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1 Coherence and Decoherence, Tunneling and Hopping

1 Coherence and Decoherence, Tunneling and Hopping

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Molecular Electronic Junction Transport: Some Pathways and Some Ideas


work of Landauer with Imry and Buttiker [10, 11]. This in turn is based on a simple

observation – in mesoscopic physics transport junctions, or in molecular transport

junctions, there is a disparity of size scales: the molecule or quantum dot is very small

compared with the electrodes. These macroscopic electrodes, then, set the chemical

potentials, and once an electron enters one of them, it can be thought of as losing its

phase immediately, and simply becoming part of the electronic sea in that metal. This

is the fundamental Landauer idea: when voltage is placed across a transport junction,

electrons travel from one electrode to the other. They travel through the molecule or

quantum dot, on which they may reside for a long time or a short time. But once they

enter the downstream electrode, that acts as a perfect sink –all phase coherence is

immediately lost, and the electron has disappeared into the Fermi sea. This fundamental idea is crucially different from understanding a classical wire, and thinking of

conduction in terms of Ohm’s law; in that situation there is no size separation, and the

electrons are thought of as a current that generates heat and undergoes resistance as it

moves–the description is initially classical, although it can easily be made quantal. In

the Landauer/Imry/Buttiker approach, the transport is quite different – it is scattering

(indeed it is elastic scattering in the simplest picture).

This approach to understanding transport leads to the Landauer/Imry/Buttiker

formula for conductance which is

g ¼ g0



Tii ¼ 2

e2 X

Tii :

h i


Here g, g0, Tii, e, and h are respectively the conductance, the quantum of

conductance equal to 77.48 microsiemens, the transmission through channel i, the

electronic charge, and Planck’s constant. The idea that conductance can be quantized is a remarkably new one compared with ohmic behavior – Fig. 6 shows

experiments that directly demonstrate quantization of transport in atomic gold wires.

The sum in (4) runs over all the transverse channels of the system – that is, the

channels that extend from the upstream to the downstream electrode. They normally are thought of (qualitatively) in terms of the molecular orbitals on the

molecule, with appropriate modifications for mesoscopic systems.

While quantum chemistry cannot be used to work with the Landauer/Buttiker/Imry

formula as it stands, a very different approach based on nonequilibrium Green’s

functions yields a different formula (sometimes called the Caroli formula [68], or

the NEGF formula in the Landauer/Imry/Buttiker limit). It is, for the current I:

I ẳ 2e=h dETrfGE; VịGr E; VịGỵ E; VÞGa ðE; VÞgðfL ðE; VÞ À fR ðE; VÞÞ (5)

where Gr is the retarded Green’s function for electrons, G is the spectral density

(twice the imaginary part of the self energy), and ƒ is the Fermi distribution

function. This equation can be rewritten, for clarity, as


G.C. Solomon et al.

Fig. 6 (a) Conductance steps

in a Au wire as an STM tip

was retracted. (b) Electron

microscope images of gold

bridges obtained

simultaneously with the

conductance measurements in

(a). Left, bridge at step A;

right, bridge at step B.

(c) Intensity profiles of the

left and right bridges shown

in (b). The shaded area is the

intensity from the bridge after

subtraction of the background

noise. (d) Models of the left

and right bridges. The bridge

at step A has two rows of

atoms; the bridge at step B

has only one row of atoms.

The distance from P to Q

(see b) is about 0.89 nm, wide

enough to have two gold

atoms in a bridge if the gold

atoms have the nearestneighbor spacing of the bulk

crystal (0.288 nm) (Reprinted

by permission from

Macmillan Publishers Ltd:

Nature (1998) Nature 395,

780783, copyright (1998))

I ẳ 2e=h dETE; Vị fL E; Vị fR E; Vịị;


TE; Vị ẳ TrfGE; VịGr E; VịGỵ E; VịGa E; Vịg:


Here, the transmission, T, is expressed as (6b). The Landauer/Imry/Buttiker formula

(almost always called the Landauer formula) then says that the left-to-right electronic

current through a molecular transport junction is the integral of the transmission through

the molecule, weighted by the statistical requirements that the electrons begin in an

occupied level of one electrode and finish in an unoccupied level of the other electrode.

