied temperature gradient ΔT are
much less than the chemical potential difference and temperature of the electrodes, respectively) and at zero temperature,
“Landauer’s equation” 1 states
G = G0 T (EF ),
(1)
where G is the electrical conductance of the junction, T (EF )
is the transmission probability at the Fermi energy of the
electrodes, and G0 = 2e2 / h is the quantum of conductance
expressed in terms of the charge of the electron e and Planck’s
constant h.
A self-consistent scheme using a combination of Kohn–
Sham density functional theory (KS-DFT) and NEGF has
become standard for modeling single molecule transport
[32, 39–43], even though it is still subject to errors, oversimplification, and misinterpretation. Theoretically, the use
of the KS Green’s function in place of the true interacting Green’s function is an unjustified approximation (even
with the exact exchange–correlation functional, the KS resonances are not the true excitations of the system) [41];
nevertheless this approach has proven to be helpful in
describing a number of molecular junction transport experiments. Many-body techniques, such as those based on
the GW approximation [44–47] to Hedin’s equations [48],
the molecular Dyson equation (MDE) [49], or Kadanoff–
Baym equations [50–53], often predict transport properties in
close agreement with observed values [49, 54–56], although
these methods are generally computationally intractable for
large systems. The development of new transport approaches
(e.g., the hybrid DFT+Σ method from Refs. [57–59]) is
currently an active field of research. However, a scalable nonequilibrium theory which simultaneously describes both the
1 Landauer’s often cited 1957 paper [23] does not actually include the
now well-known “Landauer equation” (1) connecting conductance to the
electronic transmission (cf. comment in Ref. [38]). However, it does introduce the original notions that lead to the development of those relationships
[29, 31].
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the donor (benzene) to the acceptor (naphthalene) through
the bridge, they should in many cases be proportional to one
another. This proportionality was noticed by Nitzan [60, 61],
and his rules of thumb to compare transfer with transport
through given molecules provide substantial help in qualitative understanding, approaching quantitative comparison in
some cases [73–75].
Figure 2 Schematic representations of charge transfer and charge
transport processes. (a) In the charge transfer example, an electron tunnels from the benzene donor to the acceptor naphthalene
via the intervening bridge. (b) When the molecule is connected to
macroscopic electrodes a junction is formed and injected charges
move, for instance, from the left-hand electrode to the right-hand
electrode via the molecule. These two scenarios are very different;
however, they both involve quantum tunneling and can be related
to one another to a certain degree and under certain circumstances
[60, 61].
particle-like (Coulomb blockade) and wave-like (coherent)
aspects of transport [49] has yet to be realized.
Finally, the actual preparation of the molecules of interest has been extensive. Starting with simple π-systems and
alkanes, extending to different sorts of molecular oligomers
and possible switches, varying the end-groups linking the
molecule to electrodes [57, 62–64], creating molecules with
different features to pursue concepts in molecular transport
[64–67] – this synthetic aspect, joined with the modeling and
measurement capabilities, has led to the widespread interest
that the field of molecular electronics now enjoys. In the
first decade of the 21st century, molecular electronics made
great strides. There are many excellent reviews and presentations and textbooks, some singularly beautiful experiments,
major advances in materials fabrication and utilization, and
progressive advances in modeling methodology.
1.2 Relationships between intramolecular electron transfer and molecular transport between
electrodes In Fig. 2, we sketch these two situations. In
the electron transfer scenario of Fig. 2a, the electron moves
from the benzene to the naphthalene by tunneling through
the intervening ether. The measurement of the associated
rate constants, particularly via photoexcitation experiments,
is quite common in organic and physical chemistry laboratories, where they can be used both for fundamental
understandings and for applications toward photovoltaics and
molecular energy transfer systems [69–72]. Figure 2b shows
a comparable experiment with the addition of two macroscopic metallic electrodes bonded to the molecule via the
amine (NH2 ) end-groups. In this case we have an open quantum system, and electrons occupy scattering states which
originate in the left electrode, pass through the molecule and
the NH2 binding groups, and continue into the right electrode. Fundamentally, these are two very different physical
systems. However, because the rates for both electron transfer and electron transport depend on quantum tunneling from
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1.3 Barrier tunneling Most of the molecules that
were measured in the early days of break junction technique
could be characterized as organics, perhaps with some
small component of inorganic chemistry (like the metal
complexes shown in Fig. 3, measured using a mechanical
break junction by the group at Cornell [68]). The similarity
to intramolecular electron transfer (including mixed valence
systems that echoed Hush’s original analysis, and Taube’s
original measurements) permits direct comparison with
molecular transport through such junctions, using Nitzan’s
formulation [60, 61]. This could provide an important qualitative understanding of how structure relates to transport
through molecular junctions.
It is often true that one can simplify transport through a
molecule in terms of tunneling through a particular barrier,
along a one-dimensional pathway from cathode to anode.
An analysis of such situations was given by Simmons in
the 1960s [76], and this analysis continues to be used very
broadly. For instance, Beebe et al. used the Simmons tunnel barrier model to interpret transition voltage spectroscopy
(TVS) [77] measurements, arguing that the minimum of a
Fowler–Nordheim graph (i.e., ln(I/V 2 ) vs. 1/V ) was related
to the energy alignment between the closest molecular level
and the electrodes’ Fermi energies. Although Beebe’s interpretation has since been disputed and clarified [78, 79], TVS
has become an important experimental technique to probe
single-molecule junction properties [77, 80–85] and often
relies on the Simmons formula and its parameters characterizing the height, width, and skewness of the barrier, to
understand (at least qualitatively) molecular transport.
However, barrier tunneling is not the only way in which
charge can transfer through molecules. The one-dimensional
potential is a strong simplification and, although many
molecular systems can be analyzed using a simple barrier
tunneling model, the attention of the field has focused to
slightly more complex systems, where quantum pathways
and chemical structure become relevant.