This form is quite general, and it is the one on which almost all of the quantum

calculations of simple transport are based. It does need to be generalized to deal with

Molecular Electronic Junction Transport: Some Pathways and Some Ideas


issues like electron correlation, photonic excitation, thermal processes, decoherence

and dephasing, very strong correlation, magnetic effects, and other aspects of molecular

transport junctions – but it is the basis from which most of that work is done.

One way to think about the Landauer formula is to say “conductance is scattering” [69]. In fact, conductance is elastic scattering, because in the original Landauer

approach, all scattering is considered to be elastic – particles leave the electrode and

are scattered elastically until they make it into the other electrode (or not). Inelastic

events are not included, at least conceptually.

This language is a bit different from our ordinary understanding of conduction and

resistance, but it is the right approach for systems that are by their nature quantal, and

that have the length scale separation characteristic of transport junctions.

Mechanistically, it is a bit hard to swallow the idea that conduction through a

molecule must go by elastic scattering. For example, suppose the molecule in

question were really long – something like a DNA double helix with a hundred

base pairs. Elastic scattering through such a structure would fall off exponentially

with length, and therefore any transport that was seen could not be explained. The

model that is used to derive the Landauer equation – that is, the model that assumes

the space scale separation quoted above, and the elasticity of all collisions, can begin

to fail. This brings in a series of chemical mechanisms that occur because of the

nature of the molecules. These chemical mechanisms are well understood from

problems like conductive polymers and electron transfer in molecular systems –

they might be expected to occur in molecular transport junctions, and indeed they do.

One way to think about mechanistic change is in terms of time scales. This is

familiar from classical kinetics where (for example) the steady state assumption

assumes that the reactive intermediate is made and destroyed on exactly the same

time scale, so that (after the induction period of the chemical reaction) the rate of

the overall reaction could be found by assuming that the reactive intermediates exist

at steady state. This leads to the idea of chemical mechanisms for dynamical

processes, and to the question of time scales. The time scale problem in molecular

transport junctions is complicated, but extremely important. One time scale that is

unfamiliar to most chemists is the so-called Landauer/Buttiker time or contact time

[70]. This is conceptualized as the time that the electron actually spends in contact

with the molecule. This is not the same as the inverse of the rate, which describes

how long it takes for an electron to go from one end to the other, but rather tells

about how much time the electron is actually “on” the molecule – when it can

contact other molecular degrees of freedom such as the vibrations through the

electron/vibration interaction [71, 72]. A simple argument based on the uncertainty

principle (that can be supported by scattering theory analysis) is that this Landauer/

Buttiker contact time is given approximately by

tLB ¼






Here the variables are n, which is the dimensionless length of the system in terms

of subunits and DEg ; which is the gap energy between the Fermi level of the


G.C. Solomon et al.

electrode and the relevant molecular energy level. This formula looks like the

uncertainty principle multiplied by a length, which seems reasonable. The uncertainty principle part is slightly counterintuitive: it says that the higher the injection

barrier, the smaller the contact time. This is only unexpected because, if one were to

talk about rates, the higher the barrier, the slower the rate, and therefore the longer

the rate time. Conceptually, one gets around this by thinking of the Landauer/

Buttiker contact time as describing how long the electron is under the barrier – in

the original analysis this could be tested by looking at a spin flip within the barrier,

as modulated by the presence of the tunneling electron.

Qualitatively, for a characteristic transport molecule like an alkane thiol or a

small ring system, the gap is more than 1 V, the contact time is less than 1 fs, and

there is simply not enough time for strong interaction between the electrons and the

vibrations. But as resonance is approached, the time tLB can approach the period of

molecular vibrations or motions, which can then enter into resonance. This mechanistic change is important – once the resonance regime is approached, the scattering

is certainly not elastic, the behavior does not occur simply by tunneling, thermalization is possible, vibrational subpeaks should be seen in the transport, and the

mode of transport is closer to the hopping mechanism seen in conductive polymers

than to the tunneling mechanism also seen in conductive polymers [71].