1.4 Interference, intramolecular, and intermolecular Interference effects are ubiquitous in quantum
mechanics, appearing whenever “which way” information
cannot be determined. The simplest example of interference
through a molecular junction comes from a structure called
a “stub” resonator in electrical engineering. The molecular
manifestation of such a devices is sketched in the top panel
of Fig. 4a, where the two distinct transport pathways are
indicated by the black and red lines. In this system, the
indirect transport path between the electrodes (red line) is
π radians out-of-phase with direct path (black line) at the
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J. P. Bergfield and M. A. Ratner: Forty years of molecular electronics
Figure 3 (a) A schematic of the terpyridinyl metal complexes
used in Ref. [68] is shown along with (b) the I–V curves of a
[Co(tpy–(CH2 )5 –SH)2 ]2+ for several gate voltages Vg (red and black
correspond to 20.4 and 21.0 V, respectively). (c) A “Coulomb diamond” image – a map of the differential conductance (G = dI/dV )
as a function of the gate and bias voltages. At zero bias, the conductance is peaked when the gate voltage is tuned into resonance
with a transition from an N to an N + 1 state (here, ∼0.35 V). In
the center of the diamonds (dark pink) charge fluctuations may be
neglected, whereas at finite bias between the particle and hole-lines
(light pink) multiple charge states contribute to the transport. Conductance peaks correspond to 500 nS. Adapted by permission from
Macmillan Publishers Ltd: Nature [68], copyright 2002.
mid-gap energy (E = 0). When these two coherent channels
combine they interfere and the resulting transmission spectrum presents a deep hole at this anti-resonance (i.e., a node),
as indicated in the top right-hand column of the figure. The
phase shift between paths can be engineered, for example,
by substituting the methylene group with an oxygen. This
is characteristic of molecular pathway interference, and can
be directly measured utilizing break junction techniques
[92, 94] (see Fig. 5). The cross-conjugated species shown in
the lower panel of Fig. 4a also has a low conductance value
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Figure 4 The calculated transmission function through (a) two
“stub” resonator molecules, and (b) two cyclic molecules. (a) Transport paths 1 and 2 interfere destructively in the center of the
fundamental gap of the molecule (E = 0). The cross-conjugated
molecule shown in the lower panel exhibits a qualitatively similar destructive interference feature [86–88]. (b) The archetypal
meta and para configured benzene junctions show destructive and
constructive interferences, respectively, at the mid-gap energy.
Interference in these cyclic structures can be thought of as a transport analog of a Mach–Zehnder interferometer, where the two
unique Feynman paths between the electrodes replace the optical paths [89, 90]. These spectra were calculated using a Hückel
level description of the molecular electronic structure and a lead–
molecule coupling strength of 0.5 eV. Although this method neglects
electron correlation completely, qualitatively similar spectra are
found even when correlations (e.g., within a model π-system) are
included exactly [49, 88, 91]. It is intriguing that simple topological rules should remain valid even when strong electron–electron
correlations are included.
near the particle–hole symmetric point, essentially because
it is a molecular manifestation of the stub resonator [86–88].
As another example, Fig. 4b shows transport through two
closely related molecules. The meta-connected ring provides
two different pathways whose lengths are different. Once
again there is a strong negative interference and the meta
linkage does not conduct as well as the para linkage, as
indicated in the lowest panel of the figure. Recently, this
effect was observed in stilbene derivatives [95] (see Fig. 6).
Analyzing the roles of interference (both constructive
and destructive) is a very active part of molecular electronics at the moment, both theoretically [86, 88–90, 96–101]
and experimentally [92, 94, 95]. Recently, Solomon et al.
[102] worked out an intuitive approach to map the local
current in a molecule based on current continuity between
planes perpendicular to the inter-electrode axis. Their method
illustrates the intricate cancellation of transport necessary
for complete destructive interference. In unpublished work,
Chen and colleagues have used real time density functional
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Figure 5 The differential conductance (G = dI/dV ) is shown on
a logscale as a function of bias voltage (lower four panels) for
several molecules (top left panel) using a conducting AFM device
(indicated schematically in the top right panel). These spectra are
experimental evidence of quantum interference in the transport
through cross-conjugated molecules; in linear-response, G is essentially the transmission function (in units of G0 ) evaluated at the
Fermi energy of the electrodes (cf. Eq. (1)). The destructive interference feature shown in the AQ-MT panel, for example, appears
to be similar to the cross-conjugated model calculations shown in
Fig. 4. These measurements were performed for junctions operating at room temperature. Adapted by permission from Macmillan
Publishers Ltd: Nature [92], copyright 2012.
theory methods, along with the NEGF approach, to plot how
the currents temporally build up in the molecules, starting at
zero voltage and then increasing. Their work highlights the
interplay between decoherence and the mixing of molecular transport pathways, and offers a new perspective on how
interferences develop. Understanding how interference features are affected (or not affected) by chemical structure,
molecular vibrations, dephasing, and electron–electron interactions is still a major challenge and opportunity in the field
of molecular electronics.
1.5 Molecular motion in junctions One can imagine a molecule strung between two electrodes, and then
lowered in temperature such that all of the phonons in the
electrodes, and the vibrons in the molecule, are in their
ground states. Zero point motion would still be permitted,
but the evolution of the geometry would be very small, and
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Figure 6 The force exerted F and transmission T during a simulated pulling experiment. Typical structures are shown in the top
portion of the figure. Reprinted with permission from [93]. Copyright 2011 American Chemical Society.
the results of a repeated measurement would probably form
very narrow histograms, or perhaps truly single structures.
Upon raising the temperature, however, molecular motions
in the junctions will soon come into play. The geometry will
evolve, and the molecule could switch between different minima on its potential energy surface. Indeed, the molecule
could be “coaxed” to do so either by vibrational excitation
from a photon field or by molecular manipulation, in particular involving pulling apart of the electrodes. Depending on
the particular molecule, the pulling may permit it to explore
very different regions of geometry space, and therefore to
exhibit very different transport [93, 103, 104]. The rate of
pulling (i.e., the rate at which the two electrodes are separated with the single molecule stretched between them) has
a great effect on what can happen – pulling too fast can lead
to molecular fracture, while pulling slowly enough permits
the molecule to explore all of the orientational and geometric
space, constrained only by the attachment of the molecular
termini to the two electrodes.
Figure 7 shows some results of simulations on single
molecules. The structural peaks in the pulling diagrams
(which plot either the force or the transmission as a function of time or inter-electrode displacement) can vary richly
as the molecules become more complex, and as the space
that they explore becomes more complicated. These pulling
experiments not only help to characterize molecular geometric change and binding, but can also be used for storage of
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Figure 7 Using a modified conducting atomic force microscope (AFM), shown schematically in panel (a), Aradhya et al. experimentally
measure the force and conductance simultaneously. The large discrepancy in conductance in the vicinity of the final step in force (see
black arrow in panels b and c) is a direct observation of the para versus meta interference effect discussed in Fig. 4b. Reprinted (adapted)
with permission from [95]. Copyright 2012 American Chemical Society.
energy at the nanoscale (utilizing what is effectively a Carnot
cycle [105]), and to understand biological species such as titin
[106, 107] and DNA [108].
degree of freedom which has become an invaluable tool, for
instance, to confirm the formation of a molecular junction
[123].
1.6 Inelastic electron tunneling spectroscopy If
a single molecule is placed between two electrodes, the temperature is taken to a low value, and the voltage is slowly
swept starting from zero, interesting vibrational properties
can manifest themselves. For example, as the voltage across
a junction approaches V = ω/e (where is Planck’s constant, ω is a vibrational frequency that couples with the
extension of the molecule, and e is the electron charge),
the current could either increase (because of the availability of a new channel) or decrease (because the transport was
very close to the quantum of transport if the channel was
effectively perfect, and the vibration provides an interfering
structure that reduces the current) [109, 110].
Inelastic electron tunneling spectroscopy (IETS) experiments [111] of this type were first introduced for molecular
junctions by the Reed group in the early 21st century [112];
their utilization to provide information on molecular mechanisms and geometries have been a featured part of molecular
electronics [110, 113–116], and some new measurements on
beautifully characterized systems from the Ho laboratory
[117–119] await our understanding. IETS can also be used,
in favorable situations, to predict the actual pathways that
the electrons take as they move through the molecule [120].
This topic has been generalized [121], and maps have been
drawn to show how the electrons travel through the molecule
on their journey from anode to cathode. For most simple
organic molecules, the electronic charging energy Ec Γ
(the lead–molecule coupling energy Γ ) meaning that the
Landauer–Büttiker contact time is small and the electron–
phonon coupling is weak [122]. Although the inelastic
contribution to transport is typically small (i.e., only visible in d2 I/dV 2 vs. V ), molecular vibrations introduce a new
1.7 Broken symmetry The discussion so far has dealt
with the simplest possible structure – a single molecule strung
between two electrodes. Some extensions of this are among
the most exciting areas in molecular electronics at the present
moment. The first of these is quite simple. Suppose that two
molecules are caught in the junction at the same time: Is the
current going through the two molecules the same as twice
the current through a single molecule? Previous theoretical
studies have shown that the transport through two [124–126]
or more [127, 128] molecules connected in parallel (which
interact via their joint linkage to the electrodes, or directly
with each other) can result in transport that is larger than,
equal to, or smaller than twice the transport through a single molecule. This effect can be understood in analogy with
an optical double-slit experiment, where quantum coherence
can mix the transport paths, giving twice the conductance
for complete constructive interference and close to zero for
complete destructive interference. Although this cooperative effect was not observed in a previous parallel molecular
wire experiment [129], it was recently observed in singlemolecule junctions composed of a molecule with parallel
branches (i.e., 2,11-dithia(3,3)paracyclophane) [130].
Other forms of broken symmetry are perhaps even more
significant. Molecular spintronics (using unpaired spins on
the molecule, or on the electrodes) is beginning as a field
of its own [131]. Spintronic devices are technologically
attractive because spin excitations are generally the lowenergy excitations of a system (compared with charge
excitations) and spin quasiparticles interact weakly with
one another, making them ideal for low-power logical circuits or possibly for quantum computing applications [132].
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However, in order to distinguish spin species, time-reversal
symmetry must be broken (e.g., via a magnetic field or
chemical structure). The role of molecular chirality on spin
transport has been investigated [133–136], as has excitation
by polarized light [137], molecules with asymmetric twists
(e.g., in DNA [137–140]), and transport through molecular
radicals with an applied magnetic field [141]. The realization
that, for elements beyond the first row of the periodic chart,
the Rashba Hamiltonian suggests that the spin orbit mixing will be even greater than might otherwise be expected,
coupled with the availability of specially designed molecules
with chiral properties and spin localization, promises to make
this a vibrant part of the molecular electronics field going
forward.
1.8 Noise Because most transport measurements are
truly single molecule phenomena, noise is definitely
a factor
√
to consider. Generally, fluctuations scale as 1/ N, where N
is the number of entities in the observation. This suggests that
single molecules will have large noise signatures. At low temperatures, the fundamental source of noise through a (fixed)
nanostructure is the shot noise: variations in the arrival time
of the electrons which result from the granularity of charge
[142]. In quantum systems, the shot noise can be used to
determine the charge and distribution of the quasiparticles
responsible for transport. The shot noise is also sensitive to
electron–electron interactions, the number of transmission
channels through a junction [142], and inelastic contribution to transport [143]. In their pioneering work, Djukic and
Van Ruitenbeek [144] examined the shot-noise in a singlemolecule Pt–D2 –Pt junction and determined the number of
channels supported by the junction. In a subsequent work,
Kiguchi et al. [16] measured the shot-noise in Pt–benzene–Pt
junctions. Although the theoretical connection between the
number of channels and the shot-noise is well established
in the elastic limit [142], the connection between molecular structure, shot-noise, and the number of channels is a bit
more subtle [145, 146].
Unlike the conductance, which may be described in terms
of one-body response functions, noise is a two-body observable and requires a two-body response function to describe in
general. Two-body response functions are beyond the scope
of existing effective single-particle theories, such as DFT, and
require alternative theoretical techniques. In the full counting
statistics (FCS) [147, 148] approach, the full response of a
junction is characterized by its cumulants. For example, using
a single-particle theory to describe a single-channel junction, it can be shown that the first cumulant is related to the
junction transmission function, while the second cumulant is
related to the shot-noise suppression. FCS is being pursued
in several laboratories, as are other sophisticated analyses of
the noise problem (e.g., using the Bethe–Salpeter equation).