Many other time parameters actually enter – if the molecule is conducting through

a polaron type mechanism (that is, if the gap has become small enough that polarization changes in geometry actually occur as the electron is transmitted), then one

worries about the time associated with polaron formation and polaron transport.

Other times that could enter would include frequencies of excitation, if photo processes are being thought of, and various times associated with polaron theory. This is a

poorly developed part of the area of molecular transport, but one that is conceptually


The Landauer formula assumes elastic processes. If the electrons move coherently (that is without any loss of energy or of phase) they will tunnel; if the energy

gap through which they must tunnel becomes relatively small, they can tunnel a

long way. Generally, the conduction in the tunneling regime is written as

g ¼ k0 eÀbx


where k0 is a constant depending on the system, x is the distance between the

electrodes, and b is the decay parameter corresponding to tunneling through a given

molecular system.


Pathways and Analysis

The orbital description of electrons in molecules suggests that it should be possible

to map the actual physical pathways by which electrons transfer through a molecule

Molecular Electronic Junction Transport: Some Pathways and Some Ideas


between two electrodes, or at least identify the parts of a molecule responsible for

mediating the electronic interaction between the two electrodes. Some of these

pathways have been roughly described on the basis of inelastic electron tunneling

spectroscopy – this is discussed in Sect. 6. However, a more general and useful

analysis (this time based on theory rather than experiment) has been developed in

terms of channels [73–80]. The most recent extension of the channels idea is based

on continuity: if one imagines planes perpendicular to the line between the two

electrode tips, then the current through all such planes must be identical at steady

Fig. 7 Local transmission description of transport through an extended alkane (top left), a para

linked di(thioethyne) benzene species (top right), and a meta-linked benzene species (lower figures

and panels). In the two upper cases, transport goes through a single simple pathway in the alkane,

and through two symmetrically disposed pathways in the para-benzene – this gives a relatively flat

conductance or transmission spectrum as a function of voltage or energy. In the meta-benzene,

different interference features occur (at roughly À2.5, 0.2, and 3.4 eV). The interference patterns

shown near these features are characterized by ring-current reversal moving from one side of the

interference feature to the other. Reproduced from [81]


G.C. Solomon et al.

Fig. 8 Local transmission pictures in a superposed benzenoid structure. As the two rings change

geometry from an eclipsed pseudo para geometry (upper left) through an eclipsed pseudo meta

geometry to a slip-stacked structure to a single tunneling pathway, the transmission at the Fermi

energy increases by roughly a factor of ten. Reprinted with permission from G. C. Solomon et al.

J. Am. Chem. Soc. (2010) 132, 7887–7889. Copyright 2011 American Chemical Society

state. Based on these understandings, Solomon and coworkers [81] have made use

of an analysis in which the electron motion between all possible atomic pairs in a

molecular junction can be calculated. The input into this calculation can be done

using any model for the electronic transport, from simple extended-Huckel type

models to full NEGF/DFT analyses. Figure 7 shows an analysis of the transport in

benzenoid structures – note the dependence upon the energies (different pathways

at different energies, at different interferences also) and on the geometry of meta vs

para linkage. Figure 8 shows similar analysis of a more complicated problem,

involving a strongly distorted, p-stacked molecular entity. In these pictures, the

thickness of a line indicates the amount of charge flowing through that line in steady

state at a particular geometry. These pathways ideas, developed on the basis of a

number of earlier contributions [82–88], are very helpful in understanding, rather

than simply calculating, electron transport in junctions.

5 Benzene Dithiol: An Exemplary Case

Since the first measurement reported by the Reed/Tour groups in 1997 [23], the

derivative of benzene with thiol groups at the 1,4 position (usually called benzene

dithiol) has become the standard case for the discussion of molecular transport

junctions. That measurement by the Reed group was made with a mechanical break

junction, and reported both the zero-voltage and the voltage-dependent conductance. Specific values were given for both, and the cartoons in the paper suggested

that the thiol group lost its hydrogens, and that the sulfur atoms were uniquely

coordinated to the gold electrodes. Since it was entirely a measurement paper, there

was no discussion of possible binding geometries. This important paper was one of

Molecular Electronic Junction Transport: Some Pathways and Some Ideas


the first reported single-molecule transport measurements, and therefore has been

instrumental in the entire area.