Once again, this offers an opportunity based on a few early
and pioneering measurements and models, to utilize noise
measurements to tell us more about structure and function of
molecular transport junctions.
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1.9 Chemical reactions in molecular junctions
The idea of single molecule spectroscopy underlies a great
deal of the interest, and many of the possible useful
and creative measurements, in molecular charge transport.
Since molecular structure and energetics determine transport
through single molecule junctions, it is not surprising that if
the molecules undergo any sort of chemical reaction, the conductance will change. For example, the “Coulomb diamond”
image shown in the bottom panel of Fig. 3 is a kind of reaction fingerprint, showing both the non-equilibrium transport
at finite bias and also how the conductance changes as the
molecule is reduced.
Computationally, such analyses go back more than two
decades [149]. Simple chemical reactions such as a charge
transfer interaction between an aromatic molecule and an
electron donor, like I2 , are responsible for Mulliken charge
transfer bands in the optical spectrum. The interaction is
a weak one, but it is typical of non-covalent bonding in
molecular adducts. The computational suggestion that such
a modification of the electronic structure of the aromatic
by interaction with iodine vapor would actually change the
transmission, and therefore the conductance, was made long
ago on the basis of simple Landauer theory [149]. Similarly, as yet unpublished work utilizing the full DFT + NEGF
approach has examined the interactions of univalent cations
(alkali metal ions) on molecular hosts. The change in transport spectrum is sufficient to distinguish among protons, Li,
Na, and K. Far more interesting are photoisomerizations.
Both experimental and theoretical work [150] have examined a series of dyes developed by Tsujioka and Irie [151],
in which photoisomerization using one wavelength switches
from a closed ring to an open ring structure, and its reversal can be attained by photoexcitation at a lower frequency.
Again, simple Landauer-type arguments can be used (and
have been used) to understand such experiments, and the
theory/experimental agreement is satisfactory [152].
In principle, such reactions could form the basis of a
light detection device, but molecular adsorptions are quite
weak. Therefore, such transmission modifications with photons might be useful for linked devices and new logic gates,
especially since they seem to be robust to multiple measurements, and the relatively sparse one-photon excitation
spectrum of most organics makes the transition quite binary
(either it absorbs and switches, or it does neither). Extensions
of these experiments to deal with processes such as photopolymerization, photodecay, photo-driven dynamic states,
and photobinding could all be undertaken.
2 Into the future The advent of molecular electronics
followed directly from the development of new techniques –
the Landauer and non-equilibrium Green’s function (NEGF)
[23–37, 39–42] technique for computation, and the break
junction technique for measurement. Future theoretical
approaches which describe important many-body phenomena, such as the complementary nature of the (quasi)particles
responsible for transport, or the Kondo effect [153] (a manyelectron phenomenon arising from correlations between a
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localized unpaired spin and the metal electrodes’ conduction
electrons, which has been observed in molecular junctions
[68]) will certainly be useful to describe important aspects
of future experiments. However, these techniques will not
necessarily be required for molecular electronics to progress
away from a fascination with the very simple question (which
has dominated the last four decades) of how current travels
through molecular junctions. Instead, the future will hold
more complex and more interesting issues, issues that utilize
the particular behavior of molecules when subjected both
to electromagnetic forces and to mechanical forces (as in
the mechanical break junction). Computationally, the use of
different simplifying model approaches, such as the Hubbard (or extended Hubbard, or Pariser–Parr–Pople) methods
[154, 155], or Holstein models, or the (very useful) extended
Hückel [156] and tight-binding DFT [157] schemes, will be
a key advantage.
Prognostication is difficult, but six particular issues we
believe will certainly be a part of the future of molecular
electronics include:
2.1 Broken symmetry This is a continuation of the
discussion above – when the symmetry of free-space is broken (e.g., by the polarization of photons, molecular structure,
or the application of a voltage, temperature gradient, or magnetic field), the transport rules that normally follow from
existing DFT + NEGF methods do not necessarily work any
longer. This could have major implications for fundamental
understanding. For instance, with ferromagnetic electrodes
and a non-zero spin molecule, exactly how do the majority
and minority carriers proceed, and how does that change with
field strength, geometry, and temperature? Moreover, broken
symmetry could result in substantial changes in transport,
which are not deducible from simple arguments concerning barriers. The Pauli exclusion principle is in play here,
and therefore one could imagine using broken symmetry to
completely block the transport of the minority or majority
carriers. Dealing with these problems of spin behavior, especially if they are coupled with polarization and/or chirality,
will require extensive expansion of the simple methodologies
now used for calculation of transport. Some preliminary work
in this area has already been published, but both experiment
and modeling will almost certainly extend their investigation
of broken symmetry situations.
2.2 New variables The independent variables in simple transport are the structure of the molecule and its
environment, and the nature of the external potential or
potentials applied to the molecule. The beginnings of thermoelectric behavior in molecular junctions have already
appeared, and are discussed in Section 3. Similarly, spintronics measures the transport of spins, while memristors
correspond to two-terminal non-volatile memory devices
based on resistance switching – there has been extensive
work trying to develop this since it was first proposed by
Chua [158]; the HP laboratories have developed what may be
a semiconductor-based memristor [159], and Chen’s group
© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
in Singapore has actually demonstrated memristor behavior
in a molecular junction using the biomolecule ferritin as the
functional component [160].
Other new variables would certainly include mixed photonic excitation (which could provide time-dependent pulses
to drive transport in different directions), magnetic fields
across the device, ultrasonic excitation, and non-equilibrium
chemical reactions resulting in oscillatory behavior. While
not all of these will be examined, these new variables
will once again require further understanding, appropriate
experiments, and appropriate extension of current theoretical
models.
2.3 Dynamic molecular modification in the junction While there already exist several analyses of photo
switching molecules within junctions [161–163], and of the
effects of binding of molecules within junctions, much more
can be done in this area. Dynamical modification on different
time scales and with different mechanisms (polarized light,
other molecules, vibrational excitation, etc.) can lead to quite
different behaviors, and this could be useful in detection, but
also in understanding precisely how we can modify the transport by modifying physical and chemical properties. Given
the simplicity of break junctions, it is fair to anticipate that we
will see much more of this fundamentally chemical behavior
being controlled and measured.