Questions about what was being measured, and the geometries of what was

being measured, began immediately. It was suggested that perhaps the junction

contained two molecules, one bound to one electrode, and the other to the

counterelectrode, with a sort of p-type stacking in between them [89]. A large

number of calculations using different methods were published. These modeling

activities suggested that different interactions of the molecule with the gold could

produce substantially different transport. Since the measurement from the Reed lab

was the standard, that value has been enshrined.

There are almost 100 papers that discuss benzene dithiol’s conductance. As the

point about geometric distributions became well understood, it was realized that

statistical analysis was extremely useful. Accordingly, electrochemical break junction

techniques, both in their original form of crashed electrodes being separated to form

the gap or in the newer electrochemistry form, in which a gap is created and then

electrochemically modified, have proliferated. The important thing is that statistical

measurements can be made [24, 90], with hundreds or thousands of data points. Not

surprisingly, distributions are observed (as the earlier computations had suggested).

The closest thing to a unique measurement was reported by the group at

Columbia University/Brookhaven National Laboratory [91]. They used amine

rather than thiol end groups. Both the narrowness of the experimental distributions

and very nice theoretical work integrating molecular dynamics and transport

calculations [33] suggested that the amine likes to bind to a coordinatively unsaturated site on a single gold atom, so the narrowness of the distribution here is greater

than is typical for thiols.

Some measurements showing high conduction for benzene itself and some

benzene derivatives are best explained by a geometry very different from the

extended one first suggested by Reed, and serve as the basis for much calculation.

In the measurements from Ruitenbeek’s laboratory, the conductance is close to the

atomic unit of conductance [92]. The simplest way to explain this phenomenon is

that the molecule is oriented perpendicular to the interelectrode coordinate, and

electrodes are very near one another. So the molecule really does not assist

substantially in the transport, although it can be seen in the IETS spectra.

In a 2007 overview [7], simple NEGF/DFT calculations were compared with

reported experiments, and the outlier was benzene dithiol. It is now clear that (particularly with small molecules) geometry dependence can (indeed must) give distributed

values for the conductance. This is entirely in keeping with the understanding of single

molecule spectroscopy [93, 94] that is demarked by such phenomena as blinking (in

many cases) and spectral wandering (in essentially all cases). These arise from

fluctuations, be they fluctuations of charge density or fluctuations in geometry of

the environment in which the molecule is measured. From the viewpoint of fundamental understanding, these fluctuational quantities are well described by simple

statistical mechanics – fluctuations scale as the inverse square root of the sample

number, so that with millions of samples, an average number can be readily agreed


G.C. Solomon et al.

upon. A small number of measurements would be expected to give a fairly wide

distribution of observed behaviors – this is indeed seen [28] in benzene dithiol, and

probably should be seen (and has been) in many other molecules.

6 Inelastic Electron Tunneling Spectroscopy

In the Landauer/Imry limit, the transport through the junction is due to elastic

scattering. If the gap between the injection energy and the frontier orbital resonance

is large, the Landauer/Buttiker contact time is very small, so that the charge is

present on the molecule for a very short time. This means that its interaction with

any vibration will be weak, because there just is not time to complete a full

vibrational period before the charge has gone into the electrode sink.

There will be vibronic interactions in any molecular system, because the charged

states will always have a different geometry from the uncharged ones. This means

that the charge on the molecule will cause the geometry of the molecule to change,

and that will be reflected in a vibrational side peak in the transport spectra. The

simplest and most useful measurement to make on such systems is inelastic electron

tunneling spectroscopy [8] (IETS), in which one measures the second derivative of

the current with respect to the voltage, and plots that (divided by its value at a

reference voltage) as a function of voltage. Figure 9 shows both the schematic

behavior. This experiment, first reported in molecular junctions by Reed and

coworkers [95] and by Kushmerick and coworkers [96] in 2004, is a significant

way to investigate molecules in junctions.