2.4 Decoherence and control Coherence within the
molecular entity can result in substantial change that is not
explicable in terms of the standard barrier tunneling picture. Such interference behaviors are often subsumed in the
wonderful approach taken historically by physical organic
chemistry [164]. But there are many open questions here.
Why do these single particle ideas seem robust even in some
many-particle problems? How does decoherence happen?
What forms of external decoherence can be applied, and
what do they do? Can we use coherent processes to control
reactions, and use decoherence to revert? This is one of the
most active areas in single-molecule transport at the moment,
and it promises to provide significant behaviors both in simple transport and in more complicated situations, such as
thermoelectricity.
2.5 The bio universe While transport has been measured in DNA and in some organic molecules, applications to
true biological systems remain largely unexplored. Electron
transfer processes are crucial in the world of enzymes, and the
nature of homeostatic control throughout biology depends on
fluxes and therefore might be better understood if biomolecular systems were subjected to transport measurement and
understanding. Given the complexity of charge motions in
biology, measurements of single molecule transport might
be effective in deducing mechanisms of redox processes and
of very long range electron transfer, especially in protein
systems.
Biological systems are complex, making it unlikely that
any truly ab intio quantum mechanical description will
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become feasible in the near future. Instead, it seems plausible
that new empirical approaches will be found based on the
(yet to be discovered) emergent rules governing biological
processes [165].
2.6 Devices The world of the 21st century depends
crucially on semiconductor devices, and almost all of those
semiconductor devices depend crucially upon charge transport. The transistor, the chip, and the digital computer have
changed both science and the world in remarkable ways,
and this will continue. Molecular devices are imbued naturally with symmetries and other properties which may
be harnessed to address future challenges. In Section 3 of
this overview we discuss thermoelectrics, one particularly
promising direction for molecular systems, since their transport can exhibit significant variations, which Mott suggested
[166, 167] may result in strong thermoelectric behavior.
Other devices for measuring chemical reactions, for detection, for analysis, and for memory might well be developed
based on the understandings of molecular electronics.
3 Molecular thermoelectrics Thermoelectric (TE)
devices are highly desirable since they can directly convert between thermal and electrical energy without the need
for any mechanical components or, for example, the emission of chlorofluorocarbons [170, 171]. Electrical power can
be applied to a TE device to either heat or cool adjoining
reservoirs (Peltier effect) or alternatively, the flow of heat
(e.g., from a factory or car exhaust) can be converted into
usable electrical power (Seebeck effect). In addition to the
desire for efficient thermoelectric materials, there is also a
growing interest in understanding fundamental issues associated with heat transport and the thermoelectric effect in
single-molecule junctions [172].
When a temperature gradient is applied across a material,
charges diffuse from the hot to cold end until the field built
up by the accumulation of charge is sufficient to cancel the
thermodynamic current. The thermopower (or Seebeck coefficient S) is a two-terminal, linear-response property defined
in the limit of zero charge current as
S = − lim
ΔT →0
ΔV ,
ΔT I=0
(2)
where ΔT is the applied temperature gradient, ΔV is the
potential difference, and I is the electrical current between
the electrodes. Away from any molecular resonances, a junction’s thermopower may be approximated using the so-called
Mott formula 2 [166, 167, 173]
SM (μ, T ) = −
π 2 kB
∂ ln G (μ, T )
kB T
,
3 |e|
∂E
(3)
2 In their original work, Mott and colleagues considered the zerotemperature limit [166, 167].
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where μ and T are the chemical potential and temperature of
the electrodes, respectively, kB is Boltzmann’s constant, e is
the magnitude of the electron charge, and G is the electrical
conductance of the junction. Although Eq. (3) is not exact
(e.g., it breaks down near transmission nodes [49, 174]), it is
still tremendously useful and typically very accurate.
The natural unit of thermopower is entropy per unit
charge (kB /e), highlighting the physical interpretation of the
magnitude of the thermopower as a measure of the coupling strength between heat (entropic) and charge degrees
of freedom. Measurements of S are equivalently reported in
μV/K (∼0.012kB /e). In contrast to the conductance, which
can only be positive, the thermopower has a sign which
indicates the nature of the charge carriers: positive values for hole-dominated transport, and negative values for
electron-dominated transport. The thermopower is therefore
an important probe of the energy level alignment between
the molecule and electrodes [175, 176]. For example, if the
work function (WF) of the electrodes is closer to the ionization potential (∼HOMO level) of the adsorbed molecule, the
thermopower will be positive and (with an accurate description of the molecular energy levels) the energy mismatch can
be found. Finally, Eq. (3) also shows us a way to maximize
S: maximize variations in G.
Recently, the thermoelectric response of a singlemolecule junction was measured using a modified scanning
tunneling microscope (STM) [168, 169, 177–180], similar to
the device first developed by Xu and Tao [181] to measure single-molecule conductances. In these experiments, the
modified STM is repeatedly moved up and down over a substrate prepared with adsorbed molecules, quickly forming,
measuring, and breaking thousands of molecular junctions.
The thermoelectric voltage ΔV produced in response to the
applied temperature gradient ΔT is measured each cycle, and
histograms of ΔV as a function of ΔT are used to determine
S. Experiments using a modified atomic force microscope
(AFM) report similar results [182]. A schematic representation of this experimental method is shown in Fig. 8a, and
several representative traces of the thermoelectic voltage
measured through Au–1,4-benzenedithiol–Au junctions are
shown in Fig. 8b. Histograms built from similar experimental
measurements [169] are shown in Fig. 8c. These experiments
are performed at room temperature.
There are several remarkable aspects of these experiments. First, the thermoelectric voltages are largely
insensitive to the details of the lead–molecule coupling [175],
remaining nearly constant even as the Au tip deforms and
moves over ∼50 Å (notice the 0.5 nm reference bar)! Second,
even under these conditions the thermopower is sensitive to
chemical structure. For example, the junction transport can be
adjusted from hole-dominated for Au–1,4-benzenedithiol–
Au to electron-dominated when the thiol (SH) end-groups
are replaced by cyanide (CN). These experiments show that
the thermopower can be used to probe chemical symmetry,
and that the chemical symmetry can be used to engineer
efficient thermoelectric materials. Finally, unlike the conductance where the signal decays with molecular length, the
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J. P. Bergfield and M. A. Ratner: Forty years of molecular electronics
Figure 8 (a) A schematic diagram of a single-molecule thermopower experiment using a modified Au STM tip. The tip is moved up and
down over a Au substrate with adsorbed molecules, forming many molecular junctions. For each junction, the thermoelectric voltage ΔV
is measured in response to the applied temperature gradient ΔT = Tcold − Thot , and the thermopower Sjunction is deduced. Experimentally,
Tcold is often just the ambient temperature (i.e., room temperature). (b) A representative trace of ΔV as a function of the tip-surface distance
for several values of ΔT . (c) The histograms constructed from the repeated junction formation and thermoelectric voltage measurement
are used to determine the molecular thermopower. The effect of substituent and end-groups are clearly visible in the thermoelectric
response, even though they may not be from the conductance alone. Panel (b) is reprinted with permission from Ref. [168]. Copyright
2007 American Chemical Society. Panel (c) is reprinted with permission from Ref. [169]. Copyright 2008 American Chemical Society.
Seebeck coefficient of molecules has been shown to generally
increase with molecular length [59, 168, 177, 180, 183, 184].
These experiments provide evidence that in the ballistic
regime, the thermopower does not strongly depend on the
number of molecules that bridge the two electrodes. Therefore, thermoelectric materials based on molecules where the
transport is predominantly coherent can be designed in terms
of single-molecule responses; this is in stark contrast to the
conductance, which generally increases with the number of
molecules bridging the electrodes. Certain molecular junctions are also predicted to exhibit large quantum enhanced
thermoelectric response near constructive and destructive
transport interferences [91, 174] and many-body spectral features [100], and to be enhanced by sharp molecular spectral
features [185], suggesting even more exciting experiments
to come.
3.1 Thermoelectric device performance Often,
the efficiency of a TE device is characterized by the dimensionless figure-of-merit
ZT =
S 2 GT
,
κel + κph
© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
(4)
constructed with the rationale that an efficient TE device
should simultaneously maximize the electrical conductance
G so that current can flow without much Joule heating,
minimize the thermal conductance κ = κel + κph in order
to maintain a temperature gradient across the device, and
maximize the Seebeck coefficient S to ensure that the coupling between the electronic and thermal currents is as
large as possible [170, 171]. If a TE material were found
exhibiting ZT ≥ 4, it would constitute a commercially viable
green-energy solution for many heating and cooling problems at both the macro- and nano-scales, with no operational
carbon footprint [170].
Equation (4) is the mathematical form of the engineering rule: an efficient thermoelectric material is a phonon
glass and an electron crystal. In many bulk materials, such
as Bi2 Te3 , the approach has been to maximize G, and S (by
maximizing variation in G, see Eq. 3), and to minimize κph .
The minimization of κel is often not considered in bulk systems, since the Wiedemann–Franz (WF) law κel /G = LT is
quite accurate [186], where L is the Lorenz number. After
decades of intense research, ZT has only changed marginally
[187]. Currently, the best TE materials available in the laboratory exhibit ZT ∼ 3, whereas in commercially available
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TE devices ZT ∼ 1, owing to major material engineering,
packaging and fabrication challenges [170, 171].
Although both the phonon flux and the electronic current carry heat in a molecular junction, in small organic
molecules the electron–phonon coupling is typically weak
and the Debye frequency of the metallic electrodes [188]
is typically incommensurate with the molecular vibration
spectrum. Therefore, in linear-response and near room temperature, the heat current in these junctions is carried
predominantly by the electrons. The theory of phonon heat
transport through a nanoscale constriction has been studied extensively [172, 189–194] and is beyond the scope of
this article. Recently, indirect power dissipation experiments
were used to estimate κ ∼ 40 pW K−1 [195]. In an earlier
study, Wang et al. [196] deduced κ ∼ 50 pW K−1 for a single alkanethiol using thermal conductance measurements of
a monolayer. These observations are encouraging and suggest that molecular electronics may provide a way to address
longstanding technological challenges in the field of thermoelectrics. Recently, scanning probe calorimetric devices with
picowatt sensitivity and nanometer spatial resolution have
been developed [197, 198], hinting at the exciting prospect
of observing the energy transport through a single molecule.
Thermodynamically, a system’s response is characterized by the efficiency η with which heat can be converted into
usable power P and the amount of power that can be generated. Since ZT is an ad hoc rule-of-thumb metric; it is not
expected to describe the thermodynamic response in general.
However, in linear-response we can derive the following
ηmax
√
1 + ZT − 1
= ηC √
,
1 + ZT + TTcold
hot
(5)
where ηC = 1 − Tcold /Thot is the Carnot efficiency, and Tα is
the temperature of reservoir α. When ZT → ∞, ηmax → ηC .
A recent investigation of the nonlinear heat transport in
a molecular junction using both many-body and effective
single-particle approaches found that variations of ZT are in
good qualitative agreement with η (see Fig. 9), but that P
requires an accurate description of electron–electron interactions [174]. In some ring-based molecular junctions, the
thermodynamic efficiency shows additional enhancement at
finite bias [174], suggesting a possible subject for further
investigation.
In general, ZT and η are difficult to maximize because the
transport of heat and charge are highly correlated with one
another [199–201], a fact that becomes more pronounced at
the nanoscale where the number of degrees of freedom available is small. However, a key advantage of many molecular
junction devices is that the transport is essentially ballistic;
the wave-like (coherent) nature of the transport is dominant. In the vicinity of interference features, S is predicted
to show dramatic enhancement [91] that scales with the
order of the interference feature [174], giving large values
for both ZT and η. Moreover, quantum interference effects
can be engineered with atomic precision via synthetic chemwww.pss-b.com
Figure 9 The calculated linear (top half) and nonlinear (bottom
half) electrical and thermodynamic response of a Au–1,3benzenedithiol–Au junction (π-electron model only) with lead
temperatures T1 = 300 K and T2 = 250 K. (Top half) The transmission probability T (E) and figure-of-merit ZT el are shown as
a function of the gate electrode’s chemical potential (fixed by the
voltage of the third electrode). Near the transmission node at the
mid-gap energy (μ0 ), ZT el is substantially enhanced [174]. (Bottom half) The Carnot-normalized efficiency η/ηC , and electrical
power output P are shown as functions of bias voltage ΔV and gate
potential μ. Variations in ZT el and η are generally similar (near
ΔV ≈ 0) while P remains constant, showing that ZT el is not a complete characterization of a thermoelectric device’s thermodynamic
response. Note that the non-equilibrium thermodynamic response
is maximized at finite bias. Reprinted with permission from [174].