When the gap is large, the sketch in Fig. 9 shows that a second channel will open

when there is a vibrational resonance – that is, when eV ¼ ho, with o one of the

vibrational frequencies of the molecule. This is vibronic resonance, and energy will

transfer from the momentum of the tunneling electrons into the vibrations of the

molecule. The interaction is quite weak (because the tunneling time is so short);

Fig. 9 Schematic of the inelastic electron tunneling phenomenon. From M. Galperin et al.

Science (2008), 319, 1056–1060. Reprinted with permission from AAAS

Molecular Electronic Junction Transport: Some Pathways and Some Ideas


IETS spectra are usually reported at very low temperatures, and careful data

management is required to see the IETS features.

The interpretation of IETS is helpful in understanding molecular junctions. Several

workers have developed techniques for doing so [97–102], some based on quite

complex analyses of the full Green’s function [99–101], others based on a much

simpler analysis in which the fact that the response is so weak is used as the basis for

perturbative expansion[98]. The results of these analyses fit the spectra well.

From these analyses, a number of major advances have followed. First, the presence

of the molecular vibrations indicates that the molecules are indeed in the junction, and

that the transport is passing through them. Second, the pathway of the current through

the molecule can to some extent be determined based on which vibrational modes are

enhanced – as is not surprising, if the electron density between atoms on which a

particular normal mode exhibits large amplitude is not substantially modified upon

charging, then that molecular mode will be silent in the IETS spectra. This leads to a set

of propensity rules [103–106] that have helped substantially in interpreting both IETS

spectra and (more interestingly) the actual geometries of the junction.

As has been stated several times, the geometry problem in junctions is difficult.

Several papers have utilized the differences in the IETS calculated spectrum at

different trial geometries to compare with the experimental spectrum, and thereby

to deduce the true geometry of the structure. Figure 10 shows some results by Troisi

[107], in which he was able to deduce the angle between the molecular backbone

and the electrode, based on agreement with the IETS spectrum.

It is also possible to deduce pathways in a more adventurous way by noting

which modes are enhanced, doing the normal coordinate analysis to find out where

those modes have their maximum amplitudes, and arguing that this describes the

pathway for the electron going through the molecule. An example is shown in

Fig. 11, also from Troisi’s work [108].

It was noted early by Reed and others that the IETS spectrum could exhibit both

absorption and emission peaks – that is, the plots of Fig. 9 could have positive

excursions and negative excursions called peaks and dips. The simple analysis

suggested in Fig. 9 implies that it should always be absorptive behavior, and therefore

that there should always be a peak (a maximum, an enhancement) in the IETS

spectrum at the vibrational resonances. It has been observed, however, that dips

sometimes occur in these spectra. These have been particularly visible in small

molecules in junctions, such as in the work of van Ruitenbeek [92, 109] (Fig. 12).

Here, formal analysis indicates that, as the injection gap gets smaller, the existence of

an inelastic vibrational channel does not contribute a second independent channel to

the transport, but rather opens up an interference [100]. This interference can actually

impede transport, resulting in a dip in the spectrum. Qualitatively, this occurs because

the system is close to an electronic resonance; without the vibrational coupling the

conductance is close to g0, and the interference subtracts from the current.

These IETS features have been observed. The technique is a very good one for

addressing certain aspects of a molecular structure in the junction, and the molecular pathways. Of all the areas of molecular transport, this one is probably the most

quantitatively accurate for comparison with experiment.


G.C. Solomon et al.

Fig. 10 Estimation of the tilt angle for an alkane between gold electrodes, determined by fitting the

computed IETS spectrum with the experiment (panel b below). Result is a 40 degree tilt angle

perpendicular to the plane of the carbon chain, as illustrated in the lighter shade structure in the sketch

(b) above. Sketch (a) above and panel (a) below refer to the alkane tilted in the plane of the carbon

chain. The structures in sketch (a) do not fit so well an those in (b), suggesting the methyl group

position shown in (b) above. From [107]. Reproduced by permission of the PCCP Owner Societies

Fig. 11 IETS analysis of transport through an etheric naphthalene molecular wire. Central panel

left shows the computed (red) and the experimental (black) IETS spectrum. The normal

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