Copyright 2010 American Chemical Society.
istry [92, 95]. Considering these unique aspects of transport
through molecules, in addition to the observed low heat conductivity, insensitivity to the lead–molecule interface, and
wide range of material properties, it is easy to be optimistic
about the future of molecular thermoelectric materials.
3.2 Quantum theory of electronic heat transport
In this section, we briefly outline the equations necessary
to describe the heat transport in a junction composed of M
macroscopic metallic electrodes bonded to an arbitrary interacting nanoscale system. First, we derive an exact expression
for the heat (and charge) current in an interacting nanostructure [91]. Then, together with the corresponding expression
for the electrical current [25, 26], we show how to find expressions for the linear and nonlinear thermoelectric response of
a junction.
The Hamiltonian of this open quantum system may be
partitioned as
H = Hmol +
M
α
Hlead + HTα ,
(6)
α=1
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α
where Hmol is the molecular Hamiltonian, and Hlead
and HTα
are the lead and tunneling Hamiltonians for lead α, respectively. Each electrode is modeled as a bath of non-interacting
Fermions.
Following Ref. [91], the starting point for our derivation
of the heat current is the fundamental thermodynamic identity
at constant volume
T dS = dE − μ dN,
(7)
where T , S, E, and N are temperature, entropy, internal
energy, and particle number, respectively. Applying the identity to electrode α, one finds
IαQ ≡ Tα
dSα
d (α) d
=
Hlead − μα Nα ,
dt
dt
dt
(8)
where IαQ is the heat current flowing from the molecule
into electrode α, and Tα and μα are the temperature and
chemical potential, respectively, of electrode α. . The time
derivatives on the r.h.s. of Eq. (8) may be evaluated using
standard quantum mechanics to obtain
i α
Hlead , H − μα [Nα , H]
i †
†
(εkσ −μα ) Vnk dnσ
ckσ −Vnk∗ ckσ
dnσ .
=
k∈α
IαQ = −
(9)
n, σ
Following the same general procedure as Ref. [25], we can
express the correlation functions in the second line of Eq. (9)
using non-equilibrium Green’s function (NEGF) formalism
[25–28, 30, 32–37, 40] in conjunction with Dyson’s equation.
This leads to the following general result [91]
Iα(ν) = −
i
h
dE(E − μα )ν ×
Tr Γ α (E) G< (E) + fα (E) G(E)−G† (E)
,
(10)
where Iα(1) = IαQ gives the heat current [91] and −eIα(0) is
the Meir–Wingreen [25, 26] expression for the charge current, fα (E) is the Fermi–Dirac distribution for electrode α,
Γ α (E) is the tunneling-width matrix describing the bonding
between the molecule and electrode α, and G(E) and G< (E)
are Fourier transforms of the retarded and Keldysh “lesser”
Green’s functions
number of electrodes; the sum over electrodes appears in
the tunneling self-energies used to construct the junction’s
Green’s functions.
Equation (10) is exact. However, the quantum manybody problem of a macroscopic number of electrons and
vibrations is not tractable in general, meaning that approximations for the junction’s Green’s functions are often
necessary. Although a detailed review of the theories used
to approximate G(E) is beyond the scope of this article,
Green’s function approaches may roughly be categorized
into two classes: many-body perturbation theories, such as
those based on the GW [56, 45–47, 202, 203] approximation
to Hedin’s equations [48], the molecular Dyson equation
[49], or the Kadanoff–Baym equations [50–53], and effective single-particle methods such as those based on the
Kohn–Sham scheme of density functional theory (KS-DFT)
[32, 39–42].
Within the NEGF framework, the junction’s Green’s
function G(E) is typically phrased as a perturbative series
in terms of non-interacting Green’s functions G0 (E) (often
derived from an effective single-particle theory) and a selfenergy Σ(E). Using Dyson’s equation, G(E) = G0 (E) +
G0 (E)Σ(E)G(E), a few physical processes (described by Σ)
may be included to infinite order. An alternative approach is
to solve the mathematical problem of the relevant molecular system (e.g., the π-system) exactly (or approximately)
and to treat electron hopping between molecule and electrodes as a perturbation [194, 204–210]. In a sense, density
matrix approaches are complementary since all processes are
included to infinite order. Transport through small organic
molecules is challenging since both wave-like (resonant
tunneling) and particle-like (Coulomb blockade) aspects of
charge transport can be important. Molecular electronics
encompasses a wide range of physical systems and it should
be mentioned that no theory is ideal for every problem.
Rather, each approach has advantages and disadvantages
which should be weighed in the context of each specific
investigation.
3.2.1 Elastic transport and linear response In
many cases of interest, elastic processes dominate transport.
For example, the room temperature transport through small
organic molecules bonded to metallic electrodes is predominantly elastic; the small inelastic current arising from the
electron–phonon interaction is only expected to become relevant at large bias [116, 101, 211]. In this regime, the Coulomb
self-energy used to construct G(E) may be simplified and the
currents may be cast in a form analogous to the multi-terminal
Büttiker formula [212, 213]
†
Gnσ,mσ (t) = −iθ(t){dnσ (t), dmσ
(0)},
<
nσ,mσ G
(t) = id
†
mσ (0) dnσ (t).
(11)
Both G(E) and G< (E) arise from time-ordered Green’s
functions on the Keldysh time-contour [28], so any prescription for calculating G(E) also yields G< (E) without further
approximations. Note that Eq. (10) is still for an arbitrary
© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1
h β=1
M
Iα(ν) =
dE (E − μα )ν Tαβ (E) fβ (E) − fα (E) ,
(12)
where β labels the electrodes, and the transmission function
is given by [214]
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Tαβ (E) = Tr Γ α (E)G(E)Γ β (E)G† (E) .
2261
(13)
Although Eq. (12) resembles the non-interacting result
[215, 216], it follows directly from Eq. (10) and is valid
even for strongly interacting systems, provided the inelastic
contribution to transport is negligible.
As mentioned earlier the thermodynamic response of a
junction is given by the energy conversion efficiency η and
usable power produced P, which may be expressed in terms
of Eq. (12) for a two-terminal device as [174]
P = I (0) (μ1 − μ2 ) ,
(14)
P
η = (1) .
I1 (15)
In deriving Eqs. (14) and (15), we have assumed T1 > T2 . The
transmission function, ZT el , η, and P are shown in Fig. 9 for
a many-body calculation of a model π-system describing a
Au–1,3-benzenedithiol–Au junction. In the vicinity of the
destructive interference feature near the middle of the fundamental gap energy μ0 , ZT el and η are enhanced. However,
P is only enhanced near the molecular addition resonances
(labeled HOMO, LUMO), suggesting that ZT el is not a
complete characterization of thermoelectric device performance. Finally, the thermoelectric response shows additional
enhancement at finite bias.
In the linear response regime, ΔT T and |eV | μ,
and
∂f0
E−μ
∼
fα (E) = f0 (E) + −
Δμα +
ΔTα , (16)
∂E
T
where f0 (E) is the equilibrium (i.e., zero-bias) Fermi distribution with chemical potential μ and temperature T .
Equation (12) (with ν = 0, 1) may be further simplified and
written in matrix form as
Iα(0)
Iα(1)
=
⎛
⎝
β
L(0)
αβ
1
T
L(1)
αβ
L(1)
αβ
1
T
L(2)
αβ
⎞
⎠
μβ − μα
,
Tβ − T α
(17)
where the Onsager linear-response function
L(ν)
αβ (μ, T ) =
1
h
∂f0
dE(E − μ)ν −
Tαβ (E).
∂E
(18)
We may then use the L functions to compactly encode a
number of important transport properties
G = e2 L(0) ,
S=−
1 L(1)
,
eT L(0)
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(19)
(20)
1
κel =
T
ZT el =
(1) 2 L
,
L −
L(0)
−1
L(0) L(2)
,
(1) 2 − 1
L
(2)
(21)
(22)
where G is the electrical conductance, S is the thermopower,
κel is the electronic contribution to the thermal conductance,
and ZT el is the electronic contribution to ZT . Mott’s formula
(3) is recovered from Eq. (20) by performing a Sommerfeld
expansion for T (E) (i.e., an expansion of T (E) around μ to
first order) [173, 217]. The linear-response transport coefficients of an interacting system thus have a structure identical
to that of a non-interacting system, except that Tαβ (E) must
be calculated using the interacting Green’s functions.
The formulation of transport outlined here provides a
comprehensive framework to investigate heat and charge
transport at the nanoscale. The role of electron–electron correlations, multiple electrodes, molecular vibrations, etc. can
all be described (in principle) exactly. In the next section,
we discuss some potential future directions of molecular
thermoelectrics.
3.3 The future of single-molecule thermoelectric
transport Molecular heat transport and thermoelectrics
are currently active fields of study [172]. Reliable, singlemolecule measurements of thermopower [168, 169, 177–
180] (cf. Fig. 10) have shown that efficient thermoelectric devices can be engineered using synthetic chemistry.
The small phonon contribution to the heat current and
enhancements due to interference effects, multiple terminal
configuration, and finite bias all suggest a bright future for
molecular device development.
Recent advances in thermal microscopy techniques
[197, 198, 218–222] allow direct observation of energy transport in nonequilibrium nanoscale junctions. This allows a
number of fundamental thermodynamic properties to be
probed directly, for example, the nonequilibrium temperature distribution in a quantum system subject to a thermal
gradient or voltage bias. Although the theoretical problem
of how to define a local temperature is not new [223], these
recent experiments have sparked a renewed interest in the
topic [224–230] and motivated investigations into a number of fundamental issues such as: What is the meaning
of temperature at the atomic (or sub-atomic) scale? How
does Fourier’s law of heat conduction emerge from quantum
heat transport [172, 230–232]? What fundamental limits are
there on the tip-environment coupling? Are there any quantum interference effects on the local temperature distribution
[230, 233]?
In general, the temperature distributions of fermionic
(electronic) [223, 224, 230] and bosonic (phonon or photon)
[116, 216, 234–237] degrees of freedom do not correspond to
one another away from equilibrium. This point is especially
important in small organic molecules, where the electron–
phonon coupling is weak, or in scanning tunneling devices
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References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
Figure 10 Experimental data showing that state-of-the-art techniques can be used to simultaneously measure the conductance and
thermal current with high precision. Reprinted with permission from
[178]. Copyright 2011 American Chemical Society.
[12]
[13]
[14]
operating in the tunneling regime, where there is no phonon
heat transport. Establishing the relationships among the temperatures of these distinct degrees of freedom will likely
lead to new insights into the nature of the non-equilibrium
transport problem.
Ballistic heat transport measurements have been performed in mesoscopic systems [238] and with the advent
of nanometer precision, picowatt sensitive calorimeters
[197, 198], it is likely that experiments will begin to probe the
quantum regime of heat transport; the quantum of heat conductance [190, 191, 239] κ0 = π2 kB2 T/3h ≈ 0.284 nW K−1 at
300 K. In this exciting regime, the relationships among
molecular symmetry, electron correlation, molecular vibrations, noise, and quantum interference can all be investigated
by probing the interplay between the quantum heat and
charge transport.
4 Conclusions Even after 40 years of progress, the
molecular electronics field is still extremely active and exciting. The unique and complex nature of the transport problem
in an open quantum system – whose macroscopic properties
are influenced by microscopic molecular properties – ensures
a bright future, for fundamental research, and possible device
application development.
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Acknowledgements This work was supported as part of
the Non-Equilibrium Energy Research Center (NERC), an Energy
Frontier Research Center funded by the U.S. Department of Energy,
Office of Science, Basic Energy Sciences under Award No. DESC0000989. Partial support was also provided by the NSF (CHE1058896). We thank Prof. T. Frauenheim for the invitation to write
this article, which is intended as an overview with blurred focus on
a few important aspects of molecular electronics.
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